Fatigue life prediction of composites and composite structures [Second edition]

Table of contents :
Front Cover......Page 1
Fatigue Life Prediction of Composites and Composite Structures......Page 4
Copyright......Page 5
Contents......Page 6
Contributors......Page 12
Preface......Page 14
1.1. Introduction......Page 16
1.2.1. Overview......Page 20
1.2.2. Fatigue test parameters......Page 22
1.2.3. Fatigue nomenclature......Page 23
1.3.1. Overview......Page 24
1.3.2. Fatigue life prediction theories......Page 26
1.3.2.1. Macroscopic failure theories-Also designated empirical theories......Page 27
1.3.2.2. Strength and stiffness degradation fatigue theories......Page 31
1.3.2.3. Multiscale modeling......Page 34
1.3.2.4. Diverse fatigue considerations......Page 36
1.3.3. Fatigue life prediction under complex irregular loading......Page 38
1.4. Conclusions-Future prospects......Page 47
References......Page 51
Part One: Fatigue life behavior and modeling......Page 60
2.1. Introduction......Page 62
2.2. Fatigue experiments......Page 63
2.2.2. Grips......Page 64
2.3. Measurements and sensors......Page 65
2.4.2. Frictional heating......Page 67
2.5. Specimens......Page 68
2.5.3. Tabs......Page 69
2.5.4. Planform......Page 70
2.5.6. Thickness......Page 71
2.6. S-N diagrams......Page 72
2.6.1. Statistical description of fatigue data......Page 73
2.6.2. Censoring and run-outs......Page 74
2.7. S-N formulations......Page 75
2.7.1. Two-parameter S-N curve......Page 76
2.7.3. Strength-based S-N curve......Page 78
2.7.4. Statistical formulations......Page 79
2.7.5. S-N curves that take into account the R-value......Page 80
2.7.7. Final notes on S-N curves and the CLD......Page 86
References......Page 88
Further reading......Page 90
3.1. Introduction......Page 92
3.2. Major residual strength models from the literature......Page 93
3.2.1. Broutman and Sahu......Page 94
3.2.2. Reifsnider and Stinchcomb......Page 97
3.2.3. Schaff and Davidson......Page 98
3.2.5. Other residual strength models......Page 99
3.3. Fitting of experimental data......Page 100
3.4. Prediction results......Page 108
3.5. Conclusions and future trends......Page 109
References......Page 111
4.1. Introduction......Page 114
4.2.1. Material and specimens......Page 118
4.2.2.1. Dynamic mechanical analysis......Page 119
4.2.2.3. Pure creep experiment......Page 120
4.2.2.4. Fatigue experiments......Page 121
4.2.2.5. Instrumentation......Page 122
4.3.2. Quasistatic behavior......Page 123
4.3.3. Pure creep behavior......Page 124
4.3.4.1. Fatigue life......Page 127
4.3.4.2. Hysteresis loops-stiffness degradation-energy dissipation......Page 131
4.3.4.3. Self-generated temperature-Damage evolution......Page 137
4.3.5. Failure analysis......Page 142
4.4. Conclusions and outlook......Page 144
References......Page 145
5.1. Introduction......Page 150
5.2.2. Normalized stiffness degradation approach......Page 151
5.2.3. Normalized fatigue life model......Page 154
5.2.4. Modeling strategy for the filled fibrous composites with nanoparticles......Page 157
5.2.5.1. The Halpin-Tsai micromechanics model......Page 158
5.2.5.2. Nielsen micromechanics model......Page 159
5.2.6.1. The Nano-NSDM based on the Halpin-Tsai model......Page 160
5.2.7.1. Fatigue life prediction for epoxy resin modified by silica nanoparticles......Page 161
5.2.7.2. Fatigue life prediction for GFRP with nanoparticles......Page 162
5.2.7.3. Fatigue modeling of CSM/epoxy composites......Page 166
Fatigue damage accumulation model......Page 167
Tests results-Model evaluation......Page 168
5.2.7.4. Fatigue life of thermoplastic nanocomposites......Page 170
Experimental procedure......Page 175
5.3. Fatigue life prediction based on the micromechanical-energy method......Page 183
5.3.1. The energy method......Page 184
5.3.2. Modeling strategy (Nano-EFAT model)......Page 185
5.3.3. Tests results......Page 186
5.3.4. Evaluation of the Nano-EFAT model......Page 187
5.4. Displacement-controlled flexural fatigue behavior of composites with nanoparticles......Page 188
5.4.2. Specimen preparation......Page 190
5.4.3. Calculation of the bending stress......Page 192
5.4.5.1. Static bending strength......Page 195
5.4.5.2. Cyclic flexural bending fatigue life......Page 198
5.4.5.3. Dispersion and morphology analysis......Page 199
5.5. Conclusions and outlook......Page 202
References......Page 203
Further reading......Page 208
6.1. Introduction......Page 210
6.2.1. Influence of matrix nature on fatigue behavior: TP vs TS composites......Page 211
6.2.2. Influence of reinforcement architecture......Page 212
6.2.3. Autogenous heating in polymer matrix composites under fatigue loading......Page 214
6.2.4. Influence of viscous effects on fatigue accumulated damage......Page 215
6.2.5. Creep-fatigue interaction in TP composites......Page 216
6.2.6. Influence of stress concentration on the fatigue behavior of TP laminates......Page 217
6.3.1. Materials and specimens description......Page 218
6.3.2. Experimental set-up......Page 219
6.4.1. Matrix-dominated fatigue behavior [12, 14, 81, 82]......Page 220
6.4.2. Fiber-dominated fatigue behavior [13]......Page 227
6.4.3. Autogenous heating under fatigue loading......Page 233
6.4.4. Creep-fatigue interaction in TP-based composites [14]......Page 234
6.4.5. Influence of matrix nature on the strain energy release rate in notched laminates [15, 16]......Page 240
6.5. Conclusions and outlook......Page 243
References......Page 245
7.1. Introduction......Page 254
7.2. Assessment of existing approaches for fatigue of composites......Page 255
7.3. Aspects of fatigue behavior of thick laminates......Page 260
7.4. Composite material characterization for failure parameters......Page 264
7.4.1. Fatigue failure criteria......Page 265
7.4.2. Determination of fatigue strengths of the composite material......Page 266
7.5. Failure criteria and failure modes in progressive damage......Page 268
7.6. Material degradation due to fatigue damage......Page 270
7.7. Progressive damage development and progression......Page 272
7.8. Application to a thick composite laminate......Page 275
7.9. Conclusions......Page 280
References......Page 281
8.1. Introduction......Page 284
8.2.1. Stress analysis......Page 285
8.2.2. Matrix multi-cracking......Page 287
8.2.3. Interface debonding......Page 288
8.2.4. Fibers failure......Page 290
8.2.5. Hysteresis theory......Page 291
8.2.5.1. Interface partial debonding......Page 292
8.2.5.2. Interface complete debonding......Page 294
8.2.5.3. Hysteresis loops......Page 295
8.2.6. Life prediction method......Page 296
8.3.1. Effects of fatigue peak stress......Page 298
8.3.2. Effects of fatigue stress ratio......Page 301
8.3.3. Effects of matrix crack spacing......Page 303
8.3.4. Effects of fiber volume fraction......Page 313
8.4.1. Unidirectional CMCs......Page 320
8.4.2.2. Cross-ply SiC/MAS at 1093C in air atmosphere......Page 325
8.4.3.2. 2D SiC/Si-N-C at 1000C......Page 330
8.4.3.3. 2D SiC/[Si-B4C] at 1200C in air and in steam atmospheres......Page 333
8.4.5.1. 3D C/SiC at elevated temperature......Page 336
8.4.5.2. 3D SiC/SiC at elevated temperature......Page 342
References......Page 345
Further reading......Page 348
9.1. Introduction......Page 350
9.2.1. 3D printing equipment......Page 351
9.2.2. Specimen preparation......Page 353
9.2.3. Experimental setup......Page 354
9.3. Experimental analysis......Page 355
9.4. Statistical analysis......Page 357
9.5. Discussion......Page 360
References......Page 362
10.1. Introduction......Page 364
10.2. Theoretical background......Page 367
10.2.1. Artificial neural networks......Page 368
10.2.2. Adaptive neuro-fuzzy inference system......Page 370
10.2.3. Genetic programming......Page 373
10.2.4. The gene expression programming algorithm......Page 376
10.3. Modeling examples......Page 378
10.3.1.1. Material #1, GFRP multidirectional laminate with stacking sequence [0/(±45)2/0]T......Page 380
10.3.1.3. Material #3, multidirectional glass/epoxy laminate with a stacking sequence [(±45/0)4/±45]T......Page 381
10.3.2. Application of the methods......Page 383
10.3.2.1. Artificial neural networks......Page 384
10.3.2.2. Adaptive neuro-fuzzy inference system......Page 386
10.3.2.3. Genetic programming......Page 388
10.3.2.4. Gene expression programming......Page 389
10.4. Comparison to conventional methods of fatigue life modeling......Page 391
10.5. Conclusions and future prospects......Page 394
References......Page 395
Part Two: Fatigue life prediction and monitoring......Page 400
11.1. Introduction......Page 402
11.2.1. Classic fatigue life prediction methodology......Page 403
11.2.1.1. Cycle counting......Page 404
11.2.1.2. Representation of CA fatigue data......Page 405
11.2.1.3. Assessment of the mean stress effect......Page 406
11.2.1.4. Fatigue failure criterion......Page 408
11.2.1.5. Damage summation......Page 409
11.2.2.2. Acquiring strength degradation data......Page 410
11.2.2.4. Modeling strength degradation......Page 412
11.2.2.5. Load sequence effects......Page 414
11.2.2.7. Advantages and disadvantages of strength degradation compared to classic methodology......Page 416
11.3.1. Multidirectional glass/epoxy laminate with stacking sequence [(±45/0)4/±45]T......Page 418
11.3.2. Multidirectional glass/polyester laminate with stacking sequence [0/(±45)2/0]T......Page 423
11.4.1. Cycle counting......Page 424
11.4.2. Interpretation of fatigue data-S-N curves......Page 425
11.4.3. Constant life diagrams......Page 427
11.4.4. Lifetime predictions......Page 429
11.5. Concluding remarks and future prospects......Page 432
References......Page 435
12.1. Introduction......Page 440
12.2. Constant fatigue life (CFL) diagram approach......Page 443
12.3.1. Symmetric and asymmetric Goodman diagrams......Page 444
12.3.2. Shifted Goodman diagram......Page 446
12.3.3. Inclined Goodman diagram......Page 447
12.4.1. Piecewise linear CFL diagram......Page 449
12.4.2. Symmetric and asymmetric Gerber diagrams......Page 450
12.4.3. Shifted asymmetric and symmetric Gerber diagrams......Page 452
12.4.4. Inclined Gerber diagram......Page 453
12.4.5. Bell-shaped CFL diagram......Page 454
12.4.6.1. Formulation......Page 456
12.4.6.3. Procedure for predicting the S-N curves for any stress ratios with the help of the anisomorphic CFL diagram......Page 458
12.5. Prediction of constant fatigue life (CFL) diagrams and S-N curves......Page 459
12.5.1. Application to the fiber-dominated fatigue behavior of composite laminates......Page 460
12.5.2. Application to the matrix-dominated fatigue behavior of composite laminates......Page 463
12.6. Extended anisomorphic constant fatigue life (CFL) diagram......Page 466
12.7. Conclusions......Page 470
12.8. Future trends......Page 471
References......Page 474
13.2. Definition of crack initiation......Page 480
13.3.1. Fatigue crack initiation literature......Page 481
13.3.2. Multiscale fatigue crack initiation criterion......Page 483
13.3.3. Validation of crack initiation model......Page 488
13.3.3.1. Calculation of local stress in the matrix around the fibers......Page 489
13.3.3.2. Application to C_TS_1......Page 492
13.3.3.3. Application to G_TS_1......Page 493
13.3.3.4. Application to C_TS_2......Page 497
13.3.3.5. Application to C_TP_1......Page 499
13.4. Discussion of validation results......Page 503
13.5. Conclusion and future challenges......Page 505
References......Page 506
14.1. Introduction......Page 510
14.2.1. Ply response under quasi-static monotonic loading......Page 513
14.2.2. Loading-unloading-reloading (L-U-R)......Page 514
14.2.3. Stiffness degradation......Page 518
14.2.3.1. Pre-failure material models......Page 519
14.2.3.2. Post-failure material models......Page 521
14.3. Failure onset conditions......Page 522
14.4. Strength degradation due to cyclic loading......Page 524
14.5. Constant life diagrams and S-N curves......Page 532
14.6. FAtigue DAmage Simulator (FADAS)......Page 534
14.6.1. Calculations under VA cyclic stresses......Page 538
14.6.2. Computational procedure......Page 540
14.6.3. Validation of numerical predictions: Experimental data......Page 541
14.6.4.1. Constant amplitude fatigue of MD laminate loaded on-and off-axis......Page 543
14.6.4.2. Life prediction of [±45]S specimens under CA and spectrum loading......Page 546
14.6.4.3. Residual strength of [±45]S coupons......Page 548
14.7. Conclusions......Page 549
References......Page 550
15.1. Introduction......Page 554
15.2. Theoretical background: Classical laminate theory for fatigue-life prediction......Page 555
15.2.1. Stiffness measurement under quasi-static and fatigue loads......Page 558
15.2.3. Poisson´s ratios measured in quasi-static and fatigue tests......Page 561
15.3. Fatigue experiments......Page 562
15.4. Damage mechanisms and stiffness progresses depending on fiber volume content and mean stress......Page 564
15.4.1. UD 90 degree with 30% and 55% fiber volume content......Page 565
15.4.3. UD 0 degree with 30% and 55% fiber volume content......Page 567
15.5. Application of the predictive method......Page 570
15.5.1. Quasi-static input parameters for CLT......Page 571
15.5.2. Cycle-dependent moduli and Poisson´s ratios as input for CLT......Page 572
15.5.3. Calculated stiffness degradation of multidirectional lay-up by CLT vs experiment......Page 574
15.6.1. Damage mechanisms......Page 577
15.6.2. Specimen geometry and embedded layers......Page 578
15.7. Conclusions and future perspectives......Page 579
References......Page 582
16.1. Introduction......Page 584
16.2.1. Loads on the wind turbine rotor......Page 585
16.2.2. Load distribution on the wind turbine blade......Page 586
16.2.3. Stresses in the blade material......Page 587
16.2.4. Fatigue loading......Page 588
16.2.5. Deflection of the wind turbine blade......Page 590
16.3. DTU 10MW reference turbine......Page 591
16.3.2. Blade tip deflection......Page 593
16.4. The load-carrying composite in a wind turbine blade......Page 594
16.4.1. Fatigue testing......Page 595
16.4.2. Fatigue measurement on a quasi-unidirectional glass fiber composite......Page 597
16.5. Nondestructive fatigue damage characterization methods......Page 601
16.5.1. Transilluminated white light imagining (TWLI) in situ characterization method......Page 602
16.5.2. 3D micro-X-ray computer tomography technique (XCT) ex situ characterization method......Page 603
16.6. Fatigue damage evolution during tension/tension fatigue......Page 606
16.6.1. Stage I: Initial stiffness drop studied using the in situ TWLI test-setup......Page 607
16.6.2. Stage II: Gradual stiffness degradation......Page 609
16.6.3. Stage III: Final damage localization......Page 612
16.8. Stiffness degradation during tension/compression fatigue......Page 615
References......Page 617
Part Three: Applications......Page 620
17.1. Introduction......Page 622
17.2.1. Existing models......Page 624
17.2.2. Fatigue damage accumulation under stationary loading......Page 626
17.2.3. Fatigue damage accumulation under non-stationary loading......Page 628
17.3.1. Uncertainty modeling of external loading......Page 629
17.3.2. Uncertainty modeling of material properties......Page 630
17.4.1. Fatigue reliability under stationary loading......Page 633
17.4.2. Method I: Moments matching approach......Page 635
17.4.3. Method II: FORM approach......Page 636
17.4.4. Fatigue reliability under non-stationary loading......Page 637
17.5.1. Numerical example 1......Page 639
17.5.2. Numerical example 2......Page 640
17.5.3. Experimental validation......Page 643
Conclusion......Page 646
References......Page 647
18.1. Introduction......Page 650
18.2. Engineering software for fatigue life modeling/prediction......Page 653
18.3.1. Fatigue life prediction method for laminates......Page 655
18.3.2. Experimental work......Page 658
18.3.3.1. Test results for quasi-static loading......Page 659
18.3.4. Results of fatigue tests......Page 660
18.3.5.1. FE analysis......Page 662
18.3.5.2. Fatigue life prediction......Page 663
18.3.6. Example: Multifunctional truck cross member......Page 668
18.3.8. Nomenclature......Page 670
18.4.1. Datasets description......Page 671
18.4.2. Multidirectional laminate [0/(±45)2/0]T......Page 672
18.4.3. CCfatigue software application......Page 675
18.4.4. Variable loading fatigue lifetime prediction......Page 680
18.4.5. Multiaxial fatigue lifetime prediction......Page 686
18.4.6. Summary......Page 690
References......Page 691
19.1. Introduction......Page 696
19.2. Framework of developed modeling technique......Page 699
19.3. Loading......Page 701
19.3.1. External conditions......Page 702
19.4. Static analysis......Page 703
19.5.1. Stress analysis......Page 707
19.5.2. Failure analysis......Page 708
19.5.4. Sudden degradation rules......Page 709
19.5.5. Evaluation of accumulated fatigue damage modeling......Page 710
19.6. Stochastic characterization of the wind flow......Page 713
19.7. Stochastic implementation on fatigue modeling......Page 715
19.7.1. Sources of cyclic loadings......Page 717
19.7.2. Stochastic analysis......Page 718
19.7.3. Results and discussion......Page 719
19.8. Summary and conclusion......Page 721
19.8.1. Main contributions......Page 722
References......Page 723
Further reading......Page 725
20.1. Introduction......Page 726
20.2. Structural health monitoring......Page 729
20.3. Non homogeneous hidden semi-Markov model......Page 730
20.4.1. Data processing and feature extraction......Page 733
20.4.2. Data fusion process......Page 735
20.5.1. Strain data feature extraction......Page 736
20.5.2. AE feature extraction......Page 739
20.5.4. RUL estimations......Page 742
20.5.5. Performance metrics......Page 745
20.6. Conclusions......Page 750
References......Page 751
Index......Page 756
Back Cover......Page 766

Citation preview

Fatigue Life Prediction of Composites and Composite Structures

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Woodhead Publishing Series in Composites Science and Engineering

Fatigue Life Prediction of Composites and Composite Structures Second Edition

Edited By

Anastasios P. Vassilopoulos Senior Scientist (MER) Civil, Environmental and Architectural Engineering de rale de Lausanne Ecole Polytechnique Fe Lausanne Switzerland

An imprint of Elsevier

Woodhead Publishing is an imprint of Elsevier The Officers’ Mess Business Centre, Royston Road, Duxford, CB22 4QH, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, OX5 1GB, United Kingdom © 2020 Elsevier Ltd. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-08-102575-8 (print) For information on all Woodhead publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Matthew Deans Acquisition Editor: Gwen Jones Editorial Project Manager: John Leonard Production Project Manager: Joy Christel Neumarin Honest Thangiah Cover Designer: Christian Bilbow Typeset by SPi Global, India

Contents

Contributors Preface 1

Fatigue life modeling and prediction methods for composite materials and structures—Past, present, and future prospects Anastasios P. Vassilopoulos 1.1 Introduction 1.2 Experimental characterization of composite materials 1.3 Fatigue life prediction of composite materials and structures—Past and present 1.4 Conclusions—Future prospects References

Part One 2

3

xi xiii

Fatigue life behavior and modeling

1 1 5 9 32 36

45

Phenomenological fatigue analysis and life modeling Rogier P.L. Nijssen 2.1 Introduction 2.2 Fatigue experiments 2.3 Measurements and sensors 2.4 Test frequency 2.5 Specimens 2.6 S-N diagrams 2.7 S-N formulations 2.8 Future trends References Further reading

47

Residual strength fatigue theories for composite materials N.L. Post, J.J. Lesko and S.W. Case 3.1 Introduction 3.2 Major residual strength models from the literature 3.3 Fitting of experimental data

77

47 48 50 52 53 57 60 73 73 75

77 78 85

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Contents

3.4 Prediction results 3.5 Conclusions and future trends References 4

5

6

7

Creep/fatigue/relaxation of angle-ply GFRP composite laminates Anastasios P. Vassilopoulos 4.1 Introduction 4.2 Experimental procedure 4.3 Experimental results and discussion 4.4 Conclusions and outlook Acknowledgments References Fatigue behavior of nanoparticle-filled fibrous polymeric composites M. Esmkhani, M.M. Shokrieh and F. Taheri-Behrooz 5.1 Introduction 5.2 Fatigue life prediction based on the micromechanical and normalized stiffness degradation approaches 5.3 Fatigue life prediction based on the micromechanical-energy method 5.4 Displacement-controlled flexural fatigue behavior of composites with nanoparticles 5.5 Conclusions and outlook References Further reading High-temperature fatigue behavior of woven-ply thermoplastic composites B. Vieille and L. Taleb 6.1 Introduction 6.2 Literature review 6.3 TP- and TS-based composites in fatigue: An experimental study 6.4 Discussions on the fatigue behavior of TP vs TS laminates 6.5 Conclusions and outlook References Fatigue behavior of thick composite laminates Rajamohan Ganesan 7.1 Introduction 7.2 Assessment of existing approaches for fatigue of composites

93 94 96

99 99 103 108 129 130 130

135 135 136 168 173 187 188 193

195 195 196 203 205 228 230 239 239 240

Contents

7.3 7.4

Aspects of fatigue behavior of thick laminates Composite material characterization for failure parameters 7.5 Failure criteria and failure modes in progressive damage 7.6 Material degradation due to fatigue damage 7.7 Progressive damage development and progression 7.8 Application to a thick composite laminate 7.9 Conclusions References

8

9

10

Fatigue damage and lifetime prediction of fiber-reinforced ceramic-matrix composites Longbiao Li 8.1 Introduction 8.2 Theoretical analysis 8.3 Results and discussion 8.4 Experimental comparisons 8.5 Conclusions and outlook Acknowledgments References Further reading Fatigue behaviors of fiber-reinforced composite 3D printing Astrit Imeri and Ismail Fidan 9.1 Introduction 9.2 Materials and specimen preparations 9.3 Experimental analysis 9.4 Statistical analysis 9.5 Discussion 9.6 Conclusions and outlook Acknowledgments References Computational intelligence methods for the fatigue life modeling of composite materials Anastasios P. Vassilopoulos and Efstratios F. Georgopoulos 10.1 Introduction 10.2 Theoretical background 10.3 Modeling examples 10.4 Comparison to conventional methods of fatigue life modeling 10.5 Conclusions and future prospects References

vii

245 249 253 255 257 260 265 266

269 269 270 283 305 330 330 330 333 335 335 336 340 342 345 347 347 347

349 349 352 363 376 379 380

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Contents

Part Two Fatigue life prediction and monitoring 11

12

13

14

Fatigue life prediction under realistic loading conditions Anastasios P. Vassilopoulos and Rogier P.L. Nijssen 11.1 Introduction 11.2 Theoretical background 11.3 Experimental data 11.4 Life prediction examples—Discussion 11.5 Concluding remarks and future prospects References Fatigue life prediction of composite materials under constant amplitude loading M. Kawai 12.1 Introduction 12.2 Constant fatigue life (CFL) diagram approach 12.3 Linear constant fatigue life (CFL) diagrams 12.4 Nonlinear constant fatigue life (CFL) diagrams 12.5 Prediction of constant fatigue life (CFL) diagrams and S-N curves 12.6 Extended anisomorphic constant fatigue life (CFL) diagram 12.7 Conclusions 12.8 Future trends 12.9 Source of further information and advice Acknowledgments References Prediction of fatigue crack initiation in UD laminates under different stress ratios R.D.B. Sevenois and W. Van Paepegem 13.1 Introduction 13.2 Definition of crack initiation 13.3 Predicting fatigue crack initiation 13.4 Discussion of validation results 13.5 Conclusion and future challenges 13.6 Sources of further information and advice Acknowledgments References A progressive damage mechanics algorithm for life prediction of composite materials under cyclic complex stress T.P. Philippidis and E.N. Eliopoulos 14.1 Introduction 14.2 Constitutive laws 14.3 Failure onset conditions

385 387 387 388 403 409 417 420

425 425 428 429 434 444 451 455 456 459 459 459

465 465 465 466 488 490 491 491 491

495 495 498 507

Contents

14.4 Strength degradation due to cyclic loading 14.5 Constant life diagrams and S-N curves 14.6 FAtigue DAmage Simulator (FADAS) 14.7 Conclusions Acknowledgments References 15

16

Stiffness-based approach to fatigue-life prediction of composite materials Julia Maier and Gerald Pinter 15.1 Introduction 15.2 Theoretical background: Classical laminate theory for fatigue-life prediction 15.3 Fatigue experiments 15.4 Damage mechanisms and stiffness progresses depending on fiber volume content and mean stress 15.5 Application of the predictive method 15.6 Applicability of predictive models—General considerations 15.7 Conclusions and future perspectives References The fatigue damage evolution in the load-carrying composite laminates of wind turbine blades Lars P. Mikkelsen 16.1 Introduction 16.2 Loads on the load-carrying laminates in wind turbine blades 16.3 DTU 10MW reference turbine 16.4 The load-carrying composite in a wind turbine blade 16.5 Nondestructive fatigue damage characterization methods 16.6 Fatigue damage evolution during tension/tension fatigue 16.7 Stiffness degradation during compression/compression fatigue 16.8 Stiffness degradation during tension/compression fatigue 16.9 Summary and outlook References

Part Three Applications 17

Probabilistic fatigue life prediction of composite materials Y. Liu and S. Mahadevan 17.1 Introduction 17.2 Fatigue damage accumulation 17.3 Uncertainty modeling

ix

509 517 519 534 535 535

539 539 540 547 549 555 562 564 567

569 569 570 576 579 586 591 600 600 602 602

605 607 607 609 614

x

Contents

17.4 Methods for probabilistic fatigue life prediction 17.5 Demonstration examples Conclusion References 18

19

20

Computational tools for the fatigue life modeling and prediction of composite materials and structures Anastasios P. Vassilopoulos, Julia Maier, Gerald Pinter and Christian Gaier 18.1 Introduction 18.2 Engineering software for fatigue life modeling/prediction 18.3 FEMFAT laminate approach 18.4 Description of CCfatigue and case studies 18.5 Conclusions and outlook References Fatigue life prediction of wind turbine rotor blades M.M. Shokrieh and R. Rafiee 19.1 Introduction 19.2 Framework of developed modeling technique 19.3 Loading 19.4 Static analysis 19.5 Fatigue damage criterion 19.6 Stochastic characterization of the wind flow 19.7 Stochastic implementation on fatigue modeling 19.8 Summary and conclusion References Further reading In-situ fatigue damage analysis and prognostics of composite structures based on health monitoring data Dimitrios Zarouchas and Nick Eleftheroglou 20.1 Introduction 20.2 Structural health monitoring 20.3 Non homogeneous hidden semi-Markov model 20.4 Prognostics framework 20.5 Case study 20.6 Conclusions References

Index

618 624 631 632

635

635 638 640 656 676 676 681 681 684 686 688 692 698 700 706 708 710

711 711 714 715 718 721 735 736 741

Contributors

S.W. Case Virginia Tech, Blacksburg, VA, United States Nick Eleftheroglou Structural Integrity & Composites Group, Delft University of Technology, Delft, The Netherlands E.N. Eliopoulos University of Patras, Patras, Greece M. Esmkhani Composites Research Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran Ismail Fidan Tennessee Tech University, Cookeville, TN, United States Christian Gaier Magna Powertrain, Engineering Center Steyr GmbH&CoKG, St. Valentin, Austria Rajamohan Ganesan Concordia Centre for Composites (CONCOM), Department of Mechanical, Industrial and Aerospace Engineering, Concordia University, Montreal, QC, Canada Efstratios F. Georgopoulos Technological Educational Institute (T.E.I.) of Peloponnese, Kalamata, Greece Astrit Imeri Tennessee Tech University, Cookeville, TN, United States M. Kawai University of Tsukuba, Tsukuba, Japan J.J. Lesko Virginia Tech, Blacksburg, VA, United States Longbiao Li College of Civil Aviation, Nanjing University of Aeronautics and Astronautics, Nanjing, PR China Y. Liu Clarkson University, Potsdam, NY, United States S. Mahadevan Vanderbilt University, Nashville, TN, United States Julia Maier Materials Science and Testing of Polymers, Montanuniversitaet Leoben, Leoben, Austria

xii

Contributors

Lars P. Mikkelsen Department of Wind Energy, Composite Mechanics and Structures, Technical University of Denmark, Lyngby, Denmark Rogier P.L. Nijssen Knowledge Centre Wind Turbine Materials and Constructions, Wieringerwerf, The Netherlands T.P. Philippidis University of Patras, Patras, Greece Gerald Pinter Materials Science and Testing of Polymers, Montanuniversitaet Leoben, Leoben, Austria N.L. Post Virginia Tech, Blacksburg, VA, United States R. Rafiee Faculty of New Science and Technologies, University of Tehran, Tehran, Iran R.D.B. Sevenois Department of Materials, Textiles and Chemical Engineering, Ghent University, Tech Lane Ghent Science Park Campus A; SIM vzw, Ghent, Belgium M.M. Shokrieh Composites Research Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering; Mechanical Engineering Department, Iran University of Science and Technology, Tehran, Iran F. Taheri-Behrooz Composites Research Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran L. Taleb INSA Rouen Normandie, Groupe de Physique des Materiaux (UMR CNRS 6634), Rouen, France W. Van Paepegem Department of Materials, Textiles and Chemical Engineering, Ghent University, Tech Lane Ghent Science Park Campus A, Ghent, Belgium Anastasios P. Vassilopoulos Ecole Polytechnique Federale de Lausanne (EPFL), Composite Construction Laboratory (CCLab), Lausanne, Switzerland B. Vieille INSA Rouen Normandie, Groupe de Physique des Materiaux (UMR CNRS 6634), Rouen, France Dimitrios Zarouchas Structural Integrity & Composites Group, Delft University of Technology, Delft, The Netherlands

Preface

Between 2010, the date when the first edition of this volume has been released, and today significant efforts have been allocated to the investigation of the behavior of composite materials under fatigue loading conditions. A great deal of scientific publications emerged, discussing the behavior of a variety of composite material systems under a variety of loadings and environmental conditions. The fatigue behavior of polymer-based thermoset and thermoplastic composites, ceramic, and metal matrix composites, as well as that of new types of 3D printed composite material systems, has been thoroughly examined, and relevant models have been established, with new methods/techniques for the damage identification and monitoring assisting in these procedures. The second edition of this volume aims to address and catch up with contemporary research developments in the field. The main structure of the first edition has been retained, although some of the old chapters have been removed and 11 new chapters have been implemented. In the first part of the new volume, the reader can now find information regarding the fatigue behavior of nanoparticle-filled, thermoplastic, 3D printed composites, as well as that of thick composite laminates used in an abundance of engineering applications of several domains. The loading effects on the fatigue behavior of fiber reinforced composite laminates are also discussed together with the presentation of some of the available methods for the fatigue life modeling of composite materials and structures. The second part of the new volume, referring to the fatigue life prediction, has also been updated with five new chapters. In addition to the basic chapters discussing the classical methods for life prediction under constant amplitude and variable amplitude loading, methods for the prediction of the fatigue crack initiation in unidirectional laminates, as well as methods for the fatigue life prediction based on fatigue stiffness measurements, are presented. Information about the newer techniques for the damage evolution monitoring, such as the 3D X-ray computed tomography scanning technique and the acoustic emission monitoring are presented, as well as some of the efforts for the implementation of fatigue life prediction methodologies in computational tools. It is hoped that the book will continue to appeal to those interested in the topic and provides a good starting document to those starting their research career in this very interesting and promising field. The contribution of the authors of the new chapters, as well as the work done for the update of the previous edition’s chapters, is highly appreciated. The support from the

xiv

Preface

Elsevier Team, and especially that from Mr. John Leonard who followed the project since the beginning, was valuable in order to complete this volume. Anastasios P. Vassilopoulos Ecole Polytechnique Federale de Lausanne (EPFL), Lausanne, Switzerland

Fatigue life modeling and prediction methods for composite materials and structures—Past, present, and future prospects

1

Anastasios P. Vassilopoulos Ecole Polytechnique F
ed
erale de Lausanne (EPFL), Composite Construction Laboratory (CCLab), Lausanne, Switzerland

1.1

Introduction

One of the first fundamental facts of which human being becomes aware of is that nothing lasts forever. Life may come to a sudden end or last longer, but still for only a finite period. This latter case is normally supplemented by a reduction in efficiency, known as aging. This human life experience is directly reflected in materials science; under a high load, a structure or a component can fail at once, whereas it can effectually sustain lower loads. On the other hand, the same structure or component can also fail under lower loads if they are applied over longer time frames in a constant (creep) or alternating (fatigue) way. The phenomenon of the degradation of properties of a material due to the application of loads that fluctuate over time is called fatigue and the resulting failure is called fatigue failure. Independent of the material, fatigue is caused due to time-varying stresses, below the material strength. Fatigue is localized at geometrical/material discontinuities and fatigue failure is a sequence of events including damage initiation (crack for metals, crack(s), delaminations, and other damage types for composites) damage propagation and final fracture. Fatigue was identified as a critical loading pattern a long time ago by the scientific community. The first manuscript where fatigue was referred to is probably the book written in 1841 by Jean-Victor Poncelet, a French mathematician and mechanical engineer. In that book entitled “Introduction a` la m
echanique industrielle physique ou exp
erimentale” Poncelet mentioned that any spring subjected to a push-pull force will, eventually, break under a load which is far smaller than the static breaking load [1]. He was the first one to address the term fatigue to describe this phenomenon. In the English language, the term has been introduced in 1854 by the British Engineer F. Braithwaite in his paper “On the fatigue and consequent failure of metals” in the proceedings of the Institution of Civil Engineers (ICE) [1]. Several experimental campaigns were performed in the United Kingdom in the following years, to investigate the phenomenon, mainly led by unexpected failures of railway axles.

Fatigue Life Prediction of Composites and Composite Structures. https://doi.org/10.1016/B978-0-08-102575-8.00001-2 2020 Elsevier Ltd. All rights reserved.

2

Fatigue Life Prediction of Composites and Composite Structures

Nevertheless, earlier, in 1829 the German mining engineer W.A.S. Albert was the first to carry out fatigue tests on metallic conveyor chains [2, 3] and report his observations. Subsequently, numerous failures that could not be explained on the basis of the known theory were attributed to fatigue loading. With the development of the railways, in the mid-19th century, the failure of wagon axles was such a frequent occurrence that it attracted the attention of engineers. Between 1852 and 1870 a German engineer, August W€ ohler, realized the first extended experimental program on the fatigue of metallic materials [2]. The program comprised full-scale fatigue tests on wagon axles but also specimen tests under cyclic loading patterns of tensile, bending, and torsional loads. W€ ohler constructed a test rig on which he could test wagon axles under bending moments that were developed by loads suspended from the ends of the axles. The developed stresses were recorded together with the number of rotations up to failure. The results were drawn on the σ- plane to formulate the first S-N curve, which, however, was restricted to the representation of experimental data, without proposing any mathematical formulation to describe this behavior. These first attempts to analyze the fatigue behavior of materials and structures were based on experience with constructions operating under real loading conditions. Failures that could not be explained by existing theories were designated fatigue failures. As from 1850, engineers recognized fatigue as a critical loading pattern that could be the reason for a significant percentage of structural failures and it was thereafter widely accepted that fatigue should not be neglected. However, as mentioned in the work of Sch€ utz [4], knowledge concerning certain methods was very advanced in one location, while a few kilometers away it was nonexistent. It was not until 1946 when the term fatigue was incorporated in the dictionary of the American Society for Testing and Materials (ASTM), when the E9 committee was founded to promote the development of fatigue test methods [5]. Today, it is documented that the majority of structural failures occur through a fatigue mechanism and, as mentioned in Ref. [6] after extensive study by the US National Institute of Standards and Technology, approximately 60% of 230 examined failures were associated with fatigue. This percentage was higher (between 80% and 90%) in another study carried out by the Battelle Institution [7]. During the following years, numerous experimental programs were conducted for the characterization of the fatigue behavior of several structural materials of that time. As technology developed and new test frames and measuring devices were invented, it became more and more straightforward to conduct complex fatigue experiments and measure properties and characteristics, something which some years earlier would not have been possible. As a result, almost all failure modes were identified and many theoretical models were established for modeling and eventually predicting the fatigue life of several different material systems. Significant support to this has been provided by the new monitoring, and damage identification techniques, such as the Digital Image Correlation (DIC) [8] and the X-ray tomography (see, e.g., Chapter 16). Nevertheless, fatigue theories, even today, are basically, empirical. Fatigue model equations dealing with the damage and remaining life are, literally, “fitted” to existing experimental data coming from material testing in order to calibrate their parameters and produce a reliable simulation of the exhibited behavior and/or prediction of the fatigue life under different loading conditions, or material configuration.

Fatigue life modeling and prediction methods for composite materials and structures

3

Although composite materials are designated as fatigue insensitive, especially when compared to metallic ones, they suffer from fatigue loads as well. The introduction of composite materials in a wide range of applications obliged researchers to consider fatigue when investigating a composite material and engineers to realize that fatigue is an important parameter that must be considered in calculations during design processes. Initially, composites were used as replacements for previous “conventional” materials such as steel, aluminum or wood, and later as “advanced” materials that allow engineers to adopt a different approach to design problems, propose alternative design concepts (based on the free formability and lightweight characteristics of composites) and redesign structures. Unfortunately, the situation regarding the fatigue behavior of composite materials is different from that of metallic ones. Therefore, the already developed, and validated, methods for the fatigue life modeling and prediction of “conventional” materials cannot be directly applied to composite materials. Moreover, the large number of different material configurations resulting from the multitude of fibers, matrices, manufacturing methods, lamination stacking sequences, etc., makes the development of a commonly accepted method to cover all these variances difficult. As mentioned in Ref. [9] “obviously, it is difficult to get a general approach of the fatigue behavior of composite materials including polymer matrix, metal matrix, ceramic matrix composites, elastomeric composites, Glare, short fiber reinforced polymers, and nanocomposites.” One way of dealing with the fatigue of composite materials is to undertake extended experimental programs and then develop analytical, mathematical, expressions in order to model fatigue life and be able to reproduce experimental results. Numerous experimental programs have been realized over the last three to four decades and very comprehensive databases have been constructed. Some of these are limited, refer to specific materials, and have been mainly executed in order to assist the development of a theoretical model, e.g., Refs. [10–15], but others, like [16, 17], are more extensive and cover a wide range of materials for specific applications. Along with the aforementioned experimental works, a considerable number of theoretical models have been developed to model the fatigue behavior of the examined composites and consequently predict their behavior under unknown loading conditions, e.g., Refs. [10, 12, 18–22]. Lately, experimental programs were developed to investigate special loading cases, such as the creep-fatigue interaction in composites (e.g., Refs. [23–27]), the two-dimensional fatigue crack propagation in laminates, and sandwich structures (e.g., Ref. [28]) as well as to cover newly introduced concepts and materials, such as 3D printed composites (Chapter 9, Ref. [29]), and the fatigue behavior of Thick (Chapter 7, Ref. [30]) and thin ply composite laminates [31, 32]. The behavior of composite materials is sensitive to the loading pattern due to their cyclic- and time-dependent mechanical properties [24, 25]. The time-dependent mechanical properties are related to the rheological deformation of the material when it is subjected to an external load while the cyclic-dependent mechanical properties are mainly attributed to the damage formation and accumulation [25, 33, 34]. The timeand cyclic-dependent mechanical properties of laminated composites usually interact with each other, and the degree of interaction, depends on the loading spectrum, and the material [23, 26, 35–37]. In continuous fatigue, the interaction between the timeand cyclic-dependent mechanical properties were studied by monitoring the evolution

4

Fatigue Life Prediction of Composites and Composite Structures

of fatigue hysteresis loops [23, 26, 36, 38] assigning the shift of the fatigue hysteresis loops to cyclic creep effects and the slope of the loops to degradation of the fatigue stiffness due to damage [23]. The interaction between the time- and cyclic-dependent mechanical properties was also studied by applying more complicated loading patterns in which continuous-fatigue was interrupted in different ways [24–26, 39]. Interruption of cyclic loading could also change the fatigue life depending on applied stress level and material type, as was presented in Refs. [24, 25]. Vieille et al. [26] showed that fatigue life could be extended with prior creep in angle-ply carbon/PPS thermoplastic composite depending on the loading conditions at temperatures higher than the glass transition temperature (Tg). Similarly, in angle-ply thermoset graphite/epoxy composite, it was shown that sustained periods of static loads have significant retardation effects on damage propagation and extended fatigue life [39]. Recently, it was shown that the effect of creep on the fatigue life of angle-ply thermoset laminates could be positive or negative deepening on the applied stress level and the hold time [25]. A literature search (www.scopus.com) with keywords “fatigue” and “composites” in the disciplines “Engineering,” “Materials science,” “Energy,” and “Multidisciplinary” gave ca. 9500 research articles in the field, with more than 85% of them published after 1980, and around 400 articles per year after 1995. The same search today gives more than 18,300 research articles, an average of ca. 800 per year since 2010. This immense increase shows, among others, the high interest of researchers, engineers, and industry in this field. Despite this explosive production of scientific publications in the field, countless unresolved topics exist in the domain of composite fatigue. Typical areas requiring further investigation concern the S-N and constant life diagram (CLD) formulations for the interpretation of existing fatigue data, nonlinear damage accumulation rules that can take the load sequence effects into account, cycle-counting methods that do not scramble the load sequence of the applied load time series, consideration of nonproportional stress components in a biaxial/multiaxial loading case, the exploitation of material behavior at very high cycle regimes, the development of methods that take into account the stochastic nature of the phenomenon, etc. Successful elaboration of the aforementioned areas would allow the development of reliable procedures for addressing composites fatigue. Researchers have attempted to address these, and other, topics in order to model or predict the fatigue behavior of composite materials of interest. However, the terms “modeling” and “prediction” have often been misused, usually by adopting the term “prediction” when a “modeling” is performed with the aim of interpolating between known fatigue data. The term “prediction” must be used when extrapolation is performed outside the existing database in terms of prediction of the behavior of the same material under new loading conditions, e.g., spectrum loading based on constant amplitude fatigue data, or extension of the modeling to low or high cycle fatigue regimes, when data exist in the range between 103 and 106 cycles, or even prediction of the behavior of other material systems based on models derived for a specific material. Experimental work is the fundamental first step of any investigation aiming to describe the behavior of composite material, model the failure mechanisms, and

Fatigue life modeling and prediction methods for composite materials and structures

5

predict its fatigue behavior under new, “unseen,” loading patterns. In the following section, the basic considerations for the design of an experimental program are discussed.

1.2

Experimental characterization of composite materials

1.2.1 Overview Extensive experimental programs concerning composite materials and structures have been carried out over the last four decades and a significant amount of quasistatic and fatigue data has been collected. The motivation for an experimental investigation is directly related to the aim of the experimental output and can be classified based on several criteria. In the following, an attempt to describe and classify experimental works based on the specimen size and relevant objectives will be presented: l

l

l

Specimen testing for research purposes #1—usually refers to tests on standardized specimens to investigate material behavior and characterize the damage development process. Representative examples can be found in Refs. [40–42] where the authors attempted to characterize the laminate behavior and explain the macroscopically measured strength or stiffness degradation based on observation of failure surfaces, or in Refs. [43–45], where fractography was used in order to describe the Mode I interlaminar fracture toughness of multidirectional laminates [43], angle-ply carbon/nylon laminates [44], and unidirectional and angle-ply glass/polyester DCB specimens [45]. In another case, [23–25] the authors applied nonstandardized fatigue loading spectra in order to investigate the effect of hold times and interrupted loading on the fatigue behavior of angle-ply GFRP laminates. Specimen testing for research purposes #2—usually refers to tests on standardized specimens in order to characterize the material and develop theoretical models for the description of its behavior. Unlike the previous category, this kind of testing program focuses mainly on macroscopic observations and data acquisition, e.g., measurement of stiffness degradation, residual strength or the derivation of S-N curves. Representative examples can be found in Refs. [10, 14] where the authors created their own fatigue databases in order to develop multiaxial fatigue failure criteria, or in Ref. [46] where the authors created their database in order to evaluate existing multiaxial fatigue failure theories. The laminated material is usually considered as being a homogeneous orthotropic medium and its experimental characterization, i.e., static and fatigue strength and stiffness fluctuations, should be performed for both material principal directions and in shear (Chapter 14). Nevertheless, such practices necessitate a huge amount of experiments when multiple materials and laminate configurations are used in engineering structures. Therefore, in such cases, lamina-to-laminate approaches seem more appropriate, although requiring the development of additional calculation modules able to take into account the implications in local stress fields, stress redistribution in neighboring plies, and damage propagation as a function of the loading cycles [47, 48]. This experimental approach can go lower in scale and be based on fatigue data on the fiber/matrix level, together with information about microscopic defects progression with loading. In such cases, multiscale models must be developed to allow the estimation of the fatigue life of structural elements (composite laminates) when only the fatigue performance of their constituents is known, introducing, however, significant uncertainties as discussed in Chapter 17. Specimen testing for engineering purposes—including tests on predetermined standardized specimens of different materials and/or different specimen configurations aimed at material

6

l

l

Fatigue Life Prediction of Composites and Composite Structures

selection and optimization of specimen configuration. Often, experimental programs of this class are realized within the framework of an industrial application, such as the wind turbine rotor blade industry for which a significant number of fatigue databases exist, e.g., Refs. [16, 17], or the aerospace industry. In such cases, usually a building block testing approach [49] is followed in order to avoid uncertainties led by the theoretical models needed for a layer-tolaminate approach. Component testing for research purposes—aimed at the development of analytical models for the modeling and subsequent prediction of the fatigue life of the examined components. Representative works in this class include those on adhesively bonded and bolted joints that are used as structural components in a wide range of applications, e.g., Refs. [50–54]. Specimen/component/full-scale testing for design verification—refers to experimental programs performed in order to validate the design of a structure, normally based on quasistatic load cases. The design verification is usually performed by applying representative constant amplitude fatigue loading, e.g., at the serviceability limit state, defined as the load that produces a maximum deformation equal to the span/600 for an FRP bridge deck in Ref. [55], or by using accelerated testing for simulation of the long-term behavior of the examined materials. Full-scale testing is performed in order to validate the design of a prototype (verify that its lifetime is at least as long as expected) and measure the developed damage throughout this lifetime. An example of full-scale testing for the validation of the design of a composite sandwich vehicular bridge deck is described in Ref. [56]. The authors performed a series of experiments, from small-scale specimen testing to full-scale component experiments in order to confirm that the GFRP sandwich deck with structural balsa core fulfills all the requirements concerning serviceability, ultimate limit state, and fatigue. Thanks to full-scale testing, size or free edge effects (introduced when shifting from the specimen to the full-scale application) are eliminated and credible results regarding the fatigue life of the final structure can be obtained. The high cost and time limitations are the disadvantages of this type of test.

Although full-scale experimental procedures can provide valuable data for the evaluation of the integrity and the structural performance of engineering structures, they are very expensive. Therefore, the development of virtual testing tools, able to simulate the fatigue performance of engineering structural elements/engineering structures based on information from specimen experimental campaigns (at different scales) is highly desirable. Depending on the selected type of the experimental program, the researcher must determine the termination criterion. This can be a failure criterion referring to material rupture, e.g., Refs. [10, 14], or related to predetermined stiffness degradation, e.g., Ref. [57] or even a design objective, e.g., exceed a certain number of cycles without failure, e.g., Refs. [54, 55] or without obvious damage (e.g., any significant crack). The recorded output is also related to the aim of each experimental program. Simple and relatively cheap measurements are required when the objective is the derivation of the S-N curves of the examined material, e.g., Refs. [10–12, 14]. However, more expensive configurations are needed when the output must provide information about strain development during the fatigue life (need for strain gages or clip gages, e.g., Ref. [17]), crack propagation measurements using crack gages, e.g., Refs. [52, 58], and even more sophisticated measurements, such as acoustic emission (AE) as described in Refs. [59, 60].

Fatigue life modeling and prediction methods for composite materials and structures

7

1.2.2 Fatigue test parameters There are a number of steps that have to be followed for the design of a fatigue-testing program. Decisions must be taken concerning several parameters that affect the test results to a greater or less extent. Among these, the following can be identified: l

l

l

l

Loading pattern: Although the majority of existing experimental databases contain data obtained from constant amplitude testing, experimental programs may contain more complicated loading patterns such as block loading and variable amplitude loading and are not necessarily limited to uniaxial loading. In fact, in the case of anisotropic composite materials, multiaxial stress states can be developed even by the application of uniaxial loading on the specimens, see, for example, Ref. [61]. The necessity for these experimental programs is multiple. Block loading is used to investigate the sequence effect on the fatigue life of composite materials, see, for example, Ref. [62]. Load sequence effects, especially the effect of creep and/or recovery on the fatigue performance of composite laminates have been investigated also by using interrupted loading patterns as described in Refs. [24–26, 39] and Chapter 4. Variable amplitude loading is performed in order to investigate the material behavior under representative, realistic, loading cases and provide reliable fatigue life prediction results, e.g., Ref. [63]. It also allows fatigue life prediction methodologies to be validated via the comparison of theoretical predictions with experimental data and occasionally serves as a mean for design verification. Multiaxial loading (due to its complexity, normally of constant amplitude) is used for the investigation of material behavior under complex stress states and for the development of models for the prediction of fatigue life under such conditions, e.g., Refs. [21, 22]. Control mode: In general a fatigue test can be performed under load or under displacement (or strain) control. The former case results in material failure after a number of applied load cycles. Since the load is kept constant, deformation is increased with load cycles due to the damage accumulated in the material. This type of control model is preferred for the derivation of standard S-N curves, e.g., Refs. [10–12, 14], to examine the sequence effects on fatigue life, e.g., Ref. [62], and also for the application of spectrum loading patterns on a specimen or structural component [63]. On the other hand, the selection of the displacement control mode leads to smoother damage development. The load is continuously decreased during fatigue loading under displacement control and therefore the examined material does not fail suddenly. In this case, other types of failure criteria must be established and serve as test termination criteria. This kind of control mode is preferred in fatigue fracture testing where stable crack propagation that can be recorded during fatigue life is desirable, e.g., Ref. [64]. Stress ratio: The type of the applied load, whether tensile or compressive or even a combination of both, can be easily determined by the stress ratio, i.e., the ratio of the minimum over the maximum applied cyclic stress (R ¼ σ min/σ max). Composite materials behave differently under tension and under compression since different mechanisms are developed under these loading patterns. When tensile loads are applied, fatigue failure is mainly fiber-dominated, whereas under compression the role of the matrix, fiber misalignment, material defects, etc. is more pronounced. Therefore, the successful design of a testing program premises correct selection of the examined loading cases, in keeping with the application for which the material is intended. Testing frequency/strain rate: A limiting factor in the fatigue testing of composite materials is frequency. In contrast to metals, the fatigue life of composite materials is considerably affected by the testing frequency. Researchers in this subject agree that the dependence of fatigue life on loading frequency is due to the heating of the material at higher frequencies, or creep fatigue at lower frequencies, or the interaction of both [65–67]. Mechanical energy dissipated during each stress-strain hysteresis loop is transformed into heat and subsequently results in a greater localized temperature rise in the material. When the energy is such that it cannot be rejected into the environment and produces temperatures close to or even higher

8

l

l

Fatigue Life Prediction of Composites and Composite Structures

than the glass transition temperature of the matrix, which is a viscoelastic material, fatigue life is considerably decreased. This is a common phenomenon observed at elevated test frequencies, although usually different for each material system, depending on the matrix material, fiber orientation in the laminate layers, the geometry of the specimens, etc. Standards concerning the derivation of S-N curves for composite materials, e.g., ASTM D 3479-76 [68] or DIN 65586 [69], refer to continuous loading, under the same constant amplitude load, until failure. As far as testing frequency is concerned, no directions are given, and the only prerequisite is that no significant changes in temperature must be noted. Keeping a constant frequency during a fatigue test results in different strain rates for different stress levels. On the other hand, to achieve a constant strain rate for all applied cyclic stress levels, the user must select multiple frequencies, i.e., lower frequencies for low stress levels and higher frequencies for the higher stress levels. A procedure for defining the test frequency was established in Ref. [70]. It was proposed that the testing frequency at a certain load level can be determined by using a constant energy approach (energy-load*ε2). However, real engineering structures made (entirely or partly) of polymeric matrix composite materials, like airplanes, helicopter, or wind turbine rotor blades, etc., do not operate continuously; they stop and “rest” without any load being applied for a large percentage of their lifetime. Therefore, it is questionable as to whether fatigue design allowables used in design can be determined under continuous loading conditions or must be derived for conditions as similar as possible to the actual operational conditions [24, 71]. Waveform: The shape of the applied waveform can also affect fatigue results. The sinusoidal waveform is the most commonly used since it can be easily generated, even in simple testing rigs, and can be assumed as being the more realistic one compared to other types of loading that represent sudden changes, like the triangular, step (square), and sawtooth waveforms. For the same maximum applied load, waveforms that possess higher mean loads (like the square) and also allow for the application of the maximum load for a longer period (creep-fatigue) are more damaging for the material, especially in low cycle fatigue regimes where the creep fatigue effect is more pronounced. More sophisticated loading waveforms can be selected (see, e.g., Refs. [24–26, 39, 72, 73] and Chapter 4, in order to investigate fatigue/creep/recovery effects on the fatigue life of composite materials. A characteristic comparison between sinusoidal, triangular, and square waveforms for the GFRP laminate is presented in Ref. [74] (see Fig. 2.4 in Ref. [74]). Testing temperature: The majority of testing programs in the literature refer to experimental results obtained under ambient environmental conditions. This is because this type of test is simpler, less expensive, and provides basic information about material fatigue behavior. However, in reality, structures are subjected to combined thermomechanical loading patterns, see, e.g., Refs. [52, 74], and therefore knowledge regarding their behavior under similar conditions becomes essential for credible design. In general, higher testing temperatures, especially in the range of the glass transition temperature of the composite, decrease the fatigue (and static) strength of composite materials.

1.2.3 Fatigue nomenclature Conventionally, in fatigue, the following abbreviations are used: CA: constant amplitude loading H-L: high-low combination in a two-stage block loading pattern L-H: low-high combination in a two-stage block loading pattern L-H-L…: multiple block loading patterns describing the load sequence VA, irregular, spectrum, or random: refers to the loading under a variable amplitude fatigue spectrum

Fatigue life modeling and prediction methods for composite materials and structures

9

σ max ¼ maximum applied cyclic stress σ min ¼ minimum applied cyclic stress σ m ¼ mean stress σ a ¼ cyclic stress amplitude Δσ ¼ cyclic stress range R-ratio: the ratio of minimum over maximum cyclic stress. This ratio defines the loading patterns that might be of: T–T: tension-Tension loading, when 0  R < 1 C-C: compression-compression loading, when 1 < R < +∞ T-C or C-T: combined tension-compression loading when ∞ < R < 0 Special case: R ¼ 1 when the compressive stress amplitude is the same as the tensile stress amplitude and the mean stress equals zero. This is known as reversed loading. f: test frequency, measured in Hz, or loading cycles per second

The basic fatigue terminology together with representative constant amplitude and variable amplitude loading patterns are schematically presented in Figs. 1.1–1.3.

1.3

Fatigue life prediction of composite materials and structures—Past and present

1.3.1 Overview The introduction of composite materials in a wide range of structural components attracted the attention of the scientific community and led engineers to reconsider fatigue as a primary loading pattern that dominates failure, even for structures where traditionally fatigue was not considered an issue. There is a long list of reasons for smax

1 cycle

Cyclic stress or strain

sa

smean

Δs

smin

T=1/f

Time

Fig. 1.1 Basic fatigue terminology.

10

Fatigue Life Prediction of Composites and Composite Structures

Fig. 1.2 Representative constant amplitude loading patterns.

Fig. 1.3 Example of irregular fatigue time series.

which accurate fatigue life prediction is critical for composite structural components or structures; some of them are listed below: l

Composites are used for critical structural components and nowadays they participate as an equal material candidate, compared to the traditionally used steel, aluminum, or concrete, in

Fatigue life modeling and prediction methods for composite materials and structures

l

l

l

11

emerging structures. This development changes the common perception concerning the sensitivity of each structure to fatigue. For example, a concrete bridge is not fatigue-sensitive since the dead loads are significantly higher than the live loads. However, fatigue becomes an issue for a lightweight composite bridge. On the other hand, the use of composites as the main structural material in emerging structures that must bear significant fatigue loads during operation, such as airplanes, means that accurate fatigue life modeling is essential. The hitherto followed practice for product development was based on an iterative process whereby a prototype was built and tested against real, or realistic, loading patterns. However, this process is costly and time consuming. The ability to simulate the fatigue behavior of the material, structural component, and/or structure reduces the cost and in addition, allows the development of a wider range of products without the need for increasing the number of physical prototypes. The durability of composite structures is also an important factor. The danger of evaluating durability on the basis of static strength calculations is that the durability impact of cyclic loads is likely to be disregarded. The introduction of fatigue life prediction methodologies into durability simulation procedures allows the assessment of the durability performance early in the product development process and the establishment of clear recommendations for guiding major design choices. Unidirectional composite materials are in general brittle and behave linearly under load. Their failure is sudden, without any prior notice, and the accurate modeling of their behavior and prediction of their fatigue life are therefore of major importance.

1.3.2 Fatigue life prediction theories Mathematical models were developed in order to describe fatigue damage analytically and finally predict the fatigue lifetime of composite materials. Based on previous experience of fatigue life prediction for metallic materials, similar measurable material characteristics were selected, according to the case studied, to form the fatigue damage metric. Material damage was then measured based on the degradation of that quantity with loading cycles. Different approaches have been adopted, based on different damage metrics for measuring fatigue damage accumulation. The aim of such studies was to establish a process requiring a minimum of experimental data, while reliably predicting the condition of the material. A fatigue life prediction methodology is usually based on the development of empirical relationships between the applied loads and the fatigue lifetime of the examined materials at different scales depending on the followed approach. The objective of the theoretical approaches is to replace, in a way, the need for excessive experimental campaigns and develop virtual testing environments, which, after validation can be used to satisfy a multitude of tasks. Virtual testing serves for reducing physical prototypes and complicated experimental campaigns. Nevertheless, the existence of validated fundamental modeling and analysis methods does not mean that testing is not necessary. It only implies that less validation against actual data will be required at complex and large-scale structural levels [75], as described in Chapter 18. The drawback of virtual testing is the lack of confidence due to the lack of validation, as well as the lack of certification processes for virtual testing environments [75], especially given the scarcity of structural failure data against which to benchmark.

12

Fatigue Life Prediction of Composites and Composite Structures

Empirical, phenomenological methods and especially progressive damage methods have not yet received enough verification and validation to be used for structural analysis [76]. Theoretical models can be classified into two major categories [77]. The first consists of those theories that are based on macroscopic failure criteria and theoretical formulations to predict life under constant or variable amplitude loading. Theories in this category do not take into account the experimental observation of the damage mechanisms and their development during fatigue loading. The characteristic of the second group of theories is that they are all based on actual damage measurements during fatigue life. Thus, a damage metric exists that is used as an indicator of damage accumulation. According to the damage metric, these theories can be further classified into subcategories: strength degradation fatigue theories, where the damage metric is the residual strength after a cyclic program; stiffness degradation fatigue theories, where stiffness is conceived as the fatigue damage metric; and finally, actual damage mechanism fatigue theories based on the modeling of intrinsic defects in the matrix of the composite material that can be treated as matrix cracks. The ideal fatigue theory is described by Sendeckyj in Ref. [77] as one based on a damage metric that accurately models the experimentally observed damage accumulation process, takes into account all pertinent material, test and environmental variables, correlates the data for a large class of materials, permits the accurate prediction of laminate fatigue behavior from lamina fatigue data, is readily extendable to twostage and spectrum fatigue loading and takes into account data scatter. These requirements cannot be met simultaneously for many reasons [77] and theoretical models that address only some of them have been introduced. For predicting the fatigue life of structural components made of composites, at least two alternative design concepts could be used: the damage tolerant (or fail-safe) and the safe-life design concepts. In the former, it is assumed that a damage metric, such as crack length, delamination area, residual strength or stiffness can be correlated to fatigue life via a valid criterion. The presence of damage is allowed as long as it is not critical—i.e., it cannot lead to sudden failure. In the latter—safe-life design situations—cyclic stress or strain is directly associated with operational life via the S-N or ε-N curves. The structure is allowed to operate since no damage is observed, e.g., before the initiation of any measurable cracks.

1.3.2.1 Macroscopic failure theories—Also designated empirical theories This is the broadest group of theoretical models for the modeling and prediction of the fatigue behavior of composite materials. Mainly because of the simplicity of the necessary measurements during fatigue loading, this type of modeling was adopted far more rapidly than the others. August W€ohler, as far back as the 1850s, conceived the idea of representing cyclic stress against the number of cycles to failure in order to quantify the results of his experimental program. The only input required with regard to experimental data consists of pairs of a number of cycles up to failure and the corresponding alternating stress or strain parameter. The S-N or ε-N curve of the material is then determined under the applied loading condition.

Fatigue life modeling and prediction methods for composite materials and structures

13

However, another half century went by before the introduction of the first mathematical model to describe this relationship. In 1910 Basquin stated that the lifetime of the material increases as a power law when the external load amplitude decreases. The value of the exponent, which denotes the slope of the S-N curve, is related to the examined material. It is now documented that for composite materials this exponent ranges between 7 and 25, being higher (less steep curve) for unidirectional carbon and lower (steeper curve) for multidirectional glass fiber composites. Although laws of this kind are empirical and do not have any explicit physical meaning, they follow the accumulation of the microscopic damage of the examined material which they finally describe [77a]. The choice of a particular (and appropriate) fatigue theory for composite materials is based on the material’s behavior under the given loading pattern and the experience of the user. For uniaxial loading, the established S-N curves for metals were initially adopted for composites as well. Traditionally, the S-N data are fitted by a semilogarithmic or logarithmic equation. Other types of S-N curve formulations are usually employed to take into account the statistical nature of fatigue data, e.g., Refs. [78, 79]. Recently, artificial intelligence methods and soft computational techniques have been introduced for the fatigue life modeling of composite materials. Artificial neural networks [80], adapted neuro-fuzzy inference systems [81], and genetic programming (GP) [82] have proved to be very powerful tools for modeling the nonlinear behavior of composite laminates subjected to cyclic and constant amplitude loading. They can be used to model the fatigue life of several composite material systems, and compare favorably with other modeling techniques. Finding the appropriate S-N curve type for the examined material is not simple since there is no rule governing this selection process. The best S-N curve type is the one that can best fit the available fatigue data. It is nowadays accepted that power curve-like equations can be better adapted than linear equations for composite material fatigue data [82]. Echtermeyer et al. [83] concluded that the fatigue data for glass fiber-reinforced composites can be well described by common strain-life (ε-) fatigue standard curves, as long as some fibers in the material run in the load direction for reversed loading, flexural loading, and tensile loading with similar R-ratios. These constraints, and the numerous assumptions imposed by the mathematical models boosted the development of material-independent methods for the derivation of S-N curves, like the recently introduced computational techniques presented in Refs. [80–82]. The resulting S-N curves do not follow any specific mathematical form; they simply follow the trend of the available data, each time giving the best estimate of their behavior, without taking the mechanics of each material system into account [82]. As presented in Fig. 1.4, the application of different methods for derivation of the S-N curve of the examined material leads to different types of curves. For the examined case, the model proposed by Sendeckyj (wear-out model) [78] and the curve estimated using GP [82] seem to be more accurate than the linear regression and Whitney [79] models. The aforementioned fatigue theories are limited to uniaxial loading and do not take into account the effect of other stress components on fatigue life. This assumption seems reasonable for the highly anisotropic composite materials and has even been

14

Fatigue Life Prediction of Composites and Composite Structures

GP Whitney Wear out model Linear regression Exp. data

smax (MPa)

360

240

120

0

1

2

3

4

5

6

7

Log(N)

Fig. 1.4 Application of different S-N curve formulations for description of constant life fatigue behavior of multidirectional glass-epoxy laminate with stacking sequence [(45/0)4/45]T [82].

adopted by the scientific community in the past. For example, a composite wind turbine rotor blade was treated by state-of-the-art design codes, e.g., Refs. [84, 85], as a typical beam-like structure in which fatigue life calculations are limited in that they consider only the action of the normal stress component in the beam axis direction. However, regardless of the effect of various model parameters on the accuracy of the theoretical predictions, a strong effect of the transverse normal and shear stress components on fatigue life has been recorded, e.g., Ref. [63]. The fatigue behavior of a multidirectional composite laminate was evaluated under multiaxial cyclic stress states in Ref. [63]. For on- and off-axis loading, even a few degrees of misalignment yielded substantially different theoretical predictions. This phenomenon proves the significant role of transverse to the fiber normal and shear stress components on the fatigue life of the investigated laminate, since even slight deviations of the loading direction affect the values of the transverse normal stress σ 2 and shear stress σ 6 components in the principal material system, and drastically change the expected number of passes of the applied load spectrum. A misalignment of 3°, for example, from the ideal axial loading direction produces a plane stress state with σ 1 ¼ 99.7% of σ x, σ 2 ¼ 0.27% of σ x, and σ 6 ¼ 5.23% of σ x. However, despite the low values of transverse normal and shear stress components, life prediction is apparently affected as shown, for example, in Fig. 1.5 for specimens cut at 0 degree and tested under the applied spectrum. According to the example presented in Fig. 1.5, life is reduced by almost 50% due to 3 degrees of off-axis loading. Similar conclusions were drawn for CA loading [14] where a life reduction of the order of 25%–40% was predicted, depending on the value of the stress ratio R, due to the presence of transverse normal and shear stresses.

Fatigue life modeling and prediction methods for composite materials and structures

MWX 0 degrees 0 degrees 1 degrees 3 degrees

400

s max (MPa)

15

340 smax =310 MPa

280 8.29 passes

220

10-1

100

101 Spectrum passes

17.07 passes

102

Fig. 1.5 Effect of load misalignment on life prediction of specimens cut at 0–3 degrees and tested under irregular fatigue spectrum.

Under complex stress states, multiaxial limit state functions are introduced, usually being generalizations of static failure theories to take into account factors relevant to the fatigue life of the structure, i.e., number of cycles, stress ratio, and loading frequency. Due to the fact that the damage-tolerant fatigue design of composite structures is still in its infancy and that much more research is needed to establish reliable methodologies with general applicability, most of the industrial applications for this type of material lie in “safe life” components. Numerous attempts at generalizing a multiaxial static failure theory to take fatigue into account can be found in the literature, e.g., Refs. [10, 18, 46, 86–88]. However, most of these studies treat the subject of life prediction only partially under an irregular plane state of stress. They concentrate mostly on the introduction and validation of fatigue strength criteria suitable for CA multiaxial proportional loading without addressing the question of life prediction under irregular load spectra. An exception to the above is presented in a series of publications [89–92] in which a complete life prediction methodology, for even a 3D state of stress, is established. This approach is based on a “ply-to-laminate characterization” scheme, dealing with damage accumulation under VA loading by adopting appropriate residual strength engineering models. Another paradigm of a complete life prediction methodology under irregular plane stress histories and an experimental database for a glass fiber-reinforced polyester (GFRP) [0/45]S laminate was presented in Refs. [21, 22]. Fatigue strength-allowable values in the various material symmetry axes were derived based on the “direct characterization” approach for a number of different CA loading cases. Experimental verification of the entire methodology is performed by means of VA complex stress tests in Ref. [63]. The application of multiaxial fatigue life prediction methodologies allows the efficient and reliable prediction of the lifetime of a composite material or structural

16

Fatigue Life Prediction of Composites and Composite Structures

300

Δs hp (MPa)

100

–100 N=1 N = 10 3 N = 10 4 N = 10 5 N = 10 6

–300 –300

–200

–100

0 100 Δsax (MPa)

200

300

Fig. 1.6 FTPF predictions vs experimental data from woven GRP cylindrical specimens loaded under hoop and axial stresses [21].

component under complex stress states. For example, the fatigue failure loci of woven glass polyester cylinders loaded under biaxial hoop, σ hp, and axial, σ ax, stresses can be derived by using the failure tensor polynomial in fatigue (FTPF, Ref. [21]), see Fig. 1.6. As shown in this figure, the theoretical predictions are very well corroborated by the available experimental data (retrieved from Ref. [46]), proving the validity of the applied multiaxial fatigue strength theory.

1.3.2.2 Strength and stiffness degradation fatigue theories Strength and stiffness degradation fatigue theories have been introduced in order to model and predict the fatigue life of composite materials by taking into account the actual damage state, expressed by a representative damage metric of the material status. The damage metric is usually the residual strength or the residual stiffness. Failure occurs when one of these metrics decreases to such an extent that a certain limit is reached. Residual strength theories assume that failure occurs when the residual strength of the material reaches the maximum applied cyclic stress level. On the other hand, stiffness degradation theories are not linked to the macroscopic failure (rupture) of the examined material but rather to the prediction of its behavior in terms of stiffness degradation. Failure can be determined in various ways, e.g., when a predetermined critical stiffness degradation level is reached, or when stiffness degrades to a minimum stiffness designated by the design process in order to meet operational requirements for deformations, or even as a measure of the actual cyclic strains—e.g., failure occurs when the cyclic strain reaches the maximum static strain [93].

Fatigue life modeling and prediction methods for composite materials and structures

17

Both residual strength and residual stiffness methods have advantages and disadvantages. As mentioned in Ref. [94] residual strength prediction has been the subject of numerous investigations during recent decades for several good reasons: modeling the loss of strength of a material after cyclic loading can be a powerful tool in the development of life prediction schemes, especially when dealing with variable amplitude or spectrum loading, since it offers a physically meaningful alternative to empirical damage accumulation rules such as the Palmgren-Miner rule. In addition, knowledge regarding residual strength in the case of composite structures designed and certified against an envelope of extreme static loads, also undergoing cyclic loading, enables the designer to certify their capacity to bear the design load during their operational life. According to state-of-the-art design practices, large safety factors take into account the loss of strength. Reliable prediction of static strength degradation would help to reduce such safety factors and ultimately achieve full use of the composite material. Residual strength fatigue theories can be successfully used in predicting the fatigue life of composite materials by means of a progressive damage modeling process, e.g., Ref. [95]. Residual stiffness fatigue theories were developed to overcome two of the major weaknesses of the residual strength fatigue theories: residual strength exhibits minimal decreases with the number of cycles until a stage close to the end of a lifetime when it begins to change rapidly. Chou and Croman described this behavior as “sudden death” in their work in 1979 [96]. On the other hand, stiffness exhibits greater changes during fatigue life. Another superior characteristic of the residual stiffness fatigue theories in comparison with their counterpart lies in their nondestructive inspection option of fatigue damage growth. However, while stiffness can be easily and continuously monitored, stiffness changes as a function of fatigue loading cannot be easily predicted [78] since this involves the accurate modeling of the damage accumulation process. A stiffness degradation model was introduced in Refs. [97–99] and has also been presented in the first version of this book (Chapter 4, 2010) in order to demonstrate qualitatively that cumulative damage rules are not needed when applying residual stiffness models in terms of damage growth rate equations to the problem of variable amplitude loading. Residual stiffness fatigue theories can also be implemented in order to derive fatigue design curves that do not correspond to failure but to a certain percentage of stiffness degradation. This concept was introduced by Salkind in 1972 [100], when he suggested drawing a family of S-N curves, being contours of a specified percentage of stiffness loss, to present fatigue data. The same concept was adopted later [57] in correlation with the statistical analysis of fatigue data to develop the so-called stiffness controlled curves (Sc-N). An empirical model was presented and validated in Ref. [57] for the determination of design curves, which do not correspond to fatigue strength, but to a predetermined value of stiffness reduction by using only a portion of the fatigue data, needed for the determination of a reliable S-N curve. Furthermore, S-N curves at any reliability level were determined after a statistical analysis of each set of fatigue data. It was shown that these two kinds of design curves were comparable. A heuristic procedure was established to define the so-called Sc-N curves providing information on both the

18

Fatigue Life Prediction of Composites and Composite Structures

allowable stiffness degradation and the probability of survival. The process is schematically presented in Figs. 1.7 and 1.8. The authors derived both the Sc-N curves and the traditional S-N curves for different reliability levels. Comparing these two kinds of fatigue design curves it was concluded that they could be correlated in the following sense. To any survival probability level, PS(N), corresponds a unique 1.0

F(EN /E1)

0.95

0.5 0.50

0.87

0.0

0.6

0.7

0.8 EN/E1

0.96

0.9

1.0

Fig. 1.7 Sampling distribution of stiffness degradation data, R ¼ 0.1, specimen cut at 15 degrees off-axis [57].

110 Exp. data 50% & 95% reliabililty Stiffness based

smax (MPa)

95

80

0.87

0.96

65

50

104

105 N

106

Fig. 1.8 Sc-N vs S-N curves. R ¼ 0.1, specimen cut at 15 degrees off-axis [57].

Fatigue life modeling and prediction methods for composite materials and structures

19

stiffness degradation value, EN/E1, which can be determined from the cumulative distribution function, F(EN/E1), of the respective data. It is this value of EN/E1 for which F(EN/E1) ¼ PS(N), see Fig. 1.7. Observing the two different curves derived as stated above, it was concluded that they are similar for all cases considered in Ref. [57], with the Sc-N being generally slightly more conservative. Therefore, a Sc-N curve, providing information on both survival probability and residual stiffness, can be used in the design process. The derivation procedure of an Sc-N curve is schematically demonstrated in Figs. 1.7 and 1.8 for a selected data set. In Fig. 1.8 both design curves, for 50% and 95% survival probability, are plotted together along with experimental failure data. It can indeed be observed that Sc-N and S-N curves from each set lie very close and that the former type of design curve is slightly more conservative. Using a design allowable the Sc-N at EN/E1 ¼ 0.96, as derived from Fig. 1.7, a reliability level of at least 95% is guaranteed while stiffness reduction will be less than 5%. A stiffness-based fatigue life prediction methodology was presented lately, proposing the modification of the classical lamination theory in order to be able to calculate the fatigue-induced stiffness decrease of multidirectional CFRP laminates [101]. The potential of the proposed method has been examined by comparing theoretical results to experimental data from unidirectional and multidirectional CFRP laminates [102]. The obtained experimental data on the layer level were used as input parameters for the software-based fatigue life prediction method “FEMFAT Laminate” which, following a lamina-to-laminate procedure, enables the assessment of fiber and inter fiber fracture even for nonproportional loading. Fatigue life of a unidirectional 45 degrees laminate and a multilayer composite was calculated and the accuracy of the theoretical predictions was validated by comparisons to experimental results. The FEMFAT laminate is described in details in Chapter 18. A significant drawback of residual strength and stiffness fatigue theories is their inability to take into account complex loading patterns and predict the lifetime of a composite material when multiaxial stress fields develop. An alternative approach to this problem, addressing the effect of complex loading profiles on the lifetime prediction, is presented in Chapter 14.

1.3.2.3 Multiscale modeling During the last two decades, a lot of efforts have been allocated to the development of progressive damage analysis (PDA) methodologies for fatigue life modeling and prediction. A certain number of such methodologies is based on multiscale modeling, implementing combinations of the abovementioned approaches, i.e., residual strength, stiffness based, etc. The concept of each multiscale modeling procedure is schematically shown in Fig. 1.9. Composite material systems generally exhibit a range of behaviors on different length and time scales, and any multiscale problems have to be addressed by appropriate methods [103]. Multiscale modeling approaches can provide an alternative to extensive testing, however, these approaches are also plagued by the large input data set requirements to calibrate the model parameters at the different scales. It is clear

1D orientation

2D orientation

3D orientation

Fig. 1.9 Schematic representation of multiscale modeling.

Matrix

Fiber

RVE

90°

–θ

90°

0°

–θ

+θ

0°

+θ

y z

θ

x

Shear Web: Infusion Prepreg SPRINT Corecell

Spar: Glass Carbon Airstream

Root: SPRINT Infusion Prepreg

Infusion Core Prepreg Core

Shell: SPRINT Prepreg Infusion

Priming: UV Gelcoat CR3400 SPRINT IPT

Finishing: PU Paint Epoxy Geleoat

Structural adhesive: SP340 SP340LV

20 Fatigue Life Prediction of Composites and Composite Structures

Fatigue life modeling and prediction methods for composite materials and structures

21

that there is a need for methods that require the minimum of experimental data for the parameter calibration and the fatigue degradation prediction of multiple material configurations of interest, as described in Ref. [104]. In this context, multiscale fatigue life modeling theories are very promising, since, in theory, they can respond to the industry demand for reduction of physical prototypes and derivation of virtual testing protocols for fatigue simulations of engineering structures. Nevertheless, as experience has shown, and, as it is well documented in Ref. [76] and other works, a long way needs to be covered before having reliable multiscale modeling methodologies in the form of virtual testing tools for life prediction of engineering composite structures.

1.3.2.4 Diverse fatigue considerations A common characteristic of all the aforementioned theoretical models is that they refer to deterministic material properties and use deterministic loading and material parameters. However, the phenomenon of fatigue is of a stochastic nature, and in general cases, it is not possible to estimate the fatigue loading or material properties in a deterministic way. Probabilistic fatigue analysis is based on the hypothesis that every parameter affecting fatigue life is described by a distribution instead of a deterministic value. Therefore, the loading of the structure can follow a Normal or Weibull distribution, the strength of the material can follow a Weibull distribution and so on. In contrast to the deterministic approach, in which each material property has a value, for example, and each S-N curve is normally derived for the mean or median lifetime, in probabilistic fatigue analysis the material properties have a value for a certain level of reliability and S-N curves for assumed reliability levels are used. The material properties are the most common sources of uncertainty in a fatigue life calculation. This is because usually, only a limited number of experimental data exist but also because the same material may encounter a wide range of environments and loading patterns during its lifetime. The test conditions in the laboratory cannot be made to reflect every possible combination of these two parameters. The use of probabilistic models for the description of the fatigue behavior of composite materials and quantification of the uncertainties in fatigue life prediction form the subjects of Chapter 17. In addition to the difficulties encountered during the adoption of an uncertainty modeling technique that is required to address the stochastic nature of both the material properties and external loading, a main reason for the delay in the development of probabilistic fatigue failure theories is that a potentially accurate model would be extremely computationally and experimentally expensive, see Chapter 17. Assistance with this problem can be provided by the novel computational techniques that have emerged as one of the most powerful modeling tools in a number of scientific domains. In artificial intelligence, stochastic programs work by using probabilistic methods to solve or partially solve problems. In engineering, artificial neural networks and GP have been used for the optimization of design methods and manufacturing processes, e.g., Ref. [105]. The modeling of the fatigue life of composite materials and structures is a topic that has been addressed by this type of analytical tools only during recent years. Until recently, artificial neural networks

22

Fatigue Life Prediction of Composites and Composite Structures

Exp. data Predicted data

0 degree

100

sa (MPa)

80

60 15 degrees

40

20

45 degrees 75 degrees 90 degrees

0

103

104

105

106

N

Fig. 1.10 Experimental data vs ANN modeling. Tension-Tension fatigue, R ¼ 0.1. Coupons cut at 0, 15, 45, 75, and 90 degrees; 30% used for training set, 70% for test set [106].

constituted the only method used for the fatigue life modeling of composite materials and structures, e.g., Ref. [106] (see Fig. 1.10), used also in order to model the effect of mean stress on fatigue life and derive CLDs for short fiber composite materials, e.g., Refs. [80, 107] (see Fig. 1.11). New tools like GP and adaptive neuro-fuzzy inference systems have been applied lately [81, 82] with the same goal and were used in addition 160

sa (MPa)

120

N = 10 4 N = 10 5 N = 10 6 N = 10 7 PWL CLD ANN, CLD

R = –1

R = 0.1

R = 10

80

R = 0.5

40

0 –300

–150

0 sm (MPa)

150

300

Fig. 1.11 Piecewise linear constant life diagrams for GFRP multidirectional laminate [80].

Fatigue life modeling and prediction methods for composite materials and structures

23

in order to predict the fatigue behavior of structural joints under thermomechanical loading patterns [105].

1.3.3 Fatigue life prediction under complex irregular loading All the aforementioned fatigue theories were established mainly on the basis of relatively simple constant amplitude loading experiments. In reality, a procedure for the estimation of the fatigue life of a composite material or composite structure that operates in the “open air” must comprise a number of steps, since the real loading is never (or almost never) of constant amplitude. Moreover, the real loading is not known in a deterministic way. Wind turbine rotor blades, for example, are safe-life designed composite structures. The nominal lifetime for cost-effectiveness of a rotor blade is a 20–30 years interval during which the blade experiences in the order of 108–109 loading cycles of variable amplitude and mean value. The load spectrum of a rotor blade is composed of different stochastic and deterministic cases defined by strong loading fluctuations due to varying wind speed, atmospheric turbulence, and topography effects. Operational peculiarities in a wind turbine such as stall, grid, or control failure, etc., also give rise to additional strong excitations of the rotor blade. Although reliable computational tools to deal with such problems have been developed, e.g., Ref. [108], fatigue design calculations are even more difficult and of questionable validity for FRP composites due mainly to the lack of a reliable nonlinear damage accumulation rule and limited knowledge concerning the fatigue properties of specific FRP laminates, especially at such high cycle loading levels. It is obvious therefore that the application of any fatigue life prediction methodology is founded on assumptions as from its first stage, the load case definition. In order to apply any fatigue life prediction methodology, the following steps must be implemented. The derivation of the most representative load time series is the first step, normally based on field measurements and aeroelastic calculations, see Fig. 1.12.

Fig. 1.12 Load case definition.

24

Fatigue Life Prediction of Composites and Composite Structures

The resulting stress time series for each stress component can be proportional, or independent of each other (having different frequencies), depending on the excitation that causes the loads. The derivation of the material fatigue properties is the second step, normally based on extensive experimental programs under quasistatic, constant, block, and variable amplitude loading patterns. This stage includes material selection and preparation of the specimen (see Fig. 1.13). In the case of constant amplitude fatigue testing, the expected output is the number of cycles under each selected loading level and condition, e.g., application of tensile or compressive loads or even a combination of both. When compressive loads are present the risk of buckling of the usually thin specimens is high. Therefore, antibuckling devices like the one shown in Fig. 1.14 are used to protect the specimen from premature failure. The adoption of methods for the interpolation (modeling) and extrapolation (prediction) of the static and fatigue data follows. During this step, deterministic or stochastic theoretical models can be employed. The use of the selected models (S-N curves, CLDs, residual strength models, residual stiffness models, etc.) permits interpretation of the fatigue data and estimation of the fatigue life of the materials, theoretically, under any applied loading pattern. The method of estimating S-N curves and

Fig. 1.13 Example of material architecture and specimen dimensions.

Fatigue life modeling and prediction methods for composite materials and structures

25

Fig. 1.14 Antibuckling device according to DIN 65586.

constructing CLDs based on constant amplitude fatigue data is schematically presented in Fig. 1.15. Three different tests are presented on Fig. 1.15A, each corresponding to different stress levels and resulting in a different number of cycles to failure. As expected, the lower the stress level, the longer the fatigue life of the examined material. Interpolation between the collected fatigue data results in the S-N curve of the material under the selected fatigue conditions, R-ratio, frequency, environment, etc. Data from several S-N curves obtained under different loading patterns (tensile, compressive, and combinations of both) can be plotted together on the σ m–σ a plane and form the so-called CLD in order to take into account the effect of mean stress on the fatigue life of the material. This diagram helps in the derivation of S-N curves under different R-ratios than those derived experimentally. Constant life (CL) lines connecting data from different S-N curves are then estimated. The CL lines can be linear or nonlinear depending on the selected CLD formulation. A comprehensive review and performance assessment of the most recent and commonly used CLDs is presented in Ref. [109]. A relatively new formulation for predicting an asymmetric, piecewise nonlinear CLD was introduced [110]. The development of the new model is based on the representation of the fatigue data on the stress ratio (R)-stress amplitude (σ a) plane instead of the conventional representation on the mean stress (σ m)-stress amplitude (σ a) plane. Comparisons of data from different material systems proved that the PNL model offers better performance than most conventional CLD models found in the literature [110].

26

Fatigue Life Prediction of Composites and Composite Structures

sa1 S-N curve

sa

sa2 sa3

n1

(A)

n2

N

n3

R = –1

sa1 n1 R= 0

sa

R = inf (sm, sa)

n2

sa 3

(B)

(sm, sa )

sa 2

n3

sm

Fig. 1.15 Derivation of S-N curves and construction of CLD.

The application of different formulations for the derivation of the CLDs of multidirectional glass-polyester composite laminate [17] is presented in Fig. 1.16 for demonstration purposes. Details concerning the applied methods can be found in Refs. [109, 110]. The selection of a cycle counting algorithm to summarize the irregular load vs time histories and provide the number of occurrences of cycles of various sizes allows

Fatigue life modeling and prediction methods for composite materials and structures

Used exp. data Pred. CL lines

250 R = –1 200

R = –2.5

27

R = –0.4

sa (MPa)

R =10

150

R = 0.1

100

50

0

(A)

–200

0 sm (MPa)

250

sa (MPa)

400

Used exp. data Pred. CL lines

R = –1 R = –2.5

200

150

200

R= –0.4

R = 10

R = 0.1

100

50

(B)

0

–300

–100

100 sm (MPa)

300

Used exp. data Pred. CL lines

250 R = –1 200

sa (MPa)

R= –2.5 150

R = –0.4

R = 10

R = 0.1

100

50

0

(C)

–300

–100

100 sm (MPa)

300

Fig. 1.16 Results of different CLD formulations for N ¼ 103–107 (linear-a, piecewise linear-b, Harris-c, Kawai-d, Boerstra-e, Piecewise nonlinear-f ) [109, 110]. (Continued)

28

Fatigue Life Prediction of Composites and Composite Structures

250

Used exp. data Pred. CL lines

R= –1

200

R = –0.4

sa (MPa)

R = –2.5 150

R = 10

R = 0.1

100

50

0

–300

(D)

–100

250

100 sm (MPa)

300

Used exp. data Pred. CL lines

R= –1

200

R = –0.4

sa (MPa)

R = –2.5 150

R = 10

R = 0.1

100

50

0

(E)

–300

–100

250

100 sm (MPa)

Used exp. data Pred. CL lines

R = –1

200

R= –0.4

sa (MPa)

R = –2.5

300

150

R = 0.1

100

R = 10

50

0

(F) Fig 1.16, cont’d

–300

–100

100 sm (MPa)

300

Fatigue life modeling and prediction methods for composite materials and structures

29

better management of fatigue data and theoretical models. It is impossible to perform tests to cover all the potential loading cases that a structure may have to face during its operational lifetime. The conversion of the variable amplitude fatigue load cases to constant amplitude loads compensates for this deficiency. Simple methods exist— known as one-parameter techniques—e.g., level crossing counting or peak counting methods, but they are not applicable in the fatigue analysis of composite materials since they do not take into account the significant mean stress effect on lifetime. Rainflow counting, range-pair, and range-mean counting methods seem more appropriate for the analysis of composite material fatigue data, giving similar cycle counting results for most practical applications, see Chapter 11. However, the aforementioned cycle counting methods present a number of deficiencies: rainflow counting cannot be used for cycle-by-cycle analysis and therefore it is difficult to apply this method together with a residual strength fatigue theory. Moreover, rainflow counting algorithms categorize counted cycles, according to their range and mean values, without considering the location of the counted cycle in the time series, and put cycles of the same category in blocks (usually called bins). This process masks any possible load sequence effect and “imposes” the use of the linear Miner damage summation rule. On the other hand, range-pair and range-mean counting mask the presence of large and damaging cycles. Based on the previous comments, it may be concluded that, according to the application and material, the appropriate cycle counting technique has to be very carefully selected. A comprehensive discussion regarding counting techniques can be found in Ref. [111]. A schematic representation of the rainflow counting method is presented in Fig. 1.17. When the stress field is uniaxial, the situation is straightforward. However, in most real cases, a multiaxial stress field develops in the material’s principal coordinate system, even under uniaxial loading patterns, due to the anisotropy of the composite materials themselves (see Fig. 1.18). Consider as an example the case of a 20-m wind turbine rotor blade made of GRP. Under a realistic load case (power production at rated wind speed), the plane stress

Cycle definition

10

n7 R7 R3

5 R2

Δs 7

R4

ï

s m6

nR66

R1

s m1

0

Bin #5

n1 -5

R5

n3 n2

n4

Δs 7

n5

Fig. 1.17 Schematic representation of rainflow counting method.

30

Fatigue Life Prediction of Composites and Composite Structures

s 1(t)

F(t)

s 2(t)

s 6(t)

s 6(t)

s 2(t)

F(t) s 1(t)

F(t) s1(t) s2(t) s 6(t)

Fig. 1.18 Complex stress state developed in principal coordinate system of off-axis uniaxially loaded composite material.

field developed in leading-edge inboard regions is an example of that presented in Fig. 1.19. The part of the finite element method (FEM) model shown corresponds to the cylindrical root and the transition region bridging to the aerodynamic section of the blade. Also shown is part of the inner structure, while grayscale shades correspond to different values of shear flow. Magnified is a typical element and the corresponding in-plane stress resultants, Ni (i ¼ 1, 2, 6). It is indeed observed that the transverse normal and shear components are only small fractions of the axial stress resultant, albeit not negligible in fatigue life calculations as shown above (see Fig. 1.5). When a plane stress state, rather than only a stress component, is present, the calculation of the allowable number of cycles for each applied load case is based on the use of multiaxial fatigue failure criteria. This process is schematically presented in Fig. 1.20. The final step of every fatigue life prediction methodology involves estimation of the fatigue life. This step can be based on damage rules used to quantify the damage accumulated during loading, or on residual strength or stiffness criteria that define failure based on the change of the selected damage metric. To date, there is no widely accepted theoretical model that can guarantee accurate lifetime estimation for any examined composite material system, composite component, or composite structure under any variable amplitude loading. The estimated lifetime can be accurate, conservative or nonconservative (usually called optimistic) according to the examined material, load complexity, available experimental data, selected solvers for each of the steps of the methodology, etc. Unique solutions do not exist; there are more than five “widely accepted” methods for deriving the S-N curves, all of them proved to be good methods for the modeling of a series of composite materials [82, 110]. There are more than three “widely accepted”

Fatigue life modeling and prediction methods for composite materials and structures

31

NXX-STRS. RESLT. VIEW : –1025791. RANGE : 1062700. (Band * 1.0E4) 205.8 176.1 146.5 116.8 87.18 57.53 27.88 –1.765 –31.41 –61.06 –90.71 –120.4 –150.0 –179.7 –209.3 EMRC–NISA/DISPLAY APR/12/01 13:42:10 ROTX Y 6.7 ROTY –10.7 ROTZ 11.2

X Z

6

1

1

1

2

Fig. 1.19 Detail from GRP rotor blade root. Typical state of in-plane stress resultants.

S-N curves and CLD s 1 (t)

s1(t)

Multiaxial failure criterion

Nallowable

s 2(t) s 6(t)

Fig. 1.20 Derivation of allowable number of cycles for uniaxial and multiaxial stress states.

methods for the derivation of the CLDs [110]. There are a significant number of cycle counting methods [111] and fatigue failure criteria for composites under multiaxial stress states [10, 18, 20–22, 86–92]. Finally, there are number of strength, e.g., Refs. [59, 94–96], and stiffness, e.g., Refs. [57, 93, 97–99], degradation theories and linear

32

Fatigue Life Prediction of Composites and Composite Structures

or nonlinear damage accumulation rules for the summation of partial damage and estimation of the fatigue life of the examined material. Nevertheless, no commonly acceptable universal fatigue life prediction methodology has yet been adopted.

1.4

Conclusions—Future prospects

The future of the topic looks prosperous, although very challenging. Several problems have been solved, while others still await a commonly accepted solution. Within the next few years, the scientific community is expected to come up with a credible methodology for the life prediction of engineering FRP structures which will allow rapid and reliable prototyping without the need for the fabrication and testing of numerous physical prototypes. Therefore, the commercialization of a wide range of reliable products will be feasible within shorter time periods and at reduced costs. This process is well assisted by developments in computer science, where computational power has been increased dramatically during the last decades. Although this increase is now slowed down, it is today feasible to use even personal computers to “run” progressive damage models incorporated in finite element analysis software and estimate the fatigue life of an entire structure within hours. Nevertheless, the uncertainty in all steps of any fatigue life prediction methodology hinders the adoption of such methods. Fatigue life prediction methodologies have been presented in the past in order to address the problem with regard to composite materials—however, they are all hampered by the same deficiency. At least one of their steps was inaccurate, thereby rendering the entire method inaccurate. For example, a simplified approach was presented by Nyman in 1996 [112] based on the reasonable assumption that the S-N curves for the laminae as part of the laminate can be different compared to the S-N curves for the layers themselves, due to interlaminar stresses. The author tried to establish a method for derivation of the fatigue curves by using a Tsai-Hill criterion. However, a semi-log representation of the constant amplitude fatigue data was used in the sequel, together with Goodman diagram for the mean stress effect, but both methods subsequently proved to be inaccurate, e.g., Refs. [110, 111]. Other proposed methodologies, e.g., Ref. [63], are hindered by the use of the linear Miner’s damage rule for damage accumulation. It is therefore obvious that certain reliable solutions have to be adopted for specific fatigue life prediction tasks, some of which are listed below. l

The debate on the representation of the constant amplitude fatigue data that began a century ago, with Basquin’s equation, is still alive and well today. Yet in 2010 there is no universal theoretical model able to accurately describe the constant amplitude fatigue behavior of any composite material (in terms of S-N curve) under any thermomechanical loading condition. It is accepted that a log-log representation is sufficiently accurate for most commonly used composite laminates. However, other methods have been proposed to take into account the low cycle fatigue range as well, e.g., Ref. [78], and consider the statistics of the data sample, [78, 79, 113]. The use of quasistatic strength data for the derivation of fatigue curves (as fatigue data for 1 or ¼ cycle) is also arguable. No complete study on this subject exists. Previous publications, e.g., Ref. [114], showed that quasistatic data should not be a part of the

Fatigue life modeling and prediction methods for composite materials and structures

l

l

l

33

S-N curve, especially when they have been acquired under strain rates much lower than those used in fatigue loading. The use of quasistatic data in the regression leads to incorrect slopes of the S-N curves as presented in Ref. [114]. On the other hand, excluding quasistatic data although improved the description of the fatigue data, introduced errors in the lifetime predictions when the low cycle regime is important, as, for example, for loading spectra with a few high-load cycles. As reported by Kawai in Chapter 12, the fatigue behavior of composites becomes more complicated when the influence of temperature is added. To the authors’ knowledge, there is no method in the literature that can address thermomechanical loads and predict the fatigue behavior for general thermomechanical loading patterns, although a lot of research efforts have been devoted to the characterization of the fatigue behavior of adhesively bonded composite joints and composite laminates under different temperature and humidity environments. A small number of modeling approaches have been published, e.g., Refs. [115, 116]. However, in order to accommodate a significant number of parameters that affect the fatigue life of FRP joints or laminates, these phenomenological models adopt a great many assumptions. Therefore, their applicability could not be validated on the data of different material systems. Novel computational techniques seem able to address this problem, as presented in Refs. [105, 117], as well as in Chapter 18. The reduction of the irregular loading spectra to constant amplitude loading blocks is a critical step in a life prediction methodology. To date, among the proposed methods, the rainflow counting together with the range-mean and the range-pair counting methods have been proved the most appropriate for composite materials. However, as presented in Chapter 11, they both present advantages and disadvantages. Rainflow counting cannot be used for cycle-by-cycle analysis and it is therefore difficult to apply this method together with a residual strength fatigue theory. On the other hand, range-pair and range-mean counting mask the presence of large and damaging cycles. A new method presented in Ref. [111] is actually a mixture of both rainflow and range-mean counting. This method, known as “rainflow-equivalent-range-mean counting” has been accurately used together with a strength degradation model in Chapter 11 to produce accurate fatigue life prediction results for a multidirectional composite laminate under variable amplitude loading. Further verification of the method on other types of composite materials and other loading spectra is needed. The mean stress effect on the fatigue life of composite materials and structures is significant. As presented in a number of publications [118–120], this effect can be accurately modeled by means of the so-called CLDs. The most recent CLD formulations were presented by Kawai and Koizumi [119] and in a series of publications by Vassilopoulos et al. [110, 121]. Although from a theoretical point of view the classic representation of the CLD is rational, it presents a deficiency when seen from the engineering point of view. This deficiency is related to the region close to the horizontal axis, which represents loading under very low stress amplitude and high mean values with a culmination for zero stress amplitude (R ¼ 1). The classic CLD formulations require that the CL lines converge to the ultimate tensile stress (UTS) and the ultimate compressive stress (UCS), regardless of the number of loading cycles. However, this is an arbitrary simplification originating from the lack of information about the fatigue behavior of the material when no amplitude is applied. In fact, this type of loading cannot be considered as being fatigue loading, but rather creep of the material (constant static load over a short or long period). [16, 121, 122] Although modifications have been introduced to take into account the time-dependent material strength, their integration into CLD formulations requires the adoption of additional assumptions, see, e.g., Refs. [12, 118].

34 l

l

l

l

Fatigue Life Prediction of Composites and Composite Structures

Although the first S-N theoretical formulation was introduced by Basquin in 1910, it was Palmgren in 1924 and 1936 [113] who realized that “…the variation of fatigue test data is such that there is no pronounced accumulation of values around a mean value, and the minimum life is clearly not governed by laws to more than a trifling extent. Consequently, neither the average nor the minimum life provides a satisfactory basis for a definition”. More than 80 years later, the problem of the statistical analysis of composite material fatigue data still remains unsolved. New methods are proposed to address it, e.g., Ref. [123], where the authors proposed a stochastic damage accumulation model for symmetric composite laminates subjected to fatigue loading. The model is a hybrid strength/stiffness degradation model which utilizes basic probability—stress-life information (P—S-N curves) in conjunction with the evolutionary distribution of stresses on the critical load-bearing plies of a laminate to determine the evolutionary probability of failure of a laminate. In Chapter 17, a general methodology for stochastic fatigue life prediction under variable amplitude loading is proposed, which combines a nonlinear fatigue damage accumulation rule with a stochastic S-N curve representation technique. Engineering structures like bridges or wind turbine rotor blades can experience between 108 and 109 loading cycles during their operational lifetime. On the other hand, the usual range of recorded fatigue data is between 103 and 106–107 loading cycles. It is obvious that a model to extrapolate material behavior is needed in order to estimate the fatigue life of materials in real loading environments. However, existing fatigue models usually fail to accurately model this effect since they have no information concerning this cycle regime. Accelerated testing is used in laboratories to compare the predicting ability of existing fatigue models to realistic loading, but this type of testing masks the effect of the existence of low cycles in real operational loads on fatigue life. Recently a number of researchers tried to perform high cycle fatigue tests [16, 124, 125]. The authors of [124] performed tests on very small diameter impregnated strands with only sufficient fibers to be representative of the behavior of larger specimens. The load was applied by using low-frequency audio speakers (woofers) as actuators that can handle frequencies in the range of 300 Hz. Hosoi et al. [125] tested standardized quasiisotropic carbon fiber-reinforced plastic laminates with a stacking sequence of [45/0/45/90]S and reaching a frequency of 100 Hz for more than 108 loading cycles. According to the authors, the frequency does not affect fatigue behavior as long as the temperature remains well below the glass transition temperature of the examined material. These two publications prove that it is possible to perform high frequency tests and validate existing models (or develop new ones) for interpretation of the fatigue data at the high cycle fatigue regime as well. It is also a known fact that engineering structures used in open-air applications undergo more complicated loading spectra than the constant or block loading conditions usually applied in laboratories and used thereafter for the description of the fatigue behavior of the examined material. The behavior of composite materials is also known to be affected by load sequence [62, 126], overloads [127, 128], and changes in loading [63, 129]. A considerable amount of time has been spent on addressing this problem and quantifying the effects of spectrum loading on the fatigue behavior of several material systems, e.g., Refs. [16, 17, 63], and structural components [126–129]. Other researchers worked on the development of methods for accelerated testing in order to simulate lengthy loading spectra, e.g., Ref. [130]. Innovative measuring techniques can also assist in the health monitoring of a structure during operational life in order to achieve fail-safe, cost-effective designs. AE has been used in the past in combination with artificial neural networks to characterize damage in CFRP composite laminates, e.g., Ref. [131], or for the assessment of the normal and shear strength degradation in FRP composite materials during constant and variable amplitude fatigue

Fatigue life modeling and prediction methods for composite materials and structures

l

l

35

loading, e.g., Refs. [59, 60]. Optical fiber Bragg grating (FBG) sensors have been used for the characterization of the residual stresses in single fiber composites [132] and the monitoring of hygrothermal aging effects in epoxy resins [133]. DIC techniques have been lately applied successfully for the monitoring of complete strain fields on specimens surface during fatigue loading. The potential of DIC on monitoring composite materials in fatigue has been validated in among others in Ref. [134], where DIC was used in order to capture early stage damage and obtain an improved understanding of the physics of failure under complex fatigue loading conditions, as well as in Ref. [135] which discusses the application of the DIC technique for the fatigue testing of GFRP material in order to capture anisotropic damage. In addition to DIC surface deformation/strain measurements, X-ray tomography is used more for the identification of damage caused by fatigue loading in the volume of composite materials as is well described in Chapter 16, as well as, e.g., in Ref. [136]. As mentioned, a drawback of stiffness and strength degradation theories is their inability to take into account multiaxial property degradation. They focus on the properties of the material in the principal loading direction. However, efforts have recently been devoted to the development of models that also take other material elastic constants into account, such as the degradation of the Poisson’s ratio, the transverse stiffness and the in-plane shear modulus or to the use of strength degradation theories together with phenomenological modeling for estimation of the material properties during fatigue loading (see Chapter 14). The fatigue life prediction of composite materials can be characterized as an articulated procedure since a number of subproblems must be solved sequentially to produce the final result. However, dealing with each of the steps of such a methodology in depth is very demanding and time-consuming. On the other hand, the validity of the proposed solutions for each step could be effectively evaluated as part of the entire methodology. A limited number of commercial software solutions, addressing the fatigue of composites are available, (see Chapter 18) but they mainly provide tools for work carried out in the enclosed environment of a large private company. In many cases, however, the needs of universities, research centers, and smaller firms are generally different from those that can be met by these software products. This implies customization, which is very difficult, if not impossible from a commercial point of view. For the current state of engineering software packages, modifying or extending the code requires that users have intimate knowledge of data structures, their procedures and the extent to which they affect specific portions of the code. The possibility of reusing codes from other sources is also limited. This is because data structures vary widely between programs. On the other hand, in-house software development, like [117, 137–139], forces researchers to devote part of their research time to developing and testing software instead of concentrating on their main scientific tasks. The internet has provided a robust real-time mechanism for communication and interaction. In the near future, the development and deployment of the next generation internet (http://www.ipv6.org/), will establish the infrastructure for a worldwide communication network even faster and broader than most of today’s local networks. Along the same lines as the next generation internet, one of the major motivations to develop software for the fatigue life prediction of engineering structures is to provide an inexpensive, web-accessible tool for the interpretation of composite material fatigue data and the design of engineering structures that could be further modified by its own users in order to serve specific needs.

The problem is known and solutions do exist. The way to achieve them is to be “openminded” and work with a collaborative spirit. Knowledge must be spread freely, without limits, and efforts should be appropriately acknowledged in accordance with basic ethical rules. Experience has proved that Prof. Tsai’s dictum to “Think composite” is,

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Fatigue Life Prediction of Composites and Composite Structures

after 40 years, still appropriate. In addition to this, researchers should start thinking “interdisciplinary.” Novel interdisciplinary concepts for the development of new methodologies for the fatigue life prediction of composite materials and structures have to be introduced and adopted. Current computational power permits the development of methodologies that 10 years ago would have been inconceivable. Recent developments in artificial intelligence and evolutionary computational techniques allow the rapid establishment of models to solve or partially solve multiparametric problems, such as the fatigue life prediction of composite materials. The scientific community already has mature knowledge regarding different topics in the field but the composition of a universal fatigue life prediction methodology, based on hybrid models, is yet to be accomplished.

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[16]

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[58] Y. Zhang, A.P. Vassilopoulos, T. Keller, Fracture of adhesively–bonded pultruded GFRP joints under constant amplitude fatigue loading, Int. J. Fatigue 32 (2010) 979–987. [59] T.P. Philippidis, T.T. Assimakopoulou, Strength degradation due to fatigue–induced matrix cracking in FRP composites: an acoustic emission predictive model, Compos. Sci. Technol. 68 (15–16) (2008) 3272–3277. [60] T.P. Philippidis, T.T. Assimakopoulou, Using acoustic emission to assess shear strength degradation in FRP composites due to constant and variable amplitude fatigue loading, Compos. Sci. Technol. 68 (3–4) (2008) 840–847. [61] T.P. Philippidis, A.P. Vassilopoulos, Fatigue of glass fiber reinforced plastics under complex stress states, in: J. Wessel (Ed.), The Handbook of Advanced Materials: Enabling New Designs, Wiley–Interscience, 2004. [62] W. Van Paepegem, J. Degrieck, Effects of load sequence and block loading on the fatigue response of fiber–reinforced composites, Mech. Adv. Mater. Struct. 9 (1) (2002) 19–35. [63] T.P. Philippidis, A.P. Vassilopoulos, Life prediction methodology for GFRP laminates under spectrum loading, Compos. A Appl. Sci 35 (6) (2004) 657–666. [64] A.J. Brunner, N. Murphy, G. Pinter, Development of a standardized procedure for the characterization of interlaminar delamination propagation in advanced composites under fatigue mode I loading conditions, Eng. Fract. Mech. 76 (18) (2009) 2678–2689. [65] C.R. Saff, Effect of load frequency and lay–up on fatigue life of composites, in: T.K.O’Brien (Ed.), Long–Term Behavior of Composites, 1983, pp. 78–91. ASTM STP 813. [66] C.T. Sun, W.S. Chan, Frequency effect on the fatigue life of laminated composite, in: S. W. Tsai (Ed.), Composite Materials: Testing and Design (Fifth Conference), 1979, pp. 418–430. ASTM STP 674. [67] A. Rotem, Load frequency effect on the fatigue strength of isotropic laminates, Compos. Sci. Technol. 46 (2) (1993) 129–138. [68] ASTM D 3479–76,n.d. Standard Test Method for Tension–Tension Fatigue of Oriented Fiber, (resin matrix composites) [69] DIN 65 586, n.d.Aerospace; Fiber Reinforced Plastics; Fatigue Strength Behavior of Fiber Composite Materials Under One–Stage Loading. [70] O. Krause, Frequency Effects on Lifetime, DLR, doc. OB_TC_N003 rev. 1, 19, vol. 12, 2002. [71] A.P. Vassilopoulos, T.P. Philippidis, Effect of interrupted cyclic loading on fatigue life of composites, in: Advances in Composite Technology, Comp 03, 5th International Symposium on Advanced Composites, 2003. Corfu Imperial Hotel, Corfu, Greece. [72] L.J. Broutman, S.K. Gaggar, Fatigue behavior of epoxy and polyester resins, Int. J. Polym. Mater. 1 (4) (1972) 295–316. [73] W.A. Herman, R.W. Hertzberg, J.A. Manson, The influence of loading history on fatigue in engineering plastics, J. Mater. Sci. 25 (1) (1990) 434–440. [74] G.D. Sims, Fatigue test methods, problems and standards, in: B. Harris (Ed.), Fatigue in Composites, Woodhead Publishing Ltd., 2003, pp. 36–62. [75] C.S. Lopes, C. Gonzalez, O. Falco, F. Naya, J. LLorca, B. Tijs, Multiscale virtual testing: the roadmap to efficient design of composites for damage resistance and tolerance, CEAS Aeronaut. J. 7 (2016) 607–619. [76] S.P. Engelstad, S.B. Clay, Comparison of composite damage growth tools for fatigue behavior of notched composite laminates, J. Compos. Mater. 51 (15) (2017) 2227–2349. [77] G.P. Sendeckyj, K.L. Reifsnider (Ed.), Life Prediction for Resin–Matrix Composite Materials, Fatigue of Composite Materials, Composite Materials Series, vol. 4, Elsevier, 1991.

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[77a] F. Kun, H.A. Carmona, J.S. Andrade Jr., H.J. Herrmann, Universality bahind Basquin’s law of fatigue, Phys. Rev. Lett. 100 (2008) 094301. [78] G.P. Sendeckyj, Fitting models to composite materials fatigue data, in: C.C. Chamis (Ed.), Test Methods and Design Allowables for Fibrous Composites, 1981, pp. 245–260. ASTM STP 734. [79] J.M. Whitney, Fatigue characterization of composite materials, in: Fatigue of Fibrous Composite Materials, 1981, pp. 133–151. ASTM STP 723. [80] A.P. Vassilopoulos, E.F. Georgopoulos, V. Dionyssopoulos, Artificial neural networks in spectrum fatigue life prediction of composite materials, Int. J. Fatigue 29 (1) (2007) 20–29. [81] A.P. Vassilopoulos, R. Bedi, Adaptive neuro–fuzzy inference system in modeling fatigue life of multidirectional composite laminates, Compos. Mater. Sci. 43 (4) (2008) 1086–1093. [82] A.P. Vassilopoulos, E.F. Georgopoulos, T. Keller, Comparison of genetic programming with conventional methods for fatigue life modeling of FRP composite materials, Int. J. Fatigue 30 (9) (2008) 1634–1645. [83] A.T. Echtermeyer, E. Hayman, K.O. Ronold, Comparison of fatigue curves for glass composite laminates, in: R.M. Mayer (Ed.), Design of Composite Structures Against Fatigue–Applications to Wind Turbine Rotor Blades, Antony Rowe Ltd., 1996. [84] Draft IEC 61400–1, Ed.2 (88/98/FDIS), Wind Turbine Generator Systems–Part 1: Safety Requirements, (1998). [85] G. Lloyd, Rules and regulations, IV–non–marine technology, Part 1–Wind Energy (1993). [86] T. Fujii, F. Lin, Fatigue behavior of a plain–woven glass fabric laminate under tension/ torsion biaxial loading, J. Compos. Mater. 29 (5) (1995) 573–590. [87] M.–.H.R. Jen, C.H. Lee, Strength and life in thermoplastic composite laminates under static and fatigue loads. Part I: experimental, Int. J. Fatigue 20 (9) (1998) 605–615. [88] E.W. Smith, K.J. Pascoe, Biaxial fatigue of a glass–fiber reinforced composite. Part 2: Failure criteria for fatigue and fracture, in: M.W. Brown, K.J. Miller (Eds.), Biaxial and Multiaxial Fatigue, Mechanical Engineering Publications, London, 1989, pp. 397–421. EGF3. [89] L.B. Lessard, M.M. Shokrieh, Two–dimensional modeling of composite pinned–joint failure, J. Compos. Mater. 29 (5) (1995) 671–697. [90] Shokrieh MM, Lessard LB, Poon C. Three–dimensional progressive failure analysis of pin/bolt loaded composite laminates, in Bolted/Bonded Joints in Polymeric Composites, AGARD CP 590, 1997, vol. 7.1–7.10. [91] M.M. Shokrieh, L.B. Lessard, Multiaxial fatigue behavior of unidirectional plies based on uniaxial fatigue experiments–I. Modeling, Int. J. Fatigue 19 (3) (1997) 201–207. [92] M.M. Shokrieh, L.B. Lessard, Multiaxial fatigue behavior of unidirectional plies based on uniaxial fatigue experiments–II. Experimental evaluation, Int. J. Fatigue 19 (3) (1997) 209–217. [93] W. Hwang, K.S. Han, Fatigue of composites—fatigue modulus concept and life prediction, J. Compos. Mater. 20 (2) (1986) 154–165. [94] V.A. Passipoularidis, T.P. Philippidis, Strength degradation due to fatigue in fiber dominated glass/epoxy composites: a statistical approach, J. Compos. Mater. 43 (9) (2009) 997–1013. [95] V.A. Passipoularidis, T.P. Philippidis, Residual strength after fatigue in composites: theory vs. experiment, Int. J. Fatigue 29 (12) (2007) 2104–2116.

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[96] P.C. Chou, R. Croman, Degradation and sudden death models of fatigue of graphite/ epoxy composites, in: Composite Materials: Testing and Design (5th Conference), 1979, pp. 431–454. ASTM STP 674. [97] W. van Paepegem, J. Degrieck, A new coupled approach of residual stiffness and strength for fatigue of fiber–reinforced composites, Int. J. Fatigue 24 (7) (2002) 747–762. [98] W. van Paepegem, J. Degrieck, Calculation of damage–dependent directional failure indices from the Tsai–Wu static failure criterion, Compos. Sci. Technol. 63 (2) (2002) 305–310. [99] W. van Paepegem, J. Degrieck, Coupled residual stiffness and strength model for fatigue of fiber–reinforced composite materials, Compos. Sci. Technol. 62 (5) (2002) 687–696. [100] M.J. Salkind, Fatigue of composites, in: H.T. Corten (Ed.), Composite Materials Testing and Design (Second Conference), 1972, pp. 143–169. ASTM STP 497. [101] J. Brunbauer, G. Pinter, Fatigue life prediction of carbon fibre reinforced laminates by using cycle-dependent classical laminate theory, Compos. Part B 70 (2015) 167–174. [102] J. Brunbauer, C. Gaier, G. Pinter, Computational fatigue life prediction of continuously fibre reinforced multiaxial composites, Compos. Part B 80 (2015) 269–277. [103] C. Soutis, P.W.R. Beaumont (Eds.), Multiscale Modeling of Composite Material Systems—The Art of Predictive Damage Modelling, first ed., Woodhead Publishing Ltd, 2005. [104] P. Davidson, A. Hasanyan, Multiscale Fatigue Modeling of Composites, AIAA SciTech Forum, 2019. San Diego, California. [105] A.P. Vassilopoulos, T. Keller, Modeling of the fatigue life of adhesively–bonded FRP joints with genetic programming, in: Proceedings of the ICCM17, Edinburgh, 2009. [106] A.P. Vassilopoulos, E.F. Georgopoulos, V. Dionyssopoulos, Fatigue life of multidirectional GFRP laminates under constant amplitude loading with artificial neural networks, Adv. Compos Letters 15 (2) (2006) 43–51. [107] R.C. Silverio Freire, A.D. Do´ria Neto, E.M.F. De Aquino, Comparative study between ANN models and conventional equation in the analysis of fatigue failure of GFRP, Int. J. Fatigue 31 (5) (2009) 831–839. [108] V.A. Riziotis, S.G. Voutsinas, Fatigue loads on wind turbines of different control strategies operating in complex terrain, J. Wind Eng. Ind. Aerod. 85 (3) (2000) 211–240. [109] A.P. Vassilopoulos, B.D. Manshadi, T. Keller, Influence of the constant life diagram formulation on the fatigue life prediction of composite materials, Int. J. Fatigue 32 (4) (2010) 659–696. [110] A.P. Vassilopoulos, B.D. Manshadi, T. Keller, Piecewise non–linear constant life diagram formulation for FRP composite materials, Int. J. Fatigue 32 (10) (2010) 1731–1738. [111] R.P.L. Nijssen, Fatigue Life Prediction and Strength Degradation of Wind Turbine Rotor Blade Composites, PhD thesis, TU Delft, 2006 (ISBN–13:978–90–9021221–0). [112] T. Nyman, Composites fatigue design methodology: a simplified approach, Compos. Struct. 35 (1996) 183–194. [113] S. Shimizu, K. Tosha, K. Tsuchiya, New data analysis of probabilistic stress–life (P–S– N) curve and its application for structural materials, Int. J. Fatigue 32 (3) (2010) 565–575. [114] R.P.L. Nijssen, O. Krause, T.P. Philippidis, Benchmark of lifetime prediction methodologies, in: Optimat blades technical report, 2004. OB_TG1_R012 rev.001. http://www. wmc.eu/public_docs/10218_001.pdf. [115] M. Kawai, N. Maki, Fatigue strength of cross–ply CFRP laminates at room and high temperatures and its phenomenological modeling, Int. J. Fatigue 28 (10) (2006) 1297–1306. [116] M. Kawai, T. Taniguchi, Off–axis fatigue behavior of plain woven carbon/epoxy composites at room and high temperatures and its phenomenological modeling, Compos. A Appl. Sci. 37 (2) (2006) 243–256.

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[117] A.P. Vassilopoulos, R. Sarfaraz, B.D. Manshadi, T. Keller, A computational tool for the life prediction of GFRP laminates under irregular complex stress states: influence of the fatigue failure criterion, Comput. Mater. Sci. 49 (3) (2010) 483–491. [118] H.J. Sutherland, J.F. Mandell, Optimized constant life diagram for the analysis of fiberglass composites used in wind turbine blades, J. Solar Energy Eng. Trans. ASME 127 (4) (2005) 563–569. [119] M. Kawai, M. Koizumi, Nonlinear constant fatigue life diagrams for carbon/epoxy laminates at room temperature, Compos. A–Appl. Sci. 38 (11) (2007) 2342–2353. [120] B. Harris, A parametric constant–life model for prediction of the fatigue lives of fiber– reinforced plastics, in: B. Harris (Ed.), Fatigue in Composites, Woodhead Publishing Limited, 2003, pp. 546–568. [121] R. Sarfaraz, A.P. Vassilopoulos, T. Keller, Experimental investigation and modeling of mean load effect on fatigue behavior of adhesively-bonded pultruded GFRP joints, Int. J. Fatigue 44 (2012) 245–252. [122] D.D. Sambrosky, J.F. Mandell, D.A. Miller, Creep/fatigue response of resin infused biaxial (double bias) glass fabric laminates in reversed loading, in: 32nd ASME Wind Energy Symposium—SciTech Forum and Exposition, 2014 (National Harbor, MD; United States; Code 102895). [123] J.D. Rowatt, P.D. Spanos, Markov chain models for life prediction of composite laminates, Struct. Safety 20 (2) (1998) 117–135. [124] J.F. Mandell, D.D. Samborsky, L. Wang, N.K. Wahl, New fatigue data for wind turbine blade materials, J. Solar Energy Eng. Trans. ASME 125 (4) (2003) 506–514. [125] A. Hosoi, N. Sato, Y. Kusumoto, K. Fujiwara, H. Kawada, High–cycle fatigue characteristics of quasi–isotropic CFRP laminates over 108 cycles (initiation and propagation of delamination considering interaction with transverse cracks), Int. J. Fatigue 32 (1) (2010) 29–36. [126] R. Sarfaraz, A.P. Vassilopoulos, T. Keller, Block loading fatigue of adhesively bonded pultruded GFRP joints, Int. J. Fatigue 49 (2013) 40–49. [127] S. Erpolat, I.A. Ashcroft, A.D. Crocombe, M.M. Abdel–Wahab, A study of adhesively bonded joints subjected to constant and variable amplitude fatigue, Int. J. Fatigue 26 (11) (2004) 1189–1196. [128] S. Erpolat, I.A. Ashcroft, A.D. Crocombe, M.M. Abdel–Wahab, Fatigue crack growth acceleration due to intermittent overstressing in adhesively bonded CFRP joints, Compos. A Appl. Sci. 35 (10) (2004) 1175–1183. [129] R. Sarfaraz, A.P. Vassilopoulos, T. Keller, Variable amplitude fatigue of adhesivelybonded pultruded GFRP joints, Int. J. Fatigue 55 (2013) 22–32. [130] Y. Miyano, M. Nakada, J. Ichimura, E. Hayakawa, Accelerated testing for long–term strength of innovative CFRP laminates for marine use, Compos. B Eng. 39 (1) (2008) 5–12. [131] T.P. Philippidis, V.N. Nikolaidis, A.A. Anastassopoulos, Damage characterization of carbon/carbon laminates using neural network techniques on AE signals, NDT&E Int. 31 (5) (1998) 329–340. [132] F. Colpo, L. Humbert, J. Botsis, Characterization of residual stresses in a single fiber composite with FBG sensor, Compos. Sci. Technol. 67 (2007) 1830–1841. [133] D. Karalekas, J. Cugnoni, J. Botsis, Monitoring of hygrothermal ageing effects in an epoxy resin using FBG sensor: a methodological study, Compos. Sci. Technol. 69 (2009) 507–514.

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[134] T. Skinner, S. Datta, A. Chattopadhyay, Biaxial Fatigue Damage Behavior in Carbon Fiber Reinforced Polymer Composites, AIAA SciTech Forum, 2019. San Diego, California. [135] M.L. Aparna, G. Chaitanya, K. Srinivas, J.A. Rao, Fatigue testing of continuous GFRP composites using digital image correlation (DIC) technique: a review, Mater. Today Proc. 2 (2015) 3125–3131. [136] P. Wagner, O. Schwarzhaupt, M. May, In-situ X-ray computed tomography of composites subjected to fatigue loading, Mater. Lett. 236 (2019) 128–130. [137] T.P. Philippidis, A.P. Vassilopoulos, K.G. Katopis, S.G. Voutsinas, THIN/PROBEAM: a software for fatigue design and analysis of composite rotor blades, Wind Eng. 20 (5) (1996) 349–362. [138] D.J. Lekou, T.P. Philippidis, PRE–and POST–THIN: a tool for the probabilistic design and analysis of composite rotor blade strength, Wind Energy 12 (7) (2009) 676–691. [139] Manshadi BD, Vassilopoulos AP, Keller T. A computational tool for the fatigue life prediction of composite materials, in Proceedings of the 2nd International Conference on Material and Component Performance under Variable Amplitude Loading, (Darmstadt, Germany, 23–26 March 2009).

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Part One Fatigue life behavior and modeling

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Phenomenological fatigue analysis and life modeling

2

Rogier P.L. Nijssen Knowledge Centre Wind turbine Materials and Constructions, Wieringerwerf, The Netherlands

2.1

Introduction

For life prediction and fatigue analysis, a thorough knowledge is required on the fatigue behavior of a material. This chapter addresses the description of fatigue behavior of a material in terms of S-N curves and constant life diagrams (CLDs). The fatigue analysis referred to in this chapter is based on experimental characterization of structures and materials through repetitive loading until a macroscopically observable failure mode occurs: usually fracture resulting in the inability to carry the applied load. Hence the term “phenomenological” in this chapter’s title. A strong relation exists between the models described in this chapter and those in the next. First of all, S-N curves and CLDs are required for residual strength calculations and stiffness degradation models. These models add information on strength and stiffness that is not present in most S-N diagrams, but S-N information is required for their successful application, as stiffness and strength degradation is typically expressed as a function of fatigue life. On a smaller scale, fracture mechanics and micromechanical modeling attempt to describe how stiffness and strength are affected by growing cracks, fiber-matrix debonds, delaminations, kinking, and buckling of fibers, and the interaction of failure modes present in composites, until final structural failure occurs. Effectively, these models predict S-N curves with as input basic material properties. Genetic algorithms also predict S-N curves, albeit via another route. This chapter focuses on constant amplitude fatigue life description in terms of S-N curves. Many parameters affect the measured fatigue characteristics of a composite, and tests are designed carefully to measure the fatigue behavior that represents as closely as possible the fatigue behavior of the material in a structure. Once fatigue data are available, suitable mathematical treatment is required to use the data in design calculations. For design cases where the number of cycles exceeds by far the number of cycles attained in the experiments, appropriate extrapolation must be formulated. The concepts described in this chapter are applicable to general composites and fatigue. However, the chapter is written from the background of wind turbine rotor blade materials research. These structures are subjected to a very large number of load cycles; low-cycle fatigue will not be specifically addressed in this chapter. Fatigue Life Prediction of Composites and Composite Structures. https://doi.org/10.1016/B978-0-08-102575-8.00002-4 © 2020 Elsevier Ltd. All rights reserved.

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Furthermore, rotor blades operate under strongly varying loads. The S-N models and CLDs discussed provide a basis for life prediction under load spectra containing fatigue loads of various types. One of the main characteristics that are relevant for this type of material are that safelife design is preferred (inspection and maintenance of wind turbine rotor blades should be minimized). Furthermore, the cost of raw material and manufacturing should be low. This typically results in relatively coarse material, and structures with some variation in properties within the structure and from product to product. It should be added that in recent years, high demand for rotor blades has led to continuous improvements in production speed, e.g., though automation, and consistency is improving.

2.2

Fatigue experiments

Fatigue experiments are the basis for life prediction and validation of these predictions. In fact, most of the state-of-the-art design software is based on identifying the loads (strains, stresses) in a certain point on the structure, and quantifying the resulting damage through interpolation and extrapolation of the S-N curves and CLDs described in this chapter. An example of commonly used fatigue test set-ups is shown in Fig. 2.1. As any test result, the location of a point in the stress-life space should be viewed in the context of the many parameters affecting the fatigue life of a specimen. Choices made in the test set-up and procedure, and variations in test conditions, can significantly affect the results.

Fig. 2.1 Example of fatigue test set-up.

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For static testing, test specimen dimensions, equipment, test conditions, and measurements are described in detail in numerous standards or guidelines provided mostly by ISO, ASTM, and many national institutes. In the case of fatigue testing of composites, standardization is much more limited. Some guidelines exist for tension fatigue testing and tension-compression testing using mainly specimens and the methodology described in relevant static testing [1, 2]. An informal note on requirements for appropriate S-N curve testing is circulating around test laboratories [3]. The lack of standardization of fatigue testing gives rise to issues when comparing fatigue strengths and lives under different experimental circumstances. A great number of variables affect fatigue life, most of which are free for the experimentalist to manipulate at will, and a large part of which have not been considered in models. This makes extracting conclusions from different sets of fatigue data difficult. Some of the variables affecting fatigue life are discussed in the following.

2.2.1 Control mode Most test frames can be set to load control or displacement control (cyclic loads or displacements are kept constant, respectively). In some cases, strain control is also possible. For laminates that feature a significant stiffness reduction during their fatigue life, the chosen control mode can have a big impact on the test results. It is important to choose the relevant control mode for the intended use of the laminate. In case of load control, typically the grip displacement will increase during testing (decreasing specimen stiffness) and the specimen will fail destructively once it can no longer bear the applied load. This control mode is most commonly used in fatigue testing, especially for obtaining fatigue data for load-bearing laminates. Displacement control or strain control can be very useful for monitoring crack growth or stiffness decrease in fatigue. It is possible that the specimen does not fail destructively, as decreasing stiffness leads to decreasing loads. The failure criterion should then be formulated in terms of, e.g., stiffness decrease or crack length. Displacement-controlled tests, however, have the disadvantage that damage or sliding in the grips can incorrectly be associated with stiffness degradation in the specimen. In cases where damage accumulates outside the measurement area during fatigue, strain control using the measurement signal of strain sensors at the gauge section can be employed. Unless specifically mentioned in the description test set-up, reported strain-life data can usually be assumed to be actually obtained through load-controlled testing (converting stress to strain by using the modulus of elasticity). It is more correct in these cases to express the data in terms of initial strain vs life, as the stiffness of a laminate coupon can decrease significantly in fatigue, especially if it has a large off-axis fiber content.

2.2.2 Grips Grips come in various sizes and fastening mechanisms, either mechanical (using bolts) or hydraulically operated clamps. The influence of the grips depends strongly on the planform and fiber content of the laminate (see Section 2.5). For dog-boned

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(waisted) specimens, grip influence is less significant than for prismatic specimens and specimens with a high fraction of fibers in the loading direction. Most test frames allow one to set the overall hydraulic grip pressure. As a rule of thumb, grip pressure should be high enough to prevent sliding of the specimen from the grips, but not much higher. The higher the grip pressure, the more through-thethickness load is introduced into the grip area, resulting in a more unfavorable multiaxial stress state. Wear of grip faces should be checked periodically, as this might change the grip pressure distribution, thereby apparently lengthening the gauge length, and affecting fatigue life in loading types that include compression (see also Section 2.4).

2.2.3 System stiffness and alignment After proper alignment of the grips, the stiffness of the test frame should be such that the grip alignment is not compromised by the application of high loads and possible asymmetry in the specimens. Poor alignment reduces the measured strengths because it lowers the specimen’s resistance to buckling. Then the maximum load measured represents the buckling strength, not the material strength.

2.2.4 Loading rate For static testing of load-bearing materials in most structures, high loading rates are typically avoided because they tend to yield an optimistic representation of strength and stiffness, while requiring high data acquisition rates. In fatigue, loading frequency should be low enough to prevent any significant influence on fatigue life, but as high as possible to reduce testing time (see Section 2.4).

2.2.5 Specimen geometry Apart from its implications for specimen manufacturing, the geometry of the specimen has a significant influence on measured performance (see Section 2.5).

2.3

Measurements and sensors

Loads, displacements, strains, and specimen surface temperature (or internal temperature if this can be measured in a noninvasive manner) should be measured continuously, and the number of fatigue cycles should be counted. Loads and displacements in most test frames are measured by a fixed load cell and an LVDT (linear variable displacement transducer) built into the servo-hydraulic actuator. Separate strain measurement and temperature measurement devices are typically used to document stiffness and check for temperature exceedance. For strain measurements strain gauges are used, as well as strain-gauge based extensometers. A very relevant disadvantage of strain gauges is that they suffer from fatigue to a much larger extent than most composite laminates. This is understandable, since metal foil strain gauges also have an S-N curve, which is steeper than that of

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composites. Moreover, they experience considerable drift over the lifetime of the strain gauge (typically 1000–10,000 cycles). The drift, a gradual increase or decrease in measured strain, occurs as the strain gauge is damaged itself in fatigue, changing the electrical resistance of the strain gauge. Thus, they are really only appropriate for measurement of the initial modulus of the specimen. Some solutions exist for long-term strain measurements. Strain-gauge-based extensometers are much less sensitive to fatigue than the single-use strain gauges themselves. On the other hand, they often require a larger free gauge length and are much more expensive to buy. For this reason, additional measures are often taken to protect the extensometer after the specimen fails. In case explosive failure of the specimen is expected (for instance, in specimens with a very long gauge length subjected to high loads), removal of the extensometer should be considered prior to failure. These measures may preclude any potential stiffness changes at the end of fatigue life from being noticed. Both for bonded strain gauges and for extensometers, some care should be taken in choosing the gauge length of the strain measurement device, in relation to the gauge length of the specimen and the microstructure of the material. For a short specimen gauge length, irregular strain fields can be expected, for example in the area near the grips. For short gauge lengths, this can lead to an unjustified “averaging” of measured strain when using a large strain gauge length. On the other hand, the strain gauge should be large enough to average the measured strain in both the tows and matrix-rich areas between the tows, i.e., it should cover a couple of tows in the material’s structure. Noncontact optical strain measurement techniques have become more popular. An example of full-field strain imaging is the digital image correlation technique (DICT), which uses postprocessing of video images of a specimen to calculate displacements (hence strains) in up to three dimensions. An advantage of this method is that nonuniform strain fields can be observed, and a better choice can be made over which area the strain should be averaged. Optical fibers (glass fibers, with an optical grating etched into the material, reflecting light transmitted through this fiber) have the potential of measuring strains inside a composite material, as they can be embedded in the laminates during production. These sensors are highly durable and give reliable results, provided that temperature compensation is taken care of. Measuring strains using the displacement readings from the test frames should be avoided, especially in fatigue experiments, as it might produce inaccurate results for various reasons. First of all, the displacement range during most tests is much smaller than the stroke of the test machine (i.e., the full scale range of the sensor). Thus, without modifications, the resolution is far less than the resolution of a strain gauge. More importantly, the displacement readings also include deformation of the specimen outside the gauge section. Strain calculation requires modeling of the specimen, which is not desirable and might not be possible to a sufficient extent. Many specimens have nonuniform strain fields, so that the “strain” measured using the actuator displacement sensor is an average value of the strain over the specimen. Furthermore, many specimens are equipped with tabs to protect the specimen surface from grip damage. Wear during the test usually causes part of the tab and gauge

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section to move with respect to the grips. The adhesive bond between tabs and specimen may partly disintegrate during the fatigue test, causing additional displacement in a load-controlled test, particularly in rectangular test specimens. Another downside is that the stiffness of the whole system, including the test machine, is measured when using the LVDT signal. The inaccuracies decrease as the stiffness of the specimen becomes smaller with respect to the stiffness of the test frame, the specimens become longer, and the strain concentrations that lead to wear in the grips become smaller. Cycle counting is performed by software through peak detection algorithms, although mechanical cycle counters can be used as well. During data acquisition for fatigue testing it is acceptable to record a summary of data at intervals, e.g., record the average maximum and range of the last x cycles, at intervals of y > x cycles (provided that the absolute maxima and minima are reported if they exceed the average by a specified threshold value, typically 1%–2%). The size of these intervals can be changed during fatigue tests to limit file size. It is advisable to also retain periodic records of full cycles (i.e., measured at a data acquisition rate of ca. 1/100th of the period), to enable checking of the applied loads and displacements, such as whether the specimen is still loaded using the sine-shaped load that is required, etc. Also, these records can be used for the calculation of hysteretic damping and dynamic modulus, provided that there is no phase lag between measurements. Therefore, all sensors should preferably be fed through the same electrical circuit. For example, if the strain signal is directly fed into the acquisition system, whereas the load signal is looped through the control unit, their “electronic” distance is different, causing a time delay which will show as hysteresis.

2.4

Test frequency

As indicated, the frequency of tests should be chosen as high as possible without influencing life significantly. For larger strain ranges, a lower number of cycles and lower frequencies should typically be chosen (and vice versa). Testing frequency can influence fatigue life in the following three ways.

2.4.1 Creep/time at mean stress effects Mishnaevsky and Brøndsted [4] reported that through deduction based on the kinetic concept of strength, fatigue life dependency on test frequency can be described. Longer fatigue lives are predicted for higher frequencies. In practice, heating will limit the test fatigue frequency considerably.

2.4.2 Frictional heating As the sections of the specimens inside the grips and tabs are partly allowed to slide, some frictional heating can occur. This depends on how much the grip faces are allowed to follow the Poisson contraction/expansion during tensile/compressive

Phenomenological fatigue analysis and life modeling

53

loading. In addition, internally, frictional heat can be generated when delaminated layers move relative to each other or, on a smaller scale, by fibers and matrix sliding with respect to each other after formation of fatigue damage.

2.4.3 Viscoelastic heating Internally, a second mechanism can cause heating, i.e., viscoelastic hysteresis. (Internal) Frictional heating and viscoelastic heating can hardly be quantified separately in temperature measurements, see e.g., Ref. [5] for the effect of viscoelastic heating in thick laminates. An appropriate frequency scheduling method was described in Ref. [6], where frequency scheduling was employed using the strain range as a criterion for frequency. First, a maximum allowable temperature rise was chosen (10°C above room temperature). Then, a reference test was run in the low-cycle, high-strain regime, monitoring surface temperature. The maximum frequency was adjusted so that the surface temperature rise was within acceptable limits. For other load levels, the frequency was proportional to the square of the change in strain, according to f ¼ fref (εref/ε)2 [7]. This means that for different R-values, the frequencies for identical nominal fatigue lives were therefore different. The loading frequency for short fatigue lives (1000 cycles) was in the order of 1–2 Hz; for 1 million cycles nominal fatigue life, the frequency ranged between 3 and 8 Hz. In day-to-day laboratory practice, these frequencies limit the nominal target number of cycles to 1 million cycles. Choosing load levels such that 10 million cycles can be expected means that a test frame must be running for an average of 1 month; the typical one decade of scatter means that a test frame can easily be unavailable for other tests twice as long.

2.5

Specimens

The overall controversy in choice of specimen is the representation of an undisturbed cut-out of the structure vs testability. The boundary conditions of a small region of the structure under investigation are very different from those in a specimen in a test machine; the small region in the blade is loaded by axial or biaxial remote loads, has no sides, is subjected to a limited loading frequency, and does not suffer from damage due to the load introduction, etc. Most specimens are loaded in the longitudinal direction, where the longitudinal stresses in the specimen are introduced through shear and friction, and the specimens have sides and—usually—no curvature. Measured material performance is affected significantly by the dimensions of the tested specimen. The specimen shape can be tailored to specific shear, tensile, or compression loading, meaning that the specimen is designed to fail by the maximum values for the required parameters. In quasistatic testing this is common practice, resulting in a large variety of specimen shapes, each standardized and proven to be useful for obtaining specific material parameters, such as tensile, compressive, and shear strengths and

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Fatigue Life Prediction of Composites and Composite Structures

modulus. Comparison of performance of different specimen shapes does reveal differences caused by width or size effects, or the effects of stress/strain concentrations, or sensitivity to buckling (e.g., Ref. [8]). For fatigue, the same specimens can be used as those used in quasistatic tests, tailored for a particular loading type. For instance, for tension-dominated fatigue, long slender specimens can be used; while short thick specimens can be employed for compression-dominated fatigue. For validating general fatigue models, however, it is recommended to define a single “universal” specimen geometry and testing procedure in order to enable variable amplitude loading with both tension and compression loadings. This philosophy was followed in some exemplary wind turbine material research projects [6, 9, 10], as shown in Fig. 2.2. In that case, the choice of specimen dimensions is based on many conflicting arguments, some of which are indicated below.

2.5.1 Manufacturability and batch size Specimens should be manufacturable in as simple a manner as possible, requiring widely available tooling. This reduces the chance of batch-to-batch variation and allows for quick set-up of an experimental qualification program. In practice, this implies preferred absence of notches and thickness changes. It is recommended to obtain specimens from a single large batch of laminate, which helps to ensure that all specimens were manufactured under the same conditions. For large laminate plates, it is also advisable to retain the original position and orientation of the specimen within the plate, enabling possible research into effects of direction of infusion, mold time, etc.

2.5.2 Maximum load The minimum cross-sectional area, fiber volume fraction, fiber orientation, and composite constituent ultimate strengths (and sometimes ambient test conditions) determine the maximum load required to break the specimen.

2.5.3 Tabs The role of tabs is to protect the laminate surface, to increase friction between the specimen and the grips, and to make the overall specimen thickness larger, which allows for some measurement equipment to be inserted on the specimen side, and

Fig. 2.2 “Universal” specimen used in Refs. [8, 9].

Phenomenological fatigue analysis and life modeling

55

for improving end loading. A large tab section area is recommended to gradually introduce the axial loads into the specimen, and to enable small grip pressures. Tabs are usually bonded separately to the specimen. The tab thickness and bondline thickness should be well controlled to ensure that the tabs are parallel. However, tab thickness should be small, e.g., smaller than specimen thickness, to avoid overheating in the tabbed region during cyclic testing.

2.5.4 Planform In isotropic materials, the center of the specimen often has a reduced cross-sectional area to ensure localized failure and failure at the location of the strain gauges. Much of the composite structural material research is done on unidirectional laminates, which are not suited for reducing the cross-sectional area without significantly increasing the gauge section length or adding layers in other directions. Because of a nonuniform in-plane shear distribution, most in-plane “tailored,” “waisted,” or “dog-bone”-shaped specimens develop axial cracks starting at the sides of the minimum cross-section and growing toward the clamped region, essentially reducing them to a rectangular specimen during a fatigue test (see Fig. 2.3). Reducing the cross-sectional area through gradually building up the laminate toward the grips might be more useful, although delaminations are likely to occur. For large-scale production of test specimens this is also generally too labor-intensive. One of the most important implications of testing unidirectional laminates using a rectangular specimen is that the most likely failure location is in the grips. In fact, the combination of large strain gradients and Poisson contraction in the region where the tabs meet the gauge section typically results in a tab disbond gradually growing into the grip area, and ending with partial or complete fracture of the specimen in the tab (see Fig. 2.4). Damage progression in this region is usually associated with abrasion of

Fig. 2.3 Axial cracks in dog-bone specimen during fatigue.

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Fatigue Life Prediction of Composites and Composite Structures

Fig. 2.4 Schematic representation of tabbed end of specimen in grips, showing axial cracks between tabs and laminate, developed during fatigue.

the adhesive between tab and laminate, damage of the laminate surface, frictional heat generated in the disbonded area, redistribution of the grip loads, shear loads, and increased apparent gauge length. These mechanisms are particularly hard to measure and model, but it is suspected that grip failures are especially contributing to the difference between what the material is actually capable of and its measured fatigue performance.

2.5.5 Length and gauge length Small length and width are generally desirable to obtain as many specimens as possible from a limited amount of available laminate. The specimen gauge length should be small enough to avoid buckling, but large enough to allow uniform strain distributions across most of the gauge section, and to allow application of strain and temperature sensors. The latter typically precludes the use of an antibuckling guide.

2.5.6 Thickness Increasing specimen thickness allows for a longer gauge length at constant Euler buckling sensitivity. Smaller thickness facilitates better uniform strain distributions and higher cyclic frequencies, since heat generated internally can be dissipated to the surface more quickly. In composites, the thickness of a specimen is a discrete function of the number of layers. For cost-effective composites, layer thickness is in the order of 1 mm (cf. 0.1 mm for aerospace-grade composites).

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57

2.5.7 Width The width of a specimen should be significantly larger than the characteristic width of the laminate microstructure; in practice this is larger than four or five tows in a multilayered laminate. Also, the width should be smaller than the gauge section length, e.g., one-third or smaller, although for compression loading or reversed loading the width/gauge length ratio can be 1.

2.6

S-N diagrams

The parameters “S” and “N” are used as generic terms. In the case of “S” this means that the term can indicate cyclic loads, stresses, and strains (within each of these categories a choice can be made between amplitude, mean, maximum, minimum, or absolute values of maximum, or minimum, etc.). The generic term “N” is usually number of cycles to failure, but can also be used to indicate number of cycles to a predefined stiffness change, or number of times a fixed load sequence has been repeated. Since none of the above parameters for S completely defines the constant amplitude waveform, in general the S-N curves are usually plotted for a constant value of R. The definition of R is the ratio of minimum over maximum S, where compression has a negative sign. The value of R is a unique description of the loading type (see Fig. 2.5). The first observation that should be made is that the S-N diagram, with S and N usually plotted on the ordinate and abscissa, respectively, is in principle incorrectly oriented. As in all mathematical plots, the independent variable (the input into the specimen: S) should be plotted on the horizontal axis, whereas the dependent variable (output: N) should be plotted on the vertical axis. For unclear reasons, the S-N curve has been plotted the wrong way around, and for historical reasons it is acceptable to leave it like that. In any case, in regression analyses performed for design, life should be treated as the dependent variable. Regression with the incorrect dependent variable will lead to an S-N curve that has a different slope and intercept from the correct curve, possibly Low-cycle, high load regime

S

R = Smin /Smax

s Smax Saverage Time

Fig. 2.5 S-N diagram.

Smin

High-cycle, low load regime

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Fatigue Life Prediction of Composites and Composite Structures

leading to an overestimation or underestimation of performance, its severity depending on the degree of scatter and extrapolation. In S-N curve definition and extrapolation, the life-axis can be divided into three regions: low cycle (corresponding to high strains and stresses), high cycle (corresponding to low strains and stresses), and the region in between. The corresponding number of cycles depends on the application, but for structures which can expect many load cycles during operation, the low-cycle regime is up to 1000 cycles, and the high-cycle regime starts at around 1 million cycles.

2.6.1 Statistical description of fatigue data The S-N curves aim at characterizing and predicting mean fatigue life at a given S. However, for design, it is important to know the fatigue life for which a certain percentile of the population can be expected to still be intact. This requires the characterization of scatter, and the derivation of confidence limits. On the life axis, scatter in fatigue tests is 0.5–2 decades, depending mostly on loading type and material. In tensile fatigue, scatter is usually much smaller than in compression. In general, there seems to be a relation between the slope of the S-N curve and scatter: flat S-N curves show more scatter than steep ones. This opens possibilities for the theory that fatigue life is directly related to static strength; scatter on the strength axis is related, via the slope of the S-N curve, to scatter on the life-axis. Also, the scatter seems to be somehow proportional to the number of possible dominant damage (initiation) mechanisms in fatigue. This explains why composites exhibit more scatter than most metallic materials (multiple crack vs single crack), and why tensile fatigue shows less scatter than compressive fatigue (final failure in compression fatigue is a combination of fatigue, and buckling/delamination). Consequently, this implies that very low scatter in the life coordinate could mean that a single damage mechanism is driving failure. In some cases, this could mean that the test is poorly designed; damage induced by severe grip misalignment or gripping abrasion has been known to result in S-N curves with very low scatter. Although low scatter is desirable for formulating design calculations, its origins need to be taken into account. Characterization of the scatter in fatigue data is the first step in deriving design data from experimental fatigue results. Typically, the designer utilizes values of fatigue life, which are characterized by a probability of failure, and a confidence value. The probability of failure is associated with a value of N, above which a certain percentage of the specimens can be expected to fail. The confidence level is a measure of how accurate the probability of failure can be obtained from a sample of the population. A combination of failure probability and confidence is a confidence bound, or tolerance bound, which can also be associated with a value for N. A schematic representation of tolerance bounds is displayed in Fig. 2.6. The tolerance bound can be formulated as: Ntolerance ¼ Naverage  KP,C,N ðSÞ  s

(2.1)

Phenomenological fatigue analysis and life modeling

59

C P Tolerance bound

S Mean S–N curve KP,C,n Percentile of distribution

N

Fig. 2.6 Statistics of the S-N curve.

where Ntolerance ¼ generic life (tolerance bound); Naverage ¼ generic life (average described by S-N formulation); K ¼ factor depending on P, C, and N; P ¼ probability of failure; C ¼ confidence level; S ¼ generic load input; and s ¼ standard deviation. The K-factor can be found from Monte Carlo simulations or look-up tables [11, 12]. The K-factor depends largely on the distribution function that is chosen or derived to describe the scatter at a certain load level, as well as the number of specimens that are available at a load level or in the entire S-N data pool. In principle, the uncertainty near the lower and upper bounds of the S-N curve is larger, hence the tolerance bounds should be located further away from the data there. Therefore, the K-factor depends also on N in the above equation. Often, tolerance bounds are typically formulated as bounds parallel to the median fatigue curve, dropping the dependency on N in K in the expression above.

2.6.2 Censoring and run-outs In general, “censoring” means leaving out test results. There can be several reasons for doing so. Data from invalid tests should be excluded from the dataset. Validity of the experiment can be lost for many reasons: tab failures, obvious defects in specimen, faulty test parameters, etc. In some cases it is wise to discard data if the influence of an aberration of test parameters on fatigue life is unclear. It is possible that the test was nominally executed with correct conditions and no obvious defects. However, especially for long-running tests, it is possible that the test was interrupted and/or the specimen was taken out and reinserted into the machine. Limited reports are available on effects of interrupting fatigue tests on fatigue life, e.g., Refs. [13, 14]. When reinserting a test specimen, small changes in specimen orientation may influence the rate of development of damage already present in the specimen.

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Fatigue Life Prediction of Composites and Composite Structures

Therefore, reinsertion must depend on the number of cycles already tested. For a small number of cycles relative to the expected number of cycles, discarding a specimen is less of a problem (in those cases, not much machine time was invested anyway). For specimens tested in a modular fixture which can be taken out of the test frame and reinserted without changing the orientation of the specimen within the fixture, reinserting a specimen is less of a concern. For practical reasons, a long-running fatigue test must sometimes be stopped. This can be done at a predetermined number of cycles or as machine availability dictates. Such a point in the S-N curve must be marked as a “run-out.” Treatment of run-outs in S-N curve determination should be evaluated from case to case.

2.6.3 Extrapolation of fatigue curves As rules of thumb, different minimum requirements in numbers of cycles and cycle range apply to S-N curves. In most structural applications and laminates, fatigue life is most relevant from around 1000 cycles. The minimum number of cycles at the lowest load/strain levels should be 1,000,000 in most cases. For specific applications, more knowledge on fatigue behavior might be required. In the wind industry, design load cycles are in the range of 10–100 megacycles. However, only limited data are available in this cycle range. Bach [14] presented high-cycle fatigue data obtained from wind turbine laminates at R ¼  1 and R ¼ 0.1, up to >107 cycles. Other tests to even higher numbers of cycles have been done on wind turbine laminates to check the most appropriate S-N curve formulation, for instance by testing ordinary coupons in series for a time period of over a year [15]. For high-cycle, high-frequency testing up to the 1 billion cycle range, an effective and inventive solution was demonstrated [16], which used microphones as actuators to essentially perform constant displacement amplitude tests with strands. Some high-cycle data on wind turbine blade laminate coupons are included in Fig. 2.7. For design, the strain levels in operation are often so low that they cannot reasonably be used in testing, because of time constraints. As will be discussed, many S-N formulations can be used to describe S-N data within the load range of the experimental dataset, as is demonstrated in Fig. 2.8, which was published in extended form in Ref. [12]. All S-N formulations shown describe the data acceptably well. However, when extrapolation to the region of 107–108 cycles would be required, the prediction would depend severely on the choice of S-N curve and inherent extrapolation. This emphasizes that the quality of S-N curve extrapolation is highly important.

2.7

S-N formulations

Various S-N formulations exist, of different types and with different backgrounds. An overview is given in this chapter. A summary is found at the end of the chapter.

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61

0.8

Smax

0.6

0.4

S=|s max| (MPa) (R = 0.5), normalized with STT Wind turbine glass/epoxy laminate regr. Wind turbine glass/epoxy laminate (log N = –11.297.log S+1.485) 0.2 102

103

104

105

106

107

108

N 0.8

Smax

0.6

0.4

S=|s max| (MPa) (R = 0.9), normalized with STT Wind turbine glass/epoxy laminate regr. Wind turbine glass/epoxy laminate (log N = –14.344.log S+2.908) 0.2 102

103

104

105

106

107

108

N

Fig. 2.7 High-cycle fatigue data for a wind turbine laminate at R 5 0.5 and R ¼ 0.9 [16].

2.7.1 Two-parameter S-N curve The minimum number of parameters required to derive a curve through experimental data is two: a parameter describing the intercept with one of the axes, and a slope parameter. Classically, the logarithm of constant amplitude fatigue life N is assumed to be linearly dependent on the governing stress/strain S, or its logarithm. The two most used formulations of the S-N curve are: log N ¼ a log S + b,or, equivalently : N ¼ 10b Sa

(2.2)

62

Fatigue Life Prediction of Composites and Composite Structures 2300 1800

Smax

1300

800

Constant amplitude data Equivalent static data Static data Sendeckyj Ioglog (incl. static) Iinlog (incl. static) Epaarachchi Kohout and Veˇchet

300 100

101

102

103

104

105

106

107

N

Fig. 2.8 Extrapolation of S-N curves.

(see Fig. 2.9) and log N ¼ c + dS

(2.3)

where N ¼ generic life, S ¼ generic load input, and a-d ¼ fitting parameters. The parameters of the S-N curve are found by linear regression using the life variable as the dependent variable. Especially the first expression (log-log, power law) is used prevalently, as extrapolation to high-cycle fatigue is more accurate for many composites.

log S

S = 10–b/a

b↓ S = 10b

log N

Fig. 2.9 Log-log S-N curve.

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63

Both static and fatigue data can be used to calculate the parameters of the S-N curve. This means that static data are included in the regression analysis, taking static strength as the maximum load that leads to 1 cycle until failure.a For the above expressions, it is recommended to regard fatigue behavior and static behavior as unrelated, and limit the regression to the fatigue data. As a result, the above relations do not generally describe fatigue behavior in the low-cycle region outside the actual data accurately, as demonstrated in Ref. [17]. In this reference, S-N curves were fitted through all data or only through fatigue data for R ¼ 0.1, R ¼  1, and R ¼ 10. In all cases, including static data led to a discrepancy between the resulting S-N curve and the data; on the other hand, excluding static data from the regression generally led to poor prediction of static strength based on fatigue only.

2.7.2 Adding parameters Adding parameters allows for tailoring the S-N curve description for different cycle regions. In Ref. [18], two points were defined on the S-N curve. Between these points, the S-N curve is a log-log curve. Outside these points, their formulation allows for flattening or steepening of the curve, following an additional pair of parameters (Fig. 2.10):
1 a b ðN + BÞC S ¼ 10 ðN + C Þ

(2.4)

where B, C ¼ fitting parameters and a, b ¼ fitting parameters from Eq. (2.2).

2.7.3 Strength-based S-N curve Other S-N formulations exist which take into account low-cycle fatigue and/or a fatigue limit or slope change in the high-cycle region. An example is the formulation proposed by Sendeckyj [19], who generates an S-N curve based on the concept of a Fig. 2.10 Log-log S-N curve with additional parameters.

B=C

log S

B↓

C↑ log N

a

It can even be argued that this should be 0.25 cycles to failure, or 0.75 cycles to failure, if the load is sinusoidal starting with tension, and failure mode is tensile or compressive, respectively.

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Fatigue Life Prediction of Composites and Composite Structures

Fig. 2.11 Strength-based S-N curve.

E>1 E=1

log S

E j Smin j, ultimate compressive strength (UCS) if jSmin j > jSmax j, and F ¼ fitting parameter After analyzing literature data, Appel and Olthoff [26] proposed a simple R-value dependency (Fig. 2.14): F ¼ 0:095  0:015R

(2.10)

This expression works well for 1 < R < 1. For mean stresses below zero, the following adaptation is proposed here: F ¼ 0:095  0:015  ð1=RÞ

(2.11)

The resulting CLD shows a discontinuity at R ¼  1 that is proportional to the ratio of UCS to UTS. The CLD consists mostly of parallel lines (Fig. 2.15). Another approach is to relate the fatigue behavior at any R-value to a reference R-value. Based on Refs. [27, 28], it was suggested in Ref. [29] to use a fatigue

S

Fig. 2.14 Simple R-value dependency.

R↑

Samp

log N

N↑

Smean

Fig. 2.15 CLD for S-N curve of Fig. 2.14.

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Fatigue Life Prediction of Composites and Composite Structures

formulation which, although not explicitly mentioned in any of these references, reduces to a parallel-line CLD.  Seq ¼

Σni Sm i Σni

1=m  1=m 1 M

(2.12)

where m, M ¼ fitting parameters and i ¼ loading type. This formulation effectively calculates an equivalent S, which can be related to N via the reference S-N curve, usually at a reference R-value, e.g., R ¼ 0.1. This CLD is shown in Fig. 2.16. In Ref. [30], a rather comprehensive model was composed for the full description of fatigue life by integrating various models from the literature. Starting with the assumption that testing temperature is constant, and based on strength degradation of the specimen (assuming, furthermore, that the static strength was determined under the same strain rate as fatigue life), the following S-N formulation was found: US  Smax ¼ γGðR, Smax , USÞ

 1 δ N 1 δ f

(2.13)

where γ, δ ¼ fitting parameters depending on material, G ¼ function of R, S, and static strength, and f ¼ cyclic loading frequency. Based on Refs. [19, 31], the following formulation for G was postulated: GðR, Smax , USÞ ¼ US1λ Sλmax ð1  RÞλ

(2.14)

Further, the exponent was related to R and fiber orientation: λ ¼ 1:6  ψ sin ϑ

(2.15) Samp (R = –1)

Rreference N=1

N=? UCS Smean

Fig. 2.16 Equivalent S formulation CLD.

UTS

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69

where ψ ¼ function of R: 1/R for 1 < R < ∞ (compression-compression) and R for ∞ < R < 1 (the remaining loading types); θ ¼ smallest angle between fibers and loading direction. Substituting, rearranging and lumping parameters yields:  Smax k1 δ US  Smax ¼ p½Smax  ðN  1 Þ US k ¼ 1:6  ψ j sin θj 1 p ¼ γ ð1  ψ Þk δ f

(2.16)

Lumping parameters even further, this can be rewritten into: H ¼ Nδ  1 γ

(2.17)

Then, a method that is described in Ref. [30] was used to determine the parameters γ and δ by linear regression. In Refs. [16, 32], Eq. (2.16) was simplified to:
US  Smax ¼ pSmax

Smax US

q ð N r  1Þ

(2.18)

The parameters p and q were used there as fitting parameters, employing iterative routines to find the appropriate values for a particular dataset. The S-N model is illustrated in Fig. 2.17. Due to the formulation of ψ, the CLD shows a discontinuity around R ¼ ∞ (see Fig. 2.18). In Ref. [33], two-parameter log-log S-N curves were used, but the slope was made dependent on Smean instead of R. This was named the “multislope” model. An example of the associated CLD is shown in Fig. 2.19. The approach was developed separately from earlier work published in Ref. [34]. In this work, the isolife curves are continuous

Fig. 2.17 S-N model of Eq. (2.18).

p↓

S

r↑

q↑

log N

Fatigue Life Prediction of Composites and Composite Structures

Samp

70

Smean

Samp

Fig. 2.18 CLD from model of Eq. (2.16).

Smean

Fig. 2.19 Multislope CLD (S-N curves with slope that depends on Smean).

bell-shaped functions between tensile and compressive strength in the CLD (not necessarily symmetric). For both models, the data pooling scheme for finding the model parameters was similar. All the experimental data are used in the regression defining the curve parameters. So, whereas in a piecewise continuous CLD the fatigue behavior at a particular R would be determined only by the closest R-values for which data are available, in the Behesty/Boerstra models in principle all test results can work together to determine the fatigue behavior at any point in the CLD. In Boerstra’s model, the derivations

Phenomenological fatigue analysis and life modeling

71

of the CLD left and right of the R ¼  1 line are separated, which is more practical and makes sense, since fatigue mechanisms in tension and compression can expected to be physically different and therefore should probably not be mixed when constructing the CLD. Despite the potential in physically correctly representing fatigue behavior, there is still some lack of physical argumentation for the choice of some model parameters. For instance, the formulation shown in Fig. 2.19 is based on the “apex” of the CLD being located on the “mean stress equal to zero” plane. Therefore, it is continuous only in the tension or compression plane, and there is a discontinuity at Smean ¼ 0; furthermore, it has been observed for various materials that the symmetry line of the CLD is not on the Smean ¼ 0 axis. An important implication of these methods is that a multiparameter nonlinear regression algorithm is required to determine the model parameters and the result is hence dependent on the initial and boundary conditions.

2.7.6 Strength-based S-N curves with R-value dependency Kassapoglou [35] described formulations to predict S-N curves by using the statistical properties of static data. Essentially, different regions of the CLD were classified on the basis of R-value and associated with a single S-N formula. This formula describes life as a function of only the maximum stress and the properties of the static strength distribution. N¼

1 for R < 0 ðSmax =βT ÞαT + ðSmin =βc Þαc

 αT  αc βT β N¼ for 0  R < 1, N ¼ c for R > 1 S S

(2.19)

(2.20)

where α, β ¼ shape and scale parameters of the Weibull distribution of static strength and T, C ¼ subscripts indicating tensile and compression strength, respectively. To plot the CLD, some of the equations need to be solved iteratively to find the stress value as a function of life. In the original document, predictions were compared to S-N curves only in the constant R-value domain, but Fig. 2.20 is an example of what a complete CLD looks like using the proposed set of expressions. When static properties in tension and compression are equal, the CLD is symmetric with respect to R ¼  1.

2.7.7 Final notes on S-N curves and the CLD l

Some of the CLDs shown in the previous sections show parallel lines in a large part of the fatigue domain, rather than constant life lines that converge to static strengths. Already in Refs. [36, 37], it was noted that predictions became much more accurate when based on a CLD with such parallel lines. In the region close to R ¼ 1, where the fatigue loading resembles creep rather than fatigue (high mean, low amplitude), fatigue data and creep response

Fatigue Life Prediction of Composites and Composite Structures

Samp

72

Smean

Fig. 2.20 CLD associated with Ref. [34].

l

l

l

l

could be related [16]. As more detailed CLDs have become available in the recent years for unidirectional dominated wind turbine laminates [16, 38], the notion that constant life lines converging to UTS may not be realistic has become more obvious, at least in the tensile region. S-N curves and CLDs are usually experimentally determined and used in models for loadbearing laminates. The focus in this chapter has been on these unidirectional-dominated composites. As design becomes “leaner,” fatigue characteristics of other laminates become increasingly relevant. These CLDs can have a distinctly different shape or modeling requirements. For nonload-bearing transversely reinforced laminates, the top of the CLD has been reported to be located in the compression-dominated region. Publications of CLDs describing shear fatigue strengths are rare. An important aspect of evaluating and validating structural design is to take into account the failure mode. The structural failure mode cannot always be described by coupons. Therefore, full-scale validation of the structure remains necessary. Complementary to this, testing of subcomponents is advisable. Very generally, a subcomponent is a detail of the structure that is smaller in size for easier handling, and can be loaded in such a way that one or more failure modes are evoked, which are representative for the full-scale structure. The principles of S-N curve construction and CLD analysis are equally applicable to subscale or full-scale structures. Furthermore, additional experimental work is often necessary, assessing material or structural performance under variable amplitude loading, using load sequences or spectra that are representative for the application. As fatigue data become available at more R-values, better fatigue life predictions can be made [13, 32]. For calculating life for interpolated S-N curves, the method depends on the S-N curve formulation. For linear-logarithmic S-N curves, the fatigue life can be analytically determined at any point in the CLD from the bounding R-value S-N curves. For log-log S-N curves, an iterative procedure can be used. Both methods are detailed in Ref. [39]. Other authors have proposed a method for obtaining the prediction analytically [40].

Phenomenological fatigue analysis and life modeling

2.8

73

Future trends

The state-of-the-art in life prediction on high-cycle composite fatigue was, until relatively recently, based on a limited description of fatigue behavior of a laminate, like the linear Goodman diagram. It is currently shifting to more elaborate and realistic formulations of the CLD, derived from tests performed in loading types other than zero-mean cyclic loads. For load bearing, predominantly unidirectional laminates, an increasingly large database is being created, allowing for such a detailed description of the fatigue behavior. For other laminates and subcomponent structures, such as multiaxially loaded laminates, bondlines, sandwiches, spar structures, etc., accurate fatigue characterization is becoming more important as designs become more structurally efficient. For these materials and subcomponents, detailed CLDs are also required if the structure’s design load contains strongly varying load components. More extensive standardization of fatigue specimens and subcomponents is required to facilitate the maintenance of a consistent database, or interconsistent databases. Experimentally deriving a CLD from constant amplitude fatigue tests and fatigue models, nevertheless, is a procedure that is both time consuming and limited in the sense that it is only descriptive. It typically does not include any analysis of the physical mechanisms that drive fatigue. Although significant developments in the tools, models, and test data have been made with regard to micro-mechanical and multiscale modeling, see, for a recent overview of methods, Ref. [41], current micro-mechanical models are not sufficiently mature to reliably predict fatigue behavior of a laminate or laminated structure. Future efforts should therefore be aimed at expanding the description of constant amplitude fatigue behavior, through extended experiments at more loading conditions including variable amplitude, more laminate types and orientations, and more diverse environmental conditions, as well as through validating life prediction methods using these data. Success of (numerical) prediction methods will be validated more and more on subcomponent structures, as a preliminary to full-scale structural verification. On the opposite length scale, micromechanical models have the potential of more efficiently focusing the experimental effort associated with fatigue testing.

References [1] ISO International Standard, Fibre-reinforced Plastics—Determination of Fatigue Properties under Cyclic Loading Conditions, ISO 13003:2003, International Organization for Standardization, Geneva, 2003. [2] ASTM D 3479/D 3479M-96-02, Standard Test Method for Tension–Tension Fatigue of Polymer Matrix Composite Materials, 1996. [3] Germanischer Lloyd, Wacker, Requirements for the Determination of the Gradient of the S/N Curve, 2001 (unofficial note). [4] L. Mishnaevsky Jr., P. Brøndsted, Micromechanical modelling of strength and damage of fiber reinforced composites, in: Annual report on EU FR6 project upwind, 2007.

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[5] F. Lahuerta Calahorra, Thickness effects in composite laminates in static and fatigue loading, PhD thesis, Delft University of Technology, 2017. [6] A.M. van Wingerde, D.R.V. van Delft, L.G.J. Janssen, et al., Optimat blades: results and perspectives, Proc. EWEC 2006 (2006). [7] O. Krause, in: Testing Frequency for Dynamic Tests, OB report OB_TC_N003, doc. no. 10061_001, 2002. [8] D.A. van Leeuwen, R.P.L. Nijssen, T. Westphal, E. Stammes, in: Comparison of Static Shear Test Methodologies; Test Results and Analysis, Proc. Global Wind Power 2008, vols. 29–31, 2008. [9] UPWIND, Finding Design Solutions for Very Large Wind Turbines, https://wmc.eu/ upwind.php. Accessed April 2019. [10] ‘INNWIND, Innovation in Wind Energy’, Innwind p/o Energy Research Centre of the Netherlands, Petten, the Netherlands, http://www.innwind.eu/ (accessed April 2019). [11] H.J. Sutherland, P.S. Veers, The development of confidence limits for fatigue strength data, Proc. ASME/AIAA 2000 (2000) 413–423. [12] Nijssen, R.P.L., ‘Fatigue Life Prediction and Strength Degradation of Wind Turbine Rotor Blade Composites’, PhD thesis, Delft University of Technology and Knowledge Centre, 2006, ISBN 978-90-9021221-0, or Sandia National Laboratories report SAND20067810P, October 2007. [13] A.V. Movahedi-Rad, T. Keller, A.P. Vassilopoulos, Interrupted tension-tension fatigue of angle-ply GFRP composite lamiantes, Int. J. Fatigue 113 (2018) 377–388. [14] P.W. Bach, High cycle fatigue investigation of windturbine materials, in: Proc. ECWEC, 1988, 1988, pp. 337–341. [15] D.R.V. van Delft, H.D. Rink, P.A. Joosse, Fatigue behaviour of fibreglass wind turbine blade material in the very high cycle range, Proc. Wind Energy Conver. 1993 (1993) 281–286. [16] J.F. Mandell, D.D. Samborsky, N.K. Wahl, H.J. Sutherland, in: Testing and Analysis of Low Cost Composite Materials Under Spectrum Loading and High Cycle Fatigue Condition, Conference Paper, ICCM14, Paper No. 1811, SME/ASC, 2003, p. 10. [17] R.P.L. Nijssen, O. Krause, T.P. Philippidis, in: Benchmark of Lifetime Prediction Methodologies, OPTIMAT report OB_TG1_R012, doc, 2004. no. 10218. [18] J. Kohout, S. Veˇchet, A new function for fatigue curves characterization and its multiple merits, Int. J. Fatigue 23 (2001) 175–183. [19] G.P. Sendeckyj, Fitting models to composite materials fatigue data, Proc. Test Methods Design Allow. Fibrous Compos. (1979) 245–260. [20] C.W. Kensche, Influence of composite fatigue properties on lifetime predictions of sailplanes, Presented at XXIV OSTIV Congress, Omarama, New Zealand, 1995. [21] J.M. Whitney, Fatigue characterization of composite materials, in: Fatigue of Fibrous Composite Materials, ASTM STP 723, American Society for Testing and Materials, 1981, , pp. 133–151. [22] H.T. Hahn, R.Y. Kim, Fatigue behavior of composite laminate, J. Compos. Mater. 10 (1976) 156–180. [23] Y. Liu, S. Mahadevan, Probabilistic fatigue life prediction of multidirectional composite laminates, Compos. Struct. 69 (2005) 11–19. [24] Lekou, D.J., Probabilistic Strength Assessment of FRP Laminates—Verification and Comparison of Analytical Models, UPWIND report (deliverable D3_3_2) Via https:// wmc.eu/upwind.php, 2019. [25] P.W. Bach, in: Fatigue Properties of Glass- and Glass/Carbon-Polyester Composites for Wind Turbines, Energy Research Centre of the Netherlands, report ECN-C-92-072, 1992.

Phenomenological fatigue analysis and life modeling

75

[26] Appel, N., Olthoff, J., ‘Voorontwerpstudie NEWECS-45, Polymarin report’ (in Dutch). [27] W.D. Dover, Variable amplitude fatigue of welded structures, Fract. Mech. Curr. Status Future Prosp. (1979) 129–147. [28] S. Amijima, T. Tanimoto, T. Matsuoka, in: A Study on the Fatigue Life Estimation of FRP Under Random Loading, Fourth International Conference on Composite Materials, ICCM-IV, Tokyo, 1982. [29] P. Brøndsted, S.I. Andersen, H. Lilholt, Fatigue damage accumulation and lifetime prediction of GFRP materials under block loading and stochastic loading, in: S. I. Andersen, P. Brøndsted, H. Lilholt et al. (Eds.), Proc. 18th International Symposium on Materials Science: Polymeric Composites—Expanding the Limits, 1997, pp. 269–278. [30] J.A. Epaarachchi, P.D. Clausen, An empirical model for fatigue behavior prediction of glass fibre-reinforced plastic composites for various stress ratios and test frequencies, Compos. Part A 34 (2003) 313–326. [31] R.W. Hertzberg, J.A. Manson, Fatigue of Engineering Plastics, Academic Press, New York, 1980. [32] H.J. Sutherland, J.F. Mandell, in: Optimized Goodman Diagram for the Analysis of Fiberglass Composites Used in Wind Turbine Blades, ASME/AIAA Wind Energy Symposium, 2005, paper AIAA-2005–0196. [33] G.K. Boerstra, The multislope model: a new description for the fatigue strength of glass fibre reinforced plastic, Int. J. Fatigue 29 (2007) 1571–1576. [34] M.H. Beheshty, B. Harris, T. Adam, An empirical fatigue-life model for high-performance fibre composites with and without impact damage, Compos. Part A 30 (1999) 971–987. [35] C. Kassapoglou, Fatigue life prediction of composite structures under constant amplitude loading, J. Compos. Mater. 41 (22) (2007) 2737–2754. [36] D.R.V. van Delft, G.D. de Winkel, P.A. Joosse, in: Fatigue Behaviour of Fibreglass Wind Turbine Blade Material Under Variable Amplitude Loading, Proc. AIAA/ASME Wind Energy Symposium, 1997, pp. 180–188. no. AIAA-97-0951. [37] R.P.L. Nijssen, D.R.V. van Delft, A.M. van Wingerde, Alternative fatigue lifetime prediction formulations for variable-amplitude loading, J. Solar Energy Eng. 124 (4) (2002) 396–403. [38] R. Nijssen, T. Westphal, E. Stammes, D. Lekou, P. Brøndsted, in: Rotor Structures and Materials—Strength and Fatigue Experiments and Phenomenological Modelling, Proc. European Wind Energy Conference, 2008. Brussels Expo. [39] N.K. Wahl, Spectrum Fatigue Lifetime and Residual Strength for Fiberglass Laminates, PhD thesis, Montana State University, Bozeman, MT, 2001. [40] T.P. Philippidis, A.P. Vassilopoulos, Life prediction methodology for GRFP laminates under spectrum loading, Compos. Part A 35 (2004) 657–666. [41] I. Barcelos Rocha, Numerical and Experimental Investigation of Hygrothermal Aging in Laminated Composites, PhD thesis, Delft University of Technology, 2019.

Further reading [42] UPWIND Material Database, accessed January 2009, UPWIND project website, https:// wmc.eu/optimatblades_optidat.php. Accessed April 2019.

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Residual strength fatigue theories for composite materials☆

3

N.L. Post, J.J. Lesko, S.W. Case Virginia Tech, Blacksburg, VA, United States

3.1

Introduction

The prediction of fatigue damage and fatigue life for composite materials has been the subject of many investigations during recent years. Hwang and Han [1] suggested four requirements for a universal fatigue damage model: 1. It should explain fatigue phenomena at an applied stress level. 2. It should explain fatigue phenomena for an overall applied stress range: – During a cycle at a high applied stress level the material should be more damaged than that at a low applied stress level. – If it is true that failure occurs at each maximum applied stress level, then the final damage (damage just before failure) at a low applied stress level should be larger than that at a high applied stress level. 3. It should explain multi-stress level fatigue phenomena. 4. It is desirable to establish the fatigue damage model without an S-N curve.

An excellent review of work in this area of fatigue life predictions has been given by Liu and Lessard [2]. In this chapter, they divided the models used to predict fatigue life into three classes: residual strength degradation, modulus degradation, and damage tolerance approaches. According to Huston [3], most of the life prediction methods for polymeric composite materials are based on the residual strength degradation. However, he suggested that theories for fatigue failure based on the reduction of stiffness have one significant advantage over the remaining strength theories: remaining life can be assessed by non-destructive techniques. Also, Huston suggested that less testing needs to be conducted for stiffness-degradation-based models. Despite these potential advantages, residual strength-based theories have found favor because of their relative simplicity, as well as one key physical argument: that in stress-controlled tests, the residual strength-based models provide a clear explanation for failure that the stiffness-based models do not—failure occurs when the instantaneous value of the strength is equal to the instantaneous value of the applied stress. In comparison with damage accumulation models, the residual strength models also have an advantage—because the damage is evaluated in terms of a physical quantity (the

☆

This chapter is a reprint of the chapter originally published in the first edition of Fatigue Life Prediction of Composites and Composite Structures.

Fatigue Life Prediction of Composites and Composite Structures. https://doi.org/10.1016/B978-0-08-102575-8.00003-6 © 2020 Elsevier Ltd. All rights reserved.

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Fatigue Life Prediction of Composites and Composite Structures

residual strength), the models can be fit or verified experimentally at intermediate points during the fatigue lifetime. In general, residual strength models are based on the assumption that the residual strength is a monotonically decreasing function of cycles applied [4]. They also enforce that the initial strength is equal to the static strength, and that under constant amplitude fatigue loading, the residual strength at failure (n ¼ N) is equal to the applied constant amplitude load. In addition, many residual strength models have been developed with the goal of modeling/predicting the distribution of residual strengths and fatigue lifetimes. Such predictions are most often developed by integrating some rate of degradation relationship into the statistical distributions of initial strength and ultimate lifetimes. The common method of connection is the assumption that there is a one-to-one correspondence between the probability rank of the initial strength distribution and the rank of fatigue lifetime. Such a relationship was termed the strength-life equal rank assumption (SLERA) by Chou and Croman [5], and had originally been suggested by Hahn and Kim [6]. Because the residual strength evolution equations are monotonic, this implies that the residual strength curves do not cross. While such an argument makes intuitive sense, it cannot be verified experimentally, as both the static strength measurement and the fatigue lifetime measurement are destructive tests—there is no way to make the same measurement on the same composite article.

3.2

Major residual strength models from the literature

In this section, an overview of the major residual strength models in the literature is presented. A more recent review of fatigue lifetime approaches for composites materials has been presented by Post et al. [4]. In it, the authors present a convenient means of describing many of the residual strength based failure theories has been presented by Sarkani et al. [7]: dσ r AnA1 ¼ B C1 dn Cσ r

(3.1)

where σ r is the residual strength, n is the number of fatigue cycles, and A, B, C are “constants” that may depend upon the applied fatigue stress level. Integration of Eq. (3.1) for the case of constant amplitude fatigue, and enforcing that the residual strength is equal to the ultimate strength at zero cycles and the applied stress at failure, results in

n A σ Cr ¼ σ Cult  σ Cult  σ Cp N

(3.2)

where σ ult is the ultimate strength, σ p is the peak stress during the fatigue cycle, and N is the number of cycles to failure that corresponds to σ u.

Residual strength fatigue theories for composite materials

79

This integrated form will provide the basis for subsequent discussions, which are an abbreviated version of that which appeared in Post et al. [4]; additional information may be found in the full discussion.

3.2.1 Broutman and Sahu The earliest residual strength model in the literature of which the current authors are aware was that introduced by Broutman and Sahu [8]. In this work, Broutman and Sahu examined the tensile fatigue behavior of E-glass reinforced epoxy. Their experimental work consisted of 35 quasistatic tests, followed by 134 constant amplitude fatigue tests conducted at a fatigue R-ratio of 0.5 (R ¼ 0.05). The results of these tests are summarized in Table 3.1. Subsequent to these tests, two-stress-level block fatigue loading tests were conducted. In these two-stress-level tests, samples were fatigued under constant amplitude loading conditions for a predetermined number of cycles. The stress amplitude was then adjusted to a second constant amplitude, and the samples cycled until failure occurred. Broutman and Sahu assumed (and then compared with limited experimental data) that the residual strength at a given fatigue stress amplitude varied linearly with the number of cycles, so that σ r ¼ σ ult 

n X
i σ ult  σ ip N i i

(3.3)

(This assumption corresponds to A ¼ 1, C ¼ 1 in Eq. (3.1)). For the special case of twostress-block loading, it is possible to show that Eq. (3.3) results in 

 1  Fa1 n1 n2 + ¼1 1  Fa2 N1 N2

(3.4)

where Fa1 ¼ σ 1/σ ult, Fa2 ¼ σ 2/σ ult, and N1, N2 are the number of cycles to failure corresponding to σ 1 and σ 2, respectively. The results of the Broutman-Sahu experimental study of two-stress-block loading are summarized in Table 3.2 for the case in which A ¼ 1. The form of Eq. (3.4) makes it easy to compare with the linear damage accumulation (Miner’s rule) result [9] Table 3.1 Broutman-Sahu constant amplitude fatigue data (E-glass/epoxy, R ¼ 0.05) σ 1, MPa

Fa 5 σ/σ ult

N (median)

95% confidence limit

No. specimens

386 338 290 241

0.862 0.754 0.646 0.538

493 2470 14,700 172,200

420–570 2170–2820 12,100–17,590 139,200–213,000

35 31 37 31

Source: L.J. Broutman, S. Sahu, A new theory to predict cumulative fatigue damage in fiberglass reinforced plastics. In: Composite Materials: Testing and Design (Second Conference), ASTM STP 497. American Society for Testing and Materials, 1972.

Table 3.2 Two-stress-level data of Broutman and Sahu [8] n2, Residual strength A, Eq. (3.1) σ 1, MPa

σ 2, MPa

386 386 386 386 386 386 338 338 338 338 290 290 241 241 241 241 241 241 290 290 290 290 338 338

241 241 290 290 338 338 241 241 290 290 241 241 290 290 338 338 386 386 338 338 386 386 386 386

Fa1 5 σ 1/σ ult

Fa1 5 σ 2/ σ ult

n1/ N1

n2 (exper.)

n2 (Miner’s rule)

0.75

1.00

1.25

5

100,000

0.862 0.862 0.862 0.862 0.862 0.862 0.754 0.754 0.754 0.754 0.646 0.646 0.538 0.538 0.538 0.538 0.538 0.538 0.646 0.646 0.646 0.646 0.754 0.754

0.538 0.538 0.646 0.646 0.754 0.754 0.538 0.538 0.646 0.646 0.538 0.538 0.646 0.646 0.754 0.754 0.862 0.862 0.754 0.754 0.862 0.862 0.862 0.862

0.507 0.203 0.507 0.203 0.507 0.203 0.405 0.101 0.405 0.101 0.680 0.136 0.290 0.116 0.290 0.116 0.290 0.116 0.680 0.136 0.680 0.136 0.405 0.101

192,000 183,000 5840 11,970 1250 1635 86,000 162,500 8670 8000 96,500 110,800 3730 9490 931 804 0 124 293 1290 0 355 297 503

84,895 137,243 7247 11,716 1218 1969 102,459 154,808 8747 13,215 55,104 148,781 10,437 12,995 1754 2183 350 436 790 2134 158 426 293 443

154,666 165,180 12,567 13,846 1889 2237 142,036 164,678 11,030 13,785 90,036 155,767 8625 12,270 814 1808 0 208 0 1925 0 259 63 386

146,008 161,713 11,784 13,532 1766 2188 135,005 162,924 10,558 13,667 82,426 154,245 9140 12,476 1127 1933 16 302 56 1987 0 322 138 404

138,877 158,858 11,182 13,291 1680 2154 130,022 161,682 10,247 13,589 77,527 153,265 9427 12,591 1286 1996 118 343 225 2021 0 351 177 414

103,578 144,724 8522 12,226 1354 2023 110,698 156,863 9163 13,319 61,164 149,993 10,204 12,902 1658 2145 311 420 664 2109 89 412 269 437

84,896 137,244 7247 11,716 1218 1969 102,459 154,808 8747 13,215 55,104 148,781 10,437 12,995 1754 2183 350 436 790 2134 158 426 293 443

The original Broutman-Sahu model is obtained from the A ¼ 1.00 column of the table.

Residual strength fatigue theories for composite materials

81

n1 n2 + ¼1 N1 N2

(3.5)

Thus, there are two key related features of the Broutman and Sahu analysis: (1) there is a clear sequence of loading effect on the predicted lifetimes, and (2) the results predicted using linear degradation in residual strength with fatigue cycles are not the same as those predicted by linear damage accumulation. In some cases, no experimental cycles occur when the stress level is changed. This result is observed experimentally for the cases in which the stress changes from 241 to 386 MPa and when the stress level changes from 290 to 386 MPa. In Table 3.3, the Miner’s sum calculations for the Broutman and Sahu data are presented. Broutman and Sahu pointed out that for high-low fatigue loading, the calculated Miner’s sum could be greater than unity, and that for low-high loading, the calculated Miner’s sum could be less than unity. However, in the Broutman-Sahu analysis, the Table 3.3 Miner’s sum calculations for the two-stress-level data of Broutman and Sahu [8] Miner’s sum A, Eq. (3.1) Fa1 5 σ 1/ σ ult

Fa2 5 σ 2/ σ ult

n1/N1

0.75

1.00

1.25

5

100,000

0.862 0.862 0.862 0.862 0.862 0.862 0.754 0.754 0.754 0.754 0.646 0.646 0.538 0.538 0.538 0.538 0.538 0.538 0.646 0.646 0.646 0.646 0.754 0.754

0.538 0.538 0.646 0.646 0.754 0.754 0.538 0.538 0.646 0.646 0.538 0.538 0.646 0.646 0.754 0.754 0.862 0.862 0.754 0.754 0.862 0.862 0.862 0.862

0.507 0.203 0.507 0.203 0.507 0.203 0.405 0.101 0.405 0.101 0.680 0.136 0.290 0.116 0.290 0.116 0.290 0.116 0.680 0.136 0.680 0.136 0.405 0.101

1.405 1.162 1.362 1.145 1.272 1.109 1.230 1.057 1.155 1.039 1.203 1.041 0.877 0.951 0.619 0.848 0.290 0.538 0.680 0.915 0.680 0.661 0.533 0.883

1.355 1.142 1.309 1.124 1.222 1.089 1.189 1.047 1.123 1.031 1.159 1.032 0.912 0.965 0.746 0.899 0.323 0.729 0.703 0.941 0.680 0.788 0.685 0.921

1.313 1.126 1.268 1.107 1.187 1.075 1.160 1.040 1.102 1.025 1.130 1.026 0.931 0.973 0.810 0.924 0.530 0.812 0.771 0.954 0.680 0.848 0.763 0.941

1.108 1.043 1.087 1.035 1.055 1.022 1.048 1.012 1.028 1.007 1.035 1.007 0.984 0.994 0.961 0.984 0.921 0.968 0.949 0.990 0.860 0.972 0.951 0.988

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

The original Broutman-Sahu model is obtained from the A ¼ 1.00 column of the table.

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Fatigue Life Prediction of Composites and Composite Structures

residual strength at the end of the two-step loading process is independent of the order of loading. (This feature was not discussed in the Broutman-Sahu reference.) There are some key observations that come out of the Broutman-Sahu work. First of all, there is a considerable amount of material characterization required to calibrate the model because of the statistical variation in the initial material strength (and according to the strength-life equal rank assumption, the resulting fatigue lifetime). Secondly, the residual strength approach can provide improved estimates of the fatigue lifetimes under the block loading condition compared with the Miner’s rule approach with no additional experimental characterization. In essence this added accuracy for the block-loading case comes for free, as long as the variation of residual strength with fatigue cycles is approximately linear. Finally, while the predicted lifetime depends upon the order of loading for this two-block case, the residual strength is independent of the loading order.

3.2.2 Reifsnider and Stinchcomb Reifsnider and Stinchcomb [10] proposed a “critical element model” for the residual strength of composites under fatigue loading. In this model, the residual strength is assumed to vary as Fr ðnÞ ¼ 1 

ðn

ð1  FaðnÞÞA

0


n A1
n  d N N

(3.6)

where Fr is the normalized residual strength for the critical element, and Fa is the corresponding value of the failure criterion. The critical element is that portion of the composite laminate whose failure directly implies failure of the composite laminate. (For additional information on the implementation of this model, the reader is referred to Reifsnider and Case [11] and Post et al. [12]). As a starting point, the case in which Eq. (3.6) is applied at a laminate level using the maximum stress failure criterion is considered. In such a case Post et al. [12] recast Eq. (3.6) in the form of " Fr ¼ 1 

X i

ð1  Fai Þ

1=A

 #A ni Ni

(3.7)

Eq. (3.7) is identical to Eq. (3.2) if C ¼ 1 and A is a constant for the case in which the entire composite is smeared together. This approach greatly simplifies the analysis, as load redistribution effects within the composite do not need to be considered. However, it also requires that laminates made of the same composite material but having different stacking sequences be characterized individually. Wahl et al. [13, 14], Wahl and Mandell [15], Nijssen et al. [16], and Post et al. [17] used Eq. (3.7) to varying degrees of success in describing constant R-ratio spectrum fatigue loading. An alternative approach [12] used the change in elastic modulus recorded during fatigue testing to estimate the stress in the critical element.

Residual strength fatigue theories for composite materials

83

Because there is often data available for the two-block loading case (such as the Broutman-Sahu data) it is of interest to examine the predictions of Eq. (3.7) for the two-block loading case. This result is given by 

1  Fa1 1  Fa2

1

A n1

N1

+

n2 ¼1 N2

(3.8)

There is an obvious similarity to the Broutman-Sahu result given in Eq. (3.4) as well as the Miner’s rule result of Eq. (3.5). Indeed, for Miner’s rule and the Reifsnider and Stinchcomb residual strength analysis to give consistent results, the value of the residual strength parameter A must approach infinity. Thus, the use of Miner’s rule implies sudden-death material failure (at least in the context of residual-strength based theories). For all cases where A > 0, Eq. (3.8) implies that for a low stress level followed by a high stress level, the lifetime is always less than the Miner’s rule result, and for a high stress level followed by a low stress level, the resulting lifetime is always less than the Miner’s rule result. (Cases where A < 0 correspond to physically unreasonable results where the strength increases with the number of fatigue cycles.)

3.2.3 Schaff and Davidson Schaff and Davidson [18, 19] used an evolution equation for the residual strength in the form of σ r ¼ σ ult  ðσ ult  σ a Þ


n A N

(3.9)

where A is assumed to vary with the magnitude of σ a as well as with the R-ratio. To account for the observed statistical spreading of residual strength data, they applied a linear reduction to the Weibull shape parameter at each stress level. This method essentially results in a complex curve fit for the statistical strength distribution. A more mathematically consistent model has been proposed by Yang and Liu [20]. In their first paper, Schaff and Davidson applied the model of Eq. (3.9) to twostress-level repeated block loading fatigue. They noted that in some cases, repeatedly changing the stress level seemed to cause more fatigue damage than applying the same loads in longer blocks. To account for this effect, which they termed the “cycle mix effect,” they introduced a cycle mix factor, CM, to reduce the residual strength whenever the magnitude of the mean stress increased. This cycle mix factor was given by  2   Δσ a Δσ m Δσ m CM ¼ Cm σ ult σr

(3.10)

where Cm is a constant that must be determined from experimental data on a repeat block fatigue loading with different numbers of cycle mix events. Schaff and Davidson found this model appeared to fit their limited dataset well.

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Fatigue Life Prediction of Composites and Composite Structures

In their second paper, Schaff and Davidson applied their model to spectrum loading cases using two sets of data. In both cases, they performed curve fitting of the constant lifetime diagram and a linear interpolation to determine the value of A for each loading cycle. Because they did not have sufficient information to evaluate CM, they assumed that it did not have a significant impact. (Thus, it was not possible to evaluate the effectiveness of this CM parameter in describing spectrum loading behavior.) Nijssen et al. [16] did evaluate the impact of treating A as a function of applied stress. They found that a large data set is required to develop an empirical functional form for this variable. Thus far, it has only been possible to develop the value for a limited number of materials at a limited number of stress levels, so it is not possible to draw general conclusions about the overall form the parameter takes. Nijssen et al. [16] showed that improved predictions were possible for two-stress-level block loading when the values of A were found from constant amplitude residual strength tests for each stress level. Such a result is not surprising, given that additional parameters have been introduced into the modeling.

3.2.4 Hahn and Kim Hahn and Kim [6] as well as Hashin [21] and Whitney [22] used residual strength equations equivalent to Eq. (3.2) where A ¼ 1 and C is an unknown parameter fitted to experimental data (and potentially a function of stress level), so that for the constant amplitude case, the result is

n  σ Cr ¼ σ Cult  σ Cult  σ Cp N

(3.11)

3.2.5 Other residual strength models Other residual strength models have been presented in the literature. In particular, the discussion above has focused on deterministic representations of residual strength changes. Yang and Liu [20] assumed that the initial strength distribution conformed to a two-parameter Weibull representation, and that the residual strength varied according to  H σp, f , R dσ r ¼ dn Cσ r ðnÞC1

(3.12)

where f is the fatigue frequency. The predicted distribution of fatigue lifetime is then given by 2

 !α=C 3 n + SCp =H σ p 5  FN ðnÞ ¼ 1  exp 4 σ C0 =H σ p

(3.13)

Residual strength fatigue theories for composite materials

85

where σ 0 and α are the Weibull location parameter and the shape parameter for the initial strength distribution, respectively. Eq. (3.13) is then in the form of a threeparameter Weibull distribution where the characteristic fatigue lifetime is given by  N0 ¼ σ C0 =H σ p

(3.14)

A similar approach was employed by Sendeckyj [23].

3.3

Fitting of experimental data

In general, variable amplitude fatigue models require empirical fitting of parameters for a given material. Therefore, predictions made with that model can only be applied to the laminate for which they were fitted. Thus, comparison of the models can take place where the data required for fitting all of the selected models is available, and at least one spectrum case that can be used for verification of the predictions is available on a given material system. Post et al. [4] analyzed data sets containing fatigue life under stress-controlled tests similar to those typically available for structural design applications. Four different material systems were considered, covering a wide range of E-glass/polymer matrix composites typical of those used for wind turbine blade and naval architecture. These data sets were considered because they each include a statistically significant number of measurements of the initial strength, constant amplitude fatigue to failure when subjected to various R-ratios, and residual strength at various fractions of lifetime during constant amplitude loading. Some of the data sets also contain various cases of block loading. Each data set contains several examples of spectrum loading that enable validation of the predictive capabilities of each model. The first material data set, denoted DD16, is part of the DOE/MSU database [24]. The tests performed are detailed in Wahl and Mandell [15], and various analyses of the data are provided in Mandell et al. [25, 26], Nijssen et al. [16], Wahl et al. [13, 14], and Wahl and Mandell [15]. Validation spectra available for the DD16 data set include fatigue to failure under the standard WISPERX spectrum and two modified versions of WISPERX where the valleys of each cycle were adjusted to enforce a constant R-ratio of R ¼ 0.1. Following Post et al. [4], the Wahl and Mandell [15] notation for these spectra is used: WISPK includes all WISPERX cycles peaks, but adjusts the valleys to require that R ¼ 0.1, while WISXR01 includes only the tension-tension WISPERX cycles and forces resulting valleys to R ¼ 0.1. The second and third material data sets are MD2 (R0400 geometry) and UD2 (I1000 geometry). These are part of the Optimat Blades database, publicly available online in Excel format. Spectrum predictions were made for the WISPER and WISPERX standard spectra fatigue to failure experimental results also available in the data set. The final data set was collected at Virginia Tech, with the details and test results available in Post [27]. This material consists of 10 layers of woven roving E-glass (Vetrotex 324) with a [0/+45/90/45/0]s stacking sequence (denoted by the warp direction) in a rubber toughened vinyl ester matrix (Ashland Derakane 8084). This

86

Fatigue Life Prediction of Composites and Composite Structures

material system is characteristic of those used in US Navy ship construction. The variable amplitude fatigue loading data for this material system includes Rayleighdistributed loading with 95% autocorrelation (a measure of the degree of load ordering: see Post et al. [28]) with a nominal value of the fatigue R-ratio, R ¼  1, and the same peaks with the following valleys forced to R ¼ 0.1 called RAY95 and RAY95R01, respectively. In the paragraphs that follow, approaches for fitting the experimental data to provide inputs to the residual strength models are discussed. Since the ultimate goal for practical loading situations is to be able to analyze spectrum loading, each of the residual strength models described above requires a separate empirical model for determining the total number of cycles to failure, Ni, under a constant amplitude stress equivalent to the current applied cycle in the spectrum characterized by the peak stress, σ p, and the valley stress, σ v. Wahl et al. [14] and Wahl and Mandell [15] examined the results of fatigue predictions for two residual strength models and Miner’s rule. In doing so, they considered exponential and power law representations, both including and excluding the initial strength data in determining the fitting parameters. They found (not surprisingly) that there can be significant differences in the resulting spectrum fatigue predictions, because the shape of the extrapolated S-N curve in the longer lifetime region is different—often substantially so. While the selection of the S-N fits is important, particularly because of the practical need to extrapolate fatigue lives beyond those measured experimentally, the selection of fits should be on the basis of their ability to represent the constant amplitude data rather than as a “tuning” method in the spectrum fatigue predictions, if the goal is prediction rather than representation. Sendeckyj [23] identifies a number of the equations used to represent S-N data as σ ult =σ a ¼ N S

(3.15)

σ a ¼ σ ult  b log N

(3.16)

σ range ¼ a + b=N x

(3.17)

σ range ¼ a + b=N x  c=Ay

(3.18)

σ a =σ ult ¼ a + b=ð log N Þx

(3.19)

where A ¼ ð1  RÞð1 + RÞ ¼ σ range =σ mean R ¼ σ min =σ max

(3.20)

and the subscripts a, range, mean, max, and min to σ are used to denote the stress amplitude, stress range, mean stress, maximum stress, and minimum stress, respectively. The remaining quantities in the equations are experimentally determined material constants. Eq. (3.15) is the classical power-law representation that leads to

Residual strength fatigue theories for composite materials

87

a straight line on a log-log plot. Eq. (3.16) has been used by Mandell [29] to characterize short fiber glass-reinforced composite materials. (In his work, Mandell has suggested that b  0.1σ ult.) Sendeckyj [23] points out that Eq. (3.17) was used by Albrecht [30] for a large number of metallic systems. Eq. (3.18), an extension of Eq. (3.17) that includes mean stress effects, was used by Sims and Brogdon [31] to represent glass/epoxy fatigue data. Finally, Reifsnider and Jen [32] used Eq. (3.19) to characterize a number of composite materials. For example purposes, the constant amplitude fatigue data are fitted using Eq. (3.15) cast in the form of

a N ¼ B σ p

(3.21)

where B and a are fitting parameters, N is the number of cycles to failure, and σ p is the maximum stress in the applied cyclic loading. Alternatively, Eq. (3.21) may be expressed as



log N ¼ alog σ p + b

(3.22)

where a and b ¼ log B are the slope and intercept of a log-log plot of stress versus cycles to failure. (In performing the fitting, we note that the number of cycles to failure, N, is the dependent variable with the stress, σ p, as the independent variable rather than vice versa.) Eq. (3.22) can be fitted using standard linear regression tools. Johannesson et al. [33] point out that this expression coincides with the maximum likelihood method assuming log-normally distributed errors in the fatigue lifetime data. The typical literature approach is to use the {kσ pk, kσ vk} for σ p in Eq. (3.22) and thus to calculate N in terms of the minimum stress σ v for R < 0 and R > 1 (for cases in which the largest compressive stress magnitude is greater than the largest tensile stress magnitude). However, the absolute value of the maximum stress as indicated in Eq. (3.22) may be used because it will simplify the mathematics of interpolation on the constant lifetime diagram. Thus, the values of b that are calculated will be different from those reported elsewhere, and the plotted S-N curves for R > 1 appear to be at much lower stresses. However, as long as a consistent approach is applied, this choice will not impact the outcome of the calculations. The S-N curves for each of the four material systems studied are given in Figs. 3.1–3.4. In addition the resulting values for the parameters a and b are listed in Table 3.4. To apply the models for loading conditions in which the R-ratio is not a value that has been characterized experimentally, it is necessary to use the values that have been measured with some scheme to arrive at values for other R-ratios. Some authors (e.g., Epaarachchi and Clausen [34], Fatemi and Yang [35], Schaff and Davidson [19]) have chosen to curve-fit an equation to the constant lifetime diagram, such as that illustrated in Figs. 3.5–3.8. However, because of the shape of the constant lifetime plot, it is unlikely that any function will be able to fit the data available for all data sets. One possible solution is the artificial neural network approach suggested by Vassilopoulos et al. [36]. However, this is more complicated than may be desirable

88

Fatigue Life Prediction of Composites and Composite Structures

Maximum stress, ||

p ||

(MPa)

1000

R

100

0.9 0.8 0.7 0.5 0.1 2 –0.5 –1 –2 10

10

1 1

10

100

1000 10,000 100,000 1,000,000 10,000,000 Cycles to failure, N

Fig. 3.1 S-N data for the DOE/MSU DD16 material data set [4].

Maximum stress, ||

p ||

(MPa)

1000

100

R 0.5 0.1 2 –0.4 –1 10

10

1 1

100

10,000

1,000,000

Cycles to failure, N

Fig. 3.2 S-N curves for the Optimat MD2 material data set [4].

100,000,000

Residual strength fatigue theories for composite materials

89

Maximum stress, || p || (MPa)

1000

100

10

R 0.1 –1 10

1 1

100

10,000 1,000,000 Cycles to failure, N

100,000,000

Fig. 3.3 S-N curves for the Optimat UD2 material data set [4].

Maximum stress, || p || (MPa)

1000

100

10

R 0.1 –1 10

1 1

10

100

1000 10,000 100,000 1,000,000 10,000,000 Cycles to failure, N

Fig. 3.4 S-N curves for the Virginia Tech VT8084 material data set [4].

for many situations. Post et al. [4] have suggested using linear interpolation on the constant lifetime plot in order to determine lifetimes for a particular set of mean stress and stress amplitude values. This process is illustrated graphically in Fig. 3.9. Here R1 and R3 are R-ratios for which S-N data has been collected. The stress amplitude, σ a, and the mean stress, σ m, correspond to an R-ratio at which experimental data has not been collected. A line is then drawn which passes through the points on the line

90

Fatigue Life Prediction of Composites and Composite Structures

Table 3.4 Average tensile and compression strength values, and S-N curve fit parameters

Material

σ ult (tension), MPa

σ ult (compression), MPa

DD16

602.9

401.2

MD2

555.6

459.8

UD2

800

500.9

VT8084

346.8

299.2

R-ratio

Total tests

a

b

2 10 2 1 0.5 0.1 0.5 0.7 0.8 0.9 2 10 1 0.4 0.1 0.5 10 1 0.1 10 1 0.1

15 52 32 35 21 98 66 23 27 24 9 28 65 28 47 15 47 153 57 57 66 61

11.91 18.02 11.72 8.56 7.89 9.99 10.6 9.44 11.35 22.18 15.51 29.19 9.35 7.58 9.27 10.54 8.19 9.23 8.40 14.6 8.34 6.88

30.95 29.68 29.26 23.90 22.51 28.54 30.95 28.27 33.77 62.93 40.89 47.72 25.65 22.29 27.03 30.94 17.93 26.67 26.17 22.08 20.83 19.03

Source: N.L. Post, S.W. Case, J.J. Lesko, Modeling the variable amplitude fatigue of composite materials: a review and evaluation of the state of the art for spectrum loading. Int. J. Fatigue 30 (2008) 2064–2086.

400 R = –2

R = –1

R = –0.1

R = –0.5

200 105 107 0 –600

–400

R = 0.5

N 103

R=2

a

(MPa)

R = 10

–200

0 m

10

104

R = 0.7

6

(MPa)

R = 0.8 R = 0.9 200

400

600

Fig. 3.5 Constant lifetime plot for the DOE/MSU DD16 material data set [4].

800

Residual strength fatigue theories for composite materials

91

400 R = –1

R = 10

R = 0.1 R = –0.4

200

103

R=2

R = 0.5

a

(MPa)

N

105

104

10

6

107

0 –600

–400

–200

0 m

(MPa)

200

400

600

800

Fig. 3.6 Constant lifetime plot for the Optimat MD2 material data set [4]. 600 R = –1

R = 10

(MPa)

400

R = 0.1

N

a

104

200

103

106

105 107

0 –600

–400

–200

0 m

200 (MPa)

400

600

800

Fig. 3.7 Constant lifetime plot for the Optimat UD2 material data set [4].

100

R = 10

R = 0.1

R = –1

R=2

N

104

103

a

(MPa)

200

106 0 –600

105

107 –400

–200 m

0 (MPa)

200

400

600

Fig. 3.8 Constant lifetime plot for the Virginia Tech VT8084 material data set [4].

1000

92

Fatigue Life Prediction of Composites and Composite Structures a

a

=

a,1 m,1

– –

a,3 m,3

m

+

a,1

–

a,1 m,1

– –

a,3 m,3

m,1

R1

R2 N a,1

N

a

R3 N

a,3

m,1

m

m,3

m

Fig. 3.9 Linear interpolation scheme used by Post et al. [4] on constant lifetime plots to determine lifetimes for R-ratios not tested experimentally.

corresponding to R1 and R3 at a lifetime N, where N is the lifetime that corresponds to σ a and σ m. In their analysis, Post et al. solve numerically for the appropriate value of N using a trial-and-error process. Further details are available in Post et al. [4]. Once the S-N data has been established, it is then necessary to evaluate the other model parameters that may appear, such as the A and C values in Eq. (3.2). The easiest place to begin is with the Broutman-Sahu model given by Eq. (3.3). This model requires only the initial strength data in addition to the fatigue lifetimes. The other residual strength models require additional parameter determinations. For example, the Post et al. model given by Eq. (3.7) requires the fitting of the parameter A, and the Han and Kim model given by Eq. (3.11) requires the fitting of C. For these types of models, the parameters A and C which impact the shape of the residual strength curve are fitted to residual strength data collected by interrupted constant amplitude fatigue tests where residual strength was measured. A challenge that arises in the experimental tests is that premature failures may occur. (Such failures are predicted by models of the type given by Eq. (3.13).) Exclusion of these premature failures from the residual strength has the effect of biasing the calculated distribution to higher residual strengths. To avoid this bias, a two-parameter Weibull distribution may be fitted to the residual strength distribution at each residual strength measurement point using the approach described by Yao and Himmel [37]. This method considers premature failures in calculating the median rank of the surviving specimens’ strength and thus estimates the entire residual strength distribution including the “imaginary” residual strength that is calculated to be below the applied stress for those specimens that failed prematurely. Then the mean residual strength may be calculated from the Weibull distribution as

Residual strength fatigue theories for composite materials

93

  1 σ r ¼ σ r0 Γ 1 + α

(3.23)

where Γ is the Euler gamma function and α and σ r0 are the shape and location parameters. The calculated mean strength is then used to fit the residual strength model parameters by minimizing the least-squares error in residual strength between the model and experiment for the relevant mean residual strength points.

3.4

Prediction results

Using this approach, Post et al. [4] evaluated the performance of 12 lifetime prediction models using the available spectrum-loading data for a combination of residual strength and damage accumulation models. These results are summarized in Fig. 3.10. Here we will focus on the results for the residual strength models. The simple residual strength model developed by Broutman and Sahu [8] gave more conservative predictions of fatigue lifetime than Miner’s rule (PM) in all cases because failure occurred at the highest load point in each spectrum. Since Miner’s rule generally over-predicted fatigue lifetime, the Broutman-Sahu results were generally better. The Reifsnider and Stinchcomb model (RS1) performed better than Broutman-Sahu

PM BF Dam age rules

HR OH*

Data set

BS RS1 RS2* Residual strength m odels

RS3 RS4* RS5 INT Y1

–1

–0.5

0

m ax (|| ||), exp . M Pa N

VT8084

– RAY95R01

– 184

VT8084

– RAY95

– 108

– 916,000

VT8084

– RAY95

– 127

– 266,000

– 150,000

– 9,200,000

UD2

– WISPER

– 350

UD2

– WISPER

– 375

– 4,790,000

UD2

– WISPER

– 248

– 2,740,000

MD 2

– WISPER

– 284

– 5,940,000

MD 2

– WISPER

– 355

– 678,000

DD16

– WISXR01

– 204

– 1,380,000

DD16

– WISXR01

– 237

– 204,000

DD16

– WISPK

– 255

– 532,000

DD16

– WISPERX

– 260

– 915,000

0.5 M od el err or = M e = log

Spectrum

1

1.5

2

N mod el N experim ent

Fig. 3.10 Comparison of fatigue lifetime prediction model performance for models evaluated by Post et al. [4].

94

Fatigue Life Prediction of Composites and Composite Structures

for the VT8084 material, particularly for tensile loading, but performed worse than Broutman-Sahu for the DD16, MD2, and UD2 materials under WISPER-type spectra. The Han and Kim model (RS3) performed similarly to RS1. Overall, the simplicity and relatively good accuracy of the Broutman-Sahu model suggest that it should be used in place of the Miner’s rule result. It requires no additional characterization, and provides improved accuracy. It is more difficult to assess the performance of the remaining models—they require additional data to calibrate, and those additional data requirements do not appear to translate into improved prediction accuracy—at least not for the cases studied.

3.5

Conclusions and future trends

In this chapter, some of the basic residual strength model implementations have been discussed. In spite of the long history associated with these types of models, progress continues to be made, albeit somewhat slowly. As indicated, the application of fatigue models to real data sets without allowing the models to be “adjusted” after the fact highlights many of the difficulties associated with them. The phenomenological nature of the models and the scatter inherent in all fatigue data, but particularly that of composite materials, limits general conclusions that can be drawn. At a very minimum, it appears that the Broutman-Sahu approach offers improved predictions over the Miner’s rule approach without requiring additional characterization testing to be performed. However, because the predictions given by the Broutman-Sahu analysis lead to shorter lifetimes, there is a natural disincentive for designers to adopt it—if they are already struggling to meet safety margins that require different knockdowns for fatigue and statistical variation by themselves, they are not likely to be interested in approaches that require further knockdowns. For this reason, ongoing efforts are focused not only on improving the accuracy of the residual strength predictions, but also on providing designers with systematic calculations of knockdown factors that incorporate fatigue degradation and statistical variability of the material. The long-term goal is to combine the understanding of environmental effects, fatigue, and fatigue spectrum in a unified model to predict material behavior under realistic design conditions. Post et al. [38] suggest that the use of such understandings could be reduced to a series of knockdown factors, ki, or partial safety factors that could be used in an allowable strength design approach as suggested here, where the applied stresses have to be smaller than the reduced strength. σ app  σ ult

Y

ki i  σ ult kfatigue kstatistical kmoisture ktemperature kaging kUV kweathering kscaleup ⋯

(3.24)

To examine a reasonably realistic condition, Post et al. examined a representative 30-year ship history spectrum, created by NSWC-CD [39]. In this spectrum, a 5-year load spectrum is followed by 25 repetitions of a 1-year load spectrum (so that

Residual strength fatigue theories for composite materials

95

the load levels in the 5-year spectrum are greater in magnitude than those in the 1-year spectrum), and applied it to the VT8084 material. To calculate a design knockdown factor, they assume that the initial strength distribution for the composite may be represented by a two-parameter Weibull distribution so that   α  σ Pf ¼ 1  exp  β

(3.25)

For a given specified (desired) probability of failure, they can solve for the initial compression strength at this specified probability of failure 



1 XPf ¼ β ln 1  Pf



1 α

(3.26)

where it is noted that α and β may be determined with a required confidence level based upon the available experimental data. (For the E-glass/vinyl ester composite studied here, the values of α and β are 26.2 and 305 MPa, respectively.) A probability of failure of 10% corresponds to a value of 280 MPa for XPf. The residual strengths are then calculated as σr ¼1 XP f

ð    σ a 1j dn j 1 N XP f

(3.27)

where σ a is the applied stress in each block of loading, and N is determined from log N ¼ A log ðFaÞ + B

(3.28)

where Fa ¼ σ peak =Xt,c

(3.29)

σa XP f

(3.30)

with Fa ¼

The scaled values of the stresses in the spectrum are then adjusted so that the composite is predicted to just survive. This scaling factor is then a function of (1) the spectrum itself, (2) the properties of the composite, and (3) the allowed probability of failure. For the particular values studied here, the corresponding scaling (knockdown factor) is that the maximum stress is allowed to be 73% of the median compression strength for a probability of failure of 10%. This level is significantly higher than the current Navy design practice, and so perhaps offers some opportunity to encourage designers to adopt the approach.

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Fatigue Life Prediction of Composites and Composite Structures

References [1] W. Hwang, K.S. Han, Cumulative damage models and multi-stress fatigue life prediction, J. Compos. Mater. 20 (1986) 125–153. [2] B. Liu, L.B. Lessard, Fatigue and damage-tolerance analysis of composite laminates: stiffness loss, damage-modelling, and life prediction, Compos. Sci. Technol. 51 (1994) 43–51. [3] R.J. Huston, Fatigue life prediction in composites, Int. J. Press. Vessel. Pip. 59 (1994) 131–140. [4] N.L. Post, S.W. Case, J.J. Lesko, Modeling the variable amplitude fatigue of composite materials: a review and evaluation of the state of the art for spectrum loading, Int. J. Fatigue 30 (2008) 2064–2086. [5] P.C. Chou, R. Croman, Residual strength in fatigue based on the strength-life equal rank assumption, J. Compos. Mater. 12 (1978) 177–194. [6] H.T. Hahn, R.Y. Kim, Proof testing of composite materials, J. Compos. Mater. 9 (1975) 297–311. [7] S. Sarkani, G. Michaelov, D.P. Kihl, D.L. Bonanni, Comparative study of nonlinear damage accumulation models in stochastic fatigue of FRP laminates, J. Struct. Eng. 127 (2001) 314–322 851. [8] L.J. Broutman, S. Sahu, A new theory to predict cumulative fatigue damage in fiberglass reinforced plastics, in: Composite Materials: Testing and Design (Second Conference), American Society for Testing and Materials, 1972. ASTM STP 497. [9] M.A. Miner, Cumulative damage in fatigue, J. Appl. Mech. 67 (1945) A159–A164. [10] K.L. Reifsnider, W.W. Stinchcomb, A critical element model of the residual strength and life of fatigue loaded composite coupons, in: Composite Materials: Fatigue and Fracture, American Society for Testing and Materials, 1986. ASTM STP 907. [11] K.L. Reifsnider, S.W. Case, Damage Tolerance and Durability of Material Systems, John Wiley & Sons, New York, 2002. [12] N.L. Post, J. Bausano, S.W. Case, J.J. Lesko, Modeling the remaining strength of structural composite materials subjected to fatigue, Int. J. Fatigue 28 (2006) 1100–1108. [13] N. Wahl, D. Samborsky, J. Mandell, D. Cairns, Spectrum fatigue lifetime and residual strength for fiberglass laminates in tension, in: ASME Wind Energy Symposium, ASME/AIAA, 2001. [14] N. Wahl, D. Samborsky, J. Mandell, D. Cairns, Effects of modeling assumptions on the accuracy of spectrum fatigue lifetime predictions for a fiberglass laminate, in: ASME 2002 Wind Energy Symposium, AIAA/ASME, 2002. [15] N.K. Wahl, J.F. Mandell, Spectrum Fatigue Lifetime and Residual Strength for Fiberglass Laminates, Contractor Report SAND2002-0546, Sandia National Laboratories, Albuquerque, NM, 2001. [16] R.P.L. Nijssen, D.D. Samborsky, J.F. Mandell, D.R.V. Vandelft, Strength degradation and simple load spectrum tests in rotor blade composites, in: ASME Wind Energy Symposium, 2005. [17] N.L. Post, J. Cain, K.J. Mcdonald, S.W. Case, J.J. Lesko, Residual strength prediction of composite materials: random spectrum loading, Eng. Fract. Mech. 75 (2008) 2707–2724. [18] J.R. Schaff, B.D. Davidson, Life prediction methodology for composite structures. Part I—constant amplitude and two-stress level fatigue, J. Compos. Mater. 31 (1997) 128–157. [19] J.R. Schaff, B.D. Davidson, Life prediction methodology for composite structures. Part II—spectrum fatigue, J. Compos. Mater. 31 (1997) 158–181.

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[20] J.N. Yang, M.D. Liu, Residual strength degradation model and theory of periodic proof tests for graphite/epoxy laminates, J. Compos. Mater. 11 (1977) 176–203. [21] Z. Hashin, Cumulative damage theory for composite materials: residual life and residual strength methods, Compos. Sci. Technol. 23 (1985) 1–19. [22] J.M. Whitney, Fatigue characterization of composite materials, in: Fatigue of Fibrous Composite Materials, American Society for Testing and Materials, 1981. ASTM STP 723. [23] G.P. Sendeckyj, Life prediction for resin-matrix composite materials, in: K.L. Reifsnider (Ed.), Fatigue of Composite Materials, Elsevier Science Publishers, Amsterdam, 1990. [24] J.F. Mandell, DOE/MSU Composite Material Fatigue Database, Sandia National Laboratories, Albuquerque, NM, 2004. [25] J.F. Mandell, D.D. Samborsky, D.S. Cairns, Fatigue of Composite Materials and Substructures for Wind Turbine Blades, Sandia National Laboratories, Albuquerque, NM, 2002. [26] J.F. Mandell, D.D. Samborsky, L. Wang, N.K. Wahl, New fatigue data for wind turbine blade materials, in: 41st Aerospace Sciences Meeting and Exhibit, 2003. [27] N.L. Post, Reliability based design methodology incorporating residual strength prediction of structural fiber reinforced polymer composites under stochastic variable amplitude fatigue loading, in: N.L. Post (Ed.), Engineering Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA, 2008. [28] N.L. Post, J.J. Lesko, S.W. Case, Fatigue durability of E-glass composites under variable amplitude loading: the importance of load sequence, in: 2008 European Wind Energy Conference, Brussels, 2008. [29] J.F. Mandell, Fatigue behavior of short fiber composite materials, in: K.L. Reifsnider (Ed.), Fatigue of Composite Materials, Elsevier Science Publishers, Amsterdam, 1990. [30] C.O. Albrecht, Statistical evaluation of a limited number of fatigue test specimens, in: Fatigue Test of Aircraft Structures, American Society for Testing and Materials, 1962. ASTM STP 338. [31] D.F. Sims, V.H. Brogdon, Fatigue behavior of composites under different loading modes, in: K.L. Reifsnider, K.N. Lauraitis (Eds.), Fatigue of Filamentary Composite Materials, American Society of Testing and Materials, 1977. ASTM STP 636. [32] K.L. Reifsnider, M.-H. Jen, Composite Flywheel Durability and Life. Part II: Long-Term Materials Data, Lawrence Livermore National Laboratory, 1982. [33] P. Johannesson, T. Svensson, J. Demare, Fatigue life prediction based on variable amplitude tests—methodology, Int. J. Fatigue 27 (2005) 954–965. [34] J.A. Epaarachchi, P.D. Clausen, An empirical model for fatigue behavior prediction of glass fibre-reinforced plastic composites for various stress ratios and test frequencies, Compos. Part A 34 (2003) 313–326. [35] A. Fatemi, L. Yang, Cumulative fatigue damage and life prediction theories: a survey of the state of the art for homogeneous materials, Int. J. Fatigue 20 (1998) 9–34. [36] A.P. Vassilopoulos, E.F. Georgopoulos, V. Dionysopoulos, Artificial neural networks in spectrum fatigue life prediction of composite materials, Int. J. Fatigue 29 (2007) 20–29. [37] W.X. Yao, N. Himmel, A new cumulative fatigue damage model for fibre-reinforced plastics, Compos. Sci. Technol. 60 (2000) 59–64. [38] N.L. Post, J.J. Cain, J.J. Lesko, S.W. Case, Design knockdown factors for composites subjected to spectrum loading based on a residual strength model, in: ICCM-17: 17th International Conference on Composite Materials, Edinburgh, 2009. [39] R. Speckart, N.L. Post (Ed.), Engineering Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA, 2008.

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Creep/fatigue/relaxation of angle-ply GFRP composite laminates

4

Anastasios P. Vassilopoulos Ecole Polytechnique F
ed
erale de Lausanne (EPFL), Composite Construction Laboratory (CCLab), Lausanne, Switzerland

4.1

Introduction

Damage accumulates in the volume of composite materials during fatigue loading and eventually leads them to failure. Several damage mechanisms, including fiber breakage and matrix cracking, debonding, transverse-ply cracking, and delamination, are activated either independently or synergistically during fatigue loading; the predominance of one or the other is strongly affected by both material variables and the sequence and duration of the damage events [1, 2]. The presence of these damage mechanisms results in the degradation of the materials properties under service loading conditions. The type of the individual damage mechanisms and the sequence of their occurrence are usually dependent on, among others, the loading pattern, the material type, the laminate architecture, and the environmental conditions [2–7]. Despite the complexity and multitude of the damage mechanisms, a common failure process for unidirectional, cross-ply, and angle-ply laminates has been identified [3–5, 8–10]. This resembles a three-stage damage progression process, starting with the damage formation in the matrix with multiple crack development, followed by the evolution of matrix cracks reaching the vicinity of the fibers and matrix/fiber debonding and delaminations, and ending with fiber breakage when damage accumulated during the previous stages becomes saturated [8–10]. In addition to fatigue loading, time-dependent phenomena, such as creep and relaxation, play an important role in the durability of composite structures [11–15]. The susceptibility to creep in laminated composites, even at room temperature, originates from the viscoelastic nature of their polymeric matrices. It has been shown that laminated composites cyclically creep under fatigue mean stresses [11, 14], while they are significantly recovering when the fatigue loading is interrupted [13]. Structural changes in polymer-based composite materials, depending on loading conditions and material type, can be due to damage growth and cyclic creep [11–15]. Today, several main composite components of engineering structures are manufactured from materials whose performance may be matrix dominated. Typical examples are shear webs and the aerodynamic parts of wind turbine rotor blades that consist mainly of biaxial composite laminates. These structural components may Fatigue Life Prediction of Composites and Composite Structures. https://doi.org/10.1016/B978-0-08-102575-8.00004-8 © 2020 Elsevier Ltd. All rights reserved.

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Fatigue Life Prediction of Composites and Composite Structures

undergo severe stiffness degradation early in their lifetime due to fatigue loading, resulting in a decrease in their ability to bear extreme loads, which may lead to failure due to elastic instability. When angle-ply laminates are considered, multiaxial stress states develop in the principal material system, even under uniaxial loading [16]. The extent of the effect of each stress component on the development of the damage mechanisms mainly depends on the laminate lay-up [16]. The evolution of structural changes affects the mechanical behavior [3, 17–21], thermal behavior [22–27], and optical properties [28–31] of polymer matrix composites. In terms of mechanical behavior, the evolution of structural changes with the number of cycles decreases the fatigue stiffness and residual strength due to the damage growth, and increases the average fatigue strain owing to cyclic creep [11, 17, 18]. The fatigue damage also increases the self-generated temperature [22, 23] and changes the heat capacity and thermal conductivity of the material [24]. The distribution and magnitude of the self-generated temperature are related to the distribution of fatigue damage in the material [25], and, therefore, the variation and distribution of self-generated heating during the cyclic loading can be used to characterize the structural performance of the examined composite materials [26]. Moreover, it has been shown that the formation of damage during cyclic loading changes the optical properties [28–31] of polymer matrix composites. Reflectance, for example, has been successfully employed for the identification of damage accumulation in Ref. [29]. Cyclic dissipated energy is another parameter that is affected by structural changes during the fatigue process. Measurement of the cyclic stress and strain allows the derivation of the stress-strain hysteresis loops [11, 12, 32, 33]. For materials with a purely elastic behavior, the elastic energy per loading cycle is equal to the total energy and no dissipation is measured. However, for the majority of materials, energy is dissipated at each load cycle. The hysteresis area is a measure of the total dissipated energy per cycle; the slope of each stress-strain hysteresis loop corresponds to the fatigue stiffness as it is explicitly described in Refs. [12, 22, 33]. During a load-controlled fatigue experiment, the hysteresis loops can shift, indicating the presence of creep, and the evolution of the average strain per cycle can be monitored to describe creep behavior [11, 22, 33–36]. The sources of energy dissipation in laminated composites are damage formation and growth, the viscoelasticity of the material and the self-generated hysteretic heating [37]. The presence of damage causes the dissipation of energy due to the following mechanisms: fiber bridging, broken fibers, crack tip plasticity, and friction of unbounded regions in the matrix and fiber/matric interface [37]. As a result of the viscoelastic nature of the polymeric matrix, the secondary bonds create more frictional forces between the polymeric chains, which results in more energy dissipation [38]. In addition, due to internal friction, the temperature of the specimen increases and causes thermal energy dissipation [39]. Internal friction is a process in which heat is generated as a result of the resisting frictional force between the two sides of cracks in a solid material while it undergoes deformation [40]. Engineering structures operating in open-air applications, such as wind turbine rotor blades, airplanes, and bridge decks, are subjected to different irregular loading profiles, in most cases including interrupted loading at high or low stress levels [17].

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In contrast to the existing works regarding the effects of the creep-fatigue interaction on the fatigue life of composites, the effects of load interruptions (recovery and relaxation mechanisms) have hardly been investigated, see, for example, Refs. [13, 14]. As reported in Ref. [17], interruption of fatigue loading of multidirectional [0/(
45)2/0]T GFRP laminates at the stress ratio, (R ¼ σ min/σ max), of 0.1 and at one particular (relatively high) stress level resulted in a considerable extension of lifetime when the fatigue load was interrupted for specific times. Improvement of the measured fatigue life was also reported for engineering polymers, such as epoxy and polyester, when the fatigue experiments were interrupted periodically as described in Refs. [27, 41], again without sufficient information about the experimental conditions or any analysis regarding the reasons for this life extension. In contrast, data points of interrupted fatigue (for measuring the crack density) at one stress level showed slightly lower life, although in the scatter of the S-N curve derived by continuous fatigue loading as presented in Ref. [42]. However, in Ref. [42] and other studies, e.g., [43, 44] fatigue loading was interrupted for measurement purposes having no objective to investigate the interruption effect on the fatigue life. Nevertheless, the fatigue life was assumed unaffected by the interruptions, an assumption that seemingly needs to be reconsidered, depending on the examined materials and the loading conditions. If structures are composed of fiber-reinforced polymer (FRP) composites, the sensitivity of these materials to such types of loading patterns has to be taken into account due to their cyclic- and time-dependent mechanical properties [11, 45–47]. As mentioned above, the degradation of the cyclic-dependent mechanical properties of laminated composites occurs via several damage mechanisms, such as matrix cracking, debonding, delamination, transverse-ply cracking, and fiber failure, that are activated, either independently or synergistically, during fatigue loading. The predominance of one or another of the aforementioned damage mechanisms is dependent on both material variables and loading conditions as described in Refs. [1, 2]. In addition, due to the presence of the viscoelastic polymer matrix in a variety of laminated composites, the time-dependent mechanical properties of the material are also affected during fatigue loading [12, 13, 15, 48]. The time-dependent material behavior can be studied by observing the creep, recovery, and stress relaxation behaviors of the material [13–15, 48]. Cyclic-dependent and time-dependent phenomena can interact with each other during the fatigue loading, and the degree of their interactions and the dominance of one over the other depend on the loading spectrum and material type [13, 15, 45–47]. Creep and fatigue were recognized as critical mechanisms for the durability of composites since very early [49], however, at that time very little or nothing was published about either mechanism. Since then, significant amount of information has been acquired regarding the pure creep [50, 51], the pure fatigue [50–52], as well as (although not always being the objective of the relevant studies) the combined creep-fatigue behavior [11, 14, 53, 54] of polymer matrix composite materials. Laminated composites are susceptible to creep, even at room temperature, due to the viscoelastic nature of their polymeric matrix. Apart from the typical creep deformation, it has been well documented that the laminated composites cyclically creep under fatigue mean stresses [11–15, 22].

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Investigation of the creep-fatigue behavior of composite materials and structures can be performed either by segregating the creep and fatigue components of loading and adding their effects (in a linear on nonlinear way) or by assuming both effects simultaneously by subjecting the component to a loading profile—usually a trapezoidal waveform—with a measurable time under load [50, 51, 53–55]. The second sounds like a more realistic scenario for the description of the “creep-fatigue loading,” while “pure creep” or “pure fatigue” loadings are considered as special cases of it [52, 53]. Pure creep is considered to be a time-dependent phenomenon that can be investigated by loading the material under static fatigue conditions, i.e., under stress ratio, R ¼ σ min/σ max, equal to 1. Pure fatigue loading is considered as the cyclic loading under tensile stresses with zero minimum level (R ¼ 0) [11, 45], although this profile possesses a certain amount of constant (mean) cyclic stress during loading. Alternatively, reversed loading (R ¼  1) was assumed to be the “pure” fatigue loading, since the mean cyclic stress is zero for this case [56]. Nevertheless, in many applications, a structural component is subjected to loading profiles that are neither pure creep, nor pure fatigue; rather they have elements of both forms of loading [11, 14, 22, 47, 57]. There is little information in the literature regarding the effect of creep on the fatigue life of composites because of the complexity of such effects and the difficulty to clearly separate the two phenomena. Several works exist regarding the frequency effects on the fatigue life of composite materials, see, e.g., Ref. [58–62], investigating however indirectly the effect of creep. It is documented that any frequency increase improves the fatigue life of several material systems, since the time under load (cyclic creep) decreases [63–65], as long as the increased frequency does not result in loading rates, high enough to produce considerable temperature rises that soften the matrix and deteriorate the fatigue life [14, 61]. Nevertheless, fewer works investigate the creepfatigue interaction in composite materials, e.g., [2, 14, 50, 63, 66], all agreeing that creep and fatigue are mutually influencing phenomena [15]. In some studies, the effect of creep on fatigue life appears to be beneficial [2, 15, 63], while in others it is shown that creep can act synergistically with fatigue mechanisms to accelerate damage accumulation [50, 66]. However, it is commonly accepted that in a great extent, creep plays significant role for the high-cycle fatigue regimes, especially at low frequencies, when the time under load is increased [55, 63]. In such cases, creep is dominant, and time dependent failure occurs, while the failure is mainly cycle dominated at the high-cycle fatigue regimes [14, 18, 22, 66]. Although creep and fatigue coincide in, eventually, any fatigue-loading profile, the physics of damage are different for cyclic creep and fatigue, since both phenomena involve different types of damaging mechanisms and processes [57, 66, 67]. Very limited experimental evidence exists on this topic, since usually creep and fatigue effect on the damage development are not decomposed and their common effect on the fatigue life is, evidently, investigated. Nevertheless, it is speculated that cyclic loading causes mainly stiffness degradation, and does not affect the viscoelastic response of glass fiber-reinforced polymers (GFRPs), which is a creep-related mechanism [68]. However, the interaction between these two mechanisms during loading (and damage accumulation) cannot be overseen. This chapter summarizes the research results on the study of load sequence effects on the fatigue life of angle ply composite laminates under tension-tension continuous

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and interrupted fatigue loading. The objective is to identify the different damage patterns caused by the different loading profiles and demonstrate the synergistic effects of fatigue loading, creep cyclic, and interrupted loading segments (combination of fatigue-loading profiles that can be seen in operation of actual structures) on the material’s performance.

4.2

Experimental procedure

4.2.1 Material and specimens An experimental database derived at the Composite Construction Laboratory (CCLab) of the Swiss Federal Institute of Technology (Ecole Polytechnique F
ed
erale de Lausanne—EPFL) during the last years will be used for the demonstration of the load sequence effects on the fatigue life and damage development of angle-ply composite laminates. The material of the study is a unidirectional E-glass fiber (EC 9-68) with the area density of 425 g/m2 and layer thickness of ca. 0.45 mm. This fabric comprises a finish bonding agent, which provides better adhesion to the matrix. The low-viscosity resin, Biresin CR83, mixed with the hardener Sika CH83-2, in a ratio of 3:1 was used for impregnation of the fabrics. Biresin CR83 is an epoxy resin system, provided by SIKA, with extremely low viscosity designed specifically for the infusion process for the production of high-performance fiber reinforced composite parts and molds The fabrication of the laminates was performed on a plastic substrate, which was coated with a release agent to prevent resin from bonding to the surface after fabrication. Laminates of dimensions 500  500  2.3 mm3 and stacking sequence of [
45]2s were fabricated by vacuum assisted hand lay-up, using a vacuum pump with a pressure of 0.9 bar. Each laminate was kept in a vacuum bag for 24 h under laboratory conditions (22
2°C, 40%
10% RH), and subsequently placed in an oven at 70°C for 8 h to complete the curing process. The achieved fiber content was 62% by volume as determined by performing burn-off experiments as described in ASTM D 3171-99 [69]. Rectangular glass/epoxy [
45]2s composite specimens with the average dimensions of 250  25  2.3 mm3 (length  width  thickness) were cut from the laminates using a water-jet cutting machine, and prepared, according to ASTM D3039 [70] as shown in Fig. 4.1. Two aluminum tabs with dimensions of 45  25  4 mm3 were glued to each specimen end with a cyanoacrylate glue for gripping purposes (Fig. 4.1).

Fig. 4.1 Geometry of fatigue specimen.

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4.2.2 Experimental set-up and instrumentation Different experiments were performed on the same specimen type in order to characterize the material performance under different loading profiles, as well as to examine its temperature-related performance via dynamic mechanical analysis (DMA). Quasistatic, pure creep, constant amplitude fatigue, interrupted fatigue, and creepfatigue-loading patterns were employed for this investigation. All experiments, except the creep loading, were performed on an Instron 8800 hydraulic universal testing rig of 100-kN capacity with an accuracy of
0.01 kN. The experimental set-up is shown in Fig. 4.2.

4.2.2.1 Dynamic mechanical analysis DMA was performed to verify that the self-generated temperatures did not exceed the Tg,onset of the material [71]. A TA Instruments Q800 dynamic mechanical analyzer in single cantilever configuration was used to perform the experiment. A specimen of 35.0  10.0  2.3 mm3 (length  width  thickness) was cut with the same glass fiber layout as in the fatigue experiments. Three experiments were carried out on three different specimens in the temperature range of 15–160°C with a heating rate of 5°C/min in an air atmosphere, and the same maximum loading rate at the top and bottom specimen edges as in the fatigue experiments.

Fig. 4.2 Experimental set-up.

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4.2.2.2 Quasistatic experiments Quasistatic experiments under standardized rates [70] and at rates similar to those achieved under the fatigue loading were performed in order to obtain the strength of the examined material and investigate the rate effects. Standardized quasistatic tensile experiments were carried out under displacement control at a rate of 1 mm/min, while additional experiments were realized under load control at the same rate of 30.5 kN/s as the rate applied to the fatigue loading. The tensile properties of the material were determined based on the measurements of three specimens per loading rate.

4.2.2.3 Pure creep experiment The creep experiments were performed by using in-house designed and fabricated creep frames, schematically shown in Fig. 4.3. The creep load was applied by hanging different weights from the lever arm (point A in Fig. 4.3A), in order to derive creep stresses between 49 and 70 MPa, the same range as the maximum cyclic stresses developed during the fatigue experiments (see below). The measurement of the longitudinal creep displacement was performed by two vertical displacement transducers (LVDTs) with a measurement accuracy of 0.02 mm, attached on both sides of the specimen as shown in Fig. 4.3B.

Fig. 4.3 Creep experimental set-up.

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4.2.2.4 Fatigue experiments Constant amplitude fatigue experiments were performed by applying load controlled sinusoidal-loading patterns of different amplitude and mean values in order to cover a wide range of fatigue lives from ca. 200 to around 106 fatigue cycles. A minimum of four specimens were examined at each stress level. The stress ratio, R ¼ σ min/σ max, was kept constant at 0.1 in order to apply tensile cyclic loads to the specimens. All fatigue experiments were performed in a chamber at a constant temperature of 20°C and two fans were used to circulate the air inside the chamber and cool the specimen. The same _ of 30.5 kN/s was used for all tests. This loading rate was derived after loading rate, F, preliminary experiments to avoid excessive temperature increases during the fatigue loading. Accordingly, different frequencies were selected to keep the loading rate constant for all stress levels. The frequency, f, of each load level with cyclic load amplitude, Fa, was calculated by the following equation: F_ ¼ 4Fa f

(4.1)

Interrupted fatigue and creep-fatigue-loading profiles were also applied to the same specimen configuration to investigate the load sequence effects on the fatigue life and damage development of the examined angle-ply composite laminates. The applied loading patterns are schematically presented in Fig. 4.4.

Fig. 4.4 Schematic representation of loading pattern in interrupted and creep-fatigue experiments.

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The interrupted-loading profile (Fig. 4.4A) comprises repetitive loading blocks of constant amplitude, interrupted in regular intervals by “zero-load” segments up to the specimen’s failure. For each experiment, the load was increased until the mean value was reached, after 60 s. Subsequently, the constant amplitude fatigue loading was applied for a predetermined number of cycles (20% of the average continuous fatigue life estimated for the same stress levels in continuous fatigue loading), followed by a zero-load segment lasting for 2 h. The duration of 20% of the fatigue life in each loading block was selected in order to obtain a significant number of interruptions in each fatigue experiment and a reasonable duration of the fatigue experiments. The transition from the mean fatigue loading to the zero-load intervals was achieved in 1 s. The cyclic loading was performed in the range of maximum stress levels of 47–68 MPa based on ASTM D7791-17 [72], similar to the stress levels selected for the continuous fatigue experiments. For the creep-fatigue-loading profile (Fig. 4.4B), the load was increased until the mean value was reached, after 60 s. Subsequently, the constant amplitude fatigue loading was applied for a predetermined number of cycles (20% of the average continuous fatigue life estimated for the same stress levels in continuous fatigue loading), followed by constant load at σ max of interval lasting for 2 and 48 h. The transition from the mean fatigue loading to creep part at σ max was achieved in 1 s. The cyclic loading was performed in the range of σ max was from 47 MPa up to 70 MPa based on ASTM D7791-17 [72], similar to the stress levels used in the continuous fatigue-loading experiments as well as those selected for the pure creep experiments.

4.2.2.5 Instrumentation In order to monitor the mechanical, thermal, and optical changes in the specimen during and after the fatigue process, different instrumentations were used. During the fatigue experiments, the cross heads displacement, load, and number of cycles were recorded at a frequency depending on the stress levels, i.e., from once per 10 cycles at high stress levels to once per 1000 cycles at low stress levels. The variation of the longitudinal strain was measured by a high-resolution video-extensometer (a Point Gray—Grasshopper3 camera with a resolution of 1936  1216 Mpixels and a Fujinon HF35SA-1 35 mm F/1.4 lens) with a frequency of acquisition of 160 fps. A videoextensometer measured the changes of position of two lines marked on the specimens’ surfaces (see Fig. 4.1) and the corresponding strain values were calculated. An LED white light with negligible heat emission was projected onto the sample surface to enhance measurement accuracy. Approximately 25 load and displacement measurements were recorded per cycle to estimate the strain fluctuations, fatigue stiffness, and hysteresis loops throughout the fatigue life of all the examined specimens. An infrared (IR) thermal camera with an accuracy of 0.1°C, and optical resolution of 160  120 pixels was also employed during the fatigue experiments to record the evolution of the specimen’s surface temperature during the fatigue experiments. In order to detect the damage development at a macroscale level, photographs of the translucent specimens were taken at regular intervals (depending on the life expectancy) with a digital camera with maximum aperture f/2.8 and focal length range of 24–70 mm zoom range

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during loading. Damage in the material volume was attributed to changes in the reflectance, which indicated new interfaces in the path of the transmitted light [12]. A bright white light source was positioned behind the specimens to assist this procedure. After failure, the failure surfaces were observed using a digital handheld Dino-Lite microscope, AD7013MZT, with the magnification of 20  and resolution of 2592  1944 pixels, and photos were taken for all examined specimens.

4.3

Experimental results and discussion

4.3.1 Dynamic mechanical analysis The variation of the storage modulus versus temperature is presented in Fig. 4.5. It can be seen that the storage modulus in the glassy region up to 40°C was almost constant and then started rapidly decreasing. The glass transition region for the examined material is around 70–100°C with the Tg,onset defined at around 80°C.

4.3.2 Quasistatic behavior Typical stress-strain curves of the examined laminates under the two different loading conditions are shown in Fig. 4.6. The specimens exhibited a rate-dependent,

Fig. 4.5 Evolution of storage modulus as function of temperature.

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Fig. 4.6 Quasistatic stress-strain curves and typical hysteresis loops at different stress levels.

Table 4.1 Quasistatic tensile properties under different loading conditions. Type of tensile experiment Displacement control (1 mm/min) Load control (30.5 kN/s)

YS (MPa)

UTS (MPa)

Strain at failure (%)

46
1.9

97
6.7

5.8
0.7

78
1.4

122
1.4

5.0
1.2

nonlinear, stress-strain response. When the loading rate was increased, the value of yield stress (YS) and ultimate tensile stress (UTS) increased, although the strain to failure slightly decreased due to the viscoelastic behavior of the polymeric matrix. The tensile properties of the material per loading rate are presented in Table 4.1.

4.3.3 Pure creep behavior The creep curves showing the evolution of the creep strain, ε, as a function of time is shown in Fig. 4.7 at 20°C. At all stress levels, typical creep curves were obtained under

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Fatigue Life Prediction of Composites and Composite Structures

8

Creep-70 Creep-68 Creep-64 Creep-58

6

Creep-53 Creep-49 e (%)

4

2

0

0

400

800

1200

15,500

16,200

Time (h)

Fig. 4.7 Creep curves at different stress levels. Table 4.2 Overview of creep experiments and results. No

Code

σ max (MPa)

tf (h)

1 2 3 4 5 7

Creep-70-a Creep-68-a Creep-64-a Creep-58-a Creep-53-a Creep-49-a

70.0 68.0 64.2 58.2 58.2 48.6

213.5 322.6 507.4 1007.9 15,768.0 17,500.0 (on going)

various constant stress levels; after an elastic strain, there was a primary creep stage in which the strain increased fast with a deceasing trend. Then, the creep curves arrived to the second stage as strain increased steady stately and the strain rate became minimum. At the end, the strain increased in the third creep stage, which ended up with specimen failure. Corresponding creep life of specimens is tabulated in Table 4.2. By considering all creep curves at different stress levels, it is seen that the elastic strain and the minimum creep rate increased with increasing stress level; however, the failure strain did not change significantly. The evolution of the creep strain at σ max ¼ 70 MPa and corresponding specimen translucency are shown in Fig. 4.8. No damage was visually observed in the specimen

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111

Fig. 4.8 Creep curve at σ max ¼ 70 MPa with corresponding representation of light transmittance for different percentages of creep life; (Creep-70).

during the primary creep; however, when at the steady state region, damage gradually appeared at different locations along the fibers and propagated uniformly along the specimen volume. Creep failure occurred at the third stage after concentrated damage development.

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4.3.4 Fatigue behavior 4.3.4.1 Fatigue life The experimental results from all applied fatigue-loading patterns are presented in Tables 4.3–4.5 for continuous cyclic loading, experiments interrupted at zero load and experiments interrupted at the maximum cyclic load, respectively. The specimen denomination is in the following form: the first term is Conf, Intf, or Cref, denoting either continuous fatigue, interrupted at zero load (Intf ), or interrupted Table 4.3 Continuous fatigue experimental results. No.

Code

σ max (MPa)

Frequency (Hz)

Nf

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

Conf-0.1-70-a Conf-0.1-70-b Conf-0.1-70-c Conf-0.1-70-d Conf-0.1-68-a Conf-0.1-68-b Conf-0.1-68-c Conf-0.1-68-d Conf-0.1-64-a Conf-0.1-64-b Conf-0.1-64-c Conf-0.1-64-d Conf-0.1-58-a Conf-0.1-58-b Conf-0.1-58-c Conf-0.1-58-d Conf-0.1-53-a Conf-0.1-53-b Conf-0.1-53-c Conf-0.1-53-d Conf-0.1-53-e Conf-0.1-53-f Conf-0.1-53-g Conf-0.1-53-h Conf-0.1-53-i Conf-0.1-53-j Conf-0.1-49-a Conf-0.1-49-b Conf-0.1-49-c Conf-0.1-49-d Conf-0.1-47-a Conf-0.1-47-b Conf-0.1-47-c Conf-0.1-47-d

70.1 70.1 70.1 70.1 68.0 68.0 68.0 68.0 64.2 64.2 64.2 64.2 58.2 58.2 58.2 58.2 53.4 53.4 53.4 53.4 53.4 53.4 53.4 53.4 53.4 53.4 48.6 48.6 48.6 48.6 47.4 47.4 47.4 47.4

3.95 3.95 3.95 3.95 4.35 4.35 4.35 4.35 4.68 4.68 4.68 4.68 5.06 5.06 5.06 5.06 5.53 5.53 5.53 5.53 5.53 5.53 5.53 5.53 5.53 5.53 6.10 6.10 6.10 6.10 6.33 6.33 6.33 6.33

1964 1103 468 1353 1969 1007 591 3689 2593 2304 1161 4112 6785 22,321 18,028 15,063 49,151 29,832 22,583 79,623 146,112 35,673 84,699 101,069 29,666 61,855 289,050 194,115 460,664 277,227 533,634 1,198,627 430,211 545,689

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Table 4.4 Interrupted fatigue experiments. No.

Code

σ max (MPa)

Frequency (Hz)

NBlock

Nf

nb

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Intf-0.1-68-a Intf-0.1-68-b Intf-0.1-68-c Intf-0.1-68-c Intf-0.1-64-a Intf-0.1-64-b Intf-0.1-64-c Intf-0.1-64-d Intf-0.1-58-a Intf-0.1-58-b Intf-0.1-58-c Intf-0.1-58-d Intf-0.1-53-a Intf-0.1-53-b Intf-0.1-53-c Intf-0.1-53-d Intf-0.1-53-e Intf-0.1-53-f Intf-0.1-53-g Intf-0.1-53-h Intf-0.1-53-i Intf-0.1-53-j Intf-0.1-49-a Intf-0.1-49-b Intf-0.1-49-c Intf-0.1-49-d Intf-0.1-47-a Intf-0.1-47-b Intf-0.1-47-c Intf-0.1-47-d

68.0 68.0 68.0 68.0 64.2 64.2 64.2 64.2 58.2 58.2 58.2 58.2 53.4 53.4 53.4 53.4 53.4 53.4 53.4 53.4 53.4 53.4 48.6 48.6 48.6 48.6 47.4 47.4 47.4 47.4

4.35 4.35 4.35 4.35 4.68 4.68 4.68 4.68 5.06 5.06 5.06 5.06 5.53 5.53 5.53 5.53 5.53 5.53 5.53 5.53 5.53 5.53 6.10 6.10 6.10 6.10 6.33 6.33 6.33 6.33

365 365 365 365 500 500 500 500 3100 3100 3100 3100 9000 9000 9000 9000 9000 9000 9000 9000 9000 9000 64,000 64,000 64,000 64,000 146,000 146,000 146,000 146,000

1205 4668 3349 7085 5961 3943 6225 7542 22,854 68,215 46,055 23,319 108,580 75,054 209,355 57,463 131,124 150,026 85,250 30,706 52,349 91,984 603,211 475,331 472,542 229,854 1,314,482 1,360,621 521,024 469,501

3 12 9 19 11 7 12 15 7 22 14 7 12 8 23 6 14 16 9 3 5 10 9 7 7 3 8 3 8 2

fatigue loading by holding the load at the maximum cyclic stress for certain time periods (Cref ). The second term indicates the stress ratio, the third term the maximum fatigue stress, and the last term the identification specimen number for each stress level. As mentioned, above, the cyclic loading was performed in the range of maximum stress levels of 47–68 MPa. The material behavior under the maximum cyclic stress level was still linear, as the quasi-static YS derived for material under load control at a rate of 30.5 kN/s has been estimated to 78
1.4 MPa, while the UTS to 122
1.4, see Fig. 4.6. The derived fatigue data exhibited the typical response for this kind of material; the S-N curves are shown in Fig. 4.9, where the maximum cyclic stress level is plotted against the number of fatigue cycles to failure, Nf. For the interrupted fatigue results

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Table 4.5 Creep-fatigue experiments. No Code

σ max (MPa)

Frequency (Hz)

NBlock

tcreep (h)

Nf

nb Failure

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

70.0 70.0 70.0 70.0 70.0 70.0 68.0 68.0 68.0 68.0 68.0 68.0 64.2 64.2 64.2 64.2 64.2 64.2 58.2 58.2 58.2 58.2 58.2 58.2 53.4 53.4 53.4 53.4 53.4 53.4 48.6 48.6 48.6 48.6 48.6 48.6 47.4 47.4 47.4 47.4 47.4 47.4

3.95 3.95 3.95 3.95 3.95 3.95 4.35 4.35 4.35 4.35 4.35 4.35 4.68 4.68 4.68 4.68 4.68 4.68 5.06 5.06 5.06 5.06 5.06 5.06 5.53 5.53 5.53 5.53 5.53 5.53 6.10 6.10 6.10 6.10 6.10 6.10 6.33 6.33 6.33 6.33 6.33 6.33

230 230 230 230 230 230 365 365 365 365 365 365 500 500 500 500 500 500 3100 3100 3100 3100 3100 3100 9000 9000 9000 9000 9000 9000 64,000 64,000 64,000 64,000 64,000 64,000 146,000 146,000 146,000 146,000 146,000 146,000

2 2 2 2 48 48 2 2 2 2 48 48 2 2 2 2 48 48 2 2 2 2 48 48 2 2 2 2 48 48 2 2 2 2 48 48 2 2 2 2 48 48

1840 1150 1150 1150 230 230 6126 2655 2732 1460 1825 730 2500 4220 5731 4000 2000 1000 14,527 43,023 25,466 15,959 10,589 24,385 42,815 87,654 63,288 43,277 74,391 30,704 160,255 917,136 364,394 145,809 153,769 423,859 353,689 319,752 1,095,359 350,072 1,356,496 362,532

8 5 5 5 1 1 16 7 7 4 5 2 5 8 11 8 4 2 4 13 8 5 3 7 4 9 7 4 8 3 2 14 5 2 2 6 2 2 7 2 9 2

Cref-0.1-70-2-a Cref-0.1-70-2-b Cref-0.1-70-2-c Cref-0.1-70-2-d Cref-0.1-70-48-a Cref-0.1-70-48-b Cref-0.1-68-2-a Cref-0.1-68-2-b Cref-0.1-68-2-c Cref-0.1-68-2-d Cref-0.1-68-48-a Cref-0.1-68-48-b Cref-0.1-64-2-a Cref-0.1-64-2-b Cref-0.1-64-2-c Cref-0.1-64-2-d Cref-0.1-64-48-a Cref-0.1-64-48-b Cref-0.1-58-2-a Cref-0.1-58-2-b Cref-0.1-58-2-c Cref-0.1-58-2-d Cref-0.1-58-48-a Cref-0.1-58-48-b Cref-0.1-53-2-a Cref-0.1-53-2-b Cref-0.1-53-2-c Cref-0.1-53-2-d Cref-0.1-53-48-a Cref-0.1-53-48-b Cref-0.1-49-2-a Cref-0.1-49-2-b Cref-0.1-49-2-c Cref-0.1-49-2-d Cref-0.1-49-48-a Cref-0.1-49-48-b Cref-0.1-47-2-a Cref-0.1-47-2-b Cref-0.1-47-2-c Cref-0.1-47-2-d Cref-0.1-47-48-a Cref-0.1-47-48-b

Creep Creep Creep Creep Creep Creep Fatigue Fatigue Fatigue Creep Creep Creep Creep Fatigue Fatigue Creep Creep Creep Fatigue Fatigue Fatigue Fatigue Fatigue Fatigue Fatigue Fatigue Fatigue Fatigue Fatigue Fatigue Fatigue Fatigue Fatigue Fatigue Fatigue Fatigue Fatigue Fatigue Fatigue Fatigue Fatigue Fatigue

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Fig. 4.9 Experimental fatigue data and S-N curves.

Table 4.6 S-N model parameters. Loading pattern

σ0

1/k

Conf Intf Cref (2 h) Cref (48 h)

106.01 115.99 115.17 95.98

0.0620 0.0671 0.0687 0.0534

(Tables 4.4 and 4.5) NBlock denotes the number of fatigue cycles in each loading block, tcreep the applied hold time, Nf the cycles to failure, while nb the completed fatigueloading blocks. In interrupted fatigue experiments, the number of cycles to failure was determined by excluding the interrupted part of the loading pattern and summing-up the number of cycles in the fatigue-loading segments until failure. All specimens loaded at the highest stress level (σ max ¼ 70 MPa), failed during the creep segment of the loading profile. This was changing as the load and the hold time were decreased; at σ max of 68 and 64 MPa with 2 h of hold time, failure during the fatigue and the creeploading phase was observed. However, the period of 48 h was proved sufficiently long to lead the specimens to failure during the creep segment. For lower loads, corresponding to cyclic stress levels lower than 58 MPa, all failures occurred during the constant amplitude fatigue loading, irrespective of the hold time.

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Different models can be used for the simulation of the fatigue life behavior under the different loading profiles as explained explicitly in Refs. [73, 74]. Nevertheless, a classical power law relationship can be used for the simulation of the fatigue behavior: 1 σ max ¼ σ 0 Nf k

(4.2)

in which σ 0, 1/k are model parameters derived by linear regression analysis, after fitting Eq. (4.2) to the available experimental fatigue data. The derived fatigue model parameters for all three S-N curves are given in Table 4.6. A clear life improvement can be seen for those specimens where the fatigue loading was interrupted by zero load segments. The effect was more pronounced at high stress levels, with lifetime increases reaching 126% at the level of 68 MPa on average, but becoming less at the high cycle regime, showing a 34% average increase at the level of 47 MPa. The effect of interruption almost disappeared for stress levels lower than 40 MPa, where all derived curves lay inside the confidence interval bounds of the curve for the continuous fatigue shown with long-dashed lines. This result may be slightly affected by the fact that at lower stress levels, significantly fewer fatigue blocks were completed, i.e., fewer interruptions occurred on average than at higher stress levels, see Table 4.4. The situation is a bit more complicated for the creep-fatigue-loading though. The curve corresponding to the 48-h creep load intervals was less steep than the others due to the creep failure observed in high stress levels when long creep intervals were introduced to the loading profiles. Nevertheless, for shorter creep intervals, the fatigue life was improved compared to the continuous fatigue loading, especially at the high stress levels where the two creep-fatigue curves are obviously outside the confidence bounds of the continuous fatigue-loading S-N curve. The effect of hold time was less obvious at low stress levels where all loading profiles, continuous and creep-fatigue loading profiles, resulted to similar lifetime—always considering the experimental scatter.

4.3.4.2 Hysteresis loops—stiffness degradation—energy dissipation As mentioned in the introduction, the hysteresis area is a measure of the total dissipated energy per cycle; the slope of each stress-strain hysteresis loop corresponds to the fatigue stiffness. Measurement of the cyclic stress and strain allows the derivation of the stress-strain hysteresis loops [11, 12, 32, 33]. For materials with a purely elastic behavior, the elastic energy per loading cycle is equal to the total energy and no dissipation energy is measured. However, for the majority of materials, energy is dissipated at each load cycle. The hysteresis area is a measure of the total dissipated energy per cycle; the slope of each stress-strain hysteresis loop corresponds to the fatigue stiffness as it is explicitly described in Refs. [22, 33]. During a load-controlled fatigue experiment, the hysteresis loops can shift, indicating the presence of creep, and the evolution of the average strain per cycle can be monitored to describe creep behavior [11, 22, 33]. During a load-controlled fatigue experiment, the hysteresis loops can

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shift, indicating the presence of creep, and the evolution of the average strain per cycle can be monitored to describe the creep behavior. Typical definitions regarding the hysteresis loops and relevant measurements are summarized in Fig. 4.10. The applied fatigue-loading patterns induced cyclic stresses in the linear region of the quasistatic stress-strain curves derived by applying the same loading rate, as shown in Fig. 4.6. Typical hysteresis loops resulted after the application of the continuous fatigue loading up to failure are presented in Fig. 4.11 for high (68 MPa) and low (47 MPa) nominal maximum cyclic stresses. The hysteresis area (corresponding

smax

Cyclic stress

Hysteresis area

Shifting E1

EN eav

smin

Δecyclic Cyclic strain

Fig. 4.10 Schematic representation of hysteresis loops and definitions [12].

Fig. 4.11 Variation of hysteresis loops under continuous cyclic loading at (A) σ max ¼ 68 MPa (Conf-0.1-68-a) and at (B) σ max ¼ 47 MPa (CF-0.1-47-c) [12].

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Fatigue Life Prediction of Composites and Composite Structures

80

80

60

60

s (MPa)

s (MPa)

to the dissipated energy per cycle) as well as the fatigue stiffness (as explained above, and demonstrated in Fig. 4.10, the slope of each hysteresis loop) and the cyclic creep exhibited measureable changes with fatigue cycles for both high and low cyclic stresses. The behavior of all other specimens, at any stress level, was similar. The evolution of the hysteresis loops during the fatigue life for the specimens loaded under the interrupted by zero load segments-loading spectra, is shown in Fig. 4.12, while the corresponding graphs for the creep-fatigue loading are presented in Fig. 4.13. The effect of the load sequence is obvious when comparing the results shown in Figs. 4.11–4.13. Creep-fatigue-loading profiles (Fig. 4.13) cause higher creep strains, while on the contrary, load interruptions (zero load—see Fig. 4.12) provoke lower creep strains due to the recovery of the material as documented in Ref. [13].

40 N=1 100 500 1000 1500 2500 3500

20

0

0

2

(A)

N=103 1×104 6×104 2×105 4×105 4.7×105

20

0 0.0

6

4

40

0.5

1.0

(B)

e (%)

1.5

2.0

2.5

e (%)

80

80

60

60

s (MPa)

s (MPa)

Fig. 4.12 Variation of hysteresis loops under interrupted (at zero load) cyclic loading at (A) σ max ¼ 64 MPa (Intf-0.1-64-b) and (B) σ max ¼ 49 MPa (Intf-0.1-49-b) [13].

40 N=1 230 460 690 920 1150

20

0

0

(A)

2

4 e (%)

6

40 N=1 1.4×105 2.9×105 4.3×105 7.3×105 1.1×106

20

8

0

0

(B)

1

2

3

4

e (%)

Fig. 4.13 Variation of hysteresis loops under interrupted (at the maximum load) cyclic loading at (A) σ max ¼ 70 MPa (Cref-0.1-70-b) and (B) σ max ¼ 47 MPa (Cref-0.1-49-c) [14].

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Actually, while the cyclic stress instantly dropped to zero, the strain showed a timedependent decrease. It is well documented that specimen recovery as a result of the viscoelastic nature of the polymeric matrix is responsible for the observed timedependent decrease of the specimen strain [48]. In the next loading block, the strain of the fatigue cycles only gradually increased, again due to the viscoelastic behavior; i.e., in the first 600 cycles the strain was lower than in the last cycles of the previous loading block. Fig. 4.14 thus shows, for three loading blocks, how the hysteresis loops were back-shifted on the strain axis from the last cycle after interruption to the first cycle of the subsequent reloading. In addition, it is seen that the hysteresis loops of the first loading cycles were not closed due to the considerable strain at the first cycle caused by the opening of the already existing cracks. The fatigue stiffness (i.e., the slope of each hysteresis loop as explained above), as well as the hysteretic energy dissipation (the area of each loop), exhibited measureable changes throughout the lifetime. In Fig. 4.15, the fatigue stiffness, EN, normalized by the stiffness of the first cycle, E1, is plotted against the normalized fatigue life for all specimens loaded under continuous constant amplitude fatigue. Fitted lines per stress level in the same color as the corresponding measurements were added to facilitate the results analysis. Stiffness degradation, irrespective of stress level, followed the same pattern. An initial steep decrease during the first 10%–15% of the lifetime was followed by a steady-state stiffness decreasing trend up to the failure of all specimens. This behavior was partly different from that exhibited by similar composite materials, see, e.g., [8, 18–21], where a

Fig. 4.14 Back-shift of hysteresis loop from last loop before interruption (solid line) to first loop (dashed line) of subsequent reloading (Intf-0.1-64-b) [13].

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Fatigue Life Prediction of Composites and Composite Structures

Fig. 4.15 Variation of normalized fatigue stiffness vs normalized number of cycles at different stress levels.

third segment of steep drop before failure was observed. Nevertheless, similar observations have also been reported in other experimental works, see, e.g., [75] for [
45]2S angle-ply carbon-epoxy specimens at a stress ratio of 0.1. A certain, although not very consistent, trend was further observed in Fig. 4.15—as cyclic stresses decreased, fatigue stiffness and failure stiffness decreased further, i.e., more damage was accumulated during the lifetime, as was also reported elsewhere [17, 19–21]. Similar behavior is observed for the stiffness degradation when the load is interrupted at zero level, as shown in Fig. 4.16. At all stress levels, a similar trend of stiffness degradation was observed as a result of damage formation and growth, i.e., an initial steep decrease during the first 10%–15% of the lifetime followed by a steady-state stiffness decrease up to specimen failure. In addition, the magnitude of stiffness degradation until failure increased with decreasing stress level, showing that more damage was accumulated in the material volume during longer lifetime at lower cyclic stresses. At high-cyclic stress levels, due to the severity of damage, the stiffness dropped at a high rate; however, gradual and well-distributed damage growth at low stress levels caused the fatigue stiffness to decrease at a lower rate at the same (early) age. Nevertheless, in this case, the fatigue stiffness was partially restored after each loading interruption; the restoring effect however decreased with decreasing stress level. This restoring of material stiffness can be attributed to the recovery of the time-dependent stiffness component of the viscoelastic polymeric matrix [12, 13, 48]. The situation is much different when creep-fatigue loading is applied, as presented in Fig. 4.17 where fatigue stiffness evolution of specimens loaded under creep-fatigue are is compared to that of specimens under continuous loading for high and low

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Fig. 4.16 Normalized fatigue stiffness vs normalized number of cycles.

Fig. 4.17 Normalized fatigue stiffness vs normalized number of cycle loaded under different loading patterns at two stress levels of (A) σ max ¼ 68 MPa and (B) σ max ¼ 47 MPa.

stresses, respectively. Initially, for all loading patterns, the stiffness decreased steeply, with this reduction been more pronounced for lower stresses due to the higher capacity of the specimen to accumulate damage at lower stress levels as explained in Refs. [12, 13]. The stiffness of specimens under continuous fatigue loading continued to decrease at a more moderate rate until failure. However, the specimens under creep-fatigue-loading patterns were creeping during the hold time (more at high stress levels and for longer hold times). Under this creep strain, a fiber realignment occurred,

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Fatigue Life Prediction of Composites and Composite Structures

and the internal stress increased [76]. Due to this reason, the specimens’ fatigue stiffness was restoring after each fatigue load interruption, while decreasing again during the next fatigue-loading block. Stiffness restoring, especially after the first fatigue block, was more evident for higher stress levels and longer hold times. On the other hand, at high stress levels and for longer hold time, creep damage formation was shortening the fatigue life (see Fig. 4.9). The effect of the interruption, either at zero, or at high stress levels was also visible on the strain evolution and the self-generated temperature as discussed in details in Refs. [12–14]. The result of these effects was the different damage development in the same specimens depending on the applied loading spectrum as discussed in the following.

4.3.4.3 Self-generated temperature—Damage evolution During the cyclic loading, the formation of any type of cracks in the matrix and fiber matrix debonding, mainly when the crack surface was perpendicular to the beam of light, caused light scattering and changed the specimen transparency [12, 31, 77]. Therefore, darker regions in the photos correspond to decreased light transmittance due to greater damage formation. As shown in Fig. 4.18, both for high and low stress levels under continuous fatigue loading, damage is observed along the fibers, at 45 degrees with respect to the specimen longitudinal axis. Damage is more concentrated in the specimen loaded at higher 25%

60%

90%

98%

100%

s max = 68 MPa

0%

s max = 47 MPa

(A)

(B)

50o C

30 o C

10 o C

Fig. 4.18 Light transmittance and self-generated temperature in different percentages of fatigue life at (A) σ max ¼ 68 MPa (Conf-0.1-68-c) and (B) σ max ¼ 47 MPa (Conf-0.1-47-c).

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cyclic loads (Fig. 4.18A) while it is more evenly distributed over the specimen volume for lower stress levels (Fig. 4.18B). This happened because, at low stress levels, the matrix and interface at the localized damage region were still able to transfer stresses to adjacent areas and the material had the ability to develop and spread additional damage in the volume, and therefore dissipate more energy before failure. Failure occurs in the region where significant damage is observed. In some rare cases two such regions were detected, with one being more critical and specimen failure occurred at that point. During fatigue loading the temperature of the specimens increased due to the selfgenerated heating as a result of internal friction. The surface temperature profiles of specimens, with σ max of 68 and 47 MPa, at different percentages of specimen lifetime, are shown in Fig. 4.18A and B, respectively. Due to the formation of hotspots, the temperature distribution remained nonuniform throughout the lifetime. In all cases, failure occurred at, or close to, the location where hotspots were detected on the specimen surface. The self-generated temperature pattern was more uniform for lower stress levels, see Fig. 4.18B, while more obvious hotspots were detected at higher stress levels, as shown in Fig. 4.18A. In addition, hotspots were observed earlier in the lifetime for the specimens subjected to higher stress levels. The evolution of the specimen transparency and the self-generated temperature across the surface under the interrupted at zero loading are presented in Fig. 4.19 25%

60%

90%

98%

100%

90%

98%

100%

smax = 68 MPa

0%

smax =47 MPa

(A)

(B)

0%

25%

60%

50°C

30°C

10°C

Fig. 4.19 Light transmittance and self-generated temperature for different percentages of fatigue life at (A) σ max ¼ 68 MPa (Intf-0.1-68-d) and (B) σ max ¼ 47 MPa (Intf-0.1-47-a).

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for high and low stress levels. Similar behavior as that exhibited by the specimens under the continuous fatigue loading was observed in that case. With an increasing number of cycles, damage gradually appeared along the fibers, at around 45 degrees with respect to the specimen longitudinal axis, which was thus attributed to matrix/ fiber interface debonding. Damage was more concentrated in the specimen loaded at higher cyclic loads while it was more evenly distributed throughout the specimen volume for lower stress levels. During the early stages of the fatigue life, a uniform distribution of surface temperature was observed. However, by increasing the number of cycles, the distribution of the temperature became more uneven, and hotspots, i.e., regions of high temperature concentrations, started appearing, which had an oval shape oriented in the fiber direction. As under continuous loading, such hotspots were more obvious and significantly more intense for specimens at high stress levels and formed in the same regions where the light transmission was reduced. The highest maximum temperature was 47°C, which remained clearly below Tg,onset (78°C) measured by DMA, and the material thus remained in the glassy state. Nevertheless, when comparing the transparency and temperature distributions on the specimens’ surfaces in Fig. 4.18 (continuous fatigue) and Fig. 4.19 (interrupted fatigue), especially those for high stress levels, Figs. 4.18A and 4.19A, it can be concluded that when the load was interrupted the damage and the self-generated temperature were more uniformly distributed across the specimen’s surface. Only small differences were observed at low stress levels though. Due to the viscoelastic nature of the matrix, repeated material stiffening was observed at the beginning of each loading block, following each load interruption, as can be observed in Fig. 4.16 as well. This stiffening was delaying the damage formation [13]. In addition, unloading of the damaged viscoelastic matrix led to crack blunting, i.e., the local stress intensity in the craze zone at the crack tip was significantly reduced and did not increase again at the same rate as the reloading and thus delayed the crack growth in the following cycles [13, 58, 78–80]. This delay further enabled the initiation and growth of new cracks at other locations and thus led to the observed distribution of damage throughout the whole specimen volume. These two crack-growth retardation mechanisms, i.e., repeated material stiffening and crack blunting, thus explained the significant delay in stiffness degradation and damage accumulation in specimens subjected to interrupted loading and therefore their longer life observed in Fig. 4.9, especially at high stress levels. Figs. 4.20 and 4.21 show the evolution of specimens’ translucency and the corresponding self-generated temperature at different percentages of the specimens’ fatigue life for high and low stress levels, respectively. Photos were taken during the cyclic phase of the loading to ensure the monitoring of the corresponding temperature evolution. At high stress level and for the 2 h hold time (Fig. 4.20A), up to 25% of fatigue life, no distinguishable concentrated damage zone was observed in the specimen, and a uniform distribution of surface temperature was measured. As cyclic loading continued, damage gradually appeared along the fibers, at around 45 degrees with respect to the specimen longitudinal axis. It is seen that the damage was propagated across the specimen volume and several damage zones were formed. The damage zones became more evident during the last creep-loading period, and finally creep

Creep/fatigue/relaxation of angle-ply GFRP composite laminates

25%

60%

5th cyclic blocks 99%

Creep

100%

Creep fatigue (2 h)

0%

125

40°C

25°C

10°C

(A)

0%

60%

1st cyclic block 99%

Creep

100%

Creep fatigue (48 h)

40°C

25°C

(B)

10°C

Fig. 4.20 Light transmittance and self-generated temperature for different percentages of fatigue life at σ max ¼ 70 MPa (A) tcreep ¼ 2 h (Cref-0.1-70-c) and (B) tcreep ¼ 48 h (Cref-0.1-70-f ).

failure occurred, initiated at the location of the most severe damage zone. With increasing the creep time to 48 h (Fig. 4.20B), no specific concentrated damage zone was observed up to the end of the first cyclic loading, because of the limited number of fatigue cycles. However, during the subsequent creep loading, damage gradually formed and propagated across the specimen volume creating concentrated damage zones leading to creep failure. At low stress level (Fig. 4.21), independent of the hold time, the damage formed uniformly and propagated across the specimens while faint hotspots formed at the beginning of the fatigue life. The comparison of specimens’ translucency between short and long hold time at this stress level showed that the concentrated damage zone was a little more severe as a result of more applied loading blocks in the loading pattern with 2-h creep time, although no significant differences can be observed. Typical patterns of the self-generated temperature across the surface of specimens are presented in Figs. 4.20 and 4.21 for high and low stress levels, respectively. At high stress levels, during the early stages of the fatigue life, a uniform distribution of surface temperature was measured. By increasing the number of cycles, the distribution of the temperature became slightly uneven, and faint hotspots, started

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Fatigue Life Prediction of Composites and Composite Structures

Creep fatigue (2 h)

0%

Creep fatigue (48 h)

(A)

(B)

25%

60%

90%

98%

100% 40°C

25°C

10°C 40°C

25°C

10°C

Fig. 4.21 Light transmittance and self-generated temperature for different percentages of fatigue life at σ max ¼ 47 MPa (A) tcreep ¼ 2 h (Cref-0.1-47-d) and (B) tcreep ¼ 48 h (Cref-0.1-47-f ).

appearing in the concentrated damage zone. The observed hotspots were oriented along the fiber direction in which the internal friction was significant. At this stress level failure occurred during the creep loading and a sudden increase of temperature at the moment of failure was detected. At low stress levels at either loading pattern, a faint hotspot was formed in each specimen at the beginning of fatigue life. Nevertheless, in contrast to the high stress level, with increasing number of cycles, the temperature of the hotspot increased, and failure occurred at that region. Specimens under pure creep showed uniformly distributed damage throughout their volume (see Fig. 4.8). Those specimens loaded under continuous fatigue profiles (see Fig. 4.18) developed more concentrated damage zones. Specimens loaded under the interrupted creep-fatigue loading showed an intermediate damage state, more similar to pure creep specimens with less damage accumulation. The fiber alignment due to the creep loads at the hold periods, reduced local stresses that cause crack initiation as was also described in Ref. [63, 80, 81], reducing the severity of damage zones. At the low stress level (Fig. 4.21), independent of the hold time, the damage formed uniformly and propagated across the specimens while a faint hotspot was formed in each

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127

specimen in the middle of fatigue life. The comparison of specimens’ translucency between the long and the short hold times at this stress level showed that the concentrated damage zone was a little more severe as a result of more applied loading blocks in the loading pattern with 2-h creep time.

4.3.5 Failure analysis Two main mechanisms have been observed at the final failure of all specimens; fiber breakage and fiber pull out, as described in Fig. 4.22. As can be seen in the microscopic photo of Fig. 4.22A, fibers from fiber bundles were broken when the specimen failed. In contrast to this, fiber bundles were pulled out, without showing any broken fibers (see Fig. 4.22B), in cases where this mechanism dominated specimen’s failure. The fatigue fracture surfaces of selected specimens examined at different stress levels under continuous or interrupted loading are shown in Fig. 4.23. For all cases, a diagonal damage pattern, following the 45 degrees fiber direction, was observed. For higher stress levels, Fig. 4.23A and E, the failure was characterized by extensive fiber pull-out, a consequence of the significant deterioration of the matrix/fiber interfaces, which hindered the transfer of the stresses from the matrix to the fibers at the concentrated damage zones. By decreasing the fatigue stress level, a mixed-mode failure with fiber pull-out and fiber breakage was observed with predominant fiber breakage at the lowest stress level, as shown in Fig. 4.23B–D and F–H. The presence of fiber breakage

(A)

3 mm

(B)

(C)

250 µm

1000 µm

Fig. 4.22 Dominant observed failure mechanisms (A) mixed failure mode of specimen (Intf-0.1-64-a). Zoom showing characteristic fiber breakage (B), and fiber pull out (C).

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Fatigue Life Prediction of Composites and Composite Structures

Fig. 4.23 Fatigue fracture surfaces under different loading patterns and stress levels, (A) Intf0.1-68-a, (B) Intf-0.1-58-a, (C) Intf-0.1-53-c, and (D) Intf-0.1-49-c, (E) Conf-0.1-68-c, (F) Conf-0.1-58-c, (G) Conf-0.1-53-c, and (H) Conf-0.1-49-a. Entire specimen width is shown.

at low stress levels was attributed to the more uniform and less severe damage distribution, which allowed the matrix and interface to transfer stresses to the fibers. The failure modes of specimens loaded under interrupted and continuous fatigue, did not exhibit any significant differences, as shown in Fig. 4.23; only the necking was more pronounced under continuous fatigue loading (see, e.g., Fig. 4.23E). The fracture surfaces of specimens loaded under pure creep and creep fatigue, shown in Fig. 4.24, reveal again a diagonal damage pattern, along the 45 degrees fiber direction. At high stress levels, both for creep and creep-fatigue loading, the failure was characterized by extensive fiber pull-out, a consequence of the significant deterioration of the matrix/fiber interfaces, which hindered the transfer of the stresses from the matrix to the fibers at the concentrated damage zones. At moderate and low stress levels, a mixed-mode failure, characterized by fiber pull-out and fiber breakage was observed with predominant fiber breakage at the lowest stress level, as shown in Fig. 4.24B–D. The presence of fiber breakage at low stress levels was attributed to the more uniform and less severe damage distribution, which allowed the matrix and interface to transfer stresses to the fibers [12, 13]. In the case of mixed mode failure, the fiber breakage was observed in the central region of the cross-section area, and fiber pull-out was formed near the edges of the specimen. This was attributed to the longer fiber interface at the middle of the specimen able to transfer more stress to the fibers. Close to the edges, the fibers were more susceptible to pull-out.

Creep/fatigue/relaxation of angle-ply GFRP composite laminates

129

Fig. 4.24 Fracture surfaces under different loading patterns and stress levels, (A) Creep-70, (B) Creep-64, (C) Creep-58, (D) Creep-53, (E) Cref-0.1-70-2-d, (F) Cref-0.1-64-2-c, (G) Cref-0.1-58-2-c, (H) Cref-0.1-49-2-a, (I) Cref-0.1-70-48-a, (J) Cref-0.1-64-48-a, (K) Cref-0.1-58-48-a, and (L) Cref-0.1-49-48-a.

4.4

Conclusions and outlook

Angle-ply, glass/epoxy composite laminates are used in several engineering applications, facing several different loading profiles during their operational life. Their fatigue behavior is usually investigated by performing standard constant amplitude loading spectra, although there is a strong evidence in the literature that this kind of material is very sensitive in frequency changes and the loading pattern type. This sensitivity has been proved in this chapter as well, by showing the experimental results from a fatigue-testing program containing data from continuous constant

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Fatigue Life Prediction of Composites and Composite Structures

amplitude loading, as well as interrupted fatigue-loading profiles on angle-ply, (
45)2s, glass/epoxy composite laminates. Periods of zero load, when the material recovers, or periods of load hold times, during which the material creeps were considered during the application of the interrupted-loading patterns. Several different measurements were performed for the study of the specimens’ fatigue behavior under the different loading conditions. The results obtained suggest that fatigue design allowables, if determined based on continuous fatigue, can be conservative when used for the description of the material’s fatigue behavior under loading profiles with interruptions, either at zero stress levels, or at high stress levels. Constant amplitude fatigue behavior is not representative of actual material response in structures operating in open air, especially when loads provoking high stress levels are involved. The work presented in this chapter is very recent and is limited to a material system and a narrow range of loading conditions, and it is obvious that additional efforts should be allocated for the investigation of this topic. Rigorous experimental investigations need to be performed and simulation models to emerge in order to implement, in life prediction processes, parameters that have been neglected by researchers in the past. The findings of this work should be complimented by similar findings regarding the performance of other material systems (presenting less or more fiber dominated fatigue behavior) under loading profiles creating creep/fatigue/relaxation effects. The damage mechanisms and the damage accumulation process under such conditions should be thoroughly examined in order to reveal the physical background behind each of the exhibited mechanisms. This complete study can lead to updates of standards and norms regarding the derivation of fatigue design allowables for thermoset composite material systems widely used in a wide range of engineering applications.

Acknowledgments The work presented in this chapter has been financially supported by the Swiss National Science Foundation (Grant No. 200021_156647).

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Fatigue behavior of nanoparticlefilled fibrous polymeric composites

5

M. Esmkhani, M.M. Shokrieh, F. Taheri-Behrooz Composites Research Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran

5.1

Introduction

Fatigue is a major cause for catastrophic failure in materials. The fatigue behavior of composite and nanocomposite materials was studied by many researchers. For composite materials, three principal approaches were used to predict fatigue life: residual strength [1], residual stiffness [2, 3], and empirical methodologies [4]. In each category, phenomenological, mechanistic, statistical, and mixed methods were utilized by different authors [5–8]. Recently, many researchers have focused on the experimental works and added nanofillers into epoxy polymers and reported improvement on fatigue behavior of nanocomposites [9–26]. There are numerous remarkable efforts in the literature that study the effect of nanoparticles on the mechanical properties of epoxy nanocomposites; like tensile strength and stiffness [27–33]. The electromechanical response (electrical resistance change method) as a damage index of quasi-isotropic carbon fiber-reinforced laminates under fatigue loading was investigated by Vavouliotis et al. [27]. Effect of adding carbon nanotube (CNT) into the glass/epoxy composite was investigated by them and the electrical resistance method was used as a damage control parameter in dynamic fatigue loading condition. A direct correlation between the mechanical loading and the electrical resistance change was established for the investigated specimens [28]. The fatigue behavior and lifetime of polyimide/silica hybrid films were investigated by Wang [29] to evaluate the fatigue property of this class of hybrid films, where the stress-life cyclic experiments under tension-tension fatigue loading were carried out. A semiempirical model [29] was proposed based on the fatigue modulus concept to predict the fatigue life of this class of hybrid films. An exponential model of fatigue stiffness degradation was suggested [30, 31] to predict the fatigue life of matrix-dominated polymer composite laminates based on the nonlinear stress/strain response of most polymers or matrix-dominated polymer composites under uniaxial tension. The fatigue strength assessment of a short-fiber composite using the specific heat dissipation based on an energy approach [32] was proposed and validated to analyze the fatigue strength of the plain and weakened rounded notched specimens made of short-fiber-reinforced plastics. Grimmer [33] showed that the addition of small volume fractions of multiwall carbon nanotube (MWCNT) to the matrix resulted in a Fatigue Life Prediction of Composites and Composite Structures. https://doi.org/10.1016/B978-0-08-102575-8.00005-X © 2020 Elsevier Ltd. All rights reserved.

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significant increase in high-cycle fatigue life. Rosso et al. [34] employed the welldispersed silica nanocomposites for tensile and fracture tests, indicating that the addition of 5 vol.% of silica nanoparticles could improve the stiffness and fracture energy to 20% and 140%, respectively. Guo and Li [35] performing compressive loading on SiO2/epoxy nanocomposites under different loading rates revealed that the compressive strength of composites with silica nanoparticles was higher than that of the pure epoxy at higher strain rates. They showed that there was no clear connection between the compressive strength and the nanoparticle contents at lower strain rates. Johnsen et al. [36] increased the fracture energy of epoxy polymers nanosilica (NS) particles. A set of sudden material property degradation rules, such as stiffness degradation, for various failure modes of a unidirectional ply under a multiaxial state of static and fatigue stress was developed by Shokrieh and Lessard [37]. A comprehensive survey in the available literature reveals the lack of an intensive model to predict the property degradation and fatigue life of the nanoparticle-filled fibrous polymeric composites with thermosets or thermoplastics matrix. In this chapter, after a deep review of the available models, a fatigue model to predict the stiffness reduction of nanoparticle/fibrous polymeric composites has been presented. The model was evaluated in different circumstances based on various industrial applications of composite and nanocomposite materials.

5.2

Fatigue life prediction based on the micromechanical and normalized stiffness degradation approaches

5.2.1 Material properties degradation It is assumed that the major effective reason for material properties’ reduction is due to matrix degradation while in cyclic load conditions nanofillers remain unchanged under different states of stress. Due to the crack propagation in the composite, material properties of composites are changed by a set of sudden material property degradation. The sudden material property degradation rules for some failure modes of a unidirectional ply under a bi-axial state of stress are available in the literature [38, 39] and shown in Fig. 5.1. The degraded ply is modeled by an intact ply of lower material properties. A complete set of sudden material property degradation rules for all various failure modes of a unidirectional ply under a multiaxial state of static and fatigue stress was developed by Shokrieh and Lessard [37]. The lack of models to predict the property degradation of nanoparticle-filled epoxy composites can be observed by a comprehensive survey in the available literature. In this chapter, it is tried to develop a fatigue model to predict the stiffness reduction of polymeric composites with nanoparticles as reinforcement.

5.2.2 Normalized stiffness degradation approach The residual stiffness of the material is also a function of the state of stress and the number of cycles. Since the residual stiffness can be used as a nondestructive measure

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137

Fig. 5.1 Degraded ply is modeled by an intact ply of lower material properties [37].

of the damage evaluation, the stiffness degradation models have been developed by many investigators. By using the normalization technique, all different curves for different states of stress can be shown by a single master curve. Shokrieh and Lessard [37] developed a method of normalization and the present study follows the mentioned method to predict the stiffness degradation of nanocomposite materials using the micromechanics approaches. An epoxy matrix under a constant uniaxial fatigue loading, and under static loading, or equivalently at n¼ 0.25 cycles (a quarter of a cycle) in fatigue is considered for static stiffness of the neat matrix of composites. By increasing the number of cycles, under constant applied stress, σ, the fatigue stiffness, E(n), decreases. Finally, after a certain number of cycles, which is called the number of cycles to failure (Nf), the magnitude of the stiffness decreases to a critical magnitude (Ef). At this point, the composite fails catastrophically. The stiffness degradation of a unidirectional ply is shown in Fig. 5.2. The aforementioned critical value for stiffness Ef can be expressed by the following equation: Ef ¼

σ εf

(5.1)

The average strain to failure, εf, is assumed to be constant and independent on the state of stress and number of cycles. This assumption was used by many authors [40–43] and was experimentally verified in the present study. It should be mentioned that for different states of stress, the stiffness degradation of the composites is different. The same as for the residual strength case, under a highlevel state of stress, the residual stiffness as a function of a number of cycles is nearly constant and it decreases drastically at the number of cycles to failure (Fig. 5.2). However, at the low-level state of stress the residual stiffness of the composite, as a function of the number of cycles, degrades gradually. In practice, designers must deal with a wide range of states of stress varying from low to high. Therefore, similar to the strength degradation case, a model to present the residual stiffness behavior of composite materials under a general state of stress is essential.

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Fig. 5.2 Stiffness degradation under different states of stress [37].

To present the residual stiffness as a function of the number of cycles in a normalized form, the following equation was developed [37]: 2 Eðn, σ, K Þ ¼ 41 

!λ 31γ   log ðnÞ  log ð0:25Þ 5 σ σ
 Es  + εf εf log Nf  log ð0:25Þ

(5.2)

where E(n, σ, K)¼residual stiffness, Es ¼ static stiffness, σ ¼ magnitude of applied maximum stress, εf is an average strain to failure, n ¼ number of applied cycles, Nf ¼ fatigue life at σ, λ, γ ¼ experimental curve-fitting parameters. By using the normalization technique [37], all different curves for different states in Fig. 5.2 collapse to a single curve that has been shown in Fig. 5.3.

Fig. 5.3 Normalized stiffness degradation curve [37].

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5.2.3 Normalized fatigue life model The effect of mean stress (σ max + σ min)/2 on fatigue life was presented efficiently by using the constant life (Goodman-type) diagrams [44]. The establishing and interpolation of the constant life diagram data in a traditional form is a tedious task. However, there are analytical methods [45–49] for predicting the effect of mean stress on the fatigue life based on a limited number of experiments. In a paper by Adam et al. [45], an analytical method has been proposed to convert and present all data from a constant life diagram in a single two-parameter fatigue curve, which can reduce the number of required experiments drastically. In this study, this model is called the normalized fatigue life model. The normalized fatigue life model was modified by Harris and his coworkers for more general cases [48, 49]. In the following, this model is explained in detail. Introducing the nondimensional stresses by dividing of the mean stress σ m, the alternating stress σ a, and the compressive strength σ c by the tensile strength σ t, where q ¼ σ m/σ t, a ¼ σ a/σ t, and c ¼ σ c/σ t, an empirical interaction curve may be derived [48, 49]: a ¼ f ð1  qÞu + ðc + qÞv

(5.3)

where f, u, and v¼ curve-fitting constants, alternating stress, σ m ¼ (σ max + σ min)/ 2 ¼ mean stress, q ¼ σ m/σ t, a ¼ σ a/σ t, and c ¼ σ c/σ t. A typical curve for fatigue life of 106 cycles is shown in Fig. 5.4. The bell-shaped curve is the fatigue life curve. Experimental results by Gathercole et al. [49] showed that their previous quadratic model [46] is inappropriate for the constant life curve especially in both low and high mean stress regions (Fig. 5.4). Therefore, they introduced a power law model (Eq. 5.4) that produces a bell-shaped curve, which corresponds closely to the material behavior under the fatigue loading. In a paper by

Fig. 5.4 Typical constant life diagram [37].

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Gathercole et al. [49], it was shown that the exponents u and v determine the shapes of the left and right wings of the bell-shaped curve. However, it was also shown that the degree of curve-shape asymmetry was not very great, therefore, they assumed u and v are equal and are linear functions of the fatigue life Nf. u ¼ v ¼ A + B log Nf

(5.4)

where A and B are the curve-fitting constants. By substituting Eq. (5.4) into Eq. (5.3), the following equation is obtained: a ¼ f ½ð1  qÞ + ðc + qÞA + B log Nf

(5.5)

The following example helps to explain the normalized fatigue life model. To predict the fatigue life, the following steps must be performed. First, the σ max  log Nf curve for different stress ratios should be established experimentally (Fig. 5.5). Different symbols in Fig. 5.5 represent different applied stress ratios. It is obvious that testing in more states of stress results in more accurate results. Then, by rearranging Eq. (5.5), the following equation is derived and shown graphically in Fig. 5.6: u¼

lnða=f Þ ¼ A + Blog Nf ln ½ð1  qÞðc + qÞ

(5.6)

In Figs. 5.5–5.7, this procedure has been applied to numerical data from the paper by Adam et al. [45]. In Fig. 5.5, the original fatigue data are presented. Then, based on the data from Fig. 5.6, setting f ¼ 1.06 (suggested by Gathercole et al. [49]) and Eq. (5.7), u ¼ ln(a/f )/ ln[(1  q)(c + q)] vs log Nf curve is extracted (Fig. 5.6), from which A and B are found.

CFRP

s max, GPa

1.5

1.0

0.5

1

2

3

4

5

Log Nf

Fig. 5.5 S-log Nf curve, σ t ¼1.91 (GPa), and σ c ¼ 1.08 (GPa) [45].

6

7

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141

Fig. 5.6 u vs log Nf curve [37].

Fig. 5.7 Predicted constant life diagram [37].

In Fig. 5.7, based on all previous information, the constant life diagram for a different number of cycles to failure is predicted. Fig. 5.7 is generated by knowing A, B, and f, three constants which can be determined from a relatively small quantity of tests, as demonstrated in this example. Thus, the method is very useful for reducing the number of experiments for the characterization of materials.

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5.2.4 Modeling strategy for the filled fibrous composites with nanoparticles A comprehensive survey in the available literature reveals the lack of models to predict the property degradation of nanoparticle-filled fibrous epoxy composites. In this chapter, a fatigue model is developed to predict the stiffness reduction of nanoparticle/ fibrous polymeric composites. For this purpose, a schematic framework of the modeling strategy is shown in Fig. 5.8. The model is an integration of two major components: the micromechanical (such as the Halpin-Tsai or Nielsen models) and the normalized stiffness degradation approaches. The model is able to predict the final fatigue life of nanoparticle/fibrous polymeric composites under general fatigue loading conditions. As shown in Fig. 5.8, in the first step, the model predicts the equivalent stiffness of nanoparticle/epoxy nanocomposites using a micromechanical approach with pure epoxy resin and nanoparticles parameters. Then, the normalized stiffness degradation model for fibrous polymeric composites under fatigue loading was used to predict the stiffness degradation of fibrous polymeric composites. While the archived equivalent stiffness is used for stiffness of fibrous polymeric composites. By coupling of the normalized stiffness degradation model of fibrous polymeric composites and the micromechanical approach, the normalized stiffness degradation model for nanoparticle/fibrous polymeric composites under fatigue loading was developed.

Pure epoxy resin

Nanoparticles

Micromechanical Models

Equivalent stiffness of nanoparticle/epoxy nanocomposites

Normalized stiffness degradation model for fibrous polymeric composites under fatigue loading

Normalized stiffness degradation model for nanoparticle/fibrous polymeric composites under fatigue loading

Fig. 5.8 A schematic flowchart of the present model.

Fatigue behavior of nanoparticle-filled fibrous polymeric composites

143

5.2.5 Micromechanical models 5.2.5.1 The Halpin-Tsai micromechanics model There are several models for the prediction of elastic properties of nanocomposite materials, based on the geometry, the orientation of the filler, the elastic properties of the filler and matrix, such as the Halpin-Tsai model [50–52], the Nielsen model [53–56], the Mori-Tanaka model, the Eshelby model, etc. The Halpin-Tsai model is based on the self-consistent field method, although often considered being a semiempirical also a simple but accurate model that takes into account the shape and the aspect ratio of the reinforcing particles. In this model, the quality of the bonding and fillers arrangements was not considered. It can predict the modulus of a composite material (Ec), containing nanoparticles as a function of the modulus of the polymer (Em) containing no nanoparticles and of the modulus of the particles (Ep). The predicted modulus of the nanoparticle-modified epoxy polymer (Ec) is given by the following equation: w 1 + 2 ηVp t Ec ¼ Em 1  ηVp

(5.7)



   Ep Ep w where, η ¼ 1 = +2 t Em Em

(5.8)

where w/t is the shape factor and Vp is the volume fraction of particles. The equivalent modulus can be derived for nanoparticles in different types based on the particles shapes. For instance for the plane shape (such as graphene nanoparticles), the predicted modulus of filled composite (EC) is given by Eq. (5.9). 

   ðEG =Em Þ  1 ðEG =Em Þ  1 1 + ½ðw + lÞ=t 1+2 VG VG EC 3 5 ðEG =Em Þ + ½ðw + lÞ=t ðE =E Þ + 2    G m  ¼ + ðEG =Em Þ  1 ðEG =Em Þ  1 8 Em 8 1 1 VG VG ðEG =Em Þ + ððw + lÞ=tÞ ðEG =Em Þ + 2 (5.9) where EG is the elastic modulus of plane nanoparticles and Em is the composite modulus without nanoparticles. Moreover, L, W, and t are the length, width, and thickness of the plane nanoparticles, respectively. In terms of using carbon nanofibers (CNFs) as particles, the Halpin-Tsai equations are represented as the following equation:      L ðECNF =Em Þ  1 ðECNF =Em Þ  1 1+2 1+2 VCNF VCNF Enc 3 5 d ECNF =Em + 2  ðL=d Þ ðE =E Þ + 2    CNF m  ¼ + ðECNF =Em Þ  1 ðECNF =Em Þ  1 8 Em 8 1 1 VCNF VCNF ½ECNF =Em Þ + 2  ðL=dÞ ðECNF =Em Þ + 2 (5.10)

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Fatigue Life Prediction of Composites and Composite Structures

where L and d are the length and diameter of CNFs [50, 53, 57]. Furthermore, the Halpin-Tsai model can be implemented to predict the stiffness of chopped strand mat (CSM)/polymer composites (ECSM,nc) reinforced with nanoparticles, as a function of the stiffness of the pure matrix (Em), the stiffness of CSM (ECSM), and the stiffness of nanoparticles (Ep) in static loading conditions. The predicted stiffness of the nanocomposites (ECSM,nc) reinforced with CSM and nanoparticles is given by the following equation:   1 + ξ1 η1 VCSM + ξ2 η2 Vnanoparticles ECSM, nc ¼ Em (5.11) 11 η1 VCSM  η2 Vnanoparticles where VCSM and Vnanoparticles are the volume fractions of CSM and nanoparticles, respectively. Moreover, ξ is the shape factor of CSM and nanoparticles. Also, η1 and η2 are defined as below: ECSM =Esm  1 ECSM =Esm + ξ1 Enanoparticles =Esm  1 η2 ¼ Enanoparticles =Esm + ξ2 η1 ¼

(5.12)

where ECSM and Enanoparticles are the stiffness of CSM and nanoparticles under static loading conditions. Moreover, ξ1 and ξ2 are shape factors of CSM and nanoparticles, defined as ξ1 ¼ 2

l1 l2 , ξ2 ¼ 2 d1 d2

(5.13)

5.2.5.2 Nielsen micromechanics model The Nielsen model predicts Young’s modulus of nanocomposite materials, especially for spherical particles. In his model, the effect of slippage between the nanoparticle and matrix was evaluated using a coefficient factor. The Nielsen model used to predict the modulus of a material containing nanoparticles (Ec) as a function of the modulus of the polymer containing no nanoparticles (Em), and of the modulus of the particles (Ep). The predicted modulus of the nanoparticle-modified epoxy polymer (Ec) is given by the following equation [55, 58]: EC ¼

1 + ðKE  1Þβ Vf Em 1  μ β Vf

(5.14)

where KE is the generalized Einstein coefficient and β and μ are the constants. The constant β is given by     EP EP 1 + ð K E  1Þ (5.15) β¼ Em Em

Fatigue behavior of nanoparticle-filled fibrous polymeric composites

145

It should be noted that β is identical to h in the Halpin-Tsai model when a shape factor of ζ ¼ (KE  1) is used. The value of μ depends on the maximum volume fraction of particles (νmax) that can be incorporated and calculated by the following equation:

 1  νf  μ¼1+ νmax νf + ð1  νmax Þ 1  νf (5.16) νmax where values of νmax have been published by Nielsen and Landel [55] for a range of particle types. Nielsen and Landel [59] quoted a value of νmax ¼ 0.632 for such random close-packed, non-agglomerated spheres and this value is used in the present model. The value of KE varies with the degree of adhesion of the epoxy polymer to the particle. For the epoxy polymer with a Poisson’s ratio of 0.5 which contains dispersed spherical particles: (a) KE ¼ 2.5 if there is “no slippage” at the interface (i.e., very good adhesion), or (b) KE ¼ 1.0 if there is “slippage” (i.e., relatively low adhesion) [55]. However, the value of KE is reduced when the Poisson’s ratio of the polymer is less than 0.5 [56]. These earlier studies have also found that at relatively high values of vf, above about 0.1 of silica nanoparticles, the Nielsen “slip” model gave the best agreement with the measured values. However, the present study found that relatively low values of vf (i.e., at values of vf below about 0.1) the Halpin-Tsai and the Nielsen “no-slip” models show better agreement. Thus, an overall conclusion is that the measured modulus of the different silica nanoparticle-filled epoxy polymers approximately lay between an upper-bound value set by the Halpin-Tsai and the Nielsen “no-slip” models, and a lower-bound value set by the Nielsen “slip” model, with the last model being the more accurate at relatively higher values of vf [59].

5.2.6 The normalized stiffness degradation model for nanocomposites (Nano-NSDM) 5.2.6.1 The Nano-NSDM based on the Halpin-Tsai model By coupling of the normalized stiffness degradation approach (Eq. 5.2) for fibrous polymeric composites under fatigue loading and the Halpin-Tsai micromechanics model, the normalized stiffness degradation model for nanoparticle/fibrous polymeric composites under fatigue loading in form of Eq. (5.17) was developed.     EP EP 2 1 + 2w=t 1 + 2w=t Vf Em Em 41      EC ¼ EP EP 1 1 + 2w=t Vf Em Em    EP EP 1 + 2w=t 1 + 2w=t Vf σ Em Em +      εf EP EP 1 1 + 2w=t Vf Em Em

!λ 31γ   log ðnÞ  log ð0:25Þ 5 σ
 Es  εf log Nf  log ð0:25Þ



(5.17)

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Fatigue Life Prediction of Composites and Composite Structures

5.2.6.2 The Nano-NSDM based on the Nielsen model After a combination of the normalized stiffness degradation approach for fibrous polymeric composites under fatigue loading with the Nielsen micromechanics model, the normalized stiffness degradation model for nanoparticle/fibrous polymeric composites under fatigue loading in form of Eq. (5.18) is developed. 

   EP EP 1 + ðKE  1Þ Vf Em Em    
 EP EP νf + ð1  νmax Þ 1  νf 1 + ðKE  1Þ Vf Em Em

1 + ðKE  1Þ

EC ¼


 1  νf  1 1+ νmax νmax 

2 41 

log ðnÞ  log ð0:25Þ
 log Nf  log ð0:25Þ

!λ 31γ ! σ σ 5 Es  + εf εf

    EP EP 1 + ðKE  1 Þ 1 + ðKE  1Þ Vf Em Em
      
 1  νf  EP EP 1 1+ νmax νf + ð1  νmax Þ 1  νf 1 + ð K E  1Þ V f Em Em νmax (5.18)

5.2.7 The evaluation of Nano-NSDM In this section, the evaluation of the Nano-NSDM is performed under different following circumstances.

5.2.7.1 Fatigue life prediction for epoxy resin modified by silica nanoparticles In this research, the silica nanoparticles have been employed to modify the epoxy resin. In general, the dimensions of these particles are in micron ranges. However, with the advance of nanotechnology as well as the processing techniques, various types of particles in nanoscales have recently been developed and utilized as reinforcement in polymeric composites [26]. Experimental results obtained from tension-tension fatigue tests on a bulk epoxy confirm the reduction of the epoxy laminated composite stiffness, caused by the presence of cracks, which can effectively be compensated by silica nanoparticles. In a work by Manjunatha et al. [26], 185 g/mol of epoxy resin, LY556, bisphenol A (DGEBA), supplied by Huntsman, Duxford UK and the silica (SiO2) nanoparticles supplied by Nanoresins, Geesthacht, Germany were used. For the spherical silica nanoparticles used in the present work, the aspect ratio is unity and hence, w/t ¼ 1 was used. Also, in this case study, νmax ¼ 0.632, νf ¼ 0.048 so

Fatigue behavior of nanoparticle-filled fibrous polymeric composites

147

the values of KE is 2.5 and the Nielsen “no-slip” model was implemented. So Eq. (5.15) will be changed in form of the following equation:     ESio2 ESio2 2 1+2 1 + 2 Vf Em Em     41  EC ¼ ESio2 ESio2 1 1 + 2 Vf Em Em     ESio2 ESio2 1+2 1 + 2 Vf σ E E  m   m  +  ESio2 ESio2 εf 1 1 + 2 Vf Em Em

log ðnÞ  log ð0:25Þ
 log Nf  log ð0:25Þ

!λ 31γ   5 Es  σ εf

(5.19)

Typical stiffness variation curves obtained at σ max ¼ 225 MPa are shown in Fig. 5.9A. In general, all materials exhibit a stiffness reduction with fatigue cycles, as has been previously observed in FRPs [60, 61]. The stiffness reduction was quite steep and very significant in GRP nanocomposites. By using the neat resin (NR), data of Fig. 5.9A and using the normalized technique, curve-fitting parameters, λ and γ are obtained (see Fig. 5.9B). The other properties of the GRP composites and NS are shown in Tables 5.1 and 5.2. For evaluating of the accuracy of derived equations for the NR, according to Fig. 5.10, the trend for stiffness of NR vs the number of cycles is depicted and good agreement with experimental data was obtained. This compatibility shows that obtained curve-fitting parameters are suitable for this composite. Fig. 5.11 shows that the reported stiffness vs the number of cycles for 10 wt% silica nanoparticle-filled epoxy polymers is in a good agreement with the modified normalized stiffness degradation models. Also, the behavior of the modified Nielsen “noslip” model is nearer to the result of the experiment compared with the modified Halpin-Tsai model. In Table 5.3, the results and value of the errors for each modified model are presented.

5.2.7.2 Fatigue life prediction for GFRP with nanoparticles As a next verification of the model, the results presented in Ref. [26] are considered. The fatigue limit, that is, the maximum applied stress for a life of 106cycles of the neat epoxy is about 95 MPa. The presence of silica particles in the matrix raises this fatigue limit by about 15% to 110 MPa. At this state of stress, the normalized stiffness degradation curve was generated and after 2,000,226 cycles, the stiffness of modified composites with nanoparticles (Ec) and the tensile strength were obtained equal 5.54 GPa and 111.92 MPa, respectively (see Table 5.4). On the other hand, the results presented in Ref. [20] as shown in Fig. 5.12 are considered. It was assumed that the curve-fitting parameters represented in Fig. 5.9A are again valid. This assumption has a negligible error. The final results, after applying the

148

Fatigue Life Prediction of Composites and Composite Structures 1.00 NR NRR NRS

Normalised stiffness

0.95

0.90

0.85

0.80

0.75 0

500

(A)

1000

1500

2000

2500

Number of cycles, N Normalized stiffness degradation curve for LY556 epoxy resin 1 0.9 0.8

Normalized stiffness

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

(B)

0

0.2

0.4 0.6 Normalized number of cycles

0.8

1

Fig. 5.9 (A) The stiffness variation during fatigue cycling in the GFRP composites at σ max ¼ 225 MPa, (i) neat resin (NR), (ii) resin with 9 wt% rubber microparticles (NRR), (iii) resin with 10 wt% silica nanoparticles (NRS), and (iv) resin with a “hybrid” matrix containing both 9 wt% rubber and 10 wt% silica particles (NRRS) [26]. (B) Normalized stiffness degradation curve for LY 556 epoxy resin, λ ¼ 3.348, γ ¼ 0.357, and εf ¼ 0.02 [62].

Fatigue behavior of nanoparticle-filled fibrous polymeric composites

149

Table 5.1 Properties of GFRP composites [26] Tensile properties Material

Condition

Strength, Xt (MPa)

Modulus, E (GPa)

GFRP

Without nanoparticles

365

17.5 0.6

Table 5.2 Properties of Silica nanoparticles [26] Tensile properties Mean diameter, D (nm)

Modulus, E (GPa)

20

85

18 Em(n)- Manjunatha et al. (2009) Em(n)- Model

Em (GPa)

17

16

15

14

0

200

400 n (cycles)

600

800

Fig. 5.10 Stiffness reduction for neat epoxy resin, σ max ¼ 225 MPa [62].

developed model at 22,143 cycles, are presented in Table 5.5. The tensile strength for 10 wt% SiO2 /LY556 epoxy nanocomposites calculated by the model is equal to 29.24 MPa (in comparison to the 28.8 MPa presented in the reference). Therefore, the accuracy of the developed model is acceptable. The results are expressed in Table 5.5 [62].

150

Fatigue Life Prediction of Composites and Composite Structures 20 Manjunatha et al. (2009) The Nano-NSDM based on Halpin-Tsai model The Nano-NSDM based on Nielsen model

19

Ec (GPa)

18

17

16

15

14 0

500

1000

1500

2000

n (cycles)

Fig. 5.11 The stiffness reduction of 10 wt% silica epoxy nanocomposites, σ max ¼ 225 MPa [62]. Table 5.3 Results and value of error for modified models [62]

n (cycles)

Em(n): Present model

Ec(n): Experimental results [26]

Ec(n): HalpinTsai model

Error (%)

Ec(n): Nielsen model

Error (%)

11 118 182 300 589 705

17.19 16.00 15.67 15.24 14.58 14.40

18.82 17.58 17.18 16.88 16.22 16.13

19.82 18.45 18.06 17.56 16.80 16.59

5.33 5.00 5.11 4.00 3.56 2.84

18.85 17.61 17.25 16.79 16.09 15.90

0.19 0.18 0.39 0.55 0.79 1.43

Table 5.4 Results of the neat epoxy and 10 wt% nano-silica modified GFRP composites, σ ¼ 95 MPa [62] Material

Condition

GFRP

0 wt% nanoparticles 10 wt% nano silica-modified epoxy [26] 10 wt% nano silica filled epoxy, based on the Halpin-Tsai model

Strength (MPa) 95 110 111.92

Fatigue behavior of nanoparticle-filled fibrous polymeric composites

151

Maximum Stress, smax (MPa)

45 Bulk Epoxies R = 0.1 n = 1 Hz 20°C, 55% RH

40 35 30 25

Neat epoxy (EP) Epoxy+Rubber (ER)

20

Epoxy+Silica (ES) Epoxy+Rubber+Silica (ERS)

15 1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

No. of cycles, Nf

Fig. 5.12 Stress vs lifetime (S-N) curves of neat epoxy and nanoparticle-filled epoxy [20]. Table 5.5 Properties of nanoparticle-filled GFRP composites [62] Material

Condition

Bulk Epoxy

0 wt% nanoparticles, Experimental Result [20] 0 wt% nanoparticles, Present work

GFRP without nanoparticles GFRP with nanoparticles GFRP with nanoparticles

10 wt% nano-silica-modified epoxy, Experimental result [20] 10 wt% nano-silica filled epoxy, presentwork based on the HalpinTsai model

Strength (MPa)

Error (%)

25.0

–

25.3

1.2

28.8

–

29.2

1.5

5.2.7.3 Fatigue modeling of CSM/epoxy composites There are many applications for CSM composites in various industries because of their intrinsic properties. In a published paper by the present authors [63], the normalized stiffness degradation method was utilized for CSM/epoxy composites to predict the fatigue life. Moreover, the fatigue damage accumulation model developed by Ye [64], for E-glass fabric in CSM form with isophthalic polyester resin, was not applied for CSM/epoxy composites. So, in this research, the capability of this model for epoxy matrix composites was also investigated. A series of tests in tension-tension fatigue condition at room temperature was carried out at different load levels to evaluate the capabilities of both models with experimental observations [63].

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Fatigue Life Prediction of Composites and Composite Structures

Fatigue damage accumulation model Damage accumulation and cyclic degradation are critical issues for design and life prediction of composites under fatigue loading conditions. The fatigue damage theory for the CSM composites based on the phenomenological aspects of damage accumulation in composite materials has been assessed by Ye et al. [64]. They assumed that the residual stiffness is a monotonically decreasing function of the fatigue cycle and depends on the damage variable as follows: D¼1

E Es

(5.20)

where E is the current stiffness and Es is the initial static stiffness of the material. The characteristics of fatigue crack growth and a damage accumulation law for composites can be defined as  2 n dD σ ¼ C max dN D

(5.21)

where C and n are material constants that can be determined by testing specimens at various applied stresses. Using the definition of the damage variable in Eq. (5.20), the damage development in the composites is shown in Fig. 5.13 [65, 66]. Due to the inherently heterogeneous microstructure of the chopped fiber-reinforced composites, various kinds of stress concentrators exist in the material, including fiber/matrix interfaces, fiber ends, process-induced defects, laminate stress-free edges or discontinuities, and residual stress concentrations developed during the curing process [64]. The presence of these stress concentrators results in significant energy storage.

Fig. 5.13 Fatigue damage development in composite materials [65, 66].

Fatigue behavior of nanoparticle-filled fibrous polymeric composites

153

The energy storage tends to approach an energy balanced state, as indicated by the feature of entropy in nonlinear irreversible thermodynamics [64]. In the early stage (stage I) of fatigue damage development, rates of energy dissipation and material degradation are rapid. In this stage, fatigue damage appears predominantly in the form of matrix cracking along fibers in plies of composite laminates [67] and in addition, fiber-matrix interface debonding in the case of chopped fiber composites. The fibers aligned ahead of a matrix crack are obstacles to the crack growth. Therefore, it is more difficult to break a fiber than to break the matrix. Hence, during stage II of fatigue damage development, energy dissipation and material degradation rates decrease when the matrix cracks reach fibers aligned at some angle in front of them. In the final stage (stage III) of fatigue damage development, sudden coalescence of micro-cracks and severity of interaction, in addition to the rapid growth of the most favorable cracks, lead to the catastrophic fracture of composites [64]. Finally, the predicted modulus after N cycles can be expressed by the following expression: h i ðn + 1Þ E ¼ 1  fNCðn + 1Þg1=n + 1 σ 2n= Es max

(5.22)

It can be used to predict the number of cycles required to reach a given stiffness reduction for a known fatigue maximum applied stress. In this relation, C and n, as material constants have to be identified by testing specimens after measuring the stiffness reduction at various applied stresses [63].

Tests results—Model evaluation A series of tests in static loadings was carried out to determine tensile properties of fabricated composites using CSM short glass fibers as isotropic fiber-reinforced composites. The stress-strain relationship of a sample is presented in Fig. 5.14. In order to evaluate the fiber weight fraction, the burn-off test was performed to obtain the glass fiber content. In order to maintain statistical reliability, minimum of four samples were tested in each step. The mean tensile strength of the four samples was achieved equal to 158 MPa. The Young’s modulus of fabricated composites was around 8.9 GPa. In addition, the tension-tension fatigue tests were conducted at three different stress levels. The maximum stresses were chosen to be 50%, 60%, and 70% of the ultimate tensile strength of the specimen. The fatigue tests were carried out under a loadcontrol condition at a frequency of 2 Hz. During all fatigue tests, the stress ratio (minimum applied stress/maximum applied stress) was set at 0.1 under room temperature condition. The applied maximum stress and the number of cycles to failure from experiments for CSM/epoxy composites under the load control fatigue loading are demonstrated in Table 5.6. In the normalized stiffness degradation approach for CSM polymeric composites under fatigue loading to predict the fatigue life of CSM/epoxy composites, λ, γ experimental curve-fitting parameters from Eq. (5.2) have to be determined. By using the normalization technique and obtaining parameters as discussed in the previous

154

Fatigue Life Prediction of Composites and Composite Structures 160 140

Stress (MPa)

120 100 80 60 40 20 0 0.00

0.01

0.02

0.03

0.04

Strain

Fig. 5.14 The stress-strain curve of CSM/epoxy composites [63]. Table 5.6 Sample of experimental results of CSM/epoxy composites under tension-tension fatigue loading at room temperature condition, R ¼0.1, strain to failure ¼0.0187 [63] Stress level (%)

Applied stress (MPa)

Number of cycles to failure (cycles)

70 60 50

111 95 79

6121 19398 36481

sections, all different curves for different states of stress collapse to a single curve as shown in Fig. 5.15. The obtained value of λ and γ is 7.348 and 0.852, respectively. Moreover, the empirical parameters of C and n for the fatigue damage accumulation approach (Ye’s Model, [64]) are found from experiments equal to 2E  43 and 8.1256, respectively. Finally, during the stage II of the fatigue life (4000–24,000 cycles), the normalized stiffness reduction of CSM/epoxy composites at 79 MPa stress loading or 50% of the ultimate strength has been achieved and demonstrated. Then, the results of both models have been compared with achieved experimental results by Shokrieh et al. [63] and represented in Fig. 5.16. Evaluation of models indicates there was a better correlation between the normalized stiffness degradation technique and the experimental results in comparison with the Ye model [64]. In addition, the results show that the normalized stiffness degradation technique is properly applicable for CSM/

Fatigue behavior of nanoparticle-filled fibrous polymeric composites

155

1 0.9

Normalized stiffness

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0.5

0.55

0.6

0.7 0.75 0.8 0.85 0.65 Normalized number of cycles

0.9

0.95

1

Fig. 5.15 Normalized stiffness degradation curve for CSM/epoxy composites, λ ¼ 7.348, γ ¼ 0.852, and εf ¼ 0.0187 [63]. 1

E/Es

0.8

0.6

Experimental data Normalized Stiffness Degradation Model Lin Ye Model, C = 2E-43, n = 8.1256

0.4 4000

9000

14000 n (cycles)

19000

24000

Fig. 5.16 Stiffness reduction of CSM/epoxy composites, stress level 50% of UTS (comparison between models and experimental data [63]).

epoxy composites and the application of this approach is not limited to unidirectional ply composites [63].

5.2.7.4 Fatigue life of thermoplastic nanocomposites In order to evaluate the present model for thermoplastic nanocomposites, experimental results (Fig. 5.17) of Ramkumar and Gnanamoorthy [68] are used. They investigated the effect of adding nanoclay on the temperature rise and the modulus drop during

156

Fatigue Life Prediction of Composites and Composite Structures

35

Fig. 5.17 Temperature rise during fatigue testing in PA6 and PA6NC specimens tested at 30 and 22 MPa [68]. Temperature rise (K)

30 25

PA6-30MPa PA6NC-30MPa PA6-22MPa PA6NC-22MPa

20 15 10 5 0 1.E+00

1.E+01 1.E+02 1.E+03 No. of cycles

1.E+04

1.E+05

the biaxial cyclic loading for thermoplastic modulus behavior of the neat matrix and clay/polyamide-6 (PA6) nanocomposites (PA6NC). The major reason for the modulus reduction of PA6NC and neat PA6 under fatigue loading conditions are assumed to be due to the temperature rise and thermal softening phenomena. The investigated temperature rise during the fatigue testing in PA6 and PA6NC specimens are presented in Fig. 5.17 at 30 and 22 MPa [68]. Ramkumar and Gnanamoorthy [68] employed commercial grades of PA6 pellets and hectorite clay (bentone) nanoparticles in micron dimensions to modify the PA6 thermoplastic matrix. The clay was organically modified with a hydrogenated tallow quaternary amine complex. PA6 pellets and 5 wt% clay nanofillers were mixed in their research to make PA6NC and measured tensile properties of PA6 and PA6NC shown in Table 5.7 [68]. In Fig. 5.18, the normalized modulus drop as a function of the number of cycles for the PA6 thermoplastic matrix without nanoparticles at 30 and 22 MPa stress magnitudes is demonstrated to verify the present model for PA6. Then, the normalization technique was applied and the normalized stiffness degradation curve for PA6 thermoplastic resin without nanoparticles was implemented [69]. Next, the curve-fitting parameters, λ and γ were obtained as shown in Fig. 5.19 equal to 3.984 and 1.169, respectively. For evaluating the accuracy of the derived equations for the neat PA6 thermoplastic resin without nanoparticles, the trend of the stiffness vs the number of cycles is depicted in Fig. 5.20 and a good agreement with experimental data Table 5.7 Tensile properties of PA6 and PA6NC [68] Material

Tensile modulus (MPa)

Tensile strength (MPa)

PA6 PA6NC

720 2310

34 52

Fatigue behavior of nanoparticle-filled fibrous polymeric composites

157

1.2

Modulus (normalized )

1.0

0.8

0.6

PA6NC, 30 MPa PA6NC, 22 MPa PA6, 30 MPa PA6, 22 MPa

0.4

0.2 1e+0

1e+1

1e+2

1e+3

1e+4

1e+5

No. of cycles Fig. 5.18 Modulus drop (normalized) as a function of the number of cycles for PA6 and PA6NC specimens tested at 30 and 22 MPa stress magnitudes [68].

1

Normalized stiffness

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2

0.2

0.3

0.6 0.7 0.4 0.5 Normalized number of cycles

0.8

0.9

Fig. 5.19 Normalized stiffness degradation curve for polyamide-6 (PA6) thermoplastic resin without nanoparticles,λ ¼ 3.984, γ ¼ 1.169, and εf ¼ 0.52 [69].

158

Fatigue Life Prediction of Composites and Composite Structures

Fig. 5.20 Stiffness reduction for PA6 thermoplastic resin without nanoparticles [69].

was obtained. This compatibility shows that obtained curve-fitting parameters are suitable for this thermoplastic resin. The major capability of the present model is the fatigue life prediction of thermoplastic-filled with two-dimensional (2D) nanoparticles composites under fatigue loading condition based on the experimental data of the neat thermoplastic resin without nanofillers. For this purpose, the properties of the nanoparticles and the neat thermoplastic matrix are replaced in the present model. The properties of the hectorite clay were used from the available data in the literature [70], both aspect ratio and width are 100 nm. The stiffness of hectorite clay was found equal to 120 GPa by means of an inverse technique and the Halpin-Tsai micromechanical model (Eq. 5.9) while subjected to static loading conditions. The present model was also verified at 22 and 30 MPa of stress magnitudes. First, the stiffness vs the number of cycles for 5 wt% PA6NC at 30 MPa stress magnitude was found and shown in Fig. 5.21. It shows that the normalized stiffness degradation vs the number of cycles for 5 wt% hectorite PA6NC, calculated by the present model, is in a good agreement with experiments. The behavior of the Nano-NSDM based on the Halpin-Tsai model completely covers the result of the experiment. For more elaboration, the results and error value of 2D nanoparticles are presented in Table 5.8. For instance, while the number of cycles was 474 cycles, according to the experimenf tal results, EPA6NC was 1.88 GPa. The predicted stiffness by the present model for 5 wt% PA6NC material is 1.87 and only 0.39% error is observed. The error percent at the number of cycles between 0.25 and 100 is more than 1% which confirms the primary assumptions of the normalized stiffness degradation technique and the capability of the established model in this region. Furthermore, the present model was applied for the reinforced nanocomposite at 22 MPa of stress magnitude to show

Fatigue behavior of nanoparticle-filled fibrous polymeric composites

159

Fig. 5.21 The stiffness reduction for 5 wt% clay/PA6 nanocomposites (PA6NC), σ max ¼ 30 MPa [69].

Table 5.8 Results and value of error for modified model [69] (PA6NC), σ max ¼ 30 MPa

n (Cycles)

Experiments PA6NC [68] Efnc (GPa)

Nano-NSDM based on The Halpin-Tsai Model Efnc (GPa)

Error (%)

3 10 31 102 474 1010

2.12 2.13 2.08 2.03 1.88 1.79

2.07 2.07 2.05 2.00 1.87 1.78

2.40 2.69 1.39 1.31 0.39 0.86

the capability of the model in another stress level. According to Ramkumar and Gnanamoorthy [68], the stiffness of 5 wt% PA6NC in this state, while the number of cycles equal to 800 cycles, was 1.9 GPa. To apply Nano-NSDM, it was assumed that curve-fitting parameters in the normalized stiffness degradation techniques are valid. Then, the stiffness vs the number of cycles for this nanocomposite predicted by the present model is equal to 1.8 GPa. A comparison shows there is a good consistency between the obtained results and experimental observation. It means that the developed model is also applicable to this stress level [69].

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Fatigue Life Prediction of Composites and Composite Structures

5.2.7.5 Fatigue life of nanoparticles/CSM/polymer hybrid nanocomposites In this step, to evaluate the capability of the present model (Nano-NSDM) in the calculation of the fatigue life of nanoparticles/CSM/polymer hybrid nanocomposites, the CNF is selected as the nanoparticle and experiments are carried out in the present work.

Experimental procedure The ML-526 epoxy resin based on bisphenol-A was used in the present study. The ML-526 epoxy was selected because of its low viscosity and extensive applications in composites industries. The curing agent was HA-11 (polyamine). The ML-526 epoxy resin and polyamine hardener HA-11 were supplied by Mokarrar Engineering Materials Co., Iran. The E-glass fabric in form of CSM was supplied by Taishan Fiberglass Inc., China. The randomly distributed fibers have an average diameter of approximately 13 μm; around 5 cm lengths and a surface density of 450 g/m2. The CNF was supplied by Grupo Antolin SL, Spain. The CNF has an average diameter of approximately 20–80 nm, and the length is about 30 μm. Fig. 5.22 shows the scanning electron microscopy (SEM) and the transmission electron microscopy (TEM) images of the procured CNF. The specifications of CNF, E-glass CSM, and the resin are presented in Tables 5.9–5.11.

Fig. 5.22 (A) The scanning electron microscopy and (B) the transmission electron microscopy images of carbon nanofiber, prepared by Grupo Antolin SL, Spain [71]. Table 5.9 Specifications of carbon nanofiber [71] Parameter

Value

Fiber diameter (TEM) (nm) Fiber length (SEM) (μm) Tensile modulus (GPa) Aspect ratio

20–80 30 550 375

Fatigue behavior of nanoparticle-filled fibrous polymeric composites

161

Table 5.10 Specifications of the E-glass CSM fiber [63] Parameter

Value

Fiber diameter (μm) Fiber length (mm) Tensile modulus (GPa) Aspect ratio

10–20 25–50 71 2500

Table 5.11 Properties of ML-526 epoxy resin [71] Physical properties

Mechanical properties

Viscosity at 25 °C (Centipoise)

Glass transition temperature (°C)

Tensile modulus (GPa)

Tensile strength (MPa)

1190

72

2.6

60

57

165

R76

R76

33

19

13

R76

3

R76

To fabricate CSM/epoxy composite specimens, the hardener was added to the epoxy resin at a ratio of 15:100 and stirred gently by using a mechanical stirrer (Heidolph RZR2102) for 5 min at 100 rpm. The stirring at low speed was very important to avoid any undesirable bubble formation. The CSM/epoxy composite specimens were manufactured using the hand layup process. Six layers of E-glass CSM were cut into a sheet of dimensions of 210 200 3 mm. Then, layers were stacked with ML-526 epoxy resin and impregnated at room temperature. A roller was used to release the trapped air and voids. Later, samples were kept under 12 kPa static pressure to get the trapped bubbles out. The fabricated sheet was also pre-cured under the static pressure for 48 h. For post-curing, the sheet was placed in an oven for 2 h at 80°C and further 1 h at 110°C. Finally, the test specimens were cut in accordance with type 1 in ISO 527-4 standard by the water jet cutting process. The drawing of the test specimen is shown in Fig. 5.23. In order to prepare composite specimens with CNF particles, all previous steps were kept as same as before, but the following processes were followed before adding the hardener. First, epoxy resin was mixed with 0.25 wt% of CNF and stirred for 10 min at 2000 rpm and then the mixture was sonicated via 14 mm diameter probe

Fig. 5.23 Drawing of the test specimen (dimensions in mm).

162

Fatigue Life Prediction of Composites and Composite Structures

160

Sonication time (min)

140 120 100 80 60 40 20 0 0.0

0.2

0.6 0.4 0.8 Filler Content (wt%)

1.0

1.2

Fig. 5.24 The sonication time vs filler content (wt%) [71].

sonicator (Hielscher UP400S) at an output power of 200 W and 12 kHz frequency. The approach was used to disperse the CNF in bisphenol A-based thermosetting epoxy resin. Time for sonication depends on the filler contents and has been defined on the basis of experiments until fillers remain intact. In Fig. 5.24, the suitable time for sonication vs the filler contents is reported. It is worth mentioning that during the sonication, the mixture container was kept both by the aid of an ice bath to prevent the overheating of the suspension. The universal testing machine, STM-150 made by Santam Co., Iran was utilized to perform the tensile tests in accordance with DIN EN ISO 527–4 standard. The crosshead speed for the tensile test was set at 2 mm/min. For fatigue loading, servo hydraulics Instron 8802 uniaxial fatigue testing machine was used. Furthermore, a series of tests under static loadings were carried out to determine the tensile properties of fabricated composites as isotropic fiber-reinforced composites. The stress-strain behavior of the CSM/epoxy composite specimen is presented in Fig. 5.25. In order to evaluate the fiber weight fraction, the burn-off test was performed to obtain the glass fiber content. In order to maintain the statistical reliability, four samples were tested in each step. The mean values of the tensile strength and Young’s modulus of CSM/epoxy composites and 0.25 wt% CNF/CSM/epoxy hybrid nanocomposites are demonstrated in Tables 5.12 and 5.13. The tension-tension fatigue tests were conducted at different stress levels. The applied maximum stresses were chosen as different percentages of the ultimate tensile strength of the specimen. The fatigue tests were carried out under the load-control condition at a frequency of 2 Hz. During all fatigue tests, the stress ratio (minimum applied stress/maximum applied stress) was set at 0.1 under room temperature condition. The applied maximum stress and the number of cycles to failure from experiments for CSM/epoxy composites with 0.25 wt% CNF/CSM/epoxy hybrid nanocomposites are demonstrated in Tables 5.14 and 5.15.

Fatigue behavior of nanoparticle-filled fibrous polymeric composites

163

160 140

Stress (MPa)

120 100 80 60 40 20 0 0.00

0.01

0.02 Strain

0.03

0.04

Fig. 5.25 The stress-strain behavior of the CSM/epoxy composite specimen.

Table 5.12 Tensile strength and Young’s modulus of CSM/epoxy composites Mechanical properties Experimental Young’s modulus (GPa)

Model-estimated Young’s modulus (GPa)

Ultimate tensile strength (MPa)

E-glass CSM weight fraction (%)

8.9 0.2

8.82

1502.7

50.5

Table 5.13 Tensile strength and Young’s modulus of 0.25 wt% CNF/CSM/epoxy hybrid nanocomposites Mechanical properties Experimental Young’s modulus (GPa)

Model-estimated Young’s modulus (GPa)

Ultimate tensile strength (MPa)

E-glass CSM weight fraction (%)

11.8  0.2

11.91

183.5  2.7

50

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Fatigue Life Prediction of Composites and Composite Structures

Table 5.14 Number of cycles to failure for CSM/epoxy composites at R ¼ 0.1 Applied stress (MPa)

R0 (%) (Applied stress/UTS)×100

Mean number of cycles to failure (cycles)

150 105 94 90 75

100 70 63 60 50

1 1274 3577 25034 148467

Table 5.15 Number of cycles to failure for 0.25 wt% CNF/CSM/epoxy hybrid nanocomposites at R ¼ 0.1 Applied stress σ a (MPa)

R0 (%) (σ a/UTS)×100

Mean number of cycles to failure (cycles)

183.50 128.45 120.00 110.10 91.75

100 70 65 60 50

1 332 431 8270 47366

Moreover, the trends of normalized modulus degradation (dynamic modulus divided by static Young’s modulus) of CSM/epoxy composites during fatigue loading condition were monitored under R0 ¼ 50 % , 60 % , and 70% and shown in Figs. 5.26–5.28. By means of the normalization technique and obtained parameters as discussed in the previous sections, all different curves for different states of stress (R0 ¼ 50 % , 60 % , and 70%) collapse to a single curve as shown in Fig. 5.29. The obtained values of λ, γ are 2.473 and 9.07, respectively. For evaluating of the accuracy of derived equations for CSM/epoxy composites, according to Fig. 5.30, the trend of the stiffness reduction for CSM/epoxy composites vs the number of cycles was depicted and a good agreement with experimental data was obtained. This compatibility shows that obtained curve-fitting parameters are suitable for this composite. As similar to the previous steps, all experiments were followed again for 0.25 wt% CNF/CSM/epoxy composite under R0 ¼ 50% state of stress and the trend of the normalized modulus degradation (dynamic modulus divided by static Young’s modulus) of 0.25 wt% CNF/CSM/epoxy composites during fatigue loading condition was monitored (Fig. 5.31). By means of the obtained parameters value of λ, γare 2.473 and 9.07, the stiffness vs the number of cycles for 0.25 wt% CNF/CSM/epoxy composite are found as shown in Fig. 5.32. It is observed that a vital agreement with the experimental and predicted results by Nano-NSDM exists. The observed error (%) is represented in Table 5.16.

Fatigue behavior of nanoparticle-filled fibrous polymeric composites

165

1.02

Normalized modulus (E/Es)

1 0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84

R' = 50% 0

20,000

40,000

60,000 n (cycles)

80,000

100,000

120,000

Fig. 5.26 The normalized modulus (E/Es) vs number of cycles of CSM/epoxy composites, R0 ¼ 50, σ a ¼ 75 MPa, and Fr ¼ 2 Hz. 1.06

Normalized modulus (E/Es)

1.04 1.02 1 0.98 0.96 0.94 0.92 0.9

R' = 60%

0.88 0.86

1

10

100 n (cycles)

1000

10,000

Fig. 5.27 The normalized modulus (E/Es) vs the number of cycles of CSM/epoxy composites, R0 ¼ 60, σ a ¼ 90 MPa, and Fr ¼ 2 Hz.

The bottleneck of Nano-NSDM is that in each state of stress, at least one experiment is needed. In order to solve this restriction, by choosing another stress state such as R0 ¼ 70, the S-N curve can be experimentally obtained. Then, without performing new tests, the model is able to predict the number of cycle to failure in each stress state and the model can be applied without any limitation (see Fig. 5.33). As

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Fatigue Life Prediction of Composites and Composite Structures

Normalized modulus (E/Es)

1.02 1.01 1 0.99 0.98 0.97 0.96 0.95

R' = 70%

1

10

100

1000

n (cycles)

Fig. 5.28 The normalized modulus (E/Es) vs the number of cycles of CSM/epoxy composites, R0 ¼ 70, σ a ¼ 105 MPa, and Fr ¼ 2 Hz.

1 0.9

Normalized stiffness

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

R' = 50%, 60%, 70% 0

0.1

0.2

0.3 0.6 0.4 0.7 0.5 Normalized number of cycles

0.8

0.9

1

Fig. 5.29 The normalized stiffness degradation curve CSM/epoxy composites, R0 ¼ 50 % , 60 % , and 70%, Fr ¼ 2 Hz, λ ¼ 2.473 , γ ¼ 9.07, and εf ¼ 0.017.

Fatigue behavior of nanoparticle-filled fibrous polymeric composites

167

Fig. 5.30 The verification of the stiffness reduction for CSM/epoxy composite, R0 ¼ 50, σ a ¼ 75 MPa, and Fr ¼ 2 Hz.

Normalized modulus (E/Es)

1.2 1 0.8 0.6 0.4 R' = 50%

0.2 0

0.25 wt% CNF/CSM/Epoxy composites

1

10

100

1000

10,000

100,000

n (cycles)

Fig. 5.31 The normalized modulus (E/Es) vs number of cycles of 0.25 wt% CNF/CSM/epoxy composite, R0 ¼ 50, σ a ¼ 92 MPa, and Fr ¼ 2 Hz.

previously mentioned, based on the experimental observations, it is necessary to state that the average strain to failure (εf) can be considered to be a constant and independent on the state of stress and number of cycles. For instance, as depicted in Fig. 5.33, the S-N curve for CSM/epoxy composite and 0.25 wt% CNF/CSM/epoxy composites are obtained and the number of cycles to failure can be anticipated in each applies stress, generally. The material constants, A and B are as shown in Table 5.17.

168

Fatigue Life Prediction of Composites and Composite Structures

11 R' = 50%

10.5

Effib, nc (GPa)

10 9.5 9 8.5 8

Experimental results, 92 MPa The Nano-NSDM based on the Halpin-Tsai model, 92 MPa

7.5 7

10,000

0

20,000

30,000

40,000

50,000

n (cycles)

Fig. 5.32 The verification of stiffness reduction for 0.25 wt% CSM/epoxy composite, R0 ¼ 50, σ a ¼ 92 MPa, and Fr ¼ 2 Hz.

Table 5.16 Results and value of error for Nano-NSDM for 0.25 wt% CSM/epoxy composite, R0 ¼ 50, σ a ¼ 92 MPa, Fr ¼ 2 Hz

n (cycles)

Ec(n): Experimental results Effib, c (GPa)

Nano-NSDM based on The Halpin-Tsai model Effib, c (GPa)

Error (%)

1000 2500 5000 7500 10000 20000

10.46 10.21 10.01 9.87 9.83 9.48

10.23 10.09 9.95 9.84 9.76 9.45

2.25 1.14 0.62 0.32 0.69 0.34

5.3

Fatigue life prediction based on the micromechanicalenergy method

In this section, a novel method is developed in order to predict the fatigue life of nanoparticle/CSM/polymer hybrid nanocomposites. The established model will be a combination of the micromechanics and the energy method. The special feature of the present model is the capability of predicting the fatigue life of hybrid nanocomposites by means of the experimental fatigue data of the same composites without adding any nanoparticles. A survey of the literature shows the energy method has not been fully developed for CSM composites. Thus, in this section, a novel model based on the

Fatigue behavior of nanoparticle-filled fibrous polymeric composites

169

Applied stress (MPa)

200 180

0.25 wt% CNF/CSM/Epoxy nanocomposites

160

CSM/Epoxy nanocomposites

140 120 100 80 60 40 20 0 1

10

100

1000

10,000

100,000

1,000,000

Number of cycles to failure, Nf

Fig. 5.33 The S-N curve for CSM/epoxy composite and 0.25 wt% CNF/CSM/epoxy composites R0 ¼ 50 % , 60 % , and 70%, Fr ¼ 2 Hz, and εf ¼ 0.017 Table 5.17 A and B material constants for Neat CSM/epoxy composite and 0.25 wt% CNF/CSM/epoxy composites σ Applied 5A ln(Nf) +B

A

B

Neat CSM/epoxy composites 0.25-wt%-CNF/CSM/epoxy composites

148.25 185.06

6.187 8.147

micromechanics and the energy method is developed in order to predict the fatigue life of CSM polymeric composites. The fatigue life of nanoparticle/CSM/polymer hybrid nanocomposites can also be predicted by the present model. Later, a series of tests in tension-tension fatigue conditions at room temperature for CSM/epoxy composites and for 0.1 wt%, and 0.25 wt% CNF/CSM/epoxy hybrid nanocomposites are carried out at different load levels to evaluate the capabilities of the present model [72].

5.3.1 The energy method The energy method is a simple approach and can be used for laminated composites with different fiber orientations and loading conditions. Kachanov, Lemaitre, and Ladeveze [73–75] presented the relationship between the strain energy density and the damage in composites. Ellyin and El-Kadi [76, 77] used the strain energy density as a damage function for composite materials. A major portion of the useful life of a composite structure component involves subcritical damage accumulation, which was

170

Fatigue Life Prediction of Composites and Composite Structures

finally manifested in various combinations of the matrix cracking, fiber-matrix debonding, delamination, and fiber breakage failure modes. A precise characterization of a composite material would require knowledge of the way the energy dissipates throughout the inhomogeneous structure as damage is being accumulated [76]. The strain energy density is a parameter which can be related to this damage process [77]. Shokrieh and Taheri [78] developed a unified fatigue life model based on the energy method for unidirectional polymer composite laminates. Their proposed model is capable of predicting the fatigue life of unidirectional composite laminates over a range of positive stress ratios in various fiber orientation angles. It was shown that the total input energy was directly related to fatigue life and can be expressed using a power law relation as follows:
 ΔW ¼ g Nf ! ΔW ¼ kN αf

(5.23)

where k and α are materials constants and ΔW is the total input energy and Nf is the fatigue life (the number of cycles to failure). For the elastic plane stress condition, the strain energy density can be expressed in terms of stresses and strains as follows: W¼

 1
σ x εx + σ y εy + τxy γ xy 2

(5.24)

For the uniaxial loading condition: h i ΔW ¼ Sxx Δσ 2x =2ð1  Rx Þ2

(5.25)

where Δσ indicates the stresses range: Δσ ¼ σ max  σ min

(5.26)

Rx (the stress ratio) and Sxx (the compliance component) are defined as below: max Rx ¼ σ min , Sxx ¼ x =σ x

1 Ex

(5.27)

5.3.2 Modeling strategy (Nano-EFAT model) A schematic framework of the modeling strategy is shown in Fig. 5.34. The model is an integration of two major components: the micromechanical (such as Halpin-Tsai model) and the energy method and called Nano-EFAT model. The novel model is able to predict the fatigue life of nanoparticle/CSM/polymer hybrid nanocomposites. By coupling of the energy method for composite specimens without nanoparticles and the micromechanical Halpin-Tsai approach, a comprehensive model for nanoparticle/CSM/polymer hybrid nanocomposites under fatigue loading was developed. As shown in Fig. 5.34, in the first step, according to the micromechanical model (Eq. 5.11), the stiffness of the nanocomposites (EsCSM, nc) is obtained by using the

Fatigue behavior of nanoparticle-filled fibrous polymeric composites

Stiffness of pure epoxy resin

Stiffness of nanoparticles

Stiffness of CSM

171

Fig. 5.34 The schematic flowchart of the Nano-EFAT model [72].

Micromechanical Models

Equivalent stiffness of nanoparticle/CSM/polymer hybrid nanocomposites

Energy model for composites under fatigue loading condition without nanoparticles

Fatigue life of nanoparticle/CSM/polymer hybrid nanocomposites

moduli of the pure matrix (Esm), nanoparticles (Esp), and CSM (EsCSM) measured under static loading conditions. Later, the material constants (i.e., α and k) as experimental curve-fitting parameters of composite specimens without nanoparticles are needed under tension-tension fatigue loading condition in order to find the fatigue life of nanoparticle/CSM/polymer hybrid nanocomposites. According to the inherent property of CSM composites like an isotropic material, stiffness and ΔW can be simplified and able to estimate the fatigue life of nanoparticle/ CSM/polymer hybrid nanocomposites by having the applied stress Δσ and the stress ratio R [72]. ΔW ¼

Δσ 2 2  ð1  RÞ2  EsCSM,nc

¼ kN f α

(5.28)

5.3.3 Tests results A series of tests under static loadings are carried out to determine the tensile properties of fabricated composites as isotropic reinforced composites. In order to maintain the statistical reliability, four samples were tested in each step. The mean values of the tensile strength and Young’s modulus of CSM/epoxy composites and 0.1 wt%

172

Fatigue Life Prediction of Composites and Composite Structures

Table 5.18 The tensile strength and Young’s modulus of 0.1 wt% CNF/CSM/epoxy hybrid nanocomposites [72] Mechanical properties Experimental Young’s modulus (GPa)

Model-estimated Young’s modulus(GPa)

Ultimate tensile strength (MPa)

E-glass CSM weight fraction (%)

10.8 0.2

10.6

160.5  2

50

CNF/CSM/epoxy hybrid nanocomposites are demonstrated in Tables 5.12, 5.13, and 5.18 [72]. The tension-tension fatigue tests were conducted at different stress levels. Applied maximum load levels were chosen to develop maximum stresses at different percentages of the ultimate tensile stress of the specimen. The fatigue tests were carried out under the load-control condition at a frequency of 2 Hz. During all fatigue tests, the stress ratio (minimum applied stress divided by maximum applied stress) was set at 0.1 under room temperature condition. The applied maximum stress and the number of cycles to failure from experiments for neat CSM/epoxy composites and reinforced with 0.25 wt% and 0.1 wt% CNF particles are demonstrated in Tables 5.14, 5.15, and 5.19, respectively. By comparing Tables 5.14 and 5.15, it can be found about 300% increase in fatigue life of 0.25 wt% CNF/CSM/epoxy hybrid in some cases by adding 0.25 wt% CNF to CSM/epoxy composites. Also, by comparing Tables 5.14 and 5.19, it can be found about 150% increase in fatigue life of 0.1 wt% CNF/CSM/epoxy hybrid in some cases by adding 0.1 wt% CNF to CSM/epoxy composites.

5.3.4 Evaluation of the Nano-EFAT model In this section, the capability of the present model has been evaluated by experimental results. In the first step, the present model was evaluated by the experimental data of the CSM/epoxy composites. In the next step, the model was evaluated by the experimental data of 0.1 wt% and 0.25 wt% CNF/CSM/epoxy hybrid nanocomposites. Table 5.19 Number of cycles to failure for 0.1 wt% CNF/CSM/epoxy hybrid nanocomposites at R ¼ 0.1 [72] Applied stress (MPa)

Mean number of cycles to failure (cycles)

160.50 120.40 100.20 89.70

1 224 4804 31182

Fatigue behavior of nanoparticle-filled fibrous polymeric composites

173

In order to use the model, material constants α and k are needed to be calculated. By using the test data of the CSM/epoxy composite specimen without adding any nanoparticle (Table 5.14), applying them to Eq. (5.28) and by using a simple curve fitting (Fig. 5.35), α and k are obtained (Table 5.20). The modulus of nanocomposites (EsCSM, nc) was calculated using the Halpin-Tsai model equal to 10.6 GPa for 0.1 wt% and 11.91 GPa for 0.25 wt% CNF/CSM/epoxy hybrid nanocomposites. Then, material constants α, k (shown in Table 5.20) and s ECSM, nc were used in Eq. (5.28) and the results were plotted in Fig. 5.35. This figure presents a comparison of the results obtained by the present model with experimental results (Tables 5.15 and 5.19) for 0.1 wt% and 0.25 wt% CNF/CSM/epoxy hybrid nanocomposites. The capability of the present model in the simulation of the fatigue life of the 0.1 wt% and 0.25wt% CNF/CSM/epoxy composites is clearly shown in Fig. 5.35.

5.4

Displacement-controlled flexural fatigue behavior of composites with nanoparticles

The flexural fatigue behavior of composites and nanocomposites has been carried out by many researchers [79–82]. For composites under the displacement-controlled

Fig. 5.35 Obtaining α and k using a curve-fitting method for CSM/epoxy composites and evaluation of the present model with experimental data for 0.1 wt% and 0.25 wt% CNF/CSM/ epoxy hybrid nanocomposites [72]. Table 5.20 Material constants α and k for CSM/epoxy composites Parameter

Value

α k

0.1135 1.2980  106

174

Fatigue Life Prediction of Composites and Composite Structures

fatigue loading, Paepegem and Degrieck [79] developed an experimental setup for bending fatigue test. They adopted a residual stiffness model which describes the fatigue damage behavior of the composite material [80]. Also, Paepegem et al. [81] used a finite element approach for composites fatigue life prediction. El Mahi et al. [82] studied the flexural fatigue behavior of the sandwich composite materials using three-point bend test and the derived approach permitted them to predict the fatigue life of the sandwich composite materials while avoiding a large number of experiments, normally required in fatigue testing. A survey in the available literature reveals that the addition of nanoparticles can improve the fatigue behavior of composites under displacement control loading and has been carried out by many researchers [83–86]. Ramkumar and Gnanamoorthy [83] studied the stiffness and flexural fatigue life improvements of polymer-matrix reinforced nanocomposites with nanoclay. They described the effect of adding nanoclay fillers on the flexural fatigue response of PA6. Rajeesh et al. [84] considered the influence of humidity on the flexural fatigue behavior of commercial grade PA6 granules and hectorite clay nanocomposites. Timmaraju et al. [85] considered the influence of the environment on the flexural fatigue behavior of polyamide 66/hectorite nanocomposites. They also found the effect of initial absorbed moisture content on the flexural fatigue behavior of polyamide 66/hectorite nanocomposites conducted under deflection control method using a custom-built, table-top flexural fatigue test rig at a laboratory condition [86]. A survey in the literature also reveals that the presence of multi-nanoparticles in composites improves the properties of nanocomposites. Some researchers used hybrid fillers in order to have a perfect potential of both fillers. For instance, as a first group, a combination of microrubber and NS has been used to improve the fracture toughness and fatigue behavior of [87–91]. Liang and Pearson [87] used two different sizes of NS particles, 20 and 80 nm in diameter, and carboxyl-terminated butadiene acrylonitrile (CTBN) which was blended into a lightly cross-linked, DGEBA/piperidine epoxy system in order to investigate the toughening mechanisms. It was shown that the addition of a small amount of NS particles into CTBN caused an increase in the fracture toughness. Manjunatha et al. [19, 26, 88–91] investigated the fatigue behavior of reinforced composites by adding a combination of micro rubber and NS particles into the epoxy matrix in several states. For instance, they [19] studied the tensile fatigue behavior of modified micron-rubber and NS particle epoxy polymers. They [26] also addressed the tensile fatigue behavior of a glass-fiber-reinforced plastic (GFRP) with the participation of rubber microparticles and silica nanoparticles. They [88] also observed the enhanced capability to withstand longer crack lengths, due to the improved toughness together with the retarded crack growth rate, to enhance the total fatigue life of the hybridmodified epoxy polymer. Also, Manjunatha et al. [89] enhanced the fatigue behavior of fiber-reinforced plastic composites by means of 9 wt% of rubber microparticles and 10 wt% of silica nanoparticles and showed the fatigue life under WISPERX load sequence was about 4–5 times higher than that of the neat composites. Manjunatha et al. [90] also used another hybridization of carboxyl-terminated butadieneacrylonitrile rubber microparticles and silica nanoparticles to increase the tensile fatigue behavior of GFRP composites at a stress ratio equal to 0.1. Manjunatha et al. [91] conducted the fatigue crack growth test on a thermosetting epoxy polymer

Fatigue behavior of nanoparticle-filled fibrous polymeric composites

175

which was hybrid modified by incorporating 9 wt% of CTBN rubber microparticles and 10 wt% of silica nanoparticles. The fatigue crack growth rate of the hybrid epoxy polymer was observed to be significantly lower than that of the unmodified epoxy polymer. In the next category, applying CNTs with different nanoparticles as hybrid fillers were taken into account in the literature [23, 92–94] to improve the fatigue behavior, mechanical and electrical properties of reinforced composites. Boeger et al. [23] used silica and MWCNT hybrid nanoparticles to increase the high-cycle fatigue life of epoxy laminates and finally reported that the life was increased by several orders of magnitude in a number of load cycles. Fritzsche et al. [92] investigated the CNT-based elastomer-hybrid-nanocomposites prepared by melt mixing and showed promising results in electrical, mechanical, and fracture-mechanical properties. Witt et al. [93] improved mechanical properties such as tensile strength and strain to failure of a conductive silicone rubber composite using both CNTs and carbon black (CB). Al-Saleh and Walaa Saadeh [94] fabricated nanostructured hybrid polymeric materials based on CNTs, CB, and CNFs and investigated electrical properties and electromagnetic interference shielding effectiveness in the X-band frequency range. The other various hybrid nanoparticles were discussed in the literature are considered here as the last category [95, 96]. Jen et al. [95] applied hybrid magnesium/carbon fiber to increase the fatigue life of nanocomposite laminates. On the other hand, applying CNT and graphite nanoplatelets (GNPs) to epoxy nanocomposites was conducted by Jing Li et al. [96]. It was represented that the flexural mechanical and the electrical properties of the NR were marginally changed by the hybridization. The present survey reveals that the effect of hybrid particles is mostly positive and can improve the static and dynamic properties of composites. However, it is figured out that in case of displacement control fatigue loading condition, there is a lack of research on the hybrid nanofillers/epoxy nanocomposites. Therefore, in the chapter, the flexural fatigue behavior of graphene/carbon-nanofiber/epoxy hybrid nanocomposites under displacement control flexural loading is investigated and compared with those of the pure epoxy resin [97].

5.4.1 Materials specification In this section, ML-526 epoxy resin as matrix and CNF as reinforcement nanoparticles were selected. Moreover, the graphene nanoplatelets (GPL) were synthesized by a stirring grinding driven by changing the magnetic field as shown in Fig. 5.36 [97]. Physical properties of synthesized graphene powders are shown in Table 5.21. The TEM image of the synthesized GPL powder is shown in Fig. 5.37. The D, G, and 2D bands of Raman spectra of the synthesized GPLs powder are demonstrated in Fig. 5.38.

5.4.2 Specimen preparation The polymer reinforced with 0.5 wt% of hybrid graphene/carbon-nanofiber was prepared as described below. First, epoxy resin was mixed with 0.25 wt% CNF and stirred for 10 min at 2000 rpm and then the mixture was sonicated via 14 mm diameter probe

176

Fatigue Life Prediction of Composites and Composite Structures

Fig. 5.36 GPL synthesis method. (A, B) Still condition and (C) moving condition [97].

Table 5.21 GPL nanoparticles specifications Nanoparticle

Diameter (nm)

Thickness (nm)

Specific surface area (m2/g)

GPL

40–120

3–5

500

Fig. 5.37 The transmission electron microscopy (TEM) of the synthesized graphene nanoplatelets [97].

sonicator (Hielscher UP400S) at an output power of 200 W and 12 kHz frequency. The mixture was sonicated for 60 min. It is worth mentioning that during the sonication, the mixture container was kept with the aid of ice bath to prevent the overheating of the suspension to keep the temperature around 40°C. Second, the suspension was mixed with 0.25 wt% GPL under the same condition within 30 min by the sonication. After sonication, the hardener at a ratio of 15:100 was added to the mixture and stirred gently for 5 min. Then, it was vacuumed at 1 mbar for 10 min to remove any trapped air. Six samples were prepared and cured at room temperature for 48 h and followed by 2 h at 80°C and 1 h at 110°C for post-curing. The approach was used to disperse GPL/CNF hybrid nanoparticles into epoxy resin, is adapted from a combination of supplementary research [98]. Time for the sonication depends on the filler contents

Fatigue behavior of nanoparticle-filled fibrous polymeric composites

177

350 300

1339.4 cm–1

1580.7 cm–1

Intensity (a.u.)

250 2677.0 cm–1

200 150 100 50 0

1200

1600

2000

2400

2800

3200

–1

Wave number (cm )

Fig. 5.38 Raman spectra of synthesized graphene nanoplatelets, D, G, and 2D bands [97].

and has been defined based on the experiments until fillers remain intact. For CNF fillers, Shokrieh et al. [98] investigated the suitable time for sonication vs contents of the filler and pointed out for 0.25 wt% CNF materials, the optimum value of sonication with regard to Fig. 5.39 was found around 90 min with the same compartment and conditions. Also, the optimum sonication time for 0.25 wt% GPL was equal to 30 min. In addition, to inspect the dispersion state of nanofillers, a new technique based on SEM, which utilizes the burn-off test, was introduced to visualize the dispersion state of nanofillers [99].

5.4.3 Calculation of the bending stress In this section, high-cycle fatigue properties of nanocomposites are measured by a modified cantilever beam bending test. A typical fatigue life test specimen for the cantilever beam bending test is shown in Fig. 5.40. The specimen was designed based on ASTM: B593-96 standard and the method presented by Ramkumar and Gnanamoorthy [83]. The wide end of the specimen was clamped to a bed plate, while the narrow end is cyclically deflected (see Fig. 5.40A). To catch reliable results of the flexural fatigue strength, the gage area of the specimen is designed based on the stress concentration concept (Fig. 5.40B). The stress concentration of the critical location of the specimen has the maximum magnitude; therefore, the failure will start in this area. For the wedge-shaped beam as

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Fatigue Life Prediction of Composites and Composite Structures

Fig. 5.39 The viscosity (mPa.s) vs sonication time (min) of 0.25 wt% CNF/epoxy nanocomposites [98].

an applied specimen, the cross section is not uniform and defined by means of a parameter called “local B” according to Eq. (5.29) (Fig. 5.41): BðxÞ ¼

B0 ðL0  xÞ L0

(5.29)

where L0 is the length of the specimen and B0 is the width at the base of the wedgeshaped beam. Therefore, the magnitude of the second moment of area of the cross section depends on the position along the x-axis as Eq. (5.30): I ðxÞ ¼

B0 ðL0  xÞH3 12L0

(5.30)

where H is the thickness of the beam. Finally, the maximum tension or compression stress at a given cross section for small displacements within elastic deformation behavior is calculated according to the following equation [100]: σ max ¼ 

z0  E  H L0 2

(5.31)

where σ max is the maximum stress, H is the thickness of the beam, z0 is the displacement at point x¼ L0, and E is Young’s modulus. The relation between the displacement z0 at the tip and the maximum stress σ max for a small deformation is linear.

Upper clamp

Bolt

Specimen

Lower clamp

Cyclic displacement

(A) 41

44.1

Span=30

9.44

10.4

(B)

23.9

Fig. 5.40 (A) A schematic of specimen clamping procedure. (B) A schematic picture of the bending fatigue specimen [97].

B0

Fig. 5.41 A schematic view of a beam, with coordinates [97].

F H

L0

M+

x z

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Fatigue Life Prediction of Composites and Composite Structures

5.4.4 Test equipment 5.4.4.1 Static testing instruments The Santam universal testing machine STM-150 was utilized to perform bending tests in accordance with the ASTM D790 standard. The cross-head speed for bending tests was 16 mm/min. To analyze hybrid nanoparticles, gold sputtered samples were used. The field-emission scanning electron microscopy (FESEM) photographs were taken by using Zeiss-Germany Sigma microscope.

5.4.4.2 Experimental setup for flexural bending fatigue The pure epoxy and reinforced polymer specimens are mounted into a fixed cantilever, constant deflection type fatigue testing machine. The machine called BFM-110 was designed and manufactured based on a developed version of a testing machine designed by Paepegem and Degrieck [79] and shown in Fig. 5.42. The specimen was held at one end, acting as a cantilever beam and cycled until a complete failure was achieved. The number of cycles to failure was recorded as a measure of the fatigue life during the test. Generally, the shaft of the motor has a rotational speed of 0–1450 rpm. The power is transmitted via a V-belt to the second shaft, provides a fatigue testing frequency between 2 and 20 Hz and gives the possibility to investigate the influence of the frequency in this range of values. The power transmission through a V-belt ensures the motor and the measuring system are electrically isolated. The second shaft bears a crank-linkage mechanism, as shown in Fig. 5.42. Hence, the sample is loaded as a cantilever beam. The amplitude of the imposed displacement is a controllable parameter and the adjustable crank allows choosing between single-sided and fully reversed bending, that is, the deflection can vary from zero to a maximum deflection in one direction, or in two opposite directions, respectively. The maximum deflection was measured by a displacement dial gauge at the back of the lower clamp. The number of cycles to failure should be counted directly for each test specimen. A counting signal was generated once per cycle by a PES-R18PO3MD reflector speed sensor which was supplied by IBEST Electric, LTD., China. The counting signal was transferred to the counter fabricated by RASAM Madar electronic Co., Iran. In this setup, there are two parallel stands with counting system implemented separately. To stop the counter of speed sensors, at the bottom of each specimen a thin wire as an electrical contact was used and after failure, the damaged specimen drops down and disconnects the wire and stops the counting system. Therefore, after the failure of both specimens, control system acts and turns off the main current of the machine completely.

5.4.5 Results and discussion 5.4.5.1 Static bending strength The calculation of the maximum bending stress was performed using Eq. (5.31). This expression was valid when the specimen is subjected to a small and linear deformation. The static bending strength of GPL/epoxy nanocomposites for 0.25 wt% of GPL content was found 118 MPa and it was 121 MPa for 0.25 wt% CNF/epoxy

Fatigue behavior of nanoparticle-filled fibrous polymeric composites

181

Main frame

V-belt

Counter Linkage

Speed sensor

Frequency inverter

Dial gauge Nanocomposites specimen

Strain gauge

Connection wire

Fig. 5.42 The experimental setup for the displacement controlled flexural bending fatigue loading [97].

nanocomposites [98]. Also, the static flexural modulus of 0.25 wt% graphene/epoxy nanocomposites was demonstrated at 3.4 GPa and for 0.25 wt% of CNF content was 3.18 GPa [98]. While, the static bending strength and modulus of the neat epoxy resin were 110 MPa and 3 GPa, respectively. For 0.25 wt% of GPL plus 0.25 wt% of CNF (i.e., 0.5 wt% of GPL/CNF) hybrid nanoparticles/epoxy nanocomposites based on the ASTM D790 standard, static tests to measure the flexural strength and stiffness have been conducted and eventually, 123 MPa for the strength and 3.43 GPa for the stiffness were found. The flexural strength and stiffness were presented in Figs. 5.43 and 5.44.

182

Fatigue Life Prediction of Composites and Composite Structures 130

Flexural strength (MPa)

120

110

123

121 118

110

100

90

80

Pure epoxy resin

0.25 wt.% CNF

0.25 wt.% GPL 0.5 wt.% Hybrid

Fig. 5.43 The flexural strength (MPa) for pure epoxy resin, 0.25 wt% GPL, 0.25 wt% CNF, and 0.25 wt% of GPL plus 0.25 wt% of CNF hybrid nanoparticles epoxy nanocomposites [97].

3.6 3.40

Flexural stiffness (GPa)

3.4 3.18

3.2 3.0

3.43

3.00

2.8 2.6 2.4 2.2 2.0 Pure epoxy resin

0.25 wt.% CNF

0.25 wt.% GPL 0.5 wt.% Hybrid

Fig. 5.44 The flexural stiffness (GPa) for pure epoxy resin, 0.25 wt% GPL, 0.25 wt% CNF and 0.25 wt% of GPL plus 0.25 wt% of CNF hybrid nanoparticles epoxy nanocomposites [97].

Fatigue behavior of nanoparticle-filled fibrous polymeric composites

183

5.4.5.2 Cyclic flexural bending fatigue life There is not a special standard test method for epoxy matrix and epoxy-based nanocomposites under flexural bending stress in fatigue. The ASTM: B593-96 standard and a publication of Ramkumar and Gnanamoorthy [83] are used for copper alloy spring material, filled thermoplastic nanocomposites, respectively. In the experiments presented in this chapter, the loading frequency was 5 Hz. The effective length of the specimen subjected to the bending is 32.84 mm. The drawing and picture of the specimen used in the current research are shown in Fig. 5.45. Although the BFM-110 testing machine is capable of applying the reversal bending fatigue loading, however, in the present study, the specimens were subjected to zero-bending fatigue loading conditions. The flexural bending stress vs the number of cycles for the neat epoxy resin, 0.25 wt% GPL nanoparticles, 0.25 wt% CNF nanoparticles, and 0.5 wt% of GPL/CNF hybrid nanoparticles epoxy nanocomposites, at a frequency equal to 5 Hz is illustrated in Fig. 5.46. The strength ratios (the bending stress normalized by the bending strength) vs the number of cycles to failure is presented in Fig. 5.47. For instance, the experimental observations show that at the strength ratio equal to 43% by using 0.5 wt% of hybrid nanoparticles; 37.3-fold improvement in flexural bending fatigue life of the neat epoxy was observed. While, enhancement of adding only graphene or CNF was 27.4-and 24-folds, respectively. The enhancement of the fatigue life for composites in the presence of nanofillers has been stated in the literature. For instance, Ramkumar and Gnanamoorthy [83] expressed that the nanoclay addition can be attributed to enhanced modulus coupled with reduced dissipation factor and improved surface hardness. The fibrillated 50.8

R2

10.4

R8 .2

2.84

23.9

0 R1

.85

17

9.44

50°

Fig. 5.45 Drawing of the specimen (dimensions in mm) [97].

30

81.8

41.4

t = 3.5

184

Fatigue Life Prediction of Composites and Composite Structures

60.0 Neat epoxy resin

58.0

0.25 %wt GPL / Epoxy nanocomposites 0.25 %wt CNF / Epoxy nanocomposites

Flexural stress (MPa)

56.0

0.25 wt% GPL + 0.25 wt% CNF / Epoxy hybrid nanocomposites

54.0 52.0 50.0 48.0 46.0 44.0 42.0 1000

10,000

100,000

1,00,0000

Number of cycles to failure

Fig. 5.46 Bending stress vs the number of cycles to failure for neat epoxy resin, 0.25 wt% GPL, 0.25 wt% CNF, and 0.25 wt% of GPL plus 0.25 wt% of CNF hybrid nanoparticles epoxy nanocomposites, at a frequency equal to 5 Hz [97].

appearance of the filled nanocomposite fracture surface suggests that the addition of the nanofiller promotes the toughening and influences the crack propagation characteristics of the pure polymer. Ramanathan et al. [101] argued that the micrometer-size dimensions, high aspect ratio and 2D sheet geometry of graphene make them effective in deflecting cracks in bending. In addition, hydrogen-bonding interaction or an enhanced nanofiller-polymer mechanical interlocking due to the wrinkled morphology of graphene is additional factors that can contribute to composite reinforcement. Whilst, Rafiee et al. [102] pointed out that this enhancement may be related to their high specific surface area, enhanced nanofiller-matrix adhesion/interlocking arising from their wrinkled (rough) surface.

5.4.5.3 Dispersion and morphology analysis The observation of the fracture surfaces of the damaged specimens can explain the improvement of the fatigue life of the hybrid nanocomposites. The fracture surfaces of specimens 0.25 wt% GPL/epoxy, 0.25 wt% CNF/epoxy, and 0.5 wt% of GPL/CNF hybrid nanoparticles epoxy nanocomposites are presented in Fig. 5.48. As depicted in Fig. 5.48A, dispersion of GPL was observed and shown that the dispersion was not

Fatigue behavior of nanoparticle-filled fibrous polymeric composites

185

50 Neat epoxy resin 0.25 %wt GPL / Epoxy nanocomposites

48

0.25 %wt CNF / Epoxy nanocomposites

Flexural strength ratio (%)

0.25 wt% GPL + 0.25 wt% CNF / Epoxy hybrid nanocomposites

46

44

42

40

38 1000

10,000

100,000

1,000,000

Number of cycles to failure

Fig. 5.47 The flexural stress ratio (%) vs the number of cycles to failure for neat epoxy resin, 0.25 wt% GPL, 0.25 wt% CNF, and 0.25 wt% of GPL plus 0.25 wt% of CNF hybrid nanoparticles epoxy nanocomposites, at a frequency equal to 5 Hz [97].

sufficient and the stiffness of reinforced composites with GPL improved and the strength was not influenced like the stiffness. Also, as shown in Fig. 5.48B, the fracture surface of 0.25 wt% CNF/epoxy nanocomposites was monitored and found that the dispersion of CNF into epoxy resin was appropriate as observed and was efficient for improving mechanical properties. Fig. 5.48C and D shows that a combination of both nanofillers in the fractured surface and dominant failure mechanism is pullout which leads to a higher strength for nanocomposites. The GPL increased the stiffness of the nanocomposites and the pullout of the CNT increases the strength. Therefore, hybridization of these nanoparticles promotes the toughening and influences the crack propagation characteristics of the pure polymer, which in turn causes a significant improvement in fatigue life of the nanocomposites. In concluding this section, the effect of adding hybrid nanoparticles into epoxy resin was investigated and it was found that hybrid particles can improve the static and dynamic properties of composites. For 0.25 wt% of GPL plus 0.25 wt% of CNF (i.e., 0.5 wt% of GPL/CNF) hybrid nanoparticles/epoxy nanocomposites was achieved 123 MPa. While the static bending strength of GPL/epoxy nanocomposites for 0.25 wt% of GPL content was found 118 and 121 MPa for 0.25 wt% CNF/epoxy nanocomposites. The flexural fatigue behavior of graphene/carbon-nanofiber hybrid nanocomposites under displacement control flexural loading was investigated and

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Fatigue Life Prediction of Composites and Composite Structures

Fig. 5.48 FESEM of the fractured surface, (A) 0.25 wt% GPL/epoxy nanocomposites, (B) 0.25 wt% CNF/epoxy nanocomposites, and (C) and (D) 0.5 wt% of GPL/CNF hybrid nanoparticles epoxy nanocomposites [97].

results were compared with pure epoxy resin, pure epoxy resin with the presence of GPL or CNF nanofillers. Due to the addition of hybrid nanoparticles, a remarkable improvement in fatigue life of epoxy resin was observed in comparison with results obtained by adding 0.25 wt% graphene or 0.25 wt% CNF into the resin. Also, the strength ratio (the bending stress normalized by the bending strength) vs the number of cycles to failure for the neat epoxy resin, 0.25 wt% GPL nanoparticles, 0.25 wt% CNF nanoparticles, and 0.5 wt% of GPL/CNF hybrid nanoparticles epoxy nanocomposites, have been investigated. The experimental observations show that at the strength ratio equal to 43% by using 0.5 wt% of hybrid nanoparticles; 37.3-fold improvement in flexural bending fatigue life of the neat epoxy was observed. While, enhancement of adding only graphene or CNF was 27.4- and 24-folds, respectively. The improvement of the fatigue life of hybrid nanocomposites can be explained by a closed look at the fracture surface. The GPL increased the stiffness of nanocomposites and the pullout of the CNT increases the strength. Therefore, hybridization of these nanoparticles promotes the toughening and influences the crack propagation characteristics of the pure polymer which in turn causes a significant improvement in fatigue

Fatigue behavior of nanoparticle-filled fibrous polymeric composites

187

life of nanocomposites. In addition, based on the literature, the addition of the nanofiller promotes toughening and influences the crack propagation characteristics of the polymer without nanoparticles. Also, they attribute to enhance the modulus coupled with a reduced dissipation factor and improved the surface hardness [97].

5.5

Conclusions and outlook

To conclude this section, it needs to be stressed that using nanoparticles in most of the industries have positive attraction and opened a contemporary world as like as an industrial revolution for designers to prevent sudden failure due to fatigue. Nanofillers are playing as reinforcement materials in polymeric matrices and changing the border of industrial applications dramatically. For instance, nanofillers inside polymer matrix composites are comprising a majority of aerospace applications in structures, coating, tribology, structural health monitoring, electromagnetic shielding, and shape memory applications [103]. Another sample can be addressed on engine mounts which suffer from fluctuating forces result in sudden failure due to fatigue and nanoparticles can compromise the border [104]. Therefore, modeling this phenomenon is always attractive for researchers and has a vast prospect. In this chapter, after a deep review of the available models for predicting composites fatigue life, a fatigue model was developed to predict the stiffness reduction of nanoparticle/fibrous polymeric composites. The model called Nano-NSDM and evaluated in different circumstances as the following cases based on various industrial applications of composite and nanocomposite materials: – – – – –

Fatigue life Fatigue life Fatigue life Fatigue life Fatigue life

prediction prediction prediction prediction prediction

of an epoxy resin modified by silica nanoparticles. of GFRP with nanoparticles. of CSM/epoxy composites. of thermoplastic nanocomposites. of nanoparticles/CSM/polymer hybrid nanocomposites

Consequently, the results obtained by the model are in very good agreement with experiments. Furthermore, by means of a combination of the micromechanics and energy methods, a new model named Nano-EFAT was derived. The model has the capability of predicting the fatigue life of hybrid nanocomposites by means of the experimental fatigue data of the same composites without adding any nanoparticles. In order to evaluate the model, a series of tests have been performed. The results obtained by the model are in very good agreement with the experimental data. Finally, due to easy access for industries and low tests cost, the displacementcontrolled flexural bending fatigue method was taken into consideration. Then, the behavior of nanocomposites was considered using BFM-110 experimental test setup. By means of the testing device, effects of adding synthesized graphene nanosheets and CNFs (independently and simultaneously) on the flexural fatigue behavior of epoxy polymer were investigated and a remarkable improvement in fatigue life of epoxy resin was observed at the presence of graphene and CNF nanoparticles.

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Fatigue Life Prediction of Composites and Composite Structures

Finally, the following outlines are represented to have an outlook on future works: – – – – –

Fatigue life prediction of nanoparticle-filled fibrous polymeric composites based on the residual strength approach. Progressive fatigue damage modeling for nanoparticle-filled fibrous polymeric composites. Multiscale numerical simulation of fatigue behavior of nanoparticle-filled fibrous polymeric composites. The effect of stress ratio on fatigue behavior of nanoparticle-filled fibrous polymeric composites. Promotion of Nano-EFAT model for fatigue behavior of nanoparticle-filled fibrous laminated composites.

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[59] T.H. Hsieh, A.J. Kinloch, K. Masania, A.C. Taylor, S. Sprenger, The mechanisms and mechanics of the toughening of epoxy polymers modified with silica nanoparticles, Polymer 51 (2010) 6284–6294. [60] R. Talreja, Fatigue of Composite Materials, Technomic, Lancaster, PA, USA, 1987. [61] S.W. Case, K.L. Reifsnider, Fatigue of composite materials, in: I. Milne, R.O. Ritchie, B. Karihaloo (Eds.), Comprehensive Structural Integrity. Vol. 4: Cyclic Loading and Fatigue, first ed., Elsevier Science, Amsterdam, 2003. [62] M.M. Shokrieh, M. Esmkhani, Fatigue life prediction of nanoparticle/fibrous polymeric composites based on the micromechanical and normalized stiffness degradation approaches, J. Mater. Sci. 48 (2013) 1027–1034. [63] M.M. Shokrieh, M. Esmkhani, F. Taheri-Behrooz, Fatigue modeling of chopped strand mat/epoxy composites, Struct. Eng. Mech. 50 (2) (2014) 231–240. [64] L. Ye, On fatigue damage accumulation and material degradation in composite materials, Compos. Sci. Technol. 36 (1989) 339–350. [65] K.L. Reifsnider, The critical element model: a modeling philosophy, Eng. Fract. Mech. 25 (1986) 739–749. [66] S.S. Wang, E.S.M. Chim, Fatigue damage and degradation in random short-fiber SMC composite, J. Compos. Mater. 17 (1983) 114–131. [67] A.L. Highsmith, W.W. Stinchcomb, K.L. Reifsnider, Effect of fatigue induced defects on the residual response of composite laminates, in: Effects of Defects in Composite Materials, ASTM, 1984, pp. 194–216. STP-836. [68] A. Ramkumar, R. Gnanamoorthy, Axial fatigue behavior of polyamide-6 and polyamide-6 nanocomposites at room temperature, Compos. Sci. Technol. 68 (2008) 3401–3405. [69] M.M. Shokrieh, M. Esmkhani, F. Taheri-Behrooz, A novel model to predict the fatigue life of thermoplastic nanocomposites, J. Thermoplast. Compos. Mater. 28 (11) (2015) 1496–1506. [70] S.S. Ray, M. Okamoto, Polymer/layered silicate nanocomposites: a review from preparation to processing, Prog. Polym. Sci. 28 (2003) 1539–1641. [71] H.R. Shahverdi, F. Vahedi, M.M. Shokrieh, M. Esmkhani, Effects of carbon nanotubes on mechanical and electrical properties of epoxy resin, J. New Carbon Mater. 29 (6) (2014) 1–7. [72] M.M. Shokrieh, M. Danesh, M. Esmkhani, A combined micromechanical-energy method to predict the fatigue life of nanoparticles/chopped strand mat/polymer hybrid nanocomposites, Compos. Struct. 133 (2015) 886–891. [73] J. Lemaitre, J.L. Chaboche, Mechanics of Solid Materials, Cambridge University Press, Cambridge, England, UK, 1990. [74] L.M. Kachanov, Introduction to Continuum Damage Mechanics, Martinus–Nijhoff, Dordrecht, The Netherlands, 1986. [75] P.P. Ladeve`ze, A.A. Gasser, O.O. Allix, Damage mechanisms modeling for ceramic composites, ASME J. Eng. Mater. Technol. 116 (1994) 331–336. [76] F. Ellyin, H. El-Kadi, A fatigue failure criterion for fiber reinforced composite laminate, Compos. Struct. 15 (1990) 61–74. [77] F. Ellyin, H. El-Kadi, Predicting crack growth direction in unidirectional composite laminae, Eng. Fract. Mech. 36 (I) (1990) 27–37. [78] M.M. Shokrieh, F. Taheri-Behrooz, A unified fatigue life model based on energy method, Compos. Struct. 75 (2006) 444–450. [79] V.W. Paepegem, J. Degrieck, Experimental setup for and numerical modeling of bending fatigue experiments on plain woven glass/epoxy composites, Compos. Struct. 51 (2001) 1–8.

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[80] P. Paepegem, V.W. and Degrieck, J., A new coupled approach of residual stiffness and strength for fatigue of fibre reinforced composites, Int. J. Fatigue 24 (7) (2002) 747–762. [81] V.W. Paepegem, J. Degrieck, Fatigue degradation modeling of plain woven glass/epoxy composites, Compos. Part A 32 (10) (2001) 1433–1441. [82] A. El-Mahi, M.K.H. Farooq, S. Sahraoui, A. Bezazi, Modeling the flexural behavior of sandwich composite materials under cyclic fatigue, Mater. Des. 25 (2004) 199–208. [83] A. Ramkumar, R. Gnanamoorthy, Effect of nanoclay addition on the displacementcontrolled flexural fatigue behavior of a polymer, J. Mater. Sci. 45 (15) (2010) 4180–4187. [84] K.R. Rajeesh, R. Gnanamoorthy, R. Velmurugan, Effect of humidity on the indentation hardness and flexural fatigue behavior of polyamide 6 nanocomposites, Mater. Sci. Eng. A 527 (12) (2010) 2826–2830. [85] M.V. Timmaraju, R. Gnanamoorthy, K. Kannan, Effect of environment on flexural fatigue behavior of polyamide 66/hectorite nanocomposites, Int. J. Fatigue 33 (4) (2011) 541–548. [86] M.V. Timmaraju, R. Gnanamoorthy, K. Kannan, Influence of imbibed moisture and organoclay on tensile and indentation behavior of polyamide 66/hectorite nanocomposites, Compos. Part B 42 (3) (2011) 466–472. [87] Y.L. Liang, R.A. Pearson, The toughening mechanism in hybrid epoxy-silica-rubber nanocomposites (HESRNs), Polymer 51 (21) (2010) 4880–4890. [88] C.M. Manjunatha, N. Jagannatha, K. Padamalatha, A.C. Taylor, A.J. Kinloch, The fatigue and fracture behavior of micron-rubber and nano-silica particles modified epoxy polymer, Int. J. Nanosci. 11 (3) (2012). 1240002-1-7. [89] C.M. Manjunatha, R. Bojja, N. Jagannathan, A.J. Kinloch, Enhanced fatigue behavior of a glass fiber reinforced hybrid particles modified epoxy nanocomposite under WISPERX spectrum load sequence, Int. J. Fatigue 54 (2013) 25–31. [90] C.M. Manjunatha, S. Sprenger, A.C. Taylor, A.J. Kinloch, The tensile fatigue behavior of a glass-fiber reinforced plastic composite using a hybrid-toughened epoxy matrix, J. Compos. Mater. 44 (17) (2010) 2095–2109. [91] C.M. Manjunatha, N. Jagannatha, K. Padamalatha, A.C. Taylor, A.J. Kinloch, The effect of micron-rubber and nano-silica particles on the fatigue crack growth behavior of an epoxy polymer, Int. J. Nanosci. 10 (4) (2011) 1095–1099. [92] J. Fritzsche, H. Lorenz, M. Kl€uppel, CNT based elastomer-hybrid-nanocomposites with promising mechanical and electrical properties, Macromol. Mater. Eng. 294 (9) (2009) 551–560. [93] N. Witt, Y. Tang, L. Ye, L. Fang, Silicone rubber nanocomposites containing a small amount of hybrid fillers with enhanced electrical sensitivity, Mater. Des. 45 (2013) 548–554. [94] H. Al-Saleh Mohammed, H. Saadeh Walaa, Hybrids of conductive polymer nanocomposites, Mater. Des. 52 (2013) 1071–1076. [95] M.H.R. Jen, C.K. Chang, Y.C. Sung, F.C. Hsu, Experiments and Simulations, DEStech Publications Inc, 2012, pp. 175–179. [96] J. Li, P.S. Wong, J.K. Kim, Hybrid nanocomposites containing carbon nanotubes and graphite nanoplatelets, Mater. Sci. Eng. A 483–484 (2008) 660–663. [97] M.M. Shokrieh, M. Esmkhani, A.R. Haghighatkhah, Z. Zhao, Flexural fatigue behavior of synthesized graphene/carbon-nanofiber/epoxy hybrid nanocomposites, Mater. Des. 62 (2014) 401–408. [98] M.M. Shokrieh, M. Esmkhani, F. Vahedi, H.R. Shahverdi, Improvement of mechanical and electrical properties of epoxy resin with carbon nanofibers, Iran. Polym. J. 22 (10) (2013) 721–727.

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[99] M.M. Shokrieh, A. Saeedi, M. Chitsazzadeh, Evaluating the effects of multi-walled carbon nanotubes on the mechanical properties of chopped strand mat/polyester composites, Mater. Des. 2014 (56) (2014) 274–279. [100] K. Berchem, M.G. Hocking, A simple plane bending fatigue and corrosion fatigue testing machine, Meas. Sci. Technol. 17 (2006) 60–66. [101] T. Ramanathan, A.A. Abdala, S. Stankovich, D.A. Dikin, M. Herrera-Alonso, R.D. Piner, D.H. Adamson, H.C. Schniepp, X. Chen, R.S. Ruoff, S.T. Nguyen, I.A. Aksay, R. K. Prud’Homme, L.C. Brinson, Functionalized graphene sheets for polymer nanocomposites, letter abstract, Nat. Nanotechnol. 3 (2008) 327–331. [102] M.A. Rafiee, J. Rafiee, Z. Wang, H. Song, Z.Z. Yu, N. Koratkar, Enhanced mechanical properties of nanocomposites at low graphene, ACS Nano 3 (12) (2009) 3884–3890. [103] V.T. Rathod, J.S. Kumar, A. Jain, Polymer and ceramic nanocomposites for aerospace applications, Appl. Nanosci. 7 (8) (2017) 519–548. [104] N. Mohamad, K.I. Karim, M. Mazliah, H.E.A. Maulod, A.J. Razak, M.A. Azam, M. S. Kasim, R. Izamshah, Fatigue and mechanical properties of graphene nanoplatelets reinforced Nr/Epdm nanocomposites, J. Phys. Conf. Ser. 1082 (2018).

Further reading [105] B.B. Johnsen, A.J. Kinloch, R.D. Mohammed, A.C. Taylor, S. Sprenger, Toughening mechanism of nanoparticle-modified epoxy polymer, Polymer 48 (2007) 530–541.

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High-temperature fatigue behavior of woven-ply thermoplastic composites

6

B. Vieille, L. Taleb INSA Rouen Normandie, Groupe de Physique des Mat
eriaux (UMR CNRS 6634), Rouen, France

6.1

Introduction

In the early 1980s, thermoplastic (TP)-based composites proved to be relevant to compete with thermosetting (TS)-based composites in many industrial applications [1]. In relation with composites consisting of conventional TS matrices (such as epoxies, polyesters, and vinyl esters), high-performance TP (e.g., polyphenylene sulfide— PPS, polyether ether ketone—PEEK, Polyether imide—PEI, etc.) composites are characterized by significant improvements in terms of manufacturing (reduced manufacturing time, low-cost processes such as stamping and welding), better fire resistance, impact behavior, and damage tolerance. In addition, composite systems with highly ductile and very tough matrix under monotonic conditions were expected to resist to damage initiation and failure under fatigue conditions. Unfortunately, TP-based composite materials are likely to show higher fatigue sensitivity because of self-heating (resulting from cyclic deformations) and more extensive damage. In other words, the high fracture toughness of TP-based composites does not necessarily lead to high fracture resistance under cyclic loading (especially under displacement control) and improved fatigue life [1–4]. With the emergence of high-performance TPs, the question of TP composites fatigue-performance is still open particularly under high-temperature fatigue conditions that enhance both material toughness and ductility. Further growth of TP composites therefore strongly depends on the knowledge of their long-term behavior (fatigue and creep). Though the literature on the fatigue behavior of TS-based composites is very abundant, there are far fewer studies dealing with the fatigue response of TP laminates [5]. Under fatigue loadings, it is well established that metals and polymer matrix composite (PMC) materials accumulate damage in different ways depending on materials ductility and toughness [1]. In metals, failure usually occurs after the propagation of a single macroscopic crack whose growth depends on both their fracture toughness and ductility. In composite materials, failure takes place after accumulation of multiple damage modes: crazing and cracking of the matrix, fiber/matrix debonding, fiber fracture, transverse-ply cracking, and delamination. These different modes can operate independently or interactively from each other, and their onset strongly depends on Fatigue Life Prediction of Composites and Composite Structures. https://doi.org/10.1016/B978-0-08-102575-8.00006-1 © 2020 Elsevier Ltd. All rights reserved.

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both material properties (ductility and toughness) as well as loading conditions (amplitude, frequency, temperature), making it complex to determine damage chronology during fatigue. The nature of the constitutive elements (matrix and fibers), the orientation of fibers within the plies, laminates’ thickness, and architecture (unidirectional- or woven-ply) therefore significantly influences the fatigue behavior of PMCs. As a result, time-dependent and ductile behaviors play a significant role in the fatigue behavior of PMCs depending on matrix nature [6–11]. In this chapter, attention is first directed toward the fatigue response and damage mechanisms in nominally defect-free (unnotched) TP-based composites, with a particular attention paid to the influence of matrix ductility and toughness in different configurations: (i) matrix-dominated behavior (angle-ply laminates) [12], (ii) fiberdominated behavior (quasi-isotropic laminates) [13], and (iii) behavior resulting from the creep-fatigue interaction in angle-ply laminates [14]. The second part of the chapter investigates the behavior of notched laminates in order to understand the specific contribution of stress concentration on damage mechanisms and fatigue life [15, 16].

6.2

Literature review

6.2.1 Influence of matrix nature on fatigue behavior: TP vs TS composites In composite systems, the matrix materials are usually considered as the strength limiting components. Moreover, because of the pronounced viscoelastic nature of polymeric matrices, the fatigue performance of polymeric composites is usually considered a matrix-related phenomenon [17]. The effects of matrix toughness on fatigue behavior of carbon fiber-reinforced composite laminates were extensively studied in the 1980s and 1990s [18–20]. In particular, the transferability of TP matrix toughness and fatigue resistance to delamination toughness and fatigue growth in composites has been investigated in UD-ply laminates [4, 21–29], in woven-ply laminates [6, 30, 31], and at high-temperatures lower than Tg [7–9, 32, 33]. As it was underlined in Ref. [1], even the use of a high-toughness matrix (e.g., PEEK) does not eliminate both transverse and longitudinal cracking [6, 19]. When comparing carbon fiber TP (PEEK) composites with carbon fiber TS (epoxy) composites subjected to monotonic and cyclic loadings [10, 34], the crack growth resistance is roughly maintained in fatigue conditions, but the improvements in fatigue thresholds in terms of the stress intensity factor range for the onset of crack propagation is far from the measured improvements in fracture toughness under monotonic conditions. The fracture toughness observed under monotonic conditions for TP-based composites cannot directly be transferred to fatigue conditions and low load amplitudes. Thus, the fatigue behavior of PMCs significantly depends on polymer matrix ductility and toughness [3, 20, 25, 35, 36]. The introduction of ductile bands (also known as softening strips) in fiber-reinforced composites acts as a barrier to the propagation of cracks [37], and thus contributes to the improvement of the intrinsic toughness of woven-ply composites [6, 8, 12, 20, 38], more particularly in C/PPS laminates [39–43]. The fatigue behavior of five-harness satin weave carbon-fiber-reinforced PPS, with a [(0,90)]4s lay-up, was

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investigated at room temperature [39–45], under on-axis and off-axis tensile fatigue loadings, with a stress ratio R ¼ 0. Franco et al. performed fatigue tensile tests (8 Hz—R¼ 0.1) at 80°C to understand the relationship between fatigue resistance and environmental conditioning, and conducted a fractographic analysis on the fracture surfaces [43]. They concluded that temperature is the most detrimental factor to fatigue life. In addition, considering the ductile nature of the PPS matrix at 80°C (close to its Tg), the observation of failure surfaces shows that the matrix is characterized by a plastic deformation aspect, suggesting that the PPS matrix yields until fatigue failure. With regard to the material itself, C/PPS has a very brittle behavior under on-axis tensile fatigue loadings as there is no prior sign of failure (little stiffness degradation or permanent strain) [42]. When woven-ply C/PPS laminates are subjected to off-axis fatigue loadings, the dominant damage mode is not a gradual increase in the number of broken fibers, but rather seems to be related to the matrix and viscous effects [40]. Fiber breakage is primarily observed at the beginning (when the weakest fibers break), and at the end of the test when a catastrophic failure happens. From microscopic observations, it appears that damage mechanisms primarily consist of matrix cracking and meta-delamination instead of fiber breakage [42]. In woven-ply laminates consisting of warp and weft yarns, meta-delamination refers to inter-yarn debonding at the yarn cross-over regions. The frequency effect (2 and 5 Hz) on the fatigue lifetime has also been studied, as well as heating by measuring temperature at the surface of specimens. It turned out that higher frequencies yield shorter fatigue life at low stress levels, and this effect seems to decrease as the applied stress increases [45]. Regarding the offaxis laminate [(+45,45)]4s, the authors undertook fatigue tests at 1 and 2 Hz. Surprisingly, a small frequency increase tends to yield an increase in the specimen’s life [40]. The fatigue behavior can be described by three primary steps: (i) the run-in of the fatigue test where a certain amount of permanent deformation occurs without an increase in temperature; (ii) a steady-state phase where there is a gradual increase in permanent elongation without an increase in temperature; and (iii) the end-of-life where there is a sudden growth in both temperature (higher than Tg) and permanent elongation. This type of behavior is confirmed by [11] in woven carbon/bismaleimide laminates at different temperatures. The authors also showed that the hysteresis loops become bigger during the last phase of fatigue life, suggesting significant energy dissipation [40]. A creep test was also performed to verify if the fatigue permanent elongation results from fatigue damage, time-dependent effects or a combination of both. Thus, the strain increase during the fatigue test was shown to be primarily caused by fatigue damage. Finally, there are very few references available in the literature on the fatigue behavior of PMCs when temperature is near to or higher than their glass transition temperature Tg [8, 9, 46]. Besides, they all examine the fatigue behavior of UD-ply laminates.

6.2.2 Influence of reinforcement architecture As far as the fatigue performance is concerned, it seems interesting to associate woven fabrics with highly ductile TP matrices [6, 43]. The result is even more noticeable when service temperature is higher than the material’s glass transition temperature [47], even in Q-I laminates whose behavior is fiber dominated. For the past 40 years,

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the fatigue behavior of PMCs has been extensively studied [1, 5], but most of the references deal with unidirectional-ply-reinforced composites or woven-ply TS composites [48]. Their fatigue performance is well known in the case of in-plane stress loadings, but information is limited when laminates are subjected to through-thethickness loadings (such as three points bending, short beam shear tests or impact loadings). In addition, with the generalization of composite materials in many industrial fields such as aeronautics, composite manufacturers have developed 3D composite structures able to sustain multiaxial stress loadings. Woven-ply-reinforcement laminates comply with these requirements and can be used to manufacture parts with a complex shape by means of stamping or thermoforming processes. Thus, the contribution of reinforcement architecture to monotonic and fatigue behaviors has been addressed in TS-based-reinforced laminates [6, 37, 49], but specific damage chronology in TP-based woven-ply composites subjected to fatigue loading is still an open question [11, 42]. However, the fatigue damage mechanisms are similar to those observed in the case of monotonic loadings at both mesoscopic and macroscopic scales [6]. At the mesoscopic scale, fiber-matrix debonding and matrix cracking operate in the weft fiber bundles (Fig. 6.1B), resulting in the formation of macroscopic cracks by coalescence. These cracks will grow in the transverse direction in matrix-rich areas or in the longitudinal direction at the interface between warp and weft fiber bundles within then same ply (Fig. 6.1C). This phenomenon also known as meta-delamination is influenced by the presence of transverse cracks (Fig. 6.1D). Such coupling strongly depends on the undulation of weft bundles over warp bundles in the crimp zone where the matrix-rich areas are preferentially located in woven-ply laminates [37]. Damage is usually initiated at this scale [50, 51].

Fig. 6.1 Evolution of fatigue damage in the warp direction in woven-ply laminates subjected to tension-tension fatigue [50].

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6.2.3 Autogenous heating in polymer matrix composites under fatigue loading According to Constable et al. [52], hysteretic effects can be observed in TP polymers because of their strong nonlinear behavior (including ductile and time-dependent behaviors as discussed in the following section), These effects, coupled with the poor thermal diffusivity of polymers, can result in significant increases in temperature coming along with cyclic deformations. These thermal effects are enhanced at high strain rates. The resulting thermal softening degrades the mechanical properties and may even dominate over any intrinsic mechanical fatigue effects [52, 53]. The interactions between thermal effects and mechanical fatigue are complex, and significantly depend on the fatigue loading conditions (temperature, stress amplitude, stress ratio, and frequency) as well as the laminates stacking sequence. Thus, these thermal effects may exacerbate the fatigue failure for given conditions, whereas they are not detrimental for the fatigue life in other cases. The thermal softening effects can be reduced during cyclic loading by imposing low strain rates (from low-frequency tests), which in turn may promote time-dependent behaviors. From the macroscopic standpoint, the mechanical energy brought to the material during mechanical loading is partly dissipated in the formation of damage, but a sizeable part is dissipated as heat [3]. More specifically, the energy losses in PMCs subjected to fatigue loadings are characterized by hysteresis loops which primarily depend on matrix ductility and fatigue frequency. These loops can be observed on the stress-strain responses of laminates. According to Peterman et al. [54] there are three possible modifications of the hysteresis loops during a fatigue test. In the case of a creep-dominant behavior, a mean strain increase will occur instead of a change in axial stiffness (macroscopic consequence of damage). When changes in the modulus are detected, the loops will exhibit degradation (damage evolution) or stiffening (fiber rotation). Internal damage development is attended by a rise in energy dissipation per fatigue cycle, expressed by an increase in damping, resulting in size and/or shape modification. Such hysteretic behavior results in the generation of internal heat, which is supposed to be transferred to the surface by conduction, and then to be dissipated to the surrounding by convection and radiation [17, 19]. However, the thermal conductivity of carbon-fiber composites is generally low, and not all the heat generated has sufficient time to dissipate through the laminates thickness at high frequency, which results in a more rapid increase in temperature also known as autogenous heating [55]. Thus, enhanced autogenous heating in fatigue at high frequency can lead to softening of the material and premature failure. This is of particular importance for composites with TP matrices under cyclic loading as a temperature rise is prevalent on matrix ductility [1, 56]. As pointed out by Gamstedt et al. [3], because of the ductility and high fracture toughness of many TPs, a considerable amount of heat is formed and dissipated in TP-based composites experiencing inelastic deformations [3, 52]. If the heat cannot be dissipated away from TP-based composites, it can make these materials sensitive to cyclic loading. The typical service temperature of TPs would preferably be below the glass-transition temperature, because they lose most of their stiffness at T > Tg.

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Thus, TPs used for high-temperature applications should therefore generally have high glass-transition temperatures. Finally, in laminates with a quasi-isotropic layup (whose behavior is primarily fiber dominated), hysteretic losses and therefore viscous effects are usually lower than the ones generated in angle-ply laminates whose behavior is matrix dominated.

6.2.4 Influence of viscous effects on fatigue accumulated damage As pointed out by Dillard et al. [57], although engineers are often aware of the viscous nature of typical polymeric materials, it is easy to lose sight of this time dependence when these plastics are reinforced with advanced fibers (such as graphite or glass) to form a high-performance structural material [47, 55, 57–60]. Fibers reinforced polymers consisting of viscous matrices exhibit little creep or stress relaxation in the fiber direction. Under transverse or off-axis loading, the same composite material is characterized by strong time-dependent behaviors. Quasi-isotropic laminates (combining 0, 45, and 90 degree oriented fibers within the plies) may also exhibit creep, stress relaxation, and delayed failure. TP-based composites are, by nature, sensitive to the loading conditions (especially the loading frequency), even in laminates with a stacking sequence whose behavior is fiber dominated. Their time-dependent behavior and damage mechanisms under cyclic loading are affected by the autogenous heating [3, 12, 43, 48, 55, 61]. The presence of matrix-rich regions in composite laminates makes it essential to evaluate the contribution of time-dependent behaviors (viscoelasticity and viscoplasticity) to the fatigue behavior of TP-based laminates at high frequencies. In fiber-reinforced epoxy laminates with an orthotropic lay-up, longer fatigue lives have been reported as frequency increases [62]. Moore studied the frequency’s influence on UD C/PEEK laminates with quasi-isotropic sequences for 0.5 and 5 Hz under tensiletensile fatigue solicitation, and found that an increase in frequency results in a shorter fatigue life [27]. The same results were found on C/PEEK angle-ply laminates between 1 and 10 Hz [12, 22, 23, 62, 63]. A few authors observed significant temperature increases at the specimen surface as frequency increases in laminates subjected to off-axis loadings [22, 48]. This temperature can even exceed the material Tg, decreasing dramatically the laminate’s mechanical behavior leading to a shorter fatigue life. This effect is more noticeable in highly ductile TP-based laminates. It also appears that, at high-loading levels, the increase in loading frequency will result in an increase and accumulation of cyclic deformation due to local heating and stress heterogeneity, as evidenced by the increase in damage rates [64]. From these works, it appears clearly that the fatigue life of TP-based laminates decreases as loading frequency increases. At the same frequency, the fatigue life decreases as the loading level increases, with more degradation at higher frequencies. At such frequencies, creep deformation rates are increased with temperature [12], because the specimens heat up and the matrix eventually becomes more compliant and yields at lower stresses. The temperature might even exceed the glass transition temperature locally in regions of high stress concentration [1], making essential to evaluate the contribution of viscous effects to the fatigue behavior of TP-based laminates at high frequencies. However, fatigue-induced damage and viscous effects may not readily be separated in cyclic loading of TPs [3]. Indeed, because of the complexity of the

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viscous effects, and because the behavior patterns are not yet fully elucidated, the number of works which deal with the interaction between fatigue damage and viscous effects is still very limited [54]. In fiber-reinforced epoxy laminates, high frequencies lead to a significant accumulation of cyclic creep due to local heating, and a decrease in stiffness, resulting in more elongation and shorter fatigue lives [21, 62–64]. However, the effect of stress intensities reduction in the matrix by plastic or viscoplastic deformation is more or less pronounced depending on the applied stress level. The plastic deformations due to initial load, cyclic loading, and cyclic creep are competitive. It seems that cyclic creep occurs once the mean stress in fatigue exceeds the threshold stress for monotonic creep, for which a plastic deformation is induced (residual permanent deformation). Thus, the crack propagation in cyclic loading can be attributed to creep at higher load levels, whereas cycle-dependent crack growth prevails at lower load amplitudes in short-fiber reinforced TP-based laminates [65, 66]. These works clearly point out the importance of the polymer matrix low-cycle fatigue properties for composite structures subjected to high cyclic stresses [1].

6.2.5 Creep-fatigue interaction in TP composites As shown in the previous section, time-dependent behaviors play a major role in the fatigue response of PMCs, and more specifically in TP-based composites. For many composite structures, cyclic fatigue is the critical loading condition. However, there are many applications in which static fatigue (creep) can play a critical role in the long-term performance of the structure [57, 67–71]. The interaction of creep and fatigue comprises the concerted action of creep and fatigue mechanisms under repeated loading. The mechanical properties of polymers and polymer composites are time dependent, and this may reflect in the fatigue properties [54]. Several authors have investigated the effect of creep strain on the fatigue life of C/epoxy angle-ply laminates [18, 20, 21, 54, 72–76]. From these works, it can be concluded that creep is more important during a cyclic loading with a nonzero average applied stress, because the matrix undergoes time-dependent deformation in addition to cycledependent irreversible deformation. In order to study the influence of creep on the fatigue behavior of polymer-based composites, Sun and Chim [75] have performed interrupted tension-tension fatigue tests under constant amplitude sinusoidal loadings (R ¼ 1/15 and f ¼ 10 Hz). Between two fatigue periods, specimens were held statically under the maximum applied stress (70% σ ult) during various times. It was concluded that the fatigue life increases as the time under static load (cyclic creep) increased. In order to investigate the influence of both creep stresses and creep-recovery times on the fatigue behavior, Petermann conducted a study dealing with the creep-fatigue interaction, and reached the same conclusions on UD C/epoxy laminates with angle-plied (AP) stacking sequences [54]. In order to explain the fatigue retardation due to creep in polymer-based composites, Jones et al. [72] concluded that the crack growth rate under cyclic loading following each static loading period slowed down significantly. Such an effect is viewed as cracks blunting, and the permanent strain is explained to be the result of creep (depending on the static hold time) in the matrix, more particularly in the case of TP-based composites. On the one hand, the development of plastic deformations

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reduces the stress intensity at the cracks tips. On the other hand, a retarded reduction of stress intensities is due to local viscoelastic and viscoplastic deformations at cracks tips. Dillard et al. [57] have shown that the residual strength of creep loaded laminated composites is consistently higher than the original static strength by as much as 30%, due to local stress relaxation and improved fiber alignment. They also concluded that the residual strength of specimens can possibly exceed the original strength and subsequently decay to a point at which delayed failure will eventually result. Thus, the fatigue life tends rather to be prolonged by prior enduring loads which are more beneficial than short-time prior loads [54, 76]. Such mechanisms may delay the initiation—propagation of matrix cracks and delamination in the failure zone to be, and ultimately resulting in an extension of fatigue life. The delayed cracking is supported by the observations of Lafarie-Frenot and Touchard for monotonic tensile loadings [77]. In addition, it also appears that low frequencies produce larger creep strains than high frequencies at the same stress level. Thus, a low-frequency cyclic loading preceding a high-frequency cyclic loading has a delaying effect on the subsequent fatigue degradation rate in TP-based laminates [78]. Conducted experiments involving a switching of frequency (10 Hz ! 1 Hz and 1 Hz ! 10 Hz) supported these conclusions [75]. However, the effect of stress intensity reduction in the matrix by plastic or viscoplastic deformation is more or less pronounced depending on the applied stress level. The plastic deformations due to initial load, cycling loading and cyclic creep are competitive. It seems that cyclic creep occurs once the mean stress in fatigue exceeds the threshold stress for monotonic creep, for which a plastic deformation is induced (residual permanent deformation). Thus, the crack propagation in cyclic loading can be attributed to creep at higher load levels, whereas cycle-dependent crack growth prevails at lower load amplitudes in short-fiber reinforced TP-based laminates (PPS and PEEK) [65, 66]. These works clearly point out the importance of the polymer matrix low-cycle fatigue properties for composite structures subjected to high cyclic stresses. Substantial databases exist on many factors (geometrical, constituents, environmental, type of loading) contributing to the mechanical behavior of [45 degree] laminates [54]. The factor related to constituents suffers from a lack of knowledge with regard to the influence of plasticity and viscous effects on the fatigue behavior of [45 degree] laminates. Because of the complexity of the viscous effects, and because the behavior patterns are not yet fully elucidated, the number of works that deal with the interaction between fatigue and viscous effects is still very limited, particularly when composite materials are subjected to fatigue loadings at T > Tg.

6.2.6 Influence of stress concentration on the fatigue behavior of TP laminates As many composite structures contain notches for joining purposes and open cutouts for access, most applications involve stress concentration and nonuniform stress states [15, 16, 79–83]. Thus, the development of stress concentrations in composite structures remains of great concern and many studies investigated the effect of notches on strength [84–90]. Complex damage mechanisms result from the presence of a stress

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concentration, causing a wide range of phenomena, such as stress and/or strain gradients, which are not present in unnotched components. The stress and deformation distribution in the vicinity of stress risers have a strong influence on the fatigue crack formation and propagation [91]. Hence, the development of damage in notched composite laminates requires special attention [92]. Because of damage development around the notch during long-term loading, the magnitude of the stress redistribution near the notch dominates the fatigue response of notched composite laminates. Indeed, the residual strength increases before decreasing due to the local damage (matrix cracking or delamination) or the local matrix plasticity contribution to the relaxation of the overstress near the notch by means of stress redistribution, resulting in increasing the fatigue life of PMCs [92]. To evaluate the possible advantages of using a tough TP resin as a matrix in a notched composite material, many authors have compared the monotonic and fatigue behavior of TP-based and TS-based laminates [15, 20, 84–90, 93–101]. These works concluded that the ability of a material to reduce the stress concentration near the notch at room temperature is less effective in C/TP than in C/epoxy laminates, due to the lower degree of stress-relieving damage formation around the notch. The experimental observations reported in the literature indicated that fatigue damage in C/epoxy laminates consists of a combination of matrix cracking, longitudinal splitting, and delamination, which attenuate the stress concentration and suppress fiber fracture in the notch vicinity. In the literature, current methods to evaluate the modulus degradation of TP-based composites are discussed including viscoelastic/plastic and continuum damage models [102]. According to Harris [103], one of the most difficult issues for users of composites is the question of whether or not fracture-mechanics concepts may be used in fatigue design and life-prediction purposes. The crack-stopping ability of composites, which results from their inhomogeneity on a fine scale (the fiber/matrix interface) and on a gross scale (laminated structure) makes it difficult in many cases to apply a fracture mechanics approach to fatigue testing and design. To quantify the growth of fatigue cracks as a function of the cycles N in notched laminates, a few researchers have successfully use a fracture-mechanics approach, based on the Paris power law [16, 104–106]. Indeed, in the case of tensile fatigue loadings, Δ K refers to the range of mode I stress intensity factors with a fatigue stress ratio R ¼ σ min =σ max . Although the Paris law is empirical, it remains one of the most useful expressions in the analysis of fatigue crack growth for a vast spectrum of materials and fatigue test conditions [91]. It is important to note that stable fatigue crack growth occurs at stress intensity factor levels Kmax ¼ Δ K(1  R) that are well below the quasi-static fracture toughness KIc.

6.3

TP- and TS-based composites in fatigue: An experimental study [12–16, 81, 82]

6.3.1 Materials and specimens description The studied composite materials were carbon fabric-reinforced laminates consisting of two different matrices: a semicrystalline high-performance PPS (TP) and an amorphous epoxy one (TS). The toughened PPS resin (Fortron 0214) and the epoxy resin

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(914) were supplied by the Ticona and Hexcel Companies, respectively. The wovenply pre-preg, supplied by the SOFICAR Company, consists of five-harness satin weave carbon-fiber fabrics (T300 3K 5HS). The volume fraction of the fibers is 50% in C/PPS and C/epoxy laminates. A dynamic mechanical thermal analysis (DMTA) analysis showed that the glass transition temperature is 95°C in C/PPS, whereas it is 190°C in C/epoxy. The experimental method used to determine the value of Tg is based on the loss modulus [107]. The pre-preg plates are hot pressed according to two different stacking sequences: l

l

a quasi-isotropic (QI) lay-up: [(0, 90)/(45)/(0, 90)/(45)/(0, 90)/(45)/(0, 90)] whose thermomechanical response is fiber-dominated, an angle-ply (AP) lay-up: [(45)]7 whose thermomechanical response is matrix-dominated.

6.3.2 Experimental set-up All the fatigue tests were performed using a 100 kN capacity load cell of an MTS 810 servo-hydraulic testing machine at room moisture, in force-controlled mode. The temperature control system (oven and temperature controller), provided a stable temperature environment during the test. Monotonic tensile mechanical properties (see Table 6.1) were determined according to standards EN6035 and EN6031 [108, 109]. Tensiontension fatigue (R¼ 0) tests were conducted at two frequencies (1 and 10 Hz), and at three stress levels: 70%, 80%, 90% of σ ult in QI laminates, 50%, 60%, 70% of σ ult in AP laminates. The test temperature (120°C) was chosen because advanced aeronautics structures, and particularly nacelles require high-performance fiber-reinforced PMCs, which can be used at temperatures up to about 120°C. Due to the time limitation, the number of fatigue cycles was limited to 1 million. Three specimens were tested in each configuration. During cyclic loadings, temperature was monitored at the surface of the specimens by means of a thermocouple, in order to observe the effect of frequency and stress level on the material’s heating resulting from both damage and time-dependent effects. During the fatigue and creep tests, longitudinal strains were measured with an extensometer attached to the samples by means of knife edges associated with springs (gauge length l0 ¼ 25mm). In addition, a fractographic analysis (microscopic observations and scanning electron microscope observations of failed specimens) was Table 6.1 Longitudinal tensile properties at 120°C for both materials and both staking sequences C/PPS QI Longitudinal stiffness (GPa) Ultimate strength σ u (MPa) Strain to failure εu (%)

C/Epoxy

AP

QI

AP

40.49 0.30

1.35  0.05

43.29  0.65

4.19 0.08

4726.02

159 3.92

505 3.72

175 8.98

1.28 0.04

27.34  0.27

1.28  0.04

5.85 0.11

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conducted in order to understand the fatigue damage mechanisms specific to each material. Optical microscopy investigations were performed with an olympus microscope, and scanning electron microscopy (SEM) investigations were performed with a Leo 1530 Gemini Zeiss microscope. Samples were previously gold coated with an International Scientific Instruments PS-2 Coating Unit, and observed in SEM mode using 5 kV HT, and a secondary electron signal detector.

6.4

Discussions on the fatigue behavior of TP vs TS laminates

6.4.1 Matrix-dominated fatigue behavior [12, 14, 81, 82] Most of the industrial applications based on multidirectional laminates (e.g., quasiisotropic laminates whose fatigue will be discussed in the following section) contain a considerable number of [45 degree] angle-ply plies to bear shear loads, to control stress concentrations and the damage behavior [54]. A crucial role is thereby taken over by the 45 degree oriented plies, making relevant to initially consider the fatigue behavior of AP laminates [9, 10, 17, 22–24, 40, 48, 49]. Depending on the number of 45 degree oriented plies in QI laminates, the ultimate properties of PMCs are more or less frequency-dependent, and this may influence the fatigue properties. In addition, the microscopic observations of edges in AP laminates show many matrix-rich regions (see Fig. 6.2), resulting from the nonplanar interply structure of woven plies. From the tension-tension fatigue behavior of woven-ply laminates standpoint, these pure matrix regions will play a significant role depending on the ductility behavior of polymer matrix and the testing conditions (temperature, frequency). In AP laminates subjected to tensile loadings, the matrix-dominated response results in an elastic-ductile behavior in both composite systems (see Fig. 6.3). The highly ductile behavior of PPS is exacerbated because the test temperature (120°C) is higher than its Tg. Both materials have virtually the same strength at 120°C (see Table 6.1), but the elongation at failure is a lot more significant in C/PPS laminates with respect to C/epoxy laminates 27.3% vs 1.3% respectively). Fatigue tests were conducted at three stress levels where plastic deformation of the laminates is more or less significant. Contrary to C/epoxy laminates, the very low yield strength of

Fig. 6.2 Edges microscopic observations of AP laminates showing lots of matrix-rich regions (C/PPS).

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Fig. 6.3 Monotonic tensile responses of AP laminates at 120°C: (A) C/PPS and (B) C/epoxy.

C/PPS laminates at T > Tg leads to an early plasticization of the matrix (60 vs 11 MPa). The stress level corresponding to 0.6 σ ult was chosen to study the fatigue behavior in the region where fibers rotation can be estimated and significantly depends on matrix ductility (7 vs 26 degree at failure in C/epoxy and C/PPS laminates, respectively). In order to apprehend more precisely the contribution of damage to the fatigue behavior, changes in damage accumulation as a function of test frequency have been investigated at 60% σ ult for both materials. The tension-tension fatigue behavior of both materials is characterized by longitudinal stress-strain loops whose shape is different from one material to the other: a “banana” shaped loop which seems to be frequency-dependent in C/PPS laminates, and an elongated ellipse shaped loop which seems to be virtually frequency-independent in C/epoxy laminates (see Fig. 6.4). The area and the inclination of the loops depend on the applied stress level, the stress ratio R and the cycle in the fatigue life. From a general standpoint, frequency virtually does not influence the fatigue life of C/PPS laminates, whereas it dramatically decreases the fatigue life (100%) of epoxy-based laminates, particularly as applied stress increases (see Fig. 6.5). Both materials have the same reinforcement, but differ from the nature of their matrix. It suggests that ductility significantly influences the high-temperature fatigue behavior of polymer-based laminates. From the stress-strain response standpoint, the stress borne by AP laminates increased linearly in the elastic zone (Phase 1—see Fig. 6.6), as the displacement compatibility between matrix and fibers led to the

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Fig. 6.4 Influence of frequency on the stress-strain responses of AP laminates subjected to tensile fatigue loadings at T¼ 120°C and 60% σ ult: (A) C/PPS and (B) C/epoxy.

Fig. 6.5 Diagrams S-N of C/PPS and C/epoxy AP laminates subjected to fatigue tensile loadings: influence of frequency on fatigue life.

progressive rotation of the latter with the applied shear strain and increased the axial stresses borne by the fibers (Phase 2—see Fig. 6.6). In epoxy-based laminates, the low ductile behavior of epoxy matrix minimizes the fibers rotation (about 7 degree at failure). The development of interface fracture

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Loading PHASE 1 Weft fibers

Unloading

PHASE 2

PHASE 3

Warp fibers

Necking

Plasticization in pure matrix regions

(A)

PHASE 4

Locking of fibers rotation

Rotation of fibers bundles

(B)

Fig. 6.6 Different phases of the mechanical response of [(+45/45)]7 laminates subjected to tensile loadings at 120°C: (A) loading and (B) unloading.

significantly modified the in-plane shear behavior as it hindered the load transfer between matrix and fibers, and precipitated fracture by the coalescence of interface cracks. The observation of the fracture surface indicates that the specimen failure is associated with the onset of a shearing band oriented at 45 degree with respect to the loading direction during phase 3 [110]. In the case of C/PPS laminates, the highly ductile behavior of PPS matrix minimizes the development of interface fracture, and promotes the rotation (about 26 degree at failure). Such mechanisms cause necking that is clearly observable on stress-strain curves (see Fig. 6.3). Fatigue tests conducted at high frequencies (typically 10 Hz) are expected to minimize viscous effects on the fatigue behavior. During cyclic loadings, the mechanical energy introduced to the material can be gradually dissipated at each cycle. On stressstrain curves, the energy losses are represented by hysteresis loops whose shape clearly depends on both frequency and matrix nature (see Fig. 6.7). C/PPS laminates display “banana”-shaped loops consisting of two parts. The first part is associated with the loading phase during which the large rotation of fibers comes along with the large plastic deformation of PPS matrix mostly in matrix-rich regions (see Fig. 6.7A). Secondary stiffening associated with the locking of rotating fibers (Phase 3—see Fig. 6.6) can be observed on the C/PPS loops from stresses reaching 40% σ ult (see black dotted lines on Fig. 6.7A). This suggests that there is a threshold orientation from which laminates regain stiffness and start damaging. The second part is related to the unloading phase (Phase 4—see Fig. 6.6) during which fibers reorientate, and the opening of the loops seems to be ascribed to the viscoelastic behavior (even more noticeable at low frequency). C/epoxy laminates display loops whose shapes are either a wide ellipse at 10 Hz or a narrow ellipse at 1 Hz (see Fig. 6.7B). For both frequencies, the loops’ shape

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Fig. 6.7 Comparison of the shape of stress-strain loops depending on the test frequency at 80% of fatigue life and at 60% σ ult: (A) C/PPS and (B) C/epoxy.

is associated with reduced fibers rotation (lower than 5 degree) and damage accumulation which are consistent with the low ductility of epoxy matrix during loading phase. In addition, secondary stiffening is not observed on C/epoxy loops because fibers rotation is reduced during loading phase, hence justifying that fibers disorientation is reduced, and the viscoelastic response of the matrix is minimized during the unloading phase. To evaluate the fatigue damage growth as a function of both frequency and applied stress, the concept of damage accumulation was used. This very simple approach, based on the changes in macroscopic properties (axial stiffness in QI laminates and mean strain in AP laminates), can provide an acceptable approximation of the macroscopic damage development in the laminates during the test. In AP laminates, the evolution of the axial stiffness results from the competing effects of macroscopic damage (decreasing stiffness) and fiber rotation (increasing stiffness) [54]. Both of these competing effects are less pronounced for small strains but significant for the large strain response, which is in agreement with the conclusions drawn in Ref. [111]. Thus, the mean strain is more meaningful for describing the fatigue degradation process than the axial stiffness [11, 76, 112].

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In order to evaluate the damage accumulation during fatigue tests, a damage variable based on the features of the stress-strain loops during cyclic loadings can be defined. The basic idea of many authors was to discuss the changes in axial stiffness that occur during cycling, in order to establish relationships between stiffness and accumulated damage [23, 113–115] in matrix-dominated laminates [36, 49] and woven-ply laminates [42, 116, 117]. Therefore, the axial stiffness of composite materials E(N) at cycle N is often used to give an estimation of the fatigue damage due to cyclic loading thanks to a damage variable d(N) for each cycle N defined by d ðN Þ ¼ 1 

Eð N Þ E0

(6.1)

where E0 is the initial longitudinal stiffness value of the undamaged material. However, the final accumulated damage is 1  Ef/E0 instead of unity when the material fails, where Ef failure longitudinal stiffness. Another expression of the damage variable d(N) ensures that its value varies from zero initially, and increases toward unity when failure occurs [36] d ðN Þ ¼

E0  EðN Þ E0  Ef

(6.2)

However, the measured dynamic stiffness is not a good indicator of damage accumulation as discussed in Ref. [11]. As it was underlined above, for composites that exhibit significant fiber rotation or a creep response due to a sustained positive mean stress during cyclic loading, the mean strain is more meaningful for describing the fatigue degradation [11, 76, 112]. The mean strain εmean(N) can be calculated on each cycle N from the stressstrain loops such as εmean(N) ¼ (εmax(N) + εmin(N))/2. Thus, a similar expression for the accumulated damage d(N) can be obtained from the value of εmean(N) on each cycle: d ðN Þ ¼

εmean ðN Þ  εmean ð0Þ
εmean Nf  εmean ð0Þ

(6.3)

where εmean(0) and εmean(Nf) are the initial and final mean strain, respectively. This definition also ensures that the damage variable d(N) varies in the range from 0 to 1. From this definition, it is therefore possible to compare the changes in damage accumulation during fatigue tests at different test frequencies (see Fig. 6.8). In both materials, damage grows more rapidly as applied stress increase. The effect of frequency depends on both the applied stress level and the studied material. In C/PPS laminates, a low frequency seems to accelerate damage accumulation at high stress levels. At intermediate stress level (50%), a low frequency seems to slow down damage accumulation. At low stress level, it appears that frequency has virtually no influence. In addition, early damage (after a thousand cycles) is much more important at low frequency, which also true in the case of epoxy-based laminates. In C/epoxy laminates, a high frequency seems to accelerate dramatically damage accumulation at high stress levels, whereas it seems to slow down damage accumulation at low stress level.

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Fig. 6.8 Influence of frequency and applied stress level on the damage accumulation d(N) during fatigue tests depending on test frequency and applied stress level: (A) C/PPS and (B) C/epoxy.

For both materials, even though the damage growth and the damage mechanisms are different, the fatigue behavior consists of three discrete stages: (i) In the initial phase of cyclic loading, damage accumulates rapidly under the form of microcracks which may initiate in multiple locations (particularly in epoxy matrix), but preferably at the interfaces between fibers and matrix in the crimp region where the weft fiber bundles undulate over the warp fiber bundles (ii) The second stage is characterized by a steady damage growth rate and little damage accumulation (iii) Damage (mostly debonding and interlaminar cracks) generalize rapidly during the last stage ultimately resulting in extensive delamination, fiber bundles pull-out, and breakage of rotated fibers in epoxy-based laminates at 1 Hz.

Due to the presence of matrix-rich regions resulting from the nonplanar interply structure of woven plies, the results show that polymer matrix ductility (exacerbated at T > Tg) play a significant role in the fatigue behavior of woven-ply PMCs by modifying the damage mechanisms and their chronology. Thus, the fatigue behavior of C/PPS laminates at T> Tg is affected by fibers alignment coming along with plastic

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deformation of the matrix during loading phase, as well as the disorientation of fibers and the viscoelastic response of the matrix during unloading phase (particularly at low frequency), rather than to fatigue damage accumulation. On the contrary, the fatigue response of epoxy-based laminates at T< Tg is a result of fatigue damage accumulation. At high frequency, the autogenous heating (discussed in the next section) increases matrix ductility in pure matrix regions, resulting in the development of larger plastic deformations which locally release stress concentration. It may delay the initiation-propagation of matrix cracks, and debonding, ultimately resulting in an extension of fatigue life. Thus, a frequency increase virtually does not influence the fatigue life of C/PPS laminates, except at high stress levels for which the R-ratio remains higher than 0 throughout the test, hence justifying an increase in fatigue life. However, a frequency increase dramatically decreases the fatigue life of epoxy-based laminates (100%), because of a more extensive damage and a faster damage growth.

6.4.2 Fiber-dominated fatigue behavior [13] In QI laminates, the thermomechanical response is dominated by the 0 degree on-axis fibers resulting in an elastic-brittle behavior (see Fig. 6.9). The dotted line corresponds to a purely elastic response, and emphasizes a behavior characterized by a very low

Fig. 6.9 Monotonic tensile responses of QI laminates at 120°C: (A) C/PPS and (B) C/epoxy.

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213

degree of nonlinearity. The conclusions drawn in Ref. [79] suggest that this slightly nonlinear behavior is not associated with the gradual failure of 0 degree oriented fibers during loading, but with a localized plastic behavior of 45 degree-oriented plies. However, such behavior is limited by the 0 degree plies and plasticity cannot develop. At 120°C, the ductile behavior of the PPS matrix is enhanced, but the laminate response, being primarily fiber dominated, is therefore not seriously influenced by the matrix response at high temperature (see Fig. 6.9A). From the fatigue thermomechanical response standpoint, the stress-strain diagrams exhibit a quasi-linear behavior. There are no significant hysteresis loops and there is no noteworthy change in the loop shape. However, it can be noted that every test displays a small shift of the loops along the strain axis in the initial stage of fatigue life (about 0.2%) and reduced stiffness degradation. In particular, frequency has virtually no effect on loops’ shape and shifting (see Fig. 6.10) due to the, dominated by the fiber’s time-independent response laminate behavior. Both materials have the same reinforcement, but differ in the nature of their matrix. PPS has a highly ductile behavior and is time-dependent at 120°C (> Tg), whereas epoxy matrix has a more brittle

Fig. 6.10 Influence of frequency on the stress-strain responses of Q-I laminates subjected to tensile fatigue loadings at T¼ 120°C and 80% σ ult: (A) C/PPS and (B) C/epoxy.

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Fatigue Life Prediction of Composites and Composite Structures

Fig. 6.11 Edges microscopic observations of QI C/PPS laminates showing lots of matrix-rich regions.

behavior. The laminate fatigue response, presented in Fig. 6.10, shows that frequency has an effect on the behavior of the QI laminates resulting from the presence of a large number of matrix-rich regions (see Fig. 6.11). Though the thermomechanical behavior is primarily fiber-dominated, it seems that these plain matrix regions play an important role depending on the ductility and the time-dependent behavior of the polymer matrix. In epoxy-based laminates whose behavior is rather brittle, matrix micro-cracking can be initiated in these regions causing debonding at the fiber/matrix interface, inter-ply cracking and, ultimately, delamination [48]. Unlike C/epoxy, those regions will endure a localized plasticity at 120°C in C/PPS laminates, delaying the onset of cracks, thanks to the enhanced ductility of the TP matrix. In addition, when inter-ply cracking is initiated, it will propagate until it meets with the matrix-rich regions, resulting in a substantial crack growth resistance of the material. Frequency has different effects on each material’s fatigue life (see Fig. 6.12). In the case of C/PPS, an increase in frequency and applied stress resulted in decreasing the fatigue life. In contrast, C/epoxy specimens tested at low frequency and low applied stress were found to fail faster. The results obtained at 90% σ ult have to be considered carefully, because there is a significant standard deviation on the number of cycles to failure. Even though the fatigue tests on C/PPS laminates at 70% σ ult and 1 Hz still

Fig. 6.12 Diagrams S-N of C/PPS and C/epoxy QI laminates subjected to fatigue tensile loadings: influence of frequency on fatigue life.

High-temperature fatigue behavior of woven-ply thermoplastic composites

215

have to be undertaken, the above observations suggest that there will be no failure before 1 million cycles under these loading conditions. As introduced in Section 3.1, the damage variable d(N) can be estimated in QI laminates from Eq. (6.2), when the initial and the final stiffness (corresponding to the first and the last stress-strain loops, respectively) are known. Such normalization is particularly interesting to determine the state of damage at any time N of fatigue life, that is, with respect to ultimate failure. From this definition of accumulated damage, it is therefore possible to compare the changes in damage accumulation during fatigue tests depending on test frequency (see Fig. 6.13). The calculation has been carried

Fig. 6.13 Changes in damage accumulation d(N) in C/PPS and C/epoxy laminates subjected to tensile fatigue loadings at 120°C depending on test frequency and maximum applied stress.

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Fatigue Life Prediction of Composites and Composite Structures

out by using the secant modulus’ method [113] between the extremum points of each loop. This method was compared with the classical modulus evaluation described in standard EN6035 [108] and was found to be equivalent, validating this approach. In both materials, damage grows more rapidly as applied stress increases. The effect of frequency depends on both the applied stress level and the studied material. In C/PPS laminates, for every applied stress level, a high frequency seems to accelerate damage accumulation. In contrast, in C/epoxy laminates at high stress levels, low frequencies seem to be more detrimental to damage accumulation. However, such an effect seems to decrease with the applied stress. It seems that, for low stress levels (70% σ ult), frequency has virtually no influence. The tests conducted at 90% σ ult have to be considered with caution in both materials since the fatigue behavior is very close to the monotonic one. In order to discuss the damage chronology, several fatigue tests were undertaken and stopped at different stages (20%–40%–60%–80%) of the fatigue life [13]. These tests were carried out at 80% σ ult and 10 Hz, to find a compromise between a long fatigue life and a reduced duration. In each specimen, two longitudinal and transverse cuts were made to investigate the damage accumulation in the material. From these microscopic observations, damage chronology can be identified in both materials. In C/PPS laminates, the early life (from virgin state to 15% of fatigue life) is characterized by the onset of a few longitudinal and transverse yarn cracks (see Fig. 6.14) due to the coalescence of localized fibers/matrix debonding in 45 degree plies [119]. The next stages are characterized by a generalization of the intra-yarn cracks as well as the onset of intra-ply cracking in 45 degree plies due to the coalescence of the yarn cracks. During the last stage of fatigue life, propagation of the intra-ply cracks along

Fig. 6.14 Illustration of the different damage mechanisms observed during fatigue test [48, 118].

High-temperature fatigue behavior of woven-ply thermoplastic composites

217

weft and warp fibers interface can be observed, as well as some “meta-delamination” in 0 degree/90 degree plies and much localized edge effects; ultimately failure is associated with the breakage of 0 degree fibers. In C/epoxy laminates, the fracture surfaces display an extensive pull-out of broken bare fiber bundles through the thickness. Such damage arises with important delamination, and perfectly debonded plies, highlighting the reinforcement architecture, especially at 10 Hz. Damage (mostly inter-ply cracking and delamination) extends in the specimen far from the fracture surface. It shows a gradual scenario of damage accumulation by intra-ply and inter-ply growth during fatigue loading. At lower frequencies (1 Hz), a similar damage scenario can be observed with the same type of mechanisms (inter-ply and intra-ply cracking—see Fig. 6.14). However, the damage area seems to be much more localized around the fracture area with a few inter-ply cracking along the specimen’s surface. The difference between the two frequencies can be explained by localized plasticization of the matrix, probably in the rich resin areas at low frequency. In both materials, these cracks are important because crack tips are known to act as onset sites for delamination that might lead to fiber breakages. As these cracks propagate, they may interact with matrix-rich regions, resulting in substantial crack growth resistance (see Fig. 6.15) and a better resistance to delamination, subsequent to the

Fig. 6.15 Matrix plasticization around the cracks tip in matrix-rich areas (at the crimps where warp fiber bundles undulating over weft fiber bundles).

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Fatigue Life Prediction of Composites and Composite Structures

interlaminar fracture toughness, as it was highlighted in the literature [120, 121]. From the experimental results of the present study, matrix-rich areas resulting from the nonplanar weave structure may act differently, depending on matrix ductility, on damage mechanisms, hence the fatigue behavior of woven-ply laminates. On the one hand, these regions will act as “bridges” for cracking, allowing a fast propagation throughout the laminates in brittle matrix composites. On the other hand, these regions will act as crack “barriers,” slowing down the crack propagation in ductile matrix composites. This effect is emphasized when temperature is higher than Tg, as it exacerbates timedependent effects in the matrix.

6.4.3 Autogenous heating under fatigue loading The amount of energy loss and hysteresis heat that is generated during cyclical loadings depend on a number of factors [28]; the loading conditions (loading level and frequency), the plastic-viscoelastic and viscoplastic behaviors of the material (depending on the matrix nature and its Tg), the heat transfer characteristics (such as the material’s thickness, geometry, and thermal conductivity), and the material damage characteristics, as manifested by the rate of damage development and growth under fatigue loading. As fatigue proceeds, a number of damage types start to develop and progress within the material, and at the same time, creep, and relaxation effects can operate in the matrix-rich regions of the laminates. The generated heat leads to thermal softening which reduces the ability of the matrix to transfer the load to the reinforcing fiber [56]. Damage also influences heat generation by increasing the effective stress acting on the undamaged regions of the material and by increasing the hysteretic losses [11, 24, 39, 63]. Indeed, cracks appear between warp and weft fiber bundles, and between plies, resulting in internal friction during cyclical loading which leads to a temperature increase. During fatigue tests conducted on QI and AP laminates, the surface temperature can be measured at the specimen surface with a surface-mounted thermocouple. It is reasonable to consider that the temperature rise is more significant at the specimen core, as it has been shown in PEEK-based laminates by Xiao and Al-Hmouz [24]. Thus, it should not change the interpretation of the results about the influence of ductility in matrix-rich regions on the fatigue behavior of woven-ply PMCs. In QI laminates, the temperature increase is limited to 20°C (see Fig. 6.16), and it therefore does not influence dramatically the fatigue behavior of both materials. In AP laminates, the autogenous heating depending on matrix ductility is more significant in C/PPS than in C/epoxy laminates as the surface temperature is higher (see Fig. 6.17). Considering that the test temperature is 120°C, the temperature increases up to about 200°C (i.e., twice as high as the Tg) at the specimen surface in C/PPS laminates. The gradual decrease in temperature around 30,000 cycles at 10 Hz and 50% σ ult was due to the detachment of the thermocouple at specimen’s surface, because of the large rotations. In C/epoxy laminates, the temperature increase is limited to 175°C at the specimen surface, suggesting that the temperature at the core is probably higher than the Tg of C/epoxy (190°C). Three primary phases can be observed on these curves:

High-temperature fatigue behavior of woven-ply thermoplastic composites

219

Fig. 6.16 Influence of frequency and applied stress level on the temperature gradient at the surface of QI laminates: (A) C/PPS and (B) C/PPS vs C/epoxy at 80% σ ult. l

l

l

The run-in of the fatigue test where a certain amount of permanent deformation occurs with rapid increase in temperature. A steady-state phase where there is a gradual increase in both temperature and permanent deformation. The end-of-life where there is a sudden growth in both temperature and permanent deformation in C/epoxy laminates. In C/PPS laminates, the last phase (just before failure) cannot be observed due to the loss of contact between the thermocouple and specimens surface, because of the extensive delamination coming along with the large rotation of fibers. The autogenous heating of specimens is of the utmost importance, as it exacerbates the ductility of matrix-rich areas, which has a significant impact on the fatigue behavior of AP laminates. Firstly, the fibers rotation comes along with the plasticization of the matrix, and the plies move relatively to one another in a scissoring action allowing correspondingly more or less large plastic deformations (see Fig. 6.6).

6.4.4 Creep-fatigue interaction in TP-based composites [14] The creep-fatigue interaction is particularly interesting for the industry because different kinds of preloads (Improper handling, storage accidental loading, tool drop, uncertainties of the service loads, proof loads, etc.) may occur accidentally or on

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Fig. 6.17 Influence of frequency and applied stress level on the temperature gradient at the surface of AP laminates: (A) C/PPS and (B) C/epoxy.

purpose during the life of a structure. The consequences of those preloads on the structure’s fatigue life are not well known. Most of the works available in the literature deal with the creep-fatigue interaction of PMCs at room temperature. However, considering the strong influence of time-dependent effects on the fatigue behavior of PMCs, testing temperature is also of the utmost importance, particularly at T > Tg when viscoelasticity and viscoplasticity are exacerbated [33, 78]. Indeed, the fracture toughness increases at T > Tg, and the crack propagation in high-temperature fatigue can be classified into two types: time dependent and cycle dependent. This section is directed toward the improvement of the creep-fatigue interaction understanding in TP-based composites at high temperature. Significant portions of creep strains characterize C/PPS laminates with an AP lay-up when they are subjected to creep loads at T > Tg [82], and their marked time-dependent effects may significantly change the fatigue behavior. For this purpose, an experimental fatigue campaign was performed on AP laminates with different types of prior creep/recovery loadings. The timedependent response of PMCs is associated with the inherent time-dependent behavior of the polymeric phase and the temporal behavior of carbon fibers due to fiber-matrix debonding, interlayer delamination. Since all the later damage mechanisms are also time-dependent and occur cojointly with polymeric creep, it is difficult to separate out the individual contributions of the composite constituents. Thus, in order to

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examine the influence of viscous effects (viscoelasticity and viscoplasticity) and damage on the fatigue behavior, creep-recovery tests were conducted at stresses lower or higher than a damage threshold. Such an approximate threshold was roughly determined from load-unload tests performed at 120°C on AP laminates at a load rate 0.3 MPa/s [122]. The load-unload tests consist in performing a gradual tensile loading, and in evaluating the loss in laminates’ stiffness from which the approximate damage threshold can be obtained: σ damage ¼ 80 MPa ¼ 50 % σ ult. A reference fatigue life is defined from the average value of fatigue life (50,612 cycles) obtained from several fatigue tests at 1 Hz and 50% σ ult [9]. Since this value is an average, the “virgin” reference (fatigue life equal to 38,185 cycles) chosen in the figures for comparison purposes represents the fatigue behavior of specimens subjected to no prior creep. In order to investigate the creep-fatigue interaction at T > Tg, creep-recovery fatigue (CRF) tests have been conducted at 120°C. These tests consist in subjecting AP laminates to the three different prior load cases (CR1, CR2, CR3) before they were subjected to fatigue tests at 1 Hz and 50% σ ult: creep-fatigue interaction at T > Tg, CRF tests have been conducted at 120°C. These tests consist in subjecting AP laminates to the three different prior load cases (CR1, CR2, and CR3) before they were subjected to fatigue tests at 1 Hz and 50% σ ult: l

l

l

CR1: Creep loadings at 120°C for 24 h at different loads (ranging from 25% to 60% σ ult) followed by a recovery period of 48 h. CR2: 24 h multisteps creep loadings (gradually increasing from 25% to 50% σ ult) at 120°C followed by a recovery period of 48 h. CR3: 1000 s multisteps creep-recovery loading at different loads (ranging from 25% to 60% σ ult) and at 120°C.

As far the influence of prior creep on fatigue life is concerned, load cases CR1 and CR3 show that, compared to the “virgin” reference fatigue life, the fatigue life is virtually unchanged (cf. Fig. 6.18), when the applied stress is lower than the damage threshold. When the creep stress is higher than the damage threshold, a significant increase in the fatigue life in the range of 64%–348% is observed depending on the applied creep stress (load cases CR1 and CR3). In addition, the increase in fatigue life is +114% at 100 MPa for the load case CR3 (3000 s hold time), whereas it reaches +348% at 100 MPa for a 24-h hold time (load case CR1). The load case CR2 leads to +80% in the fatigue life, whereas the fatigue life following a single creep recovery at 80 MPa remains unchanged. These observations suggest that longer hold times also contribute to the improvement of the fatigue life (in the case of stress-controlled loadings). The mechanism responsible for the increase in the fatigue life will be further discussed. As a conclusion, the present results clearly confirm the weak trends already observed in the literature about the creep-fatigue interaction: compared to the reference data, the fatigue life is significantly extended with prior creep depending on loading conditions. The fatigue life seems to be improved for creep load levels such as σ creep  σ damage, and when hold time is significantly longer. In order to further investigate the creep-fatigue interaction, the changes in the fatigue mean strain εamplitude ¼ εmax  εmin as a function of the number of cycles N can be compared between a virgin specimen and specimens subjected to load cases

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Fig. 6.18 Influence of prior creep-recovery loadings (CR1, CR2, and CR3 load cases) on the fatigue life of AP C/PPS laminates during constant amplitude fatigue tests (50% σ ult, 1 Hz, 120°C).

(CR1, CR2, and CR3). The initial strain amplitude is slightly higher for the virgin specimen (about 1%). The strain amplitude of the virgin specimen is always higher than that of the creep-recovery specimens for the same cycle number (see Fig. 6.19). All curves display virtually constant strain amplitudes for the first 100 cycles in semilogarithmic presentation. Then, they increase linearly for approximately one decade of fatigue life (until about 1000 cycles). For all specimens, an inflexion of the curves can be observed after about 1500 cycles. From this inflexion point, the strain amplitude gradually increases, and accelerates when approaching failure. The effect of a prior creep is such as it more or less slows down the evolution of

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Fig. 6.19 Comparison of strain amplitude in AP C/PPS laminates during constant amplitude fatigue tests (50% σ ult, 1 Hz, 120°C): (A) with and without prior CR1 loading and (B) with and without prior CR2 or CR3 loadings.

the strain amplitude, depending on creep stress (see Fig. 6.19A), but more significantly on hold time (see Fig. 6.19B). Indeed, the curve associated with load case CR2 is always lower the curve associated with load case CR3 (40–100–40 MPa). In order to evaluate the damage accumulation during fatigue tests, a damage variable d(N) based on the fatigue mean strain εmean ¼ (εmax + εmin)/2 obtained from stress-strain loops during cyclic loadings can be used as introduced in Eq. (6.3). From this definition, it is therefore possible to compare the changes in damage accumulation during fatigue tests depending on load cases (see Fig. 6.20). Damage growth d(N) is always faster in virgin specimens, and prior creep loadings result in slowing down the damage growth, regardless the load cases. These results confirm the conclusions drawn in the literature [33, 54, 72, 78]: the development of plastic or viscoplastic deformations during prior creep loadings minimizes stress intensities in the matrix. The plastic deformations are locally very large in C/PPS laminates, more particularly in matrix-rich regions where they proved to be barriers for fatigue cracks [12], and prevent PPS matrix from cracking. It also worth noticing that fibers bundles align

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Fig. 6.20 Comparison of damage accumulation d(N) in AP C/PPS laminates during constant amplitude fatigue tests (50% σ ult,1 Hz, 120°C): (A) with and without prior CR1 loading and (B) with and without prior CR2 or CR3 loadings.

along with the plasticization of the matrix, and the fatigue behavior shifts from a matrix-related mode to a fiber-related mode. Both mechanisms (stress intensities reduction and fibers rotation) may ultimately result in increasing the fatigue life after creep. These results showed that preloads (creep loading in the present case) influence the fatigue behavior of C/PPS laminates at T > Tg, when matrix ductility and timedependent effects are exacerbated. Thus, different types of preloads (for various creep stresses and creep times) are found to significantly increase the fatigue life of laminates compared to the virgin state for long creep steps or high stress levels. Simultaneously, the strain accumulation seems to slow down after a long time creep preload, as if the time-dependent mechanisms were “evacuated” during this preload. The same conclusion can be drawn for damage accumulation d(N) when the prior creep stresses are higher than the damage threshold or when the hold time is long enough, inducing significant plastic or viscoplastic deformations. Such deformations originally result from a significant rotation of fibers bundles, a distinctive phenomenon that can be observed in angle-ply laminates. In addition, prior creep loadings at high stress levels will also possibly initiates different types of damage

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(essentially fiber/matrix debonding) within laminates. Large plastic deformations coming along with the alignment of the fibers, the reoriented fibers eventually bear a more significant portion of the mechanical load. This mechanism contributes to a localized relaxation of the stresses, leading to extended fatigue life. Finally, the effect of test frequency needs to be further investigated, as it may also influence the portion of cyclic creep because the time-dependent mechanisms are preferentially activated at low frequency.

6.4.5 Influence of matrix nature on the strain energy release rate in notched laminates [15, 16] The influence of stress concentrations on the thermomechanical behavior of composite structures depends on scaling effects [123, 124], but also on the ability of a material to accommodate overstresses near the notch [79, 80, 86, 87]. This ability is closely associated with matrix ductility, as ductility may delay matrix cracking in TP and TS-based laminates [84–90]. More specifically, a study was conducted on the fatigue of notched carbon fiber-reinforced polymers with ductile PPS versus brittle epoxy matrix, with underlying damage mechanisms observed at micro- and mesoscopic levels [15, 16]. Through the comparison with a brittle matrix system (C/epoxy) tested at T < Tg, highly ductile C/PPS laminates were tested at T> Tg to investigate the influence of matrix ductility on the redistribution of stresses and resulting damage mechanisms near the notch. SEM and X-ray observations provide experimental evidence to support the discussion on the specific role of matrix ductility in the delay of matrix cracking, and its contribution to the slowing down of inter- and intra-laminar fatigue cracks propagation during fatigue loading. This mechanism seems to increase material toughness, as these matrix-rich regions at the microscopic level act as a crack barrier, resulting in localized damage near the notch. According to Reifsnider [92], for laminates that have off-axis plies, such as the QI stacking sequence, the first damage mode observed is usually matrix cracking. Although the toughness and ductility of the matrix material may accelerate or retard the initiation of such cracks, even to the extent of suppressing matrix crack formation entirely for quasi-static loading in some cases [15], cyclic loading is known to cause matrix cracks in virtually every laminated high-modulus continuous fiber composite material system. Indeed, there is frequently a significant micro-geometric contribution to the fatigue effect. For example, micro-cracks cause local stress concentrations, which cause further damage, etc. In notched laminates, these individual geometric details dominate the problem, and linear elastic fracture mechanics (LEFM) may be used to describe fatigue degradation as the propagation of a single dominant crack in composite materials subjected to monotonic loadings. In the early 1970s, in their equivalent flaw model, Waddoups et al. [125] assumed the existence of intense energy (inherent flaw) regions whose size c is modeled as a through crack of constant length, developing at the edges of the notch in a direction transverse to the loading direction [16]. For isotropic and homogeneous materials, the problem of symmetrical cracks emanating from a circular notch was solved by Bowie et al. [83, 126–129].

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The damage mechanisms observed in notched PPS- and epoxy-based laminates are associated with specific redistributions of stresses ahead of the notch, resulting in a more uniform net section stress, a mechanism also known as wear-in Refs. [97–100]. When notched laminates are subjected to static-loading conditions [79, 80], there is competition between damage and plastic zone formation mechanisms (enhanced at elevated temperature), and both mechanisms contribute to the decrease in stress concentration near the notch. In C/epoxy laminates, notch-tip blunting (splitting) is known to reduce the stress concentration effect of the notch, thereby reducing the laminate’s notch sensitivity. In quasi-isotropic C/PPS laminates, plastic deformation cannot develop in the 45 degree-oriented plies near the notch, even though the matrix ductile behavior is enhanced at 120°C. This effect is too localized to release stresses in the fiber bundles near the notch. When notched laminates are subjected to cyclic-loading conditions, fatigue damage accumulates and propagates, leading to final failure, a mechanism known as wear out [97–100]. At high temperature, both mechanisms depend on matrix ductility and viscosity, and they operate simultaneously to rule the fatigue behavior. The mechanisms of stress redistribution near the notch have been investigated by means of X-rays (XR) observations. It appears that fatigue damage is concentrated in the vicinity of the notch, which causes a more or less localized damage depending on matrix ductility and toughness. The analysis of damage mechanisms suggests that matrix ductility modifies the stress distribution near the notch. In C/PPS laminates, the damage area is much more localized near the notch (see Fig. 6.21A), whereas damage is widespread in C/epoxy laminates (see Fig. 6.21B). A sequence of energy-absorbing events (fiber breakage, plastic deformation, matrix cracking) occurs in a region surrounding the notch tip. In C/PPS laminates, damage is concentrated near the notch, and mostly consists of locally extensive delamination and longitudinal splitting. Damage growth appears to be steady and slow

Fig. 6.21 XR observations of notched QI laminates subjected to fatigue tensile loadings (10 Hz, 80% σ u, 120°C) before failure: (A) C/PPS and (B) C/epoxy.

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Fig. 6.22 Changes in the fatigue damage accumulation and the strain energy release rate in notched QI laminates subjected to fatigue tensile loading (10 Hz, 80% σ u, 120°C) depending on matrix nature: (A) C/PPS and (B) C/epoxy.

(see Fig. 6.22A) due to local plastic deformation in matrix-rich areas. In C/epoxy laminates, faster damage growth (see Fig. 6.22B) and widespread damage (longitudinal splitting and matrix cracking in 45 degree plies) are observed due to the low ductility and toughness of epoxy matrix. The idea here is to discuss the processes of energy absorption near the notch in C/epoxy (brittle) and C/PPS (highly ductile) laminates subjected to cyclic loadings. The absorption of mechanical energy is quantified in terms of the strain energy released during translaminar failure initiated from an existing notch (a central notch). In order to quantify the influence of matrix nature on fatigue-accumulated damage, a damage variable d(N) based on changes in stiffness for each fatigue cycle was used (see Eq. 6.2). A model derived from a Paris law and a fracture mechanics criterion were combined to evaluate the fatigue crack growth, but also to compare the changes

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in the strain energy release rate G and the macroscopic damage during cyclic loading [16]. In the following example, the central circular notch has a 3.2-mm diameter d such that the ratio of the notch diameter over specimen width is d/w ¼ 0.2. Compared with the unit cell size in the woven architecture (a five harness satin weave is 7.5 mm wide), the notch diameter is about 3 times lower. Notched QI laminates have been subjected to fatigue tensile loading (10 Hz, 80% σ u, 120°C). Once the stress intensity factor ΔK ¼ Kmax is computed as a function of the number of the cycles N, it is therefore possible to compute the evolution of the strain energy release rate GI vs N for different applied stress levels and both materials (see Fig. 6.11). In C/PPS laminates, the evolution of GI implies that fracture energy is virtually constant as fatigue-loading proceeds, and suddenly increases when approaching the end of the fatigue life. In C/epoxy, fracture energy gradually increases along with the growth of damage within the laminates. It also appears that macroscopic damage d(N) and the strain energies released during fracture are well correlated in both cases (see Fig. 6.22). In C/PPS laminates, the evolution of GI implies that fracture energy is virtually constant as fatigue-loading proceeds, and suddenly increases when approaching the end of the fatigue life. In C/epoxy, fracture energy gradually increases along with the growth of damage within the laminates. In composite materials with a highly ductile matrix (C/PPS), the ability to arrest fatigue cracks through local plastic deformation contributes to the overall improvement in energy absorption capability and thus its fracture toughness. Indeed, the crack arrest or blunting by longitudinal splitting or matrix plastic deformation along the fiber direction leads to a substantial reduction in the stress concentration ahead of the crack, enabling the fibers to sustain higher levels of load prior to fracture. Finally, the energy absorption capability of composites (associated with the strain energy release rate) is significantly enhanced at T > Tg due to the high ductility of the PPS matrix. Ultimately it results in a fatigue behavior virtually independent of the applied stress level under high temperatures T > Tg.

6.5

Conclusions and outlook

With the emergence of high-performance TPs in the 1980s, the fatigue-performance of unidirectional-ply TP based composites has already been addressed by the scientific community to understand why the high fracture toughness of TP composites under monotonic conditions does not necessarily result in high fracture resistance under fatigue loading. Compared to composites with brittle TS matrix, the fatigue life of UD composites with ductile and tough TP matrix is usually shorter. It is well established that matrix nature and the quality of the adhesion at fiber/matrix interface control gradual damages. It also appears that fatigue-induced damages are significantly different in UD- and woven-ply TP laminates. The woven-ply architecture is of importance in the fatigue response of TP composites as they are characterized by matrix-rich regions resulting from the nonplanar interply structure of woven plies. Thus, it appears that TP matrix ductility and toughness (exacerbated at T > Tg) play a significant role in the fatigue performance of woven-ply laminates by modifying the damage mechanisms and their chronology.

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In this context, the main objective of this chapter was to show why woven-ply TP composites present different fatigue behaviors compared to TS composites, with a focus on the high-temperature fatigue damage mechanisms and their relation to the fatigue life. More specifically, the question is to better understand how temperature affects the fatigue behavior, and how creep effects may improve the fatigue performance of PMCs. Thus, the specific influence of temperature on the fatigue behavior (damage mechanisms and fatigue life) of PMCs needs to be investigated in the following configurations: (i) (ii) (iii) (iv)

matrix-dominated behavior (angle-ply laminates) fiber-dominated behavior (quasi-isotropic or orthotropic laminates) creep-fatigue interaction (angle-ply laminates) influence of stress concentration on fatigue behavior.

As far fatigue is concerned, the real difference between TP and TS composites is the significant autogenous heating. It is one of the specificities of the fatigue behavior of TP composites due to hysteretic effects resulting from their strong nonlinear behavior (including ductile and time-dependent behaviors). The generated heat leads to thermal softening which reduces both the fiber/matrix interfacial strength and the ability of the matrix to transfer the load to neighboring fibers. In some cases (e.g., angle-ply laminates at high frequency and at test temperatures higher than Tg), thermal softening dramatically degrades the mechanical properties and may even dominate over any intrinsic mechanical fatigue effects. Cyclic loadings with a nonzero average applied stress inducing creep damage and deformation in addition to fatigue damage accumulation, autogenous heating also exacerbates creep effects. Particularly in angle-ply laminates in which creep and fatigue are mutually influencing phenomena. Thus, the fatigue life can be significantly extended with creep depending on loading conditions (frequency, applied stress, and surroundings temperature). Indeed, the accumulation of plastic and time-dependent deformations in fatigue seems to slow down or even to stop damage propagation within TP laminates. Such deformations are associated with the alignment of fibers, eventually bearing a more significant portion of the mechanical load. This mechanism contributes to the reduction of stress intensities at cracks tips, ultimately resulting in increasing both fatigue life and maximum strain at failure. Even in laminates with a fiber-dominated behavior (e.g., quasi-isotropic laminates), TP composites may also exhibit creep, stress relaxation, and delayed failure. As a result, they are also very sensitive to the loading conditions (especially the loading frequency). In notched composites, local damage mechanisms and subsequent propagation at microscopic level are also controlled to a large extent by matrix ductility and toughness, which therefore drive the extension of fatigue damage at both meso- and macroscales. These mechanisms seem to increase material toughness hence justifying the structural fatigue performance of TP composites. On the whole, woven-ply TP composites are promising for structural applications under high-temperature fatigue-loading conditions. By means of a specific phenomenon (autogenous heating) and due to their intrinsic properties (fracture toughness, high degree of ductility, and viscous behavior), the woven-ply architecture allows TP composites to delay and reduce fatigue-induced damage.

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There are still open questions regarding the fatigue behavior of woven-ply composites. Firstly, it is difficult to quantify the temperature increase resulting from the heat induced by cyclic loading, whatever the loading conditions and the laminates lay-up are. From a technical standpoint, the core temperature of the laminates cannot be easily measured. Thus, numerical modeling should be considered to examine the temperature gradient evolution within the mesostructure during fatigue loading. The knowledge of temperature throughout the laminates is a key point to explain its local influence on matrix mechanical properties as well as on damage mechanisms. Secondly, the influence of temperature on matrix transverse crack density and the interfacial adhesion should be further investigated as they are closely associated with damage initiation and subsequent propagation. Finally, the fatigue after impact should specifically be addressed as impact is one of the most critical conditions in service for load-carrying parts.

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Fatigue behavior of thick composite laminates

7

Rajamohan Ganesan Concordia Centre for Composites (CONCOM), Department of Mechanical, Industrial and Aerospace Engineering, Concordia University, Montreal, QC, Canada

7.1

Introduction

In modern mechanical, civil infrastructure, and aerospace engineering applications, composite laminates of different types and sizes are being used. The manufacturing processes used for producing these laminates are constantly evolving. A wide spectrum of laminate types and geometries and constituent materials are to be designed and developed for use in adverse service environments, and a wide spectrum of manufacturing and assembling methods have to be used to produce and assemble them into final engineering product satisfying prescribed geometric tolerances and damage tolerant specifications. In many engineering applications, thick laminates have to be used as structural parts and they have to be assembled and fastened together with other structural parts using metallic fasteners such as steel or aluminum bolts. These thick composite laminates are subjected to high cycle fatigue loadings in service. Helicopter yoke structure and robotic structure can be cited as example applications in this regard. The mechanical behavior of thick laminates has been studied in adequate details in the literature during the recent past and it is well known to be distinctly different from that of thin laminates. In thick laminates, considerable through-the-thickness normal and shear stresses get developed under the action of axial, flexural, and torsional loadings. As a consequence, in thick composite laminates, the failure mode of delamination that is due to the interlaminar shear stresses and the out-of-plane normal stress is more dominant. In addition, in thick laminates, significant thermally induced residual stresses are present and they get developed during their manufacturing. The heat generation and transfer inside the thick laminate during fatigue loading would be different from that in the thin laminate since the low thermal conductivity of composite layers would play a major role. The use of metallic fasteners in thick composite laminates results in severe stress concentrations and contact stress distributions. These, in turn, alter the failure development and progression processes. Therefore, there is a need to characterize the fatigue behavior of thick composite laminates taking into account these distinct and critical features of such thick laminates. The present chapter concerns with the fatigue behavior of thick composite laminates. In the present chapter, the distinct aspects of the fatigue behavior of thick composite laminates compared to that of thin laminates are described. Limitations of existing approaches of fatigue behavior characterization in adequately and accurately Fatigue Life Prediction of Composites and Composite Structures. https://doi.org/10.1016/B978-0-08-102575-8.00007-3 © 2020 Elsevier Ltd. All rights reserved.

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characterizing the fatigue behavior of thick laminates are highlighted. An appropriate and efficient approach for characterizing the fatigue behavior of thick composite laminates is presented and explained in detail. An example application of this approach to a thick composite laminate used in an aerospace engineering application such as a helicopter yoke structure is discussed.

7.2

Assessment of existing approaches for fatigue of composites

Fatigue behavior of composite material has since long been considered mainly at the ply-level material response and at the thin-laminate-level response. A vast amount of literature is available in this regard. In such existing works, the approaches developed and used to characterize and quantify the fatigue behavior of composite materials and laminates can be grouped into three major categories [1] that are, (i) Fatigue Life Modeling and Prediction, (ii) Phenomenological and Empirical Modeling, and (iii) Progressive Damage Modeling. These approaches have been developed considering different types of composite material plies and thin laminates made of such plies, and in many cases, based on the inspiration obtained from the approaches that had been used for characterizing the fatigue behavior of metallic materials. Fatigue failure criteria have been developed as an extension and further adaptation of corresponding static failure criteria. In the first approach of Fatigue Life Modeling and Prediction, the individual material degradation mechanisms are not directly concerned with, rather the determination of stress-life relationships or Goodman-type charts based on experimental data is concerned with and the failure criteria or the residual strength determination is established based on these relationships, for the specific composite laminate. The prediction of the stress-life relationships of unidirectional laminates with arbitrary ply orientations is performed based on that of a laminate with fibers aligned in one orientation by suitable characterization methods. Such works are mainly based on a deterministic framework. The statistical methodology has been used but to a very limited extent. These works consider in each case specific constituent materials, ply stacking sequence, loading condition and more importantly a specific laminate thickness. The distinction between different individual failure mechanisms and to some extent their interactions have been incorporated. Use of in-plane stress state assumption, which is acceptable for individual ply and thin laminate, is inherent in most of these works. For the fatigue life prediction, based on a suitable theory of stress interactions, such as the quadratic approximation theory, the failure criteria were expressed in terms of the stress-life relationships for specific failure modes of the laminate such as fiber failure, matrix failure, and delamination. The fatigue life models developed in Refs. [2–6] belong to this approach. A fatigue life model based on the microstructural level interactions between the fibers and the matrix, as well as the interfacial bonding, through a micromechanical analysis, has also been developed [7]. The prediction of the S-N curve of unidirectional laminate with arbitrary ply orientations based on that of a laminate with fibers

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aligned in one orientation has been considered in Ref. [8]. All these works consider a deterministic methodology to the fatigue problem. The statistical methodology has also been used but to a limited extent [9]. The prediction of stress-life curves for variable amplitude loading based on limited experimental data has been conducted through the use of constant life diagrams and a suitable nonlinear constant life formulation [10]. For the fatigue life prediction, the earliest criterion that considered the dynamic state of plane stress loading was proposed by Hashin and Rotem [2]. In this work, a quadratic approximation theory has been proposed and the failure criterion was expressed in terms of three S-N curves that included the variation of tensile off-axis fatigue failure stress for different off-axis angles, the variation of transverse normal failure stress, and the variation of shear failure stress as a function of the number of cycles. These variations have to be determined from fatigue testing of off-axis unidirectional specimens under uniaxial oscillatory load. An extensive series of tests have been performed that demonstrated good agreement of the proposed failure criterion with experimental data. Hashin [3] continued to develop further the quadratic approximation to unidirectional reinforced composites subjected to three-dimensional cyclic stress state, in terms of quadratic stress polynomials and based on the consideration of the transverse isotropy of the material. Two failure modes, fiber failure, and matrix failure, have been distinguished in this work. However, in order to be able to make this distinction between failure modes only the state of completely reversed cyclic loading has been considered. The stress interactions proposed by Hashin do not always fit the experimental results, especially in the case of matrix or fiber in compression [11]. Moderate transverse compression is known to increase the apparent shear strength of a composite ply, which seems to be not predicted by Hashin criteria as well as desired. Moreover, Hashin’s fiber compression criterion does not account for the effects of in-plane shear, which is known to reduce significantly the effective compressive strength of a composite ply. Several researchers have over the years proposed appropriate modifications to Hashin’s criterion so as to improve the predictive capabilities. For instance, the modifications proposed by Sun [12] and Puck [13] are in this regard. In composite laminates, especially in thick laminates, an additional failure mode which takes place is the delamination of adjacent layers due to the interlaminar shear stress and the out-of-plane normal stress. In addition, especially in thick laminates, most frequently, thermally induced residual stresses are present and significant. These two key aspects of thick laminates have not been considered in Hashin’s failure criteria. Other and similar works on fatigue life prediction mentioned in the above consider in each case specific constituent materials, ply stacking sequence, loading condition, and more importantly a specific laminate thickness. The distinction between different individual failure mechanisms and to some extent their interactions have been incorporated in the model development. Use of in-plane stress state assumption is inherent in such works. In the case of thick laminates, the use of such an assumption is not valid since considerable through-the-thickness normal and shear stresses exist in thick laminates, especially the interlaminar shear stresses.

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The second approach of Phenomenological and Empirical Modeling endeavors to take into account, at least to some extent, the degradation of an appropriate and susceptible mechanical property of the ply material, especially the stiffness or the strength. Residual stiffness models characterize the degradation of the stiffness of composite ply through nondestructive testing and evaluation techniques. Residual strength models have been established based on strength reduction as cyclic loading proceeds. The strength of the material may remain almost constant until a certain number of fatigue cycles and after that, the strength may be abruptly reduced to a very small magnitude. Alternately, after reaching a threshold number of fatigue cycles, the strength may decrease gradually and monotonically as a function of cyclic loading frequency and stress ratio. This approach has limitations that (a) a large number of parameters have to be determined based on extensive experimental data, (b) the models were not evaluated for a wide range of loading conditions and frequencies, and (c) the experimental data used correspond only to thin laminates. Residual stiffness models [14–17] characterize the degradation of the elastic stiffness of the composite material ply through nondestructive testing and evaluation techniques. Residual strength models have been established based on strength reduction due to dispersed and accumulative fatigue damage. The strength of the material may remain almost constant or gradually change until a certain number of fatigue cycles mainly due to dispersed damage accumulation and after that phase, the strength may be reduced drastically and abruptly to very small magnitude mainly due to accumulated damage. Alternately, in some cases, after reaching a certain number of fatigue cycles, the strength may decrease gradually and in a continuous manner as a function of cyclic loading frequency and stress ratio [14, 18, 19] till the complete failure of the material. A hybrid model that combines the features of different approaches has also been proposed. Such a hybrid model that combines the strain failure criterion with stiffness degradation model so as to develop a phenomenological model for predicting fatigue failure has been proposed [20] especially for glass fiber reinforced composite laminates. Residual stiffness or strength models involve relatively a large number of parameters compared to stress-life models and these parameters have to be determined using appropriate curve fitting methods based on extensive experimental data. Further, each model or a small group of models have been developed and validated for specific type of loading conditions and were not evaluated for a wide range of loading conditions. Almost all of the experimental data available and used for model development and validation correspond to thin laminates and therefore, the applicability of these models for thick laminates has not yet been assessed. The third approach, Progressive Damage Modeling is based on advanced analytical modeling methodologies such as cohesive zone modeling and they have been developed considering the progression of a specific type of damage in the laminate. This approach possesses the dual capability of prediction of the number of cycles to failure and prediction of mechanical property such as stiffness and strength degradation. Two major groups of such models, predicting damage growth and residual mechanical properties, have been developed. In the former, models characterize accumulation

Fatigue behavior of thick composite laminates

243

of specific damage types, such as matrix cracks and delamination. In such works, adoption, adaption, and extension of damage evolution models that were originally proposed for metallic materials have been the routes followed. The model parameters are to be determined from experiments on specific laminate samples. Such models predict the residual mechanical properties of the laminate based on the relationships between the residual mechanical properties and the specific damage types. Extensive material testing is involved in this approach too. Fatigue tests of laminate specimens until a various number of cycles followed by a static test until failure is involved in addition to fatigue tests to the final failure of laminate specimens. In this approach two distinct phases of fatigue behavior are hypothesized to be present, that is, Damage Initiation Phase and Damage Propagation Phase. Specification and clear demarcation of these two phases are left open to speculation and/or individual judgment and preference, and one has to define themselves what kind of damage aggregation happens in the laminate that eventually leads to final failure, again leaving the specification of final failure to individual judgment and preference. For thin, moderately thick, and thick laminates, the corresponding damage aggregation processes are assumed to be the same. Advanced modeling techniques have been developed considering progressive and accumulative damage in composite laminates. They set out to establish direct or indirect relationships between the number of cycles to laminate failure and mechanical property degradation, such as stiffness and strength degradation. Two major groups of such models, models that quantify the damage growth and models that track residual mechanical properties, have been developed. In the former group, models have been proposed to characterize damage accumulation of specific damage types, such as matrix cracks and delamination [21–23]. In such works, the delamination growth rate under fatigue loading is assumed to be described by the Paris law, which was originally proposed for metallic materials, and an empirical delamination propagation model in which all the three modes of fracture, viz., Tension Mode I and Shear Modes II and III have been incorporated. The model parameters are to be determined from experiments on laminate specimens. Models that endeavor to predict the residual mechanical properties of the laminate have been developed [24–27] based on the relationships between the residual mechanical properties of the composite material and the preselected damage variables, through an appropriate failure criterion such as the modified Hashin failure criterion. Correspondingly extensive material testing is involved. In order to fully characterize a composite material, experimental results based on the three loading conditions of tension, compression, and shear on fibers and resins are needed, and further, for each combination of load and fiber or matrix, two different sets of tests are needed. One set involves the fatigue test of specimens until a certain number of cycles followed by a static test until failure, in order to determine the residual stiffness and strength. The second set involves the fatigue test to failure in order to establish the S-N curves. A set of progressive damage models have recently been developed [28–31]. May and Hallett [31] used cohesive interface elements for modeling initiation and propagation of the damage in composite laminates under fatigue loading. The damage initiation laws based on S-N curves have been applied to the interface elements that are

244

Fatigue Life Prediction of Composites and Composite Structures

located within a zone, which is called as ‘initiation zone’ of the so-called characteristic length. Fatigue damage is hypothesized to be accumulated as a function of loading cycles and to progress from the value of 0 corresponding to the state of no damage, to the value of 1 corresponding to the state of damage initiation in the form of a macroscopic crack. Once a macroscopic crack has been initiated, a suitable crack propagation model is invoked to quantify the damage accumulation based on the Paris law. The delamination crack propagation under cyclic loading is analyzed using a degradation law for the cohesive zone interface element. Development of this law was based on a detailed study of the computational model of the cohesive zone and the calculation of strain energy release rate from this zone, so as to enable a direct link with the experimental data corresponding to the Paris Law. The Paris Law is then used to calculate the required crack propagation rate. The fatigue damage accumulation parameter is defined as a function of the crack propagation rate, strain energy release rate, the instantaneous critical fracture energy and the length of the cohesive zone. The value of the fatigue damage accumulation parameter is then added to the static damage parameter of the interface element defined in Ref. [29], and the sum is defined as the parameter of the total damage accumulated. The static damage parameter has been used to track the accumulation of irreversible damage under quasi-static loading and has been defined as a function of the relative displacement corresponding to the interface failure under mixed-modes-I-and-II loading and the relative displacement corresponding to the onset of softening or the damage initiation. The final failure has been deemed to occur when the total damage parameter reaches the value of unity. The limitation of this methodology is that the stresslife curves are normally obtained by testing the specimens up to final failure. It has not been explicitly stated in this work whether the S-N curves that correspond to failure initiation or those corresponding to final failure should be used in the model. Also, the issue of how to characterize failure initiation has not adequately been addressed. It could be speculated that an appropriate event during the fatigue testing such as either sudden dropping of load bearing capacity up to a certain percentage level or reaching a certain preset value of final strain should be considered as representing the failure initiation and correspondingly prepare the failure initiation S-N curves. Moreover, the cohesive interface model used in this approach consists of two phases, Crack Initiation and Crack Propagation. For the first phase, the interface elements should be introduced to the areas of interest inside the laminate in order to model the damage initiation based on the prescribed S-N curve. After damage initiation and after reaching a certain crack size, the crack propagation phase is activated to find the crack growth rate based on a Paris Law type model. In composite laminates, at the early stages of cyclic loading, cracks are initiated in the matrix material at different locations in the laminate. Once a certain threshold of the density of cracks is reached, the corresponding state could be characterized as damage initiation. However, for the propagation phase, one has to define what kind of damage aggregation happens in the laminate that eventually leads to final failure. For thin and thick laminates, the corresponding damage aggregation processes are assumed to be the same, which is questionable.

Fatigue behavior of thick composite laminates

245

As can be observed from the above review and assessment, the suitability, efficiency, and accuracy of existing approaches to characterize and quantify the fatigue behavior of thick composite laminates have not yet been evaluated and established. Further, from the viewpoint of mechanical behavior and structural response, thick laminates exhibit distinct nature and characteristics compared to thin laminates. This, in turn, causes the fatigue behavior of thick composite laminates to be distinctly different from that of thin composite laminates.

7.3

Aspects of fatigue behavior of thick laminates

Three key aspects play a major role in constituting the behavior of thick composite laminates under fatigue loading. They are: (i) Out-of-plane normal and shear stresses in the laminate (ii) Residual stresses induced during manufacturing (iii) Temperature increase during cyclic loading

1.00

The behavior of thick laminates under fatigue loading has not so far adequately been investigated in the literature. Research work in this direction has very recently been initiated [32, 33]. High-cycle fatigue behavior of thick laminates has been studied in these works based on experimental investigations and computational simulations. Appropriate characterization methodology has also been developed. It would be worthwhile to first infer from the experimental investigations on the fatigue response of thick laminates. Relevant test results have been reported in the works of Hamidi et al. [32, 33]. The flexural fatigue response of thick laminates has been investigated in consideration of aerospace applications in which thick laminates have to be used. The dimensions of the laminate specimen are shown in Fig. 7.1. The laminate specimen has the lay-up configuration of [0]80.

2,y

4.00

Top surface

Side A 1,x Bolt row 1

3,z

3.00

0.50

Side B

Bolt row 2 0.06

1,x

0.72

1.00 2.50

5.50 10.50

Fixed end

14.00

Loading end

Fig. 7.1 Thick laminate test specimen [33]; All dimensions are given in inches.

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Fatigue Life Prediction of Composites and Composite Structures

1.00

40

0.98

38

0.96

36

0.94

34

0.92

1st Delamination+Sound

32

0.90 30

0.88

28

0.86 0.84

26

0.82

24

0.80 1

10

100

1000

10,000

100,000

Temperature (°C)

Load ratio, F/F0

An aerospace-grade material, CYCOM E773/S2 glass-epoxy composite material has been used to manufacture the laminate. The fatigue behavior of these thick composite laminate samples under cyclic bending has been studied using a special flexural fatigue test machine that was specifically designed and constructed for the research work at Concordia Center for Composites. The test fixture was designed to hold the specimen in the horizontal direction as a cantilever beam and the laminate specimen was bolted to the frame at the fixed end. At the loaded end, the specimen was clamped to the ball joint which was in turn connected to the actuator. The fatigue tests have been conducted with a cyclic displacement ratio, which is the ratio of minimum displacement over maximum displacement imposed onto the laminate specimen, of 0.1 and a loading frequency of 3 Hz. A displacement controlled fatigue test has been conducted. For preset levels of minimum and maximum deflections and for the set load frequency, the corresponding cyclic loading that needs to be applied onto the laminate specimen has been determined. This loading has been called the “load-bearing capacity” of the laminate specimen. In order to determine this capacity, the quasi-static test has been conducted on the fatigued laminate specimen after a certain number of load cycles had been applied. The fatigue behavior of the laminate specimen has been characterized by the variation of the load ratio, F/F0, which is the ratio of the instantaneous actuator force F(t) to the initial actuator force F0, both forces corresponding to that required to enforce a preset deflection level, with fatigue loading progression represented by the number of fatigue load cycles. The deflection level is a percentage of the deflection that corresponds to the static ultimate tensile strength of the laminate specimen. For the 70% deflection level, this variation is as shown in Fig. 7.2. At the time of fatigue loading, a thermal camera was recording the temperature. The variation of the maximum

22 1,000,000

Number of cycles Fig. 7.2 Variations of the load ratio and the temperature of the thick laminate test specimen for 70% deflection level fatigue loading [33].

Fatigue behavior of thick composite laminates

247

temperature on the side B surface of the laminate specimen as measured by the thermal camera is also shown in Fig. 7.2. As can be seen from this figure, between about 100 load cycles and about 5000 load cycles, the load ratio remains constant and this indicates the first stage in the fatigue response of the laminate. Subsequent to this stage, the load ratio starts to decrease gradually until about 110,000 cycles thereby indicating the second stage in the fatigue response of the laminate. A drastic reduction in load ratio is observed subsequently until around 300,000 cycles, thereby indicating the third stage in the fatigue response of the laminate. At this load cycle value, the load ratio decreases by more than 20% of the initial value, and at this point, the final failure of the laminate has been deemed to have taken place. The variation of the side surface temperature also indicates the occurrence of the three stages of the fatigue response of the laminate and the corresponding critical load cycle values. Initial gradual temperature increase corresponds to the first stage of the fatigue response of the laminate, and a gradual increase corresponds to the second stage, and a rapid temperature increase corresponds to the third stage. An audible sound has been recorded indicative of delamination initiation as and when observed on the side surface of the laminate. A very sharp increase in temperature after the occurrence of the delamination has been observed. A localized delamination failure has been observed to occur in the laminate. Damage initiation and accumulation and/or aggregation in the composite laminate cause considerable stiffness degradation in the laminate. The reduction in the load ratio of the laminate under constant amplitude displacement cycling that can be seen in Fig. 7.2, in turn, corresponds to the reduced load level required to enforce the preset constant amplitude displacement cycling, and hence reflects the stiffness degradation of the laminate. This stiffness degradation hence displays three distinct stages of fatigue damage development and progression as discussed above. The corresponding temperature changes in the laminate also display three distinct regions. A similar study of the load ratio and side surface temperature variations of the laminate specimen with a number of cycles for 75% deflection level brought out the following aspects. For this deflection level too, the fatigue behavior of the laminate evolves in three stages but with considerably lesser values of critical load cycle values bounding these three stages. A localized delamination failure has been observed to occur in the laminate. However, the location and propagation paths of the surface cracks were different. See the work of Hamidi et al. [33] for further details. The DIC contour of the out-of-plane shear strain on the side A surface has been shown in this work. The shear strain profile along a line in the thickness direction has been shown and from this profile, it has been observed that the maximum shear strain occurred at the very close proximity of the mid-plane of the laminate. The delamination has been observed to occur at the location of maximum out-of-plane shear strain. The variations of the load ratio, that is, the ratio of the instantaneous load F(t) over the initial load F0 corresponding to a particular deflection level, as a function of the load cycles has also been determined for the thick [0]80 laminate specimen made of CYCOM E773/S2 glass-epoxy composite material for different deflection levels. The load ratio has also been called a ‘load reduction ratio’ thereby signifying the increase of the compliance of the laminate. The test results are shown in Fig. 7.3. Consideration of the load ratio reduction curves for the laminate specimens tested corresponding to

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Fatigue Life Prediction of Composites and Composite Structures

Load reduction ratio, F/F0

1

0.95

61%

0.9

65% 70% 75%

0.85

0.8

1

10

100 1000 10,000 Number of cycles

100,000 1,000,000

Fig. 7.3 Variation of the load ratio of the [0]80 glass-epoxy composite laminate test specimen with fatigue loading [33].

different deflection levels of 61%, 65%, 70%, and 75% together as shown in Fig. 7.3 would point to the presence of two distinct regions of load ratio reduction and correspondingly, two distinct regions of stiffness degradation. At the initial stage of fatigue cycling the load ratio and correspondingly the stiffness of the laminate decrease gradually. This constitutes the first stage. In the second stage, a sharper decrease in the load ratio and correspondingly the stiffness of the laminate takes place, thereby indicating some kind of unstable damage progression. The threshold number of load cycles that demarcates these two stages has been observed to depend on the deflection level of fatigue cycling. Up to a certain threshold number of load cycles, the load-bearing capacity of the laminate specimen decreases in a gradual manner and after that, a steeper reduction in load- bearing capacity occurs. This threshold number of load cycles is a function of the deflection level imposed on the laminate specimen. This deflection level is a set percentage ratio of the maximum displacement imposed during cyclic loading to the ultimate displacement corresponding to the final failure of the laminate specimen under quasi-static loading. The higher the deflection level, the sharper and more abrupt is the transition from the first phase to the second phase, and also the load cycle value at which this transition occurs becomes lesser. That is, a drastic compliance change occurs at an earlier stage in the life span of the laminate. A direct application of any one of the three approaches described in the previous section would not be accurate and reliable enough especially for thick laminates. There are severe limitations associated with the existing approaches: the plane-stress state assumption is not valid for moderately thick and thick laminates; same damage aggregation or accumulation process is not applicable for laminates of different thicknesses and shapes; the number and types and characteristics of failure modes are vastly different between laminates of different thicknesses and geometries; effects of hostile

Fatigue behavior of thick composite laminates

249

service environments including humidity and temperature have not adequately been incorporated. For moderately thick and thick laminates, the use of plane stress state assumption is not valid since considerable through-the-thickness normal and shear stresses get developed in these laminates. In addition, in thick composite laminates, a dominant failure mode is the delamination due to the interlaminar shear stresses and the out-of-plane normal stress. In addition, especially in thick laminates, most frequently, thermally induced residual stresses are present and significant, and further, the nature and characteristics of these residual stresses differ between various manufacturing methods. Under cyclic loading, some part of the mechanical energy imparted onto the laminate specimen is dissipated due to viscous and structural damping that is inherent in the material. In the case of thick composite laminate since the thermal conductivity of the composite material, especially the glass fiber composite material is very low, heat is generated which in turn results in the temperature increase in the laminate specimen. The relative internal motion of crack surfaces created during the cyclic loading can also generate a certain amount of heat which results in a further temperature increase in the laminate specimen. In the case of thick laminate, this temperature increase would be significant and would need an appropriate measurement. Their influence on altering the fatigue behavior has to be taken into account in the characterization. These key aspects of thick laminates have to be considered in the modeling and characterization of the fatigue behavior of thick laminates. Perhaps, it would be appropriate and preferable to develop a hybrid or mixed approach based on the existing three approaches that were discussed in the previous sections, when the fatigue behavior of thick laminate is considered. In this regard, the modeling based on progressive fatigue damage wherein the damage is represented through appropriate mechanical property degradation and strength degradation due to fatigue cycling would be efficient and versatile when combined with an accurate computational simulation of the stress redistributions in the thick laminate. The composite material has to be characterized based on experimental data to determine the fatigue failure parameters that would be required to evaluate and quantify the above said progressive fatigue damage. Aspects of this characterization are considered and explained in the next section.

7.4

Composite material characterization for failure parameters

A test program is required to determine the parameters that are required for the characterization and quantification of the fatigue response of the thick laminate. This test program consists of two sets of tests that have to be carried out using appropriate composite material test coupons. The first set is to determine the orthotropic mechanical properties of the composite material. Quasi-static tests in accordance with the corresponding ASTM standards should be conducted for this purpose. The second set is to characterize and quantify the fatigue behavior of the composite material in terms of Stress-Life Curves and corresponding tests are to be based on relevant ASTM test standards. The material parameters that are required for use in strength degradation models of the composite material have to be determined using the test data. For

250

Fatigue Life Prediction of Composites and Composite Structures

consistency and compatibility, all the tests of the test program, as well as the tests on thick laminate specimens should be conducted at the same loading frequency. The specific tests to be conducted are to be determined based on the selection of the fatigue failure criteria that is appropriate and accurate for the thick composite laminate.

7.4.1 Fatigue failure criteria A vast majority of existing fatigue failure criteria is based on the assumption of the plane stress state in the laminate. The plane stress state is relevant and accurate only for thin laminates. In thick laminates, a three-dimensional stress state exists. Relatively few of the existing failure criteria consider the triaxial stress state. Among those, the Hashin’s three-dimensional fatigue failure criteria [3] are versatile. They have the capability to distinguish between fiber breakage and matrix cracking failure modes and also are efficient in the sense that only a set of three S-N curves need to be characterized using test specimens of simple configurations and simple loadings. These curves correspond to the tensile off-axis fatigue failure stress for different off-axis angles, transverse normal fatigue failure stress, and shear fatigue failure stress. Hashin’s three-dimensional criteria [3] have been developed considering threedimensional stress state and transversely isotropic property of the composite material and expressed in terms of quadratic stress polynomials. The material behavior in the form of S-N Curves mentioned above for simple stress loadings should be determined considering the combined stress state. The failure modes of fiber failure and matrix failure have been distinguished and corresponding fatigue failure functions have been developed. For the fiber failure mode, the fatigue failure function is given as Ff ðσ 11 , σ 12 , σ 13 , R, N Þ ¼ 1

(7.1)

For the matrix failure mode, the fatigue failure function is given as Fm ðσ 22 , σ 33 , σ 12 , σ 23 , σ 13 , R, N Þ ¼ 1

(7.2)

Based on these failure functions, Hashin’s failure criteria for reversed cycling are given as


σ 11 σ F11T

2

σ2 + σ2 + 12 213 ¼ 1 τF12

(7.3)

for fiber failure mode, and ðσ 22 + σ 33 Þ2 σ 223  σ 22 σ 33 σ 212 + σ 213 +  2 ¼ 1  F 2 +  F 2 σ 22T τ23 τF12 for matrix failure mode.

(7.4)

Fatigue behavior of thick composite laminates

251

In the above equations, σ 11, σ 22, and σ 33 denote the applied normal stresses in longitudinal (fiber), in-plane transverse, and out-of-plane transverse directions, and further, σ 12 is the on-axis in-plane shear stress, and σ 13 and σ 23 are out-of-plane shear stresses. Superscript F denotes the fatigue failure stress value, that is, the fatigue strength of the composite material, which is a function of cyclic stress ratio R, number of cycles N, and cyclic load frequency f. Subscript T denotes the tensile stress. In order to use the above failure criteria for a composite laminate, the corresponding composite material should be characterized using appropriate experimental investigations to obtain the material strength parameters involved in the criteria. The parameters of the fatigue loading, including the frequency and stress ratio of the cyclic loading, applied on test coupons should be the same as that applied in service on the composite laminate.

7.4.2 Determination of fatigue strengths of the composite material The fatigue failure stresses, that is, the fatigue strengths, of a specific composite material have to be obtained from the corresponding S-N curves. These curves have to be obtained from coupon tests following ASTM D 3039 test standard, as described in the following. To determine the Tensile Fatigue Strength in Longitudinal Direction and corresponding S-N Curve, unidirectional specimens are manufactured with fibers along the longitudinal direction. The specimen geometry is shown in Fig. 7.4. Specimens of unidirectional 8-layer laminate with 0-, 30-, and 45-degree fiber orientation angles have to be manufactured. For each off-axis angle, a total of four specimens have to be tested under quasi-static loading condition to obtain the average ultimate strength of the specimens. The quasi-static tests have to be conducted in accordance with the corresponding ASTM test standard. Using the obtained average ultimate strength, cyclic loading has to be applied using three different stress ratio values. For each stress ratio, a total of four specimens have to be tested. From the test data thus obtained for the selected off-axis angle, the S-N curves of σ F11T are obtained and using the suitable curve-fitting method, the data is expressed in the form of a 2.50''

4.00''

Glass epoxy tabs

2.50''

0.75''

0.072'' (8-Layer laminate)

Fig. 7.4 Test specimen geometry for the determination of tensile fatigue strength.

0.06''

252

Fatigue Life Prediction of Composites and Composite Structures

straight line linear equation with two parameters representing σ F11T. This test procedure is repeated for other off-axis angles using corresponding test specimens. It should be noted here that for 0 degree specimens the value of σ F11T can directly be obtained from the test data. For specimens with other fiber orientation angles, test data will only provide the strength of those off-axis specimens under cyclic loading. Appropriate stress transformation equations of laminate mechanics have to be used to obtain the value of σ F11Tfrom the test data for the composite material considered. In order to obtain the Tensile Fatigue Strength in Transverse Direction σ F22T and corresponding S-N Curve, the direct test method involves loading laminate specimens with fibers oriented perpendicular to the loading direction. However, considerable and unavoidable test data scattering would be observed when using this method. Instead, an alternative test method, an indirect method, using off-axis specimens can be used. In this indirect method, two test specimens each with an arbitrary fiber orientation angles are loaded in tension until matrix failure happens. The applied stress values at the failure of these two specimens are obtained from testing. Then using the twodimensional fatigue failure criterion for matrix failure proposed by Hashin and Rotem [2], two equations can be set up in two unknowns, one unknown variable being the tensile fatigue failure stress in the transverse direction σ F22T. The other unknown variable is shear fatigue failure stress τF12. This criterion is given as


σ 22 σ F22T

2
2 σ 12 + F ¼1 τ12

(7.5)

In the test, the axial stress σ is applied on the specimen at an angle θ with respect to the fiber orientation as shown in Fig. 7.5.

s

2

1

q

s

Fig. 7.5 Off-axis in-plane loading of unidirectional laminate test specimen.

Fatigue behavior of thick composite laminates

253

After determining the corresponding stresses σ 22 and σ 12 in the principal coordinate system in the specimen using in-plane stress transformation and substituting these values in the failure criterion given by Equation (7.5), one can get the expression 

sin 4 θ cos 2 θ sin 2 θ 1 +  2 ¼ 2  2 σ σ F22T τF12

(7.6)

Since the test is done on specimens with two different fiber orientation angles θ1 and θ2, for each specimen values of the corresponding fiber orientation angle and the applied stress at matrix failure can be substituted in the above equation, thereby yielding the following set of equations: 8 sin 4 θ1 cos 2 θ1 sin 2 θ1 1 > > +    F 2 ¼ 2 > 2 > σ F ðR, N Þ σ < 1 τ12 ðR, N Þ 22T (7.7) > sin 4 θ2 cos 2 θ2 sin 2 θ2 1 > > > 2 +  F 2 ¼ 2 : F σ2 σ ðR, N Þ τ ðR, N Þ 22T

12

In the above expressions, R denotes the cyclic stress ratio and N is the number of cycles. These two simultaneous equations can be solved to obtain both σ F22T ¼ σ F22T(R, N) and τF12 ¼ τF12(R, N). Hence the tensile fatigue strength in the transverse direction, σ F22T, is obtained. In order to obtain the Shear Fatigue Strength τF12 and corresponding S-N Curve, the direct test method is to conduct a torsion test on thin-walled tube specimens with fibers oriented along and perpendicular to the loading axis. However, manufacturing of reliable and good quality test specimens of this type is rather difficult and expensive, and further, applying properly and precisely the torsional loading is a challenging task. Instead, the indirect test method using off-axis specimens that have been described above for the determination of tensile fatigue strength in the transverse direction, σ F22T, can be used to obtain the shear fatigue strength τF12. In order to obtain the Shear Fatigue Strength τF23 and corresponding S-N Curve, the test procedure involves the use of test specimens that have fibers oriented along the thickness direction. It means that specimens should be manufactured by making transverse cuts through unidirectional specimens, and hence they will be of too small size to test. However, considering that fibers do not have considerable influence in 23 planes, it would be reasonable to assume that τF23 is the same as the shear fatigue strength of the matrix material.

7.5

Failure criteria and failure modes in progressive damage

There exist in literature many failure criteria that have originally been developed to characterize the failure of the composite ply under static loading, called herein for brevity and clarity as a static-type failure. Examples of such failure criteria include

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Fatigue Life Prediction of Composites and Composite Structures

Maximum Stress Criterion, Tsai-Wu Criterion, Tsai-Hill Criterion, and Hashin criteria. Out of these available criteria, the Hashin failure criteria [3] can be selected for the failure analysis if a mode-dependent set of failure criterion is preferred and further, criteria that are easy to apply in the progressive damage modeling are sought. These criteria have all been further developed for the characterization of failure of the composite laminate under fatigue loading, basically by replacing the ultimate static strengths of the ply material with the corresponding residual strengths of the ply material obtained from the stress-life curves of the ply material. These stress-life curves can either directly be obtained from appropriate tests on test coupons made of the composite ply following certain established and widely accepted test standards, as described in the previous section, or equivalently and approximately by using certain residual strength models that are available in the literature which were developed based on certain hypotheses and assumptions about fatigue damage development and progression, or by using an appropriate combination of these two. For the assessment of the material failure, a consistent set of failure criteria such as the Hashin fatigue failure criteria [3] are considered appropriate for each ply in the thick laminate. However, an appropriate combination of different failure criteria can also be used if necessary and relevant. For instance, the Hashin failure criterion for the mode of fiber failure in tension is given by σ 11 τ12 τ13 + + ¼ 1 , σ 11 > 0 σ F11T τF12 τF23

(7.8)

As can be seen above, the in-plane and out-of-plane shear stresses are considered to be influencing fiber failure. However, it has been shown in the existing literature that inclusion of the influence of shear stresses could make this criterion to be overly conservative [34, 35]. Based on this consideration, instead of Hashin’s criterion, for the mode of fiber failure in tension, the maximum stress criterion given below can be used. σ 11 ¼ 1 , σ 11 > 0 σ F11T

(7.9)

In a similar manner, for the mode of fiber failure in compression, one can use the following criterion: σ 11 ¼ 1 , σ 11 < 0 σ F11C

(7.10)

As for the matrix failure modes in tension and compression, one can use the following Hashin’s criteria:


σ 22 σ F22T

2
2
2 τ12 τ23 + F + F ¼ 1, σ 22 > 0 τ12 τ23

(7.11)

Fatigue behavior of thick composite laminates




σ 22 σ F22C

2
2
2 τ12 τ23 + F + F ¼ 1, σ 22 < 0 τ12 τ23

255

(7.12)

For fiber-matrix shear-out failure, the Hashin’s criterion is


σ 11 σ F11T

2
2
2 τ12 τ13 + F + F ¼ 1, σ 11 < 0 τ12 τ13

(7.13)

For the modes of delamination failure in tension and compression, the respective Hashin’s criteria are:





σ 33 σ F33T

2
2
2 τ13 τ23 + F + F ¼ 1, σ 33 > 0 τ13 τ23

(7.14)

σ 33 σ F33C

2
2
2 τ13 τ23 + F + F ¼ 1, σ 33 < 0 τ13 τ23

(7.15)

These failure criteria have to be used in the characterization and quantification of the progressive fatigue damage in the thick laminate.

7.6

Material degradation due to fatigue damage

In composite laminates due to the fatigue damage initiation, accumulation, aggregation, and progression the stiffness and the strength of the plies and in turn that of the laminate decrease continuously and monotonically through the entire fatigue cycling process. The rates of this decrease are not constant through fatigue cycling and keep increasing in a nonlinear manner. There could also be different natures of these changes in the sense that the changes could be gradual through certain phases of fatigue cycling and abrupt through certain other phases. In the progressive damage modeling of the thick composite laminate, the residual strength corresponding to each stress component of the composite ply as a function of applied load cycles needs to be taken into account. These residual strengths have to be experimentally determined. For certain such components, the experimental procedure to obtain the corresponding strength values has already been discussed in a previous section. For the residual strength values that were not or could not easily be experimentally determined, as an alternative, the material property degradation models available in the literature, such as the models presented in Refs. [24–27, 36, 37] can be used. These models quantify the strength degradation of the composite material ply during fatigue loading. The strength SF corresponding to a particular applied stress component σ (represented by σ max

256

Fatigue Life Prediction of Composites and Composite Structures

of the cyclic loading) of a unidirectional composite ply after a certain number of load cycles N of certain stress ratio R is given by 2 sF ðN, σ, RÞ ¼ 41 

log ðN Þ  log ð0:25Þ   log Nf  log ð0:25Þ

!β 31=α  S  5 S σ +σ

(7.16)

where SS denotes the static ultimate strength corresponding to the considered stress component, Nf is the fatigue life corresponding to the considered stress component and the σ max of the cyclic loading and, α and β are the scale parameters that have to be determined using experiments on test coupons as per chosen test standard and appropriate curve fitting method. The residual stiffness components corresponding to the orthotropic stiffness matrix of the composite ply as a function of applied load cycles also need to be taken into account in the progressive damage modeling of the thick composite laminate. The residual stiffness properties can also be determined using appropriate experiments or using the material property degradation models available in the literature that are consistent with the strength degradation models mentioned above. Such a model is available in Refs. [24–27, 36, 37]: 2 EF ðN, σ, RÞ ¼ 41 

log ðN Þ  log ð0:25Þ   log Nf  log ð0:25Þ

!λ 31=γ
 σ σ S 5 E  + εf εf

(7.17)

where EF is the residual stiffness after the application of N load cycles, ES is the stiffness corresponding to static loading, and εf is the average strain to failure, and further, γ and λ are the scale parameters that have to be determined using experiments on test coupons as per chosen test standard and appropriate curve fitting method. The value of Nf involved in the above residual strength and stiffness models has to be determined for each stress level and stress ratio using experiments on test coupons following relevant test standards and an appropriate modeling scheme available in the literature such as the one presented in Ref. [38] for tensile and compressive fatigue loadings u¼

  lnða=f Þ ¼ A + Blog Nf ln ½ð1  qÞðc + qÞ

(7.18)

In the above expression, u and a are model parameters to be determined using test data, and further, parameter q is the quotient of the mean stress of the applied loading over the tensile strength of the material, a is the quotient of the alternating stress of the applied loading over the tensile strength of the material, c is the quotient of the compressive strength of the material over the tensile strength of the material. For shear fatigue loading, parameter c will have to be set to unity since the material strengths corresponding to positive and negative shear stresses are the same. Further, based on previous experience on many materials, an additional logarithm will have to

Fatigue behavior of thick composite laminates

257

be used for better curve fitting of the test data for shear fatigue loading. The resulting expression will be [38]
   ln ða=f Þ u ¼ log 10 (7.19) ¼ A + B log Nf ln ½ð1  qÞðc + qÞ The stiffness degradation described in the above characterizes the damage caused to the composite material ply due to fatigue loading. This degradation is gradual. In addition to this, the composite material can fail due to the critical state of the stress components individually or collectively exceeding respectively the corresponding individual critical strengths or appropriate combinations of such critical strengths, as modeled and represented by appropriate failure criteria. This material failure results in another form of damage which is not gradual but sudden. This failure is influenced by fatigue damage in the sense that both the strength reduction and the stiffness reduction are caused by fatigue damage accumulation and aggregation in the composite material. This sudden material degradation should then also be taken into account in the progressive damage modeling of the composite laminate. Corresponding to different failure modes, at the instant of the composite material failure as determined by appropriate failure criteria, the stiffness properties of the ply are degraded as follows [39]: Fiber Failure : ½E11 , E22 , E33 , υ12 , υ23 , υ13 , G12 , G23 , G13  ! ½0, 0, 0, 0, 0, 0, 0, 0, 0 Matrix Failure : ½E11 , E22 , E33 , υ12 , υ23 , υ13 , G12 , G23 , G13  ! ½E11 , 0, E33 , 0, υ23 , υ13 , G12 , G23 , G13  Fiber=Matrix shear  out Failure : ½E11 , E22 , E33 , υ12 , υ23 , υ13 , G12 , G23 , G13  ! ½E11 , E22 , E33 , 0, υ23 , υ13 , 0, G23 , G13  Delamination Failure : ½E11 , E22 , E33 , υ12 , υ23 , υ13 , G12 , G23 , G13  ! ½E11 , E22 , 0, υ12 , 0, 0, G12 , 0, 0

However, in the computer-aided failure modeling, zero stiffness values can be represented by very small numerical values, very small relative to the nominal stiffness values or a very small fractional value that can be used for the particular computational system.

7.7

Progressive damage development and progression

The progressive damage development and progression in the thick composite laminate as the fatigue loading proceeds has to be simulated using an accurate computational structural model, instantaneous stress distributions in the laminate, material degradation models, structural stiffness modifications corresponding to material degradation, and relevant failure criteria for the composite material. In this regard, Parametric Progressive Damage Modeling using Finite Element Analysis can be performed using ANSYS Parametric Design Language (APDL). Appropriate failure criteria are required to be employed as the basis in the modeling and these criteria have to be consistent with the testing program since the parameters of the criteria have to be determined experimentally. Hence, Hashin’s 3D failure criteria are suitable for thick laminates.

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Fatigue Life Prediction of Composites and Composite Structures

As shown in the flowchart of Fig. 7.6, the Progressive Fatigue Damage Modeling (PFDM) Methodology consists of the following major steps: linear elastic stress analysis, composite material failure analysis, and incorporation of material degradation due to fatigue. After each load cycle is applied on the finite element model of the laminate, the three-dimensional stress distribution in the laminate that corresponds to the maximum load of the fatigue cycle is calculated using the material property values that correspond to the damage state at the beginning of that particular load cycle and considering boundary conditions and loading. The off-axis stress components in structural coordinates for each lamina are obtained using the stress distribution in the laminate based on the laminate theory used for the stress analysis or directly extracted from the laminate stress distribution if the three-dimensional anisotropic elasticity theory has been used for the stress analysis. The on-axis stress components for each ply are then calculated from off-axis stress components using appropriate stress transformation equations. Using the calculated values of on-axis stress components and by applying the preselected failure criteria, the failure analysis is conducted for each ply. The material points where composite material failure is deemed by the failure criteria to have happened are determined. The failed plies and the corresponding mode of failure are identified. The material property degradations of both static-type (sudden) and fatigue-type (gradual) should be incorporated at this stage. The degraded material property values should include both degraded strengths corresponding to the triaxial stress components and corresponding degraded stiffnesses of the composite ply. For each failed ply, considering the failure mode of the ply, the material properties of that ply are degraded based on the preselected material degradation models. This degradation corresponds to the static-type (sudden) failure of the ply. Appropriate static-type degradation models as explained in previous sections are to be used for this purpose. After this property degradation, the resulting laminate consisting of some plies with intact material properties and the rest with degraded material properties is considered. The material property degradation due to fatigue damage is then applied. Both the strength and stiffness properties of the ply are degraded corresponding to the gradual and continuous fatigue damage development in the laminate and using the value of a number of fatigue load cycles that have just been applied onto the laminate. Appropriate fatigue degradation models as explained in previous sections are to be used for this purpose. The resulting laminate can be called as the “updated” laminate. By using the updated laminate that has degraded material properties due to both sudden and gradual degradation, the stress redistribution is calculated for the laminate considering the next load cycle, and by repeating the entire stress analysis procedure used in the previous step. The new material points that have failed after the completion of the subsequent load cycle are determined. The stress redistribution is calculated again for the next load cycle, the failure analysis is reapplied, and the material property degradation is reincorporated as has been done in the previous step. This process is repeated until the last load cycle application is completed or the laminate has reached the final failure stage. In a finite-element-based procedure, with a very fine mesh, instead of material points, the finite elements are to be examined for their failure due to material failure.

Fatigue behavior of thick composite laminates

259

Start

Develop the Finite Element model of the laminate

Input initial material properties, pre-set displacement (loading) and support conditions Calculate stresses in all plies of the laminate in global coordinates

Apply progressive strength and stiffness degradation

Calculate stresses in all plies in material coordinates

Apply further fatigue cycles

Check for the failure of plies in the finite elements

No

Plot the model

Yes Apply stiffness degradation for the failure mode

Apply progressive strength degradation

Plot the model

No

Calculate stress redistribution and Check for failure or plies

Yes

Output Final Stress Distribution and Loading

Fig. 7.6 Progressive fatigue damage modeling methodology.

End

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Fatigue Life Prediction of Composites and Composite Structures

After the application of each load cycle, the stress components have to be determined for each laminate finite element using the results of the laminate stress analysis performed considering the boundary conditions and maximum value of the applied cyclic loading. The off-axis stress components have to be obtained for each ply in each finite element and the failure analysis of each ply has to be performed. If one or few plies of any laminate finite element is deemed to have failed in a certain failure mode, then that finite element is deemed to have failed. The strength and stiffness material properties of those plies are degraded based on the degradation procedure explained above. With the degraded ply material properties as input, the stiffness properties of the laminate finite element have to be determined. Hence, if a finite element is deemed to have failed after the completion of a particular load cycle then the elastic stiffness matrix of the element is altered corresponding to the material degradation of its constituent plies, and hence the progressive fatigue damage of the finite element and in turn that of the structure and the laminate are characterized. With the degraded laminate finite elements, the finite element model of the laminate is reconstructed into a new mesh and the stress analysis for the next load cycle is conducted in a manner similar to the one used for the previous load cycle. Obviously, an appropriate type of laminate finite element and a finer mesh are essential for the reliable and successful application of the above progressive damage modeling methodology.

7.8

Application to a thick composite laminate

Quite recently, in the works of Hamidi et al. [32, 33], the fatigue behavior of a thick composite laminate made of an aerospace-grade composite material, CYCOM S2/E773 glass-epoxy composite material has been studied based on the material and structural aspects described and discussed in the previous sections. The composite material ply has been characterized using experiments on test coupons following relevant test standards in order to obtain the required parameters involved in the material degradation models described in the previous section. The test methods described previously in Section 7.4 have been used to determine the failure strengths of the S2/E773 Glass/Epoxy composite material [32, 33]. For this purpose, 8-layer laminate specimens with 0, 30, and 45 degree fiber orientations were manufactured using hand lay-up and autoclave process. Laminates were cured as per an aerospace industry prescribed cure cycle. After manufacturing the laminates, a glass epoxy composite laminate of thickness of 1.5 mm was used as the loading tab. After attaching the tabs following a standard tabbing procedure, test specimens were cut using diamond saw and tests have been performed. The test results are shown in Figs. 7.7–7.9. A series of flexural fatigue experiments have been designed and conducted on thick 80-layer unidirectional composite laminates made of the CYCOM S2/E773 glassepoxy composite material. A special test setup has been built and relevant details are given in the work of Hamidi et al. [33]. The experimental work consisted of two major parts. In the first part of experimental work, thick unidirectional [0]80 composite laminate test specimens as shown in Fig. 7.1 were manufactured and their

Fatigue behavior of thick composite laminates

261

Cycles to failure vs % of UTS for 0 degree samples, S2 / E773,

100

% of the UTS

90 80

y = –5.028ln(x) + 102.11

70 60 50 40 30 20 100

10,000 100,000 1000 Number of cycles to failure

1,000,000

Fig. 7.7 The S-N Curves for 0 degree specimens of CYCOM S2/E773 glass/epoxy composite material [32].

100

Cycles to failure vs % of UTS for 30 degree samples, S2 / E773,

90

80 % of the UTS

% of the UTS

90

y = –3.782ln(x) + 98.65

80 70 60 50 40 30 20 100

Cycles to failure vs % of UTS for 45 degree samples, S2 / E773,

100

y = –3.543ln(x) + 94.058

70 60 50 40 30

1000

10,000

100,000 1,000,000

20 100

Number of cycles to failure

1000 10,000 100,000 Number of cycles to failure

1,000,000

40 35 30 25 20 15

y = –2.247ln(x) + 47.095

10 5 0 100

Cycles to failure vs transverse shear strength

Cycles to failure vs transverse normal strength

1000

10,000

100,000 1,000,000

Number of cycles to failure

Transverse shear strength (MPa)

Transverse normal strength (MPa)

Fig. 7.8 The S-N curves for 30 and 45 degree specimens of CYCOM S2/E773 glass/epoxy composite material [32]

250 200 150 100 50 0 100

y = –14.15ln(x) + 216.08 1000

10,000

100,000

1,000,000

Number of cycles to failure

Fig. 7.9 The S-N Curves for tensile fatigue strength in transverse direction σ F22T and shear fatigue strength τF12 of CYCOM S2/E773 glass/epoxy composite material [32].

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Fatigue Life Prediction of Composites and Composite Structures

behavior under cyclic flexural loading has been observed and characterized. Delamination in the laminate is caused by out-of-plane normal and shear stress components. When any one or a critical combination of these out-of-plane stresses reach a corresponding critical value that is characteristic of the particular laminate, delamination failure occurs in the laminate. Each out-of-plane stress is directly proportional to the corresponding out-of-plane strain. This strain has been measured during the experiments using a novel modern technique, the Digital Image Correlation (DIC) technique. In this work, appropriate DIC setup has been used to measure the full field shear strain distributions on the top and side A surfaces of the laminate, that are defined in Fig. 7.1. An infrared thermal camera was used to monitor the temperature changes in the tested laminates. Displacement control tests have been performed. The damage occurrence and progression in the laminate have taken to be indicated by three indicators: sudden drop in the actuator force, increase in the side surface temperature and presence of audible sound. Quasi-static tests were performed to determine the ultimate static displacement of sample laminates. The average static ultimate displacement, Uult was obtained and based on this value, four deflection levels were chosen for fatigue loading, that are, 61%, 65%, 70%, and 75%. Each deflection level is a set percentage of the maximum displacement applied during fatigue loading with respect to the static ultimate displacement. The minimum displacement is calculated from the maximum displacement for a preset displacement ratio of 0.1, which is the stress ratio R of the fatigue loading. In the second part of the experimental work, the material characterization has been done. Relevant tests have been performed in order to obtain the static material properties of stiffness and strength and the fatigue stress-life curves of the material. The parameters α, β, γ, and λ of the material property degradation models have been calculated using appropriate curve fitting techniques. The frequency of fatigue loading has been set to be 3 Hz for both laminate tests and material characterization using coupon tests. For further and detailed information about the manufacturing and fabrication of the test coupons, the test setup, test parameters, details about the tests, see Hamidi et al. [32, 33]. The Progressive Fatigue Damage Modeling has been conducted next. The modeling part comprised of three main parts: three-dimensional stress analysis of the laminate, failure analysis, and material property degradation. A combination of maximum stress criterion and Hashin fatigue failure criteria has been used to perform the progressive failure analysis of the tested laminates through the applied fatigue loading. The stiffness and strength properties of the elements were degraded as per the material degradation schemes that were described in the previous section. The predictions based on the progressive damage modeling have been compared with the experimental observations and data regarding the fatigue damage mechanisms and also the flexural load-bearing capacity of the considered laminates. The progressive fatigue damage modeling has been performed based on the concepts discussed in the previous sections and using commercial software. The finite element model of the laminate specimens tested was developed using APDL.

Fatigue behavior of thick composite laminates

263

The material properties required as input to this task were obtained from material characterization coupon tests, which constituted the second part of experimental work as mentioned before. Data from material characterization coupon tests provided both quasi-static and fatigue material properties after processing them through appropriate curve fitting techniques. These properties are available in the works of Hamidi et al. [32, 33] and they have been used in the finite elements-based progressive damage modeling of laminate specimens under the constant amplitude displacement loadings imposed on laminate specimens in the experimental work. All deflection levels considered in the experimental work have also been considered in the computational modeling and corresponding deflection-controlled fatigue loading was applied. The fatigue damage progression has been determined and compared with that observed in the experimental work. Samples of predicted progressive damage in the laminate specimen are shown in Fig. 7.10. Bolt holes shown on the left side of this figure correspond to the fixed end of the laminate sample. It has been observed that the majority of elements failed in the delamination mode and that this failure pattern observed agreed very well with the experimental observations results. The delamination failure leads to a sudden and major drop in the load-bearing capacity of the laminate. The progression of delamination failure takes place along the close vicinity of the mid-plane of the laminate. This has been observed

Fig. 7.10 Fatigue damage accumulation and progression in the laminate specimen for 75% deflection level [33].

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Fatigue Life Prediction of Composites and Composite Structures

in the computational simulation as well as in the testing of the laminate samples. For further details about the computational simulation and experimental work, and for detailed comparisons between computational simulation results and experimental observations, see the work of Hamidi et al. [33]. It has also been shown through the computational simulation that delamination had been the dominant failure mode and that it had occurred due to the out-of-plane shear stress as well as the out-of-plane normal stress in the laminate and at the locations in the laminate sample where these stresses reached their maximum absolute values. Throughout the fatigue loading applied, for all load cycle values and deflection levels, a good agreement of the computational simulation results with the results of experimental work has been shown. These include the results of load ratio reduction as a function of applied load cycles. One such comparison is given in Fig. 7.11, wherein the computational simulation results are identified with the label FPDM (Fatigue Progressive Damage Modeling) and the deflection level of 75% that corresponds to the simulation results and experimental data shown in the figure is also mentioned. In the figure in the expression for the load ratio, F denotes the instantaneous load to be applied on the laminate to impose the maximum deflection corresponding to the 75% deflection level and F0 indicates the load required to impose the same maximum deflection at the beginning of fatigue loading. It can be observed from the above figure that the load ratio reduction takes place in two distinct stages of gradual decrease and rapid decrease and that both the experimental data and simulation results show very good agreement with each other. Similar comparisons have been made for different deflection levels, and very good agreement of test data with simulation results have been shown [33]. It has also been observed that the number of load cycles corresponding to the transition between these two stages is a function of the imposed deflection level of fatigue cycling and that the

1.05

Load ratio, F/F0

1 0.95 0.9 FPDM - 75% Experiment - 75%

0.85 0.8 1

10

100 1000 10,000 Number of cycles

100,000 1,000,000

Fig. 7.11 The load ratio reduction curves obtained from experimental work and computational simulation [33].

Fatigue behavior of thick composite laminates

265

higher the deflection level is the severe is the reduction in the fatigue life. Furthermore, the temperature changes in the laminate through the fatigue cycling have been recorded and presented in the work of Hamidi et al. [33]. The temperature changes in the thick laminate can be observed to be significant and higher than that for thin laminates for the same fatigue loading frequency. In addition, the temperature change in the laminate also takes place in two distinct stages of gradual decrease and rapid decrease.

7.9

Conclusions

In the present chapter, the fatigue behavior of thick composite laminates has been considered. Limitations of existing approaches in adequately and accurately characterizing the fatigue behavior of thick laminates were highlighted. An appropriate and efficient approach for characterizing the fatigue behavior of thick composite laminates has been presented and explained in detail. Application of this approach to an example thick composite laminate has been described. It has been shown through computational simulation and confirmation by the experimental data that (a) the fatigue life of the thick laminate depends largely on the imposed deflection level, that is, the fatigue flexural loading level, (b) the higher the fatigue loading level is, the severe is the reduction in the fatigue life of the thick laminate through a nonlinear dependence, (c) the stiffness reduction takes place in two distinct stages of gradual decrease and rapid decrease, and (d) the number of load cycles that corresponds to the transition between these two stages is a nonlinear function of the loading level of fatigue cycling. The fatigue damage evolution in the laminate as a function of progression of fatigue cycling that was predicted by the computational simulation and confirmed by the experimental observations indicate that the load-bearing capacity of the laminate decreases continuously and monotonically due to fatigue loading, and that the fatigue damage evolution causes the stiffness and strength degradation of the thick composite laminate through a direct correlation. The experimental work revealed and the computational simulation confirmed that under fatigue loading the damage initiation and propagation in thick laminates take place in a localized and concentrated manner rather than a globally distributed damage development that occurs predominantly in thin composite laminates. In addition, it has been shown that the fatigue progressive damage modeling is a reliable computational simulation technique to study the behavior of thick composite laminates under cyclic flexural loading. In conjunction with the use of a finite element package, this technique can provide efficient and accurate parametric studies of the fatigue behavior of thick composite laminates. Hence, a hybrid or mixed approach needs to be used and further developed based on the existing three approaches for the fatigue of composites. Further, moving beyond the deterministic framework would also be useful and appropriate for the characterization of fatigue behavior of thick composite laminates.

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Fatigue damage and lifetime prediction of fiber-reinforced ceramic-matrix composites

8

Longbiao Li College of Civil Aviation, Nanjing University of Aeronautics and Astronautics, Nanjing, PR China

8.1

Introduction

Ceramic materials possess high specific strength and specific modulus at elevated temperature. But their use as structural components is severely limited due to their brittleness. The continuous fiber-reinforced ceramic-matrix composites (CMCs), by incorporating fibers in ceramic matrices, however, not only exploit their attractive high-temperature strength but also reduce the propensity for catastrophic failure. These materials have already been implemented on aero engines’ components [1, 2]. The CMC flaps for exhaust nozzles of Snecma M53 and M88 aeroengines have already been used for more than one decade [3]. The CMC turbine stator vanes have already been designed and tested in aeroengine environments under the implementation of the Ultra Efficient Engine Technology (UEET) program, to demonstate that the vane can successfully withstand the harsh engine environment conditions for up to 1000 h [4]. A CMC turbine blade has been tested for 4 h by General Electric in a modified GE F414 engine, which represents the first application of the CMC material in a rotating engine component. Incorparting the CMC turbine blades on a GE90-sized engine, the overall weight can be reduced by 455 kg, which represents approximately 6% of dry weight of a full-sized GE90-115 [4a]. The CMC combustion chamberfloating wall tiles have also been tested in the aero engine environment for 30 min, with the temperature range of 1047–1227°C and the pressure of 2 MPa [5]. Under cyclic fatigue loading of fiber-reinforced CMCs, matrix multi-cracking and fiber/matrix interface debonding occur first [6], the open and closure of matrix cracking upon each cycle is the basic fatigue damage mechanism [7]. The fiber push-out and push-back experiments have been used to explain the fatigue hysteresis mechanisms of CMCs [8]. The stress-strain hysteresis loops appear as the fiber sliding relative to the matrix in the fiber/matrix interface debonded region upon unloading and subsequent reloading [9–17]. The fatigue hysteresis loop area, i.e., the hysteresis dissipated energy, can be used as an effective tool to monitor the damage evolution in fiber-reinforced CMCs. Many researchers investigated the evolution characteristics of fatigue hysteresis dissipated energy under cyclic loading at room and elevated temperatures. Holmes and Cho [18] investigated the fatigue hysteresis loops evolution Fatigue Life Prediction of Composites and Composite Structures. https://doi.org/10.1016/B978-0-08-102575-8.00008-5 © 2020 Elsevier Ltd. All rights reserved.

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characteristics of unidirectional SiC/CAS-II composite at room temperature. At the initial stage, the hysteresis modulus decreases rapidly, and the hysteresis dissipated energy increases due to matrix multicracking, fiber/matrix interface debonding and interface wear. When the cycle number approaches a critical value, the hysteresis modulus remains constant. However, the hysteresis dissipated energy continually decreases with the increase of cycle number. Upon approaching fatigue fracture, the hysteresis dissipated energy increases rapidly. Li [10, 11] investigated the evolution of fatigue hysteresis dissipated energy of unidirectional and cross-ply C/SiC composite at room and 800°C under air. The hysteresis dissipated energy decreases with the increase of cycle number. Reynaud [19] investigated the evolution of hysteresis dissipated energy of two different types of fiber-reinforced CMCs at 600°C, 800°C, and 1000°C in inert atmosphere. First, the hysteresis dissipated energy of 2D SiC/SiC composite increases with the increase of cycle number due to interface radial thermal residual compressive stress. The second ceramic composite, [0/90]s SiC/MAS L, the hysteresis dissipated energy decreases with the increase of cycle number due to interface radial thermal residual tensile stress. Fantozzi and Reynaud [20] investigated the evolution of fatigue hysteresis dissipated energy of 2.5D SiC/[Si-B-C] and 2.5D C/[Si-B-C] composites at 1200°C under air. The hysteresis dissipated energy of 2.5D SiC/[Si-B-C] composite decreases with the increase of cycle number due to interface wear; and the hysteresis dissipated energy of 2.5D C/[Si-B-C] composite decreases significantly after 144 h static loading attributed to PyC interface recession by oxidation or by a beginning of carbon fibers recession. The objective of this paper is to develop a hysteresis dissipated energy-based damage parameter to predict the fatigue life of fiber-reinforced CMCs. Under cyclic fatigue loading, the fatigue hysteresis dissipated energy, which results from the interface frictional slip between fibers and matrix, can be employed to provide a quantitative damage assessment for fiber-reinforced CMCs. Based on the hysteresis theories considering fibers failure, the fatigue hysteresis dissipated energy and a hysteresis dissipated energy-based damage parameter changing with the increase of cycle number, considering the fatigue damage mechanisms of matrix multicracking, fiber/matrix interface debonding, interface slipping and wear, and fibers failure, have been investigated. The relationships between the hysteresis dissipated energy, hysteresis dissipated energy-based damage parameter, stress-strain hysteresis loops and fatigue damage mechanisms, have been analyzed. The effects of fatigue peak stress, stress ratio, matrix crack spacing, and fiber volume fraction on the evolution of hysteresis dissipated energy and hysteresis dissipated energy-based damage parameter as a function of cycle number, have been analyzed. The experimental fatigue life S-N curves of unidirectional CMCs is predicted using the present analysis.

8.2

Theoretical analysis

8.2.1 Stress analysis Upon first loading to the fatigue peak stress of σ max, which is higher than the initial matrix cracking stress σ mc, it is assumed that matrix cracks run across the cross section of the composites, and the fiber/matrix interface debonds. When the fatigue peak

Fatigue damage and lifetime prediction of fiber-reinforced ceramic-matrix composites

271

stress is less than the first matrix cracking stress, there will be no cracking in CMCs. To analyze the stress distributions in the fiber and the matrix, a unit cell is extracted from the CMCs, as shown in Fig. 8.1. The unit cell contains a single fiber surrounded by a hollow cylinder of matrix. The fiber radius is rf, and the matrix radius is R(R ¼ rf/V1/2 f ). The length of the unit cell is lc/2, which is just the half matrix crack space. The fiber/matrix interface debonding length is ld. At the matrix cracking plane, the fibers carry all the applied stress (σ/Vf), where σ denotes far field applied stress and Vf denotes fiber volume fraction. The BHE shear-lag model [21] is adopted to perform the stress and strain calculations in the fiber/matrix interface debonded region (x2[0, ld]) and the interface bonded region(x2[ld, lc/2]). 8 σ 2τi > > > < Vf  rf x, x 2 ½0, ld  σ f ðxÞ ¼

   > Vm ld x  ld lc > > σ mo  2 τi exp ρ , x 2 ld , : σ fo + Vf rf rf 2

(8.1)

8 V f x > > , x 2 ½0, ld  2τ > < i V m rf σ m ðxÞ ¼
     > V l ρðx  ld Þ lc > > : σ mo  σ mo  2τi f d exp  ,x 2 ld , rf Vm r f 2

(8.2)

8 τ ,x 2 ½0, ld  > < i      τi ðxÞ ¼ ρ
Vm ld ρð x  l d Þ lc > σ mo  2τi exp  , x 2 ld , : 2 Vf rf rf 2

(8.3)

where Vm denotes matrix volume fraction, τi denotes fiber/matrix interface shear stress; and ρ denotes the BHE shear-lag parameter [21].

Fig. 8.1 The unit cell of the Budiansky-Hutchinson-Evans shear-lag model.

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Fatigue Life Prediction of Composites and Composite Structures

ρ2 ¼

4Ec Gm Vm Em Ef φ

(8.4)

where Gm denotes matrix shear modulus, and   2 ln Vf + Vm 3  Vf φ¼ 2Vm2

(8.5)

σ fo and σ mo denote fiber and matrix axial stress in the interface bonded region, respectively. σ fo ¼

  Ef σ + Ef αc  αf ΔT Ec

σ mo ¼

Em σ + Em ðαc  αm ÞΔT Ec

(8.6)

(8.7)

where Ef, Em, and Ec denote fiber, matrix, and composite elastic modulus, respectively; αf, αm, and αc denote fiber, matrix, and composite thermal expansion coefficient, respectively; ΔT denotes the temperature difference between the fabricated temperature T0 and testing temperature T1 (ΔT¼ T1–T0). E c ¼ Vf E f + Vm Em

(8.8)

When fiber fails, the fiber axial stress distribution in the fiber/matrix interface debonded region and bonded region is determined using the following equation:

σ f ðxÞ ¼

8 2τi > > > < T  r x, x 2 ½0, ld  f



   > ld x  ld lc > > ,x 2 ld , : σ fo + T  σ fo  2 τi exp ρ rf rf 2

(8.9)

where, T is the intact fiber axial stress on the matrix crack plane.

8.2.2 Matrix multi-cracking When loading the fiber-reinforced CMCs, cracks typically initiate within the composite matrix since the strain-to-failure of the matrix is usually less than that of the fiber. The matrix crack spacing decreases with increases in stress above the initial matrix cracking stress σ mc and may eventually approach to saturation at stress σ sat. There are four dominant failure criteria in the literature for modeling matrix multi-crack evolution of fiber-reinforced CMCs, i.e., the maximum stress criterion, energy balance approach, critical matrix strain energy criterion, and statistical failure approach. The maximum stress criterion assumes that a new matrix crack will form whenever

Fatigue damage and lifetime prediction of fiber-reinforced ceramic-matrix composites

273

the matrix stress exceeds the ultimate strength of the matrix, which is assumed to be single-valued and a known material property [22]. The energy balance failure criterion involves calculation of the energy balance relationship before and after the formation of a single dominant crack as originally proposed by Aveston et al. [23]. The progression of matrix cracking as determined by the energy criterion is dependent on the matrix strain energy release rate. The energy criterion is represented by Zok and Spearing [24] and Zhu and Weitsman [25]. The concept of a critical matrix strain energy criterion [26] presupposes the existence of an ultimate or critical strain energy limit beyond which the matrix fails. Beyond this, as more energy is placed into the composite, the matrix, unable to support the additional load, continues to fail. As more energy is placed into the system, the matrix fails such that all the additional energy is transferred to the fibers. Failure may consist of the formation of matrix cracks, the propagation of existing cracks or fiber/matrix interface debonding. Statistical failure approach [27] assumes that the matrix cracking is governed by statistical relations, which relate the size and spatial distribution of matrix flaws to their relative propagation stress. The brittle nature of the matrix material and the possible formation of initial cracks distribution throughout the microstructure suggest that a statistical approach to matrix crack evolution is warranted in fiber-reinforced CMCs. The tensile strength of the brittle matrix is assumed to be described by the two-parameter Weibull distribution where the probability of the matrix failure Pm is determined using the following equation [27]: 
m  σm Pm ¼ 1  exp  σR

(8.10)

where σ m denotes the tensile stress in the matrix; and σ R and m denote the matrix characteristic strength and matrix Weibull modulus, respectively. To estimate the instantaneous matrix crack space with increasing applied stress, we write Pm ¼ lsat =lc

(8.11)

where lsat denotes the matrix saturation crack spacing. Using Eqs. (8.10) and (8.11), the instantaneous matrix crack space can be determined using the following equation:  lc ¼ lsat


m  1 σm 1  exp  σR

(8.12)

8.2.3 Interface debonding There are two approaches to the problem of fiber/matrix interface debonding, i.e., the shear stress approach and the fracture mechanics approach. The shear stress approach is based upon a maximum shear stress criterion in which the fiber/matrix interface debonding occurs as the interfacial shear stress approaches to the interface shear

274

Fatigue Life Prediction of Composites and Composite Structures

strength [28]. On the other hand, the fracture mechanics approach treats the fiber/ matrix interface debonding as a particular crack propagation problem in which the interface debonding occurs as the strain energy release rate of the fiber/matrix interface achieves the debonding toughness [29]. It has been proved that the fracture mechanics approach is preferred to the shear stress approach for interface debonding [30]. The fracture mechanics approach is adopted to determine the fiber/matrix interface debonded length [29]. F ∂wf ð0Þ 1 ξd ¼  4πrf ∂ld 2

ð ld 0

τi

∂vðxÞ dx ∂ld

(8.13)

where F ¼ πr2f σ/Vf, is the fiber load at the matrix crack plane. wf(0) denotes the fiber axial displacement at the matrix crack plane. v(x) denotes the relative displacement between fiber and matrix. The axial displacement of fiber and matrix, wf(x) and wm(x) are determined using the following equations: wf ðxÞ ¼

ð lc =2 x

¼

σf dx Ef

 2τi rf Vm Em σ τi  2 σ ðld  xÞ  l  x2  ld + σ + ðlc =2  ld Þ Vf Ef Ec rf E f d ρEf ρVf Ef Ec (8.14)

w m ðxÞ ¼

ð lc =2 x

¼

σm dx Em

 2Vf τi Vf τ i  2 rf σ l  x2 + ld  σ + ðlc =2  ld Þ Ec Vm Em r f d ρVm Em ρEc

ð8:15Þ

Using Eqs. (8.14) and (8.15), the relative displacement between the fiber and matrix is determined using the following equation:



vðxÞ ¼ wf ðxÞ  wm ðxÞ

¼

 rf σ τ i Ec  2 2τi Ec ld ðld  xÞ  ld  x2  + σ V f Ef V m Em Ef r f ρVm Em Ef ρVf Ef

(8.16)

Substituting wf(x¼ 0) and v(x) into Eq. (8.13), leads to the form of the following equation:
 r f Vm Em σ 2 rf τ i Ec τ2i Ec τ2i τi σ 2 ld +   σ  ξd ¼ 0 ld + 2 V m Em Ef r f ρVm Em Ef Vf Ef 2ρV 4Vf Ef Ec f Ef

(8.17)

To solve Eq. (8.17), the fiber/matrix interface debonding length is determined using the following equation:

Fatigue damage and lifetime prediction of fiber-reinforced ceramic-matrix composites


 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 r f Vm Em σ 1 rf r f Vm Em Ef ld ¼  + ξd  2 V f Ec τ i ρ 2ρ Ec τ2i

275

(8.18)

8.2.4 Fibers failure As fibers begin to break, the loads dropped by broken fibers must be transferred to the intact fibers at the cross section. Two dominant failure criteria are present in the literature for modeling fiber failure: The global load-sharing (GLS) criterion and the local load-sharing (LLS) criterion. The GLS criterion assumes that the load from any one fiber is transferred equally to all other intact fibers at the same cross-section plane. The GLS assumption neglects any local stress concentrations in the neighborhood of existing breaks, and is expected to be accurate when the fiber/matrix interface shear stress is sufficiently low. Models that include GLS explicitly have been developed, including: Thouless and Evans [31], Cao and Thouless [32], Sutcu [33], Schwietert and Steif [34], Curtin [35], Weitsman and Zhu [36], Hild et al. [37], Zhu and Weitsman [25], Curtin et al. [38], Paar et al. [39], Liao and Reifsnider [40], etc. The LLS assumes that the load from the broken fiber is transferred to the neighborhood intact fibers, and is expected to be accurate when the interface shear stress is sufficiently high. Models that include LLS explicitly have been developed, which includes Zhou and Curtin [41], Dutton et al. [42], Xia and Curtin [43], etc. The two-parameter Weibull model is adopted to describe fiber strength distribution, and the GLS assumption is used to determine the loads carried by the intact and fracture fibers [35]. σ ¼ T ð1  PðT ÞÞ + hTb iPðT Þ Vf

(8.19)

where, hTbi denotes the load carried by the broken fibers; and P(T) denotes the fiber failure volume fraction. "
 # T mf + 1 PðT Þ ¼ 1  exp  σc

(8.20)

where mf is the fiber Weibull modulus; and σ c is the fiber characteristic strength of a length δc of fiber. !
mf 1=mf + 1 1=m mf =mf + 1 σ o rf lo f lo σ o τ i ,δc ¼ σc ¼ rf τi

(8.21)

where lo is the reference length and σ o is the fiber reference strength of a length of lo of fiber.

276

Fatigue Life Prediction of Composites and Composite Structures

When fiber fractures, the fiber stress drops to zero at the break, and the stress in the fiber builds up through the stress transfer across the fiber/matrix interface shear stress. T b ðxÞ ¼

2τi x rf

(8.22)

The sliding length lf required to build the fiber stress up to its previous intact value given by the following equation: lf ¼

rf T 2τi

(8.23)

The probability distribution f(x) of the distance x of a fiber break from reference matrix crack plane, provided that a break occurs within a distance  lf, is constructed based on the Weibull statistics by Phoenix and Raj [44]. "

 #
mf + 1

1 T x T mf + 1 f ðx Þ ¼ exp  , x 2 0, lf PðT Þlf σ c lf σc

(8.24)

Using Eqs. (8.22) and (8.24), the average stress carried by the broken fiber is determined using the following equation: hTb i ¼

ð lf 0

   σ c mf + 1 1  PðT Þ Tb ðxÞf ðxÞdx ¼ T  PðT Þ T

(8.25)

Substituting Eqs. (8.25) into (8.19), it leads to the form of the following equation: ( "
 #) σ mf + 1 σ T mf + 1 c ¼T 1  exp  Vf σc T

(8.26)

The load carried by the intact fibers T at the matrix crack plane for different applied stress can be obtained by solving Eq. (8.26), and then the fiber failure volume fraction can be obtained by substituting T into Eq. (8.20). When the load carried by the intact fibers reach the maximum value, composites fail. The composite ultimate tensile strength σ UTS is given by the following equation:
σ UTS ¼ Vf σ c

2 mf + 2



1 mf + 1


 mf + 1 mf + 2

(8.27)

8.2.5 Hysteresis theory When the fiber-reinforced CMCs are under tensile loading, matrix cracking occurs first. With increasing applied stress, the amounts of matrix cracks increase, partially

Fatigue damage and lifetime prediction of fiber-reinforced ceramic-matrix composites

277

matrix cracks deflect along the fiber/matrix interface, and some matrix cracks propagate penetrate through fibers, which makes fiber fracture. The interface debonded length, which includes the effect of fiber failure, is given by the following equation [11]: ffi

2
 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rf2 Vf Vm Ef Em T rf Vm Em 1 rf σ rf Vm Em Ef T  T ξd +  ldf ¼ 2 Ec τ i 2ρ Ec τ i 2 4E2c τi 2 ρ Vf

(8.28)

When ldf < lc/2, the fiber/matrix interface partially debonds; when ldf ¼ lc/2, the fiber/matrix interface completely debonds. The hysteresis loops of two cases are discussed in the following: l

l

The fiber/matrix interface partially debonding, and the fiber sliding relative to the matrix in the fiber/matrix interface debonded region upon unloading and subsequent reloading; The fiber/matrix interface completely debonding, and the fiber sliding relative to matrix in the entire matrix crack spacing upon unloading and subsequent reloading.

8.2.5.1 Interface partial debonding When the fiber/matrix interface partially debonds, the unit cell can be divided into the interface debonded region (x2[0, ldf]) and the interface bonded region (x2[ldf, lc/2]). Upon unloading to the applied stress of σ (σ min < σ < σ max), the fiber/matrix interface debonded region can be divided into the interface counter slip region (x 2[0, y]) and the interface slip region (x 2[y, ldf]). The fiber axial stress distribution upon unloading can be determined using the following equation: 8 2τi > > σ f ðxÞ ¼ T U + x, x 2 ½0, y > > rf > > > > <

2τi ð2y  xÞ,x 2 y, ldf σ f ðxÞ ¼ T U + rf > > > >
   > > 

x  ldf τi  > U > : σ f ðxÞ ¼ σ fo + T  σ fo  2 ldf  2y exp ρ , x 2 ldf , lc =2 rf rf (8.29) where y¼

 
 r f V m Em U 1 1 ldf  T  2 ρ 2 Ec τ i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s

  rf 2 rf2 Vf Vm Ef Em T U r V E E σ f m m f   TU  ξd + Vf 2ρ 4E2c τi 2 Ec τ i 2 (8.30)

278

Fatigue Life Prediction of Composites and Composite Structures

where TU denotes the stress carried by the intact fibers at the matrix crack plane upon unloading, which satisfied the relationship of the following equation: ( ) "

mf + 1 # σ mf + 1 σ T  TU T 1 c ¼ 2T exp   1 + Pð T Þ Vf σc 2 T 2T

(8.31)

Upon reloading to σ, slip again occurs near the matrix crack plane over a distance z, which is denoted to be the interface new slip region. The fiber/matrix interface debonded region can be divided into the interface new slip region (x 2[0, z]), interface counter slip region (x 2[z, y]), and slip region (x2[y, ldf]). The fiber axial stress distribution upon reloading can be determined using the following equation: 8 2τi R > > > σ f ðxÞ ¼ T  r x,x 2 ½0, z > f > > > > 2τi > R > ðx  2zÞ, x 2 ½z, y < σ f ðxÞ ¼ T + rf

2τ > > σ f ðxÞ ¼ T R  i ðx  2y + 2zÞ, x 2 y, ldf > > > > rf 
 > > 

x  ldf τ  > > : σ f ðxÞ ¼ σ fo + T R  σ fo  2 i ldf  2y + 2z exp ρ ,x 2 ldf , lc =2 rf rf (8.32) where z¼y

 
 r f V m Em R 1 1 ldf  T  2 ρ 2 Ec τ i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi39 s
 2 2V V E E TR
r rf rf Vm Em Ef 5= σ f f m f m   TR  ξd + 2 2 ; Vf 2ρ 4Ec τi Ec τ i 2

(8.33)

where TR is the stress carried by the intact fibers at the matrix crack plane upon reloading, which satisfies the relationship of the following equation: ( "

 # σ mf + 1 σ Tm T mf + 1 c ¼ 2T exp  Vf σc T 2T ) "

mf + 1 # T R  T + Tm T 1  exp  + Pð T Þ σc 2 2T

(8.34)

Fatigue damage and lifetime prediction of fiber-reinforced ceramic-matrix composites

279

where Tm satisfies the relationship of the following equation: ( ) "

 # σ mf + 1 Tm T mf + 1 1 c 0 ¼ 2T exp   1 + Pð T Þ σc 2 T 2T

(8.35)

8.2.5.2 Interface complete debonding When the fiber/matrix interface completely debonds, the unit cell can be divided into the interface counter slip region (x2[0, y]) and interface slip region (x 2[y, lc/2]) upon unloading. The fiber axial stress distribution upon unloading can be determined using the following equation: 8 2τi > U > > σ f ðxÞ ¼ T + r x, x 2 ½0, y < f

  > 2τi lc > U > ð2y  xÞ,x 2 y, : σ f ðxÞ ¼ T + rf 2

(8.36)

where    Ef rf  U y¼ T  T  ðσ max  σ Þ 4τi Ec

(8.37)

where TU denotes the stress carried by the intact fibers at matrix crack plane upon unloading. Upon reloading, the unit cell can be divided into the fiber/matrix interface new slip region (x2[0, z]), the interface counter slip region (x2[z, y]), and the interface slip region (x2[y, lc/2]). The fiber axial stress distribution upon reloading can be determined using the following equation: 8 2τi R > > > σ f ðxÞ ¼ T  rf x, x 2 ½0, z > > > > < 2τi σ f ðxÞ ¼ T R + ðx  2zÞ, x 2 ½z, y rf > > >   > > > 2τ l > : σ f ðxÞ ¼ T R  i ðx  2y + 2zÞ, x 2 y, c rf 2

(8.38)

where    Ef rf  R z ¼ yðσ min Þ  T  T  ðσ max  σ Þ 4τi Ec

(8.39)

280

Fatigue Life Prediction of Composites and Composite Structures

where TR denotes the stress carried by the intact fibers at the matrix crack plane upon reloading.

8.2.5.3 Hysteresis loops When damage forms within the composite, the composite strain is determined from Eq. (8.40), which assumes that the composite strain is equivalent to the average strain in an undamaged fiber. 2 εc ¼ Ef l c

ð lc =2

  σ f ðxÞdx  αc  αf ΔT

(8.40)

Substituting Eq. (8.29) into Eq. (8.40), the unloading stress-strain relationship for the fiber/matrix interface partially debonding is determined using the following equation: εc_unloading ¼

    TU τ i y2 τ i 1  +4  2y  ldf 2y + ldf  lc  αc  αf ΔT Ef Ef rf lc Ef rf lc

(8.41)

Substituting Eq. (8.32) into Eq. (8.40), the reloading stress-strain relationship for the fiber/matrix interface partially debonding is determined using the following equation: εc_reloading ¼

TR τi z2 τi ðy  2zÞ2 4 +4 Ef Ef rf lc Ef rf lc      τi ldf  2y + 2z ldf + 2y  2z  lc +2  αc  αf ΔT Ef rf lc

ð8:42Þ

Substituting Eq. (8.36) into Eq. (8.40), the unloading stress-strain relationship for the fiber/matrix interface completely debonding is determined using the following equation: εc_unloading ¼

 TU τi y2 τi ð2y  lc =2Þ2  +4 2  αc  αf ΔT Ef Ef r f l c Ef rf l c

(8.43)

Substituting Eq. (8.38) into Eq. (8.40), the reloading stress-strain relationship for the fiber/matrix interface completely debonding is determined using the following equation: εc_reloading ¼

 TR τ i z2 τi ðy  2zÞ2 τi ðlc =2  2y + 2zÞ2  4 +4 2  αc  αf ΔT Ef E f rf l c Ef rf lc Ef rf l c

(8.44)

Fatigue damage and lifetime prediction of fiber-reinforced ceramic-matrix composites

281

8.2.5.4 Hysteresis-based damage parameter Under cyclic fatigue loading, the area associated with the fatigue hysteresis loops is the dissipated energy during corresponding cycle, which can be defined as the following equation: U¼

ð σmax σ min



εc_unloading ðσ Þ  εc_reloading ðσ Þ dσ

(8.45)

The fatigue hysteresis dissipated energy-based damage parameter Φ can be defined as the following equation: Φ¼

Un  Uinitial Ue

(8.46)

where Un denotes the hysteresis dissipated energy at the Nth cycle; Uinitial denotes the initial hysteresis dissipated energy at the first cycle; and Ue denotes the elastic strain energy, which is defined as 1 Ue ¼ ðσ max  σ min Þðεmax  εmin Þ 2

(8.47)

8.2.6 Life prediction method When a CMC is subjected to a cyclic loading between a valley stress and a peak stress, upon first loading to peak stress, matrix multicracking and fiber/matrix interface debonding occur. With increase applied cycle number, fibers break due to the degradation of the fiber/matrix interface shear stress and fibers strength caused by the interface wear at room temperature [6, 7, 9, 45] or the fiber/matrix interface oxidation at elevated temperature [10, 46–48]. The two-parameter Weibull model is adopted to describe the fibers strength distribution, and the GLS is adopted to determine the loads carried by intact and fracture fibers [49]. σ ¼ T ½1  PðT Þ + hTb iPðT Þ Vf

(8.48)

where P(T) is determined by the following equation: (
 )
mf T mf + 1 σ0 τi PðT Þ ¼ 1  exp  σk σ 0 ðN Þ τ i ðN Þ

(8.49)

282

Fatigue Life Prediction of Composites and Composite Structures

and hTbi denotes the load carried by broken fibers, which is determined by the following equation: ( "

 T σ k mf + 1 σ 0 ðN Þ mf τi ðN Þ T mf + 1 1  exp  h Tb i ¼ Pð T Þ T σ0 τi σk


σ0  σ 0 ðN Þ

mf

(
 ) 
mf τi T T mf + 1 σk τi exp   ð8:50Þ PðT Þ σk τ i ðN Þ σ k ðN Þ τi ðN Þ

where σ 0(N) denotes the fiber strength at the Nth cycle; and τi(N) denotes the fiber/ matrix interface shear stress at the Nth cycle. Substituting Eqs. (8.49) and (8.50) into Eq. (8.48), it leads to the formation of the following equation: ( "
 #)
mf + 1

mf  σ σk σ 0 ðN Þ m f τ i ðN Þ T mf + 1 σ0 τi ¼T 1  exp  Vf σ0 τi σk T σ 0 ðN Þ τ i ðN Þ

(8.51)

Lee and Stinchcomb [50] found that the fiber strength degraded with increasing applied cycle number under cyclic fatigue loading. σ 0 ðN Þ ¼ σ 0 ½1  p1 ð log N Þp2 

(8.52)

where p1 and p2 are empirical parameters. Evans et al. [6] proposed the interface shear stress degradation model. The relationship between the fiebr/matrix interface shear stress and cycle number is determined using the following equation:     τi ðN Þ ¼ τio + 1  exp ωN λ τi min  τio

(8.53)

where τio denotes the initial interface shear stress; τimin denotes the steady-state interface shear stress; and ω and λ are empirical parameters. Using Eqs. (8.51)–(8.53), the stress T carried by intact fibers at the matrix cracking plane can be determined for different peak stresses. Substituting Eqs. (8.52) and (8.53), and the intact fiber stress T into Eq. (8.49), the fibers fracture probability P(T) corresponding to different number of applied cycles can be determined. When the fraction of broken fibers approaches the critical value, the composite would fatigue fracture.

Fatigue damage and lifetime prediction of fiber-reinforced ceramic-matrix composites

8.3

283

Results and discussion

With the increase of broken fibers fraction, Un and Φ can change with the increase of cycle number, and be affected by the material properties, peak stress, stress ratio, and matrix crack spacing. The effects of these above-mentioned factors on the variations of Un and Φ under tension-tension cyclic loading, considering fibers failure, are investigated. The ceramic composite system of SiC/CAS is used for the case study, and its material properties are given by: Vf ¼ 40%, Ef ¼ 190 GPa, Em ¼ 90 GPa, rf ¼ 7.5 μm, αf ¼ 4 106/°C, αm ¼ 5 106/°C, ΔΤ ¼1000°C, ξd ¼ 0.4 J/m2, lc ¼ 140 μm, σ c ¼ 2.0 GPa, and mf ¼ 4.

8.3.1 Effects of fatigue peak stress The effect of fatigue peak stress on the fatigue damage evolution and fatigue life S-N curve are predicted. The fatigue stress ratio is R ¼ 0. The fatigue life S-N curve corresponding to different fatigue peak stress is shown in Fig. 8.2, in which the failure cycles corresponding to σ max ¼ 210, 220, 240, 300, and 320 MPa are 20,370, 4920, 3428, 2405, and 2171, respectively. The effect of peak stress, i.e., σ max ¼ 210, 220, 240, 300, and 320 MPa, on the fiber/ matrix interface debonded length of 2ld/lc is shown in Fig. 8.3. When the fatigue peak

Fig. 8.2 The fatigue life S-N curve corresponding to the fatigue peak stress of σ max ¼ 210, 220, 240, 300, and 320 MPa [12, 13].

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Fig. 8.3 The effect of fatigue peak stress, i.e., σ max ¼ 210, 220, 240, 300, and 320 MPa, on (A) the fiber/matrix interface debonded length vs the interface shear stress curve; and (B) the fiber/matrix interface debonded length vs the applied cycle number curve [12, 13].

stress is σ max ¼ 210 MPa, the fiber/matrix interface debonded length of 2ld/lc increases from 0.23 when the interface shear stress is τi ¼ 50 MPa to the peak value of 1.0 when the interface shear stress is τi ¼ 11.8 MPa, i.e., the A1-B1 part in Fig. 8.3A, or from the first cycle to 1639th cycle, i.e., the A1-B1 part in Fig. 8.3B. When the fatigue peak

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285

Fig. 8.4 The effect of fatigue peak stress, i.e., σ max ¼ 210, 220, 240, 300, and 320 MPa, on (A) the fatigue hysteresis dissipated energy vs applied cycle number curve; and (B) the fatigue hysteresis dissipated energy-based damage parameter vs applied cycle number curve [12, 13].

stress is σ max ¼ 320 MPa, the fiber/matrix interface debonded length of 2ld/lc increases from 0.36 when τi ¼ 50 MPa to the peak value of 1.0 when the interface shear stress is τi ¼ 19 MPa, i.e., the A2-B2 part in Fig. 8.3C, or from the first cycle to 1194th cycle, i.e., the A2-B2 part in Fig. 8.3B.

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The effect of fatigue peak stress, i.e., σ max ¼ 210, 220, 240, 300, and 320 MPa, on Un and Φ vs cycle number curves are shown in Fig. 8.4. When the fatigue peak stress is σ max ¼ 320 MPa, Un increases from 16.9 kPa at the first cycle to 156 kPa at the 2171th cycle; and Φ increases from zero at the first cycle to 0.22 at the 2171th cycle. When the fatigue peak stress is σ max ¼ 240 MPa, Un increases from 7 kPa at the first cycle to 63.2kPa at the 2470th cycle first, then decreases to 50.8kPa at the 3428th cycle; and Φ increases from zero at the first cycle to 0.17 at the 2470th cycle first, then decreases to 0.12 at the 3428th cycle. When the fatigue peak stress is σ max ¼ 210 MPa, Un increases from 4.7 kPa at the first cycle to 46.9 kPa at the 2577th cycle first, then decreases to 29.5 kPa at the 4784th cycle, and increases to 36.2 kPa at the 20,370th failure cycle; and Φ increases from zero at the first cycle to 0.169 at the 2577th cycle first, then decreases to 0.085 at the 4068th cycle, and increases to 0.103 at the 20,370th failure cycle. With the increase of peak stress, Un and Φ vs cycle number curves can be divided into three different cases, i.e., (1) at the high peak stress levels, Un and Φ increase to final fatigue fracture; (2) at the intermediate peak stress levels, Un and Φ increase to the peak value first, then decrease to final fatigue fracture; and (3) at the low peak stress levels, Un and Φ increase to the peak value first, then decrease to the valley value, and increase again to final fatigue fracture. Compared with Un, the damage parameter Φ can better reveal the fatigue damage evolution in fiber-reinforced CMCs with the increase of peak stress.

8.3.2 Effects of fatigue stress ratio The effect of stress ratio, i.e., R ¼ 0, 0.1, and 0.2, on Un corresponding to σ max ¼ 210, 220, 240, 300, and 320 MPa are shown in Fig. 8.5. When the fatigue peak stress is σ max ¼ 210 MPa, Un decreases with the increase of stress ratio as shown Fig. 8.5A, i.e., from 4.7 kPa at the first cycle to 46.9 kPa at the 2577th cycle first, then decreases to 29.5 kPa at the 4784th cycle, and increases to 36.3 kPa at the 20,370th cycle when R ¼ 0; and Un increases from 2.4 kPa at the first cycle to 30 kPa at the 2797th cycle, then decreases to 20.2 kPa at the 5170th cycle, and increases again to 23.3 kPa at the 20,370th cycle when R¼ 0.2. When the fatigue peak stress is σ max ¼ 320 MPa, Un decreases with the increase of stress ratio as shown in Fig. 8.5E, i.e., from 16.9 kPa at the first cycle to 156 kPa at the 2171th cycle when R ¼ 0; and Un increases from 8.6 kPa at the first cycle to 91.7 kPa at the 2171th cycle when R ¼ 0.2. The effect of stress ratio, i.e., R¼ 0, 0.1, and 0.2, on Φ corresponding to σ max ¼ 210, 220, 240, 300, and 320 MPa are shown in Fig. 8.6. When the fatigue peak stress is σ max ¼ 210 MPa, Φ decreases with the increase of stress ratio when interface partially debonds, and increases with the increase of stress ratio when interface completely debonds as shown in Fig. 8.6A, i.e., Φ increases from zero at the first cycle to 0.171 at the 2450th cycle first, then decreases to 0.08 at the 4982th cycle, and increases again to 0.103 at the 20,370th cycle when R¼ 0; and Φ increases from zero at the first cycle to 0.176 at the 2663th cycle first, then decreases to 0.096 at the 5429th cycle, and increases again to 0.106 at the 20,370th cycle when R¼ 0.2. When the fatigue peak stress is σ max ¼ 320 MPa, Φ decreases with the increase of stress ratio as shown in Fig. 8.6E, i.e., Φ increases from zero at the first cycle to 0.219 at the 2171th cycle when R¼ 0; and Φ increases from zero at the first cycle to 0.218 at the 2171th cycle when R¼ 0.2.

Fig. 8.5 The effect of fatigue stress ratio (i.e., R ¼ 0, 0.1, and 0.2) on the fatigue hysteresis dissipated energy vs applied cycle number curve corresponding to different fatigue peak stresses of (A) σ max ¼210 MPa; (B) σ max ¼220 MPa; (C) σ max ¼240 MPa; (Continued)

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Fig. 8.5, cont’d

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(D) σ max ¼ 300 MPa; and (E) σ max ¼ 320 MPa [12, 13].

With the increase of fatigue stress ratio, the range and extent of interface frictional slip between fibers and matrix in the interface debonded region would decrease. Un decreases with the increase of stress ratio at the same cycle number; and Φ decreases with the increase of stress ratio when interface partially debonds, and increases with the increase of stress ratio when interface completely debonds. Compared with Un, the damage parameter Φ is much more sensitive to interface debonding and interface frictional slipping for fiber-reinforced CMCs with different stress ratio.

8.3.3 Effects of matrix crack spacing The effect of matrix crack spacing, i.e., lc ¼ 140, 200, and 240 μm, on the fiber/matrix interface debonded length 2ld/lc vs the fiber/matrix interface shear stress and cycle number curves corresponding to σ max ¼ 210, 220, 240, 300, and 320 MPa are shown

Fig. 8.6 The effect of fatigue stress ratio (i.e., R ¼ 0, 0.1, and 0.2) on the fatigue hysteresis dissipated energy-based damage parameter vs applied cycle number curve corresponding to different fatigue peak stresses of (A) σ max ¼ 210 MPa; (B) σ max ¼220 MPa; (C) σ max ¼ 240 MPa; (Continued)

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Fig. 8.6, cont’d

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(D) σ max ¼ 300 MPa; and (E) σ max ¼ 320 MPa [12, 13].

in Figs. 8.7 and 8.8. When the fatigue peak stress is σ max ¼ 210 MPa, the fiber/matrix interface debonded length of 2ld/lc decreases with the increase of matrix crack spacing at the same interface shear stress when interface partially debonds, i.e., 2ld/lc increases from 0.23 when the interface shear stress is τi ¼ 50 MPa to the peak value of 1.0 when the interface shear stress is τi ¼ 11.8 MPa, i.e., the A1-B1 part in Fig. 8.7A, or from the first cycle to 1639th cycle, i.e., the A1-B1 part in Fig. 8.8A, when the matrix crack spacing is lc ¼ 140 μm; and the fiber/matrix interface debonded length of 2ld/lc increases from 0.14 when the interface shear stress is τi ¼ 50 MPa to the peak value of 1.0 when the interface shear stress is τi ¼ 6.9 MPa, i.e., the A2-B2 part in Fig. 8.7A, or from the first cycle to 2118th cycle, i.e., the A2-B2 part in Fig. 8.8A, when the matrix crack spacing is lc ¼ 240 μm. When the fatigue peak stress is σ max ¼ 320 MPa, the fiber/matrix interface debonded length of 2ld/lc increases from 0.36 when τi ¼ 50 MPa to the peak value of 1.0 when the interface shear stress is

Fig. 8.7 The effect of matrix crack spacing (i.e., lc ¼ 140, 200 and 240 μm) on the fiber/matrix interface debonded length versus the interface shear stress curve corresponding to different fatigue peak stresses of (A) σ max ¼ 210 MPa; (B) σ max ¼ 220 MPa; (C) σ max ¼ 240 MPa; (Continued)

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Fig. 8.7, cont’d

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(D) σ max ¼ 300 MPa; and (E) σ max ¼ 320 MPa [12, 13].

τi ¼ 19 MPa, i.e., the A1-B1 part in Fig. 8.7E, or from the first cycle to 1194th cycle, i.e., the A1-B1 part in Fig. 8.8E, when the matrix crack spacing is lc ¼ 140 μm; and the fiber/ matrix interface debonded length of 2ld/lc increases from 0.21 when the interface shear stress is τi ¼ 50 MPa to the peak value of 1.0 when the interface shear stress is τi ¼ 11.7 MPa, i.e., the A2-B2 part in Fig. 8.7E, or from the first cycle to 1645th cycle, i.e., the A2-B2 part in Fig. 8.8E, when the matrix crack spacing is lc ¼ 240 μm. The effect of matrix crack spacing, i.e., lc ¼ 140, 200 and 240 μm, on Un corresponding to σ max ¼ 210, 220, 240, 300, and 320 MPa is shown in Fig. 8.9. When the fatigue peak stress is σ max ¼ 210 MPa, Un decreases with the increase of matrix crack spacing when interface partially debonds, and increases with the increase of matrix crack spacing when interface completely debonds as shown in Fig. 8.9A, i.e., Un increases from 4.7 kPa at the first cycle to 46.9 kPa at the 2577th cycle, then decreases to 29.5 kPa at the 4784th cycle, and increases again to 36.3 kPa at the

Fig. 8.8 The effect of matrix crack spacing (i.e., lc ¼ 140, 200, and 240 μm) on the fiber/matrix interface debonded length vs the applied cycle number curve corresponding to different fatigue peak stresses of (A) σ max ¼ 210 MPa; (B) σ max ¼ 220 MPa; (C) σ max ¼ 240 MPa; (Continued)

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Fig. 8.8, cont’d

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(D) σ max ¼ 300 MPa; and (E) σ max ¼ 320 MPa [12, 13].

20,370th cycle when the matrix crack spacing is lc ¼ 140 μm; and Un increases from 2.7 kPa at the first cycle to 48.5 kPa at the 3160th cycle, then decreases to 39.7 kPa at the 4927th cycle, and increases again to 46.4 kPa at the 20,370th cycle when the matrix crack spacing is lc ¼ 240 μm. When the fatigue peak stress is σ max ¼ 320 MPa, Un decreases with the increase of matrix crack spacing as shown in Fig. 8.9E, i.e., Un increases from 16.9 kPa at the first cycle to 156 kPa at the 20,370th cycle when the matrix crack spacing is lc ¼ 140 μm; and Un increases from 9.8 kPa at the first cycle to 131 kPa at the 20,370th cycle when the matrix crack spacing is lc ¼ 240 μm. The effect of matrix crack spacing, i.e., lc ¼ 140, 200, and 240 μm, on Φ corresponding to σ max ¼ 210, 220, 240, 300, and 320 MPa are shown in Fig. 8.10. When the fatigue peak stress is σ max ¼ 210 MPa, Φ decreases with the increase of matrix crack spacing when interface partially debonds, and increases with the increases of matrix crack spacing when interface completely debonds, as shown in

Fig. 8.9 The effect of matrix crack spacing (i.e., lc ¼140, 200, and 240 μm) on the fatigue hysteresis dissipated energy vs the applied cycle number curve corresponding to different fatigue peak stresses of (A) σ max ¼ 210 MPa; (B) σ max ¼ 220 MPa; (C) σ max ¼ 240 MPa; (Continued)

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Fig. 8.9, cont’d

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(D) σ max ¼ 300 MPa; and (E) σ max ¼ 320 MPa [12, 13].

Fig. 8.10A, i.e., Φ increases from zero at the first cycle to 0.17 at the 2450th cycle, then decreases to 0.08 at the 4982th cycle, and increases again to 0.1 at the 20,370th cycle, when the matrix crack spacing is lc ¼ 140 μm; and Φ increases from zero at the first cycle to 0.18 at the 2988th cycle, then decreases to 0.13 at the 5332th cycle, and increases again to 0.15 at the 20,370th cycle, when the matrix crack spacing is lc ¼ 240 μm. When the fatigue peak stress is σ max ¼ 320 MPa, Φ decreases with the increase of matrix crack spacing as shown in Fig. 8.10E, i.e., Φ increases from zero at the first cycle to 0.22 at the 20,370th cycle when the matrix crack spacing is lc ¼ 140 μm; and Φ increases from zero at the first cycle to 0.216 at the 20,370th cycle when the matrix crack spacing is lc ¼ 240 μm. With increasing matrix crack spacing, the extent of fiber/matrix interface frictional slip between fibers and matrix in the interface debonded region is affected by the interface debonding, i.e., increases when interface partially debonds, and decreases when

Fig. 8.10 The effect of matrix crack spacing (i.e., lc ¼140, 200, and 240 μm) on the fatigue hysteresis dissipated energy-based damage parameter vs applied cycle number curve corresponding to different fatigue peak stresses of (A) σ max ¼210 MPa; (B) σ max ¼220 MPa; (C) σ max ¼ 240 MPa; (Continued)

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Fig. 8.10, cont’d

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(D) σ max ¼ 300 MPa; and (E) σ max ¼ 320 MPa [12, 13].

the interface completely debonds; Un and Φ decrease when the interface partially debonds, and increase when the interface completely debonds at the same cycle number.

8.3.4 Effects of fiber volume fraction The fatigue life data of different fiber volume fractions, i.e., Vf ¼ 35%, 40%, and 45%, are shown in Fig. 8.11. With the increase of fiber volume fraction, the fatigue life increases at the same applied stress level, i.e., the fatigue life increases from 1584 cycles when the fiber volume fraction is Vf ¼ 35% to 2569 cycles when the fiber volume fraction is Vf ¼ 45% at the peak stress of σ max ¼ 320 MPa. The effect of fiber volume fraction, i.e., Vf ¼ 35, 40 and 45%, on the fiber/matrix interface debonded length 2ld/lc as functions of the interface shear stress and applied

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299

Fig. 8.11 The effect of fiber volume fraction (i.e., Vf ¼ 35%, 40%, and 45%) on the fatigue life S-N curve [12, 13].

cycle number corresponding to different fatigue peak stresses of σ max ¼ 210, 220, 240, 300, and 320 MPa, are shown in Figs. 8.12 and 8.13. When the fatigue peak stress is σ max ¼ 210 MPa, the fiber/matrix interface debonded length of 2ld/lc decreases with the increase of fiber volume fraction at the same interface shear stress or applied cycle number when the interface partially debonds, i.e., 2ld/lc increases from 0.3 when τi ¼ 50 MPa to the peak value of 1 when τi ¼ 15.3 MPa, i.e., the A1-B1 part in Fig. 8.12A, or from the first cycle to 1402th cycle, i.e., the A1-B1 part in Fig. 8.13A, when Vf ¼ 35%; and 2ld/lc increases from 0.18 when τi ¼ 50 MPa to the peak value of 1 when τi ¼ 9.2 MPa, i.e., the A2-B2 part in Fig. 8.12A, or from the first cycle to 1861th cyle, i.e., the A2-B2 part in Fig. 8.13A, when Vf ¼ 45%. When the fatigue peak stress is σ max ¼ 320 MPa, the fiber/matrix interface debonded length of 2ld/lc increases from 0.47 when τi ¼ 50 MPa to the peak value of 1 when τi ¼ 25.1 MPa, i.e., the A1-B1 part in Fig. 8.12E, or from the first cycle to 916th cycle, i.e., the A1-B1 part in Fig. 8.13E, when Vf ¼ 35%; and 2Ld/L increases from 0.28 when τi ¼ 50 MPa to the peak value of 1 when τi ¼ 14.7 MPa, i.e., the A2-B2 part in Fig. 8.12E, or from the first cycle to 1437th cyle, i.e., the A2-B2 part in Fig. 8.13E, when Vf ¼ 45%. The effect of fiber volume fraction, i.e., Vf ¼ 35%, 40%, and 45%, on Un corresponding to the fatigue peak stresses of σ max ¼ 210, 220, 240, 300, and 320 MPa are shown in Fig. 8.14. When the fatigue peak stress is σ max ¼ 210 MPa, Un decreases with the increase of fiber volume fraction as shown in Fig. 8.14A, i.e., Un increases from 7.8 kPa at the first cycle to 61.3 kPa at the 2350th cycle, then decreases to 45.6 kPa at the 3428th cycle when Vf ¼ 35%; and Un increases from

Fig. 8.12 The effect of fiber volume fraction (i.e., Vf ¼ 30%, 35%, and 40%) on the fiber/matrix interface debonded length vs the fiber/matrix interface shear stress curve corresponding to different fatigue peak stresses of (A) σ max ¼210 MPa; (B) σ max ¼ 220 MPa; (C) σ max ¼ 240 MPa; (Continued)

Fatigue damage and lifetime prediction of fiber-reinforced ceramic-matrix composites

Fig. 8.12, cont’d

301

(D) σ max ¼ 300 MPa; and (E) σ max ¼ 320 MPa [12, 13].

2.9 kPa at the first cycle to 36.5 kPa at the 2810th cycle, then decreases to 22.1 kPa at the 6594th cycle, and increases again to 27.2 kPa at the 10,00,000th cycle when Vf ¼ 45%. When the fatigue peak stress is σ max ¼ 320 MPa, Un decreases with the increase of fiber volume fraction as shown in Fig. 8.14E, i.e., Un increases from 28.5 kPa at the first cycle to 174.6 kPa at the 1584th cycle when Vf ¼ 35%; and Un increases from 10.3 kPa at the first cycle to 105.6 kPa at the 2569th cycle when Vf ¼ 45%. The effect of fiber volume fraction, i.e., Vf ¼ 35, 40 and 45%, on Φ corresponding to different fatigue peak stresses of σ max ¼ 210, 220, 240, 300, and 320 MPa are shown in Fig. 8.15. When the fatigue peak stress is σ max ¼ 210 MPa, Φ decreases with the increase of fiber volume fraction as shown in Fig. 8.15E, i.e., Φ increases from zero at the first cycle to 0.19 at the 2205th cycle, then decreases to 0.11 at the 3428th cycle

Fig. 8.13 The effect of fiber volume fraction (i.e., Vf ¼ 30%, 35%, and 40%) on the fiber/matrix interface debonded length vs the applied cycle number curve corresponding to different fatigue peak stresses of (A) σ max ¼ 210 MPa; (B) σ max ¼ 220 MPa; (C) σ max ¼ 240 MPa; (Continued)

Fatigue damage and lifetime prediction of fiber-reinforced ceramic-matrix composites

Fig. 8.13, cont’d

303

(D) σ max ¼ 300 MPa; and (E) σ max ¼ 320 MPa [12, 13].

when Vf ¼ 35%; and Φ increases from zero at the first cycle to 0.15 at the 2691th cycle, then decreases to 0.076 at the 6581th cycle, and increases again to 0.093 at the 10,00,000th cycle when Vf ¼ 45%. When the fatigue peak stress is σ max ¼ 320 MPa, Φ decreases with the increase of fiber volume fraction as shown in Fig. 8.15E, i.e., Φ increases from zero at the first cycle to 0.23 at the 1584th cycle when Vf ¼ 35%; and Φ increases from zero at the first cycle to 0.17 at the 2569th cycle when Vf ¼ 45%. With the increase of fiber volume fraction, the range and extent of interface frictional slip between fibers and matrix in the interface debonded region would decrease. The parameters of Un and Φ both decrease with the increase of fiber volume fraction at the same cycle number.

Fig. 8.14 The effect of fiber volume fraction (i.e., Vf ¼ 30%, 35%, and 40%) on the fatigue hysteresis dissipated energy vs the applied cycle number curve corresponding to different fatigue peak stresses of (A) σ max ¼ 210 MPa; (B) σ max ¼ 220 MPa; (C) σ max ¼ 240 MPa; (Continued)

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Fig. 8.14, cont’d

8.4

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(D) σ max ¼ 300 MPa; and (E) σ max ¼ 320 MPa [12, 13].

Experimental comparisons

8.4.1 Unidirectional CMCs The evolution of fatigue hysteresis dissipated energy Un and fatigue hysteresis dissipated energy-based damage parameter Φ in unidirectional SiC/CAS and SiC/1723 composites under cyclic fatigue loading are investigated. The fatigue life S-N curves corresponding to different peak stresses are predicted.

Fig. 8.15 The effect of fiber volume fraction (i.e., Vf ¼ 30%, 35%, and 40%) on the fatigue hysteresis dissipated energy-based damage parameter vs applied cycle number curve corresponding to different fatigue peak stresses of (A) σ max ¼210 MPa; (B) σ max ¼220 MPa; (C) σ max ¼ 240 MPa; (Continued)

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Fig. 8.15, cont’d

307

(D) σ max ¼ 300 MPa; and (E) σ max ¼ 320 MPa [12, 13].

Evans et al. [6] performed an investigation on the tension-tension fatigue behavior of unidirectional SiC/CAS composite at room temperature. The stress ratio was 0.05 and the loading frequency was 10 Hz. The experimental and predicted interface shear stress as a function of cycle number is shown in Fig. 8.16A. The parameters of Un and Φ corresponding to σ max ¼ 420, 380, and 340 MPa are illustrated in Fig. 8.16B and C. When the fatigue peak stress is σ max ¼ 420 MPa, Un and Φ increase with the increase of cycle number to final fatigue fracture, i.e., the A1-B1 part in Fig. 8.16B and C; when the fatigue peak stress is σ max ¼ 380 MPa, Un and Φ first increases rapidly, i.e., the A2-B2

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part in Fig. 8.16B and C, then increases slowly to final fatigue fracture, i.e., the B2-C2 part in Fig. 8.16B and C; when the fatigue peak stress is σ max ¼ 340 MPa, Un and Φ first increases rapidly, i.e., the A3-B3 part in Fig. 8.16B and C, then increases slowly to final fatigue fracture, i.e., the B3-C3 part in Fig. 8.16B and C. The experimental and predicted

Fig. 8.16 (A) The interface shear stress vs applied cycle number curves; (B) the hysteresis dissipated energy vs applied cycle number curves; (Continued)

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309

Fig. 8.16, cont’d (C) the hysteresis dissipated energy-based damage parameter vs applied cycle number curves; and (D) the fatigue life S-N curves of unidirectional SiC/CAS composite at room temperature [12, 13].

fatigue life S-N curves are illustrated in Fig. 8.16D, in which the fatigue limit stress approaches to 62% of the tensile strength. The predicted fatigue life S-N curves of unidirectional SiC/CAS composite can be divided into two regions, i.e., (1) the A-B part is affected by the degradation of

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interface shear stress and fiber strength; and (2) when the interface shear stress approaches steady-state velue, the B-C part is only affected by the fibers strength degradation, as shown in Fig. 8.16D.

8.4.2 Cross-Ply CMCs 8.4.2.1 Cross-ply SiC/MAS at 566°C in air atmosphere For the cross-ply SiC/MAS composite, at the loading frequency of 1 Hz and 566°C in air atmosphere, when the fatigue peak stress is σ max ¼ 137 MPa, the theoretical fatigue hysteresis dissipated energy decreases from 5.8 kPa at the first applied cycle to 4.3 kPa at the 1000th applied cycle, and the experimental fatigue hysteresis dissipated energy decreases from 5.4 kPa at the fourth applied cycle to 4.4 kPa at the 230th applied cycle, as shown in Fig. 8.17A. The fiber/matrix interface shear stress decreases from 1.2 MPa at the fourth applied cycle to 1 MPa at the 230th applied cycle, as shown in Fig. 8.17B. With increasing applied cycles, the broken fibers fraction increases from 3% at the first applied cycle to 39.9% at the 1273th applied cycle, as shown in Fig. 8.17C. When the fatigue peak stress is σ max ¼ 120 MPa, the theoretical fatigue hysteresis dissipated energy decreases from 4.9 kPa at the first applied cycle to 2.8 kPa at the 1000th applied cycle, and the experimental fatigue hysteresis dissipated energy decreases from 4.5 kPa at the third applied cycle to 3.2 kPa at the 105th applied cycle, as shown in Fig. 8.17A. The interface shear stress decreases from 1.2 MPa at the third applied cycle to 0.8 MPa at the 105th applied cycle, as shown in Fig. 8.17B. With increasing applied cycles, the broken fibers fraction increases from 1.7% at the first applied cycle to 39.9% at the 5433th applied cycle, as shown in Fig. 8.17C. When the fatigue peak stress is σ max ¼ 103 MPa, the theoretical fatigue hysteresis dissipated energy decreases from 3.0 kPa at the first applied cycle to 2.4 kPa at the 1000th applied cycle, and the experimental fatigue hysteresis dissipated energy decreases from 2.7 kPa at the fourth applied cycle to 2.4 kPa at the 920th applied cycle, as shown in Fig. 8.17A. The interface shear stress decreases from 1 MPa at the fourth applied cycle to 0.8 MPa at the 920th applied cycle, as shown in Fig. 8.17B. With increasing applied cycles, the broken fibers fraction increases from 0.9% at the first applied cycle to 34.2% at the 19,931th applied cycle, as shown in Fig. 8.17C. The experimental and theoretical fatigue life S-N curves at the loading frequency of 1 Hz are given in Fig. 8.17D, in which the fatigue life at 566°C in air is greatly reduced attributed to oxidation of interphase and fibers.

8.4.2.2 Cross-ply SiC/MAS at 1093°C in air atmosphere For the cross-ply SiC/MAS composite, at the loading frequency of 1 Hz and 1093°C in air atmosphere, when the fatigue peak stress is σ max ¼ 137 MPa, the theoretical fatigue hysteresis dissipated energy increases from 40.6 kPa at the first applied cycle to 43.4 kPa at the fifth applied cycle, and decreases to 28 kPa at the 100th applied cycle, and the experimental fatigue hysteresis dissipated energy decreases from 43.8 kPa at the fourth applied cycle to 32.8 kPa at the 25th applied cycle, as shown in Fig. 8.18A.

Fatigue damage and lifetime prediction of fiber-reinforced ceramic-matrix composites

311

Fig. 8.17 (A) The hysteresis dissipated energy vs applied cycle number curves; (B) the interface shear stress vs applied cycle number curves; (Continued)

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Fig. 8.17, cont’d (C) the broken fibers fraction vs applied cycle number curves; and (D) the fatigue life S-N curves of cross-ply SiC/MAS composite at 566°C in air atmosphere [17].

The interface shear stress decreases from 9.8 MPa at the fourth applied cycle to 5.6 MPa at the 25th applied cycle, as shown in Fig. 8.18B. With increasing applied cycles, the broken fibers fraction increases from 7.7% at the first applied cycle to 38.4% at the 38th applied cycle, as shown in Fig. 8.18C.

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313

When the fatigue peak stress is σ max ¼ 120 MPa, the theoretical fatigue hysteresis dissipated energy increases from 33.6 kPa at the first applied cycle to 35.1 kPa at the third applied cycle, and decreases to 23 kPa at the 100th applied cycle, and the experimental fatigue hysteresis dissipated energy decreases from 34.9 kPa at the third

Fig. 8.18 (A) The hysteresis dissipated energy vs applied cycle number curves; (B) the interface shear stress vs applied cycle number curves; (Continued)

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Fig 8.18, cont’d (C) the broken fibers fraction vs applied cycle number curves; and (D) the fatigue life S-N curves of cross-ply SiC/MAS composite at 1093°C in air atmosphere [17].

applied cycle to 22.5 kPa at the 75th applied cycle, as shown in Fig. 8.18A. The interface shear stress decreases from 8.2 MPa at the third applied cycle to 3.6 MPa at the 75th applied cycle, as shown in Fig. 8.18B. With increasing applied cycles, the broken fibers fraction increases from 4.3% at the first applied cycle to 39.9% at the 371th applied cycle, as shown in Fig. 8.18C.

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When the fatigue peak stress is σ max ¼ 103 MPa, the theoretical fatigue hysteresis dissipated energy increases from 22 kPa at the first applied cycle to 26 kPa at the third applied cycle, and decreases to 7.4 kPa at the 20,000th applied cycle, and the experimental fatigue hysteresis dissipated energy decreases from 25.5 kPa at the fourth applied cycle to 6.5 kPa at the 10,608th applied cycle, as shown in Fig. 8.18A. The interface shear stress decreases from 7.6 MPa at the fourth applied cycle to 1.1 MPa at the 10,608th applied cycle, as shown in Fig. 8.18B. With increasing applied cycles, the broken fibers fraction increases from 2.2% at the first applied cycle to 39.5% at the 31,053th applied cycle, as shown in Fig. 8.18C. When the fatigue peak stress is σ max ¼ 96 MPa, the theoretical fatigue hysteresis dissipated energy decreases from 25.1 kPa at the first applied cycle to 4.3 kPa at the 30,000th applied cycle, and the experimental fatigue hysteresis dissipated energy decreases from 16 kPa at the third applied cycle to 4.4 kPa at the 23,067th applied cycle, as shown in Fig. 8.18A. The interface shear stress decreases from 6.2 MPa at the third applied cycle to 1.0 MPa at the 23,067th applied cycle, as shown in Fig. 8.18B. With increasing applied cycles, the broken fibers fraction increases from 1% at the first applied cycle to 17.6% at the 2,04,364th applied cycle, as shown in Fig. 8.18C. The experimental and theoretical fatigue life S-N curves at the loading frequency of 1 Hz are given in Fig. 8.18D, in which the predicted results agreed with experimental data.

8.4.3 2D woven CMCs 8.4.3.1 2D SiC/SiC at 800°C in air atmosphere Shi [51] investigated the cyclic tension-tension fatigue behavior of 2D SiC/SiC composite at 800°C in air atmosphere. The fatigue tests were conducted at the loading frequency of f¼ 1 Hz with a stress ratio of 0.1. The ultimate tensile strength was 215 MPa at 800°C, and the fatigue peak stresses were 0.35, 0.55, 0.7, and 0.9 tensile strength. The theoretical hysteresis dissipated energy vs interface shear stress curve is illustrated in Fig. 8.19A, in which the fatigue hysteresis dissipated energy increases from U¼ 3.6 kPa at the interface shear stress of 50 MPa to the peak value of U¼ 32.1 kPa at the interface shear stress of 4.6 MPa, then to zero kPa at the interface shear stress of zero MPa. With increasing cycle number, the experimental fatigue hysteresis dissipated energy increases from U¼ 5.5 kPa at the 5th cycle to U¼ 26 kPa at the 36,500th cycle. By comparing experimental fatigue hysteresis dissipated energy with theoretical computational values, the interface shear stress of 2D SiC/SiC at 800°C in air atmosphere can be estimated, i.e., the fiber/matrix interface shear stress decreases from 32.6 MPa at the 5th cycle to 6.8 MPa at the 36,500th cycle, as shown in Fig. 8.19B. The experimental and theoretical fatigue life S-N curves of 2D SiC/SiC at 800°C in air atmosphere are shown in Fig. 8.19C. The fatigue limit stress is 0.23 tensile strength.

8.4.3.2 2D SiC/Si-N-C at 1000°C Lee et al. [52] investigated the cyclic tension-tension fatigue behavior of 2D SiC/Si-NC composite at 1000°C. The fatigue tests were conducted at the loading frequency of f ¼ 1 Hz with a stress ratio of 0.05. The fatigue peak stresses were 0.65, 0.75, 0.9, and 0.95 tensile strength.

Fig. 8.19 (A) The hysteresis dissipated energy vs the interface shear stress curves; (B) the interface shear stress vs applied cycle number curves; and (C) the fatigue life S-N curves of 2D SiC/SiC composite at 800°C in air atmosphere.

Fig. 8.20 (A) The hysteresis dissipated energy vs the interface shear stress curves; (B) the interface shear stress vs applied cycle number curves; and (C) the fatigue life S-N curves of 2D SiC/Si-N-C composite at 1000°C.

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The theoretical hysteresis dissipated energy vs interface shear stress curve at 1000° C is shown in Fig. 8.20A, in which the fatigue hysteresis dissipated energy increases from U ¼ 35.5 kPa at the interface shear stress of 50 MPa to the peak value of U ¼ 53 kPa at the interface shear stress of 27.7 MPa, then to U ¼ 0 kPa at the interface shear stress of zero MPa. With increasing cycle number, the experimental fatigue hysteresis dissipated energy decreases from U ¼ 97.9 kPa at the first cycle to U ¼ 64.3 kPa at the 100th cycle. By comparing experimental fatigue hysteresis dissipated energy with theoretical computational values, the interface shear stress of 2D SiC/Si-N-C at 1000°C can be estimated, i.e., the interface shear stress decreases from 18.5 MPa at the first cycle to 9 MPa at the 100th cycle, as shown in Fig. 8.20B. The experimental and theoretical fatigue life S-N curves of 2D SiC/Si-N-C at 1000°C are shown in Fig. 8.20C. The fatigue limit stress is 0.28 tensile strength.

8.4.3.3 2D SiC/[Si-B4C] at 1200°C in air and in steam atmospheres Ruggles-Wrenn et al. [53] investigated the cyclic tension-tension fatigue behavior of 2D woven SiC/[Si-B4C] composite at 1200°C in air and in steam atmosphere. The fatigue tests were conducted at the loading frequency of f ¼ 1 Hz with a stress ratio of 0.05. The ultimate tensile strength was 307 MPa at 1200°C. The fatigue peak stresses were 0.32, 0.39, 0.42, and 0.45 tensile strength. The theoretical fatigue hysteresis dissipated energy vs the fiber/matrix interface shear stress curve at 1200°C in air atmosphere is shown in Fig. 8.21A, in which the fatigue hysteresis dissipated energy increases from U ¼ 4.9 kPa at the interface shear stress of 20 MPa to the peak value of U ¼ 34.8 kPa at the interface shear stress of 2.3 MPa, then to U ¼ 0 kPa at the interface shear stress of zero MPa. With increasing cycle number, the experimental fatigue hysteresis dissipated energy increases from U ¼ 8.4 kPa at the 1000th cycle to U ¼ 32.4 kPa at the 60,000th cycle. By comparing experimental fatigue hysteresis dissipated energy with theoretical computational values, the interface shear stress of 2D SiC/[Si-B4C] at 1200°C in air atmosphere can be estimated, i.e., the fiber/matrix interface shear stress decreases from 10.8 MPa at the 1000th cycle to 2.6 MPa at the 60,000th cycle, as shown in Fig. 8.21B. The experimental and theoretical fatigue life S-N curves of 2D SiC/Si-B4C at 1200°C in air atmosphere are shown in Fig. 8.21C. The fatigue limit stress is 0.32 tensile strength. The theoretical fatigue hysteresis dissipated energy vs the fiber/matrix interface shear stress curve at 1200°C in steam atmosphere is shown in Fig. 8.22A, in which the fatigue hysteresis dissipated energy increases from U¼ 4.9 kPa at the interface shear stress of 20 MPa to the peak value of U¼ 34.8 kPa at the interface shear stress of 2.3 MPa, then to U¼ 0 kPa at the interface shear stress of zero MPa. With increasing cycle number, the experimental fatigue hysteresis dissipated energy increases from U¼ 8.5 kPa at the 1000th cycle to U¼ 18.1 kPa at the 30,000th cycle. By comparing experimental fatigue hysteresis dissipated energy with theoretical computational values, the interface shear stress of 2D SiC/Si-B4C at 1200°C in steam atmosphere can be estimated, i.e., the fiber/matrix interface shear stress decreases from 10.7MPa at the 1000th cycle to 4.7 MPa at the 30,000th cycle, as shown in Fig. 8.22B. The experimental and theoretical fatigue life S-N curves of 2D SiC/Si-B4C at 1200°C in steam atmosphere are shown in Fig. 8.22C. The fatigue limit stress is 0.32 tensile strength.

Fig. 8.21 (A) The hysteresis dissipated energy vs the interface shear stress curves; (B) the interface shear stress vs applied cycle number curves; and (C) the fatigue life S-N curves of 2D SiC/[Si-B4C] composite at 1200°C in air atmosphere.

Fig. 8.22 (A) The hysteresis dissipated energy vs the interface shear stress curves; (B) the interface shear stress vs applied cycle number curves; and (C) the fatigue life S-N curves of 2D SiC/[Si-B4C] composite at 1200°C in steam atmosphere.

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8.4.3.4 2D Nextel 610/Aluminosilicate at 1000°C Zawada et al. [54] investigated the cyclic tension-tension fatigue behavior of 2D woven Nextel 610/Aluminosilicate composite at 1000°C. The fatigue tests were conducted at the loading frequency of f ¼ 1 Hz with a stress ratio of 0.05. The ultimate tensile strength was 173 MPa at 1200°C. The fatigue peak stresses were 0.78 and 0.86 of tensile strength. The theoretical hysteresis dissipated energy vs interface shear stress curve at 1000°C is shown in Fig. 8.23A, in which the fatigue hysteresis dissipated energy increases from U ¼ 16.3 kPa at the interface shear stress of 50 MPa to the peak value of U ¼ 31.5 kPa at the interface shear stress of 21.4 MPa, then to U ¼ 0 kPa at the interface shear stress of zero MPa. With increasing cycle number, the experimental fatigue hysteresis dissipated energy decreases from U ¼ 9.7 kPa at the first cycle to U ¼ 0.3 kPa at the 510th cycle. By comparing experimental fatigue hysteresis dissipated energy with theoretical computational values, the interface shear stress of 2D Nextel 610/Aluminosilicate at 1000°C can be estimated, i.e., the interface shear stress decreases from 3.4 MPa at the first cycle to 0.1 MPa at the 510th cycle, as shown in Fig. 8.23B. The experimental and theoretical fatigue life S-N curves of 2D Nextel 610/Aluminosilicate at 1000°C are illustrated in Fig. 8.23C. The fatigue limit stress is 0.56 tensile strength.

8.4.4 2.5D woven CMCs The monotonic tensile strength of 2.5D C/SiC composite is 212 MPa at room temperature, and the fatigue peak stresses are 0.85, 0.89, 0.94, 0.96, and 0.99 tensile strength [55]; and the monotonic tensile strength of 2.5D C/SiC composite is 280 MPa at 800°C in air, and the fatigue peak stresses are 0.5, 0.6, 0.7, and 0.8 tensile strength [56]. The experimental and theoretical fatigue life S-N curves at room temperature and 800°C in air are shown in Fig. 8.24A. The fatigue limit stress at room temperature is 0.85 tensile strength, and 0.28 tensile strength at 800°C in air atmosphere. The broken fibers fraction vs the applied cycle number curves under 0.85, 0.89, 0.94, 0.96, and 0.99 tensile strength at room temperature are shown in Fig. 8.24B. When σ max ¼ 0.99, 0.96, 0.94, 0.89, and 0.85 σ UTS, the composite experienced 6182, 9084, 12,173, 77,354, and 1,000,000 cycles, respectively. The broken fibers fraction vs cycle number curves under 0.5, 0.6, 0.7, and 0.8 tensile strength at 800°C in air are given in Fig. 8.24C. When σ max ¼ 0.8, 0.7, 0.6, and 0.5 σ UTS, the composite experienced 736, 3525, 8314, and 37,308 cycles, respectively. The broken fibers fraction vs cycle number curves under 0.8 tensile strength at room temperature and 800°C in air are given in Fig. 8.24D. When σ max ¼ 0.8 σ UTS, the composite experienced 1,000,000 cycles at room temperature, and 736 cycles at 800°C in air atmosphere.

8.4.5 3D woven CMCs 8.4.5.1 3D C/SiC at elevated temperature The monotonic tensile strength is 360 MPa at 1100°C in 104 Pa vacuum, and the fatigue peak stresses are 0.97 and 0.88 tensile strength [57]; the monotonic tensile strength is 304 MPa at 1300°C in 104 Pa vacuum, and the fatigue peak stresses

Fig. 8.23 (A) The hysteresis dissipated energy vs the interface shear stress curves; (B) the interface shear stress vs applied cycle number curves; and (C) the fatigue life S-N curves of 2D Nextel 610/Aluminosilicate composite at 1000°C.

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are 0.5, 0.83, 0.93, 0.98, and 0.99 tensile strength [57]; and the monotonic tensile strength is 261 MPa at 1500°C in 104 Pa vacuum, and the fatigue peak stresses are 0.98, 0.96, 0.95, 0.92, 0.9, 0.87, and 0.83 tensile strength [58]. The experimental and theoretical fatigue life S-N curves of 3D C/SiC composite at 1100°C in vacuum are illustrated in Fig. 8.25A, in which the fatigue limit is about 0.95

Fig. 8.24 (A) The fatigue life S-N curves at room temperature and 800°C in air; (B) the broken fibers fraction vs cycle number at room temperature; (Continued)

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Fig. 8.24, cont’d (C) the broken fibers fraction vs cycle number at 800°C in air; and (D) the broken fibers fraction vs cycle number at room temperature and 800°C in air atmosphere of 2.5D C/SiC composite [14].

tensile strength [57]. The broken fibers fraction vs cycle number curves under 0.88 and 0.97 tensile strength are given in Fig. 8.25B, in which the composite experienced 1,000,000 and 171,281 cycles corresponding to σ max ¼ 0.88 and 0.97 σ UTS, respectively. The experimental and theoretical fatigue life S-N curves of 3D C/SiC composite at 1300°C in vacuum are illustrated in Fig. 8.26A, in which the fatigue limit is about 0.93

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Fig. 8.25 (A) The experimental and theoretical fatigue life S-N curves; and (B) the broken fibers fraction vs cycle number under 0.88 and 0.97 σ UTS of 3D C/SiC composite at 1100°C in vacuum [14–16].

tensile strength [57]. The broken fibers fraction vs cycle number curves under 0.83, 0.93, 0.98, and 0.99 tensile strength are given in Fig. 8.26B, in which the composite experienced 1,000,000, 870,731, 17,740, and 14,008 cycles corresponding to σ max ¼ 0.83, 0.93, 0.98, and 0.99 σ UTS, respectively. The experimental and theoretical fatigue life S-N curves of 3D C/SiC composite at 1500°C in vacuum are illustrated in Fig. 8.27A, in which the fatigue limit is about 0.9

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Fig. 8.26 (A) The experimental and theoretical fatigue life S-N curves; and (B) the broken fibers fraction vs cycle number under 0.83, 0.93, 0.98, and 0.99 σ UTS of 3D C/SiC composite at 1300°C in vacuum [14–16].

tensile strength [58]. The broken fibers fraction vs cycle number curves under 0.9, 0.92, 0.95, 0.96, and 0.98 tensile strength are given in Fig. 8.27B, in which the composite experienced 1,000,000, 259,213, 25,608, 11,799, and 2493 cycles corresponding to σ max ¼ 0.9, 0.92, 0.95, 0.96, and 0.98 σ UTS, respectively.

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Fig. 8.27 (A) The experimental and theoretical fatigue life S-N curves; and (B) the broken fibers fraction vs cycle number under 0.9, 0.92, 0.95, 0.96, and 0.98 σ UTS of 3D C/SiC composite at 1500°C in vacuum [14–16].

8.4.5.2 3D SiC/SiC at elevated temperature The monotonic tensile strength of 3D woven SiC/SiC composite is 220 MPa at 1100°C, and the fatigue peak stresses are 0.6, 0.46, 0.36, 0.34, and 0.27 of tensile strength both at 1100°C in air and steam [59]; and the monotonic tensile strength of 3D woven SiC/SiC

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composite is 198 MPa at 1300°C, and the fatigue peak stresses are 0.68, 0.59, 0.55, 0.43, 0.26, and 0.23 of tensile strength both at 1300°C in air and in steam atmosphere [59] The experimental and theoretical fatigue life S-N curves of 3D SiC/SiC composite at 1100°C in air are shown in Fig. 8.28A, in which the fatigue limit stress is about 0.27 tensile strength [59]. The broken fibers fraction vs cycle number curves under 0.36, 0.46, and 0.6

Fig. 8.28 (A) The experimental and theoretical fatigue life S-N curves; and (B) the broken fibers fraction vs cycle number under 0.36, 0.46, and 0.6 σ UTS of 3D SiC/SiC composite at 1100°C in air [14–16].

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tensile strength are given in Fig. 8.28B, in which the composite experienced 68,439, 35,007, and 976 cycles corresponding to σ max ¼ 0.36, 0.46, and 0.6 σ UTS, respectively. The experimental and theoretical fatigue life S-N curves of 3D SiC/SiC composite at 1300°C in air atmosphere are shown in Fig. 8.29A, in which the fatigue limit stress

Fig. 8.29 (A) The experimental and theoretical fatigue life S-N curves; and (B) the broken fibers fraction vs cycle number under 0.26, 0.43, 0.55, 0.59, and 0.68 σ UTS of 3D SiC/SiC composite at 1300°C in air [14–16].

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is about 0.22 tensile strength [59]. The broken fibers fraction vs cycle number curves under 0.26, 0.43, 0.55, 0.59, and 0.68 tensile strength are given in Fig. 8.29B, in which the composite experienced 160,581, 58,625, 23,844, 12,925, and 812 cycles corresponding to σ max ¼ 0.26, 0.43, 0.55, 0.59, and 0.68 σ UTS, respectively.

8.5

Conclusions and outlook

Under cyclic fatigue loading, multiple fatigue damage mechanisms occur inside of fiber-reinforced CMCs, i.e., matrix cracking, interface debonding, sliding, wear and oxidation, and fibers fracture. The fatigue hysteresis loop is an important tool to predict the damage evolution and lifetime prediction of CMCs. The hysteresis dissipated energy and hysteresis-based damage parameters have already been used to predict the damage evolution and lifetime of unidirectional, cross-ply, 2D, 2.5D, and 3D CMCs at room and elevated temperatures. However, during the application of fiber-reinforced CMCs at elevated temperature, the loading may change with the applied cycles, the effects of overloading or stochastic loading on fatigue damage and lifetime will be investigated in the further study. Based on the airworthiness requirements, the failure risk of CMC components should be predicted for the maintenance schedule. The developed fatigue damage and lifetime prediction methods can also be used to calculate the failure risk of CMC components during operation.

Acknowledgments The work reported here is supported by the Fundamental Research Funds for the Central Universities (Grant Nos. NS2016070, NS2019038).

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[52] S.S. Lee, L.P. Zawada, J.M. Staehler, C.A. Folsom, Mechanical behavior and hightemperature performance of a woven NicalonTM/Si-N-C ceramic-matrix composite, J. Am. Ceram. Soc. 81 (1998) 1797–1811. [53] M.B. Ruggles-Wrenn, J. Delapasse, A.L. Chamberlain, J.E. Lane, T.S. Cook, Fatigue behavior of a Hi-NicalonTM/SiC-B4C composite at 1200°C in air and in steam, Mater. Sci. Eng. A 534 (2012) 119–128. [54] L.P. Zawada, R.S. Hay, S.S. Lee, J. Staehler, Characterization and high-temperature mechanical behavior of an oxide/oxide composite, J. Am. Ceram. Soc. 86 (2003) 981–990. [55] C.Y. Zhang, X.W. Wang, Y.S. Liu, B. Wang, D. Han, Tensile fatigue of a 2.5D-C/SiC composite at room temperature and 900°C, Mater. Des. 49 (2013) 814–819. [56] Yang FS. Research on Fatigue Behavior of 2.5d Woven Ceramic Matrix Composites. Master Thesis, Nanjing University of Aeronautics and Astronautics, Nanjing, 2011. [57] S.M. Du, S.R. Qiao, G.C. Ji, D. Han, Tension-tension fatigue behavior of 3D-C/SiC composite at room temperature and 1300°C, Mater. Eng. 9 (2002) 22–25. [58] S.M. Du, S.R. Qiao, Tension-tension fatigue behavior of 3D-C/SiC composite at 1500°C, Mater. Eng. 5 (2011) 34–37. [59] D.Q. Shi, X. Jing, X.G. Yang, Low cycle fatigue behavior of a 3D braided KD-I fiber reinforced ceramic matrix composite for coated and uncoated specimens at 1100°C and 1300°C, Mater. Sci. Eng. A 631 (2015) 38–44.

Further reading [60] D. Jacob, Fatigue Behavior of an Advanced SiC/SiC Composite With an Oxidation Inhibited Matrix at 1200°C in Air and in Steam, AFIT/GEA/ENY/10-M07 (2010). [61] W.K. Michael, Fatigue Behavior of a SiC/SiC Composite at 1000°C in Air and Steam, (2010)AFIT/GAE/ENY/10-D01. [62] L.P. Zawada, L.M. Butkus, G.A. Hartman, Tensile and fatigue behavior of silicon carbide fiber-reinforced aluminosilicate glass, J. Am. Ceram. Soc. 74 (1991) 2851–2858.

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Fatigue behaviors of fiberreinforced composite 3D printing

9

Astrit Imeri, Ismail Fidan Tennessee Tech University, Cookeville, TN, United States

9.1

Introduction

Additive manufacturing (AM) is an advanced manufacturing technology that produces parts successively by adding material layer-by-layer [1]. The term 3D printing is interchangeably used with AM. Overall, it is known that AM is an evolving technology since there is a high number of research and development activities to elevate the usability of this technology. Producing complex shapes, low material waste, and no tooling requirements are just some of the advantages of AM, compared to other conventional manufacturing processes [2]. Also, an important advantage is the production of lightweight parts [3]. In terms of material usage, it is reported that 90% of the material is usable in the AM processes [4]. Due to these advantages, AM has found applications in many industries such as aerospace, biomedicine, and motorsports [5]. Parts made vary from end-user parts to prototypes and patterns [6–8]. Usually, AM produced parts are of either polymers or metals. Currently, metallic parts built with AM are expensive, while polymer parts (mostly PLA, ABS, and PC [9]) suffer from weak mechanical and thermal properties. Recent techniques like fiber-reinforced additive manufacturing (FRAM) have made it possible to make light, strong, and low-cost additively manufactured parts [10]. Fibers used in FRAM can be classified as short and continuous fibers. Short fibers have found more application in FRAM due to the easiness of use [11]. However, a printer that is capable of printing continuous FRAM parts, Markforged Mark Two (MKF), has entered the market in recent years [12]. This printer using its dual extrusion head prints the base materials (nylon or Onyx) before reinforcing them with fibers [carbon fiber (CF), fiberglass (FG), and Kevlar]. Such a unique FRAM printer has made the fiber-reinforced composite manufacturing adaptable to various industries and provided practical solutions. Fibers as reinforcement material carry the higher loads. Hence, fiber orientation in the part is important for the mechanical properties. Testing mechanical properties of parts produced by this AM method is crucial for the utilization of parts in industry to prevent mechanical failures. Statistically, 50%–90% of all failures are due to fatigue [13]. Investigating fatigue properties of FRAM materials is critical for the design stage. However, since the technology is new, not many studies have been conducted on investigating fatigue properties of continuous FRAM. In this chapter, the conducted studies are explained in detail. Generally, fatigue studies have been conducted Fatigue Life Prediction of Composites and Composite Structures. https://doi.org/10.1016/B978-0-08-102575-8.00009-7 © 2020 Elsevier Ltd. All rights reserved.

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on conventional polymers without fiber reinforcement [14, 15]. Such AM technologies are called fused filament fabrication (FFF) or fused deposition modeling (FDM). Some studies reported the tensile, flexural, and creep properties of various FRAM materials [16]. Matsuzaki et al. and Chen et al have shown that FRAM parts are superior compared to 3D printed PLA and ABS ones, in terms of yield strength [17]. Melenka et al studied the effect of increasing the volumetric fraction of continuous fibers which resulted in an increased elastic modulus of the parts [18]. Furthermore, Dickson et al performed a more thorough investigation of the tensile properties of FRAM materials by testing nylon specimens reinforced with CF, Kevlar, and FG. It was found that the specimens fail at the shoulder of the “dog-bone” specimen and it was concluded that the reason for this was the shear force due to fiber misalignment [19]. Der Klift et al also noted that continuous fibers reinforced parts contain discontinuities of fiber in each layer although not as fragmented as short fibers [20]. Creep of FRAM materials was studied by Mohammadizadeh et al., where specimens of different materials were loaded at two different temperatures [21]. When FRAM built parts are under cyclic loading, they could face catastrophic failures in the long term. Hence, generating fatigue properties data of FRAM parts under different loading conditions of various fiber patterns could benefit and broaden the knowledge base and prevent the failure. Fatigue life of parts can be improved by defining proper manufacturing parameters. This chapter provides an insight to the most recent information on how to make stronger parts which are subjected to cyclic loadings.

9.2

Materials and specimen preparations

9.2.1 3D printing equipment The equipment used for the preparation of FRAM samples is a commercially available 3D printer, MKF [12]. This printer has dual extrusion head with two nozzles that print the base and reinforcement materials, respectively. A schematic representation of the printer is shown in Fig. 9.1. As in other AM processes, to produce a part, first a digital file is needed. A solid model, designed with a computer aided design (CAD) software tool, is converted to a Standard Tessellation (STL) file. This file is then uploaded to the slicer software. Eiger is a web-based slicer of MKF. Printing settings are fixed on the slicer, which generates the numerical code for the machine to command the movement of printer heads and filaments. Specimens at the different stages are visualized in Fig. 9.2. Filaments which come in the shape of cylindrical continuous wires are packed in spools. With the aid of stepper motors, filaments are pushed to the extruders, where the temperature varies between 265°C and 270°C, enough to put nylon or Onyx in a molten state. Matrix material solidifies immediately after its extruded. Fibers have higher melting temperatures, so they are just laid down horizontally in the interior of the part area. The extrusion heads move on the horizontal plane, where the base material is

Fatigue behaviors of fiber-reinforced composite 3D printing

337

Nylon spool

Fiber spool

Extrusion head w/ nozzles

Composite part

Printing bed and platform

Fig. 9.1 Schematic representation of the continuous FRAM.

3D CAD model (solidworks)

Layer-wise assembly (eiger)

3D printed part (markforged mark two)

Fig. 9.2 Part preparation from CAD model to the 3D printed part.

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Fatigue Life Prediction of Composites and Composite Structures

firstly printed on the periphery of the layer then it is reinforced by fibers. When the layer is concluded, the printing bed moves down so the new horizontal layer is started on the previously printed material.

9.2.2 Specimen preparation The way of preparing the specimen is critical. There are many parameters that affect the mechanical properties. Some of the parameters are already set by the printer capability, while the others are set by the user. Among those parameters are fiber volume, layer height, fiber orientation, infill pattern, infill percentage, and other slicing parameters. Kuchipudi [21] and Imeri et al. [22], the only studies currently available in the literature, focused the research on fiber orientation and fiber volume fraction. For simplifying purposes, Kuchipudi’s work will be classified as A, while Imeri et al.’s work will be classified as B. In A, fiber orientations with respect to longitudinal axis were set at 0, 45, and 90 degree, respectively. The fiber volume fractions are 0.25 and 0.50 out of total volume. Also, the fiber-reinforced material was FG. Schematic representation of the different fiber directions is shown in Fig. 9.3. The yellow lines represent the fiber reinforcement, while the black lines represent the base matrix. White boundary lines represent the shell lines and are made of the base material. In B, concentric (C) and isotropic (I) infills were used. In concentric infill, rings are added around the conferential perimeter of the specimen. In isotropic infill, fiber is added unidirectionally, linearly in a horizontal plane. Schematic representation of the two infill types is shown in Fig. 9.4. Different types of specimens were generated with 0, 1, 2, and 3 concentric rings. The angle for the isotropic infill was zero. Fiber reinforcement materials were CF, FG, and Kevlar. The nylon direction was chosen to be rectilinear due to giving better uniaxial load performance [22].

Fig. 9.3 Schematic representation of 0-, 45-, and 90-degree fiber orientations.

Fatigue behaviors of fiber-reinforced composite 3D printing

339

Fig. 9.4 Schematic representation of concentric and isotropic infill used in B study.

R = 10 mm

H = 10 mm

W = 20 mm

10T = 50 mm L = 15 mm

T = 5 mm

Fig. 9.5 ASTM specimen dimensions [23].

Specimens in A are designed according to the ASTM D3479 standard [23]. The specimens dimensions are 250
25
2.5 mm. Specimens in B are designed according to the ASTM E606 standard [24]. Dimensions of the specimens are shown in Fig. 9.5.

9.2.3 Experimental setup To perform the experiment, a servo-hydraulic test system is needed. The test system used was a closed-loop servo-hydraulic MTS810 machine with load cell capacity of 100 kN with an accuracy on the applied load under 1%. Machine has a lower moving gripping head that performs the cycles, and an upper gripping head only for gripping purposes. Fiber-reinforced nylon composites can be crushed by the gripping pressure. High pressure can crush the specimen, while low pressure can be a reason for the specimen to slip. Hence, it is important to set up a proper gripping pressure. In this study, by trial and error it was found that the ideal gripping pressure was 4 MPa. To initiate the experiment, it is required to input the testing variables. Frequency, amplitude, and average load are all controlled through MTS’s Multipurpose Testware interface. Different loads and frequencies were applied which will be given in more details in the following sections. However, it is important to note that for any load that was applied, machine tuning was needed. Tuning was performed to understand the difference between the test parameters and the received feedback.

340

9.3

Fatigue Life Prediction of Composites and Composite Structures

Experimental analysis

In A, the specimen is first tested for maximum tensile strength then the applied highest fatigue loads are 80%, 60%, and so on. The frequency used was 15 Hz and the minimum-to-maximum load ratio R ¼ 0. Specimens are tested under these loads until failure. From the experiments various results were observed. In the 0 degree infill specimens, for both 0.25 and 0.50 fiber volume fractions the failure was brittle in the direction of the fibers. The endurance limit for 0.25 infill was 244.4 MPa, since the specimen did not fail even after 4
107 cycles. For the 0.50 infill the endurance limit was at 335.3 MPa. The specimen did not break after 9
107 cycles. Increasing the fiber volumetric fraction proved to increase the fatigue strength. In the 45 degree specimens, for 0.25 and 0.50 fiber volumetric fractions, the failure was more ductile, and the material showed large viscoplastic deformations. The endurance limits were 36.2 and 60.2 MPa for 0.25 and 0.50 fiber volume fraction, respectively. A large stretch in the material at low load levels was observed, which shows that load is acting on the base matrix. Moving forward in the 90 degree specimens, the transverse properties of the material were tested. At this direction the load is carried by the base matrix mostly. Materials showed large viscoplastic deformations and failed at much lower loads and cycles compared to the other types of specimens. From the experimental results, it is understood that the direction and volumetric fraction of FG in the nylon matrix has a high influence on the fatigue properties. Strongest results were seen in the fiber direction, while the highest ductility was noticed on the 45 degree specimens. Similarly, endurance limits are highest for 0 degree, followed by 45 degree, and 90 degree specimens, correspondingly. Further, in the experimental study in B, the frequency used was 1 cycle per second with a minimum-to-maximum load ratio of 0.1. The goal in here was to understand the effect of fiber infill type, hence the experiments were truncated after 10,000 cycles. In addition, specimens that failed in the first cycle were also eliminated to be tested in the higher loads. Specimens were grouped in concentric (C), isotropic (I), and a mix of concentric and isotropic (I + C) infills with different number of rings. The first loads level was at a maximum of 3.33 kN. Results of the number of cycles at this load are shown in Table 9.1. Table 9.1 Results of the specimens from the first load level at 3.33 kN Max load (kN)

Material

Rings

Type

Test #1

Test #2

Test #3

3.33 3.33 3.33 3.33 3.33 3.33

Carbon Carbon Kevlar Kevlar Glass Glass

2 3 2 3 2 3

C C C C C C

510 10000+ 45 383 168 644

222 – 190 72 132 561

417 – 9 181 137 644

Fatigue behaviors of fiber-reinforced composite 3D printing

341

Isotropic infill specimens all passed the 10,000 cycles milestone hence were not tabulated. Among the concentric infill specimens, the ones with three number of rings endured more cycles. Adding more rings, increases the fiber volumetric fraction and hence could result in more cycles. CF-reinforced specimen outperformed the FG and Kevlar counterparts. Next, in the second load level (10.3 kN) results are shown in Table 9.2. From the first load, the specimens with three concentric rings were tested. However, regardless of material all three concentric rings failed in the first cycle, and thus, were eliminated from testing at higher loads. Similarly, for concentric and isotropic specimen with three rings, specimens failed in the first cycle. CF- and FG-reinforced specimens with two rings and isotropic infill did not break in the first cycle, however the Kevlar-reinforced specimen did. Truncating all the specimens from the previous loads, the third load level was at a maximum of 13.8 kN. Results are presented in Table 9.3. Table 9.2 Results from the second load at 10.3–1.03 kN Max load (kN)

Material

Rings

Type

Test #1

Test #2

Test #3

10.3 10.3 10.3 10.3 10.3 10.3 10.3 10.3 10.3 10.3 10.3 10.3 10.3 10.3

Glass Glass Glass Glass Glass Carbon Carbon Carbon Kevlar Kevlar Kevlar Kevlar Carbon Glass

0 1 2 3 0 1 2 3 0 1 2 3 3 3

I I+ C I+ C I+ C I I+ C I+ C I+ C I I+ C I+ C I+ C C C

750 699 19 1 10000+ 10000+ 3753 1 10000+ 54 1 1 1 1

1161 996 26 – – – 3995 1 – 285 – – – –

989 744 31 – – – 1747 – 196 – – – –

Table 9.3 Results from the third load at 13.8–1.38 kN Max load (kN)

Material

Rings

Type

Test #1

Test #2

Test #3

13.8 13.8 13.8 13.8 13.8 13.8 13.8 13.8

Glass Glass Glass Carbon Carbon Carbon Kevlar Kevlar

0 1 2 0 1 2 0 1

I I+ C I+ C I I+ C I+ C I I+ C

5 15 1 10000+ 324 1 3 1

66 24 – – 275 – 7 –

20 38 – – 209 – 6 –

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Fatigue Life Prediction of Composites and Composite Structures

At this relevantly high load, all the specimens with two rings combined with isotropic infill failed in the first cycle. Among Kevlar specimens, only the isotropic infill specimen endured a very small numbers of cycles. At this load, only CF with isotropic infill resisted more than 10,000 cycles. In the final load, (17.4–1.74 kN), all the specimens except the isotropic CF-reinforced specimen failed in the first cycle. Even, in these specimens cracking sound was heard in the first cycles.

9.4

Statistical analysis

A full factorial ANOVA design was applied to the experimental results [25]. In ANOVA, the independent variables were load, material, number or rings, and the type of infill. The dependent variable was the number of cycles to failure. A log transformation of the dependent variable was performed due to data being counts. In addition, right and left truncations were performed due to experiments being stopped at specimens that did not fail after 10,000 cycles and the ones that failed after one cycle, respectively. Thus, leading to reducing the full-factorial ANOVA to fourway ANOVA with missing data. Since ANOVA is a robust procedure, the truncated data had very little effect on the overall power of the test. The assumptions of normality and homogeneous variations were not violated because the results were within the boundaries of classical ANOVA. There was one violation of the classical method, which was resolved. Rings and type of infill were found to overlap, which means that the two variables were correlated. Rings were found to be more influential which lead to not include the type of infill in the analysis. ANOVA results are presented in Table 9.4. From the last column, it can be understood that load, material, and rings all have significant value in the number of cycles while the interaction of load and material is slightly significant (load*material). To visualize and simplify the meaning of the numbers in Table 9.4, a group of boxplots figures for the number of cycles and each dependent variable, respectively, is provided. Starting with the number of cycles and the different load is presented in Fig. 9.6. Load 1 was not included due to the specimens tested in this load were with concentric infill. The box plots for the number of cycles are regardless of material. The thicker horizontal line in the figure represents the median. It should be noted that the median line Table 9.4 Summary of ANOVA results

Load Material Rings Load*material Residuals

Df

Sum Sq

Mean Sq

F value

p value

3 2 3 2 28

46.542 33.521 27.645 4.331 13.62

15.5141 16.7607 9.2149 2.1653 0.4864

31.894 34.4567 18.944 4.4514

3.58
1009 2.82
1008 6.57
1007 0.02096

343

4 0

2

log(cycles)

6

8

Fatigue behaviors of fiber-reinforced composite 3D printing

10.3 kN

13.8 kN

17.4 kN

Fig. 9.6 Box plot of the number of cycles vs the different types of load.

4 0

2

log(cycles)

6

8

in each of the load, is higher than the maximum number of the cycles in the other load, from right to left. This result explains the importance of the load in the variance of the number of cycles. Further, number of rings in the specimens was varied with the goal of understanding their effect. Next, the number of cycles versus the number rings is presented in Fig. 9.7.

0

1

2

Number of rings

Fig. 9.7 Box plots of number of cycles vs the number of rings.

3

344

Fatigue Life Prediction of Composites and Composite Structures

In Fig. 9.7, the median of three number of rings is higher than the other counterparts. However, analyzing the number of the cycles in Tables 9.2–9.3, the three rings specimens failed in relatively low numbers. This high median number of cycles is achieved by the concentric infill specimens. In the concentric infill the three rings number specimens gave the best results. The variance of this box plot is very high as it can be seen in the Fig. 9.7. Similarly, for two number of rings, the median is mixed between isotropic and concentric infills specimens. Finally, a preliminary S-N curve is fitted. More FG-reinforced specimens with zero rings isotropic infill were tested at different loads (14.8, 12.8, and 11.3 kN). Although more data are needed for full S-N curve, a preliminary S-N curve is shown in Fig. 9.8. Nonlinear regression was applied to the given formula: load ¼ 3547∗ exp ð0:0004∗cyclesÞ

13.3 kN 11.1 kN

Load

15.6 kN

Estimate of S-N curve with 95% confidence bands

0

200

400

600

Number of cycles

Fig. 9.8 Preliminary S-N curve for FG with zero rings isotropic infill.

800

1000

Fatigue behaviors of fiber-reinforced composite 3D printing

9.5

345

Discussion

Fiber orientation and volume fraction are critical to the mechanical properties of the FRAM parts. The higher the volume fraction the better the fatigue performance. In terms of fiber direction, specimens with similar fiber volumetric fractions but with different fiber orientation gave out different experimental results. Thus, showing that fiber orientation is influential. To visualize it, the horizontal cross-section view of the specimens can be analyzed. Stress is higher in the narrower area of the specimen, hence fiber orientation in there becomes very important. The horizontal cross-section view is shown in Fig. 9.9. Addition of rings causes the isotropic infill area to decrease. This reduction of isotropic infill area results in lower failure resistance to fatigue. In general, the failure mode of the specimens at 0 degree was fiber pull-out. At different loads, different amount of fibers pulled out were noticed. In terms of angle direction, the 45 degree deformed more than the 0 and 90 degree. In 90 and 0 degree infill specimens the break was at 90 degree. In the 45 degree infill specimens the break angle was 45 degree. Regarding fiber material type, CF specimens had a brittle failure, while FG and Kevlar specimens were more ductile. Failed specimens with different fiber reinforcing materials are shown in Fig. 9.10.

Fig. 9.9 Horizontal cross-section view of specimens with zero, one, two, and three rings, top to bottom.

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Fig. 9.10 FG, CF, and Kevlar fiber-reinforced broken specimens at 10.3 kN.

Fig. 9.11 CF, Kevlar, and FG broken specimens at maximum load of 13.8 kN.

In study B in general, similar breaking patterns among different materials were noticed at the same load. For instance, in a higher load, the fiber pull-out was faster and in higher volume for all the materials. Broken specimens are shown in Fig. 9.11. Comparing Figs. 9.10 and 9.11, it can be seen the difference in the amount fiber pull-out.

Fatigue behaviors of fiber-reinforced composite 3D printing

9.6

347

Conclusions and outlook

The current research studies conducted on fatigue properties showed that 0 degree specimens perform best in uniaxial tension-tension loads, with zero ring carbon-fiber being the strongest. Also, 45 degree specimens turned out to be the most ductile. Increasing fiber volume fraction increases the fatigue performance. The information presented in this chapter can serve as a knowledge base for the fatigue properties of continuous fiber-reinforced AM parts. Since continuous FRAM is a 4-year-old technique, the conducted research is preliminary. Future studies could include testing at different orientations, mixing fiber orientations, and fiber materials. Furthermore, to improve the interfacial bonding of the fibers and base matrix different types of coatings could be applied and investigated. Also, analytical models could be developed to predict the fatigue life of FRAM components. Finally, automated fiber placement for better properties could be developed with the help of topology optimization. Specifically, the angle direction of fibers for different applications could be optimized.

Acknowledgments This work is part of a larger project funded by the Advanced Technological Education Program of the National Science Foundation, DUE #1601587. The funding provided by the National Science Foundation is greatly appreciated.

References [1] I. Gibson, D.W. Rosen, B. Stucker, Additive Manufacturing Technologies: Rapid Prototyping to Direct Digital Manufacturing, Springer, 2010. [2] G.J.M. Krijnen, 3D printing of functional structures, Vonk 34 (1) (2016) 10–20. [3] R. Huang, et al., Energy and emissions saving potential of additive manufacturing: the case of lightweight aircraft components, J. Clean. Prod. 135 (2016) 1559–1570. [4] Y. Swolfs, S.T. Pinho, Designing and 3D printing continuous fibre-reinforced composites with a high fracture toughness, in: American Society for Composites Thirty-First Technical Conference, 2016. [5] S. Rawal, J. Brantley, N. Karabudak, Additive manufacturing of Ti-6Al-4V alloy components for spacecraft applications, in: RAST 2013—Proceedings of 6th International Conference on Recent Advances in Space Technologies, 2013. [6] N.A. Russell, J. Floyd, J. Caston, M.R. Villalpando, I. Fidan, Project InnoDino: additively innovative dinosaur design and manufacturing, Int. J. Rapid Manuf. 6 (4) (2017) 262–278. [7] T. Fresques, D. Cantrell, I. Fidan, The development of a framework between the 3D printed patterns and sand-cast work pieces, Int. J. Rapid Manuf. 5 (2) (2015) 170–185. [8] J. Watson, F. Vondra, I. Fidan, The development of a framework for 3D printing, casting, and entrepreneurship, in: 2017 ASEE Annual Conference & Exposition, 2017. [9] P. Dudek, FDM 3D printing technology in manufacturing composite elements, Arch. Metall. Mater. (2013). [10] T. Page, Technology forecast for composite materials in product design, i-manager’s J. Mater. Sci. 3 (4) (2016) 5–19.

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[11] Z. Quan, et al., Additive manufacturing of multi-directional preforms for composites: opportunities and challenges, Mater. Today 18 (2015) 503–512. [12] Markforged.com, About Us: 3D Printer Manufacturer & Technology Company j Markforged [online], 2019. Available at: https://markforged.com/about/. (Accessed 20 May 2019). [13] R. Stephens, A. Fatemi, R. Stephens, H. Fuchs, Metal Fatigue in Engineering, John Wiley & Sons Inc, 2000. [14] T. Letcher, M. Waytashek, Material property testing of 3D-printed specimen in PLA on an entry-level 3D printer, in: Volume 2A: Advanced Manufacturing, 2014. [15] M. Fischer, V. Sch€oppner, Fatigue behavior of FDM parts manufactured with Ultem 9085, JOM 69 (2017) 563–568. [16] I. Fidan, et al., The trends and challenges of fiber reinforced additive manufacturing, J. Adv. Manuf. Technol. (2019) 1–18. [17] R. Matsuzaki, et al., Three-dimensional printing of continuous-fiber composites by in-nozzle impregnation, Sci. Rep. 6 (2016)23058. [18] G.W. Melenka, B.K.O. Cheung, J.S. Schofield, M.R. Dawson, J.P. Carey, Evaluation and prediction of the tensile properties of continuous fiber-reinforced 3D printed structures, Compos. Struct. 153 (2016) 866–875. [19] A.N. Dickson, J.N. Barry, K.A. McDonnell, D.P. Dowling, Fabrication of continuous carbon, glass and Kevlar fibre reinforced polymer composites using additive manufacturing, Addit. Manuf. 16 (2017) 146–152. [20] F. Van Der Klift, Y. Koga, A. Todoroki, M. Ueda, Y. Hirano, R. Matsuzaki, 3D printing of continuous carbon fibre reinforced thermo-plastic (CFRTP) tensile test specimens, Open J. Compos. Mater. 6 (2016) 18–27. [21] M. Mohammadizadeh, I. Fidan, M. Allen, A. Imeri, Creep behavior analysis of additively manufactured fiber-reinforced components, Int. J. Adv. Manuf. Technol. 99 (2018) 1225–1234. [22] M. Fernandez-Vicente, W. Calle, S. Ferrandiz, A. Conejero, Effect of infill parameters on tensile mechanical behavior in desktop 3D printing, 3D Print. Addit. Manuf. 3 (2016) 183–192. [23] D3479/D3479M-12, Standard Test Method for Tension-Tension Fatigue of Polymer Matrix Composite. 2012. [24] ASTM International, E606/E606M-12, Standard Test Method Strain-Controlled Fatigue Testing, ASTM International, West Conshohocken, PA, 2012. [25] A. Imeri, I. Fidan, M. Allen, D.A. Wilson, S. Canfield, Fatigue analysis of the fiber reinforced additively manufactured objects, Int. J. Adv. Manuf. Technol. 98 (2018) 2717–2724.

Computational intelligence methods for the fatigue life modeling of composite materials

10

Anastasios P. Vassilopoulosa, Efstratios F. Georgopoulosb a Ecole Polytechnique F
ed
erale de Lausanne (EPFL), Composite Construction Laboratory (CCLab), Lausanne, Switzerland, bTechnological Educational Institute (T.E.I.) of Peloponnese, Kalamata, Greece

10.1

Introduction

Computational engineering involves the design, development, and application of computational systems for solving physical problems encountered during the study of science and engineering. These computational systems are used to obtain solutions of mathematical models that represent particular physical processes through the use of algorithms, software, and other methods. Computational engineering has emerged as a fast-growing multidisciplinary area with connections to such scientific fields as engineering, mathematics, and computer science and is bound to play an important role in new inventions and discoveries. Computation is today regarded as an important tool for the advancement of scientific knowledge and engineering practice, in addition to the already existing tools of theoretical analysis and physical experimentation. Simulation techniques allow scientists to easily study complex natural phenomena and processes, which would otherwise be very difficult, dangerous, or even impossible. The need for greater accuracy and detail in such simulations has created the need for faster and more efficient computer algorithms and architectures. It is because of these advancements that scientists and engineers can solve highly complex problems that were once thought to be unsolvable. The development of “thinking machines” has been an objective of humanity for centuries. However, it was only in the 20th century with the development of powerful computers that scientists began to build “intelligent” machines based on recent discoveries in several scientific domains. Artificial intelligence is the branch of computer science that deals with the development of algorithms and techniques that can simulate or even recreate the capabilities of the human mind. Artificial intelligence (AI) methods such as artificial neural networks (ANNs), genetic algorithms (GAs), genetic programming (GP), and fuzzy logic techniques (FL) have been successfully used for years in different scientific fields for optimization, pattern recognition, data clustering, and signal processing.

Fatigue Life Prediction of Composites and Composite Structures. https://doi.org/10.1016/B978-0-08-102575-8.00010-3 © 2020 Elsevier Ltd. All rights reserved.

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Over the last decade these novel computational methods have been introduced in the new sectors of engineering and material science. Neural networks and GAs have initially been used as tools for optimization of design methods [1, 2], and damage characterization of composite materials [3–6]. These methods appear to offer a means of dealing with many multivariate problems for which an accurate analytical model does not exist or would be very difficult to develop. Computational intelligence techniques offer an advantage over conventional optimization techniques in that they could find maxima in multidirectional search spaces, and in that sense are very valuable for solving several multiparametric optimization problems that arise in engineering. For example, the optimization of laminate design against buckling [3], the optimization of a composite laminate to maximize its strength [4], the optimum design of bolted composite lap joints [5], or the characterization of damage in carbon/carbon composite laminates [6]. GAs and neural networks were used in conjunction with other analytical tools and experimental methods, finite element in Ref. [4], stress analysis in Ref. [5], and acoustic emission in Ref. [6] to achieve the final objective. The application of similar computer techniques for modeling material behavior under static and fatigue-loading conditions, for example, Ref. [7–11] went a step further. Subsequently, ANNs, GP, and adaptive neuro-fuzzy inference systems (ANFIS) have been used for the interpretation of the fatigue data of composite materials. A limited number of relevant articles have been published concerning the fatigue of composite materials and structures, for example, Ref. [12–25]. In Ref. [12], Lee and coworkers tried to model fatigue lives of [(
45/02)2]S and unidirectional composite laminates under constant amplitude loading at different stress ratios (the ratio of minimum over maximum cyclic stress) and under block loading. Comparison of their findings proved that, at least for the case of constant amplitude loading, ANN modeling is equivalent to, if not better than, the modeling based on other conventional modeling techniques. Two assets of ANN modeling were recognized: less computational effort and effective modeling, even for relatively small databases. However, even when ANNs are used for modeling, there is a lower limit to data set size. An ANN model can be considered as a black box and can be well trained to produce accurate output data, even for very small existing experimental data. However, cases like these should be treated with care since the use of limited data sets for training may lead to indeterminate and unstable systems as mentioned in Ref. [26]. Moreover, ANN does not offer any improvement in accuracy compared to simple regression analyses when simple relationships are modeled [27]. Al-Assaf and El Kadi [13–15] succeeded in modeling the fatigue life of unidirectional composite laminates by using different ANN paradigms and they discussed the possibility of improving modeling efficiency by using other types of ANN, besides the classic feed-forward algorithm. The good modeling efficiency of ANN was also proved in a series of articles by Vassilopoulos et al. [16, 17] in which it was pointed out that ANN could be used to reduce experimental effort and cost, as efficient modeling could be achieved even if only 50% of the available experimental data were used. ANN proved a good tool for the construction of constant life diagrams (CLDs) by using a reduced data set [17–19]. ANNs have also been used to model the fatigue life of composite structures, see for example, [20] where an ANN was used for the prediction of the fatigue life of sandwich composites

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under flexural loading and [21] where the same method was employed for the modeling of the fatigue crack growth rate of bonded FRP-wood interfaces. Other computational methods were also adopted for the analysis of the fatigue data of composite materials. A hybrid neuro-fuzzy method designated ANFIS has been used to model the fatigue life of unidirectional and multidirectional composite laminates. Results of its application to two material systems have been presented in Refs. [22, 23]. ANFIS is a combination of ANN and fuzzy logic and combines the advantages of both techniques; the ability of ANNs in adapting and learning together with the merit of approximate reasoning offered by fuzzy logic. In fatigue life modeling, ANFIS is based on linguistic rules that are dictated by experience, for example, if developed stress is low and the off-axis angle is low then the life of a laminate is long. Despite the introduction of these fuzzy rules and the combination of the advantages of two techniques, no improvement in modeling was mentioned in Refs. [22, 23] when compared to ANN results. Genetic programming, GP, was also used in this field. GP has been successfully used as a tool for modeling the fatigue behavior of composite materials, as presented by Vassilopoulos et al. in Refs. [24, 25]. This tool can be used to model the fatigue behavior of composite laminates subjected to constant amplitude loading. Since it is a material-independent method, it can be used to model the fatigue behavior of any composite material, and compares favorably with conventional modeling techniques. Interpretation of the fatigue data of composite materials seems to represent an ideal problem for the computational methods under consideration. The problem consists of the material behavior modeling under the given environmental and mechanical loading conditions. The data available in fatigue databases frequently suffice for the adequate training of the tools. This is in any case the condition for accurate and reliable modeling using the described computational methods. With a reasonable amount of fatigue data, problems like the production of an indeterminate model (when input data are limited compared to the designed model parameters) or model overfitting (when the number of input parameters is relatively large compared to the input data) can be eliminated. In fatigue life modeling, between one and four input parameters are considered, for example, cyclic stress, stress ratio, testing temperature, off-axis angle etc., and one single output parameter is set, usually the number of cycles to failure. During the training and testing process, the structure, learning algorithm, and other parameters of the neural network, GP, or ANFIS model should be optimized. The training process for an ANN involves minimizing the error between actual and predicted outputs, using the available training data, by continuously adjusting the connection weights. During the development of a GA model or a GP model, parameters such as population size, the genetic operator’s probabilities, and the termination criteria should also be well defined in order to generate a well-structured model. A number of methods have been developed to address the problem of the model parameters’ optimization, see for example, Refs. [28, 29]. A sufficiently optimal model, trained with the available experimental data, can subsequently generate accurate results for any new input data set.

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Other, “conventional” methods [25], in the form of deterministic or stochastic mathematical models, were developed for modeling the fatigue behavior of composite materials. However, the question “Which type of S-N curve?” has not yet been satisfactorily answered. Extensive literature exists on this subject, a short, but comprehensive review of which is given in Ref. [30]. As pointed out, the S-N curve-type selection, but also type of fitting (e.g., linear regression or use of Weibull statistics), could lead to extremely different inter/extrapolated results. The best choice depends mainly on the material. For example, it was shown in Ref. [31] that a log-log expression is the best function for fitting 0/
45 glass fiber-reinforced fatigue data but show pure modeling results for another glass polyester material system, see Fig. 9 of Ref. [25]. Moreover, since engineering applications are designed for an extended time period, it is imperative to assume an S-N curve that is able to accurately estimate life at the high-cycle fatigue regime as well. In this case S-N curves with different slopes in low-cycle and high-cycle fatigue regimes seem more adequate. This chapter aims to explore the possibilities of novel computational methods and examine their ability to model/predict the fatigue life of composite laminates. ANNs, adaptive neuro fuzzy inference systems, and GP will be considered. Their applicability in the modeling of the fatigue life of several different material systems will be demonstrated. The discussion will focus on the limitations of these methods, but also the advantages that they offer when compared to the conventional methods used by researchers and engineers over the last five decades.

10.2

Theoretical background

The computational techniques described in the following sections fall under the category of data-driven as opposed to model-driven approaches. These techniques build models based on available input-output mapping data, which is one of the main reasons for these techniques being so well suited to problems like the one examined here. AI methods work on available experimental data, in the form of input-output mappings that are separated to define two or three data sets usually, one designated “training set,” the second “validation set” and the third, if one exists, the test set. Every data set contains the input and the corresponding output values of the physical or artificial system under investigation. The technique that is used (neural network, ANFIS, or GP) develops a model able to describe the relationship between the inputs and the outputs in the training data set (this phase is known as training). In a second phase, the model produced is evaluated using the validation data set in order to examine the generalization ability of the produced model. In other words, the model’s ability to accurately predict (with a small error) the behavior of new (“unseen”) data, that is, data not used in training, is validated. The training and validation procedure is usually an iterative one. This procedure is completed when certain termination criteria have been fulfilled; for example a maximum number of generations or training epochs have been reached. At this stage the final model has been generated. The performance of the final model can be further evaluated using new data which form the test (or applied) set, a set of new and “unseen” data that do not belong in the training or validation sets and provide a measure of the real generalization ability of the produced model.

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10.2.1 Artificial neural networks ANNs are mathematical or computational models that are based on biological neural networks and especially on the biological neural networks of the brain. ANNs are networks consisting of many interconnected simple processing units, termed neurons or nodes, which process information using a connectionist approach to computation (in contrast to symbolic artificial intelligence). The connections between neurons are known as synapses and are characterized by a weight value. In most cases, an ANN is an adaptive system that changes its structure based on external or internal information that flows through the network during the learning phase. ANNs can be seen as nonlinear statistical data modeling tools and can be used to model complex relationships between inputs and outputs. ANNs possess many advantages that make them very suitable for real-life applications. Generally an ANN can be considered as a nonlinear mapping FANN ¼ RI ! RK from I R to RK, where I and K are, respectively, the dimensions of the input and desired output space. This function is usually a complex function of a set of nonlinear functions, one for each neuron in the ANN. The basic building block of an ANN is the artificial neuron (AN), which implements a nonlinear mapping fAN ¼ RI ! [0, 1] or [1, 1] depending on the activation function used. Generally, every neuron calculates its weighted sum of input signals and produces an output using the following formula: oAN ¼ fAN

I X

! wi ∙xi

(10.1)

i¼0

where, oAN is the output of the AN, fAN the used activation function, I the number of its inputs, xi the value of the incoming signal from some other neuron i, wi the value of the weight of the synapse that connects the neuron i with the AN, and x0 equals +1 or 1 depending on the use of a bias or a threshold term, respectively. A number of different activation functions can be used such as the linear, the step, the ramp, the sigmoid, the hyperbolic, or the Gaussian [32]. The most important characteristic that makes ANNs so appealing is their ability to learn, i.e., to adjust their connection weight values in order to perform a specific task. There are three main types of learning: l

l

l

Supervised learning, where the ANN is provided with a data set consisting of the input vectors and target (desired) outputs associated with each input vector. The aim is to adjust the weight values of the ANN so that the error between the target and the network’s actual output is minimized. Unsupervised learning, where the aim is to discover patterns or features in the input data without any external guidance or assistance. The unsupervised learning generally performs a clustering of the training patterns. Reinforcement learning, where the aim is to train by rewarding good performance and penalizing bad performance, and not by giving the target outputs.

One of the best known ANN topologies, the multilayer feed-forward network (also called multilayer perceptron—MLP), is used in this chapter. The MLP is trained by a supervised learning method, the error back-propagation (EBP) algorithm, and

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organized into layers of neurons with forward connections only. Usually an MLP has an input layer, one or more hidden layers and an output layer. The first- or input-layer consists of a number of sensory neurons (neurons that do no processing but just sense the incoming signals and pass them to the next layer). The last layer (the one that produces the final output of the network) is known as the output layer and consists of a number of computational neurons. All the other layers between the input and output layers are called hidden layers and consist of computational neurons. In contrast to sensory neurons that do not do any processing, computational neurons process their incoming signals as described above. The input signal in an MLP propagates from the input to the output layer through the hidden layers in a forward direction. These networks are usually fully connected, which means that each neuron in any layer of the network is connected to all the neurons in the next layer. The typical example of an MLP shown in Fig. 10.1 consists of one input, one hidden, and one output layer. An MLP can have more than one hidden layer and more than one neuron in the output layer. As explained above, the EBP algorithm is used to train the MLP. The EBP is the best known and widely used supervised learning algorithm and operates in two phases: l

l

The feed-forward pass in which it calculates the output value(s) of the network (it propagates the input signal forward from the input layer to the output layer). The backward pass in which it propagates an error signal backwards from the output layer toward the input layer. The weights of the MLP are adjusted in order to minimize this error signal, according to this back-propagated error signal.

Hidden layer

Input layer

Output layer

. . .

Fig. 10.1 Typical ANN topology.

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These two phases constitute a learning iteration (one iteration is also known as an epoch). The training of an MLP using EBP includes a large number of such iterations until a satisfactory performance is attained. An in-depth description of the EBP algorithm can be found in Refs. [32, 33] or any other neural network textbook. For the successful application of MLP in modeling the fatigue life of FRP composite materials, the following aspects should be considered: l

l

l

l

The preprocessing of the application data; that is, the normalization of the data, the creation of the training, validation, and test set. The selection of the most appropriate MLP architecture for the problem at hand; that is, how many inputs, hidden layers, neurons in each hidden layer will be used. The selection of the most suitable parameters for the EBP algorithm; that is, termination criterion, learning rate, and momentum. Scatter of the available experimental fatigue data is also a factor that has to be taken into account for modeling accuracy. However, data scatter would have a similar effect on any other conventional modeling method.

10.2.2 Adaptive neuro-fuzzy inference system A fuzzy logic system is unique in the sense that it is able to simultaneously handle numerical data and logic knowledge. It is a nonlinear mapping of an input data (feature) vector into a scalar output, that is, it maps numbers into numbers. Fuzzy set theory and fuzzy logic establish the specifics of the nonlinear mapping. A fuzzy logic system can be expressed mathematically as a linear combination of fuzzy functions. It is a nonlinear universal function approximator, a property that it shares with feed-forward neural networks. Adaptive neuro-fuzzy inference system, ANFIS, is a blend of ANN and fuzzy logic that combines the advantages of both methods. ANFIS was introduced by Takagi and Sugeno in 1985 [34] and further developed by Jang [35]. Zhang and Morris [36] discussed the architecture of ANFIS, which could construct a nonlinear input-output mapping, based mostly on human expertise and stipulated data pairs. Fuzzy logic methods have been used to model various highly complex and nonlinear systems based on a set of sample data and fuzzy “if-then rules.” A fuzzy inference system can model the qualitative aspects of human knowledge without employing any quantitative analyses. The following notation is common in fuzzy logic modeling and is adapted to serve the needs of this study: l

l

Linguistic variables: form the basic concept underlying fuzzy logic that is, a variable whose values are expressed in words rather than numbers. The input linguistic variables specified here for the specific problem of fatigue life modeling are the following: orientation angle (θ), stress ratio (R), maximum cyclic stress (σ max), and cyclic stress amplitude (σ a). The number of cycles to failure (N) is used as the only output variable. Membership function (MF): is the curve which defines the way each point in the input space is mapped to a membership value (or degree of membership) between 0 and 1. The membership function type can be any appropriate parameterized membership function such as triangle, Gaussian, or bell-shaped.

356 l

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Fuzzy sets: unlike a classic set, a fuzzy set does not have a crisp boundary, that is, the transition from the case of “belonging to a set” to the case of “not belonging to a set” is gradual. Normally this smooth transition is characterized by a membership function which gives flexibility to the fuzzy sets in commonly used modeling linguistic expressions. For the case studied here, a linguistic expression could be: “fiber orientation angle (θ) is close to zero” or “stress ratio (R) is high,” etc. Linguistic rules: a set of linguistic “if-then” rules applied to the defined linguistic variables. A single fuzzy “if-then” rule assumes the form “If x is A then y is B,” where A and B are linguistic values defined by fuzzy sets on the ranges X and Y, respectively. The if-part of the rule “x is A” is termed the antecedent or premise, while the then-part of the rule “y is B” is termed the consequent or conclusion. Fuzzy “if-then” rules with multiple antecedents like the following are often used:

Rule: If the fiber orientation angle is near to zero, the stress ratio is low, the stress amplitude is low and the maximum stress is low, then specimen life is long. The output resulting from the described fuzzy logic method has to be defuzzified or else converted to a crisp value by using any of the available defuzzification methods, such as the center of gravity method. The membership functions used to represent linguistic variables may have a significant effect on modeling performance as the type of MF used determines when a given rule is to be activated (in fuzzy logic “the rule is fired”). Three types of membership functions—triangular, Gaussian, and bellshaped—have been used in this study to examine MF influence on the modeling efficiency of ANFIS. Although the fuzzy inference system has a structured knowledge representation in the form of fuzzy “if-then” rules, it lacks the adaptability to deal with a changing external environment. Therefore neural network learning concepts have been incorporated into fuzzy inference systems, resulting in adaptive neuro-fuzzy modeling. The adaptive inference system is a network which consists of a number of interconnected nodes. Each node is characterized by a node function with fixed or adjustable parameters. The network “learns” the behavior of the available data during the training phase by adjusting the parameters of the node functions to fit that data. The basic learning algorithm, the EBP, is also applied to minimize a set measure or a defined error, usually the sum of squared differences between desired and actual model outputs. An ANFIS architecture based on the first order Takagi-Sugeno model is schematically presented in Fig. 10.2. It is assumed that the desired output is a function of all the input parameters. The relationship between input and output parameters is dominated by linguistic rules. Moreover, the input parameters are defined by fuzzy sets rather than crisp sets. The fuzzy inference system shown in Fig. 10.2 is composed of four layers, each involving several nodes. The output signals from the nodes of the previous layer will be accepted as the input signals in the current layer. After manipulation by the node function in the current layer, the output will serve as input signals for the subsequent layer. Layer 1: The first layer of this architecture is the fuzzy layer. Each node of this layer makes the membership grade of a fuzzy set.

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Fig. 10.2 Takagi-Sugeno ANFIS model. Layer 2: Every node in layer 2 is a fixed node, indicated by a circle, whose output is the product of all the incoming signals that is, T-norm operation: The output signal denotes the firing strength of the associated rule. The firing strength is also called the “degree of fulfillment” of the fuzzy rule, and represents the degree to which the antecedent part of the rule is satisfied. Layer 3: Every node in layer 3 is an adaptive node, indicated by a square node. The consequent parameters in this layer will be adapted in order to minimize the error between the ANFIS outputs and experimental results. Layer 4: Every node in layer 4 is a fixed node, indicated by a circle node. The node function computes the overall output by summing all the incoming signals.

This ANFIS structure represents a four-dimensional space partitioned into N1  N2  N3  N4 regions, each governed by a fuzzy “if-then” rule. In other words, the premise part of a rule defines the fuzzy region, while the consequent part specifies the output within the region. A hybrid learning algorithm is used to adapt the parameters of the first layer, known as premise or antecedent parameters, and the parameters of the third layer, referred to as consequent parameters, in order to optimize the network. The network uses a combination of back-propagation and the least squares method to estimate membership function parameters. More specifically, in the forward pass of the hybrid learning algorithm, node outputs go forward as far as layer 3 and the consequent parameters are identified by the least squares method. In the backward pass, error signals propagate backwards and the premise parameters are updated by a gradient descent method.

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10.2.3 Genetic programming Genetic programming, GP, is a domain-independent problem-solving technique in which computer programs are evolved to solve, or approximately solve, problems. GP is a member of the broad family of techniques called evolutionary algorithms. All these techniques are based on the Darwinian principle of reproduction and survival of the fittest and are similar to biological genetic operations such as crossover and mutation. GP addresses one of the central goals of computer science, namely automatic programming; which is to create, in an automated way, a computer program that enables a computer to solve a problem [37]. In GP, the evolution operates on a population of computer programs of varying sizes and shapes. These programs are habitually represented as trees, like the one shown in Fig. 10.3, where the function: f(x) ¼ 2π + ((x + 3)  3a) is represented in tree format. The operations in the “tree branches” are performed and the result is given at the “tree root.” GP starts with an initial population of thousands or millions of randomly generated computer programs composed of the available programmatic ingredients and then applies the principles of biological evolution to create a new (and often improved) population of programs. This new population is generated in a domainindependent way using the Darwinian principle of survival of the fittest, an analogue of the naturally occurring genetic operation of crossover (sexual recombination), and occasional mutation [38]. The crossover operation is designed to create syntactically valid offspring programs (given closure among the set of programmatic ingredients). GP combines the expressive high-level symbolic representations of computer programs with the near-optimal efficiency of learning of Holland’s GA. A computer program that solves (or approximately solves) a given problem often emerges from this process [38]. Six major preparatory steps should be performed before applying GP [38] in a given problem. These steps include preparation of data sets, setting-up of the model and design of the termination criteria, as explained in the following: l

Determination of the set of terminals. The terminals can be seen as the inputs to the as-yetundiscovered computer program. The set of terminals (or Terminal Set T, as it is often called) together with the set of functions are the ingredients from which GP constructs a computer program to solve, or approximately solve, the problem.

2p+((x+3)-3a)

Fig. 10.3 Tree representation in genetic programming.

Computational intelligence methods for the fatigue life modeling of composite materials l

l

l

l

l

359

Determination of the set of primitive functions. These functions will be used to generate the mathematical expression that attempts to fit the given finite sample of data. Each computer program is a combination of functions from the function set F and terminals from the terminal set T. The selected function and terminal sets should have the closure property so that any possible combination of functions and terminals produces a valid executable computer program (a valid model). Determination of the fitness measure which drives the evolutionary process. Each individual computer program in the population is executed and then evaluated, using the fitness measure, to determine how well it performs in the particular problem environment. The nature of the fitness measure varies with the problem: for example, for many problems, fitness is naturally measured by the discrepancy between the result produced by an individual candidate program and the desired result; the closer this error is to zero, the better the program. For some problems, it may be appropriate to use a multi-objective fitness measure incorporating a combination of factors such as correctness, parsimony (smallness of the evolved program), efficiency, etc. Determination of the parameters for controlling the run. These parameters define the guidelines in accordance with which each GP model is evolved. The population size, that is the number of created computer programs, the maximum number of runs, that is, of evolved program generations, and the values of the various genetic operators are included in the list of parameters. Determination of the method for designating a result. A frequently used method of result designation for a run is to appoint the best individual obtained in any generation of the population during the run (i.e., the best-so-far individual) as being the result of the run. Determination of the criterion for terminating a run. The maximum number of generations, or the maximum number of successive generations for which no improvement is achieved, values that were determined in step 4, are usually considered as the termination criteria.

GP starts with an initial population (generation 0) of randomly generated computer programs composed of the given primitive functions and terminals. Typically, the size of each program is limited, for practical reasons, to a certain maximum number of points (i.e., total number of functions and terminals) or a maximum depth of the program tree. Typically, each computer program in the population is run over a number of different fitness cases so that its fitness is measured as a sum or an average over a variety of different representative situations. For example, the fitness of an individual computer program in the population may be measured in terms of the sum of the absolute value of the differences between the output produced by the program and the correct answer (desired output) to the problem (i.e., the Minkowski distance) or the square root of the sum of the squares (i.e., Euclidean distance). These sums are taken over a sampling of different inputs (fitness cases) to the program. The fitness cases may be chosen at random or in some structured way (e.g., at regular intervals) [38]. The computer programs in generation 0 (initial population) will almost always produce a very poor performance, although some individuals in the population will fit the input data better than others. These differences in performance are then exploited by GP.

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The Darwinian principle of reproduction and survival of the fittest and the genetic operations of crossover and mutation are used to create a new offspring population of individual computer programs from the current population. The reproduction operation involves selecting a computer program from the current population of programs based on fitness (i.e., the better the fitness, the more likely the individual is to be selected) and allowing it to survive by copying it into the new population. The crossover operation creates new offspring computer programs from two parental programs that are selected based on their fitness. The parental programs in GP are usually of different sizes and shapes. The offspring programs are composed of subexpressions from their parents. These offspring programs are usually of different sizes and shapes than their parents. For example, consider the two parental computer programs (models) represented as trees in Fig. 10.4. One crossover point is randomly and independently chosen in each parent. Consider that these crossover points are the division operator (/) in the first parent (the left one) and the multiplication operator ( ) in the second parent (the right one). These two crossover fragments correspond to the underlying subprograms (sub-trees) in the two parents—the sub-trees circled in Fig. 10.4. The two offspring resulting from the crossover operation depicted in Fig. 10.5 have been created by swapping the two sub-trees between the two parents in Fig. 10.4. Thus, the crossover operation creates new computer programs using parts of existing parental programs. Since entire sub-trees are swapped, the crossover operation always produces syntactically and semantically valid programs as offspring, regardless of the choice of the two crossover points. Because programs are selected to participate in the crossover operation with a probability based on their fitness, crossover allocates future trials to regions of the search space whose programs contain parts from promising programs [38]. The mutation operation creates an offspring computer program from one parental program that is selected based on its fitness. One mutation point is randomly and independently chosen and the sub-tree occurring at that point deleted. A new sub-tree is then grown at that point using the same growth procedure as was originally used to create the initial random population (this is only one of the many different ways in which a mutation operation can be implemented) [38]. l

Fig. 10.4 Crossover operation: The two parental programs in tree representation.

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Fig. 10.5 The crossover operation: The two offspring programs after the crossover operation.

After the genetic operations have been performed on the current population, the new population of offspring (the new generation) replaces the old population (the old generation) and the generation index increases by one. Each individual in the new population is then measured for fitness, and the process is repeated over many generations until the termination criterion/criteria is/are satisfied.

10.2.4 The gene expression programming algorithm Gene Expression Programming, GEP, is an evolutionary algorithm proposed by Candida Ferreira at 2001 [39, 40], as an alternative method to overcome the drawbacks of GAs and GP. Similar to GA and GP, GEP follows the Darwinian principles of natural selection and survival of the fittest individual. The fundamental difference between the three algorithms is that, in GEP there is a distinct discrimination between the genotype and the phenotype of an individual. This difference resides in the nature of the individuals, namely in the way the individuals are represented: in GAs the individuals are symbolic strings of fixed length (chromosomes); in GP the individuals are nonlinear entities of different sizes and shapes (parse trees). In GEP the individuals are also symbolic strings of fixed length representing an organism’s genome (chromosomes/genotype), but these simple entities are encoded as nonlinear entities of different sizes and shapes, determining an organism’s fitness (expression trees/phenotype) [39–41]. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi For example [41], the mathematical expression ðða  bÞ∙ðc + d Þ Þ can be represented as the expression tree, ET, of Fig. 10.6. Where Q represents the square-root function, f ¼ {Q, ,  , +} is the set of functions and T ¼ {a, b, c, d} is the set of terminals. This kind of expression tree consists of the phenotypic expression of GEP genes, whereas the genes are linear strings encoding these complex structures. For this particular example, the linear string corresponds to l

01234567 Q∙  + abcd

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Q

* – a

+ b

c

Fig. 10.6 The expression tree of mathematical expression

d

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðða  bÞ∙ðc + dÞ Þ.

which is the straightforward reading of the expression tree from top to bottom and from left to right. These linear strings are called k-expressions, from the Karva notation. The k-expressions of GEP correspond to the region of genes that gets expressed. This means that there might be sequences in the genes that are not expressed, which is rather usual. The reason for these noncoding regions is to provide a buffer of terminals so that all k-expressions encoded in GEP genes correspond always to valid programs or expressions [41]. The genes are composed of two different parts: a head and a tail. The head contains symbols that represent both functions and terminals, whereas the tail contains only terminals. The set of functions usually includes any mathematical, or Boolean function that believed to be appropriate for the problem at hand. The set of terminals is composed of the constants and the independent variables of the problem. The head length, denoted as h, is determined by the user, whereas the tail length, denoted as t, is evaluated using the following relation: t ¼ ðnmax  1Þ∙h + 1 Where nmax is the maximum arity (the number of arguments of the function with the most arguments). For example [41], let a gene created using the set of functions F ¼ {Q, + ,  , , /} and the set of terminals T ¼ {a, b}, where nmax ¼ 2. If we choose a head length h ¼ 15, then t ¼ (nmax  1) ∙ h + 1 ¼ (2 – 1) ∙ 15 + 1 ¼ 16, which gives a gene length g ¼ 15 + 16 ¼ 31. The randomly generated string below is an example of one such gene [41]: l

0123456789012345678901234567890 ∙ b + a  aQab + == + b + babbabbbababbaaa The expression tree encoded by the above string is depicted at Fig. 10.7 [41]. In this case, the ET uses only 8 of the 31 elements that constitute the gene. Thus, despite their fixed length, each gene has the potential to code for expression trees of different sizes and shapes, with the simplest (smallest) gene composed of only one node (when the first element of the gene is a terminal), and the largest gene

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* b

+ a

– a

Q

a

Fig. 10.7 The expression tree of string b + a aQab +//+b+ babbabbbababbaaa. l

composed of as many nodes as there are elements in the gene (when all the elements in the head are functions with the maximum arity) [41, 42]. This leads to one of the main advantages of GEP: the chromosomes will always produce valid expression trees, regardless of modification, and this means that no time needs to be spent on rejecting invalid organisms, as in the case of GP [40]. In GEP, each gene encodes an ET. In the case of multigenic chromosomes, each gene codes for a sub-ET and the sub-ETs interact with one another using a linking function (any mathematical or Boolean function with more than one argument) in order to fully express the individual. Every gene has a coding region known as an open reading frame (ORF) that, after being decoded, is expressed as an ET, representing a candidate solution for the problem. While the start point of the ORF is always the first position of the gene, the termination point does not always coincide with the last position of a gene. The process of GEP algorithm begins with the random generation of the linear chromosomes (or individuals) of the initial population. Then the chromosomes are expressed as ETs and the fitness of each individual is evaluated. After that, the individuals are selected according to their fitness in order to be modified by genetic operators and reproduce the new population. The individuals of this new population are, in their turn, subjected to the same developmental process: expression of the chromosomes, evaluation, selection according to fitness and reproduction with modification. The process is repeated for a certain number of generations or until a good solution has been found. A full description of the algorithm can be found in Refs. [39, 40].

10.3

Modeling examples

One of the fundamental ways of interpreting fatigue data and modeling the fatigue behavior of composite materials and structures is the so-called stress- (or strain-) based method (see Chapter 1). The output of this technique is a phenomenological model that simply correlates the number of loading cycles that the tested material

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can sustain under a given cyclic (frequently sinusoidal) stress to certain loading parameters, such as stress ratio, testing frequency, developed stress level, etc. Test results from different specimens tested under various cyclic stress levels form the S-N curve. A mathematical model is then introduced in order to use these test results in any design process. Frequently statistical models are implemented for the analysis of fatigue data and derivation of S-N curves for certain reliability levels; see for example Refs. [43–45]. A number of different types of fatigue models (or types of S-N curves) have been presented in the literature, the most “famous” being the semilogarithmic (also called lin-log) and the logarithmic (log-log) relationships. Based on these it is assumed that the logarithm of the loading cycles is linearly dependent on the cyclic stress parameter, or its logarithm. A semi-empirical S-N formulation for the modeling of the constant amplitude fatigue behavior of composite materials and structures has been introduced in Ref. [46]. The proposed hybrid S-N formulation is based on the exponential and power law fatigue models combining them in order to improve their modeling accuracy in the low and high cycle fatigue regions. Fatigue models defined in this way do not take different stress ratios or frequencies into account, that is, different model parameters should be determined for different loading conditions. A drawback of these methods is that they are case-sensitive, since they may provide very accurate modeling results for one material system but very poor ones for another. Other types of fatigue formulations that take the influence of stress ratio and/or frequency into account have also been reported [47, 48]. A unified fatigue function that permits the representation of fatigue data under different loading conditions (different stress ratios) in a single two-parameter fatigue curve is proposed by Adam et al. [47]. In another work by Epaarachchi et al. [48], an empirical model that takes the influence of stress ratio and loading frequency into account is presented and validated against experimental data for different glass fiber-reinforced plastic composites. Although these models seem promising, their empirical nature is a disadvantage as their predictive ability is strongly affected by the selection of a number of parameters that must be estimated or even, in some cases, assumed. When the expected loading is well determined, especially when it comprises constant amplitude patterns, the task is easy and relatively inexpensive. However, when loading patterns are of variable amplitude—and even more so when they are of a stochastic nature, as in most cases of real structures operating in the environment—things become much more complicated. This is due to the theoretically unlimited number of tests that must be conducted to characterize material behavior under all possible loading conditions. The so-called CLDs are used to avoid this inconvenience. CLDs represent mean stress, σ m, vs. stress amplitude, σ a, for several loading conditions, that is, for several different stress ratios. In the previous relationship σ min and σ max denote minimum and maximum cyclic stress levels. To determine a CLD, S-N curves for at least three different stress ratios along with static strengths in tension and compression must be determined experimentally. The S-N curves at R ¼ 0.1 for tension-tension (T-T) loading, at R ¼1 for reversed tension-compression (T-C) loading, and at R ¼ 10 for compression-compression (C-C) loading are used in the majority of published works, for example, Refs. [49–52], to describe all three regions of a CLD. Any other curve that needs to be determined is calculated by linear or another type of interpolation between the known curves, as proposed in for example [52].

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The AI techniques were successfully implemented for representation of the fatigue data of composite materials and structures under constant amplitude and spectrum loading. The application of these techniques to a number of fatigue data and the discussion of the results is presented below.

10.3.1 Experimental data description A considerable amount of fatigue data for composite materials (especially composite laminates) exists in the literature and existing databases contain data covering a significant number of loading cases. Some of these databases are relatively limited and refer to a specific material system, primarily aimed at the verification of new theoretical models, for example, Refs. [53–55]. Other databases are more extensive however and were developed for the characterization of entire categories of materials primarily used in specific applications, such as databases DOE/MSU [56] and Optidat [57] for materials used in the wind turbine rotor blade industry. A new fatigue database has recently been released by Virginia Tech [51] containing experimental data from axial loading on pseudo-quasi-isotropic glass/vinyl ester specimens fabricated using the vacuum-assisted resin transfer molding (VARTM) technique. Selected material data have been used here to demonstrate the applicability of computational techniques and their potential for the modeling of fatigue behavior.

10.3.1.1 Material #1, GFRP multidirectional laminate with stacking sequence [0/(
45)2/0]T A database created by one of the authors [55] is used in this work. It refers to specimens cut at on-axis and several off-axis angles from a multidirectional GFRP composite laminate. The stacking sequence of the on-axis specimens was [0/(
45)2/0]T which is a typical material used in the wind turbine rotor blade industry. Seven different material configurations were tested as specimens were cut at seven different angles from the multidirectional laminate, namely: 0, 15, 30, 45, 60, 75, and 90 degree. Constant amplitude fatigue tests were performed at a frequency of 10 Hz using an MTS 810 servo hydraulic test rig of 250-kN capacity. In total 257 valid constant amplitude fatigue data points were collected. Tests were conducted under four different stress ratios (R ¼ σ min/σ max), two corresponding to tension-tension loading, (0.1 and 0.5), one corresponding to tension-compression loading (1) and one to compression-compression loading (10). At least three specimens were tested at each of the four or five stress levels preassigned for the determination of each S-N curve. This resulted in the existence of at least 12 and up to 18 specimens for each S-N curve data set, in a range between 1000 and 5.3 million cycles. The experimental data produced were used for the determination of 17 S-N curves corresponding to various loading patterns and material configurations. Typical input parameters for the application of computational methods to this data set are: cyclic stress level (amplitude, σ a or maximum stress, σ max), on- or off-axis angle, θ, and stress ratio, R. The number of cycles to failure, N, can be considered as the single output.

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10.3.1.2 Material #2, GFRP multidirectional laminate with the stacking sequence [90/0/
45/0]S The second example is based on experimental fatigue data retrieved from the DOE/ MSU database [56]. The material is a multidirectional laminate consisting of eight layers, six of the stitched unidirectional material D155 and two of the stitched,
45 degree, DB120. CoRezyn 63-AX-051 polyester, with the codename DD16 in the DOE/MSU database, was used as matrix material. For constant amplitude fatigue the material was tested under 12 stress ratios for a comprehensive representation of a constant life diagram. Reading counter-clockwise on the constant life diagram the following stress ratios can be identified: 0.9, 0.8, 0.7, 0.5, 0.1, 0.5, 1, 2, 10, 2, 1.43, and 1.1. The data set consists of 360 observations (valid fatigue test results) that were used for the derivation of the 12 S-N curves. The absolute maximum stress level during testing ranged between 85 and 500 MPa, while the corresponding recorded cycles up to failure were between 37 and 30.4 million. The stress ratio, R, and maximum cyclic stress, σ max, were considered as the input parameters. The number of cycles to failure, N, was also assigned as the only output parameter.

10.3.1.3 Material #3, multidirectional glass/epoxy laminate with a stacking sequence [(
45/0)4/
45]T The third material used is a multidirectional glass epoxy laminate consisting of nine plies, four with fibers in the 0 degree direction, and five stitched layers with fibers in both the 45 and 45 degree directions [57]. The stacking sequence of the laminate is [(
45/0)4/
45]T. Laminated plates were fabricated using the vacuum infusion method. Nonstandard specimen geometry was used in order to also provide uniform specimens for tensile and compressive testing. The length of the rectangular specimens was 150 mm, the length of the tabs glued to both ends was 55 mm and consequently the free length of the specimens was 40 mm to avoid buckling during compressive loading. The thickness of the specimens of this type was 6.57 mm. A total of 147 valid fatigue data points were found in the Optidat database [57] for this material tested under three different constant amplitude conditions; 47 specimens for tension-tension loading at R ¼ 0.1, 64 specimens for tension-compression loading at R¼1, and 36 specimens at compression-compression loading R ¼ 10. In this data set, the maximum stress level obtained was between 350 MPa for compressive loading and 400 MPa for R¼ 0.1. Recorded lifetime was between 150 for low-cycle fatigue, and 10.2 million cycles for longer lifetimes. Previous results [16, 17, 23] have shown that 50%–60% of available experimental data suffices for the derivation of reliable S-N curves and CLDs. As presented in Figs. 10.8 and 10.9, an arbitrarily selected proportion of 50% of the data points proved sufficient for the derivation of reliable S-N curves by implementing an ANN model (Fig. 10.8) or an ANFIS model (Fig. 10.9). Moreover, as presented in Ref. [16], after a parametric study for the examined material and testing conditions, an increase in the data used in the training set does not seem to significantly influence the accuracy of the ANN model. The same applies for the application of the ANFIS modeling,

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367

Predicted loading cycles (Log(N))

Train set Test set

6

5

4 R2 on train set: 0.87 R2 on test set: 0.87

3

3

4

5 6 Actual loading cycles (Log(N))

Fig. 10.8 Modeling ability of ANN for 50% training set, 50% test set.

Predicted loading cycles (Log(N))

Train set Test set 6

5

4 R2 on train set: 0.91 R2 on test set: 0.79

3

3

4

5 6 Actual loading cycles (Log(N))

Fig. 10.9 Modeling accuracy of ANFIS for 60% training set, 40% test set.

see Fig. 10.10, [23]. Preprocessing of the fatigue data is frequently required however. Generally, a simple normalization in order to obtain a data set in the range [0, 1] is enough, although more advanced preprocessing techniques like data clustering have proved applicable in order to produce a concise representation of system behavior [23]. Data clustering can be very beneficial for example during the derivation of an ANFIS model, since it is a means of minimizing the linguistic rules needed to define the model [23].

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25 Experimental data 50% training 60% training 90% training

sa (MPa)

20

15 50%-90% training q =75 degs R =0.1

10

3

4

5 Log(N)

6

7

Fig. 10.10 Modeling accuracy of ANFIS according to data points used to train network.

10.3.2 Application of the methods The computational techniques introduced above were implemented on the three data sets. All data were handled as follows: l

The available data sets were divided into training, validation, and test sets using a randomization technique; approximately 50% of the data were used (randomly selected) for training, while the rest were used for model validation and test of the modeling performance. The method of using training and validation sets is known as the cross-validation method and works as follows: the training set is used for the generation of a population of models of the system. These models are evaluated using the validation set and the one exhibiting the best performance is selected. a. The training set contained the data that were used for the training phase of all methods. Where applicable, maximum stress values, on/off-axis angle and stress ratio were used as input parameters, while the corresponding cycles up to failure were considered as the desired output. The process can be characterized as a nonlinear stochastic regression analysis. During the training phase each computational tool established several relationships between the input and output variables. Using an iterative process the parameters of the established relationships were adjusted in order to minimize the difference (error) between desired and actual outputs. b. The validation set contained data for the evaluation of the evolved models, after the training phase and should therefore contain patterns that were not used in the training set. It can thus appraise the generalization ability of the produced model, which is its ability to perform well with unknown data. It is also imperative that the validation set should contain patterns comprising a good representative set of samples from the training domain (and generally the domain in which the model will operate). c. The test, or applied, set was subsequently constructed, containing input data for which the output ought to be calculated by the selected model. Although these data were “new” and had not been used for training or validation of the model, they should be in the range of the

Computational intelligence methods for the fatigue life modeling of composite materials

l

369

training set (the operational range of the model), since the ability of the described computational methods for extrapolation outside the training set has not yet been validated. For the case studied, the test sets were prepared in such a way that they covered the entire range from minimum to maximum cyclic stress levels. With the sets of input and output data, an entire S-N curve can be plotted without any fitting being required. Normally, there is no need to specify the output parameter since output values would be predicted by the models. The same model can be stored and potentially used to predict other output values for a new applied input data set.

10.3.2.1 Artificial neural networks The neural network used to model the fatigue life of the first material system was a multilayer feedforward network with four inputs (θ, R, σ max, and σ a), one hidden layer with a variable number of computation neurons that use a sigmoid activation function and a single output (N) with one computation neuron using a linear activation function. This neural network was trained by the error back propagation algorithm with the use of momentum (also known as generalized delta rule) [32]. The target was to model the number of loading cycles until failure (N) as a function of the orientation angle of the fibers (θ), stress ratio (R), maximum stress applied to the specimen (σ max), and amplitude of the stress (σ a). Only three of the finally used parameters are independent, since if maximum stress and stress ratio are known, stress amplitude is also known. The reason for using a dependent variable was to check whether or not a greater number of input parameters, even if they were interconnected, could affect the ANN model. For material #1 it was shown during calculations [17] that the ANN with four inputs (θ, R, σ a, σ max) performs better than the one with three inputs (θ, R, σ max) with no significant increase in the complexity of the network. The general structure of the neural network used for material system #1 is shown in Fig. 10.11. The data set used consists of 257 observations (fatigue life experiments). In order to make these experimental data suitable for processing by the neural network, the following preprocessing was carried out: l

l

The values for θ, σ max, and σ a were normalized in [0, 1], while R values were normalized over the maximum of them, 10. The logarithmic values of the number of cycles to failure (N) were calculated and used. This was done because N has values of between 1296 and 5,269,524 loading cycles. Training a neural network with such a wide range of values will result in extremely poor modeling performance.

From the resulting data set, a training set was constructed by randomly selecting a proportion of between 30% and 50% of the available data. The remaining 50%– 70% was used as the testing set for validating the predictions. The training and test sets were selected using a simple uniform random sampling. A small proportion (10%), selected using uniform random sampling, of the training set was used as a validation set, in order to use the cross-validation method for assessing the performance of the neural networks. The results of the application of the neural network modeling technique to the experimental data concerning material #1 are given

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Fatigue Life Prediction of Composites and Composite Structures

Hidden layer

Input layer

q

Output layer

R N

s max

. . .

sa

Fig. 10.11 General structure of the ANN model used for material #1.

in Figs. 10.12 and 10.13. It is shown that even for the extreme case when only 30% of the available fatigue data is used for model training, modeling efficiency may be very high, see Fig. 10.12. Moreover, as presented in Fig. 10.13, the ability of ANN to derive CLDs of similar accuracy to those derived by using the conventional linear interpolation between known experimental data is instantly recognizable.

Actual data Predicted data

0o

sa (MPa)

90

60 15 o

30 90 o

0

3

4

5 Log(N )

6

7

Fig. 10.12 Experimental data vs. ANN modeling. Tension-tension fatigue, R ¼0.1. Coupons cut at 0, 15, 45, 75, and 90 degree; 30% used for training set, 70% for test set.

Computational intelligence methods for the fatigue life modeling of composite materials

80

R=–1 R=10

sa (MPa)

60

R=0.1

371

N=10 4 N=10 5 N=10 6 N=10 7 ANN, N=10 4 ANN, N=10 5 ANN, N=10 6 ANN, N=10 7

40

R=0.5 20

0

–110

–60

–10 40 sm (MPa)

90

140

Fig. 10.13 CLD for 45 degree off-axis coupons of material #1. Artificial neural network predictions vs. experimental data in range 104–107 loading cycles.

10.3.2.2 Adaptive neuro-fuzzy inference system The ANFIS architecture used in the present study is based on the first-order TakagiSugeno model and is schematically presented in Fig. 10.2. It is assumed that the number of cycles to failure (N) under fatigue loading is a function of the fiber orientation angle (θ), stress ratio (R), maximum stress (σ max), and stress amplitude (σ a). Therefore θ, R, σ a, and σ max were the input parameters as in the case of ANN, while the number of cycles corresponding to each combination of the four input parameters was the unique output of the ANFIS model. Data clustering was performed in order to improve the ANFIS modeling performance. The advantage of using data clustering in the proposed solution is the improvement of modeling accuracy by the development of a much simpler neuro-fuzzy model. ANFIS produced 15–17 fuzzy rules according to the size of the training data set when the fatigue data were clustered, and up to 1372 rules when no data clustering was performed. “If-then” rules consist of the premise or antecedent (“if”) part and the consequent (“then”) part and work in the following way: if the premise is true, the consequent is also true. In fuzzy logic this simple representation is slightly different. A fuzzy rule indicates that if the premise is true to some degree of membership then the consequent is also true to the same degree of membership. In the examined case the antecedent of the fuzzy rules (the “if” part) contains more than one condition for the parameters, θ, R, σ a, and σ max respectively, which should be treated simultaneously. The consequent part of the rule contains only the number of cycles to failure. The implementation of the ANFIS modeling technique derived accurate fatigue models for all the examined cases, as presented indicatively in Figs. 10.14 and 10.15. As depicted in Fig. 10.14, when 60% or more of the available experimental

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Fatigue Life Prediction of Composites and Composite Structures

Experimental data Predicted data

q =0 degs R =0.1

sa (MPa)

90

60 q =30 degs R =10

30

q =60 degs R =10

0

3

4

5 Log(N )

6

7

Fig. 10.14 ANFIS modeling accuracy for 50% training.

Experimental data Predicted data

15

sa (MPa)

30% training q =90 degs R =0.1

10

5

3

4

5 Log(N)

6

7

Fig. 10.15 ANFIS modeling accuracy for 30% training.

data were used for model training, all ANFIS predictions were very well corroborated by the experimental data. The modeling accuracy was adequate even when only small proportions of the available experimental data were used for model training, for example, 30%, see Fig. 10.15. However, this good ANFIS performance was not the rule when the training set was less than 50%.

Computational intelligence methods for the fatigue life modeling of composite materials

373

10.3.2.3 Genetic programming A GP model was also developed for the representation of the fatigue life data of the treated material. The training efficiency of the GP tool was very good when 50% or more of the available data were used to form the training set as well. As shown in Fig. 10.16, where the target output is compared to the best program output after the training process, the coefficient of multiple determination (R2) was 0.98 for the selected case of the reversed loading of material #3. For all the treated cases the training accuracy was very high, with R2 values above 0.95. It should be noted that there is no relationship between the two symbols R2 and R used in this text. The results of the application of the GP tool to the treated material are presented in Fig. 10.17, where selected S-N curves are plotted together with the experimental data for each material system on the S-N plane. For this demonstration, only one S-N curve from each data set was selected, namely that corresponding to specimens cut at 15 degree off-axis and tested under tension-tension, R ¼ 0.1 loading, from the first data set, the S-N curve for compression-compression loading under R ¼ 10 for material #2, and the S-N curve for the reversed loading of material #3. In order to plot all curves on the same graph, stress data were normalized over their maxima; 100 MPa for material #1, 408.6 MPa for material #2 and 450.4 MPa for material #3. Fig. 10.17 shows that the modeling accuracy of GP is excellent. In all the studied cases the produced curves follow the trend of the experimental data perfectly. It should be mentioned that the S-N curve predicted by the GP tool is not of a predetermined type such as power curve, polynomial, semi-logarithmic, etc. The resulting curve consists of data pairs (input and output) that can be simply plotted on the S-N plane. Although such use of the model suffices for the subsequent analysis, it should be noted that output data, even if not necessary, can easily be fitted by a second- to fourth-order polynomial equation, as shown in the same figure.

Target program output Selected program output

Log(N )

6

4

2

Data points: 32 Runs: 20 Evolved programs: 7.947.363 Elapsed time (min): 7.51

R2 : 0.976

0

0

5 10 15 20 25 30 ID number of data point used for training

Fig. 10.16 Training of GP model with experimental data for material #3.

35

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Fatigue Life Prediction of Composites and Composite Structures

1.2 Exp. data Mat 1 Exp. data Mat 2 Exp. data Mat 3

s max =5.167 - 2.121*X + 0.347*X 2 - 0.020*X 3

Normalised smax

0.9

0.6

smax =1.114 - 0.143*X + 0.007*X 2

0.3 smax =0.603 + 0.061*X - 0.032*X 2 + 0.002219*X 3

In e quations, X =Log(N )

0.0

2

4

6

8

Log(N)

Fig. 10.17 Fitted lines to predicted S-N curves together with experimental data. Predictions indicated by open symbols.

10.3.2.4 Gene expression programming Here, the task is to calculate a mathematical expression (mathematical model) of the number of cycles to failure N as a function of R and σ max. The used data set was that of material #1 (see Ref. [42] for details) and it was split to a training set consisting of 70% of the initial data set and a test set consisting of the remaining 30%. The training data set is used in order to find the best model using the GEP algorithm and the test set was used in order to appraise the generalization ability of the produced model; i.e., to evaluate the performance of the model on new data. The default parameters used to the experiments are depicted in Table 10.1. Tables 10.2–10.4 depict the results obtained using various parameter values, for several parameters like mutation probability, population size, and tournament size. The results are the mean train and test error values after executing each experiment for 100 times. The best model was found using mutation probability¼ 0.5, population size¼ 1500, and tournament size¼ 20 and it is depicted in Table 10.5. Table 10.1 Default values of the algorithm parameters Parameter

Value

Number of generations Function set

500.000 pffi {+, , *, /, ^, , abs, cos, sin, ln, exp, tan, min, max} [3, 3] 70 Two points recombination

Constants range Head size Type of recombination

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375

Table 10.2 Results for various mutation probability values Mutation probability

Mean train error

Mean test error

0.3 0.5 0.7 0.9

0.881507813989292 0.601041062152667 0.697229325457700 0.796763184654628

0.847912316065133 0.554192046787787 0.658488064214213 0.848293155315223

Table 10.3 Results for various population sizes Population size

Mean train error

Mean test error

500 1000 1500 2000

0.965429324040076 0.601041062152667 0.546792013050052 0.584963715842745

0.915014848216932 0.554192046787787 0.542416714341197 0.596317456244514

Table 10.4 Results for various tournament sizes Tournament size

Mean train error

Mean test error

8 12 16 20 24

0.607422571397248 0.616459465100258 0.726477902129997 0.383207342391752 0.548631878456645

0.595545708917424 0.613161274700709 0.701209587364539 0.483579386219554 0.521452147821475

Table 10.5 Best model found by GEP log N ¼ (((max(((normsmax)*(abs(abs((abs(abs(normR)))  (abs(min((exp((ln(abs(max ((0.7459479346411833),(0.7482939400422683)))))^((exp((normR)^1/2))*(exp(((exp (cos(((normR)^(2.526001698730898)) + ((normsmax)  (ln(exp((min ((2.8275740069528528),(0.06370620951323458))max((2.8275740069528528), (0.06370620951323458))) + (min((2.8053943509604755),((ln(normsmax))^3))max ((2.8053943509604755),((ln(normsmax))^3))))))))))^2) + (normR)))))), (2.8275740069528528))max((exp((ln(abs(max ((0.7459479346411833),(0.7482939400422683)))))^((exp((normR)^1/2))*(exp(((exp (cos(((normR)^(2.526001698730898)) + ((normsmax)  (ln(exp((min ((2.8275740069528528),(0.06370620951323458))max((2.8275740069528528), (0.06370620951323458))) + (min((2.8053943509604755),((ln(normsmax))^3))max ((2.8053943509604755),((ln(normsmax))^3))))))))))^2) + (normR)))))), (2.8275740069528528)))))))),((sin((cos(0.7482939400422683))* (0.7459479346411833)))^2)))*(abs(exp(normsmax)))) + (2.8053943509604755))^2

376

10.4

Fatigue Life Prediction of Composites and Composite Structures

Comparison to conventional methods of fatigue life modeling

Fatigue models can also be derived using conventional methods, such as linear regression, Sendeckyj’s wear-out model [43], Whitney’s Weibull statistics [44], etc. A detailed comparison of these conventional methods and GP was presented in Ref. [25]. Linear regression and ANFIS results were compared briefly in Ref. [23]. The S-N curves predicted using all the available methods are presented in Figs. 10.18–10.20 for comparison. It may be concluded that, although based on different approaches, generally speaking all fatigue models were able to adequately represent the fatigue behavior of the selected experimental data, at least for the central part of the S-N curve, for Log(N) ¼ 3 to Log(N) ¼ 6. Fig. 10.18, in which predictions for the data of material #1 (15 degree off-axis, R ¼ 0.1) are presented, shows that the GP curve “follows” the trend of the experimental data more closely than the other three fatigue models that produce a somewhat straight curve on the Log(N)-S plane. For example, when examining the stress level of 80 MPa, the experimental average number of cycles could be calculated as 77,985 and the corresponding estimated numbers from the GP curve and other methods as 63,095 and 107,152 cycles respectively. Moreover, for the stress level of 71 MPa, the GP tool estimates 380,189 loading cycles and the other methods approximately 562,341 loading cycles, while the experimental average is 421,213 loading cycles. For both examined stress levels, the GP curve underestimates actual loading cycles by a factor of 9.7%–19.1%, while the other

110 GP Whitney Wear out model Linear regression Exp. data

smax (MPa)

90

70 63095 cycle s 107152 cycle s

50

3

4

5 Log(N)

6

Fig. 10.18 Comparison of different methods for S-N data interpretation with GP predictions, material #1.

Computational intelligence methods for the fatigue life modeling of composite materials

GP Whitney Wear out model Linear regression Exp. data

360

smax (MPa)

377

240

120

0

1

2

3

4

5

6

7

Log(N )

Fig. 10.19 Comparison of different methods for S-N data interpretation with GP predictions, material #3.

50 60 o, R =–1, exp. data 60 o, 60% training, ANFIS Wear out model Linear regression

sa (MPa)

45

40

35 4

5

6

7

Log(N)

Fig. 10.20 Comparison of different methods for S-N data interpretation with ANFIS predictions, material #1.

curves overestimate the loading cycles by a factor of more than 33.5%. The weakness of linear regression and Whitney’s method is demonstrated in Fig. 10.19, where the S-N curves for the [(
45/0)4/
45]T laminate, material #3, are presented. When observed in detail, this figure shows that the GP and wear-out models are the most appropriate for interpretation of low-cycle fatigue data, Log(N) < 3, as they are capable of producing multi-slope curves. On the other hand, linear regression analysis and

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the Whitney method are based on a power curve equation for the whole fatigue cycle range. GP predictions seem to compare favorably with those produced by fatigue models. In some of the examined cases GP seems more efficient in simulating the actual trend of the experimental data, without the constraints of a specific equation type. The superiority of the computational methods over the conventional ones lies in their ability to adapt to each different data set by following its actual trend and without any constraints regarding equation types as shown in Fig. 10.20. In this figure, the fatigue model derived using the ANFIS tool is compared to that derived after linear regression analysis of the fatigue data from the first database (Material #1, specimens cut at 60 degree off-axis and tested under reversed loading). In this case linear regression analysis is clearly not the appropriate method for interpretation of the given fatigue data. The model based on the wear-out model, although better than the linear regression, is still restricted by the predetermined power curve equation. On the other hand, the ANFIS model is obviously capable of following the trend of the fatigue data and deriving a reliable S-N curve. The selected AI methods proved (see also Ref. [25]) more efficient than the conventional techniques for modeling the fatigue behavior of a number of composite laminates. Previous results showed that, when enough experimental data exist, computational methods are very efficient. However, these methods should be utilized with caution since they do not offer a magic solution to all modeling problems. A neural network, a GP model or a GEP model that produces exceptional training results is not necessarily well trained. The following considerations should be kept in mind. When very few experimental data are available for model training, the risk of overfitting is high. Overfitting is what occurs when trying to fit more parameters than the available data pairs utilized for the training of the model. It is like trying to estimate the model parameters of a parabola passing through two points. In fact there are an infinite number of parabolic curves that can cross two points. Therefore, the training process gives a perfectly trained—although misleading—model since the predictive ability of an overfitted model is very weak. The model is unable to extrapolate any reliable predictions since it is trained within a narrow range of experimental data. Thus, the predictions provided by the model (compared to the test data set) become poorer as the training set diminishes. An example of overfitting was presented in Ref. [23]. Training the ANFIS model with only 10% of the available experimental data resulted in the calculation of an R2 error of 0.99 during training. However, this apparently perfect model was unable to generalize and therefore the R2 error on the test data were 0.25. The presented AI methods have a common characteristic; they are data-driven methods in the sense that they are not based on any assumptions concerning material behavior or the adoption of mathematical models that must be employed to simulate it. Although this independence with regard to the material is an asset for their application, it can also become a disadvantage, since modeling accuracy depends on the quality of the available experimental data and cannot be improved by any physical considerations.

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10.5

379

Conclusions and future prospects

Novel computational methods such as ANNs, ANFIS, and GP have been shown to be very powerful modeling tools for the nonlinear behavior of composite laminates subjected to constant amplitude loading. They can be used to model the fatigue behavior of several different material systems. Their modeling ability compares favorably with, and is to some extent superior to, other modeling techniques. In their present form, computational methods have been used as stochastic nonlinear regression analysis tools. Their advantages compared to other conventional methods can be summarized as follows: Computational methods are stochastic nonlinear regression tools, and can therefore be used to model the fatigue behavior of any material, provided that sufficient data are available for training. Their stochastic nature is also of the utmost importance as different output is produced after every run of the model for the same input data. Following this procedure, new data sets could be created based on a specific input in order to augment limited databases. Modeling is not based on any assumptions, for example that the data follow a specific statistical distribution, or that the S-N curve is a power curve equation. Moreover, the process does not take the mechanics of each material system into account. Strictly speaking, the presented computational methods are material-independent data-driven methods that correlate input with output values in order to establish a model describing the relationship between them. In this context the proposed methods can easily be applied to any material, provided that an adequate amount of data exists. The S-N curves derived by the introduced data-driven modeling techniques do not follow any specific mathematical form. They simply follow the trend of the available data, in each case giving the best estimate of their behavior. However, as shown in previous works, output data can easily be fitted by simple second to fourth order polynomial equations. Nevertheless, although the behavior of limited data sets can easily be simulated by artificial intelligence methods, such cases should be treated with caution since there is always the risk of overfitting. Artificial intelligence methods have been successfully used for modeling the fatigue behavior of various metallic and composite material systems. However, there is to date no evidence that these methods include any predictive ability. Up until now, all these methods have clearly been proved to be very accurate tools for interpolating material behavior within a known database, but they cannot extrapolate any predictions outside the database, either for different loading conditions or for different materials. Future research should be focused in this direction. It is also anticipated that the presented computational methods can be implemented in solving more complex modeling and predicting problems. Some potential applications of the computational methods are: l

Modeling of the behavior of composite materials under block and variable amplitude loading conditions.

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Prediction of the off-axis fatigue behavior of composite laminates. Computational methods can compete with existing failure criteria insofar as they could potentially simulate the offaxis behavior of the examined material using the minimum number of experimental data. Modeling of the stiffness degradation of composite materials. Stiffness degradation during fatigue frequently exhibits a sigmoid trend. For the simulation of stiffness degradation during fatigue computational methods can compete with existing models, which often fail to accurately interpret experimental data. Initial results indicate that computational methods can be used for the derivation of models to simulate the thermomechanical behavior of composite materials. To date, there is no commonly accepted phenomenological (or other type of ) model for this task.

References [1] H.A. Deveci, H.S. Artem, Optimum design of fatigue-resistant composite laminates using hybrid algorithm, Compos. Struct. 168 (2017) 178–188. [2] J.F. Barbosa, R. Carlos Silveira Freire Junior, J.A.F.O. Correia, A.M.P. De jesus, R.A.B. Calcada, Analysis of the fatigue life estimators of the materials using small samples, J. Strain Anal. Eng. Des. 53 (8) (2018) 699–710. [3] S. Nagendra, D. Jestin, Z. G€urdal, R.T. Haftka, L.T. Watson, Improved genetic algorithm for the design of stiffened composite panels, Comput. Struct. 58 (3) (1996) 543–555. [4] J.-S. Kim, Development of a user-friendly expert system for composite laminate design, Compos. Struct. 79 (1) (2007) 76–83. [5] V. Kradinov, E. Madenci, D.R. Ambur, Application of genetic algorithm for optimum design of bolted composite lap joints, Compos. Struct. 77 (2) (2007) 148–159. [6] T.P. Philippidis, V.N. Nikolaidis, A.A. Anastassopoulos, Damage characterization of carbon/carbon laminates using neural network techniques on AE signals, NDT&E Int. 31 (5) (1998) 329–340. [7] C.S. Lee, W. Hwang, H.C. Park, K.S. Han, Failure of carbon/epoxy composite tubes under combined axial and torsional loading 1. Experimental results and prediction of biaxial strength by the use of neural networks, Compos. Sci. Technol. 59 (12) (1999) 1779–1788. [8] S. Nasiri, M.R. Khosravani, K. Weinberg, Fracture mechanics and mechanical fault detection by artificial intelligence methods: a review, Eng. Fail. Anal. 81 (2017) 270–293. [9] M.N. Rohman, M.I.P. Hidayat, A. Purniawan, Prediction of composite fatigue life under variable amplitude loading using artificial neural network trained by genetic algorithm, AIP Conf. Proc. 1945 (2018)020019. [10] Q. Ji, P. Zhu, J. Lu, Z. Liu, Study of in-plane fatigue failure and life prediction of weave composites under constant and variable amplitude loading, Polym. Polym. Compos. 24 (8) (2016) 597–608. [11] A.H. Ertas, F.O. Sonmez, Design optimization of fiber-reinforced laminates for maximum fatigue life, J. Compos. Mater. 48 (20) (2014) 2493–2503. [12] J.A. Lee, D.P. Almond, B. Harris, The use of neural networks for the prediction of fatigue lives of composite materials, Compos. A Appl. Sci. 30 (10) (1999) 1159–1169. [13] Y. Al-Assaf, H. El Kadi, Fatigue life prediction of unidirectional glass fiber/epoxy composite laminae using neural networks, Compos. Struct. 53 (1) (2001) 65–71. [14] H. El Kadi, Y. Al-Assaf, Prediction of the fatigue life of unidirectional glass fiber/epoxy composite laminae using different neural network paradigms, Compos. Struct. 55 (2) (2002) 239–246.

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[15] Y. Al-Assaf, H. El Kadi, Fatigue life prediction of composite materials using polynomial classifiers and recurrent neural networks, Compos. Struct. 77 (4) (2007) 561–569. [16] A.P. Vassilopoulos, E.F. Georgopoulos, V. Dionyssopoulos, Modeling fatigue life of multidirectional GFRP laminates under constant amplitude loading with artificial neural networks, Adv. Compos. Lett. 15 (2) (2006) 43–51. [17] A.P. Vassilopoulos, E.F. Georgopoulos, V. Dionyssopoulos, Artificial neural networks in spectrum fatigue life prediction of composite materials, Int. J. Fatigue 29 (1) (2007) 20–29. [18] R.C. Silverio Freire Jr., A. Duarte Doria Neto, E.M. Freiri de Aquino, Use of modular networks in the building of constant life diagrams, Int. J. Fatigue 29 (3) (2007) 389–396. [19] R.C. Silverio Freire Jr., A. Duarte Doria Neto, E.M. Freiri de Aquino, Building of constant life diagrams of fatigue using artificial neural networks, Int. J. Fatigue 27 (7) (2005) 746–751. [20] A. Bezazi, S.G. Pierce, K. Worden, H. El Hadi, Fatigue life prediction of sandwich composite materials under flexural tests using a Bayesian trained artificial neural network, Int. J. Fatigue 29 (4) (2007) 738–747. [21] J. Jia, J.G. Davalos, An artificial neural network for the fatigue study of bonded FRP-wood interfaces, Compos. Struct. 74 (1) (2006) 106–114. [22] M.A. Jarrah, Y. Al-Assaf, H. El Kadi, Neuro-Fuzzy modeling of fatigue life prediction of unidirectional glass fiber/epoxy composite laminates, J. Compos. Mater. 36 (6) (2002) 685–699. [23] A.P. Vassilopoulos, R. Bedi, Adaptive neuro-fuzzy inference system in modeling fatigue life of multidirectional composite laminates, Comput. Mater. Sci. 43 (4) (2008) 1086–1093. [24] A.P. Vassilopoulos, E.F. Georgopoulos, T. Keller, Genetic programming in modelling of fatigue life of composite materials, in: 13th International Conference on Experimental Mechanics-ICEM13 “Experimental Analysis of Nano and Engineering Materials and Structures” Alexandroupolis, Greece, July 1–6, 2007. [25] A.P. Vassilopoulos, E.F. Georgopoulos, T. Keller, Comparison of genetic programming with conventional methods for fatigue life modelling of FRP composite materials, Int. J. Fatigue 30 (9) (2008) 1634–1645. [26] W. Sha, Comment on the issues of statistical modelling with particular reference to the use of artificial neural networks, Appl. Catal. A Gen. 324 (1–2) (2007) 87–89. [27] W. Sha, Comment on “Modeling of tribological properties of alumina fiber reinforced zinc–aluminum composites using artificial neural network” Mater. Sci. Eng. A 327 (1-2) (2003) 334–335. [28] A. Abedian, M.H. Ghiasi, B. Dehghan-Manshadi, Effect of a linear-exponential penalty function on the GA’s efficiency in optimization of a laminated composite panel, Int. J. Comput. Intell. 2 (1) (2005) 5–11. [29] P. Nanakorn, K. Mecsomklin, An adaptive penalty function in genetic algorithms for structural design optimization, Comput. Struct. 79 (29–30) (2001) 2527–2539. [30] R. Sarfaraz, A.P. Vassilopoulos, T. Keller, Modeling the constant amplitude fatigue behavior of adhesively-bonded pultruded GFRP joints, J. Adhes. Sci. Technol. 27 (8) (2013) 855–878. [31] P.W. Bach, Fatigue Properties of Glass- and Glass/Carbon-Polyester Composites for Wind Turbines. Energy Research Centre of the Netherlands report: ECN-C-92-072, Petten, the Netherlands, 1992. [32] S. Haykin, Neural Networks: A Comprehensive Foundation, second ed., Prentice Hall international, 1999.

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[33] D.E. Rumelhart, J.L. McClelland (Eds.), Parallel Distributed Processing: Explorations in the Microstructure of Cognition, MIT Press, Cambridge, MA, 1986. [34] T. Takagi, M. Sugeno, Fuzzy identification of systems and its application to modelling and control, IEEE Trans. Syst. Man Cybernet. 15 (1985) 116–132. [35] J.R. Jang, ANFIS: adaptive network based fuzzy inference systems, IEEE Trans. Syst. Man Cybernet. 23 (3) (1993) 665–685. [36] J. Zhang, A.J. Morris, Fuzzy neural networks for nonlinear systems modeling, IEE Proc. Contr. Theory Appl. 142 (6) (1995) 551–561. [37] J.R. Koza, Genetic Programming: On the Programming of Computers by Means of Natural Selection, MIT Press, Cambridge, MA, 1992. [38] J.R. Koza, Genetic programming, in: J.G. Williams, A. Kent (Eds.), Encyclopaedia of Computer Science and Technology, vol. 39, Marcel-Dekker, New York, NY, 1998, pp. 29–43. Supplement 24. [39] C. Ferreira, Gene expression programming: a new adaptive algorithm for solving problems, Complex Syst. 13 (2) (2001) 87–129. [40] C. Ferreira, Gene Expression Programming: Mathematical Modeling by an Artificial Intelligence, second ed., Springer, 2006. [41] Wikipedia, n.d. Gene Expression Programming: https://en.wikipedia.org/wiki/Gene_ expression_programming. [42] M.Α. Antoniou, E.F. Georgopoulos, K.A. Theofilatos, A.P. Vassilopoulos, S.D. Likothanassis, A gene expression programming environment for fatigue modeling of composite material, in: 6th Hellenic Conference on Artificial Intelligence (SETN 2010), May 4-7, 2010, Athens, Greece, vol. 6040, 2010, pp. 297–302, https://doi.org/ 10.1007/978-3-642-12842-4. Published in Springer’s Lecture Notes in Computer Science. [43] G.P. Sendeckyj, Fitting models to composite materials, in: C.C. Chamis (Ed.), Test Methods and Design Allowables for Fibrous Composites, American Society for Testing and Materials, 1981, pp. 245–260. ASTM STP 734. [44] J.M. Whitney, Fatigue characterization of composite materials, in: Fatigue of Fibrous Composite Materials, American Society for Testing and Materials, 1981, pp. 133–151. ASTM STP 723. [45] T.P. Philippidis, A.P. Vassilopoulos, Fatigue design allowables of GRP laminates based on stiffness degradation measurements, Compos. Sci. Technol. 60 (15) (2000) 2819–2828. [46] R. Sarfaraz, A.P. Vassilopoulos, T. Keller, A hybrid S-N formulation for fatigue life modeling of composite materials and structures, Compos. A: Appl. Sci. Manuf. 43 (3) (2012) 445–453. [47] T. Adam, G. Fernando, R.F. Dickson, H. Reiter, B. Harris, Fatigue life prediction for hybrid composites, Fatigue 11 (4) (1989) 233–237. [48] J.A. Epaarachchi, P.D. Clausen, An empirical model for fatigue behaviour prediction of glass fibre reinforced plastic composites for various stress ratios and test frequencies, Compos. A Appl. Sci. 34 (4) (2003) 313–326. [49] C.W. Kensche, (ed.). Fatigue of Materials and Components for Wind Turbine Rotor Blades, EUR 16684 EN, Directorate-Genaral XII, Science, Research and Development, 1996. [50] de Smet BJ, Bach PW. Database Fact: Fatigue of Composites for Wind Turbines. ECN-C94-045, 1994. [51] N.L. Post Reliability Based Design Methodology Incorporating Residual Strength Prediction of Structural Fiber Reinforced Polymer Composites Under Stochastic Variable Amplitude Fatigue Loading. PhD THESIS, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 2008.

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[52] N. Gathercole, H. Reiter, T. Adam, B. Harris, Life prediction for fatigue of T800/5245 carbon-fibre composites: I. Constant amplitude loading, Int. J. Fatigue 16 (8) (1994) 523–532. [53] Z. Hashin, A. Rotem, A fatigue criterion for fiber reinforced materials, J. Compos. Mater. 7 (4) (1973) 448–464. [54] H. El Kadi, F. Ellyin, Effect of stress ratio on the fatigue of unidirectional glass fibre/epoxy composite laminae, Composites 25 (10) (1994) 917–924. [55] T.P. Philippidis, A.P. Vassilopoulos, Complex stress state effect on fatigue life of GRP laminates. Part I, experimental, Int. J. Fatigue 24 (8) (2002) 813–823. [56] J.F. Mandell, D.D. Samborsky. DOE/MSU Composite Material Fatigue Database: Test Methods Material and Analysis. Sandia National Laboratories/Montana State University, SAND97-3002, (online via https://energy.sandia.gov/energy/renewable-energy/waterpower/technology-development/advanced-materials/mhk-materials-database/), 2016. [57] R.P.L. Nijssen. OptiDAT—fatigue of wind turbine materials database. https://wmc.eu/ optimatblades.php, 2019.

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Part Two Fatigue life prediction and monitoring

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Fatigue life prediction under realistic loading conditions

11

Anastasios P. Vassilopoulosa, Rogier P.L. Nijssenb a Ecole Polytechnique F
ed
erale de Lausanne (EPFL), Composite Construction Laboratory (CCLab), Lausanne, Switzerland, bKnowledge Centre Wind Turbine Materials and Constructions, Wieringerwerf, The Netherlands

11.1

Introduction

The majority of engineering structures comprise parts that are subjected to cyclic loading patterns. In fact, most structural failures occur due to mechanisms driven by fatigue loading, whereas purely static failure is rarely observed in open-air applications [1]. Fatigue mainly affects the weak links of structures, usually laminates and joints that are used to transfer loads from one part of the structure to another. The structural integrity of these components is therefore of great importance for the viability of the entire system. The durability of a structure is also affected by environmental loads, mainly extreme temperature and humidity differences during lifetime and a structure generally undergoes a complex thermomechanical cyclic loading profile throughout its lifetime. A thorough knowledge of the fatigue behavior and property degradation under the applied thermomechanical loading profile results in appropriate design. Although an estimation of the potential loading patterns can be achieved, the random behavior of the excitation (e.g., air or traffic loads, impact loads, etc.) cannot be accurately modeled [2]. On the other hand, the vast number of loading cases that could be designated after the aeroelastic calculations for the determination of the loading renders experimental investigation impossible. Thus theoretical approaches have been developed in the past, especially over the last 40 years, for the modeling of the fatigue behavior and prediction of the fatigue life of composite materials and structures under block loading and complex, irregular loading spectra, for example, Refs. [3–8]. The fatigue behavior of a material under variable amplitude (VA) loading can be validated when experimental work under this loading is performed. However, the result will be useful and available only for the examined loading scenario and the tests should be repeated for a new applied VA spectrum. It is therefore obvious that theoretical models should be established based on simple experiments and should have the ability to generalize modeling and predictions for more complicated cases such as block and variable loading situations. One widely used approach is based on the theoretical formulation and use of a damage summation rule to predict life under VA loading without recourse to experimental Fatigue Life Prediction of Composites and Composite Structures. https://doi.org/10.1016/B978-0-08-102575-8.00011-5 © 2020 Elsevier Ltd. All rights reserved.

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observation of the damage accumulation process. The most popular and best-known example of this category, which does not always lead to accurate results however, is the linear Palmgren-Miner rule. Other summation rules were also proposed as alternatives to the Palmgren-Miner rule in order to evaluate and quantify damage accumulation and accurately predict the fatigue lifetime of glass- and carbon-fiber-reinforced plastic composites loaded under block or VA loading patterns, for example, Refs. [9, 10]. Modeling of the material behavior is based on phenomenological models of constant amplitude (CA) fatigue test results. These models rely on the cyclic stress or strain vs life relationship of the examined material, the effect of mean stress on fatigue life and also estimation of fatigue strength under complex loading patterns. Numerous methods have been proposed for the solution of each step of the aforementioned classic methodology, some of which are described in Chapters 2 and 12 of this volume. As an alternative to the classic fatigue life prediction methodology, actual damage measurements during fatigue life can be employed to establish another type of fatigue theory. A damage metric is used as an indicator of damage accumulation. According to the damage metric, these theories can be further classified into subcategories: strength degradation fatigue theories, where the damage metric is the residual strength after a cyclic program, for example, Refs. [11, 12], stiffness degradation fatigue theories, where stiffness is conceived as the fatigue damage metric for example, Refs. [13–16], and finally, actual damage mechanism—fatigue theories based on the modeling of intrinsic defects in the matrix of the composite material that can be considered as matrix cracks. The way in which these cracks propagate in composite materials can be estimated by means of linear fracture mechanics calculations. This chapter aims to provide an overview of fatigue life prediction methodologies for composite materials under VA fatigue loading. The effect of the parameters that dominate the life prediction results is discussed. The classic life prediction methodology and a residual strength-based methodology will be implemented on fatigue data from the literature to demonstrate their applicability. The advantages and disadvantages of each method will be thoroughly debated and future trends in this field are discussed in conclusion.

11.2

Theoretical background

The classic fatigue life prediction methodology that leads to the calculation of the Miner’s damage coefficient and the residual strength fatigue analysis are presented in this section.

11.2.1 Classic fatigue life prediction methodology Classic fatigue life prediction methodology can be considered as an articulated method, since a number of subproblems must be solved sequentially to produce the final result. Four-to-five basic steps can be identified in this method:

Fatigue life prediction under realistic loading conditions l

l

l

l

l

389

Cycle counting Modeling of the experimental CA fatigue behavior Interpretation of fatigue behavior for assessment of the mean stress effect Adoption of the fatigue failure criterion Damage summation

Modeling CA fatigue behavior involves determination of the S-N curves (plot of cyclic stress vs life, typically by grouping data at a single R-value (R ¼ σ min/σ max) (incidentally, depending on the R-value, an S-N curve can be constructed from data obtained at varying mean and amplitude). Interpretation of the fatigue behavior for the assessment of the mean stress effect results in the construction of the constant life diagram (CLD). These two processes can be treated as separate steps, but are related in the sense that the CLD is constructed from the available S-N curves, and new S-N curves could be extracted from this CLD.

11.2.1.1 Cycle counting Cycle counting is used to summarize (often lengthy) irregular load-versus-time histories by providing the number of occurrences of cycles of various sizes (see Fig. 11.1). The definition of a cycle varies with the method of cycle counting. A significant number of cycle-counting techniques have been proposed over the last 30 years, for example, Ref. [17], with rainflow counting being the most widely used. Cycle-counting methods known as one-parameter techniques, for example, levelcrossing counting or peak-counting methods, are not applicable for the fatigue analysis of composite materials since they do not consider the significant mean stress effect on lifetime. A comprehensive description of the available cycle-counting methods for the analysis of spectra applied on composite materials is given in Ref. [18]. As mentioned in Ref. [18] the history of cycle-counting methods goes back to the 1950s and 1960s when only simple range-counting or range-mean-counting methods were used. The

n7

s

n6

Ds7

m6

sm

Ds4

1

n1

n3 n2

n4

n5

Fig. 11.1 Schematic representation of the application of a cycle counting method.

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drawback of these methods is their inability to take into account the stress-strain history to which the material is subjected, and consequently their tendency to miss the largest overall load cycle in a sequence. The rainflow counting and related methods were introduced to address this problem [19–21]. Based on these algorithms stressstrain hysteresis loops are counted rather than stress range and mean values. Rainflow-counting, range-pair, and range-mean methods seem to be the most appropriate for the analysis of composite material fatigue data, giving similar cycle counting results for most practical applications [22]. However, they present a number of deficiencies: rainflow counting cannot be used for cycle-by-cycle analysis and therefore it is difficult to apply this method in combination with a residual strength fatigue theory. On the other hand, range-pair and range-mean counting mask the presence of large and damaging cycles. Based on the previous comments, it is concluded that—according to the application and material used—the appropriate cycle-counting technique should be selected very carefully. In this work rainflow counting, range-mean and range-pair-counting methods are implemented in the fatigue life prediction methodologies. The influence of the selection of the cyclic counting method is assessed via the predictive ability of the entire methodology.

11.2.1.2 Representation of CA fatigue data Traditionally, the S-N data are fitted by a semi-logarithmic or a logarithmic equation. In the first case, it is assumed that the logarithm of the number of cycles is linearly proportional to the stress parameter, while in the second, the logarithm of the number of cycles depends linearly on the logarithm of the stress parameter. The stress parameter S could refer to any cyclic stress definition, σ max (maximum stress), σ a (stress amplitude), or even Δσ (stress range). The mathematical expression of the aforementioned statement is given in the following equations: Log ðN Þ ¼ a + bS

(11.1)

Log ðN Þ ¼ c + dLog ðSÞ

(11.2)

Based on their mathematical expressions, the first model, Eq. (11.1), is called “linlog” or “exponential” and the second one, Eq. (11.2), “log-log” or “power law” as it is equivalent to a power curve of the form: N ¼ KSd

(11.3)

In the previous equations, a, b, c, d, Kare material parameters that must be determined, in pairs, by fitting one of the aforementioned fatigue models to existing experimental data. Since the stress parameter S alone cannot define the loading pattern, another parameter must always be used in the fatigue model—either another stress parameter, or the stress ratio, R ¼ σ min/σ max. Long life testing has demonstrated that between

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391

Eqs. (11.1) and (11.2), Eq. (11.2) is the preferred formulation for extrapolation to lower levels of stress, outside the range of experimental data [23, 24]. The easiest way to estimate the material parameters is via linear regression analysis (with N as the dependent variable), which can be performed even by hand calculations. The resulting S-N curve yields an estimate of the mean time to failure as a function of the corresponding stress parameter. Other types of S-N curve formulations are usually employed to take into account the statistical nature of fatigue data, for example, Refs. [25, 26]. Recently artificial intelligence methods and soft computational techniques have been introduced for the fatigue life modeling of composite materials. Artificial neural networks [27], adapted neuro fuzzy inference systems [28], and genetic programming [29] have proved very powerful tools for modeling the nonlinear behavior of composite laminates subjected to CA loading. They can be used to model the fatigue life of several composite material systems, and can be favorably compared with other modeling techniques. A comprehensive review of the available methods is presented in Chapter 2. A description of the new computational methods is presented in Chapter 5. A model for the stochastic interpretation of the fatigue data is presented in Chapter 9.

11.2.1.3 Assessment of the mean stress effect The effect of the mean stress on the CA fatigue life of the material is assessed using CLDs. CLDs offer a predicting tool for estimation of the fatigue life of the material under loading patterns for which no experimental data exist. The main parameters that define a CLD are the mean cyclic stress, σ m, the cyclic stress amplitude, σ a, and fatigue life. The diagram connects combinations of σ m and σ a which lead to the same number of cycles to failure. Lines of constant R are straight lines emanating from the origin of this σ m  σ a diagram. A typical CLD is presented in Fig. 11.2. R=–1

s(t)

sa t R= –¥ R=+¥ s(t)

R=0 C-dominated

T-dominated

T-C, R1

T-T, 0 σ m  σ mχ 2ψ χ > χ > ¼ 0,σ mχ  σ m  σ T > σa  σa 1  > σ T  σ mχ < f ðσ m , σ a Þ ¼ " #   > χ 2ψ χ > σ  σ > m m > σa  σ χ 1  > ¼ 0,σ C  σ m < σ mχ : a σ C  σ mχ 8 1R > > σ m ¼ 0, σ mχ  σ m  σ T < σa  1+R gð σ m , σ a Þ ¼ > > : σ a  1  R σ m ¼ 0, σ C  σ m < σ mχ 1+R

(12.26)

(12.27)

Note that the solutions (σ m, σ a)¼(σ Rm, σ Ra ) for different numbers of cycles to failure Nf under fatigue loading at a given stress ratio R allow us to obtain the S-N coordinates (σ Rmax ¼ σ Rm + σ Ra , Nf) for the constant amplitude fatigue loading at the given stress ratio R.

12.4.6.4 Particular cases of the anisomorphic CFL diagram Elimination of the exponent ψ χ from the formulas for the anisomorphic CFL diagram yields the inclined Gerber diagram (Eq. 12.16). Replacing the exponent ψ χ in the formulas for the anisomorphic CFL diagram with a constant value of unity, we can reduce it to the inclined Goodman diagram (Eq. 12.5). For a class of composites with the same strength level in tension and compression, the critical stress ratio has a value of 1 (R¼ χ ¼ 1), and thus σ mχ ¼ σ R¼–1 ¼ 0. In this particular case, the formulas for m the anisomorphic CFL diagram can be reduced to the following single formula:   σ a  σ R¼1 jσ m j 2ψR¼1 a  ¼ ,  σT  σm  σT σT σ R¼1 a

(12.28)

Note that the nonlinear CFL diagram predicted by Eq. (12.28) becomes symmetric about the alternating stress axis. This nonlinear CFL diagram may be interpreted as another extension of the symmetric Gerber diagram, though it is no longer parabolic over a range of fatigue life.

12.5

Prediction of constant fatigue life (CFL) diagrams and S-N curves

This section is devoted to an evaluation of the anisomorphic CFL diagram approach. It is tested for capability to predict the full shape of the CFL diagram and the S-N

Fatigue life prediction of composite materials under constant amplitude loading

445

relationships at different stress ratios, not only for the fiber-dominated fatigue behavior [25] but also for the matrix-dominated fatigue behavior [26] of carbon/epoxy laminates.

12.5.1 Application to the fiber-dominated fatigue behavior of composite laminates The effectiveness of the anisomorphic CFL diagram is evaluated for the constant amplitude fatigue behavior of a quasi-isotropic [45/90/ 45/0]2S carbon/epoxy laminate at room temperature. The anisomorphic CFL diagram is constructed according to the procedure described above in Section 12.4.6. First, the tensile and compressive strengths of the laminate are evaluated. From the static tension and compression tests on the laminate, the values σ T ¼ 781.9 MPa and σ C ¼ 532.4 MPa were obtained. Using these static strengths, the value of the critical stress ratio can be calculated χ as χ ¼ σ C/σ T ¼ 0.68. Second, the reference S-N data (σ max versus 2Nf) are derived from fatigue tests at the critical stress ratio R ¼ χ. The reference fatigue data are approximated by means of an analytical function 2Nf ¼ f(ψ χ ). Since fatigue limit was not clearly observed in the reference fatigue behavior of the [45/90/45/0]2S laminate at R ¼ χ, the following reduced form of Eq. (12.25) has been employed for this purpose: 

2Nf ¼ f ψ χ



 a 1 1  ψχ   ¼ Kχ ψ χ n

(12.29)

Through curve fitting, the material constants involved in this function were determined as Kχ ¼ 0.0015, n¼ 8.5, and a¼ 1. It is emphasized that the reference S-N relationship for the critical stress ratio should be described by means of the function defined using ψ χ , Eq. (12.29), for the [45/90/45/0]2S laminate or Eq. (12.25) in general, since the value of ψ χ that varies with the number of cycles to failure is used as the variable exponent in the piecewise-defined functions for the anisomorphic CFL diagram; see Eq. (12.23). This is all that is necessary for drawing the anisomorphic CFL envelopes (σ m, σ a) corresponding to different numbers of cycles to failure. Using the piecewise-defined functions for the anisomorphic CFL diagram, Eq. (12.23), the S-N relationship (2Nf, σ max) can readily be predicted for any constant amplitude fatigue loading. Fig. 12.14 shows comparison between theory and experiment for the [45/90/45/0]2S laminate; the dashed lines indicate the predicted anisomorphic CFL envelopes, and symbols designate the experimental CFL data. A good agreement between the predicted and observed CFL curves can be seen over the range of fatigue life. Figs. 12.15, 12.16, and 12.17 show comparisons between the predicted and observed S-N relationships under tension-tension (T-T), compression-compression (C-C) and tension-compression (T-C) fatigue loading, respectively. The solid lines in these figures indicate the predictions, and the dashed line in Fig. 12.17 indicates the reference S-N curve identified by fitting Eq. (12.29) to the fatigue data for the

446

Fatigue Life Prediction of Composites and Composite Structures

1000

T800H/Epoxy#3631 [+45/90/–45/0] 2s

R=

Experimental 101 cycles 102 cycles 103 cycles

= –0.68

a,

MPa

800

104 cycles

600

Static strength line

105 cycles

R = –1.0 R = 0.1

400

106 cycles

R = 10

200

R=2

0 –1000 –800

R = 0.5

–600 –400

–200

m,

0 200 MPa

400

600

800

1000

Fig. 12.14 Anisomorphic constant fatigue life diagram for a [+45/90/45/0]2S carbon/epoxy laminate [25]. 1000

T800H/Epoxy#3631 [+45/90/–45/0] 2s R = 0.5

max ,

MPa

800

600

R = 0.1

400 Experimental (RT) 200

0 100

R = 0.1 R = 0.5 Predicted 101

102

103

2N f

104

105

106

107

Fig. 12.15 S-N relationships predicted using the anisomorphic constant fatigue life diagram for a [+45/90/45/0]2S carbon/epoxy laminate subjected to tension-tension fatigue loading [25].

critical stress ratio. It is seen that the mean stress dependence of the S-N relationship for the [45/90/45/0]2S laminate is adequately predicted by means of the anisomorphic CFL diagram. It is important to note that all the solid lines in Figs. 12.15, 12.16, and 12.17 indicate predictions, since only the fatigue data for the critical stress ratio were used for construction of the anisomorphic CFL diagram. The anisomorphic CFL diagram approach was also successfully applied to different carbon/epoxy laminates of [0/60/60]2S and [0/90]3S lay-ups. The predicted CFL diagrams for these laminates are presented in Figs. 12.18 and 12.19.

Fatigue life prediction of composite materials under constant amplitude loading

1000

447

T800H/Epoxy#3631 [+45/90/–45/0] 2s

max ,

MPa

800 R=2

600

400 Experimental (RT) R=2

200

R = 10

R = 10 Predicted

0 100

101

102

103

2N f

104

105

106

107

Fig. 12.16 S-N relationships predicted using the anisomorphic constant fatigue life diagram for a [+45/90/45/0]2S carbon/epoxy laminate subjected to compression-compression fatigue loading [25].

1000

T800H/Epoxy#3631 [+45/90/–45/0] 2s

Experimental (RT) R = –1.0 R = = –0.68

800

max ,

MPa

R=

= –0.68

600

400

200

0 100

Predicted ( R = –1) Fitted ( R = = –0.68) 101

102

103

2N f

R = –1.0

104

105

106

107

Fig. 12.17 S-N relationships predicted using the anisomorphic constant fatigue life diagram for a [+45/90/45/0]2S carbon/epoxy laminate subjected to tension-compression fatigue loading [25].

448

Fatigue Life Prediction of Composites and Composite Structures

1000

T800H/Epoxy#3631 [0/60/–60] 2s

Experimental 101 cycles

= –0.53

R=

102 cycles 103 cycles

a,

MPa

800 600

104 cycles

R = –1.0

105 cycles 106 cycles

R = 0.1

Static strength line

400 R = 10 200

R = 0.5

R=2

0 –1000 –800

–600 –400

–200

m,

0 200 MPa

400

600

800

1000

Fig. 12.18 Anisomorphic constant fatigue life diagram for a [0/60/60]2S carbon/epoxy laminate [25].

1600 1400

T800H/Epoxy#2500 [0/90] 3s

R=

Experimental 101 cycles

= –0.44

102 cycles 103 cycles

1000

a,

MPa

1200 Static strength line

800

R = 0.1

R = –1.0

104 cycles 105 cycles 106 cycles

600 R = 10

400 200 0 –1600

R = 0.5

R=2 –1200

–800

–400

m,

0 MPa

400

800

1200

1600

Fig. 12.19 Anisomorphic constant fatigue life diagram for a [0/90]3S carbon/epoxy laminate [25].

12.5.2 Application to the matrix-dominated fatigue behavior of composite laminates The anisomorphic CFL diagram is further tested for the capability in predicting the matrix-dominated fatigue behavior of angle-ply [
θ]3S carbon/epoxy laminates. Fig. 12.20 shows the anisomorphic CFL diagram for the [
30]3S laminate with the critical stress ratio χ ¼0.56; the dashed lines indicate predictions, and symbols designate experimental results. It is seen that the predicted and observed CFL envelopes agree well with each other over the range of fatigue life. The successful application of

Fatigue life prediction of composite materials under constant amplitude loading

600

449

Experimental 101 cycles

Fatigue angle-ply T800H/Epoxy#2500 RT [±30] 3s

102 cycles 103 cycles

= –0.56

104 cycles 105 cycles

400 MPa

R = –1

106 cycles

a,

R = 0.1 200

R = 10 R = 0.5

R=2 0 –600

–400

–200

m,

0 MPa

200

400

600

Fig. 12.20 Anisomorphic constant fatigue life diagram for a [
30]3S carbon/epoxy laminate [26]. 800 700

T800H/Epoxy#2500 angle-ply [±30]3S Experimental RT

max ,

MPa

600 500

R = 0.5

400 300 200 100 0 100

Predicted Experimental R = 0.5 R = 0.1 101

102

R = 0.1

103

2N f

104

105

106

107

Fig. 12.21 S-N relationships predicted using the anisomorphic constant fatigue life diagram for a [
30]3S carbon/epoxy laminate subjected to tension-tension fatigue loading [26].

the anisomorphic CFL diagram demonstrates that consideration of the asymmetry and variable nonlinearity in CFL envelopes is decisive for accurate description of the CFL diagram for the [
30]3S laminate over the whole range of fatigue life. It also reveals that the traditional Goodman diagram cannot accurately be applied to description of the effect of mean stress on the fatigue life of the [
30]3S laminate. Figs. 12.21 and 12.22 show the predicted S-N relationships for the [
30]3S laminate under T-T, C-C,

450

Fatigue Life Prediction of Composites and Composite Structures

350 Predicted

300

R=2

MPa

200

a,

250

150

R = 10

R = –1

100 50 0 100

T300H/Epoxy#2500 angle-ply [±30]3S Experimental RT 101

102

103

2N f

104

105

106

107

Fig. 12.22 S-N relationships predicted using the anisomorphic constant fatigue life diagram for a [
30]3S carbon/epoxy laminate subjected to compression-compression fatigue loading [26].

and T-C fatigue loading. Reasonably good agreements between the predicted and observed S-N relationships have been achieved. The anisomorphic CFL diagram approach was also successfully applied to a different angle-ply laminate of [
45]3S lay-up. It is interesting that the CFL diagrams for the matrix-dominated [
30]3S and [
45]3S laminates are similar in asymmetry and variable nonlinearity to those for the fiber-dominated [45/90/45/0]2S, [0/60/60]2S, and [0/90]3S laminates. Unlike the angle-ply laminates of [
30]3S and [
45]3S lay-ups examined above, the compressive strength of the [
60]3S carbon/epoxy laminate was larger than the tensile strength. The larger strength in compression than in tension suggests that the anisomorphic CFL diagram inclines to the left of the alternating stress axis, which is demonstrated in Fig. 12.23. The dashed lines in Fig. 12.23 indicate the anisomorphic CFL envelopes of the [
60]3S laminate for different numbers of cycles to failure. It is seen that the agreement between the predicted and experimental CFL envelopes for the [
60]3S laminate is poor in the left segment partitioned by the radial line associated with the critical stress ratio χ ¼2. The discrepancy is ascribed to a significant change in mean stress sensitivity in fatigue for a range of stress ratios in the left neighborhood of the critical stress ratio. For composites in which such an appreciable change in the sensitivity to mean stress happens, it is not reasonable to assume that their fatigue performance is characterized by the representative fatigue behavior at a particular stress ratio (i.e., the critical stress ratio). In fact, similar distortion in the CFL diagram can be found in the experimental results for other composites, e.g., Sch€ utz and Gerharz [48] and Phillips [38]. Therefore, the significant change in the mean stress sensitivity in fatigue observed in the [
60]3S laminate, as well as in the composites tested by Sch€ utz and Gerharz [48] and Phillips [38], suggests

Fatigue life prediction of composite materials under constant amplitude loading

200

Experimental 101 cycles 102 cycles 103 cycles

= – 1.98

104 cycles

R = –1 100

50

105 cycles

R = 10

a,

MPa

150

Fatigue angle-ply T800H/Epoxy#2500 RT [±60] 3S

451

106 cycles R = 0.1

R=2

R = 0.5 0 –200

–150

–100

–50

m,

0 MPa

50

100

150

200

Fig. 12.23 Anisomorphic constant fatigue life diagram for a [
60]3S carbon/epoxy laminate [26].

that some extension of the anisomorphic CFL diagram should be made in order to allow for accommodating such an anomalous mean stress sensitivity in fatigue for a class of composites. Incidentally, an idea of mapping a reference CFL curve for a representative number of cycles to failure that reflects the actual shape observed by experiment to the CFL curve for any given number of cycles to failure [49] provides a solution to the problem of accurately describing highly distorted CFL diagrams for a class of composites. While it is interesting, a method based on this idea requires a model by which the S-N relationships for any mean stresses can be predicted. Since focusing on development of a method that allows predicting the S-N relationships for any mean stresses in this study, we confine our attention to modeling the nested CFL envelopes that depend on the number of cycles to failure, and further seek a CFL diagram based solution to the problem. An attempt is made in the next section.

12.6

Extended anisomorphic constant fatigue life (CFL) diagram

This section is devoted to development of an extended anisomorphic CFL diagram approach which is furnished with enhanced capability to more accurately describe the nonlinear shape of the CFL diagram and thus with more general applicability to a variety of composites with different mean stress sensitivities. The validity of the extended anisomorphic CFL diagram is demonstrated by comparing with experimental results. The anisomorphic CFL diagram approach to prediction of fatigue lives of composites assumes that the S-N relationships for any stress ratios can be predicted on the basis of only the fatigue data for the critical stress ratio. This assumption was found

452

Fatigue Life Prediction of Composites and Composite Structures

to be valid for the fiber-dominated quasi-isotropic carbon/epoxy laminates of [45/90/45/0]2S and [0/60/60]2S lay-ups and the cross-ply laminate of [0/90]3S lay-up, and for the matrix-dominated angle-ply carbon/epoxy laminates of [
30]3S and [
45]3S lay-ups. However, it was too optimistic for the angle-ply [
60]3S carbon/epoxy laminate, as observed above. The experimental CFL diagram for the [
60]3S laminate showed a considerable change in mean stress sensitivity in the left neighborhood of the critical stress ratio, and accordingly it was not accurately described using the anisomorphic CFL diagram. This unsatisfactory result reveals that relying on only the S-N data for the critical stress ratio to construct the CFL diagram over a whole range of mean stresses leads to oversimplification, especially for a class of composite laminates that exhibit higher sensitivity to mean stress in a transitional segment between the T-T and C-C dominated segments in the σ m  σ a plane. To cope with this problem, it was attempted to generalize the anisomorphic CFL diagram further without much loss of convenience [26]. If the mean stress sensitivity in fatigue becomes higher in the vicinity of the critical stress ratio, a transitional segment is assumed to appear between the two segments associated with T-T and C-C dominated fatigue failure. The transitional segment plays a role in accommodating a distortion in the CFL diagram due to a change in mean stress sensitivity, and it connects the two neighboring segments with the aid of linear interpolation. The threesegment version of the anisomorphic CFL diagram was called a connected anisomorphic CFL diagram [26]. In addition to the assumptions of the original anisomorphic CFL diagram, the following assumptions were added to formulate the connected anisomorphic CFL diagram: (B1) if an appreciable change in mean stress sensitivity is involved in the CFL diagram, a sub-critical stress ratio χ s is additionally introduced to define the transitional segment in the CFL diagram which is bounded by the critical line σ a/σ m ¼ (1  χ)/(1 +χ) and the sub-critical line σ a/ σ m ¼ (1  χ s)/(1 + χ s), the critical and sub-critical stress ratios dividing the CFL diagram into three segments; (B2) the CFL curves in the right and left segments that are partitioned by the critical and sub-critical lines, respectively, are drawn according to the procedure prescribed in the original formulation; and (B3) in the transitional segment, linear interpolation is assumed. Thus, the points of the same fatigue life located on the critical and sub-critical lines are connected by straight lines. Mathematically, the connected anisomorphic CFL diagram is described by means of a function on the domain [σ C, σ T]. The function is prescribed by different formulas depending on the position of mean stress σ m, which are given as follows: I. Tension-dominated zone:

  kT σ a  σ aχ σ m  σ mχ 2ψ χ χ  ¼ ,σ m  σ m  σ T σ aχ σ T  σ mχ

(12.30)

II. Transitional zone:



σ a  σ aχ σ m  σ mχ χ s , σ  σ m  σ mχ χ ¼ χ σ aχ  σ as σ ms  σ mχ m

(12.31)

Fatigue life prediction of composite materials under constant amplitude loading

453

III. Compression-dominated zone:

  kC σ a  σ χas σ m  σ χms 2ψ χS  ¼ , σ C  σ m < σ χmS χ χ σ as σ C  σ ms

(12.32)

where σ aχ , σ mχ , σ aχ s, and σ mχ s represent the alternating and mean stress components of the χ χs maximum fatigue stresses σ max and σ max which are associated with the fatigue loading at the critical and sub-critical stress ratios, χ(¼ σ C/σ T) and χ s, respectively. The fatigue strength ratios ψ χ and ψ χ s are associated with the critical and sub-critical stress ratios, χ χ and χ s, respectively. The critical fatigue strength ratio ψ χ is expressed as ψ χ ¼ σ max / χs σ T, while the sub-critical fatigue strength ratio ψ χ s is defined as ψ χ s ¼ σ max/σ T if χs χ  χ s  0, and ψ χ s ¼ σ min /σ C if χ s < χ. They are described by means of the monotonic continuous functions of the same form as given by Eq. (12.25) (or Eq. 12.29). Note that the exponents kT and kC, which are constant, are added to the constituent functions; they allow adjusting the transition from a straight line to a parabola, independently for the right and left halves of CFL curves, to obtain a better description of the nonlinear CFL diagram. The connected anisomorphic CFL diagram for the [
60]3S laminate is shown in Fig. 12.24, along with the experimental CFL data. Good agreement between the predicted and observed CFL curves for all the stress ratios over the range of fatigue life can be achieved. Comparisons between the predicted and observed S-N relationships for T-T, C-C, and T-C fatigue loading are shown in Figs. 12.25 and 12.26, respectively. It is seen that the S-N relationships for the [
60]3S laminate at different mean stress levels have been accurately predicted by means of the connected anisomorphic CFL diagram. These results demonstrate that insertion of a transitional zone into the anisomorphic CFL diagram greatly improves the accuracy of prediction of the full 200

a,

MPa

150

Fatigue angle-ply T800H/Epoxy#2500 RT [±60] 3S

Experimental 101 cycles 102 cycles 103 cycles

= – 1.98

104 cycles

R = –3 R = –5

100

105 cycles

R = –1

106 cycles

R = 10 50

R = 0.1 R= 2 R = 0.5

0 –200

–150

–100

–50

m,

0 MPa

50

100

kT = 0.2

150

200

Fig. 12.24 Extended anisomorphic constant fatigue life diagram for a [
60]3S carbon/epoxy laminate [26].

454

Fatigue Life Prediction of Composites and Composite Structures

150

T800H/Epoxy#2500 angle-ply Experimental RT

[±60] 3S

kT = 0.2

R = 0.5

max ,

MPa

100

50

R = 0.1

Predicted R = –1

0 100

101

102

103

2N f

104

105

106

107

Fig. 12.25 S-N relationships predicted using the extended anisomorphic constant fatigue life diagram for a [
60]3S carbon/epoxy laminate subjected to tension-tension fatigue loading [26].

300

T800H/Epoxy#2500 angle-ply [±60] 3S Experimental RT

250

R=2

150

a,

MPa

200

100 50 0 100

R = 10

Predicted Experimental R=2 R = 10 R = –5 101

102

R = –5

103

2N f

104

105

106

107

Fig. 12.26 S-N relationships predicted using the extended anisomorphic constant fatigue life diagram for a [
60]3S carbon/epoxy laminate subjected to compression-compression fatigue loading [26].

Fatigue life prediction of composite materials under constant amplitude loading

455

shape of the nonlinear CFL diagram for a given composite, and allows application to a greater variety of composites with different mean stress sensitivity. The connected anisomorphic CFL diagram requires additional fatigue data for the second critical stress ratio, i.e., the sub-critical stress ratio, which impairs the great simplicity in the original two-segment anisomorphic CFL diagram. However, it still carries significant advantages not only in construction of nonlinear CFL envelopes in an efficient manner and with a small amount of fatigue data, but also in its enhanced capability to describe a local distortion in the CFL diagram due to a significant change in mean stress sensitivity in fatigue of composites.

12.7

Conclusions

Accurate prediction of the constant amplitude fatigue lives of composites at any amplitude levels for any stress ratios is a vital prerequisite to the successful fatigue life analysis of composite structures subjected to complicated service loading. In order to meet the prerequisite, two approaches have been developed so far: (1) the approach using a master S-N relationship; and (2) the approach using a CFL diagram. The CFL diagram approach easily accommodates itself to the mean stress sensitivity observed by experiment, suggesting that the CFL diagram approach is more flexible, and thus more fruitful for most engineers, than the master S-N curve approach, especially when dealing with a non-Goodman type of fatigue behavior of composites. This chapter, therefore, focused on the CFL diagram approach and reviewed the linear and nonlinear CFL diagrams which were developed so far to account for the effect of mean stress on the fatigue lives of composites in a systematic manner. A particular emphasis has been placed on the recent progress in the CFL diagram approach that has been made by taking into account the requirements suggested by Boller [29, 30] for accurate description of the CFL diagrams for fiber-reinforced composites, and on the smooth link to the latest model, called the anisomorphic CFL diagram [24]. The anisomorphic CFL diagram is one of the most general theoretical tools to date for predicting the mean stress sensitivity in fatigue of composites, and it has been formulated by taking into account all the requirements suggested by Boller [29, 30]: (1) the asymmetry in CFL envelopes about the alternating stress axis; (2) the nonlinearity in CFL envelopes; and (3) the gradual change in shape of CFL envelopes with increasing number of cycles to failure. For a given composite, the anisomorphic CFL diagram can be constructed using only a limited amount of experimental data: (i) the static strengths in tension and compression; and (ii) the fatigue data for a particular stress ratio, called the critical stress ratio, which is equal to the ratio of the compressive strength to the tensile strength. The ease of drawing CFL envelopes with a minimal amount of experimental data is an inherent advantage of the method. The validity of the fatigue life prediction method based on the anisomorphic CFL diagram has been evaluated for the fiber-dominated and matrix-dominated fatigue behaviors of carbon/epoxy laminates. For the fiber-dominated fatigue behaviors of the [45/90/45/0]2S, [0/60/60]2S, and [0/90]3S laminates, it was demonstrated that the CFL envelopes and S-N curves predicted using the anisomorphic CFL model agree

456

Fatigue Life Prediction of Composites and Composite Structures

well with the experimental results, regardless of the type of laminate. The anisomorphic CFL diagram was also shown to be valid for the matrix-dominated fatigue behavior of the [
30]3S and [
45]3S laminates. However, it failed to accurately predict the mean stress sensitivity in the fatigue of the [
60]3S laminate. The failure was due to a significant change in mean stress sensitivity in fatigue life of the laminate, and it happened at stress ratios in a narrow range that are smaller than the critical stress ratio. To overcome the above problem, an extension of the anisomorphic CFL diagram has been attempted. The extended anisomorphic CFL diagram, which is called the connected anisomorphic CFL diagram, consists of three segments: the T-T and C-C dominated segments, and a transitional segment in between. It was demonstrated that the extended (connected) anisomorphic CFL diagram can successfully be applied to describing the CFL diagram for the [
60]3S laminate as well, and thus the S-N curves for constant amplitude fatigue loading at any stress ratios can accurately be predicted for all of the fiber-dominated and matrix-dominated carbon/epoxy laminates tested in this chapter. The extended anisomorphic CFL diagram approach is also applicable to the carbon/epoxy laminates examined by Sch€utz and Gerharz [48] and Phillips [38]. The extended method requires additional fatigue data for another reference stress ratio, called the sub-critical stress ratio, to define the transitional mean stress interval bounded by the critical and sub-critical stress ratios. This slightly impairs the efficiency of the original method, but the slight increase in inefficiency is almost cancelled by the enhanced flexibility of the extended method. The extended anisomorphic CFL diagram approach allows describing a distortion of CFL envelopes for a class of composites that is caused by a significant change in mean stress sensitivity in fatigue in a transitional mean stress interval. Once the CFL diagram for a given composite has accurately been identified over a range of fatigue life by means of the proposed method, it allows predicting the S-N curves of the composite for any constant amplitude fatigue loading, and using them in conjunction with a damage accumulation rule for evaluation of the fatigue lives of the composite for any operational load spectra [43].

12.8

Future trends

The spectra of fatigue load sustained by composite structures during service often involve a small number of large-amplitude cycles, and the maximum fatigue stress during the large-amplitude cycles may accidentally reach a high level of fraction of the static strength of the composite material employed. Phillips [38] has reported that the fatigue lives of carbon/epoxy laminates under spectrum loading are sensitive to the high load cycles involved and should not be truncated in spectrum fatigue life analysis. This explains why it is required to predict the fatigue lives of composites in a short range as well under large-amplitude cyclic loading. Therefore, the accuracy of prediction of the CFL envelopes for composites should be taken into account not only for a typical range of fatigue life Nf ¼ 104–107 but also for a short life range Nf ¼ 100–103. Another concern that has not fully been discussed so far is to evaluate the accuracy of prediction using CFL models for longer lives beyond the fatigue life that can be

Fatigue life prediction of composite materials under constant amplitude loading

457

observed by experiment. If the anisomorphic CFL diagram approach is valid for a given composite in the long life range at low stress levels, it allows prediction of the fatigue lives of composites in the long life range on the basis of the long life fatigue data only at the critical stress ratio. No fatigue testing at any other stress ratio in the long life range is required. Such an efficient implementation of fatigue analysis would be of great significance for applications in which a long life fatigue design of components subjected to variable cyclic loading becomes a critical issue. Therefore, it is worth pursuing further the applicability of the method in a range of long fatigue life. On the other hand, the anisomorphic CFL diagram for composites is affected by factors such as temperature, moisture, loading rate, and damage that change the static strengths in tension and compression and the reference S-N relationship for tensioncompression fatigue loading. The increase in test temperature of composites often results in decrease in their tensile and compressive strengths [10, 50, 51]. The degree of reduction in compressive strength due to temperature rise is different from that in tensile strength, and the former is often more significant [51, 52]. This suggests that the absolute value of the critical stress ratio, which is given by the ratio of the compressive strength to the tensile strength, tends to decrease as temperature increases, and thus the anisomorphic CFL diagram shrinks and the peak position in the σ m  σ a plane moves to the right with increasing temperature [52, 53]. Incidentally, the reduction in the in-plane compressive strengths of composites due to impact damage [54] leads to similar changes in their CFL diagrams [44]. The fatigue behavior of composites becomes more complicated with the influence of temperature added in. It has been reported that the fatigue strengths of plain weave carbon/epoxy fabric laminates at 100°C are lower than those at room temperature, not only in the fiber direction but also in off-axis directions [8]. The reduction in fatigue strength in the fiber direction at 100°C that was observed in the study was reflected by the increase in the slope of the S-N relationship. For the fatigue performance of the cross-ply carbon/epoxy laminate in the fiber direction, however, no significant difference was found in the results obtained at room temperature and 100°C [55]. In contrast, the slope of the S-N relationship for unidirectional carbon/epoxy laminates slightly increased with increasing temperature [10]. It is well known that the properties of the constituents of composites have significant influences on their fatigue performance [56, 57]. Thus, the temperature dependence of the fatigue behavior of composites reflects the change in properties of the matrix and fiber/matrix interfaces due to temperature. Khan et al. [58] have demonstrated that the thermal degradation of matrix resins at high temperature changes the temperature dependence of the fatigue resistance of composite laminates and results in even more complicated fatigue behavior. Great care is needed when dealing with the static and fatigue strengths of composites that are exposed to hygrothermal environments [59, 60]. It has been observed for two kinds of carbon/epoxy laminates, [
45/03/
45/0]S and [0/
45/02/
45/0]S, that the tensile strengths increase with increasing moisture content in contrast to a consistent reduction in the compressive strengths [61, 62]. This observation suggests that the absolute value of the critical stress ratio decreases with increasing moisture content and accordingly the anisomorphic CFL diagram inclines rightward, similar to the

458

Fatigue Life Prediction of Composites and Composite Structures

change with increasing temperature. According to Asp [63], on the other hand, moisture content and temperature produce a significant effect on the interlaminar delamination toughness, i.e., the critical strain energy release rate decreases with moisture content in mode II and mixed mode loading, and with temperature in mode II loading, whereas it slightly increases with increasing temperature in mode I loading. These experimental results imply that the growths of damage in composite laminates in hygrothermal environments are differently observed depending on a competition between moisture and temperature, especially under mode II loading condition. Shan and Liao [64] have compared the fatigue behaviors of unidirectional glass fiber reinforced and glass-carbon fiber reinforced epoxy matrix composites in wet and dry environments at 25°C, respectively. They found that while both systems are more sensitive to a wet environment, especially at low stress levels, the hybrid system containing 25% of carbon fibers shows better resistance to fatigue in water than the allglass fiber system in water over the range up to 107 cycles. The former observation is consistent with the reduction in interlaminar toughness due to the uptake of water, and the latter corresponds to the observation of the moderate degradation of fatigue performance in the wet-conditioned cross-ply [0/90]S and angle-ply [
45]S carbon/epoxy laminates [65]. These experimental results suggest that the shape of the CFL curve for a given fatigue life and its deformation with increasing number of cycles to failure in a wet environment may differ from those in a dry environment. A higher frequency of cyclic loading has been reported to have a more significant degrading effect on the fatigue performance of carbon fiber reinforced composites. Curtis et al. [66] examined the fatigue behaviors of quasi-isotropic [45/0/45/90]2S and angle-ply [
45]4S APC-2/AS4 laminates at different loading frequencies (0.5 Hz, 5 Hz), and demonstrated that the fatigue strength at 5 Hz is lower than that at 0.5 Hz, regardless of the stacking sequence of laminates. A similar reduction in fatigue performance was observed for a plain weave carbon/epoxy fabric laminate [8]. It is considered that the reduction in fatigue strength of composites with the increase in loading frequency is caused by the change in the properties of the matrix and the matrix-fiber interface due to the temperature rise in specimens during fatigue loading, although the frequency-dependent reduction in fatigue strength is not always ascribed to the reduction in static strength due to temperature rise [66]. The strength of fiber-reinforced composites degrades with time [67, 68]. Such stress rupture (or creep rupture) behavior becomes more significant in matrixdominated laminates at higher temperatures [47, 50, 69]. In constructing the anisomorphic CFL diagrams for composites, therefore, the creep strengths in tension and compression for a given total time to fracture which is equivalent to the duration of a given constant number of cycles to failure should be used in place of their initial static strengths if the reduction in strength due to creep becomes significant [70]. The necessity of considering the creep rupture strength of composites in identifying their CFL diagrams had already been pointed out by Boller [29, 30]. Consequently, when building the CFL diagrams for composites, we need to take into account the changes in their strengths that are caused by temperature, impact damage, water uptake, and time. Almost all the factors that influence the static and fatigue strengths of continuous fiber-reinforced polymer matrix composites have been

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reviewed by Schulte and Stinchcomb [51] and Agarwal and Broutman [71], respectively. The information from these articles allows us to qualitatively understand how the anisomorphic CFL diagram for a given composite is affected by the factors, but further efforts are necessary to quantify the effect of the factors on the mean stress sensitivity in fatigue of composites through experiment and to assess the validity of the theoretical methods for constructing the CFL diagrams for composites.

12.9

Source of further information and advice

The engineering methods for predicting the S-N relationships for composites under constant amplitude fatigue loading at any stress ratios have been described in the book Fatigue in Composites, edited by Harris [1], that covers the most important aspects of the fatigue of composites. Progress that has been made since the publication of this encyclopedic book will be found in the present volume. So, these two books would be excellent aids in the continued journey of developing realistic fatigue life prediction methods suitable for composites. It is also helpful to revisit pioneering articles on the subject discussed in this chapter, e.g., Boller [29, 30], and to learn the history of the early development of the CFL diagram that has been reviewed by Sendeckyj [72]. In addition to the master S-N curve and CFL diagram approaches, the cultivation of the fatigue failure criteria based on the principal residual strengths, e.g., Hashin and Rotem [73], Sims and Brogdon [74], Hashin [75], Sendeckyj [76], Kawai et al. [10], and Liu and Mahadevan [77], may inspire a different approach to constant amplitude fatigue life prediction. The guidelines for the design of wind turbines [78] and the SAE Fatigue Design Handbook [79], which describe the current technologies and procedures for fatigue design of industrial products, although the latter is intended mainly for conventional materials, will also help to clearly understand the current state of the fatigue life prediction methods for composites and to further elaborate them. In regard to the factors that should be considered for establishing a more accurate fatigue life prediction method based on a CFL diagram, the reader can obtain access to distributed information with the aid of the two main reference sources, along with the additional references provided in the previous section.

Acknowledgments The research on which this chapter is based has been carried out with my students at the University of Tsukuba, mainly with the financial support of the University of Tsukuba and the Ministry of Education, Culture, Sports, Science and Technology of Japan. The author is grateful to all the members in my lab who have contributed to the work. This chapter has been written in the course of the research work supported in part by the Ministry of Education, Culture, Sports, Science and Technology of Japan under a Grant-in-Aid for Scientific Research (No. 20360050).

References [1] B. Harris, Fatigue in Composites, Woodhead Publishing, Cambridge, UK, 2003.

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[2] G. Caprino, A. D’Amore, Flexural fatigue behaviour of random continuous-fibrereinforced thermoplastic composites, Compos. Sci. Technol. 58 (1998) 957–965. [3] G. Caprino, G. Giorleo, Fatigue lifetime of glass fabric/epoxy composites, Compos. Part A 30 (1999) 299–304. [4] A. D’Amore, G. Caprino, P. Stupak, J. Zhou, L. Nicolais, Effect of stress ratio on the flexural fatigue behaviour of continuous strand mat reinforced plastics, Sci. Eng. Compos. Mater. 5 (1) (1996) 1–8. [5] F. Ellyin, H. El Kadi, A fatigue failure criterion for fiber reinforced composite laminae, Compos. Struct. 15 (1990) 61–74. [6] M. Kawai, Damage mechanics model for off-axis fatigue behavior of unidirectional carbon fiber-reinforced composites at room and high temperatures, in: T. Massard, A. Vautrin (Eds.), Proceedings of the 12th International Conference on Composite Materials (ICCM12), Paris, 5–9 July, 1999, p. 322. [7] M. Kawai, H. Suda, Effects of non-negative mean stress on the off-axis fatigue behavior of unidirectional carbon/epoxy composites at room temperature, J. Compos. Mater. 38 (10) (2004) 833–854. [8] M. Kawai, T. Taniguchi, Off-axis fatigue behavior of plain woven carbon/epoxy composites at room and high temperatures and its phenomenological modeling, Compos. Part A 37 (2) (2006) 243–256. [9] M. Kawai, A. Hachinohe, K. Takumida, Y. Kawase, Off-axis fatigue behaviour and its damage mechanics modelling for unidirectional fibre-metal hybrid composite: GLARE 2, Compos. Part A 32 (2000) 13–23. [10] M. Kawai, S. Yajima, A. Hachinohe, Y. Takano, Off-axis fatigue behavior of unidirectional carbon fiber-reinforced composites at room and high temperatures, J. Compos. Mater. 35 (7) (2001) 545–576. [11] M. Kawai, S. Yajima, A. Hachinohe, Y. Kawase, High-temperature off-axis fatigue behaviour of unidirectional carbon fiber-reinforced composites with different resin matrices, Compos. Sci. Technol. 61 (2001) 1285–1302. [12] J. Goodman, Mechanics Applied to Engineering, Longmans, Green & Co, London, 1899. [13] M.J. Salkind, Fatigue of composites, in: Composite Materials: Testing and Design (Second Conference), 1972, pp. 143–169. ASTM STP 497. [14] S.V. Ramani, D.P. Williams, Notched and unnotched fatigue behavior of angle-ply graphite/epoxy composites, in: K.L. Reifsnider, K.N. Lauraitis (Eds.), Fatigue of Filamentary Composite Materials, 1977, pp. 27–46. ASTM STP 636. [15] M.P. Ansell, I.P. Bond, P.W. Bonfield, Constant life diagrams for wood composites and polymer matrix composites, in: Proceedings of the 9th International Conference on Composite Materials (ICCM 9), Madrid, 1993, pp. 692–699. [16] B. Harris, H. Reiter, T. Adam, R.F. Dickson, G. Fernando, Fatigue behaviour of carbon fibre reinforced plastics, Composites 21 (3) (1990) 232–242. [17] B. Harris, N. Gathercole, J.A. Lee, H. Reiter, T. Adam, Life-prediction for constant-stress fatigue in carbon-fibre composites, Phil. Trans. R. Soc. London A355 (1997) 1259–1294. [18] T. Adam, G. Fernando, R.F. Dickson, H. Reiter, B. Harris, Fatigue life prediction for hybrid composites, Int. J. Fatigue 11 (4) (1989) 233–237. [19] T. Adam, N. Gathercole, H. Reiter, B. Harris, Fatigue life prediction for carbon fibre composites, Adv. Compos. Lett. 1 (1992) 23–26. [20] N. Gathercole, H. Reiter, T. Adam, B. Harris, Life prediction for fatigue of T800/5245 carbon-fibre composites: I. Constant-amplitude loading, Fatigue 16 (1994) 523–532. [21] M.H. Beheshty, B. Harris, T. Adam, An empirical fatigue-life model for high-performance fibre composites with and without impact damage, Compos. Part A 30 (1999) 971–987.

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[22] T.P. Philippidis, A.P. Vassilopoulos, Complex stress state effect on fatigue life of GRP laminates. Part I, experimental, Int. J. Fatigue 24 (2002) 813–823. [23] T.P. Philippidis, A.P. Vassilopoulos, Complex stress state effect on fatigue life of GRP laminates. Part II, theoretical formulation, Int. J. Fatigue 24 (2002) 825–830. [24] M. Kawai, A method for identifying asymmetric dissimilar constant fatigue life diagrams for CFRP laminates, Key Eng. Mater. 61–64 (2006) 334–335. [25] M. Kawai, M. Koizumi, Nonlinear constant fatigue life diagrams for carbon/epoxy laminates at room temperature, Compos. Part A 38 (2007) 2342–2353. [26] M. Kawai, T. Murata, A modified asymmetric anisomorphic constant fatigue life diagram and application to CFRP symmetric angle-ply laminates, in: Proc. 13th US-Japan Conf. Compos. Mater. (CD-ROM), Nihon University, Tokyo, 6–7 June, 2008, pp. 1–8. [27] R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, John Wiley & Sons, New York, 1989. [28] W.Z. Gerber, Bestimmung der zul€assigen Spannungen in Eisen-Constructionen (Calculation of the allowable stresses in iron structures), Z. Bayer Archit. Ing. Ver. 6 (6) (1874) 101–110. [29] Boller K H (1957), ‘Fatigue properties of fibrous glass-reinforced plastics laminates subjected to various conditions’, Mod. Plast., 34, 163–186, 293. [30] K.H. Boller, Fatigue characteristics of RP laminates subjected to axial loading, Mod. Plast. 41 (1964) 145–150. 188. [31] H.T. Hahn, Fatigue behavior and life prediction of composite laminates, in: S.W. Tsai (Ed.), Composite Materials: Testing and Design (Fifth Conference), 1979, pp. 383–417. ASTM STP 674. [32] M.N.A. Nasr, M.N. Abouelwafa, A. Gomaa, A. Hamdy, E. Morsi, The effect of mean stress on the fatigue behavior of woven-roving glass fiber-reinforced polyester subjected to torsional moments, ASME J. Eng. Mater. Technol. 127 (2005) 301–309. [33] M. Kawai, A phenomenological model for off-axis fatigue behavior of unidirectional polymer matrix composites under different stress ratios, Compos. Part A 35 (7–8) (2004) 955–963. [34] I.P. Bond, Fatigue life prediction for GRP subjected to variable amplitude loading, Compos. Part A 30 (1999) 961–970. [35] I.P. Bond, M.P. Ansell, Fatigue properties of jointed wood composites, Part I Statistical analysis, fatigue master curves and constant life diagrams, J. Mater. Sci. 33 (1998) 2751–2762. [36] I.P. Bond, M.P. Ansell, Fatigue properties of jointed wood composites, Part II Life prediction analysis for variable amplitude loading, J. Mater. Sci. 33 (1998) 4121–4129. [37] H.J. Sutherland, J.F. Mandell, The effect of mean stress on damage predictions for spectral loading of fiberglass composite coupons, in: EWEA, Special Topic Conference 2004: The Science of Making Torque From the Wind, Delft, the Netherlands, 19–21 April, 2004, pp. 546–555. [38] E. Phillips, Effects of truncation on a predominantly compression load spectrum on the life of a notched graphite/epoxy laminate, in: Fatigue of Fibrous Composite Materials, 1981, pp. 197–212. ASTM STP 723. [39] P.W. Bonfield, M.P. Ansell, Fatigue properties of wood in tension, compression and shear, J. Mater. Sci. 26 (1991) 4765–4773. [40] R.P.L. Nijssen, Fatigue Life Prediction and Strength Degradation of Wind Turbine Rotor Blade Composites, Knowledge Centre Wind Turbine Materials and Constructions, KC-WMC, Wieringerwerf, The Netherlands, 2006.

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[41] I.P. Bond, I.R. Farrow, Fatigue life prediction under complex loading for XAS/914 CFRP incorporating a mechanical fastener, Int. J. Fatigue 22 (2000) 633–644. [42] T.P. Philippidis, A.P. Vassilopoulos, Life prediction methodology for GFRP laminates under spectrum loading, Compos. Part A 35 (2004) 657–666. [43] M. Kawai, T. Shiratsuchi, K. Yang, A spectrum fatigue life prediction method based on the nonlinear constant fatigue life diagram for CFRP laminates, in: Proc. 6th AsiaAustralasian Conf. Compos. Mater. (ACCM-6), Kumamoto University, Kumamoto, Kyushu, Japan, 23–26 September, 2008, pp. 153–156. [44] M.H. Beheshty, B. Harris, A constant-life model of fatigue behaviour for carbon-fibre composites: the effect of impact damage, Compos. Sci. Technol. 58 (1998) 9–18. [45] M.M. Shokrieh, L.B. Lessard, Multiaxial fatigue behaviour of unidirectional plies based on uniaxial fatigue experiments—I. Modelling, Int. J. Fatigue 19 (1997) 201–207. [46] V.A. Passipoularidis, T.P. Philippidis, A study of factors affecting life prediction of composites under spectrum loading, Int. J. Fatigue 31 (2009) 408–417. [47] M. Kawai, Y. Masuko, T. Sagawa, Off-axis tensile creep rupture of unidirectional CFRP laminates at elevated temperature, Compos. Part A 37 (2) (2006) 257–269. [48] D. Sch€utz, J.J. Gerharz, Fatigue strength of a fibre-reinforced material, Composites 8 (1977) 245–250. [49] G.K. Boerstra, The multislope model: a new description for the fatigue strength of glass fibre reinforced plastic, Int. J. Fatigue 29 (2007) 1571–1576. [50] M. Kawai, T. Sagawa, Temperature dependence of off-axis tensile creep rupture behavior of a unidirectional carbon/epoxy laminate, Compos. Part A 39 (2008) 523–539. [51] K. Schulte, W.W. Stinchcomb, Damage mechanisms—including edge effects—in carbon fibre-reinforced composite materials, in: K. Friedrich (Ed.), Application of Fracture Mechanics to Composite Materials, Elsevier Science Publishers, Amsterdam, 1989, pp. 273–325. [52] M. Kawai, Y. Matsuda, H. Hoshi, Y. Iwahori, Anisomorphic constant life diagrams for a woven CFRP laminate at room and high temperatures, in: Proc. 17th Int. Conf. Compos. Mater. (ICCM-17), 27–31 July, Edinburgh, UK, 2009. [53] Y. Matsuda, M. Kawai, H. Hoshi, Y. Iwahori, Nonlinear constant life diagrams for the fatigue of woven CFRP quasi-isotropic laminates at room and high temperatures, in: Proc 8th China-Japan Joint Conf Compos (CJJCC-8), 27–29 October, 2008, pp. 303–306. [54] S.R. Swanson, D.S. Cairns, M.E. Guyll, D. Johnson, Compression fatigue response for carbon fibre with conventional and toughened epoxy matrices with damage, ASME J. Eng. Mater. Technol. 115 (1993) 116–121. [55] M. Kawai, N. Maki, Fatigue strength of cross-ply CFRP laminates at room and high temperatures and its phenomenological modeling, Int. J. Fatigue 28 (10) (2006) 1297–1306. [56] M. Kawai, M. Morishita, K. Fuzi, T. Sakurai, K. Kemmochi, Effects of matrix ductility and progressive damage on fatigue strengths of unnotched and notched carbon fibre plain woven roving fabric laminates, Composites 27A (1996) 493–502. [57] O. Konur, F.L. Matthews, Effects of the properties of the constituents on the fatigue performance of composites: a review, Composites 20 (4) (1989) 317–328. [58] R.K. Khan, Z. Khan, F. Al-Sulaiman, N. Merah, Fatigue life estimates in woven carbon fabric/epoxy composites at non-ambient temperatures, J. Compos. Mater. 36 (22) (2002) 2517–2535. [59] C.J. Jones, R.F. Dickson, T. Adam, H. Reiter, B. Harris, The environmental fatigue behaviour of reinforced plastics, Proc. Roy. Soc. London (1984) 315–338. A396.

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[60] R. Selzer, K. Friedrich, Mechanical properties and failure behaviour of carbon fibrereinforced polymer composites under the influence of moisture, Compos. Part A 28 (1997) 595–604. [61] S. Kellas, J. Morton, P.T. Curtis, The effect of hygrothermal environments upon the tensile and compressive strengths of notched CFRP laminates, Part 1: Static loading, Composites 21 (1) (1990) 41–51. [62] S. Kellas, J. Morton, P.T. Curtis, The effect of hygrothermal environments upon the tensile and compressive strengths of notched CFRP laminates, Part 2: fatigue loading, Composites 21 (1) (1990) 52–62. [63] L.E. Asp, The effect of moisture and temperature on the interlaminar delamination toughness of a carbon/epoxy composites, Compos. Sci. Technol. 58 (1998) 967–977. [64] Y. Shan, K. Liao, Environmental fatigue of unidirectional glass-carbon fiber reinforced hybrid composites, Compos. Part B 32 (2001) 355–363. [65] G. Sala, Composite degradation due to fluid absorption, Compos. Part B 31 (2000) 357–373. [66] D.C. Curtis, D.R. Moore, B. Slater, N. Zahlan, Fatigue testing of multi-angle laminates of CF/PEEK, Composites 19 (6) (1988) 446–452. [67] D.A. Dillard, D.H. Morris, H.F. Brinson, Predicting viscoelastic response and delayed failures in general laminated composites, in: I.M. Daniel (Ed.), Composite Materials: Testing and Design (Sixth Conference), 1982, pp. 357–370. ASTM STP 787. [68] J. Raghavan, M. Meshii, Creep rupture of polymer composites, Compos. Sci. Technol. 57 (1997) 375–388. [69] H.F. Brinson, Matrix dominated time dependent failure predictions in polymer matrix composites, Compos. Struct. 47 (1999) 445–456. [70] P.K. Mallick, Y. Zhou, Effect of mean stress on the stress-controlled fatigue of a short E-glass fiber reinforced polyamide-6.6, Int. J. Fatigue 26 (2004) 941–946. [71] B.D. Agarwal, L.J. Broutman, Analysis and Performance of Fiber Composites, John Wiley & Sons, New York, 1990. [72] G. Sendeckyj, Constant life diagrams—a historical review, Int. J. Fatigue 23 (2001) 347–353. [73] Z. Hashin, A. Rotem, A fatigue failure criterion for fiber-reinforced materials, J. Compos. Mater. 7 (1973) 448–464. [74] D.F. Sims, V.H. Brogdon, Fatigue behavior of composites under different loading modes, in: K.L. Reifsnider, K.N. Lauraitis (Eds.), Fatigue of Filamentary Composite Materials, 1977, pp. 185–205. ASTM STP 636. [75] Z. Hashin, Fatigue failure criteria for unidirectional fiber composites, ASME J. Appl. Mech. 48 (1981) 846–852. [76] G. Sendeckyj, Life prediction for resin-matrix composite materials, in: K.L. Reifsnider (Ed.), Fatigue of Composite Materials, Elsevier Science Publishers, Amsterdam, 1990, pp. 431–483. [77] Y. Liu, S. Mahadevan, A unified multiaxial fatigue damage model for isotropic and anisotropic materials, Int. J. Fatigue 29 (2007) 347–359. [78] Risø National Laboratory (Ed.), Guidelines for Design of Wind Turbines, second ed., Risø National Laboratory, Roskilde, Denmark, 2002. [79] R.C. Rice, SAE Fatigue Design Handbook, third ed., SAE International, Warrendale, PA, 1997.

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Prediction of fatigue crack initiation in UD laminates under different stress ratios

13

R.D.B. Sevenoisa,b, W. Van Paepegema a Department of Materials, Textiles and Chemical Engineering, Ghent University, Tech Lane Ghent Science Park Campus A, Ghent, Belgium, bSIM vzw, Ghent, Belgium

13.1

Introduction

A significant part of the fatigue life of a unidirectional (UD) composite material can be attributed to the initiation of microcracks [1–3]. Thus, to achieve good predictive capabilities, the initiation of cracks must be included in fatigue life prediction methodologies. Several authors have already presented predictive capabilities of this sort. The presented approaches, however, fall short in some aspect of predicting the fatigue load. Either model parameters must be recalibrated for each new stress ratio or they need to be recalibrated for each new layup. This chapter presents a phenomenological crack initiation criterion which unifies all those aspects that were separately treated by other authors. A complete criterion for the prediction of matrix fatigue crack initiation in a UD ply under any multiaxial stress state is presented. TT, CC, and TC loading can be accounted for and any load ratio can be handled. In Section 13.3 the existing approaches and the newly formed criterion are laid out. Before that is done, however, a brief discussion is presented on the definition of crack initiation, Section 13.2, which is important for the setting of this research and future developments. Next, Sections 13.3 and 13.4 show a validation of the criterion and a discussion of the results. Section 13.5 contains conclusions and possible roadmaps for future development. Finally, some sources of further information and advice are given in Section 13.6.

13.2

Definition of crack initiation

In several works [1, 4–6], the definition of crack initiation is defined as the time required to form a crack of detectable size. This definition seems strange as it is open for interpretation. Effectively, because the quantified minimum size of a crack is dependent on the accuracy of the used experimental method, the concept of initiation is subject to the length scale the research is performed at. Thus, crack initiation could on the macroscale be defined as a visible Fatigue Life Prediction of Composites and Composite Structures. https://doi.org/10.1016/B978-0-08-102575-8.00013-9 © 2020 Elsevier Ltd. All rights reserved.

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crack of 1–2 mm while on the microscale it is the first microscopically observable debond between fiber and matrix interface. Practically, or at least until true multiscale testing and simulation is feasible, the growth of a crack from atomic to mm size is lumped into the ‘initiation’ phase. Here, the usefulness of the definition shows. It evolves with the size scale of the research and cannot get out-of-date. Although the definition of crack initiation is flexible, it is important to note that it should be carefully chosen. Lumping too many phenomena in a single parameter is destructive to experimental observations. The result could become misleading which will show in unsatisfactory predictions. The choice of an appropriate detectable crack initiation size is paramount for useful research. In this chapter, crack initiation is defined as the appearance of the first matrix crack through the thickness of a ply. This size is appropriate because the geometrical environment at this scale positions, for most of composite materials today, the initiation at the fiber-matrix interface. Additionally, only two crack initiation phenomena were observed [3]: a normal and a shear mode which simplifies the modeling problem and improves likelihood of correct predictions.

13.3

Predicting fatigue crack initiation

Based on the definition in Section 13.2 a literature review on the prediction of fatigue crack initiation is given. The essential conclusions from the existing failure criteria are then combined in a coherent way to a unified matrix crack initiation criterion in Section 13.3.2. Finally, in Section 13.3.3 the parameter identification and predictions of the criterion are compared to experimental evidence.

13.3.1 Fatigue crack initiation literature Only a limited number of experimental studies reliably quantify crack initiation as a phenomenon in fatigue [7–11]. The literature is so scarce that also studies are included where the phenomenon has only been indirectly studied. Namely, studies where fatigue crack initiation in the plies of the laminate can be assumed to occur shortly before final failure of the specimen. Such a scenario is for example present when UD laminates are subjected to off-axis loading [12–14]. In these studies, generally no matrix cracks are observed before final fatigue failure. This is also supported by the absence of any stiffness degradation, caused by microcracks, of the laminate during loading. When this is confirmed, and the final failure is sudden and brittle, the initiation of a matrix crack can be assumed to be followed by its immediate propagation resulting in catastrophic failure of the specimen. Also, Quaresimin et al. [15] show that the behavior between the initiation and propagation of fatigue cracks in multidirectional laminates subjected to uniaxial cyclic load, and tubes subjected to tension-torsion loading, is the same when the local multiaxial stress state is similar. Therefore, it is concluded that the moment of final failure of UD flat laminates subjected to off-axis loading can be used for the life to

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initiation. Literature describing the cycles to failure for off-axis tests on UD laminates are thus interrogated as well. Most of the works [7–11] are based on the classical W€ohler (S-N) curve approach where initiation is quantified based on the remote macro-stress applied to the composite ply through a linear-logarithmic relation between the maximum and/or mean (when a Goodman diagram is used) stress and the life to initiation. Multiple authors express the maximum allowable stress as a function of the number of cycles to initiation [7, 10, 13, 16, 17] σ max ¼ f ðN Þ

(13.1)

although this formulation seems easy to use, it is biased because it implies that somehow a given number of cycles results in a unique load that can be applied to a fatigue sample. It is however the applied load that drives the number of cycles to initiation. Also, it is well possible that certain combinations of mean and maximum stress result in the same fatigue life. A practical difficulty when using S-N curves is that the user must recalibrate the fit every time the fatigue load, multiaxial load condition, or layup is changed. This requires a significant amount of experimental work before good predictions can be made. Several authors have attempted to reduce the number of required tests using a variety of methods. The majority attempts to constructs a new damage parameter to quantify the fatigue life, for example, Refs. [13, 16, 18–23]. A trend is that the static strength, σ stat, of the material is used to normalize the data [16, 24, 25]. In a way, this is natural because when the applied fatigue load reaches the static strength of the material the number of cycles to failure reduces to 1. Kawai et al. [14, 24, 26–28] present a formulation which normalizes for the maximum and the mean stress through the anisomorphic constant fatigue life diagram. Consequently, recalibration for multiple stress ratios becomes unnecessary. On the other hand, Quaresimin and Carraro [9, 10] showed that improved predictions for crack initiation are obtained when the local maximum principal stress (LMPS) or local maximum hydrostatic stress (LMHS) in the matrix around the fibers is used instead of the remote macro-stress as a driving force. This eliminates the need to recalibrate S-N curves for each new multiaxial load scenario or layup. Additionally, it is shown that each driving force results in a different fracture surface morphology which indicates the existence of at least two competing crack initiation mechanisms. However, these approaches fall short in some aspect of predicting the fatigue load. Quaresimin et al. have to recalibrate the model parameters for every new stress ratio and Kawai et al. have to recalibrate for each multiaxial stress state. Also, with few exceptions, for example, Ref. [24], most approaches are only applicable to one type of loading: tension-tension (TT), compression-compression (CC), or tensioncompression (TC). An integrated approach which allows predictions and has been validated for multiple stress ratios with multiple multiaxial stress states is necessary. In the following subsection a phenomenological crack initiation criterion that unifies all those aspects that where separately treated here is presented.

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13.3.2 Multiscale fatigue crack initiation criterion The construction of the phenomenological crack initiation criterion is based on a few considerations concerning crack development up to the moment of crack initiation. First the type of fatigue model is determined. Based on the review by Sevenois and Van Paepegem [29], existing techniques for fatigue prediction can be classified as (i) fatigue life models, (ii) residual strength models, (iii) residual stiffness models, and (iv) mechanistic models. Crack initiation is the first observation of damage, that is, the first appearance of a crack of a detectable length at a certain number of cycles. No stiffness degradation nor other mechanisms of damage growth precede this phenomenon. Moreover, no experimental evidence is found in the literature which indicates strength degradation before the initiation of matrix cracks. Hence, a model of the residual strength, residual stiffness, or mechanistic type is difficult to construct and validate for this type of damage. The sole alternative is to predict the moment the first crack is observed using a fatigue life model of the S-N curve type. Using the S-N curve approach is only justified with some additional assumptions. First, although it is acknowledged that there are changes in the arrangement of the atomic and molecular structure of the material during the cycles before crack initiation, it is assumed that these changes do not significantly influence the local stress state. As such, the multiaxial stress state in the undamaged matrix during the first load cycles can be used as a driver for the entire fatigue process up to crack initiation. Second, since there are no records of whether the contributions of the first or last cycle to the development of the damage are different, it is assumed that each cycle contributes evenly to the development of this damage. This is also in agreement with the first assumption because an insignificant change in the multiaxial stress state should essentially give an equal internal damage increase. Naturally, these assumptions conflict with reality. For example, one cannot assume that the local stress state is stable until first failure in the case that a tough matrix, with high plastic deformation capability, is used. The model is thus expected to perform less for thermoplastic matrix composites. Next to that, also sources of stress concentrations other than fibers, such as voids, might influence the local stress state. The effect of these is not considered. As discussed in the previous section, using Eq. (13.1) as a basis for an S-N curve approach is not useful. The formulation is therefore inverted N ¼ f ðσ max , σ mean Þ ¼ f ðσ max , σ min Þ ¼ f ðσ max , RÞ ¼ f ðσ min , RÞ ¼ f ðσ max , Δσ Þ

(13.2)

where N is the number of cycles to initiation, σ max, σ mean, σ min stand for the maximum, mean and minimum stress of the cycle, respectively, Δσ is the difference between the maximum and minimum stress and R is the stress ratio defined as R¼

σ min σ max

(13.3)

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469

Fig. 13.1 Fatigue load parameters.

In Eq. (13.2) the life to damage initiation is given as a function of a combination of minimum, maximum, mean stress, stress amplitude, and/or stress ratio. This is in agreement with the observation of Refs. [24, 30] that minimum two characteristic parameters of the applied load are needed to provide unique predictions. Eq. (13.2) is further specified using the observation that a normalization to the static strength is usually successful. One can argue that the speed of strength degradation is related to both the ratio of the maximum stress to the static strength, σ max/σ stat, and the ratio of stress change to the required stress increase from the minimum stress to obtain static failure, Δσ/Δσ stat (see Fig. 13.1) N¼f


 σ max Δσ , σ stat Δσ stat

(13.4)

where σ stat is the static strength of the material and Δσ stat is the required stress increase (tensile or compressive) to reach the static strength, namely Δσ Tstat ¼ σ Tstat  σ min

(13.5)

Δσ Cstat ¼ σ Cstat  σ max

(13.6)

where T stands for tension load and C for compression load. The choice to use σ max is related to the requirement that the life to failure must be equal to 1 when the fatigue

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Fatigue Life Prediction of Composites and Composite Structures

load reaches the static strength of the material. The contributions of the static and cyclic part of the stress cycle, respectively σ max/σ stat and Δσ max/Δσ stat, to the fatigue life must thus form a complementary system 

Δσ σ max N ¼ f ð1  β Þ +β Δσ stat σ stat

 (13.7)

In Eq. (13.7), β is a material parameter which regulates the influence of both cyclic and monotonic parts on the fatigue life. Next to being complementary, it is conceivable that both the cyclic and monotonic part interact and amplify one another in some way. It is also observed that in the selected data for fatigue initiation, the S-N curves can be approached by a linear log-log relationship [3, 9, 11, 13, 15, 24, 31–33]. The amplification can then be quantified through a second material parameter α and a power law as follows 

Δσ σ max N ¼ ð1  β Þ +β Δσ stat σ stat

α (13.8)

This is the basic form of the initiation law that is used to estimate the life to damage initiation. It is important to note that the stresses in Eq. (13.8) are those driving a certain damage type and do not equal the far-field stress applied to the sample [10]. For TT fatigue, a damage-type driven by the LMPS and a damage-type driven by the LMHS is observed. In CC loading only an out-of-plane-shear matrix failure mode is been observed, Section 13.3.1. Thus, Eq. (13.8) should be particularized for TT and CC loading cases, resulting in three uncoupled equations, one for each specific failure type 8 " #αTT LMPS > >   Δσ TT σ max > TT TT TT > N ¼ 1  β + β > LMPS LMPS T LMPS > > Δσ Tstat,LMPS σ stat,LMPS > > > " #αTT > < LMHS TT   Δσ σ max TT TT TT NLMHS ¼ 1  βLMHS + βLMHS T > Δσ Tstat,LMHS σ stat,LMHS > > > > " #αCC > LMPS > >   Δσ CC σ min > CC CC CC > > N ¼ 1  β + β LMPS C LMPS : LMPS Δσ Cstat,LMPS σ stat,LMPS

(13.9)

where the additional designators LMPS and LMHS (with respect to Eq. 13.8) indicate the specific driving force. The specific load type is indicated by TT for TT fatigue and CC for CC fatigue. The designators T and C indicate whether the tensile or compressive static strength should be used. Applying Eq. (13.9) therefore results in three estimations for the number of cycles to initiation for each specific failure phenomenon. To obtain a unique prediction, Carraro and Quaresimin [10] take the minimum life of these criteria  TT TT NfTT ¼ min NLMPS , NLMHS

(13.10)

Prediction of fatigue crack initiation in UD

471

This approach assumes that the failure types and their respective damage phenomena are independent and do not influence one another. Indeed, a dominant local maximum principal or local hydrostatic stress cannot coexist at the same material point in the matrix when considering a constant amplitude (CA) fatigue load. From this perspective, the dominant force will always drive the entire fatigue life to initiation at that location in the material. When considering variable fatigue load, both in amplitude (VA) and in multiaxiality of stress, the situation is different because the failure modes can be alternatingly activated in specific parts of the load history. This can lead to a state where the remaining life of each individual failure mode is reduced. If the remaining life of one failure mode does not influence the remaining life of the other failure mode, Eq. (13.10) can be safely used. Otherwise, Eq. (13.10) will have to be abandoned and the synergistic effect of one dominant failure mode on the other should be combined in one approach (e.g., Miner). Currently, it is unknown whether the driving forces influence the degradation of the other damage phenomena. Being conservative, it is more appropriate to employ the Miner’s rule for fatigue loading where multiple failure types occur for the same type of loading. This has also the advantage that the predicted lives of the individual failure types are combined into one damage variable. For the prediction of CA fatigue life the impact will, compared to the usage of the “Min[]” function, be minimal since both critical stress states cannot coexist at the same time. The Miner’s rule is also preferred from the perspective of implementation in a numerical framework. This because it avoids the sharp corner on the S-N curve for stress amplitudes where the dominant failure mode switches from one to another (see Fig. 13.2). This avoids difficulties with numerical algorithms. Moreover, one

Fig. 13.2 Difference between using the minimum and Miner’s rule to predict fatigue life.

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Fatigue Life Prediction of Composites and Composite Structures

does not need to track the life to failure of each individual dominant mode separately, which demands less computational effort. The formulation becomes as follows: 1

NFTT ¼

1 TT NLMPS

+

(13.11)

1 TT NLMHS

For TT and CC loading, the types of failure can clearly be quantified. In TC loading multiple failure modes are activated simultaneously. Although these failures can be traced back to a tensile or compressive failure type, the life to failure is significantly shorter compared to the cases when only tensile or compressive load is applied [24]. There is thus an interaction between alternating tensile and compressive load cycles. Nonetheless, the stress life diagram suggests that also TC loading can be approached with a log-log like behavior [24]. Note that TC loading can be viewed as a variable amplitude load cycle of alternating tensile and compressive stress where R ¼ 0 and R ¼  ∞, it is possible to apply Miner’s rule once again. This will ensure that the failure type is either of the tensile or compressive type, whichever is more dominant. The interaction between both mechanisms, which results in a larger degradation speed than when only tensile or compressive load is applied, must be governed by an additional material parameter. This is unavoidable because the interaction between the tensile and compressive parts of the load cycle will depend on the material. To study this interaction, fatigue life to initiation data is needed for multiple negative R ratios. Unfortunately, only one dataset was available where the required data are present. This makes it difficult to study the trends across different datasets and materials. Therefore, it is not included in this study. Nonetheless, it is proposed to represent the additional degradation as a power in the Miner’s rule as follows: NfTC ¼


1 1

TT NLMPS

+

1 TT NLMHS

+

1

f ðγTC RÞ

(13.12)

CC NLMPS

The power, f(γ TC, R) is a function of the stress ratio and the material parameter γ TC. The form of f(γ TC, R) cannot be determined with sufficient confidence, due to the lack of experimental evidence in fatigue life for negative stress ratios. Based on dataset C_TS_1 (as is shown in Section 13.3.3), it seems that a function based on the ratio between the peak-to-peak cycle amplitude and the minimum stress provides a good fit to the experimental data, Fig. 13.3 2 3   6 7   σ max  σ min 1 7 f γ TC , R ¼ min 1, γ TC ¼ min 6 1, 2γ TC
4 1+R 5 σ min 1 1R

(13.13)

Prediction of fatigue crack initiation in UD

473

Fig. 13.3 Ideal and predicted value of f(γTC, R) for dataset C_TS_1, Table 13.1.

The use of the ratio of the peak-to-peak amplitude to the portion of the negative load cycle gives the formulation a physical meaning. Also, the fact that the formula can be rewritten in terms of the stress ratio alone supports the physical aspect. However, note the use of the “Min[]” function in Eq. (13.13) which restricts f(γ TC, R)  1. This restriction is necessary because a value larger than 1 would, for the combined TC loading, return a fatigue life to initiation which is longer than the life to initiation of the tensile or compressive parts individually. Allowing f(γ TC, R) > 1 would thus be unrealistic. The final form of the formulation for TC loading can be expressed as NfTC ¼

1


1

TT NLMPS

+

1 TT NLMHS

+

h i min 1, γTC σ max σmin σ min 1

(13.14)

CC NLMPS

Eqs. (13.11) and (13.14) completely define the initiation criterion for matrix cracks under fatigue loading. The formulation requires 10 parameters to be fitted from experimental evidence; 6 to predict TT loading, 3 for CC loading, and 1 for TC loading. The following section shows how this criterion is calibrated and validated using several material systems.

13.3.3 Validation of crack initiation model Several criteria are used to select the experimental data for verification and validation of the fatigue criterion. As argued in Section 13.3.1, either fatigue crack initiation has to be measured explicitly in the experiments or UD specimens, subjected to off-axis

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Fatigue Life Prediction of Composites and Composite Structures

Table 13.1 Material systems used for application of the experimental data ID

Fiber/matrix

Sources

C_TS_1 C_TS_2 C_TP_1 G_TS_1 G_TS_2

T700/2592 T800/2500EP AS4/PEEK E-glass/Epoxy E-glass/Epoxy

[24] [14, 27] [33] [3, 9, 11, 15] [13]

loading, must fail in a brittle manner. Preferably experimental data at several stress ratios are available. The selection process resulted in five experimental datasets from multiple sources and provided a variety of commonly used fibers and matrix in research and industry: 2 Carbon/Thermoset, 1 Carbon/Thermoplast, and 2 Glass/Thermoset material systems. The systems, their IDs and respective data source(s) are shown in Table 13.1.

13.3.3.1 Calculation of local stress in the matrix around the fibers The experimental data taken from literature represent the life to failure in S-N graphs as a function of the applied external load. Recall, however, that the criterion requires the LMPS and LMHS in the matrix around the fibers. As these stresses occur on the microscale, they cannot simply be measured experimentally. They can however be estimated using micromechanics. This can be done analytically [10] or using finite element analysis (FEA) with single or multi-fiber representative unit cell (RUC) models [34, 35]. The choice for the estimation of the local stresses in the matrix is up to the researchers or developers implementing the criterion. Depending on the necessary accuracy, it can be decided to include several features such as thermal residual stresses or the effect of voids. Since the main topic of this chapter is the evaluation of the fatigue crack initiation, the determination of the local stresses in the matrix for this purpose will only be limitedly discussed. Details can be found in Ref. [34]. Ideally, one would determine the local stresses using a randomly packed fiber distribution. Carraro [36] has shown, however, that the way the global stresses are combined do not differ significantly between the configuration of a random and regular square fiber packing. For this reason, and to limit computational effort, a single fiber RUC with a fiber volume fraction of 60% is used for all material systems, Fig. 13.4. The RUC uses periodic boundary conditions (PBC) according to Ref. [37]. Based on the selected materials, the constitutive properties for the individual materials, matrix, and fiber, are retrieved from literature and shown Tables 13.2 and 13.3. Note that in both tables some properties are indicated as assumed values because not all 3D material properties could be obtained from the literature. Recall that in this work the definition of crack initiation is a crack through the thickness of the ply. The source of this crack is, however, assumed to be related to the local stress around the fibers. Because of this, it is inherently assumed that the time between

Prediction of fatigue crack initiation in UD

475

Fig. 13.4 Single fiber representative unit cell.

Table 13.2 Fiber properties used in the micro-models Propertie

E-glass [10]

Carbon AS4 [37]

Carbon T700S [38–40]

Carbon T800H [41, 42]

E1 [GPa] E2 [GPa] E3 [GPa] v12 [] v13 [] v23 [] G12 [GPa] G13 [GPa] G23 [GPa]

70 70a 70a 0.22 0.22a 0.22a 29a 29a 29a

235 14 14 0.20 0.20 0.25 28 28 5.5b

230 17 17 0.20 0.20 0.25c 15 15 5.8b

294 14c 14c 0.23c 0.23c 0.25c 15c 15c 5.5b

a

Isotropic material. G23 ¼E22/2(1 + v23). Assumed value.

b c

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Fatigue Life Prediction of Composites and Composite Structures

Table 13.3 Matrix properties used in the micro-models Properties

PEEK [43]

Epoxy (general) [10]

Epoxy 2592 [44]

Epoxy 2500EP [45]

E [GPa] ν []

4.0 0.3a

3.2 0.37

2.4 0.4

3.0 0.3

a

Assumed value.

crack initiation at the fiber-matrix interface and the moment the crack spans the ply thickness is quite short. For the RUC, the matrix and fibers are assumed to behave linear elastically. Note that this assumption is only valid if the behavior of the material during the (fatigue) tests is linear elastic as well. This can be observed in experiments by inspecting the evolution of permanent strain and stiffness over the lifetime or by considering the failure pattern, which should be a single brittle crack over the entire specimen width. The experimental data used in this study was inspected for these properties from which it was found that only the material system with a thermoplastic matrix exhibits nonlinear material behavior. Accordingly, the predictions for this material combination are expected to be in less agreement with experimental data, Section 13.3.3.5. For illustration, the dominant failure stresses for a RUC loaded at a 15-degree offaxis angle and a RUC at a 90-degree off-axis angle are shown in Fig. 13.5. A numerical legend is not shown because the RUC is subjected to a unit load and the numerical values are therefore irrelevant. Both figures show high stress concentrations at the fiber-matrix interface.

Fig. 13.5 Distribution of l stress in a single fiber RUC. (A) Maximum principal stress, RUC loaded at 15 degree off-axis angle, (B) Hydrostatic stress, RUC loaded at 90 degree off-axis angle. Numerical legend is not shown due to irrelevance.

Prediction of fatigue crack initiation in UD

477

With the estimation of the local stress in the matrix around the fibers, the criterion is applied to the five selected datasets in the following subsections.

13.3.3.2 Application to C_TS_1 The dataset for C_TS_1, provided by Kawai and Itoh [24], is one of the most complete sets of fatigue life data. It contains fatigue life to failure for multiple stress ratios in TT, CC, and TC loading in laminates at multiple off-axis angles. For all specimens loaded under TT conditions, (tensile) brittle fracture occurred. For the specimens where some part of the fatigue load was compressive and the off-axis angle between 0 and 30 degree, fiber kinking was found as main failure type. For the samples with off-axis angle of 45 and 90 degree, out-of-plane shear failure was recorded. Since the criterion developed in this work is meant for the prediction of matrix crack initiation, and fiber kinking is not of that type, no fatigue life predictions will be made for the off-axis angles of 0 and 30 degree subjected to TC or CC loading. The parameters α and β in Eq. (13.9) are identified using the approach of test condition parameter elimination. The parameter α is calibrated first using the S-N data where either the minimum stress for TT or the maximum stress for CC is close to zero. σ In this case it is fulfilled that ΔσΔσstat  σstat and Eq. (13.8) reduces to N


 σ max α σ stat

(13.15)

which leaves only α as a fitting parameter. β is calibrated next using the S-N data where the minimum stress for TT or the maximum stress for CC is far from zero. In practice, this comes down to using (preferably) R ¼ 0.1 and R ¼ 10 S-N data to calibrate, respectively, αTT and αCC, and subsequently use R ¼ 0.5 and R ¼ 2 to calibrate, respectively, βTT and βCC. Note that this should be done for each driving fatigue load, LMPS and LMHS. The datasets that evidence more clearly the effects of the driving fatigue load should be used as well. Thus, a small off-axis angle (10–20 degree) is used to characterize for LMPS and a large off-axis angle (90 degree) for LMHS. When α and β for TT and CC loading are calibrated, the synergistic degradation increase, γ TC, for TC can be calibrated. This should preferably be done using S-N data for a large off-axis angle (90 degree) at the most detrimental scenario R ¼  1. An overview of the used S-N data to calibrate the several material parameters is given in Table 13.4. The results of the parameter calibration for C_TS_1 are shown in Fig. 13.6. The predictions made for the remaining off-axis angles are shown in Fig. 13.6D and Fig. 13.7. Fig. 13.7A–C show a good agreement between prediction and experiment for the off-axis angles of 15, 30, and 45 degree under TT loading. The predictions shown in Fig. 13.7C for R ¼ 0.1 at an off-axis angle of 45 degree underestimate the fatigue life. It seems that the experimental data points are aligned on a single linear log-log line which suggests that there is no difference in degradation speed between both stress ratios. Also, the predictions shown in Fig. 13.7C for the TC loading of the 45 degree off-axis angle are less satisfactory. However, as Fig. 13.7D shows, the agreement with

478

Fatigue Life Prediction of Composites and Composite Structures

Table 13.4 Preferred S-N test data to calibrate material parameters Parameter

Off-axis angle [degree]

Stress ratio R [2]

αTT LMPS αTT LMHS βTT LMPS βTT LMHS αCC LMPS βCC LMPS γ TC

15 90 15 90 90 90 90

0.1 0.1 0.5 0.5 10 2 1

the experiments for R ¼  5.12 and R ¼  10 for the off-axis angle of 90 degree in TC is excellent. Furthermore, the knee-shaped fit curve for the case R ¼  10 is noticeable. This feature indicates that the main failure mode changes from the compressive to the tensile failure type. The discrepancies between prediction and experiment are centered around the experimental data for the off-axis angle of 45 degree. Some remarks on this discrepancy are made. First, although the used criterion is based on physical damage phenomena, it is also based on several assumptions that might neglect some fatigue load influencing effects. It can also be argued that a more progressive damage development was possible because the specimens were fatigued at a frequency of 5 Hz, and the 45 degree samples are more sensitive to frictional heating. These facts could have influenced the life to damage initiation. The absence of the stress ratio effect showed in Fig. 13.7C is also an unexpected result. Nevertheless, apart from the predictions for the 45 degree off-axis angle, the agreement with experimental results is very good for all stress ratios predicted.

13.3.3.3 Application to G_TS_1 Quaresimin et al. [3, 9, 11, 15] quantified explicitly the life to matrix crack initiation. Glass/epoxy tubes with layup [0F/90U, 3/0F] (F stands for fabric and U stands for UD tape) were fatigued under tension-torsion conditions. The transparency of the specimen makes optical identification of the appearance and growth of the matrix cracks possible. These cracks appeared first in the internal 90U,3 layers. Like changing the off-axis loading angle for UD samples, the biaxiality ratio λ12 ¼ σ6 was varied to obtain various multiaxial loading conditions. The two main failure σ2 modes, LMPS dominated failure under high biaxiality ratio λ12 ¼ 2 and LMHS dominated failure for a biaxiality ratio λ12 ¼ 0,were observed. These datasets were used to TT TT TT calibrate the parameters αTT LMPS, βLMPS and αLMHS, βLMHS, see Section 13.3.3.2 for details on the procedure. The result of the parameter calibration is shown in Fig. 13.8 and Table 13.5 The fitting parameters for CC fatigue are not determined due to the absence of experimental data in this loading regime. The parameter γ TC for TC loading was calibrated using the experimental data for λ12 ¼ 0 at a stress ratio R ¼  1. Fig. 13.9A shows an

(B)

(C)

(D) 479

Fig. 13.6 Calibration and prediction of material parameters for loading on material C_TS_1. (A) TT-LMPS dominated failure, (B) TT-LMHS dominated failure, (C) CC-LMPS dominated failure, (D) TC-combined dominated failure.

Prediction of fatigue crack initiation in UD

(A)

480

(B)

(C)

(D)

Fig. 13.7 Prediction of fatigue life for fatigue loading on material C_TS_1. (A) TT-LMPS dominated failure, (B) TT-LMPS dominated failure, (C) TT-LMHS dominated failure, (D) TC-combined dominated failure.

Fatigue Life Prediction of Composites and Composite Structures

(A)

Prediction of fatigue crack initiation in UD

481

(A)

(B) Fig. 13.8 Result of calibration of material parameters for loading on material G_TS_1. (A) TT-LMHS dominated failure, (B) TT-LMPS dominated failure.

excellent agreement between prediction and experiment for TT loading at λ12 ¼ 1. Also for a glass/epoxy material combination under TT loading, the criterion can reliably predict the dominant failure type which drives fatigue life to initiation. For TC loading, Fig. 13.9B, an excellent agreement for the biaxiality ratio λ12 ¼ 1 is observed. The life to initiation predicted for λ12 ¼ 2 is slightly longer than experimentally observed. It appears that the synergistic effect of combined shear and normal, tensile, and compressive, load is insufficiently captured by the criterion. A similar observation was made for C_TS_1 where, for the laminates subjected to TC at an off-axis angle of 45°, an overestimation of the fatigue life was made (see Section 13.3.3.2),

482

Fatigue Life Prediction of Composites and Composite Structures

Table 13.5 Material properties and fatigue law parameters for the five material systems Material systems

C_TS_1

C_TS_2

C_TP_1

G_TS_1

G_TS_2

122 255 37

135 – 42

246 – 80

130 – 80

130 – 55

14.71 4.00 14.75 0.00 – – –

20.41 0.24 16.23 0.24 – – –

16.67 0.19 15.38 0.1750 – – 0.35

15.38 0.55 18.45 1.19 – – 0.40

Static properties σ Tstat,LMPS [MPa] σ Cstat,LMPS [MPa] σ Tstat,LMHS [MPa]

Fatigue model parameters αTT LMPS βTT LMPS αTT LMHS βTT LMHS αCC LMPS βCC LMPS γ TC

20.08 1.54 14.49 1.00 185.19 0.00 0.55

Fig. 13.7C. In that case, doubts on the effect of frictional heating on the fatigue life caused by a high test frequency were present. In this case, the test frequency was higher, 10 Hz, but no mention was made of self-heating of the sample prior to failure initiation. However, diffuse damage was observed in the external fabric layers prior to crack initiation in the internal 90 degree layers. The effect of damage in the adjacent plies on the life to crack initiation is, at the moment, unclear and requires further investigation.

13.3.3.4 Application to C_TS_2 Kawai and Suda measured fatigue failure data for a T800/2500 carbon/epoxy material system. Experiments were executed on UD specimens at off-axis angles of 0, 15, 30, 45, and 90 degree for R-ratios of 0.1, 0.5, and 1 [27]. An anti-buckling guide was used to prevent the specimens from buckling under compressive loads. All specimens tested in TT conditions failed in a brittle manner. Two failure modes in TC loading condition were observed: specimens tested with off-axis angle between 0 and 15 degree failed via local fiber kinking while those tested with an off-axis angle between 30 and 90 degree failed through in-plane shear mode. Compression tests at R ¼  1 are not considered in this investigation for two reasons: first, the off-axis angles lower than or equal to 30 degree are excluded because fiber kinking is the main failure mode while the criterion does not predict this (same argument as for the material system C_TS_, Section 13.3.3.2), and second, the antibuckling guide might have lengthened the life of the samples. The material parameters for TT are calibrated using the off-axis angles of 15 and 90 degree, Fig. 13.10. The predictions regarding the remaining off-axis angles can be seen in Fig. 13.11. A good agreement with experimental data for the off-axis angles of 30 and 45 degree is obtained. The prediction for the off-axis angle of 10 degree is less satisfactory. The

Prediction of fatigue crack initiation in UD

483

R R R R

(A)

l

l l

l

l

l

(B) Fig. 13.9 Prediction of fatigue life for fatigue loading on material G_TS_1. (A) TT-LMHS dominated failure, (B) TC-LMPS dominated failure.

initial static strength is overpredicted while the predictions for the fatigue experiments are rather conservative. Note the “knee” in the predictions for the off-axis angle of 30 degree, Fig. 13.11B. This is where the failure type changes from LMPS dominated failure to LMHS dominated failure as already mentioned in Section 13.3.2. The predictions for the off-axis angle of 45 degree also show a slight overestimation of the static strength. Apart from that, the high-cycle fatigue life for R ¼ 0.1 and R ¼ 0.5 is good to reasonable. Because the off-axis angle is close to 0, the discrepancy between the predictions and experiments at the off-axis angle of 10 degree could be explained

484

Fatigue Life Prediction of Composites and Composite Structures

(A)

(B) Fig. 13.10 Result of calibration of material parameters for loading on material C_TS_2. (A) TT-LMPS dominated failure, (B) TT-LMHS dominated failure.

by a combined fiber and matrix failure. This is unfortunately not explicitly mentioned by Kawai et al. [27]. Furthermore, the photographs shown by these authors suggest that failure was initiated at the tabs, where high stress concentrations can be present. The influence of the apparent tab-induced failure on the fatigue life is unknown.

13.3.3.5 Application to C_TP_1 Jen and Lee [33] provide experimental data for TT fatigue on Carbon/PEEK UD laminates. The off-axis angles tested were 15, 30, 45, 60, 75, and 90 degree. For this thermoplastic matrix, the assumption that matrix crack initiation is closely followed by

Prediction of fatigue crack initiation in UD

485

(A)

(B)

(C) Fig. 13.11 Prediction of fatigue life for fatigue loading on material C_TS_2. (A) TT-LMPS dominated failure, (B) TT-LMPS dominated failure, (C) TT-LMHS dominated failure.

486

Fatigue Life Prediction of Composites and Composite Structures

unstable crack propagation and final specimen failure (Section 13.2) might be invalidated because its visco-plastic properties can influence the fatigue crack growth rate and time to initiation. Out of curiosity, the capabilities of the matrix crack initiation criterion are tested regardless of the previous consideration. The calibration for the material parameters is done similarly as for the previous TT TT material systems and shown in Fig. 13.12 for the parameters αTT LMPS, β LMPS and αLMHS, TT βLMHS, respectively. The predictions for the off-axis angles of 30, 45, 60, and 75 degree are shown in Fig. 13.13. A good agreement for the off-axis angle of 75 degree and

(A)

(B) Fig. 13.12 Result of calibration of material parameters for loading on material C_TP_1. (A) TT-LMPS dominated failure, (B) TT-LMHS dominated failure.

(B)

(C)

(D) 487

Fig. 13.13 Prediction of fatigue life for fatigue loading on material C_TP_1. (A) TT-LMPS dominated failure, (B) TT-LMPS dominated failure, (C) TT-LMHS dominated failure, (D) TT-LMHS dominated failure.

Prediction of fatigue crack initiation in UD

(A)

488

Fatigue Life Prediction of Composites and Composite Structures

reasonable for the off-axis angle of 60 degree is observed. For the remaining off-axis angles, Fig. 13.13A and B, the criterion overestimates the time to failure. Overall, the agreement with the experiments is not as good as for the composites with a thermoset matrix. Fig. 13.13C and D show a remarkably good agreement with experiments when the LMHS becomes the dominant failure mode for the fatigue life to initiation. These results suggest that the LMPS-driven failure type is more sensitive to plasticity than the LMHS type.

13.3.3.6 Application to G_TS_2 El Kadi and Ellyin performed a fatigue life study on UD glass/epoxy laminates at offaxis angles of 19, 45, 71, and 90 degree at stress ratios of 0.0, 0.5, and 1.0 [13]. All specimens failed in a brittle matrix-dominated mode with a crack front parallel to the fiber direction. No stiffness degradation of the specimens was observed prior to failure. Testing the 45° laminates at both a frequency of 3.3 and 0.426 Hz showed no effect of load frequency on the fatigue life. The material parameters are calibrated in the same way as the previous datasets. TT Fig. 13.14A shows the calibration for αTT LMPS and βLMPS using the 19 degree off-axis TC TT angle. Fig. 13.14B shows the calibration for αLMHS, βTT where the LMHS, and γ 90 degree off-axis angle was used. The material parameters for CC were not calibrated due to the absence of experimental data. The properties are shown in Table 13.5. The prediction for the off-axis angle of 45 degree, Fig. 13.14C, shows an excellent agreement with experiments. A slightly conservative prediction for the stress ratio R ¼  1 is noted. For the off-axis angle of 90 degree, Fig. 13.14D, the quality of the agreement is equally well. However, a better agreement for R ¼  1 is noted. The observations are in line with the observations from the other glass/epoxy material system (G_TS_1, Section 13.3.3.3) which increases the applicability of this matrix crack initiation model.

13.4

Discussion of validation results

For the queried datasets it is generally observed that the predictions by the criterion agree well with the experiments. Especially for the glass/epoxy material systems, Sections 13.3.3.3 and 13.3.3.6 the agreement between experiment and prediction is excellent. For the carbon/epoxy material systems, Sections 13.3.3.2 and 13.3.3.4, the agreement is good when the specimens are subjected to TT fatigue loading. Some discrepancies occur when TC or CC loading was applied. Several causes for these discrepancies have been posed. First and foremost, the authors note that the expansion of the initiation criterion from TT to CC and TC loading conditions is based on the assumption that similar damage phenomena could be driven by similar forces, although with different damage rates. Although some evidence supports this assumption, such as the collapse of several data points on a “Master” curve when plotting the LMPS for CC data, it has not yet been proven explicitly. Due to this uncertainty, the collapse of data on the Master curve could have been

(B)

(C)

(D) 489

Fig. 13.14 Calibration and prediction of fatigue life for fatigue loading on material G_TS_2. (A) TT-LMPS dominated failure, (B) TT-combined dominated failure, (C) TT-LMPS dominated failure, (D) TT-combined dominated failure.

Prediction of fatigue crack initiation in UD

(A)

490

Fatigue Life Prediction of Composites and Composite Structures

misinterpreted. Second, the use of the Miner’s rule to join the TT and CC phenomena to one damage state, and to predict TC life based on the TT and CC predictions, can be disputed. It is not certain that the proposed interaction between the damage phenomena can be approached linearly because the synergy between these phenomena has not been found in the literature. A third shortcoming of the proposed criterion concerns the assumption of linear elastic behavior of the microstructure. In the comparison shown for C_TP_1, Section 13.3.3.5, discrepancies with the TT fatigue data are even seen. This linear elastic assumption is therefore only valid when the matrix material fractures in a brittle nature, for example, for thermoset matrix materials. The experimental data used in this research can also be a plausible cause of discrepancy. For example, the assumption that crack initiation would be shortly followed by unstable crack propagation and final fracture of the sample, Section 13.2, allowed the use of test data from off-axis tests on UD laminates. This assumption holds for TT-loaded specimens because they usually fracture in a brittle manner with a distinct matrix damage type. TC- and CC-loaded samples exhibit other additional failure types such as fiber kinking and out-of-plane buckling. It is therefore difficult to assess whether the compressive fracture is of the correct failure type. Additionally, the effect of the influence of frictional heating or the damage in the neighboring plies on the fatigue life to initiation of the investigated plies is unknown. So far, the crack initiation framework cannot deal with these types of failure. Despite the shortcomings of the framework, the criterion provides good to excellent predictions for the largest part of the experimental data selected. The framework is built on consistent logic from observed physical damage phenomena to predict the life of TT-, CC-, and TC- loaded plies to crack initiation and the purpose and the effect of each fitting parameter on the prediction is clear. Because the fitting parameters are used to regulate physical damage rates which are different for each material, they are carefully proposed as material parameters.

13.5

Conclusion and future challenges

It is shown that a coherent prediction of individual damage phenomena can be constructed for the prediction of matrix cracking under TT, CC, and TC conditions. Three assumptions for the criterion are made. First, a linear elastic constitutive relation between fibers and matrix at the microscale is assumed. Second, damage is assumed to accumulate linearly. Finally, the S-N formulation is assumed to be of the log-log type. The resulting predictions, provided that the assumptions are valid for the material used, are in good to excellent agreement with experiments. Some uncertainty still exists for the prediction of TC and CC loading due to the lack of observations of physical damage phenomena in the existing literature and the presence of other damage phenomena such as fiber kinking or buckling. In practice, the framework can reliably predict time to matrix crack initiation for composites when the material combinations involve a carbon or glass fiber with thermoset matrix. For carbon fibers with a thermoplastic matrix, the prediction is less reliable due to the visco-plastic properties of the matrix. Other combinations have not

Prediction of fatigue crack initiation in UD

491

been queried but the authors expect equally good predictions when the fracture type is of the brittle nature and the matrix exhibits limited visco-plastic behavior. Although the current framework can readily be used for prediction of matrix crack initiation, it is not complete. Mainly the synergistic effects between the damage phenomena in TT, CC, and TC loading and the effect of plasticity on fatigue crack initiation need to be further investigated. Needed in future work is the application of the failure criterion to the prediction of matrix cracking in the layers of multidirectional laminates. For this, the local static strengths, σ Tstat and σ Cstat, as measured from the UD off-axis experiments, cannot simply be reused. This because the resistance of an individual ply in a multidirectional laminate against crack formation is affected by its thickness and the orientation of the neighboring plies (the so-called in-situ effect [38, 39]). Multiple approaches to obtain the local strength of a ply can be taken. One can, for example, obtain the strengths experimentally through a number of representative experiments for which the moment of crack initiation is recorded [40, 41] or estimate the theoretical strengths using finite fracture mechanics [39, 42], or on a semiempirical basis [43]. These are only a few possible approaches to tackle these predictions. Also, at the moment the criterion is applied to point locations inside the matrix. This gives mesh dependency on the predictions. To avoid these, an initiation zone for computation, see Ref. [7], could be used. The numerical damage initiation laws would then not only be applied to the element with the highest Failure Index, but also to the elements in its direct vicinity. Finally, the effects of voids and defects from the manufacturing process, the effect of material visco-plasticity and thermal stresses need to be more thoroughly investigated.

13.6

Sources of further information and advice

The field of predicting fatigue damage for composite materials is, at the time of writing, under rapid development. Many theorems are proposed and the field is quickly evolving. To stay up to date with developments it is necessary to regularly track new literature. Interesting fora for researchers are the 3/4-yearly International Conference on Fatigue of Composites (ICFC) and the two-yearly European and International Conference on Composite Materials (ECCM and ICCM).

Acknowledgments The work leading to this publication has been funded by the SBO project “M3Strength,” which fits in the MacroModelMat (M3) research program funded by SIM (Strategic Initiative Materials in Flanders) and VLAIO (Flemish government agency Flanders Innovation & Entrepreneurship).

References [1] M. May, R. Pullin, M. Eaton, C. Featherston, S.R. Hallett, An advanced model for initiation and propagation of damage under fatigue loading—part II: matrix cracking validation cases, Compos. Struct. 93 (9) (2011) 2350–2357.

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[2] M. May, S.R. Hallett, Damage initiation in polymer matrix composites under high-cycle fatigue loading—a question of definition or a material property? Int. J. Fatigue 87 (2016) 59–62. [3] M. Quaresimin, P.A. Carraro, Damage initiation and evolution in glass/epoxy tubes subjected to combined tension-torsion fatigue loading, Int. J. Fatigue 63 (2014) (2014) 25–35. [4] M. May, S.R. Hallett, An assessment of through-thickness shear tests for initiation of fatigue failure, Compos. Part A Appl. Sci. Manuf. 41 (11) (2010) 1570–1578. [5] T.K. O’Brien, A.D. Chawan, R. Krueger, I.L. Paris, Transverse tension fatigue life characterization through flexure testing of composite materials, Int. J. Fatigue 24 (2–4) (2002) 127–145. [6] W. Fricke, A. M€uller-Schmerl, Consideration of crack propagation behaviour in the design of cyclic loaded structures, in: Eur. Struct. Integr. Soc., 23, 1999, pp. 163–172. [7] M. May, S.R. Hallett, An advanced model for initiation and propagation of damage under fatigue loading—part I: model formulation, Compos. Struct. 93 (9) (2011) 2340–2349. [8] S. Nojavan, D. Schesser, Q.D. Yang, An in situ fatigue-CZM for unified crack initiation and propagation in composites under cyclic loading, Compos. Struct. 146 (2016) 34–49. [9] M. Quaresimin, P.A. Carraro, L. Maragoni, Early stage damage in off-axis plies under fatigue loading, Compos. Sci. Technol. 128 (2016) 147–154. [10] P.A. Carraro, M. Quaresimin, A damage based model for crack initiation in unidirectional composites under multiaxial cyclic loading, Compos. Sci. Technol. 99 (2014) 154–163. [11] M. Quaresimin, P.A. Carraro, L. Maragoni, Influence of load ratio on the biaxial fatigue behaviour and damage evolution in glass/epoxy tubes under tension-torsion loading, Compos. Part A Appl. Sci. Manuf. 78 (2015) 294–302. [12] J. Awerbuch, H.T. Hahn, Off-Axis Fatigue of Graphite/Epoxy Composite, ASTM, 1981, pp. 243–273. STP 723. [13] H. El Kadi, F. Ellyin, Effect of stress ratio on the fatigue of unidirectional glass fibre/epoxy composite laminae, Composites 25 (10) (1994) 917–924. [14] M. Kawai, S. Yajima, A. Hachinohe, Y. Takano, Off-axis fatigue behavior of unidirectional carbon fiber-reinforced compoistes at room and high temperatures, J. Compos. Mater. 35 (7) (2001) 545–576. [15] M. Quaresimin, P.A. Carraro, L.P. Mikkelsen, N. Lucato, L. Vivian, P. Brondsted, B. F. Sorensen, J. Varna, R. Talreja, Damage evolution under cyclic multiaxial stress state: a comparative analysis between glass/epoxy laminates and tubes, Compos. Part B Eng. 61 (2014) (2014) 282–290. [16] Z. Hashin, A. Rotem, A fatigue failure criterion for fiber reinforced materials, J. Compos. Mater. 7 (4) (1973) 448–464. [17] A. D’Amore, L. Grassia, Constitutive law describing the strength degradation kinetics of fibre-reinforced composites subjected to constant amplitude cyclic loading, Mech. TimeDepend. Mater. 20 (1) (2016) 1–12. [18] A. Plumtree, G.X. Cheng, Fatigue damage parameter for off-axis unidirectional fibrereinforced composites, Int. J. Fatigue 21 (8) (1999) 849–856. [19] A. Varvani-Farahani, H. Haftchenari, M. Panbechi, An energy-based fatigue damage parameter for off-axis unidirectional FRP composites, Compos. Struct. 79 (3) (2007) 381–389. [20] A. Varvani-Farahani, A fatigue damage parameter for life assessment of off-axis unidirectional GRP composites, J. Compos. Mater. 40 (18) (2006) 1659–1670. [21] J. Petermann, A. Plumtree, A unified fatigue failure criterion for unidirectional laminates, Compos. Part A Appl. Sci. Manuf. 32 (1) (2001) 107–118.

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[22] F. Ellyin, Cyclic strain energy density as a criterion for multiaxial fatigue failure, in: Biaxial and Multiaxial Fatigue, International Conference on Biaxial/Multiaxial Fatigue and Fracture, 1989, pp. 571–583. [23] F. Ellyin, H. El-Kadi, A fatigue failure criterion for fiber reinforced composite laminae, Compos. Struct. 15 (1990) 61–74. [24] M. Kawai, N. Itoh, A failure-mode based anisomorphic constant life diagram for a unidirectional carbon/epoxy laminate under off-axis fatigue loading at room temperature, J. Compos. Mater. 48 (5) (2014) 571–592. [25] W. Lian, W. Yao, Fatigue life prediction of composite laminates by FEA simulation method, Int. J. Fatigue 32 (1) (2010) 123–133. [26] M. Kawai, Damage mechanics model for off-axis fatigue behavior of unidirectional carbon fiber-reinforced composites at room and high temperatures, in: Proceedings of the Twelfth International Conference on Composite Materials, 1999, p. 332. [27] M. Kawai, H. Suda, Effects of non-negative mean stress on the off-axis fatigue behavior of unidirectional carbon/epoxy composites at room temperature, J. Compos. Mater. 38 (10) (2004) 833–854. [28] M. Kawai, N. Honda, Off-axis fatigue behavior of a carbon/epoxy cross-ply laminate and predictions considering inelasticity and in situ strength of embedded plies, Int. J. Fatigue 30 (10–11) (2008) 1743–1755. [29] R.D.B. Sevenois, W. Van Paepegem, Fatigue damage modeling techniques for textile composites: review and comparison with unidirectional composite modeling techniques, Appl. Mech. Rev. 67 (2) (2015). 21401. [30] R.C. Alderliesten, Critical review on the assessment of fatigue and fracture in composite materials and structures, Eng. Fail. Anal. 35 (2013) 370–379. [31] M. Kawai, A. Hachinohe, K. Takumida, Y. Kawase, Off-axis fatigue behaviour and its damage mechanics modelling for unidirectional fibre-metal hybrid composite: GLARE 2, Compos. Part A Appl. Sci. Manuf. 32 (1) (2001) 13–23. [32] M. Kawai, T. Sagawa, Temperature dependence of off-axis tensile creep rupture behavior of a unidirectional carbon/epoxy laminate, Compos. Part A Appl. Sci. Manuf. 39 (3) (2008) 523–539. [33] M. Jen, C. Lee, Strength and life in thermoplastic composite laminates under static and fatigue loads. Part I: experimental, Int. J. Fatigue 20 (9) (1998) 605–615. [34] R.D.B. Sevenois, D. Garoz, F.A. Gilabert, S.W.F. Spronk, W. Van Paepegem, Microscale based prediction of matrix crack initiation in UD composite plies subjected to multiaxial fatigue for all stress ratios and load levels, Compos. Sci. Technol. 142 (2017) 124–138. [35] D. Garoz, F.A. Gilabert, R.D.B. Sevenois, S.W.F. Spronk, W. Van Paepegem, Material parameter identification of the elementary ply damage mesomodel using virtual micro-mechanical tests of a carbon fiber epoxy system, Compos. Struct. 181 (2017) 391–404. [36] P.A. Carraro, L. Maragoni, M. Quaresimin, A tool for the simulation of fatigue damage, in: 16th European Conference on Composite Materials, Seville, Spain, 22–26 June, 2014. [37] D. Garoz, F.A. Gilabert, R.D.B. Sevenois, S.W.F. Spronk, A. Rezaei, W. Van Paepegem, Definition of periodic boundary conditions in explicit dynamic simulations of micro- or meso-scale unit cells with conformal and non-conformal meshes, in: 17th European Conference on Composite Materials, Munich, Germany, 26–30 June, 2016. [38] A. Arteiro, G. Catalanotti, A.R. Melro, P. Linde, P.P. Camanho, Micro-mechanical analysis of the in situ effect in polymer composite laminates, Compos. Struct. 116 (2014) 827–840.

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[39] P.P. Camanho, C.G. Da´vila, S.T. Pinho, L. Iannucci, P. Robinson, Prediction of in situ strengths and matrix cracking in composites under transverse tension and in-plane shear, Compos. Part A Appl. Sci. Manuf. 37 (2) (2006) 165–176. [40] D.L. Flaggs, M.H. Kural, Experimental determination of the in situ transverse lamina strength in graphite/epoxy laminates, J. Compos. Mater. 16 (2) (1982) 103–116. [41] T.A. Sebaey, J. Costa, P. Maimı´, Y. Batista, N. Blanco, J.A. Mayugo, Measurement of the in situ transverse tensile strength of composite plies by means of the real time monitoring of microcracking, Compos. Part B Eng. 65 (2014) 40–46. [42] J.A. Nairn, The formation and propagation of matrix microcracks in cross-ply laminates during static loading, J. Reinf. Plast. Compos. 11 (2) (1992) 158–178. [43] N.V. Akshantala, R. Talreja, A micromechanics based model for predicting fatigue life of composite laminates, Mater. Sci. Eng. A 285 (1–2) (2000) 303–313. [44] G.D. Limited, CES EduPack Material Library Software Package. [45] H. Miyagawa, C. Sato, T. Mase, E. Drown, L.T. Drzal, K. Ikegami, Transverse elastic modulus of carbon fibers measured by Raman spectroscopy, Mater. Sci. Eng. A 412 (1–2) (2005) 88–92. https://doi.org/10.1016/j.msea.2005.08.037.

A progressive damage mechanics algorithm for life prediction of composite materials under cyclic complex stress☆

14

T.P. Philippidis, E.N. Eliopoulos University of Patras, Patras, Greece

14.1

Introduction

When complex stress fields are developed in composite structures operating under cyclic load, life prediction becomes a formidable task, especially in cases of irregular spectrum loading. The implementation of the numerical procedure for fatigue analysis consists of a number of distinct calculation modules, related to the main theme of life prediction. Some of them are purely conjectural or of a semi-empirical nature, e.g., the failure criteria, while others rely heavily on experimental data, e.g., S-N curves and constant life diagrams (CLDs). In cases of composite laminates under uniaxial loading, leading to uniform axial stress fields, the situation might be substantially simplified since almost all relevant procedures could be implemented by experiment. On the other hand, for complex stress states, the laminated material is considered as being a homogeneous orthotropic medium and its experimental characterization, i.e., static and fatigue strength, is performed for both material principal directions and in-plane shear. The relevant stress parameters taken into account when comparing with strength in the failure criteria are the stress resultants (Nx, Ny, Ns), as defined in classical plate theory. Such a laminate approach is a straightforward one, in predicting fatigue strength under plane stress conditions, avoiding the consideration of damage modelling, interaction effects between the plies and stress redistribution. The experimental data set required to implement the procedure, to cover variable amplitude (VA) loading for structures such as wind turbine rotor blades, consists of a number of S-N curves at various values of the stress ratio, R. Not less than three values are taken, usually R ¼ 10, 1 and 0.1. These characteristics must be derived for both principal material directions and in-plane shear, resulting in a total of at least seven S-N curves by assuming that in-plane shear fatigue strength is independent of stress ratio, R. The experimentally defined fatigue property set is unique for each laminate. The approach was implemented by Philippidis and Vassilopoulos [1–4] for a glass/polyester multidirectional ☆

This chapter is a reprint of the chapter originally published in the first edition of Fatigue Life Prediction of Composites and Composite Structures.

Fatigue Life Prediction of Composites and Composite Structures. https://doi.org/10.1016/B978-0-08-102575-8.00014-0 © 2020 Elsevier Ltd. All rights reserved.

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Fatigue Life Prediction of Composites and Composite Structures

laminate of [0/45] stacking sequence and was shown to yield satisfactory predictions for fatigue strength under complex stress conditions for both constant amplitude (CA) and VA loading. When the number of different stacking sequences in a structural element is limited, the laminate approach is a reliable alternative to a life prediction task. However, with large composite structures, composed of different components of varying structural details, numerous laminates of different stacking sequences are usually in order. The huge amount of test effort to characterize fatigue strength of all different layups prevents one from implementing a laminate approach methodology. In these cases, a ply-to-laminate approach, in which experimental characterization is performed at the basic constitutive ply level and then fatigue strength of whatever laminate configuration is numerically derived, is certainly a more effective procedure. Nevertheless, with such an option a number of additional calculation modules need to be developed, requiring both theoretical and experimental implementation. These are related to how damage is modelled in each lamina, the implications in local stress fields, stress redistribution in neighbouring plies, and finally, how damage propagates as a function of loading cycles. Relatively few works have been published on the subject, most of them in the last 15 years. Based on the internal state variable approach of Lee et al. [5] to describe stiffness degradation of a material element due to distributed damage, Coats and Harris [6] and Lo et al. [7] presented the first contributions in the field. Input data for damage progression were derived experimentally and consisted of crack surface area and crack density measurements by means of edge replicas and X-rays. Model predictions were possible only for tension-tension loading, while experimental verification so far has been provided only for stiffness degradation as a function of the applied number of cycles. The approach followed by Harris and co-workers is of the “ply-to-laminate” type in which all constitutive formulation takes place at the ply level. Prediction of life, strength or stiffness for a laminate of any stacking sequence, composed of the building ply, is in general possible. One of the most complete works of that type of approach was published by Shokrieh and Lessard [8, 9]. They have developed a method that could be used as a design tool, predicting life, residual stiffness and strength of a laminate based on ply properties. Linear stress analysis was performed, although nonlinear effects for the shear stresses were included in their failure criteria. Van Paepegem and Degrieck [10–13] and Van Paepegem et al. [14, 15] have developed a stiffness degradation-based damage mechanics model, using material properties of a cross-ply laminate and not of the UD ply. So, the model does not predict failure modes of the ply but the macroscopic failure of the cross-ply laminate. It includes a number of parameters, fitted by experiments on a specific load case (single side displacement-controlled bending) which probably depend on the stacking sequence and the load case. Sihn and Park [16] have presented an integrated design module for predicting strength and life of composite structures. Their analysis was based on micromechanics of failure by considering separately the composite constituents. Viscoelastic behaviour of the polymeric matrix was also taken into account. No experimental evidence on the validity of their method was presented. Although this type of approach, based

A progressive damage mechanics algorithm

497

on micromechanics, seems promising for the future, it also has serious disadvantages related especially to the mechanical characterization of fibre and matrix interaction and the description of damage evolution laws. Further, from an engineering point of view, composite material properties are certified at the ply level and thus before a life prediction method based on micromechanics could be used, wide agreement on characterization procedures and test methods should be sought. In this chapter, a continuum damage mechanics method is implemented in a ply-tolaminate life prediction scheme for composite laminates under cyclic CA or VA loading. According to theoretical foundations of distributed damage, e.g., as in Lee et al. [5], Ladeveze [17], Renard et al. [18] and Maire and Chaboche [19], instead of considering the geometric description of a type of defect induced by local failure, a set of appropriate stiffness degradation rules is applied, resulting in a modified stiffness tensor, i.e., an equivalent, homogeneous, continuum description, such that either the resulting strain field or the strain energy density under the same load is similar to that of the damaged medium. This effective medium description requires, besides sudden stiffness degradation due to failure onset driven by the stress at a point, also strength and stiffness degradation of a progressive character due to cycling, expressed as a function of the number of load cycles, n. Hence, residual strength and stiffness after cycling become of importance for this progressive damage mechanics approach and certainly require great experimental effort, besides efficient modelling, to cover the various loading conditions, e.g., tension-tension (T-T), tension-compression (T-C), etc., at various stress ratio values and material principal directions. To assess conditions for incipient failure in a specific mode, compatible with certain defect types and their respective stiffness degradation strategies, appropriate failure criteria considering separately the various failure modes, such as those introduced by Puck and Sch€ urmann [20, 21], Puck et al. [22] or Chang and Lessard [23], were implemented in the numerical procedure. The material model consists also of the detailed description of fatigue strength in each principal material direction and in-plane shear, always for the basic building ply, for several R values to ease the implementation of CLD formulations. A detailed step-by-step load simulation of each cycle is foreseen in the realization of the algorithm; however, in practice, albeit more accurate, this could be extremely time-consuming, especially when the routine is implemented in finite element formulations. Alternatively, calculations are performed in steps for a complete cycle and then after a block, Δn, of cycles again in steps for a detailed complete cycle and so forth. The size of cycle jump is defined by the user. Non-linear response of the unidirectional (UD) ply, especially under static in-plane shear and normal loading transverse to the fibres, is also taken into account, introducing appropriate models derived by fitting experimental data. In the numerical analysis, nonlinearity is modelled by implementing an incremental stress-strain constitutive law. An extensive comparison of life prediction numerical results with experimental data from constant (CA) or variable amplitude (VA) tensile cyclic testing of a [45]S plate and loading at various R-ratios of a multidirectional (MD) glass/epoxy laminate [(45/0)4/45]T was presented. Coupons cut from the MD plate were loaded either on-axis, where fibre-dominated failure modes were observed, or off-axis in

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various directions where matrix damage also made a significant contribution to the observed failure. Predictions of tensile residual strength after cyclic loading of [45]S coupons were also compared with test results.

14.2

Constitutive laws

The progressive damage simulator for life prediction under cyclic complex stress presented in this work was initially devised for glass/epoxy composites used in the wind turbine rotor blade industry. It relies on material data from a huge experimental effort in the frame of an EC-funded research project that resulted in a comprehensive material property database with test results from static, cyclic and residual strength experiments under axial and multiaxial loading conditions. All data is free for download from the official Optimat Blades site (http://www.wmc.eu/optimatblades.php) along with the relevant reports and publications.

14.2.1 Ply response under quasi-static monotonic loading The basic building block of all laminates considered is the UD ply, hereafter called OB_UD glass/epoxy. Besides information on mechanical property characterization that was reported in the OptiDAT database as indicated in the above, most of the data were also published by Philippidis and Antoniou [24]. In-plane mechanical properties of the UD ply were obtained through an extensive experimental program consisting of static tests, both parallel and transverse to the fibres and also in shear. The unique specimen geometry used in the characterization procedure has replaced the numerous ISO geometry coupons in all kinds of tests, static, fatigue and residual strength. Experimental data compare very well with those produced with ISO specimens, except in the case of compression along the fibres where the so-called OB-coupon suffers from buckling (ISO strength is adopted). Shear properties were still obtained through the ISO procedure—see also Philippidis and Assimakopoulou [25]. UD material performs linearly in the fibre direction as expected; however, transversely to the fibres, mainly in compression and under in-plane shear, the material behaviour is highly non-linear. To take into account this observation, incremental stress-strain theory is implemented, retaining the validity of the generalized Hooke law for each individual interval: dσ 1 ¼

dσ 2 ¼

E1 ν12 E2t dε + dε2 E2t 2 1 E2t 1  ν12 1  ν212 E1 E1 ν12 E2t E2t dε + dε2 E2t 2 1 E2t 1  ν12 1  ν212 E1 E1

dσ 6 ¼ dε6

(14.1)

A progressive damage mechanics algorithm

499

In the above equations, E1 and ν12 were considered constant throughout the static loading up to failure, while the tangential elastic moduli were given by the nonlinear constitutive relation introduced by Richard and Blacklock [26]: σi ¼ 

E ε
oi i ni  1 , i ¼ 2,6 Eoi εi ni 1+ σ oi

(14.2)

Eoi, ni and σ oi are model parameters defined by fitting the experimental data. Tangential elastic moduli were derived using the following relations: 
n2  1 + 1 n2 dσ 2 σ2 ¼ Eo2 1  E2t ¼ dε2 σ o2 
n6  1 dσ σ n6 Gt ¼ ¼ Go12 1  dε σ o6 1

(14.3)

The parameters Eo2, σ o2 and n2 were found to be different in tension and compression. Numerical values for all the above constants are summarized in Table 14.1. The predicted stress-strain curves compare favourably to the experimental data as seen in Fig. 14.1 and Fig. 14.2. Mean values of the ply in-plane failure stresses are given in Table 14.2. The respective strengths in the fibre direction, transversely and in-plane shear are denoted by X, Y and S. Average ply thickness from measurements on uniaxial UD [0] coupons was found to be 0.94 mm, while when stitched fabric [45] was used for multi-directional lay-ups, ply thickness was taken to be 0.33 mm. Therefore, attention should be paid to the definition of the MD lay-up, [(45/0)4/45]T, in which the thickness of the [0] ply is about three times that of the [45] layers.

14.2.2 Loading-unloading-reloading (L-U-R) Engineering elastic constants appearing in the constitutive relations, Eq. (14.1), are valid for monotonic loading conditions. Upon unloading, the stiffness changes and must be again defined experimentally. It was further observed that stiffness decreases upon repeated L-U-R cycles, depending on the stress level previously reached. As compiled by Philippidis et al. [27], the stiffness reduction is more severe for matrix-dominated response, e.g., in-plane shear and transverse loading to the fibres: Table 14.1 Elastic constants (in MPa), OB_UD glass/epoxy (E1 ¼ 37,950; ν12 ¼0.28)

) E(T 2t (C) E2t G12t

Eoi

σ oi

ni

15,035 15,262 5,000

75 188 67

3 2.18 1.3

60

50

30

6

[MPa]

40

20

10

Exp. Eq. 14.2

0 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

6

Fig. 14.1 In-plane shear stress-strain behaviour of OB_UD glass/epoxy.

60 Exp. Eq. 14.2

–0.025

–0.020

30

–0.015

–0.010

0 0.000

–0.005

0.005

[MPa]

–30

2

–60 –90 –120 –150 –180

2

Fig. 14.2 Transverse tension-compression response of OB_UD glass/epoxy.

Table 14.2 Strength values (in MPa), OB_UD glass/epoxy XT

XC

YT

YC

S

776

686

54

165

80

A progressive damage mechanics algorithm –0.018

–0.015

–0.012

501 –0.009

–0.006

–0.003

0 –20 –40

[MPa]

–60

2

–80

–100 –120 –140 –160 2

Fig. 14.3 Typical stress-strain cycles in static L-U-R compression transverse to the fibres.

see Fig. 14.3. In fact, no stiffness reduction due to L-U-R cycles was foreseen under loading parallel to the fibres. This type of stiffness degradation, although probably due to micro-cracking of the polymeric matrix possibly in the interface region with the fibres, and individual fibre breaks, was considered as a constitutive tensor property of the lamina that was derived by means of dedicated experiments. Strain was recorded during the L-U-R tests using strain gauges. The elastic modulus was determined as the slope of the linear regression model of each stress-strain loop. The values from a test were normalized with respect to the modulus of the first cycle and plotted against the normalized (with respect to the maximum stress of each test) stress level: see Fig. 14.4. The stiffness degradation models were determined with nonlinear regression applied on the normalized stiffness-stress data from all tests and given by:
 E2t σ 2 Gmax b2 ¼ 1  ð 1  a2 Þ Eo 2 YT  σ 6 b6 G12t ¼ 1  ð1  a6 Þ Gmax Go12 S

(14.4)

The global maximum stress reached during cycling is denoted by σ iGmax, while YT and S stand for the tensile strength transversely to the fibres and the in-plane shear strength respectively. Since values of Eo2 or Go12 presented slight variations for the different

502

Fatigue Life Prediction of Composites and Composite Structures 1

0.9

E2t /Eo2

0.8

0.7

GEV213-R0390-0109 GEV213-R0390-0279 GEV213-R0390-0287 GEV213-R0390-0106

0.6

GEV213-R0390-0289 GEV213-R0390-0293 Eq. 11.4

0.5 0

0.1

0.2

0.3

0.4

0.5

2/ 2max

0.6

0.7

0.8

0.9

1

Fig. 14.4 Modulus degradation transverse to the fibres due to compressive L-U-R cycles.

coupon tests, the respective values from Table 14.1 were implemented along with Eq. (14.4). The parameters a2 and b2 were found to be different in tension and compression. Numerical values for all the above constants are summarized in Table 14.3. When the first equation of Eqs (14.4 is used to determine the compressive elastic modulus transverse to the fibres the tensile strength, YT, should be replaced by the corresponding compressive strengths, YC. The elastic modulus as determined by Eq. (14.4) is implemented for the reloading branch. Considering a slightly greater unloading modulus, e.g., multiplying by a factor greater than unity, enables the model to take into account the permanent strains as well. The inclusion of this type of stiffness reduction in the constitutive material model is expected to affect numerical predictions, especially in cases of VA loading.

Table 14.3 Parameter values for L-U-R stiffness degradation models, Eq. (14.4)

) E(T 2t (C) E2t G12t

ai

bi

0.88 0.65 0.38

1.60 2.77 1.40

A progressive damage mechanics algorithm

503

14.2.3 Stiffness degradation In-plane stiffness of the lamina degrades for several reasons, e.g., sudden stiffness reduction due to some kind of failure occurrence or progressive stiffness reduction due to cycling. In general, the latter is nonlinear and several formulations have been proposed in the literature to describe it. As presented by Philippidis et al. [27], during the fatigue tests, load-displacement data were recorded periodically and were transformed to respective stress-strain data, e.g., see Fig. 14.5 where experimental data from tensile cyclic loading (T-T) of a [45]S coupon at a stress ratio R ¼ 0.1 are shown. The stiffness of the coupon at each cycle was determined as the slope of the linear regression model of the respective stress-strain loop. These stiffness values were normalized with respect to the stiffness of the first cycle and plotted with the normalized number of cycles with respect to the number of cycles at failure. The stiffness degradation models were determined with nonlinear regression applied to the normalized stiffness-cycle number data from all coupon tests, at various loading levels. An example for the degradation of the in-plane shear modulus G12 of the OB_UD glass/epoxy ply is shown in Fig. 14.6. The experimental data were derived from T-T cyclic tests at R ¼ 0.1. In the present FADAS implementation the regression models depend only on the fatigue life fraction, i.e., the ratio of the applied cycles versus the nominal fatigue life at the current stress level. In this way, the modulus degradation depends implicitly also on the stress ratio, R, and the maximum applied cyclic stress, σ max. The following functional forms were fitted to the experimental data: 60

50

30

x

[MPa]

40

20

10 GEV208-l1000-0045 0

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

x

Fig. 14.5 Stress-strain loops under CA fatigue of an ISO 14129 [45]S coupon.

0.008

504

Fatigue Life Prediction of Composites and Composite Structures 1.05 1 0.95

G12(n) /G12(1)

0.9 0.85 0.8 0.75 0.7 R = 0.1 level 2 R = 0.1 level 3 Eq. 11.5

0.65 0.6 0.55 0

0.1

0.2

0.3

0.4

0.5 n/N

0.6

0.7

0.8

0.9

1

Fig. 14.6 In-plane shear modulus degradation data for the OB_UD glass/epoxy. Table 14.4 Parameter values for the cyclic stiffness degradation models, Eq. (14.5)

) E(T 2 E(C) 2 G12

ci

di

0.75 0.95 0.68

3.17 0.62 1.65

 n d2 E2 ðnÞ ¼ 1  ð1  c 2 Þ E2 ð1Þ N  n d6 G12 ðnÞ ¼ 1  ð1  c 6 Þ G12 ð1Þ N

(14.5)

The parameters c2 and d2 were found to be different in tension and compression. Numerical values for all the above constants are presented in Table 14.4. Since the modulus values at the first cycle from the different coupons tested are different, E2(1) and G12(1) in Eq. (14.5) were substituted by the respective reloading stiffness values corresponding to σ Gmax, as given by Eq. (14.4). Progressive stiffness degradation for any cyclic loading type parallel to the fibres was not important and thus it was neglected in the numerical model.

14.2.3.1 Pre-failure material models In case no failure was detected, the simulated ply response and especially the description of stiffness evolution for VA cyclic loading were expressed by combining the constitutive relations presented in the above. Each stress tensor component at the kth loading

A progressive damage mechanics algorithm

505

step is examined to see whether it corresponds to loading, i.e., jσ i(k)jj σ i(k 1)j, i¼ 1, 2, 6 or else to unloading. In the former case, if σ i(k) is higher than the global maximum stress, σ iGmax, or lower than the global minimum stress, σ iGmin the initial material behaviour under quasi-static loading, presented in Section 14.2.1, is assumed, that is, a constant modulus E1 and Poisson ratio v12 while E2 and G12 are functions of σ 2(k) and σ 6(k) as expressed by Eq. (14.3). If σ i(k) lies between the global minimum and maximum stress, the reload elastic properties are used, Eq. (14.4), calculated at the global maximum or minimum stresses, degraded according to the stiffness degradation models due to cycling, i.e., Eq. (14.5). In the case of unloading, elastic properties slightly higher than in the case of reloading are used to introduce an increasing permanent strain due to cyclic loading. In the routine this is realized by multiplying by 1.00002 the reloading stiffness values for E2 and G12. The above is illustrated in Fig. 14.7: l

l

l

A–B: Initial loading. Stress is always greater than its previous global maximum value, so the non-linear material behaviour under quasistatic loading is used. B–C–D: Stress cycling under CA or VA. Stress values remain between their global minimum and maximum values, 0 and σ iGmax respectively, so the reload and unload elastic properties are used, gradually degrading with increasing number of cycles. D–E: Stress becomes greater than its previous global maximum value σ iGmax, so the initial material behaviour is assumed, etc.

As seen in Fig. 14.7, the behaviour of the material under cyclic stress is assumed to be linear elastic, its stiffness depending on the global maximum stresses reached so far

E

B

D

i

iGmax

C

A i

Fig. 14.7 Pre-failure material model for the OB_UD glass/epoxy.

506

Fatigue Life Prediction of Composites and Composite Structures

and also on the applied number of cycles. However, when the applied stress level exceeds previous maxima, nonlinear response is again recalled.

14.2.3.2 Post-failure material models Upon failure onset in some loading step, the stiffness degrades, depending on the failure mode observed, and the changes apply for the next loading step. In fibre failure (FF) under either tensile or compressive stresses the three engineering elastic constants, E1, E2 and G12, drop to zero. If matrix damage modes occur, also called inter-fibre failure (IFF) (see Section 14.3 for a detailed description), only E2 and G12 drop to zero. After fibre failure (FF), the unload behaviour for all three stress tensor components remains as in the virgin material, i.e., degraded reload values for E2 and G12 multiplied by the appropriate factors mentioned earlier to take into account residual strains. If reloading occurs before any stress tensor component has changed sign, the respective modulus, i.e., E1, E2 or G12, drops to zero. If the stress has changed sign once, the corresponding modulus remains always at zero. The above is illustrated in Fig. 14.8: l

l

l

l

l

A: Stress level at which FF mode was detected. A–B: If σ i(k) stands for loading, the corresponding engineering elastic constant drops to zero. B–C, C–D, E–F: Unloading using the unloading elastic properties. C–E: If reloading is encountered before stress has changed sign, the elastic property drop to zero. D, F: Following unloading, a stress tensor component changes sign. The corresponding elastic property drops and remains henceforth at zero.

i

A

C

D

E

F i

Fig. 14.8 Post-FF material model for the OB_UD glass/epoxy.

B

A progressive damage mechanics algorithm

507

B

D

E

2,

6

A

C

2,

6

Fig. 14.9 Post-IFF material model for the OB_UD glass/epoxy.

In case of matrix failure (IFF damage modes), E1 remains unaffected and only the normal stress transverse to the fibres and the in-plane shear component are taken into account in the stiffness degradation model. Once IFF is detected, both unload and reload properties remain as for the virgin material models presented in Sections 14.2.2 and 14.2.3, unless IFF is detected again; both engineering elastic constants E2 and G12 drop to zero. If only the value of the normal stress transverse to the fibres σ 2 or the in-plane shear stress σ 6 exceeds its value for which IFF has been predicted last time, the respective elastic property (E2 or G12) drops to zero and the process is continued. With respect to Fig. 14.9, illustrating the above, the following characteristics can be noted. l

l

l

l

A: Stress level at which IFF was first detected. A–B: Loading is continued; both E2 and G12 drop to zero. B–C–D: No IFF is predicted again. The stress component remains lower than its value at failure. The reload and unload elastic properties of the virgin material are used, gradually degraded with the number of cycles. D–E: IFF is predicted once more or stress σ 2 or σ 6 becomes equal to or greater than its value when IFF was predicted. The corresponding elastic property drops to zero.

14.3

Failure onset conditions

Predicting laminate strength under cyclic complex stress states is conceptually different from predicting failure onset under monotonic loading. The latter is a first ply failure (FPF) approach directly implementing ply stresses in a suitable limit condition and finally suggesting the layer with the maximum risk of failure. On the other hand,

508

Fatigue Life Prediction of Composites and Composite Structures

fatigue strength prediction is a last ply failure (LPF) procedure involving modelling of progressive damage, e.g., consideration of strength and stiffness degradation due to cyclic stresses when a ply-to-laminate approach is implemented, as by Shokrieh and Lessard [8, 9]. When the laminate is considered macroscopically as the anisotropic material (and not the ply), its static and fatigue strength need to be adequately characterized experimentally; then fatigue strength criteria such as those developed for example by Fawaz and Ellyin [28] or Philippidis and Vassilopoulos [29] can be used. The methodology used in FADAS is of the ply-to-laminate type with progressive damage modelling. In such an approach it is sufficient to use a static limit condition at the ply level where, however, material strength parameters are replaced by the corresponding residual strength values, which are in general functions of the number of cycles and the type of loading. For the cases studied in this work, numerical results were derived by implementing the Puck criterion in the FADAS routine. Based on the concepts first introduced by Hashin [30] for different damage mechanisms in composite materials and the Mohr-Coulomb hypothesis for brittle materials that fracture is exclusively triggered by stresses acting on the fracture plane, the criterion of Puck and Sch€ urmann [20, 21] describes several failure modes using different equations. For 2D plane stress analysis five types of damage were assumed, two related to fibre fracture (FF), one in tension and another in compression, and the other three concerning matrix failure (inter-fibre fracture or IFF). Under tension (σ 2  0), cracks open transverse to the applied normal stresses (θfp ¼ 0 degree, the angle subtended by the fracture plane and the vertical one to the layer plane), described as mode A. In compression, either closed cracks are formed transverse to the applied normal stresses (θfp ¼ 0 degree), described as mode B, or an oblique rupture occurs with the fracture plane forming an angle θfp between 45 and 55 degree, described as mode C. Fibre and matrix failure effort or stress exposure factor, fE(FF) and fE(IFF) respectively, can be calculated as follows. For fibre failure (FF) under tensile loading: fETðFFÞ ¼


  1 E1 σ1 + νf 12 mσf  ν12 σ 2  1if ½…  0 XT Ef 1

(14.6)

Respectively, for fibre failure (FF) under compressive loading: fECðFFÞ ¼


   1  E1  σ + ν m  ν 1 f 12 σf 12 σ 2   1 if ½… < 0  XC Ef 1

(14.7)

For Mode A, IFF condition, for which the fracture plane is vertical to the layer plane (θfp ¼ 0 degree): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
 σ 2
6 σ2 6 ð +Þ YT ð +Þ σ 2

+ p?k fEAðIFFÞ ¼ + 0:9fEðFFÞ + 1  p?k S S YT S  1if σ 2  0 (14.8)

A progressive damage mechanics algorithm

509

For transverse compression and moderate in-plane shear, Mode B, IFF condition, for which again the fracture plane is vertical to the layer plane (θfp ¼ 0 degree): fEBðIFFÞ

1 ¼ S

"rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #  2

6 ð Þ ð Þ 2 ðσ 6 Þ + p?k σ 2 + p?k σ 2 + 0:9fEðFFÞ (14.9)

  σ 2  R A  1 if σ 2 < 0and0     ?? σ6 σ 6c

Finally, for the explosive mode C, IFF condition, the fracture plane forms an oblique angle with the vertical to the ply plane (θfp 6¼ 0 degree): 20

3
2


6 σ Yc 2 C 4 5 @ A   + fEðIFFÞ ¼ + 0:9fEðFFÞ ðÞ Yc ðσ 2 Þ 2 1 + p?? S   σ 6  jσ 6c j  1 if σ 2 < 0and0     A σ 2 R?? σ6

12

(14.10)

Ef1 and vf12 are the elastic modulus and the Poisson ratio of the fibres. The term mσf accounts for a stress magnification effect caused by the different moduli of fibres and matrix which leads to an uneven distribution of the transverse stress σ 2 from a micromechanical point of view; in the fibres it is slightly higher than in the matrix. For the variety of parameters implemented in the above relations, guidelines and typical values were explicitly presented by Puck et al. [22]. The values used in the present version are given by: ð +Þ

ðÞ

Ef 1 ¼ 72:45 GPa, vf 12 ¼ 0:22, mσf ¼ 1:3, p?k ¼ 0:3, p?k ¼ 0:25 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Yc ðÞ ðÞ Yc ðÞ  1 + 2p?k  1, σ 6c ¼ S 1 + 2 p?? , RA?? ¼  p?k ¼ ð Þ 2 S 2 1+p

(14.11)

?k

When the criterion is used for cyclic stresses, the lamina strength values XT, XC, YT, YC and S given in Table 14.2 must be replaced by the corresponding residual strength values: see Section 14.4.

14.4

Strength degradation due to cyclic loading

Static strength degradation or residual strength after fatigue in composites has been intensively investigated during the last 30 years. Numerous research groups have developed a variety of models; an appraisal of their effectiveness has been recently presented by Philippidis and Passipoularidis [31]. The majority of the work concerns modelling of residual tensile strength in the laminate level, under axial loading and in

510

Fatigue Life Prediction of Composites and Composite Structures

most cases at a single R-ratio, usually in the tension-tension region. Very limited are the experimental data sets concerning complex stress states. The model introduced by Shokrieh and Lessard [32, 33] is perhaps one of the first phenomenological approaches to examine damage evolution and failure due to multiaxial fatigue in a composite laminate in terms of the strength and stiffness degradation of the building ply. On the other hand, only a few tests under limited loading conditions have been performed for model evaluation, without any investigation on more complex issues, e.g., residual tensile strength after compression-compression fatigue. In general, the lack of detailed experimental data, under various fatigue conditions and for a single material, has restricted the study of residual strength-based models to limited loading conditions and to specific lay-ups. To link existing knowledge and promote the modelling of static strength degradation due to stress cycling, a comprehensive experimental program was undertaken in the frame of the European research project Optimat Blades in order to study, amongst other things, the static strength, fatigue life and residual strength behaviour of wind turbine rotor blade materials. In particular, these properties have been studied in the symmetry directions of an orthotropic UD glass/epoxy material (OB_UD), i.e., along and transverse to the fibres and under in-plane shear. Regarding residual strength, both the tensile and compressive residual strength were studied in detail—for the first time—under different stress ratios of fatigue loading, for all the principal directions of the specific lamina, using a single coupon geometry for tensile and compressive tests under either static or fatigue loads, in an effort to keep results unbiased by different coupon geometries, use of anti-buckling devices, etc. The investigation considered stress ratios in the range of those experienced at different points of the blades during operation, i.e., tension-tension and reversed loading as well as compression-compression fatigue. From the processing of the experimental data, the main conclusions of Philippidis and Passipoularidis [34] were derived and formulated as guidelines for further development. First, the residual strength in both principal material directions is not affected when cyclic stress of the opposite sign is applied, i.e., tensile strength is not reduced under purely compressive cycles and vice versa. A similar trend was also observed by Nijssen [35] from tests in the fibre-dominated direction of a [(45/0)4/45]T laminate, made also of the OB_UD glass/epoxy, under various loading conditions (R-ratios). The tensile and the in-plane shear static strength experienced degradation of up to 40% when tested at a nominal life fraction of 80%. The compressive residual strength, on the other hand, did not show significant degradation in all types of loading and material directions. Concerning the many theoretical models considered in this investigation, only a few corroborated satisfactorily with the majority of the experimental data [36]. It was demonstrated in a clear manner that the complexity of a model is not related to the accuracy of its predictions. In addition, it was also proved by Passipoularidis and Philippidis [37] that life prediction results under VA loading are not very sensitive to the particular residual strength model, of those few validated, of course, that is used as damage metric.

A progressive damage mechanics algorithm

511

Therefore, two models are used herein to describe the phenomenon. For the modelling of tensile residual strength along the principal material directions, under T-T or T-C cyclic loading, as well as under in-plane shear, the linear degradation model proposed by Broutman and Sahu [38] was implemented. Besides being the simplest one available, it requires no residual strength testing while at the same time it has been proven by Philippidis and Passipoularidis [31] to produce always safe residual strength predictions under various stress conditions and lay-ups. It is described respectively by the following equations:
 n XTr ¼ XT  ðXT  σ 1 max Þ N1
 n YTr ¼ YT  ðYT  σ 2 max Þ N2
 n Xr ¼ S  ðS  σ 6 max Þ N6

(14.12)

XTr and YTr are the tensile residual strength parallel and transverse to the fibres respectively, while Sr is the residual shear strength. σ 1max, σ 2max and σ 6max are the maximum cyclic stresses applied for n cycles, and Ni, i ¼ 1, 2, 6, are the corresponding fatigue lives at the specific stress level. Although the three relations of Eq. (14.12) seem to depend only on the applied stress level, they also depend on the stress ratio through the fatigue life Ni obtained for a specific stress ratio through the constant life diagram (CLD) used: see Section 14.5. The model can be implemented once the static strength and fatigue S-N curves at arbitrary R-ratios are known. The compressive strength, both parallel and transverse to the fibres, has been shown not to degrade significantly due to fatigue, especially when the specimens were subjected to tensile cyclic stress. Nevertheless, in modelling the compressive residual strength under C-C or T-C cyclic loading, a degradation equation simulating constant strength throughout the life with a sudden drop near failure (sudden death) of the following form was implemented:
   n k

Xcr ¼ Xc  Xc  σ 1 min  N
1   n k

Xcr ¼ Xc  Yc  σ 2 min  N2

(14.13)

A summary of experimental evidence on the effectiveness of Eqs (14.12 and (14.13 in modelling the residual strength behaviour in the principal material directions of the OB_UD glass/epoxy is presented in Figs. 14.10 to 14.15. In each of these figures, static strength data, tensile, compressive or in-plane shear, are plotted on the ordinate axis which was moved to N ¼ 10 or 100 for increased resolution. The corresponding S-N curve is also shown as a dashed line, where appropriate, e.g., under T-T loading at R ¼ 0.1 it appears in the picture for the residual tensile strength whereas for compressive R ¼ 10 loading it is plotted along with the residual compressive strength.

512

Fatigue Life Prediction of Composites and Composite Structures 900 800

1max

[MPa]

700 600 500 400 300 200 1.E+01

1.E+02

1.E+03

1.E+04 N

1.E+05

1.E+06

1.E+07

650 600

abs (

1min )

[MPa]

550 500 450 400 350 300 250 200 1.E+01

1.E+02

1.E+03

N

1.E+04

1.E+05

1.E+06

Fig. 14.10 Strength degradation of OB_UD in the fibre direction under R ¼ 0.1 (tensile residual strength: top; compressive residual strength: bottom).

The solid lines appearing in Figs. 14.10 to 14.15 are theoretical predictions from Eq. (14.12) or Eq. (14.13) for the residual compressive strength. The exponent k for the latter case was set equal to 50. Different data point sets were also displayed that correspond to three different stress levels, while for each set the data correspond to coupons cycled up to 20%, 50% or 80% of their nominal life. Details of all these tests were reported by Philippidis and Passipoularidis [34].

A progressive damage mechanics algorithm

513

900 800 700

[MPa]

500

1max

600

400 300 200 100 0 1.E+01

1.E+02

1.E+03

1.E+04 N

1.E+05

1.E+06

1.E+07

700

abs (

tmin )

[MPa]

600 500 400 300 200 100 0 1.E+01

1.E+02

1.E+03

N

1.E+04

1.E+05

1.E+06

Fig. 14.11 Strength degradation of OB_UD in the fibre direction under R ¼1 (tensile residual strength: top; compressive residual strength: bottom).

Initially tests were planned for three stress ratios, both parallel and transverse to the fibres. Nevertheless, compressive tests at R ¼ 10 in the fibre direction were skipped due to the very flat S-N curve derived at this stress ratio, which made the definition of stress levels for specific fatigue lives very sensitive to even slight variations of applied load. That has also introduced uncertainty on the quality of the results and also caused many premature failures. The residual shear strength tests were performed using the ISO 14129 standard [45]S tensile coupon. For this reason only cyclic tests at R ¼ 0.1 were possible. Summarizing the above, the residual strength model in the symmetry directions of the unidirectional glass/epoxy layer, and also for in-plane shear, due to cyclic

514

Fatigue Life Prediction of Composites and Composite Structures 65 60

2max

[MPa]

55 50 45 40 35 30 25 20 1.E+02

1.E+03

1.E+04 N

1.E+05

1.E+06

1.E+03

1.E+04 N

1.E+05

1.E+06

200 180

abs (

2min)

[MPa]

160 140 120 100 80 60 40 20 0 1.E+02

Fig. 14.12 Strength degradation of OB_UD transversely to the fibres, R ¼ 0.1 (tensile residual strength: top; compressive residual strength: bottom).

loading is given by a different set of equations, depending on the value of the cyclic stress ratio, R. As is well known from the representation of constant life diagrams (CLD) in the (σ a  σ m) plane of the alternating (σ a), and mean (σ m) stress characteristics of the cyclic loading, radial lines emanating from the origin of the coordinate system correspond to stress states with constant R values. Then, with respect to Fig. 14.16, the following sets of equations are valid for purely tensile (T-T) cyclic loading:

A progressive damage mechanics algorithm

515

70 60

2max

[MPa]

50 40 30 20 10 1.E+01

1.E+02

1.E+03

1.E+02

1.E+03

N

1.E+04

1.E+05

1.E+04

1.E+05

1.E+06

180 160

[MPa]

120

2min)

100

abs (

140

80 60 40 20 1.E+01

N

1.E+06

Fig. 14.13 Strength degradation of OB_UD transversely to the fibres, R ¼1 (tensile residual strength: top; compressive residual strength: bottom).


 n XTr ¼ XT  ðXT  σ 1 max Þ , X Cr ¼ X C N1
 n YTr ¼ YT  ðYT  σ 2 max Þ , YC r ¼ Y C 0  R < 1 N2
 n Sr ¼ S  ðS  σ 6 max Þ N6

(14.14)

For negative R values, corresponding to compressive σ min and tensile σ max values where, however, the tensile stresses are of greater magnitude (T-C):

516

Fatigue Life Prediction of Composites and Composite Structures 65 60 55

[MPa]

45

2max

50

40 35 30 25 20 1.E+02

1.E+03

1.E+04 N

1.E+05

1.E+06

180 170

[MPa]

150

2min)

140

abs (

160

130 120 110 100 1.E+02

1.E+03

1.E+04 N

1.E+05

1.E+06

Fig. 14.14 Strength degradation of OB_UD transversely to the fibres, R ¼10 (tensile residual strength: top; compressive residual strength: bottom).







 n k n XTr ¼ XT  ðXT  σ 1 max Þ , XCr ¼ XC  XC  j σ 1 min j N N
 1
k 1

 n n YTr ¼ YT  ðYT  σ 2 max Þ 1R X X > > λ s s 2 2 2 2 j j i j > ωi σ Dðsi Þ + 2 ωi ωj σ Dðsi Þ σ Dðsj Þ e > : σ DT ¼ T i¼1

(17.33)

i¼1 j¼i + 1

where μD(s) and σ D(s) in Eqs 17.32 and 17.33 are the mean and standard deviation of damage in a single cycle at stress level s, which are obtained from constant amplitude loading tests. In order to calculate the time-dependent fatigue reliability, we need to assume the probability distribution of the fatigue damage DT since only the first two central moments are available. Eqs. (17.28) and (17.29) can be treated as a summation of a set of random variables. It is well known that a summation of Gaussian random variables is a Gaussian random variable. However, the distribution of the summation of non-Gaussian random variables is usually unknown. Studies for some special cases of summation of non-Gaussian random variables have been reported. Fenton [41] proposed a method to approximate the summation of a set of correlated lognormal

Probabilistic fatigue life prediction of composite materials

621

random variables as a single lognormal random variable. The method matches the mean and variance of the lognormal sum to the target random variable. It has been shown that this method is very accurate at the tail region, which is usually of the most interest for the reliability analysis. Once the distribution type of DT is known or assumed, the reliability can be directly calculated. For example, if DT follows the lognormal distribution, ln (DT) follows the normal distribution with the mean and variance determined by 8    1  > < μDT ¼ 2 ln μDT  ln μ2DT + σ 2DT 2   > : σ 2 ¼ 2 ln μ  + ln μ2 + σ 2 DT DT DT DT

(17.34)

The limit state function is defined as shown in Eq. (17.30) (or Eq. 17.31). The failure probability Pf is the damage exceedance probability, i.e., 

   μDT  ln ðψ Þ DT 1 ¼Φ Pf ¼ P ψ σ DT

(17.35)

Following the lognormal assumption of the fatigue damage, the time-dependent reliability can be expressed as   μDT  ln ðψ Þ Reliability ¼ 1  Pf ¼ 1  Φ σ DT

(17.36)

where Φ is the cumulative density function (CDF) of the standard Gaussian variable. μDT and σ DT have been determined by Eqs 17.32 and 17.33. ψ is the critical damage value determined by Eqs 17.10 and 17.11. Since the variable T is explicitly included in the mean and variance of the fatigue damage, the reliability calculated by Eqs 17.34–17.36 is time-dependent.

17.4.3 Method II: FORM approach The proposed moments matching method described above needs to assume the type of probability distribution of the accumulated fatigue damage. In order to calculate the time dependent reliability without assuming the fatigue damage distribution, another approximation method is proposed based on the first-order reliability method (FORM). The limit state function is defined in Eq. (17.30). Therefore, the probability of failure Pf is defined through a multi-dimensional integral ð

ð

Pf ¼ ⋯

fD ðD1 ðsÞ, D2 ðsÞ, …, Di ðsÞÞdD1 ðsÞdD2 ðsÞ…dDi ðsÞ

(17.37)

g0 > μ ¼ E T f ð s, T ÞD ð s Þds ¼ T f ðs, T ÞμDðsÞ ds > D T > > 0 0 > > >  ð∞  < > > > > > > > > :

σ DT ¼ Cov T ¼T

ð∞ð∞

f ðs, T ÞDðsÞds

(17.40)

f ðs1 T Þf ðs2 T Þσ Dðs1 Þ σ Dðs2 Þ eλjs1 s2 j ds1 ds2

For discrete loading, the mean value and variance of DT can be expressed as 8 ∞ X > > ¼ T f ðsi , T ÞμDðsi Þ μ > DT > > > i¼1 > > > 3 2X > ∞ < 2 2 f ð s , T Þσ i ð Þ D s i 7 6 > 7 6 i¼1 > > 2 26 7 > σ ¼ T > DT 7 6 > ∞ ∞ XX >   5 4 > λ s j j si j > > +2 f ð s , T Þf s , T σ σ e i j Dðsi Þ Dðsj Þ :

(17.41)

i¼1 j¼i + 1

For example, two-step loading is commonly used for variable loading tests under laboratory conditions. The material is first pre-cycled under stress level Sa for Ta cycles. Then the material is cycled till failure at another stress level Sb. The cycle distribution function f(s, T) can be expressed for the two-step loading as

f ðs, T Þ ¼

8 1 > > T  T a > > > > 0 < T

s ¼ Sa s ¼ Sb

a

s ¼ Sa > T > > > > T  Ta > : T > Ta s ¼ Sb T

(17.42)

For the moments matching approach, the first two central moments of the fatigue damage DT can be expressed as 8 ( T  Ta TμDðsa Þ > > > μDT ¼ > > > Ta μDðsa Þ + ðT  Ta ÞμDðsbÞ T >Ta > > > 8 > > > T 2 σ 2Dðsa Þ TT a > < > > > > ! > 2 < > > > > σ2 ¼ T2 σ2 + T  T σ 2Dðsb Þ > DT a Dðsa Þ > > > a > > > > > > > > > > : +2T ðT  T Þσ : λjsa sb j T > Ta a a Dðsa Þ σ Dðsb Þ e

(17.43)

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Fatigue Life Prediction of Composites and Composite Structures

For the FORM approach, the limit state function can be expressed as G¼

ψ  TDðsa Þ T < Ta ψ  Ta Dðsa Þ  ðT  Ta ÞDðsb Þ T > Ta

(17.44)

The time-dependent fatigue reliability can now be calculated following the same procedure described for stationary loading.

17.4.5 Time-dependent fatigue reliability and probabilistic life distribution The proposed approximation methods are simple formulations for time-dependent fatigue reliability analysis. Using these methods, the reliability at time T can be calculated. Similarly, for a given reliability level (or probability of failure), the corresponding fatigue life of the material (i.e., time T) can also be calculated. Thus, the current formulation can also be used for probabilistic fatigue life prediction. The probability of fatigue damage being larger than a critical damage amount ψ at the time instant T is equal to the probability of fatigue life being less than at the time instant T when the fatigue damage is ψ. The relationship of time-dependent failure probability and the probabilistic fatigue life distribution is shown in Fig. 17.4 schematically. Mathematically, this relation is expressed as PðDT > ψ Þt¼T  Pðt < T ÞDT ¼ψ

17.5

(17.45)

Demonstration examples

In this section, the discussed probabilistic methodologies above are demonstrated using both numerical and experimental examples. The first example is a simple numerical example under two-block loading. Parametric study is performed to investigate the effects of different approaches, and the model predictions are verified with direct Monte Carlo simulation results. The second considers continuous random external loading and demonstrates the application of the proposed methodology to a realistic situation. The third example is a validation example using measured fatigue failure data of composite laminates.

17.5.1 Numerical example 1 A numerical example is calculated and compared with direct Monte Carlo simulation to show the accuracy of the discussed moments matching approach and FORM method. Consider a two-block variable amplitude loading (S1 ¼ 666 MPa and S2 ¼ 478 MPa). The means of single cycle damage at the two stress levels are Mean(D (S1)) ¼ 1.89E  05 and Mean(D(S2)) ¼ 2.44E  06 using constant amplitude loading at each individual stress level. The standard deviations of single

Probabilistic fatigue life prediction of composite materials

625 Probabilistic life distribution

P(DT > y)t = T P(t < T )DT = y

Damage

y Damage distribution

Mean damage growth curve

T

Time

Fig. 17.4 Schematic illustration of probabilistic fatigue life distribution and time-dependent fatigue reliability.

cycle damage at the two stress levels are Std(D(S1)) ¼ 3.16E  06 and Std(D(S2)) ¼ 5.72E  07. The Monte Carlo simulation uses 106 samples at each time instant and is assumed to be the exact solution. Both the moments matching method and the FORM method are used to calculate the fatigue reliability and are compared with the direct MC simulation. Parametric studies are performed to investigate the effect of the cycle fraction at each stress level and the correlation coefficient between single cycle damage at the two stress levels. The cycle fraction effects are compared in Fig. 17.5 for four different cycle fractions of S1 with the correlation coefficient fixed at zero. The correlation effects are compared in Fig. 17.6 for four different correlation coefficients with the cycle fraction fixed at 0.5. In this numerical example, the input random variables are assumed to follow a lognormal distribution. It is shown that the approximation for the lognormal distribution is very accurate. Overall, the moments matching method and the FORM method give very good approximations.

17.5.2 Numerical example 2 A numerical example is shown here to demonstrate the applicability of the proposed method to a continuous random load spectrum. Fatigue S-N testing under constant amplitude load is shown in Fig. 17.7. The statistics of measured fatigue life are listed in Table 17.1. Suppose the material is under stationary continuous random load spectrum. Rain-flow counting is used to get the cycle distribution of the external loading. Here it is assumed to follow the lognormal distribution with a mean value of 600 MPa and a standard deviation of 30 MPa. In the numerical calculation, the

0.99

0.99 Reliability

Reliability

1

0.98 Approximation – Moments Matching

0.97

Approximation – FORM

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626

1

0.98 0.97

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0.96

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Monte Carlo Simulation 0.95

0.95 5.05

5.15

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1

1

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5

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0.96

Monte Carlo Simulation

Monte Carlo Simulation 0.95

0.95 4.6

(C)

4.75

(B)

Time (log(N))

Reliability

Reliability

(A)

5.1

4.65

4.7 Time (log(N))

4.75

4.8

4.5

(D)

4.55

4.6 4.65 Time (log(N))

4.7

Fig. 17.5 Effects of cycle distribution using the moments matching approach. Cycle distribution at the first stress level: (A) 0.2; (B) 0.4; (C) 0.6; (D) 0.8.

4.75

Fatigue Life Prediction of Composites and Composite Structures

5

0.99

0.99 Reliability

Reliability

1

0.98 0.97

Approximation – Moments Matching

0.96

Approximation – FORM

0.98 0.97

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0.96

Approximation – FORM Monte Carlo Simulation

Monte Carlo Simulation 0.95

4.7

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(A)

1

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4.8

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Monte Carlo Simulation 0.95

0.95 4.7

(C)

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1

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4.7

(B)

Reliability

Reliability

4.8

0.95

Probabilistic fatigue life prediction of composite materials

1

4.75

4.8 Time (log(N))

4.85

4.9

4.7

(D)

4.75

4.8 Time (log(N))

627

Fig. 17.6 Effects of correlation using FORM. Correlation coefficient between the two stress levels: (A) 0.0; (B) 0.4; (C) 0.6; (D) 1.0.

628

Fatigue Life Prediction of Composites and Composite Structures 2.9

Stress (log(MPa))

2.8

2.7

2.6 Experimental data Mean S–N curve 2.5 4

4.5

5

5.5

6

Fatigue life (log(N))

Fig. 17.7 Constant amplitude S-N curve data for the numerical example.

Table 17.1 Statistics of constant amplitude S-N curve data—numerical example Statistics of single cycle fatigue damage (1/N) Stress amplitude (MPa)

Mean

Std.

Distribution

478 583 666

2.44E 06 8.32E 06 1.89E 05

5.72E  07 1.79E  06 3.16E  06

Lognormal Lognormal Lognormal

continuous cycle distribution is divided into 30 equal segments. The cycle distribution is plotted in Fig. 17.8A. The prediction results using the moments matching approach, the FORM approach and the direct Monte Carlo simulation approach are plotted together in Fig. 17.8B. The results of all three methods are in very close agreement. The computational time for the moments matching approach, the FORM approach and the direct Monte Carlo simulation approach are 0.3 s, 1.4 s and 425 s, respectively. The computer has 3-GB memory and a 2.4-GHz dual-core processor. The operating system is Windows XP Pro and the computation algorithm was performed in Matlab 2007a.

17.5.3 Experimental validation In this section, the discussed probabilistic method is applied to composite materials for fatigue reliability calculation. Since both the moments matching method and the FORM method yield similar solutions, only the solution using the FORM method is discussed here. The material used is a fiberglass composite laminate

Probabilistic fatigue life prediction of composite materials

629

0.016 0.014 0.012 PDF

0.01 0.008 0.006 0.004 0.002 0 400

(A)

450

500

550

600

650

700

External loading (MPa) Cycle distribution 1

Reliability

0.99 0.98 0.97 Moments matching FORM Monte Carlo

0.96 0.95 4.7

(B)

4.75

4.8 Time (log (N))

4.85

4.9

Comparison with the Monte Carlo simulation

Fig. 17.8 Comparison between the moments matching approach and the FORM approach for continuous loading.

[44]. Constant amplitude fatigue S-N data and their statistics are shown in Fig. 17.9 and Table 17.2, respectively. Time-dependent reliability under variable loading (two-block) is plotted together with empirical cumulative density function of experimental data in Fig. 17.10. It is observed that the discussed probabilistic method gives a very good prediction of probabilistic life distribution of the composite laminate. In Section 17.2, we have discussed the use of the nonlinear damage accumulation rule. The predicted Miner’s sums under variable amplitude loading are compared with experimental results for the composite material in Fig. 17.11. From Fig. 17.11, it is shown that the proposed method gives a better prediction compared to the LDAR for two loading cases. If the LDAR is used, very non-conservative results will be obtained since the Miner’s sum is well below unity under some loading conditions.

630

Fatigue Life Prediction of Composites and Composite Structures 2.65 Experimental data Mean S–N curve

2.6

Stress (log(MPa))

2.55 2.5 2.45 2.4 2.35 2.3 2.25 1

3 5 Fatigue life (log(N))

7

Fig. 17.9 Constant amplitude S-N curve data for DD16 composite laminates.

Table 17.2 Statistics of constant amplitude S-N curve data—DD16 composite laminates Statistics of single cycle fatigue damage (1/N) Stress amplitude (MPa)

Mean

Std.

Distribution

206 241 328 414

5.48E 06 1.97E 05 0.000615 0.004569

7.17E  06 1.97E  05 0.000362 0.003278

Lognormal Lognormal Lognormal Lognormal

1 Loading 1 Prediction

Reliability

0.8 0.6 0.4 0.2 0 4

4.5

5 Time (log(N))

5.5

6

Fig. 17.10 Time-dependent reliability variation comparisons between prediction and experimental results.

Probabilistic fatigue life prediction of composite materials

631

Prediction value

1.5

1 DD16 composite laminates Miner’s rule

0.5

Experimental value

0 0

0.5

1

1.5

Fig. 17.11 Comparisons between predicted and experimental Miner’s sum for different materials.

Conclusion Two efficient fatigue reliability calculation methods are discussed in this chapter and applied to composite material fatigue analysis. They are based on a stochastic process representation of the material properties under constant amplitude loading and a non-linear damage accumulation rule. In the moments matching approach, the fatigue damage under variable amplitude loading is assumed to follow the lognormal distribution, and the first two central moments are determined analytically without approximation. This results in a simple analytical solution for either the probability distribution of the service time to failure (fatigue life) or the probability distribution of the amount of damage at any service time. In the FORM approach, no assumption is made for the damage distribution under variable amplitude loading and the statistics of the basic variables are used together with the first-order reliability method. The methods discussed above are very efficient in calculating the time-dependent reliability variation under cyclic fatigue loading compared to the simulation-based approaches. The methods also include the correlation effect of the damage accumulation under variable amplitude loading, which has been mostly ignored in previous models. Currently available models in the literature are shown to be two special cases of the proposed approach, i.e. independent random variables and fully correlated random variables. The discussed methodology has been demonstrated using numerical examples and validated using experimental data under deterministic variable amplitude loading. Due to the general format and the simplicity of the calculation, the discussed probabilistic methodology may be conveniently applied to composite structures and materials.

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References [1] J.T.P. Yao, F. Kozin, Y.K. Wen, J.N. Yang, G.I. Schueller, O. Ditlevsen, Stochastic fatigue, fracture and damage analysis, Struct. Saf. 3 (1986) 231–267. [2] A. Fatemi, L. Yang, Cumulative fatigue damage and life prediction theories: a survey of the state of the art for homogeneous materials, Int. J. Fatigue 20 (1998) 9–34. [3] T.Y. Kam, K.H. Chu, S.Y. Tsai, Fatigue reliability evaluation for composite laminates via a direct numerical integration technique, Int. J. Solids Struct. 35 (1998) 1411–1423. [4] X. Le, M.L. Peterson, A method for fatigue based reliability when the loading of a component is unknown, Int. J. Fatigue 21 (1999) 603–610. [5] M. Liao, X. Xu, Q.X. Yang, Cumulative fatigue damage dynamic interference statistical model, Int. J. Fatigue 17 (1995) 559–566. [6] H. Shen, J. Lin, E. Mu, Probabilistic model on stochastic fatigue damage, Int. J. Fatigue 22 (2000) 569–572. [7] M. Kaminski, On probabilistic fatigue models for composite materials, Int. J. Fatigue 22 (2002) 477–495. [8] K. Ni, S. Zhang, Fatigue reliability analysis under two-stage loading, Reliab. Eng. Syst. Saf. 68 (2000) 153–158. [9] F.G. Pascual, W.Q. Meeker, Estimating fatigue curves with the random fatigue-limit model, Technometrics 41 (1999) 277–302. [10] J.D. Rowatt, P.D. Spanos, Markov chain models for life prediction of composite laminates, Struct. Saf. 20 (1998) 117–135. [11] T. Shimokawa, S. Tanaka, A statistical consideration of Miner’s rule, Int. J. Fatigue (4) (1980) 165–170. [12] X. Zheng, J. Wei, On the prediction of P-S-N curves of 45 steel notched elements and probability distribution of fatigue life under variable amplitude loading from tensile properties, Int. J. Fatigue 27 (2005) 601–609. [13] V.A. Kopnov, A randomized endurance limit in fatigue damage accumulation models, Fatigue Fract. Eng. Mater. Struct. 16 (1993) 1041–1059. [14] V.A. Kopnov, Intrinsic fatigue curves applied to damage evaluation and life prediction of laminate and fabric material, Theor. Appl. Fract. Mech. 26 (1997) 169–176. [15] M.A. Miner, Cumulative damage in fatigure, J. Appl. Mech. 67 (1945) 159–164. [16] M. Kawai, A. Hachinohe, Two-stress level fatigue of unidirectional fiber–metal hybrid composite: GLARE 2, Int. J. Fatigue 22 (2002) 567–580. [17] G.R. Halford, Cumulative fatigue damage modeling—crack nucleation and early growth, Int. J. Fatigue 19 (1997) 253–260. [18] S.M. Marco, W.L. Starkey, A concept of fatigue damage, Trans. ASME 76 (1954) 627–632. [19] S.S. Manson, G.R. Halford, Practical implementation of the double linear damage rule and damage curve approach for treating cumulative fatigue damage, Int. J. Fatigue 17 (1981) 169–192. [20] W. Van Paepegem, J. Degrieck, Effects of load sequence and block loading on the fatigue response of fibre-reinforced composites, Mech. Compos. Mater. Struct. 9 (2002) 19–35. [21] E. Goodin, A. Kallmeyer, P. Kurath, Evaluation of nonlinear cumulative damage models for assessing HCF/LCF interactions in multiaxial loadings, in: 9th National Turbine Engine High Cycle Fatigue (HCF) Conference, Pinehurst, NC, 2004. [22] G.R. Halford, S.S. Manson, Reexamination of cumulative fatigue damage laws, in: Structure Integrity and Durability of Reusable Space Propulsion Systems, 1985, pp. 139–145. NASA CP-2381.

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[23] A. Vasek, J. Polak, Low cycle fatigue damage accumulation in Armci-iron, Fatigue Fract. Eng. Mater. Struct. 14 (1991) 193–204. [24] G. Cheng, A. Plumtree, A fatigue damage accumulation model based on continuum damage mechanics and ductility exhaustion, Int. J. Fatigue 20 (1998) 495–501. [25] D. Shang, W. Yao, A nonlinear damage cumulative model for uniaxial fatigue, Int. J. Fatigue 21 (1999) 187–194. [26] Y. Liu, S. Mahadevan, Stochastic fatigue damage modeling under variable amplitude loading, Int. J. Fatigue 29 (2007) 1149–1161. [27] G. Jiao, A theoretical model for the prediction of fatigue under combined Gaussian and impact loads, Int. J. Fatigue 17 (1995) 215–219. [28] R. Tovo, A damage-based evaluation of probability density distribution for rain-flow ranges from random processes, Int. J. Fatigue 22 (2000) 425–429. [29] A. Banvillet, T. Łagoda, E. Macha, A. Niesłony, T. Palin-Luc, J.F. Vittori, Fatigue life under non-Gaussian random loading from various models, Int. J. Fatigue 24 (2004) 349–363. [30] D. Benasciutti, R. Tovo, Spectral methods for lifetime prediction under wide-band stationary random processes, Int. J. Fatigue 27 (2005) 867–877. [31] T.T. Fu, D. Cebon, Predicting fatigue lives for bi-modal stress spectral densities, Int. J. Fatigue 22 (2000) 11–21. [32] S.D. Downing, D.F. Socie, Simple rainflow counting algorithms, Int. J. Fatigue 4 (1) (1982) 31–40. [33] ASTM, Standard Practices for Cycle Counting in Fatigue Analysis, E1048-85, ASTM International, 1985. [34] M. Grigoriu, On the spectral representation in simulation, Prob. Eng. Mech. 8 (1993) 75–90. [35] M. Shinozuka, G. Deodatis, Simulation of the stochastic process by spectral representation, Appl. Mech. Rev. 44 (1991) 29–53. [36] M. Loeve, Probability Theory, Springer, New York, 1970. [37] R. Ghanem, Stochastic finite elements with multiple random non-Gaussian properties, J. Eng. Mech. 125 (1999) 26–40. [38] R. Ghanem, P. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer, New York, 1991. [39] S.P. Huang, S.T. Quek, K.K. Phoon, Convergence study of the truncated Karhunen–Loeve expansion for simulation of stochastic processes, Int. J. Numer. Methods Eng. 52 (2001) 1029–1043. [40] K.K. Phoon, S.P. Huang, S.T. Quek, Simulation of non-Gaussian processes using Karhunen–Loeve expansion, Comput. Struct. 80 (2002) 1049–1060. [41] L.F. Fenton, The sum of lognormal probability distributions in scatter transmission systems, IRE Trans. Commun. Syst. CS-8 (1960) 57–67. [42] R. Rackwitz, B. Fiessler, Structural reliability under combined random load sequences, Comput. Struct. 9 (1978) 484–494. [43] A. Haldar, S. Mahadevan, Probability, Reliability, and Statistical Methods in Engineering Design, John Wiley & Sons, New York, 2000. [44] J.F. Mandell, D.D. Samborsky, DOE/MSU Composite Materials Fatigue Database: Test Methods, Materials, and Analysis, Sandia National Laboratories, Albuquerque, NM, 2003.

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Computational tools for the fatigue life modeling and prediction of composite materials and structures

18

Anastasios P. Vassilopoulosa, Julia Maier b, Gerald Pinter b, Christian Gaierc a Ecole Polytechnique F
ed
erale de Lausanne (EPFL), Composite Construction Laboratory (CCLab), Lausanne, Switzerland, bMaterials Science and Testing of Polymers, Montanuniversitaet Leoben, Leoben, Austria, cMagna Powertrain, Engineering Center Steyr GmbH&CoKG, St. Valentin, Austria

18.1

Introduction

As it is clearly described in the literature and other chapters of this volume, fatigue of composites has been recognized as a critical phenomenon activating damage mechanisms that affect the durability of composites and structures made of them, since very early, for example, Ref. [1], Chapter 1. Significant experimental programs have been established and a wide range of experimental investigations have been conducted in order to investigate the fatigue behavior of several types of composite materials, identify the mechanisms that cause failure, and eventually quantify their effects on the structural integrity of the examined components [2–5]. In parallel to the experimental investigations, an abundance of models have been established in order to simulate the exhibited behavior; actually to fit the material experimental behavior and provide mathematical representation of it in order to allow interpolation between the experimentally defined ranges. Typical examples of such models are the S-N curve algorithms that have been introduced, already since 1910 by Basquin [6, 7] or those models trying to simulate the evolution of the fatigue stiffness with loading cycles, both under constant amplitude fatigue loading patterns. In a second stage, more sophisticated models were introduced for the prediction of the material behavior under “unseen” loading cases, see, for example, Refs. [8–10] for fatigue failure criteria, or Refs. [11–13] for constant life diagrams (CLDs). Such theories are described as empirical, or phenomenological in the literature [8, 14, 15]. In contrast to the phenomenological modeling approaches, models based on the measurement and the evolution assessment of actual damage mechanisms into the material have been introduced and used in progressive damage modeling algorithms. Appropriate damage evolution laws have to be derived and calibrated, by using experimental evidence, before such models are able to provide any type of information regarding the material’s performance under cyclic loading. Models have been Fatigue Life Prediction of Composites and Composite Structures. https://doi.org/10.1016/B978-0-08-102575-8.00018-8 © 2020 Elsevier Ltd. All rights reserved.

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developed at different scales, based either on fracture mechanics [16–19] and/or the estimation of S-N curves at the laminate, for example, Refs. [4, 17], the lamina, or the fiber/matrix levels [10, 20, 21] in order to fulfill this requirement. Application of the progressive damage models is iterative; once the failure criterion (at any selected scale level) is satisfied a number of properties of the failed elements are degraded followed by the calculation of the new stress field (stress redistribution). The procedure is repeated until the failure condition is fulfilled. Depending on the method, the failure condition could be the final failure of the laminate, when a specific number of cycles is reached, or when a certain value of the damage metric is achieved, for example, 10% stiffness degradation (Chapter 15, Ref. [22]), or a predefined crack length [23]. In some cases, combinations of the aforementioned conditions are utilized in order to calibrate the model parameters and aim for more reliable fatigue behavior predictions, for example, Ref. [24]. Initially, such theories were developed mainly based on the constant amplitude and block loading fatigue data, and their application was limited to the simulation of constant amplitude fatigue behavior for several composite systems, since variable amplitude (VA) data were very limited to allow validation. However, over the last three decades, more fatigue data have been produced in laboratories concerning fiberreinforced composite materials in order to examine their behavior under realistic loading situations, including VA fatigue loading and complex environmental conditions, for example, Refs. [2–4, 25–30]. In parallel to the experimental work, theoretical models are developed for the simulation of the fatigue behavior of the examined materials under different thermomechanical loading conditions and the prediction of material fatigue life under complex stress states that may arise during the operation of a structure in the open air [8, 29, 31–33]. A fatigue life prediction methodology is usually based on the development of empirical relationships between the applied loads and the fatigue lifetime of the examined materials. Such relationships can be developed at different scales depending on the followed approach. The implementation of a numerical procedure for fatigue analysis consists of a number of distinct calculation modules, related to life prediction. Some of these are purely conjectural or of a semiempirical nature, for example, the failure criteria, presented in Refs. [17, 32], while others rely heavily on experimental data, for example, S-N curves and CLDs [7, 11, 34]. In cases of composite laminates under uniaxial loading, leading to uniform axial stress fields, the situation can be substantially simplified since almost all relevant procedures could be implemented by experiment. On the other hand, for complex stress states, the laminated material is considered as being a homogeneous orthotropic medium and its experimental characterization, that is, static and fatigue strength, is performed for both material principal directions and in-plane shear [35]. This laminate approach (also cited as the building block testing approach [36]) is a straightforward process for predicting fatigue strength under plane stress conditions, avoiding the consideration of damage modeling and interaction effects between the plies and stress redistribution, and can be reliably used when limited stacking sequence variations are present in a structural element. The building block approach, used together with analysis, is considered essential to the qualification/certification of

Computational tools for the fatigue life modeling and prediction

637

composite structures due to their sensitivity to out-of-plane loads, the multiplicity of composite failure modes and the lack of standard analytical methods [36]. The approach was implemented by Philippidis and Vassilopoulos [3, 32, 37, 38] for a glass/polyester multidirectional laminate of [0/
45] stacking sequence and was shown to yield satisfactory predictions for fatigue strength predictions under complex stress conditions for both constant amplitude and VA loading. A straightforward algorithm must be followed comprising steps dealing with the analysis of the load to determine the developed stress fields, the interpretation of the fatigue data of the examined laminates to derive the S-N curves and the corresponding CLDs, the fatigue failure criteria of the calculation of the design allowables, and finally the damage accumulation based on the selected damage rule. In large composite structures, consisting of numerous different materials and laminate configurations, a lamina-to-laminate approach seems more appropriate, although requiring the development of additional calculation modules able to take into account the implications in local stress fields, stress redistribution in neighboring plies, and finally, how damage propagates as a function of loading cycles, for example, Refs. [17, 35, 39]. According to this, more refined approach, the material properties of basic building plies need to be experimentally derived. The properties of any new laminate configuration, consisting of the basic building plies, are then estimated based on the existing theoretical procedures. The failure analysis is based on a progressive damage modeling, considering failed layers and stress redistributions in the laminate according to the applied load history. The analysis of structures based on the fatigue experimental data from substructural elements leads inevitably to lower scales where, for example, the fatigue behavior of a laminate is estimated (predicted) when information about the fatigue behavior at the level of the layer (as described in the previous paragraph) or of the fiber and the matrix is available. This approach goes one level lower in scale, when compared to the “lamina-to-laminate approach” described above, reaching the microscale level. Structural laminate properties are broken down to the ply level and eventually to the constituent material microscale level. At this basic level, the model taking microscopic defects into account, calculates the stress fields developed in both the fibers and matrix, and using appropriate failure criteria defines local failure, and depending on the method used is capable of identifying the failure mode as well. Then, based on a combination of micro-mechanics and laminate theories, the model is rebuilt up to the laminate in order to provide as output an estimation of the laminate lifetime, for example, Ref. [24]. The objective of the aforementioned theoretical approaches is to replace, in a way, the need for excessive experimental campaigns and develop virtual testing environments, which, after validation can be used to satisfy a multitude of tasks. Validated analytical/numerical procedures can be used, in between others, for: l

l

The simulation of the material’s response under selected loading patterns, in terms of S-N curves, stiffness degradation models, and CLDs. The lifetime estimation under unseen loading patterns, including irregular realistic spectra, through the use of appropriate fatigue failure criteria, and damage rules.

638 l

l

Fatigue Life Prediction of Composites and Composite Structures

The simulation of the damage progression, identifying damage modes, and predicting eventual material failure. Material selection/optimization, especially through bottom-up micro-mechanical multiscale approaches able to assist in selecting the appropriate mix of materials, as well as joining techniques.

Virtual testing serves for reducing physical prototypes and complicated experimental campaigns. Nevertheless, the existence of validated fundamental modeling and analysis methods does not mean that testing is not necessary. It rather implies that less validation against actual data will be required at complex and large-scale structural levels [20]. Development of numerical and analytical methods for life modeling and prediction of composite materials, structural elements, and complete structures goes hand in hand with experimental campaigns. Valid experimental data are always necessary in order to validate the developed theoretical methodologies in order to be able to be used for virtual testing of different loading and material configuration scenarios. The drawback of virtual testing is the lack of confidence due to the lack of validation, as well as the lack of certification processes for virtual testing environments [20], especially given the scarcity of structural failure data against which to benchmark. Empirical, phenomenological methods and especially progressive damage methods have not yet received enough verification and validation to be used for structural analysis. In a comparison of composite damage computational tools performed in 2017 [40], it was revealed that on average blind predictions of the behavior of notched laminates under constant amplitude fatigue at R ¼ 0.1, from all models, differed by 42% from the available experimental data. This error was reduced to around 18% only when a series of model parameters was recalibrated. Neither compression nor spectrum cycling loading cases were investigated in this study. This uncertainty appeals for additional efforts for the improvement of the accuracy of fatigue life modeling and prediction procedures in order to have such tools accepted for fatigue structural designs. It is obvious that independent of the adopted approach for the life estimation, implementation of the necessary steps into a computer program is indispensable. Software products have been developed to assist the research community in design processes using methods that are based either on the micromechanics modeling of fatigue life, for example, Autodesk Helius PFA [21] or on macromechanical approaches, such as the phenomenological laminate approach, for example, CCfatigue [33]. This chapter presents some of the existing computational tools for fatigue life modeling/prediction of composite materials and structures and discusses their potential to be used as reliable virtual testing tools in the future.

18.2

Engineering software for fatigue life modeling/ prediction

Computational tools for the modeling of the fatigue behavior and the lifetime prediction of composite materials have been established during the last decades. Such computer programs are either in-house developed software packages aiming to cover

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specific demands of research institutes and/or industries, or modules in commercial software packages aiming to cover a wider range of applications. In between these two, there are also tools that although have been developed with a purely research outlook, became mature enough after been validated against a series of experimental data and are in a pre-commercial development phase. In composites, the engineers, simultaneously to the structural design, have the ability to design the material itself. Therefore, the design process has to go through an optimization procedure, in which such analysis tools can play a significant role by allowing engineers and manufacturers to try several potential designs and focus on the optimal solution (or at least limited number of solutions) instead of undergoing physical testing of too many options. Computer simulation tools should be able to define where material is needed, and where is too much. They should either estimate fatigue life, or define critical load cases in order to refine the design and designate and validate accelerated fatigue testing procedures. Plenty of software frameworks able to perform in some extend fatigue life predictions are available. A family of tools is based on a multiscale progressive damage analysis approach, for example, GENOA (http://alphastarcorp.com/genoa/), the Multiscale Designer (MDS-C) (https://altairhyperworks.com/product/Multiscale-Designer), or the Fe-safe/composites (http://www.digitaleng.news/pics/pdfs/fe-safe_Composites.pdf) that uses a multi-continuum theory developed by Helius:Fatigue. (All web sites accessed on January 2019.) The idea of such tools is the development of constitutive models for the description of fatigue damage at the microscale level and then, in order to analyze largescale components through multiscale decomposition procedures. Models of this family can be accurate enough when appropriate experimental data are available in order to fit all model parameters from the micro to the macroscale. After successful parameter calibration procedures, those models can, quite accurately, predict the right type, amount and location of damage as a function of the fatigue cycles [40]. Nevertheless, when the available experimental data for calibration are limited, model predictions can be significantly different from experimental results. It is well documented today that the accuracy of the fatigue progressive damage models is still low for predicting the response of composite laminates, while their performance has yet to be validated for other loading cases, such as under compression and spectrum loading profiles [40]. A less data demanding family of models rely on macro mechanical/phenomenological approaches, for example, FOCUS6 (https://wmc.eu/focus6.php), CCfatigue, finite element fatigue (FEMFAT) (see later in this chapter). These theories are based on constitutive models for the material description at the lamina (e.g., FEMFAT), the laminate, or the structural component level (e.g., CCfatigue). The former case implements a lamina-tolaminate approach in order to use the experimental information at the lamina level in order, through the appropriate modeling, to simulate the fatigue behavior of laminates made of the same laminae. The theories of the latter case are more straightforward using the so-called “building block” approach, and can be used for design verification and life assessment of materials and structures under any random spectrum, given that enough constant amplitude fatigue data are available. Appropriate failure criteria are employed by these theories in order to take into account the effect of complex, multiaxial fatigue stress states on the fatigue life of the examined material, or structural component.

640

18.3

Fatigue Life Prediction of Composites and Composite Structures

FEMFAT laminate approach

For metallic materials, theories based on the fatigue strength, usually represented by stress vs number of cycles curves (S-N curves), are widely spread and have been implemented successfully in software tools for fatigue life prediction. One software tool is the FEMFAT developed by Magna Powertrain Engineering Center Steyr GmbH & Co KG (St. Valentin, Austria) [41]. In contrast to the studies published for composite materials so far, a very comprehensive, engineering approach is used. The real part geometry, quasi-static, and fatigue material data reflecting effects on the material behavior, the applied load-time history caused by the application, and local stresses calculated by finite element (FE) analysis are taken into account. For each node of the FE mesh, local S-N curves are predicted [42, 43]. Critical damages are calculated according to the critical plane concept [44–46]. Thereby, damage accumulation is performed for all planes at defined angles, at each node. The plane, in which the calculated damage reaches a maximum, is considered as critical. The equivalent stresses occurring in the critical planes are classified by rainflow-counting. Subsequently, damages are calculated based on the local S-N curves and accumulated to the total damage sum. This software tool has been successfully adapted for fatigue life prediction of orthotropic materials [47]. For injection molded short-fiberreinforced plastics, anisotropic material behavior and effects caused by the injection molding process can already be taken into account. The functionality of simulation chains from injection molding simulation to lifetime prediction has been presented and validated in different studies [48–51].

18.3.1 Fatigue life prediction method for laminates To meet the fatigue characteristics of continuously fiber-reinforced composites, the fatigue solver FEMFAT has been extended with a new module for lifetime estimation of laminates. Within this software tool for laminates, standard methods for the assessment of metallic parts based on the S-N curves have been adapted for laminates. In order to take the characteristic damage modes of composite materials into account, the three failure modes; fiber failure (FF), interfiber failure (IFF), and, optionally, delamination according to Puck are included in the software. For each ply of the laminate, the lifetime prediction is performed. For the assessment of FF, the stress history of the normal stress σ 1 longitudinal to the fiber orientation is calculated by linear superimposition of in general multiaxial load channels. A rainflow counting algorithm is applied to obtain an amplitude-meanrainflow-matrix of closed load cycles. Subsequently, the partial damages are analyzed by using experimentally measured material S-N curves and are linearly accumulated according to Palmgren/Miner [52, 53]. For the IFF modes, the same procedure is performed for the normal stress σ 2 transverse to the fiber orientation and for the in-plane shear stress τ21 and the respective material S-N curves in the fatigue life software. To apply Puck’s criterion also combinations of σ 2 and τ21 have to be considered. Nevertheless, for nonproportional loading, the stress vector spanned by σ 2 and τ21 may

Computational tools for the fatigue life modeling and prediction

641

change its direction with respect to time. It is difficult to apply a rainflow counting procedure in such a case. To solve this problem, a simplified version of the so-called “Critical Plane-Critical Component” approach was developed [40, 46, 54]. The stress vector is projected onto several fixed directions with the given unit ! vector v i : !

!

σ eqv ðtÞ ¼σ ðtÞ  ν i ¼ σ 2 ðtÞ cos φi + τ21 ðtÞsin φi , i ¼ 1, …, N

(18.1)

For each orientation direction, rainflow counting and damage analysis of the resulting equivalent stress can be performed without any restrictions. The direction, which delivers the maximum damage, is assumed critical for fatigue failure. It can be mathematically interpreted as the critical component of the stress vector and it defines the type of failure mode A, B, or C. To assure that each mode is covered by at least one direction, an angle of 30 degree is used between directions as default, leading to six directions in the σ 2-τ21-plane as illustrated in Fig. 18.1. For the damage analysis of the intermediate directions S-N curves are used, which are obtained by interpolation between the S-N curves for σ 2 and τ21. To consider the influence of the mean stress, CLDs are constructed from quasistatic and cyclic material parameters such as ultimate tensile strength, ultimate compressive strength, ultimate shear strength, alternating and pulsating fatigue limit for a given number of cycles as, for example, 5  106 [52, 53]. Due to the different strengths under tensile and compressive loads of composite materials caused by the anisotropic macroscopic material behavior, usually highly asymmetric CLDs are obtained for normal stresses, although for shear stresses, the CLD is symmetric. For intermediate directions, which cannot all be covered by experiments, the CLDs are interpolated with a smooth transition between tensile-compressive loading at an angle of φ ¼ 0 degree and shear loading at φ ¼ 90 degree, resulting in a constant life surface as illustrated in Fig. 18.2. The static limitations of the constant life surface at the left-hand

Modus/mode B Modus/mode C

t21

c

Modus/mode A PA⊥⎥⎥

c P ⊥⎥⎥ C

A

t 21c= R ⊥⎥⎥ 1+2 P⊥⎥⎥ – R⊥⊥ /R⊥⎥⎥

RA⊥⊥

0

s2

Fig. 18.1 Fatigue assessment of different load directions in the laminate plane.

642

Fatigue Life Prediction of Composites and Composite Structures Modus/mode B Modus/mode C

t21

c

Modus/mode A l p⊥⎥⎥

c p⊥⎥⎥

t21c = R⊥⎥⎥

c A 1 + 2p • R /R ⊥⎥⎥

⊥⊥

⊥⎥⎥

j s2

0°

ngle

100

g]

fl ⊥ j [de

s

ra Pola

Endurance limit [Mpa]

A R⊥⊥

0

t fl

90°

0 –200

0

90

Mean stress [Mpa]

Fig. 18.2 Constant life surface for IFF according to Refs. [55–57] with an interpolated constant life line (red) for a given angle φ of the considered loading direction.

side (green line in compressive domain) and right-hand side (yellow line in the tensile domain) are determined by Puck’s criterion in the software routine. The intralaminar stresses σ 1, σ 2, and τ21, which are acting in the plies, can be analyzed by the finite element method (FEM) with shell elements, whereas for interlaminar stresses σ 3, τ32, and τ31, which are acting between plies and causing delamination, solid elements are needed, which is even more expensive. For the fatigue assessment of interlaminar stresses, the same procedure is used as for intralaminar stresses with different S-N curves and CLDs. In Fig. 18.1, σ 2 is replaced by σ 3 and τ21 by τ32 and τ31, respectively. Nevertheless, it is difficult to measure S-N curves for interlaminar stresses. Interlaminar shear stresses can be generated by a three-point bending load applied to specimens [58]. According to Ref. [59] a high load in fiber direction reduces the strength for IFF by the following elliptical weakening factor with the two material parameters s and a: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   σ 1 =Rk  s 2 ηw ¼ 1  σ 1 =Rk > s: a

(18.2)

For cyclic loading, the weakening factor may change during a load cycle because σ 1 is generally a function of time. Therefore, the IFF strength is not unique for the damage

Computational tools for the fatigue life modeling and prediction

643

Fig. 18.3 Fatigue analysis procedure for a laminate structure.

analysis of the cycle. To overcome this theoretical problem, the stress components σ 2 and τ21 will be increased by the time-dependent weakening factor instead (σ0 2, σ0 21) and assessed with the original non-reduced IFF strength [58]: σ 02 ðtÞ ¼

σ 2 ðt Þ ηW ðtÞ

τ021 ðtÞ ¼

τ21 ðtÞ ηW ðtÞ

(18.3)

The fatigue analysis procedure for a laminate component is exemplified in Fig. 18.3. Up to six nested loops are required in order to cover the different scale levels of whole structures. All laminate elements resulting from the FE analysis, the nodes at the elements and the individual plies need to be considered. Furthermore, fatigue analyses have to be performed for each stress component at top and bottom side of each ply. To take the anisotropic material behavior into account, the different stress components σ 1, σ 2, and τ21 are included. To enable the consideration of various, realistic time ranges, rainflow counting is applied. For acceleration of the procedure appropriate filters have been implemented to select only highly stressed plies for the fatigue analysis. In addition, a parallel analysis on several CPUs is possible. For this case, the component is automatically divided into several parts, which are distributed to the CPUs for parallel computing. After finishing, the results are automatically merged together.

18.3.2 Experimental work Unidirectional (UD) laminae made of carbon fibers and epoxy resin were tested at angles of 0, 45, and 90 degree. Layups consisting of
45 degree and [0°/+45°/45°/90°]S layers were also investigated and both layups were symmetric referring to the middle plane. The fiber volume content of all produced specimens was 55% (measured by thermogravimetric analysis as published in Ref. [60]). UD 0 degree plies consisted of four layers; all other specimens were made of eight layers.

644

Fatigue Life Prediction of Composites and Composite Structures

The specimens’ geometry used in quasi-static tensile and tension-tension fatigue tests was 200  10  1 mm (length width  thickness) for UD specimens in fiber direction and 200  20  2 mm for all other specimens. For quasi-static compression and tension-compression fatigue tests, specimens’ geometry was 110  10  2 mm (UD 0 degree) and 110  20  2 mm, respectively. Aluminum tabs (length: 50 mm, thickness: 1 mm) were glued on both sides of all specimens. Quasi-static and fatigue tests were performed on a servo-hydraulic test machine equipped with a load frame and load cell for 100 kN by MTS Systems Corporations (Minnesota, USA) at room temperature. Hydraulic wedge pressure of 5 MPa was chosen in order to prevent slipping without damaging the specimens. Good adhesion between aluminum tabs and carbon/epoxy specimens was assured in preliminary tests. Gauge length was 100 mm for tensile and 10mm for compression loads. Quasi-static tension and compression tests were performed with a test speed of 0.5 mm/min until failure. During quasi-static tests, a digital image correlation (DIC) system by GOM (Braunschweig, Deutschland) was used for strain measurement in and transverse to fiber direction in order to calculate Poisson’s ratios. Tensile moduli were calculated according to Ref. [61]. Compressive moduli were evaluated between 0.001 and 0.003 absolute strain according to Ref. [55]. Shear moduli were evaluated as in-plane shear response in tensile tests with
45 degree specimens [56]. Tension-tension fatigue tests were performed with the R-value (¼minimum force/maximum force) of 0.1, tension-compression fatigue tests with R ¼  1 until total failure. For the creation of S-N curves, specimens were tested at four different stress levels. Maximum cyclic stresses between approximately 80% and 65% of the ultimate tensile strengths were chosen as load levels for UD 0 degree and between 60% and 35% for the tested off-axis specimens in tensiontension fatigue tests. In tension-compression fatigue tests, the maximum cyclic tensile stresses applied were between approximately 25% and 15% of the ultimate tensile strengths for UD 0 degree specimens and between approximately 65% and 25% for off-axis specimens [62]. A minimum of three specimens were tested on each stress level. Specimens tested on the lowest stress level, which did not fail, were manually stopped after approximately 2  106 cycles. Specimens’ temperatures were monitored by infrared sensors in all fatigue tests. Test frequencies were chosen between 2 and 10 Hz depending on the load amplitude and the specimens’ tendency for hysteretic heating in order to limit the temperature increase to a maximum of 5°C and consequently minimize the influence on the materials’ properties [63]. Results of fatigue tests were evaluated statistically to calculate the slope of the S-N curves k, scatter with Ts and the nominal stress amplitude after 5  106 cycles σ a,5106 according to Ref. [64].

18.3.3 Experimental results 18.3.3.1 Test results for quasi-static loading The results of quasi-static tensile and compressive tests are presented in the way they were used as input parameters for the software tools. Material parameters in fiber direction are summarized in Table 18.1, mechanical properties transverse to fiber direction are presented in Table 18.2, and shear properties evaluated with

Computational tools for the fatigue life modeling and prediction

645

Table 18.1 Quasi-static input parameters for FE analysis and fatigue-life prediction evaluated in quasi-static tension and compression tests in fiber direction Young’s modulus in fiber direction [GPa] Elastic Poisson’s ratio [] Elongation at rupture [%] Ultimate tensile strength [MPa] Ultimate compressive strength [MPa]

107.0 0.34 1.35 1550 549

Table 18.2 Quasi-static input parameters for FE analysis and fatigue-life prediction evaluated in quasi-static tension and compression tests transverse to fiber direction Young’s modulus transverse to fiber direction [GPa] Ultimate tensile strength [MPa] Ultimate compressive strength [MPa]

5.5 33 89

Table 18.3 Quasi-static shear input parameters for FE analysis and fatigue-life prediction evaluated in quasi-static tension tests with
45 degree specimens In-plane shear modulus G12 [GPa] In-plane shear strength [MPa]

3.3 74


45 degree specimens in Table 18.3. The shear moduli G13 and G23 necessary for the FE analysis could not be measured experimentally. Based on the assumption of transversally isotropic material behavior of UD plies, the shear modulus G13 was assumed to be equal to the measured shear modulus G12 [59]. The third shear modulus G23 was set as 2.0 GPa based on the relation between the three shear moduli in calculated material data for UD carbon/epoxy plies from literature [59]. In order to calculate the values of the shear moduli for the material used herein, micromechanical modeling based on the measured material parameters could be used [65].

18.3.4 Results of fatigue tests Fatigue data of UD 0 degree, UD 90 degree, and
45 degree were used as input parameters for the fatigue life software. The experimentally measured S-N curves for R ¼ 0.1 and R ¼  1 at room temperature are illustrated in terms of nominal stress amplitudes σ a vs number of cycles in Fig. 18.4. For statistical evaluation of the measured fatigue data, a linear model according to Eq. (18.4) was used. The parameters of the linear model were calculated as maximum likelihood estimators according to the ASTM E739 [64], assuming that: all fatigue life data pertain to a random sample and are independent, there are no run-outs, the S-N curve can be described by a linear model, a two-parameter log-normal distribution describes the fatigue life and the variance is constant. The maximum likelihood estimators A^ and B^ were calculated

646

Fatigue Life Prediction of Composites and Composite Structures

Fig. 18.4 S-N curves of 0, 90, and
45 degree coupons tested at stress ratios R ¼ 0.1 and R ¼  1 at room temperature until failure, which were used as input parameters for the fatigue software.

according to Eqs. (18.5) and (18.6). X and Y describe the average values of the log Si and the log Ni, respectively, taking the total sample size m into account [64]. For further calculation of arbitrary data points in the S-N diagrams, Eq. (18.7) was used where σ 1 > σ 2 and N1 < N2 and k represents the statistically evaluated slope of the S-N curve. The values for the slope k, scatter width Ts and nominal stress amplitude after 1 cycle σ a,0 and after 5  106 cycles σ a,5106 of the S-N curves are summarized in detail in Table 18.4. log N ¼ A + Blog S

(18.4)

^ A^ ¼ Y  BX

(18.5)

Xm  B^ ¼

   X  X ∗ Y  Y i i i¼1 Xm   Xi  X 2 i¼1

 σ1 ¼ σ2

N1 N2

(18.6)

 1 k

(18.7)

Tension-tension fatigue amplitudes (R ¼ 0.1) were higher for specimens in fiber direction than alternating amplitudes (R ¼  1) as a result of the quasi-static ultimate compressive strength being approximately 30% of the ultimate tensile strength

Computational tools for the fatigue life modeling and prediction

647

Table 18.4 Input S-N curves for the fatigue-life prediction Specimen

R [2]

k [2]

Ts [2]

σ a,0 [MPa]

σ a,5×106 [MPa]

UD 0 degree UD 90 degree
45 degree UD 0 degree UD 90 degree
45 degree

0.1 0.1 0.1 1 1 1

24.9 11.1 17.0 13.4 8.0 13.5

1/1.14 1/1.15 1/1.13 1/1.41 1/1.58 1/1.45

808.0 19.2 73.2 782.5 29.0 136.3

435.2 4.8 29.5 248.4 4.2 43.4

Slope k, scatter width Ts, the nominal stress amplitude after 1 cycle σ a,0 and the nominal stress amplitude after 5*106 cycles σ a,5*106 for R ¼ 0.1 and R ¼ 1 evaluated from experimental fatigue tests.

(Table 18.1). On the contrary, alternating amplitudes were higher than tension-tension amplitudes for
45 degree specimens. For specimens tested transverse to fiber direction, the mean stress did not significantly influence the fatigue strengths. For the implementation in the fatigue life prediction software, the respective maximum cyclic stresses for R ¼ 0 and R ¼  1 after 5  106 cycles and the slopes of S-N curves were required. Therefore, fatigue data measured at R ¼ 0.1 and the mean stress effect were used to calculate pulsating tension fatigue strength for R ¼ 0 of 928.2 MPa (in fiber direction) and 9.95 MPa (transverse to fiber direction). Shear fatigue strength after 5*106 required by the fatigue software was calculated by dividing the fatigue strengths of
45 degree specimen by two [56]. Consequently, pulsating shear fatigue strength for R ¼ 0 was 30.8 MPa. The alternating fatigue strengths were equal to the stress amplitudes σ a,5106 measured at R ¼  1 in Table 18.4. For verification of the described fatigue analysis method, fatigue tests with pulsating loading (R ¼ 0.1) of two additional laminate configurations for UD 45 degree and for a multilayer composite [0°/+45°/45°/90°]S were performed. Test fatigue results of the two layups used for validation are illustrated in comparison to the maximum cyclic stresses of UD 0 degree, UD 90 degree, and
45 degree in Fig. 18.5. The S-N curves measured in tension-tension fatigue tests are illustrated in terms of maximum cyclic stress vs number of cycles in order to ease the comparison with the respective quasi-static tensile strengths. Slope k, scatter width Ts, and the calculated nominal stress amplitudes after 5  106 cycles σ a,5106 of the S-N curves can be found in Table 18.5.

18.3.5 Simulation results 18.3.5.1 FE analysis For the FE analysis of deformation and stresses with the FE solver ABAQUS, the specimens were modeled with linear quadrilateral shell elements. In the ABAQUS input file, the thickness and material of each ply and also the orientation of the fibers are defined in a shell section with the attribute COMPOSITE. The used stiffness parameters are shown in Tables 18.1–18.3. To obtain correct clamping conditions,

648

Fatigue Life Prediction of Composites and Composite Structures

Fig. 18.5 S-N curves of 0, 45, 90,
45 degree, and [0°/+45°/45°/90°]S coupons tested at stress ratio R ¼ 0.1 until failure [66]. Table 18.5 Verification S-N curves for the fatigue-life prediction Specimen

R [2]

k [2]

Ts [2]

σ a,0 [MPa]

σ a,5*106 [MPa]

UD 45 degree Multilayer

0.1 0.1

9.2 13.0

1/1.12 1/1.21

56.7 360.3

10.6 109.8

Slope k, scatter width Ts, the nominal stress amplitude after 1 cycle σ a,0 and the nominal stress amplitude after 5*106 cycles σ a,5*106 for R ¼ 0.1 evaluated from experimental fatigue tests.

also the aluminum tabs were modeled with shell elements and connected to the laminate with tie contacts. For modeling of the aluminum tabs, tensile modulus of 70 GPa and Poisson’s ratio ν ¼ 0.34 were assumed. The FE mesh of the UD 0 degree specimens’ geometry consists of 2000 quadrilateral shell elements for the carbon fiberreinforced laminate and additional 2000 quadrilateral shell elements for the aluminum tabs. For UD 45 degree and UD 90 degree specimens, the number of elements is doubled according to the double width of 20 mm, leading to a total number of 8000 elements.

18.3.5.2 Fatigue life prediction For fatigue lifetime prediction with FEMFAT, an input material dataset was generated as a first step. Quasi-static material properties as presented in Tables 18.1–18.3 were used. Furthermore, the fatigue strengths for alternating and pulsating loading in and

Computational tools for the fatigue life modeling and prediction

649

transverse to the fiber orientation and for shear loading were implemented. Based on those data, CLDs were constructed. In contrast to the behavior known from metallic materials, the tensile mean stresses had a positive effect on the fatigue life for loading in fiber direction which corresponded to the high quasi-static material properties in fiber direction (Table 18.1). As a result of the different quasi-static strengths under tensile and compressive loads, the CLDs for loading longitudinal and transversal to fiber direction appeared asymmetric. In contrast to that the CLD for shear loading appeared symmetric (Fig. 18.6). Subsequently, fatigue life predictions were performed for the UD 0 degree, UD 90 degree, and
45 degree specimen in order to check the validity of the input data. The check was assessed successfully if the fatigue simulation produced the same S-N curves as the ones measured experimentally. A good fit with experimental test results could be achieved for UD 0 degree and UD 90 degree specimens as illustrated with red lines drawn in comparison to the experimental results (Fig. 18.7 and Table 18.6). For the two implemented load cases, tension-tension and tension-compression, the produced input data in and transverse to fiber direction fitted the experimental fatigue strengths very well. For
45 degree, the damage distribution along the specimens calculated by the software was not as homogeneous as for UD 0 degree and UD 90 degree specimens. The disturbed damage distribution in the area of the clamping of the specimen is illustrated in Fig. 18.8. This effect might be caused by the two different

=0

–∞

R

=

7 5

8

10 9

1

0

1000

–∞

=0

=

60

R

R

R = –1

0

4

R = –1

R

500

0 60

–∞

R

=

=0

R

30

0

R = –1

–60

0 –60

0

Fig. 18.6 Constant life diagram for loading in fiber direction (top), shear CLD (middle), and CLD for loading transverse to the fiber orientation (bottom).

Fatigue Life Prediction of Composites and Composite Structures

Nominal stress amplitude sa (log.) [MPa]

650

Fatigue tests CFRP 55% fibre volume content j = 2–10Hz, R 0.1, R-1, RT

UD 0° 55% R0.1 ±45° R0.1 UD 90° 55% R0.1

UD 0° 55% R-1 ±45° R-1 UD 90° 55% R-1

1000

R0.1

R-1

100

Result at middle position for R-1

Result near clamping position for R0.1

Result at middle position for R0.1 R0.1

10

R-1

1 102

103 104 105 Number of cycles N (log.) [cycles]

106

107

Fig. 18.7 Validation of simulated input parameters (red lines) with experimental input parameters. Table 18.6 Simulation results for UD 0 degree, UD 90 degree and
45 degree Specimen

R [2]

k [2]

σ a,0 [MPa]

σ a,5*106 [MPa]

UD 0 degree UD 90 degree
45 degree middle
45 degree clamping UD 0 degree UD 90 degree
45 degree

0.1 0.1 0.1 0.1 1 1 1

13.4 8.0 13.5 13.5 13.4 8.0 13.5

1311.5 34.4 91.2 80.9 745.5 28.2 136.1

414.8 5.0 29.1 25.8 235.8 4.1 43.4

directions of layers within the specimen, +45 degree and 45 degree, resulting in a stress introduction into the specimen different from uniaxial specimen. However, the obtained damage distribution reflected the layup of the laminates and corresponded to the failure locations monitored in the tested coupons. To address this effect, two different input parameter sets were tested, one corresponding to the area of clamping and one to a position in the middle of the specimen. Consequently, the correlation between calculation and experimental results was best in the undisturbed middle region of the specimen which was used as input for subsequent fatigue life prediction (Fig. 18.7). For
45 degree, the results depended on the position as shown in Fig. 18.8. Based on the input parameter sets validated for R ¼ 0.1 and R ¼  1, the fatigue life of the UD 45 degree specimens, in which all layers were aligned at the same angle of 45 degree, and of a multiaxial layup with the stacking sequence [0°/+45°/45°/90°]S

Computational tools for the fatigue life modeling and prediction

651

Fig. 18.8 Comparison of simulated damage distribution with the experimental failure of a
45 degree coupon at stress ratio R ¼ 0.1.

Fig. 18.9 Comparison of simulated damage distribution with the experimental failure of a UD 45 degree coupon at stress ratio R ¼ 0.1. The pink area is the area with highest damage.

were predicted subsequently. Due to the UD layup of the UD 45 degree specimen and the fixed clamping, asymmetric stress distribution resulting in inhomogeneous damage distribution was obtained (Fig. 18.9). For this case, not the shear stress was responsible for fatigue failure as supposed by intuition, but the normal stress transverse to the fiber direction for both investigated locations at the middle of the specimen and at the

Fatigue Life Prediction of Composites and Composite Structures

Nominal maximum stress σmax (log.) [N/mm2]

652

10,000

Fatigue tests CFRP 55% fibre volume content j = 3–10Hz, R 0.1, RT

UD 0° UD 45° UD 90° ±45° [0/45/–45/90/symm.]

1000

Ply-45°

Inside ply 90°

Ply-45°

100 Result at middle position Result at clamping

10 102

103

104 105 106 Number of cycles N (log.) [cycles]

107

Fig. 18.10 Comparison of simulation results (red lines) with test results for UD 45 degree and multilayered composite [0°/+45°/45°/90°]S.

clamping position. Consequently, the best correlation with experimental results was again obtained for the middle position as presented in Fig. 18.10 and Table 18.7. It can be assumed, that the stresses at the clamping position were too high because of singularity effects at the edges, where aluminum tabs were connected to the laminate. Only a very detailed modeling of notch radii with finite solid elements may increase the accuracy at such positions, which nevertheless would be too much effort in daily engineering practice. For the multilayered composite, the predictive situation was quite different. Due to the functional principle of analyzing the laminate ply by ply, interactions between the plies influencing in the total behavior of the entire laminate were not taken into account. Consequently, the fatigue life prediction assuming that the 90 degree ply at the middle plane of the [0°/+45°/45°/90°]S specimen was critical for failure which resulted in predictions underestimating the test result for total failure (red line in

Table 18.7 Simulation results for UD 45 degree and multi-layer composite [0°/+45°/45°/ 90°]S Specimen

R [2]

k [2]

σ a,0 [MPa]

σ 6a,5*10 [MPa]

UD 45 degree middle UD 45 degree clamping Multilayer 90 degree ply Multilayer 45 degree ply Multilayer +45 degree ply

0.1 0.1 0.1 0.1 0.1

8.0 8.0 8.0 8.0 8.0

70.1 53.6 266.1 422.9 507.5

10.2 7.8 38.7 61.5 73.8

Computational tools for the fatigue life modeling and prediction

653

Fig. 18.11 Experimental failure of [0°/+45°/45°/90]S carbon/epoxy coupon after fatigue test at stress ratio R ¼ 0.1.

Fig. 18.10 and Table 18.7). If continuing the fatigue life calculation after the assumed failure of the 90 degree plies,
45 and 0 degree would still carry load until the neighboring plies with 45 degree orientation and subsequently the plies with +45 degree orientation would fail, schematically drawn in Fig. 18.10. At last, only the 0 degree plies would carry the load according to the simulation. Therefore, the failure criterion for the analysis was different from test: while the initial crack in the weakest ply defined the failure for analysis, the experimental tests were performed until fracture of the whole compound (Fig. 18.11). However, it is difficult to detect the initial crack during the fatigue test in practice, because cracks may start somewhere inside the compound and can usually not be found with surface investigations [66]. For the simulation of the total fracture, stiffness degradation will have to be considered in an iterative way.

18.3.6 Example: Multifunctional truck cross member As example for a component, a truck cross member is presented made of a combination of continuous fiber-reinforced plastic and steel. Beside the function as an important structure of the truck main frame, the part represents a liquid/compressed natural gas (CNG) reservoir at the same time. The biggest challenge from a design point of view was the fixing within the truck main frame structure with steel brackets (Fig. 18.12). The cross member was made up of several functional layers. The inner layer of polyethylene high density (PEHD) makes the vessel airtight. During production, this liner is used as a core for the 90 degree winding and for the braiding process of the carbon fiber-reinforced plastic (CFRP) outer layers. The 90 degree winding carries the main pressure load from inside and the braided layers take over the longitudinal loads from the steel calottes and the forces from the cross member function. For the ABAQUS analysis, a simplified composite shell model was used consisting of five UD plies, which represents the axial yarn and the braider yarns [67].

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Fatigue Life Prediction of Composites and Composite Structures

Fig. 18.12 Truck cross member with steel brackets.

For the fatigue life assessment with FEMFAT laminate, two load cases were examined and compared with component tests: a sinusoidal bending and a sinusoidal torsion load pulsating between zero and maximum load (stress ratio R ¼ 0). The simulations gave a good agreement with regard to failure location, however, the predicted lifetime was much too low. The reason, of course, is that FEMFAT identifies failure based on the first failure in the 90 degree plies, what is not conform with the failure criterion in the component test, what is already a quite large delamination after detection. For every ply, the critical stress component was the normal stress σ 2 perpendicular to the fiber orientation. For both load cases, the critical ply was on the outside [58]. The results for torsion are shown in Fig. 18.13.

Fig. 18.13 Component testing with torsion load: observed crack (A) and simulated damage distribution for the most critical ply (B).

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18.3.7 Summary A fatigue life prediction method for laminates under random multiaxial loads, based on the S-N curves derived at the lamina level has been presented in this section. For the modeling, quasi-static and fatigue data of a UD carbon/epoxy laminate were measured longitudinal and transverse to the fiber directions under tensile and compressive loads and also for shear loading to characterize the anisotropic fatigue behavior of the material. The obtained experimental data were used as input parameters for the fatigue solver FEMFAT laminate which enables the assessment of fiber and interfiber fracture with ABAQUS composite shell elements, and recently delamination with ABAQUS composite solid elements, even for nonproportional loading. Fatigue life of a UD 45 degree laminate and a multilayer composite were calculated and validated with experimental results. As example for a structural component, a truck cross member was presented with test and simulation results. As a first step, the cyclic input material data were assessed in order to check the validity of the calculations. The simulated input parameters correlated well to the experimental inputs in and transverse to fiber direction and also for
45 degree orientation. However, clamping positions should be excluded from fatigue assessment, because finite shell element models could not represent the physical conditions adequately. Instead, solid elements can be used; however, they necessitate expensive numerical procedures and the experiences to be gained. The fatigue life of the UD 45 degree could be predicted very accurately. For the fatigue life calculation of the multilayered composites with the stacking sequence [0°/+45°/45°/90°]S, the fatigue software assumed failure as soon as the weakest 90 degree ply became critical (similar to the weakest link concept), whereas tests were performed until total failure which lead to a significant difference between simulation and test. However, it could be shown that considerations beyond this point can lead to improved predictions. For the accurate software-based prediction of total failure of the complete composite structure, additional research and development activities will be necessary. Especially stiffness degradation caused by fatigue-induced damage mechanisms must be considered in an iterative way [57, 68]. Stress and fatigue analyses could be conducted with iteratively adapted stiffness parameters. Mathematical models will have to be developed to describe the reduction of stiffness parameters in dependence of the damage evolution. In addition, delamination will have to be considered by taking into account stress components perpendicular to the laminate’s plane. However, only finite solid elements are able to deliver these stress components with sufficient accuracy.

18.3.8 Nomenclature Definitions according to [69]: R k t, R k c … tensile and compressive strength of UD lamina parallel to fiber direction. R ? t, R ? c … tensile and compressive strength of UD lamina transverse to fiber direction. Rk? … in-plane shear strength of UD lamina. RA?? … fracture resistance of an action-plane action parallel to the fiber direction against its fracture due to τ?? stressing acting on it.

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Fatigue Life Prediction of Composites and Composite Structures

p k t, p k c … inclination of (τ21, σ 2)-fracture curve at σ 2 ¼ 0. t for the range σ 2 > 0 (tension). c for the range σ 2 < 0 (compression).

18.4

Description of CCfatigue and case studies

A classical fatigue life prediction methodology based on the building block approach is presented in this section. The computer software in which this procedure has been implemented allows the estimation of the fatigue lifetime of UD or multidirectional laminates and structural components under multiaxial stress states of VA, when basic quasi-static and fatigue data are available. The procedure is capable of performing different tasks in an articulated method aiming the life prediction of each examined material under given loading patterns. The steps for the procedure include routines for cycle counting, the derivation of S-N curves, the establishment of appropriate CLDs, the estimation of the fatigue failure under multiaxial fatigue stress states, and finally the calculation of the failure index by using the linear Miner damage summation rule. A wide range of solvers has been implementing for each step of the procedure, allowing the benchmarking of selected methods and the selection of the most appropriate for each subproblem of the life prediction methodology. The efficiency of the CCfatigue software has been already validated in previous works (Chapter 11, Refs. [8, 33]) by comparisons of the theoretical predictions to available experimental data from two material systems. A third dataset for which data from quasi-static, constant amplitude, as well as VA loading patterns are available [25] is used in this chapter for the demonstration of the software and the reassessment of its predicting ability. Data from previously used datasets [8, 25] are used for the demonstration of the CCfatigue software for the fatigue life prediction under multiaxial stress states.

18.4.1 Datasets description The first experimental program used in this chapter for the demonstration of the CCfatigue software refers to the fatigue behavior of a material fabricated by a filament winding technique at
30 degree [25]. Specimens were cut from a GFRP (E-glass fibers with Epon826 epoxy resin) cylinder with a large enough diameter to supply samples that could be considered as flat. Results from quasi-static as well as fatigue loading patterns are provided. Two sets of constant amplitude fatigue data, at tensiontension, R ¼ 0.1, and reversed fatigue, R ¼  1, can be retrieved from Ref. [25]. In addition, two sets of VA fatigue results are also available; one referring to experiments under the WInd SPEctrum Reference (WISPER) spectrum [70], and a second one referring to fatigue data obtained after applying the WISPERX spectrum. The development of the WISPER standardized irregular spectrum was supervised by the International Energy Association (IEA) and a number of European industrial partners and research institutes active in the wind energy domain. The spectrum was based on the measurements of bending moment loadings at different sites and for

Computational tools for the fatigue life modeling and prediction

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different types of wind turbine rotor blades. A total number of nine different wind turbines with rotor diameters of between 11.7 and 100 m and rotor blades made of different materials such as metals, GFRP, wood, and epoxy resins were considered. The WISPER standardized spectrum consists of 132,711 loading cycles, a high percentage of which are of low amplitude. Therefore, a long period is needed for the failure of a specimen under this spectrum and it is for this reason that the WISPERX spectrum was derived. WISPERX is a short version of WISPER containing approximately 1/10th of the WISPER loading cycles, while theoretically producing the same damage as its parent spectrum. The WISPERX spectrum comprises 12,831 loading cycles since all cycles with a range of below level 17 of WISPER have been removed. All data are summarized in Table 18.8, and graphically presented in Fig. 18.14 (constant amplitude fatigue) and in Fig. 18.15 (VA under WISPER and WISPERX).

18.4.2 Multidirectional laminate [0/(
45)2/0]T Another dataset is used for the demonstration of the software ability to provide fatigue life predictions under multiaxial stress states and compare relevant theoretical models. The dataset is taken from Ref. [8] and consists of a series of S-N fatigue data from specimens cut from a multidirectional laminate at different angles. Experiments at different R-ratios were carried out to cover several cases between compressioncompression, tension-tension, and reversed fatigue loading. The material system was E-glass/polyester, with E-glass fibers being supplied by Ahlstrom Glassfibre and the polyester resin, Chempol 80 THIX, by Interchem. A systematic experimental investigation was undertaken, consisting of static and fatigue tests on straight edge specimens cut at various directions from a multidirectional laminate. The stacking sequence of the E-glass/polyester plate consisted of four layers, 2 UD, UD lamina of 100% aligned warp fibers, with a weight of 700 g/m2 as outer layers and two stitched laminae with fibers along
45 degree directions, of 450 g/m2, 225 g/m2 in each off-axis angle. Considering the UD layer fibers as being along the 0 degree direction, the layup can be encoded as [0/(
45)2/0]T. Specimens were cut using a diamond wheel at 0 degree, on-axis, and 15, 30, 45, 60, 75, and 90 degree off-axis directions. The fatigue results are presented in Figs. 18.16–18.19. Details about the experimental program and the data analysis can be found in Ref. [8] In addition to the constant amplitude fatigue experiments, 30 specimens cut at 0, 30, and 60 degree were experimentally investigated under a modified version of the WISPEX spectrum, denoted MWX in Ref. [8]. MWX is a shifted version of the original to produce only tensile loads. The lower positive stress level is the first level of the original WISPERX time series, in which level 25 is considered as zero stress level. The same number of cycles as WISPERX, that is, 12,831, is maintained. In all, 15 specimens at 0 degree were loaded at three different maximum stress levels, 10 specimens cut at 30 degree also at three different maximum stress levels, and five specimens cut at 60 degree at two different stress levels. The experimental results are presented in Fig. 18.20. The off-axis VA fatigue data will be used in the following paragraphs for the demonstration of the CCfatigue software ability to predict fatigue life of composite materials under multiaxial stress states of VA.

Table 18.8 Quasi static and fatigue data—FFA material CA, R 50

CA, R 51

WISPER

WISPERX

UTS (MPa)

σ max (MPa)

N

σ max (MPa)

N

σ max (MPa)

N

Spectrum passes

σ max (MPa)

N

spectrum passes

275.6 273.3

112.5 105.0 84.4 84.4 84.2 79.0 79.0 67.4 56.6 56.4 56.1 52.6 52.6 52.6 50.7 50.6 50.5 45.0 42.1 42.1 42.1 39.2

400 110 1000 700 1300 940 400 4900 32,000 29,900 80,300 2354 2095 1700 92,000 327,900 359,300 660,400 245,000 230,000 14,200 4,175,800

142.1 141.2 140.9 140.6 114.3 112.9 112.9 112.9 112.7 105.3 105.3 105.3 101.4 97.1 92.1 92.1 92.1 90.3 90.3 90.2 84.7 84.7

445 4200 2640 4600 89,800 46,300 35,100 31,300 6200 3940 3810 2000 125,770 768,700 198,000 112,000 46,700 2,685,200 1,841,500 227,009 602,100 567,900

217.1 217.1 194.3 194.3 194.3 194.3 194.3 194.3 194.3 194.3 194.3 182.9 171.4 171.4 171.4 171.4 171.4 171.4 171.4 171.4 171.4 160.0

232,000 194,000 871,000 761,000 725,000 725,000 675,000 536,000 437,000 327,000 149,000 3,645,000 3,380,000 2,981,000 2,319,000 1,920,000 1,671,000 1,471,000 1,390,000 1,123,000 991,000 2,531,000

1.75 1.46 6.56 5.73 5.46 5.46 5.09 4.04 3.29 2.46 1.12 27.47 25.47 22.46 17.47 14.47 12.59 11.08 10.47 8.46 7.47 19.07

171.4 171.4 171.4 171.4 171.4 171.4 171.4 171.4 171.4 148.6 148.6 148.6 148.6 148.6 148.6

438,194 289,044 258,928 135,061 122,230 77,565 70,900 66,275 33,282 986,119 598,577 422,455 263,450 221,448 160,724

34.15 22.53 20.18 10.53 9.53 6.05 5.53 5.17 2.59 76.85 46.65 32.92 20.53 17.26 12.53

39.2 39.2 36.8 36.8 36.8 36.8 36.8

4,153,400 3,975,500 2,530,000 2,200,000 1,540,000 443,000 280,000

84.5 79.0 79.0 79.0 79.0 78.7 78.7 68.0 65.8 65.8

779,200 391,000 111,000 93,600 61,000 4,286,000 136,300 11,254,000 1,310,000 596,000

148.6 148.6 148.6 148.6 148.6 137.1 137.1 137.1

11,475,000 10,281,000 8,157,000 4,344,000 3,909,000 11,398,000 9,143,000 8,450,000

86.47 77.47 61.46 32.73 29.45 85.89 68.89 63.67

660

Fatigue Life Prediction of Composites and Composite Structures

Fig. 18.14 Tensile strength and constant amplitude fatigue data—FFA material [25].

Fig. 18.15 Variable amplitude fatigue data—FFA material [25].

18.4.3 CCfatigue software application As mentioned in the previous paragraphs of this chapter, CCfatigue follows the building block approach and a classical fatigue life prediction procedure that comprises a number of sequentially executed modules. The process is based on an articulated algorithm consisting of a pre-processor for handling the input date, four to five basic steps

Computational tools for the fatigue life modeling and prediction

661

Fig. 18.16 Fatigue data and Power S-N curves under compression-compression loading, R ¼ 10 [8].

Fig. 18.17 Fatigue data and power S-N curves under reversed loading, R ¼ 1 [8].

(according to the needs), and a post-processor to produce (numerically and graphically) the output, as presented in Fig. 18.21: l

l

Cycle counting, to convert VA time series into blocks of certain numbers of cycles corresponding to constant amplitude and mean values. Interpretation of fatigue data to determine and apply the appropriate S-N formulation for the examined material.

662

Fatigue Life Prediction of Composites and Composite Structures

Fig. 18.18 Fatigue data and power S-N curves under tension-tension loading, R ¼ 0.1 [8].

Fig. 18.19 Fatigue data and power S-N curves under tension-tension loading, R ¼ 0.5 [8]. l

l

l

Selection of the appropriate formulation to take the mean stress effect on the fatigue life of the examined material into account. Use of the appropriate fatigue failure criterion to calculate the allowable number of cycles for each loading block that results after cycle counting, when multidirectional stress states develop. Calculate the sum of the partial damage caused to the material by each of the applied loading blocks estimated using the cycle counting method.

Computational tools for the fatigue life modeling and prediction

663

Fig. 18.20 Fatigue data under MWX spectrum loading [8].

Modeling constant amplitude fatigue behavior involves the determination of the S-N curves (plot of cyclic stress vs life), typically by grouping data at a single R-value. Interpretation of the fatigue behavior for the assessment of the mean stress effect results in the construction of the CLD. These two processes can be treated as separate steps, but are related in the sense that the CLD is constructed from the available S-N curves, or S-N curve fatigue data, while new S-N curves can be estimated from the CLD. In case of multiaxial stress states, for example, when laminates are loaded off-axis, it is necessary to derive CLDs for each off-axis angle, namely for each combination of the stress tensor components. An experimental campaign for a task like that is extremely laborious, only to cover a limited amount of selected cases, thus, constituting the implementation of fatigue failure criteria indispensable. Detailed information regarding the steps that concern the cycle counting methods, the interpretation of the fatigue data, CLDs, fatigue failure criteria, and damage accumulation rules, can be found in Ref. [8]. The CCfatigue software accommodates all the necessary modules to address all the aforementioned steps of a life prediction methodology through a sequential articulated procedure, as well as to perform individual calculations for the material behavior modeling. The welcome screen of the current version of the CCfatigue software is presented in Fig. 18.22. The five successive steps are shown at the left side with tabs. The user can run each step independently of the others, select the desired solver from those available in the software library, and obtain the solution. Results from different methods can easily be created for comparisons. The data structure of the output file for each step has been designed in such a way that it can be used as the input file for the following step. By adopting a modular infrastructure, each component of the software can be updated without any substantial amount of modifications to the existing software interface. New routines can be easily implemented in the software library, increasing the number

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Fatigue Life Prediction of Composites and Composite Structures

Variable amplitude spectrum

Input

Static strengths & constant amplitude experimental fatigue data

Processing

S-N Curve Interpretation of the CA fatigue data

Cycle Counting Convert VA load time series to CA blocks

Fatigue Failure Multiaxial stress states

Yes

Multiaxial fatigue failure criteria

No CLD Assess the effect of stress-ratio on the fatigue life

Design Allowables Partial damage coefficients

Damage Summation Miner or non-linear damage coefficient

Damage Index – life assessment

Output

Fig. 18.21 Flowchart of the CCfatigue software.

of solvers for each step and allowing the benchmarking of different methods established by different research groups, and strengthening collaborations. The application of the software for the life prediction of the selected materials and the demonstration of the modules for each step are presented in the following sections.

Computational tools for the fatigue life modeling and prediction

665

Fig. 18.22 CCfatigue software welcome screen.

18.4.4 Variable loading fatigue lifetime prediction The irregular loading patterns under which the material performance has to be predicted are traditionally subjected to a cycle counting analysis in order to count the occurrences of cycles of various characteristics (amplitude, mean value, etc.) in the spectrum. There are several methods for doing this analysis; four of the most commonly used are implemented in the CCfatigue software framework. The cycle counting analysis results to a matrix, containing groups of cycles with certain range and mean levels. A cumulative percentage of spectrum cycles (showing the percentage of the measured cycles with ranges more than given values) in the WISPERX spectrum is presented in Fig. 18.23. An alternative representation for the cycle counting results is shown in Fig. 18.24, where the numbers of cycles (the size of the bubbles) for certain range and mean stress values can be depicted by this bubble chart. As expected, see Fig. 18.24, >90% of the counted cycles have a stress range t j y1:tp, L > tp, M), that is, the probability that the composite material/component continues its operation after a time point t (less than life-time L) further than the present time tp. This is a definition, which is conditional on SHM data, that is, the observation sequence y1:tp. Details on the calculation of the conditional reliability function can be found in Ref. [25].
 d tj y1:tp , L > tp , M ¼ RUL

Z

∞


 R t + τj y1:tp ,L > tp ,M dτ

(20.4)

0

In prognostics, an estimate of the uncertainty that follows the mean RUL estimation is of utmost importance, in order to give a confidence of the predicted mean value. The calculation of confidence intervals is based on the calculation of the a% and (1  a)% lower and upper percentiles, respectively. It can be easily proved that the cumulative distribution function (CDF) for RUL can be defined at any time point utilizing the conditional reliability according to the following:

 Pr RULtp  tj y1:tp , M ¼ 1  R t + tp │y1:tp , M

(20.5)

718

20.4

Fatigue Life Prediction of Composites and Composite Structures

Prognostics framework

The main goal of a prognostics framework is to estimate the composite structure’s RUL by providing a probability density function. The framework consists of the training and online process, as presented in Fig. 20.2. The objective of the training process is to collect health monitoring data, extract features that characterize the degradation process, and estimate the model’s parameters. Based on the available training data features the parameters θ of the mathematical model that is utilized to provide the RUL predictions are estimated. After training the respective NHHSMMs, health monitoring data observations from an unseen case (online process) may feed the model, after similar future extraction, and obtain the mean RUL estimations (prognostics output) and the associated 90% confidence intervals.

20.4.1 Data processing and feature extraction In the framework of this study, AE and strain data are used. The available SHM data can be divided to training and testing sets. However, the raw AE and DIC data include noise so a feature extraction process is required in order to produce features with strong prognostic suitability. A set of three metrics, monotonicity, prognosability, and trendability, has been proposed in the relevant literature, which can be used as feature design properties [40–43]. Monotonicity characterizes a parameter’s general

Fig. 20.2 The prognostics framework for prediction of the remaining useful life.

In-situ fatigue damage analysis and prognostics

719

increasing or decreasing trend, prognosability measures the spread of a parameter’s failure value and finally, trendability indicates whether degradation histories of a specific parameter have the same underlying trend. In this study, the feature extraction process is based on monotonicity since a feature that is sensitive to the degradation process is desirable to have a monotonic trend [41, 42, 44]. Prognosability is excluded from the present feature extraction process since NHHSMM dictates that the last observation of the monitoring data must be unique and common for all the degradation histories. Finally, the feature extraction process does not take into account the influence of trendability. The main reason is that we wanted to benchmark the data fusion process against AE and DIC data keeping the computational complexity as low as possible. A second key element of the NHHSMM is that the monitoring data’s domain should be discrete. Different methods, such as vector quantization and clustering can be used to discretize the available monitoring data [45]. In this chapter, the unsupervised k-means algorithm is used to cluster and discretize the features extracted from the SHM data. The target of using k-means algorithm is to find the optimal number of discrete levels, which delivers features with maximum monotonicity. To quantify the monotonicity the modified Mann-Kendall (MMK) criterion is introduced in the following equation. D D X X


MMK ¼



   tj  ti  sgn y tj  yðti Þ

i¼1 j¼1, j>i D D X X


tj  ti



 100%

(20.6)

i¼1 j¼1, j>i

where y(ti) the8feature value9at time of measurement ti, D the number of measurements < 1 if x < 0 = and sgnðxÞ ¼ 0 if x ¼ 0 . : ; 1 if x > 0 The advantages of the MMK criterion, over the classical Mann-Kendal criterion [30], are explained in the following: l

l

Mann-Kendal (MK) values have not any informative meaning. For example, in the current case study the MK values’ range is (105, 4  105). However, MMK value as defined in Eq. (20.6) expresses a percentage of monotonicity in the range [1.1]. If MMK ¼ 1 the degradation history is strictly increasing, if MMK ¼  1 the degradation history is strictly decreasing. In any other case the degradation history is not strictly monotonic. Based on the MMK criterion each degradation history has the same monotonicity weight. On the other hand, the classical MK criterion is biased since a longer degradation history gives a higher MK value.

The objective of the feature extraction process as implemented in this study, is to obtain descritized degradation histories with the as high monotonicity as possible using features from AE data, DIC data, and fused ones.

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Fatigue Life Prediction of Composites and Composite Structures

20.4.2 Data fusion process The health monitoring data can be collected by using different sensing technologies. The process of extracting information from different monitoring techniques and integrate them into a consistent, accurate and reliable data set is known as data fusion and it has been already successfully applied to damage diagnostics [46–48]. In principle, data fusion can be implemented in three levels: l

l

l

raw multi-sensor data fusion, feature-level fusion, decision-level fusion.

Raw data fusion should be treated with caution as sensor recordings may have different acquisition, prefiltering and amplification settings. Additionally, raw data fusion needs to have as input commensurate data. As a result, the feature-level and decisionlevel fusion are the more commonly used. Combining features extracted from different sensors or monitoring techniques and integrating them into a single feature is known to enhance the diagnostics performance [49]. Data fusion for structural prognostics purposes has never been attempted according to the author’s best knowledge. It is expected that the prognostic performance should be improved when fusing SHM data from various monitoring techniques. The fusion scheme receives as inputs the quantized AE and DIC features, where the following equation explains the rationale behind fusing process.

ft ðDIC, AEÞ ¼

jM M i +X X j¼0

aij  DICj  AEi

(20.7)

i¼0

where ft is the fused output feature, aij are constant coefficients that control the weight of the exponential DIC and AE features’ product and M the maximum polynomial degree power that these features can use. The MMK criterion, Eq. (20.7), is adopted to enable the data fusion process and is expressed in Eq. (20.8). MMK is used as an objective function to be maximized and thus determine which polynomial degree M and constant coefficients aij give the most monotonic fused feature. "

# dk X dk  K X    X ðk Þ ðkÞ ðkÞ ðkÞ tj  ti  sgn fj ða, MÞ  fi ða, MÞ
 k¼1 i¼1 j¼1, j>i " # MMK aij , M ¼  100% dk dk K X   X X ðkÞ ðk Þ tj  ti k¼1 i¼1 j¼1, j>i (20.8)

In-situ fatigue damage analysis and prognostics

721

where K is the number of available training degradation histories (e.g., the number of (k) tested specimens), f(k) i the fused feature value at time of measurement ti for the kth specimen, dk the number of the kth specimen’s measurements and 8 9 < 1 if x < 0 = sgnðxÞ ¼ 0 if x ¼ 0 : : ; 1 if x > 0 The constant polynomial coefficients aij, for each polynomial degree M, are based on the optimization problem described in Eq. (20.9) with the monotonicity obtained by the MMK criterion as the objective function. For the aforementioned optimization problem, different optimization techniques were used, that is, Nelder-Mead, neural networks, particle swarm optimization (PSO), genetic algorithms, and OptQuest nonlinear programs (OQNLP). For this exercise, it was found that OQNLP is the most efficient optimization technique regarding the computational time of the parameters α*ij and M. The unconstrained optimization problem is formulated as

 α∗ij ¼ arg max aij MMK aij , M

(20.9)

In conclusion, the outputs of the proposed data fusion methodology are the optimum polynomial degree M and the optimum constant coefficients aij based on the MMK monotonicity, Eq. (20.9).

20.5

Case study

A laminate with [0/45/90]2s, lay-up were manufactured from a Hexcel AS4/8552 carbon-epoxy UD prepreg using hand lay-up, with a debulking procedure performed after every three plies. Afterwards, the laminate was cured using the autoclave process with a curing cycle as recommended by the manufacturer of the prepreg. The laminate was cut in rectangular specimens of 300  30 mm using a Proth Industrial liquidcooled saw and a central hole of 6 mm was drilled. Seven open-hole were used in the experimental campaign. These specimens were subjected to fatigue loading with maximum amplitude 90% of the static tensile strength (Fult ¼ 42.66 kN), R ¼ 0 and f ¼ 10 Hz. The tests were executed in a MTS 100 kN universal testing machine and they run up to failure. An AE system was used in order to perform AE measurements. Fig. 20.3 presents the schematic representation of the experimental set-up and the data acquisition process. Table 20.1 presents the cycles to failure for the tested specimens.

20.5.1 Strain data feature extraction The DIC technique enabled strain measurements in the entire surface of the specimen. Two Grasshopper3 5.0 MP Mono with Apo-Xenoplan 1.4/23 mm lenses and strain resolution of 200με were used while the analysis of the data was performed using the

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Fatigue Life Prediction of Composites and Composite Structures

MTS 100 kN Load Cell

VS900-M AE sensor #2

10V DIC trigger

MTS station

DIC cameras c c

DIC monitoring station

AE monitoring station

VS900-M AE sensor #1

34 dB amp

Vallen AMSY-6 acquisition system

Fig. 20.3 The schematic representation of the experimental set-up. Table 20.1 Cycles to failure of the open-hole specimens Coupon

Fatigue test conditions

Cycles to failure

Specimen 01 Specimen 02 Specimen 03 Specimen 04 Specimen 05 Specimen 06 Specimen 07

R¼ 0 F ¼10 Hz σ max ¼90% UTS [0/45/90]2s

63122 24239 22400 24015 13658 25101 29258

VIC-3D software supplied by the Correlated Solutions. Fig. 20.4 presents the axial strain distribution, strain in the load direction, as calculated at the maximum loading during the fatigue test of specimen02. Based on the analytical model of Lekhnitskii [50], which calculates the effect of a notch on the stress/strain distribution, the green rhomboid point (half a diameter distance for the hole center in the transverse direction), highlighted at the picture of 0 cycles, was chosen as the critical point to extract the axial strains. Fig. 20.5 presents the seven axial strain degradation histories, which were extracted for the aforementioned critical point.

In-situ fatigue damage analysis and prognostics

723

eyy [1] Lagrange 0.035 0.0328125 0.030625 0.0284375 0.02625 0.0240625 0.021875 0.0196875

0 cycles

5000 cycles

10,000 cycles

0.0175 0.0153125 0.013125 0.0109375 0.00875 0.0065625 0.004375 0.0021875 0

15,000 cycles

20,000 cycles

23,000 cycles

Fig. 20.4 Axial strain distribution of specimen02.

As discussed earlier, the final data feature should be presented in a discrete form by the clusters V that can be calculated using the MMK criterion. The MMK converges for the number of clusters V equal to 25 for the data, as presented in Fig. 20.6. Fig. 20.7 presents the final clustered axial strain data after the thresholding process.

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Fatigue Life Prediction of Composites and Composite Structures

0.04 0.035 0.03

specimen01 specimen02 specimen03 specimen04 specimen05 specimen06 specimen07

Axial strain

0.025 0.02 0.015 0.01 0.005 0

1

0

2

3

4

5

6

Fatigue life (cycles)

104

Fig. 20.5 Axial strain degradation histories of seven open-hole specimens. 1

Modified Mann-Kendal criterion

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Number of clusters (V)

Fig. 20.6 MMK monotonicity convergence of DIC data vs the number of clusters (V).

20.5.2 AE feature extraction An AMSY-6 Vallen, 8-channel AE system with four parametric input channels was used in this study. Two wide-band piezoelectric sensors, VS-900 M, with an external 34 dB preamplifier and a band-pass filter of 20–1200 kHz, were clamped on the specimens using a mechanical holder. In order to increase the conductivity between the AE sensors and the specimen, grease was applied on the surface of the sensors and pencil break tests were conducted before each experiment so as to ensure the conductivity. One parametric input channel was used to record the load and correlate it to the AE

In-situ fatigue damage analysis and prognostics

725

25

Clustered axial strain

20

specimen07 specimen01 specimen02 specimen03 specimen05 specimen06 specimen04

15

10

5

0

1

0

2

4

3

5

5.5

Fatigue life (cycles)

104

Fig. 20.7 Clustered axial strain degradation histories of seven open-hole specimens.

data. The AE acquisition threshold was set to 50 dB and the AE data set for each AE hit contained the duration (μs), rise time (μs), peak amplitude (dB), energy (1 eu ¼ 1018 J), the number of threshold crossings, and ratio rise time to amplitude. 1/A (1/amplitude) was found to have the highest monotonic observation sequences and it was selected as the AE feature to use. Similar to strain measurements, 1/A was calculated cumulatively in periodic time windows of 500 cycles. The respective degradation histories for seven specimens are shown in Fig. 20.8. 10–5

3.5

specimen01 specimen02 specimen03 specimen04 specimen05 specimen06 specimen07

3

1/A (1/dB)

2.5 2 1.5 1 0.5 0

0

1

2

3

4

Fatigue life (cycles)

Fig. 20.8 AE degradation histories of seven open-hole specimens.

5

6 104

726

Fatigue Life Prediction of Composites and Composite Structures

Although the MMK monotonicity converges for number of clusters 18, see Fig. 20.9, V ¼ 25 was selected for the AE data equal to the number of clusters for strain data. This way, the data fusion process becomes more efficient as the normalization of the AE and DIC features is avoided. Fig. 20.10 presents the final clustered AE data.

1 0.9

Modified Mann-Kendal criterion

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Number of clusters (V)

Fig. 20.9 MMK monotonicity convergence of AE data vs the number of clusters (V).

25

Clustered 1/A

20

specimen04 specimen07 specimen01 specimen03 specimen02 specimen05 specimen06

15

10

5

0

0

1

2

3

4

Fatigue life (cycles)

Fig. 20.10 Clustered AE degradation histories of seven open-hole specimens.

5

5.5 104

In-situ fatigue damage analysis and prognostics

727

20.5.3 Data fusion of AE and strain data The results of the optimization study, Eq. (20.9), are presented for various polynomial degrees M in Fig. 20.11. The MMK monotonicity converges for a polynomial degree M  5. Therefore, the polynomial degree is selected as M ¼ 5. For the selected polynomial degree M ¼ 5, Table 20.2 summarizes the optimization results regarding the constant coefficients aij. The fused features for polynomial degree M ¼ 5 and the aforementioned polynomial coefficients aij are shown in Fig. 20.12. Fig. 20.13 presents the MMK monotonicity for each SHM feature, that is, AE, DIC, and fused data and it is observed that the fused data have the highest monotonic rate. The data fusion process is presented in Fig. 20.14.

20.5.4 RUL estimations Seven degradation histories Y ¼ [y(1),y(2), …,y(7)] were available for each SHM technique (AE, DIC, and fused data). The training dataset employs six degradation

Modified Mann-Kendal

0.98

0.978

0.976

0.974

0.972

0.97

1

2

3

4

5 6 Polynomial degree (D)

7

8

10

9

Fig. 20.11 Modified Mann-Kendal value vs the polynomial degree. Table 20.2 Optimization results for M ¼ 5

DIC0 DIC1 DIC2 DIC3 DIC4 DIC5

AE0

AE1

AE2

AE3

AE4

AE5

39,955 783,606 412,001 922,063 292,6789 336,2406

953,757 1,989,894 411,5522 183,044 906,5035 0

743,892 746,044 271,9862 16,7071 0 0

471,0798 381,3022 829,036 0 0 0

882,5275 344,348 0 0 0 0

1,985,843 0 0 0 0 0

728

Fatigue Life Prediction of Composites and Composite Structures 1010

3.5 3

Fused function

2.5 2 specimen01 specimen02 specimen03 specimen04 specimen05 specimen06 specimen07

1.5 1 0.5 0

1

0

2

3

4

5

Fatigue life (cycles)

6 104

Fig. 20.12 Fused degradation histories of seven open-hole specimens.

Modified Mann-Kendal monotonicity

1 0.98 0.96 0.94 0.92 0.9 0.88

DIC

AE

FUSION

Fig. 20.13 Comparison between DIC, AE, and fusion MMK monotonicity.

histories in order to estimate the NHHSMM’s parameters θ and keeps the seventh degradation history as the testing prognostic dataset. The mean/median RUL and the 90% confidence intervals can be calculated using Eq. (20.5). The level of confidence intervals depends on the application, that is, for aerospace applications 90% and 95% are common values [37] and 90% will be adopted for this study. Figs. 20.15–20.18 present the RUL estimations of the three available SHM techniques for specimen02, specimen03, specimen04, and specimen06, respectively.

In-situ fatigue damage analysis and prognostics

Fig. 20.14 The data fusion process.

729

730

Fatigue Life Prediction of Composites and Composite Structures 10–4

3.5

Real RUL Mean AE RUL LB AE RUL of 90% Cl UB AE RUL of 90% Cl

3

Mean DIC RUL LB DIC RUL of 90% Cl UB DIC RUL of 90% Cl Mean Fusion RUL LB Fusion RUL of 90% Cl UB Fusion RUL of 90% Cl

RUL (cycles)

2.5 2 1.5 1 0.5 0

0

0.5

1

1.5

2

2.5 4

Fatigue life (cycles)

10

Fig. 20.15 Estimated mean RUL and 90% confidence intervals for specimen02. 104

3.5

Real RUL Mean DIC RUL LB DIC RUL of 90% CI UB DIC RUL of 90% CI Mean AE RUL LB AE RUL of 90% CI UB AE RUL of 90% CI Mean Fusion RUL LB Fusion RUL of 90% CI UB Fusion RUL of 90% CI

3

RUL (cycles)

2.5 2 1.5 1 0.5 0 0

2000

4000

6000

8000

10,000

12,000

14,000

16,000

18,000

Fatigue life (cycles)

Fig. 20.16 Estimated mean RUL and 90% confidence intervals for specimen03.

The RUL estimations converge quite satisfactorily with the real (experimental) RUL values. Based on the results shown in Figs. 20.15–20.18 the strain data provide the best RUL estimations, while fused data and AE provide fair results.

20.5.5 Performance metrics In order to quantify which data provide better predictions and validate or not the observations made for Figs. 20.15–20.18, various prognostic performance metrics are

In-situ fatigue damage analysis and prognostics

731

104

4

Real RUL Mean DIC RUL LB DIC RUL of 90% CI UB DIC RUL of 90% CI Mean AE RUL LB AE RUL of 90% CI UB AE RUL of 90% CI Mean Fusion RUL LB Fusion RUL of 90% CI UB Fusion RUL of 90% CI

3.5 3

RUL (cycles)

2.5 2 1.5 1 0.5 0

0

0.5

1

1.5 Fatigue life (cycles)

2

2.5

104

Fig. 20.17 Estimated mean RUL with 90% confidence intervals for specimen04.

4

10

3.5

Real RUL Mean AE RUL LB AE RUL of 90% Cl UB AE RUL of 90% Cl Mean DIC RUL LB DIC RUL of 90% Cl UB DIC RUL of 90% Cl Mean Fusion RUL LB Fusion RUL of 90% Cl UB Fusion RUL of 90% Cl

3

RUL (cycles)

2.5 2 1.5 1 0.5 0

0

0.5

1 1.5 Fatigue life (cycles)

2

2.5 104

Fig. 20.18 Estimated mean RUL with 90% confidence intervals for specimen06.

employed for the comparison. Eight prognostic performance metrics are used to evaluate the predictive performance of the three NHHSMMs trained with the different types of SHM features. Six of them are metrics widely used in the literature; precision, mean squared error (MSE), mean absolute percentage error (MAPE), median absolute percentage error (MdAPE), cumulative relative accuracy (CRA), and convergence (CEm) [51, 52]. The last two metrics monotonicity and confidence intervals distance convergence (CIDC) were very recently introduced by Eleftheroglou et al. [53].

732

Fatigue Life Prediction of Composites and Composite Structures

A brief description of these two new metrics is provided hereafter. The aforementioned prognostic performance metrics are defined in the following: 1. Precision Precision ¼

ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PD 2 ðEm ðti ÞEm ðti ÞÞ i¼1 D1

, where Em is the mean value of error Em and Em(ti) ¼

RULactual(ti) meanRUL(ti) and ti 2[1,D] is the discrete time moment when the ith SHM observation is recorded. 2. Mean Squared Error (MSE) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PD ðEm ðti ÞÞ2 i¼1 MSE ¼ . D 3. Mean Absolute Percentage Error (MAPE)

MAPE ¼ D1

PD  100Em ðti Þ  i¼1 RULactual ðti Þ.

4. Median Absolute Percentage Error (MdAPE) P  100Emd ðti Þ  MdAPE ¼ D1 D i¼1 RULactual ðti Þ, where Emd(ti) ¼ RULactual(ti)medianRUL(ti). 5. Cumulative Relative Accuracy (CRA) PD   RAðti Þ  m ðti Þ  where RA(ti) ¼ 1RULEactual CRA ¼ i¼1D ðti Þ . 6. Convergence (CEm) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CEm ¼ ðxc  t1 Þ2 + yc 2 .

where D1 X

xc ¼

D1 X

ðti + 1 2  ti 2 Þ  |Em ðiÞ|

i¼1 D1 X

2

i¼1

and yc ¼ ðti + 1  ti Þ  |Em ðiÞ|

ðti + 1  ti Þ  Em ðiÞ2

i¼1 D1 X

2

: ðti + 1  ti Þ  |Em ðiÞ|

i¼1

7. Monotonicity The prognostic’s function monotonicity can be measured based on the proposed MMK monotonicity criterion where y(ti) is replaced with meanRUL(ti). In case of the studied function, which is the RUL prediction function, the preferable value of MMK ¼ 1 since it is expecting that the composite structure’s RUL is decreasing monotonically during its lifetime. 8. Confidence Intervals Distance Convergence (CIDC) Goebel et al. [54] stated that as the amount of data increases during the fatigue life, the confidence intervals distance should converge. In order to quantify this statement, a new metric is introduced; the CIDC. This metric is an extension of the metric of convergence in Ref. [51] but in this case the centroid is under the confidence intervals distance curve. In general, lower Euclidian distance means faster convergence. Let (xc, yc) be the center of mass of the area under the confidence intervals distance curve, then the CIDC can be represented by the Euclidean distance between the (xc, yc) and the origin (t1,0), where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CIDC ¼ ðxc  t1 Þ2 + yc 2

In-situ fatigue damage analysis and prognostics

733

where D1 X

xc ¼

ðti + 1 2  ti 2 Þ  ðUCI ðiÞ  LCI ðiÞÞ

i¼1 D1 X

2

,yc ðti + 1  ti Þ  ðUCI ðiÞ  LCI ðiÞÞ

i¼1 D1 X

ðti + 1  ti Þ  ðUCI ðiÞ  LCI ðiÞÞ2

¼

i¼1 D1 X

2

ðti + 1  ti Þ  ðUCI ðiÞ  LCI ðiÞÞ

i¼1

and UCI, LCI, the upper and lower selected confidence intervals, respectively. The optimum values of the prognostic performance metrics are as follows and are presented in Tables 20.3–20.6 for all the specimens: Precision: minimum value MSE: minimum value MAPE: minimum value MdAPE: minimum value

CRA: maximum value Monotonicity: minimum value CEm: minimum value CIDC: minimum value

The best scores are highlighted and underlined. Based on the results, the model that uses strain data as health monitoring features outperforms for most of the performance metrics. Data fusion also scores better for at least one RUL estimation for all the metrics except the last one. Especially the results of the monotonicity demonstrate the potential of the data fusion to provide features with very strong monotonic behavior, which is a key factor for the success of the prognostics output.

Table 20.3 Prognostic performance metrics (Precision and MSE) Precision

SP1 SP2 SP3 SP4 SP5 SP6 SP7

MSE

AE

DIC

Fusion

AE

DIC

Fusion

10,242.2 1839.0 1599.5 4696.9 1359.8 2265.6 4201.6

10,321.1 3178.1 1517.0 1517.0 2264.5 1916.9 5444.5

12,158.0 3406.1 1803.2 4359.4 1118.1 3263.6 5976.0

313,929,177 5,157,196 49,833,377 65,596,414 123,908,090 11,177,939 26,771,274

398,746,894 17,305,198 13,801,781 13,801,781 55,856,930 3,747,730 30,611,449

400,438,288 14,521,109 17,525,406 29,438,165 46,194,571 10,439,022 36,091,801

Precision: strain and AE data score better for 3 RUL estimations each and fusion data scores better for 1 RUL. MSE: strain and AE data score better for 3 RUL estimations each and fusion data scores better for 1 RUL.

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Fatigue Life Prediction of Composites and Composite Structures

Table 20.4 Prognostic performance metrics (MAPE and MDAPE) MAPE

sp1 sp2 sp3 sp4 sp5 sp6 sp7

MDAPE

AE

DIC

Fusion

AE

DIC

Fusion

19.34 68.14 19,737 172,01 312,69 82,82 33,05

46.40 94.09 75.61 75.61 231.87 35.36 111.26

15.09 92.79 128.70 119.41 203.34 71.01 94.46

40.47 29.46 159.12 141.58 261.58 46.50 2.45

60.69 64.03 41.35 41.35 183.01 8.37 87.19

38.59 50.66 75.82 79.56 134.87 28.19 59.67

MAPE: strain data scores better for 2 RUL estimations, AE and fusion data score better for 2 RUL estimations. MDAPE: strain data scores better for 2 RUL estimations, AE and fusion data score better for 2 RUL estimations.

Table 20.5 Prognostic performance metrics (CRA and Monotonicity) CRA

sp1 sp2 sp3 sp4 sp5 sp6 sp7

Monotonicity

AE

DIC

Fusion

AE

DIC

Fusion

0.238 0.300 0.973 0.727 2.127 0.153 0.245

0.281 0.026 0.244 0.243 1.319 0.578 0.269

0.135 0.001 0.287 0.254 1.033 0.138 0.130

1 0.999 0.999 0.997 1 0.999 0.998

0.987 1 1 1 1 1 1

0.999 1 0.999 1 1 1 1

CRA: strain data scores better for 4 RUL estimations, AE data scores better for 2 RUL estimations and fusion data scores better for 1 RUL estimation. Monotonicity: strain data and fusion data performed equally for 5 RUL estimations and AE data scores better for 2 RUL estimations.

Table 20.6 Prognostic performance metrics (CEm and CIDC) CEm

sp1 sp2 sp3 sp4 sp5 sp6 sp7

CIDC

AE

DIC

Fusion

AE

DIC

Fusion

15,872 18,518 9766 15,746 6208 17,548 4491

17,405 19,171 8456 8456 5599 43,742 51,975

15,054 24,456 10,657 19,675 5670 63,208 63,950

24,486 10,898 8661 11,334 6041 11,343 12,678

21,450 10,823 7712 7712 5555 10,274 12,853

25,136 11,278 8529 11,094 5820 11,349 13,282

CEm: strain and AE data score better for 3 RUL estimations each and fusion data scores better for 1 RUL estimation. CIDC: strain data scores better for 6 RUL estimations and AE data scores better for 1 RUL estimation.

In-situ fatigue damage analysis and prognostics

20.6

735

Conclusions

The structural prognostics research field is a new dynamically rising field and it becomes more known, especially to the research and engineering community that works on the operation and maintenance, as it is a core element for the successful implementation of a condition-based maintenance framework. A great example is the very recently, funded under the European Union’s Horizon 2020 research and innovation program, real-time condition-based maintenance for adaptive aircraft maintenance planning (ReMAP—https://h2020-remap.eu/) project, grant agreement No 769288, where one of the main objectives is the development of health prognostics of aircraft composite structures using innovative data-driven machine learning algorithms. The main objective of the structural prognostics is to provide real-time estimations of the RUL of structures by blending machine learning algorithms, health monitoring data, and mechanics in order to design a prognostics framework. For the prediction of the RUL of composite structures subjected to fatigue loading, two types of prognostics frameworks have been developed. The first one uses physics-based algorithms while the second one uses data-driven algorithms. In this chapter, a data-driven probabilistic framework for the in-situ prognostics of composite structures subjected to fatigue loading was presented. The framework is able to provide real-time estimation of the RUL of composite structures by combining health monitoring data and the multistate degradation NHHSMM. Two different sources of health monitoring data, AE, and strain data, on a feature-level, were presented. Open-hole carbon/epoxy specimens were subjected to constant amplitude fatigue loading up to failure and DIC and AE techniques were employed, to monitor the fatigue tests and provide the required data. In addition, eight prognostic performance metrics were employed in order to compare the performance of the RUL estimations. A new data fusion approach was developed and the main objective was to produce hyper-features with high monotonicity. Although the degradation histories of the fused data had monotonicity higher than the monotonicity of the degradation histories of DIC and AE features, the fuse data did not provide always better estimations, indicating that the requirement of monotonicity is not enough and extra criteria should be involved. Nevertheless, the results demonstrate the potential of the proposed data fusion methodology and its evolvement by adding extra criteria, such as trendability, will enhance the performance of the fused data. In order to accommodate the phenomenon of the structural degradation over time and the belief that as the amount of data increases the confidence intervals should converge, two prognostic performance metrics, MMK monotonicity, and CIDC were very recently proposed by the authors and materialized these statements. Their results were similar to the results of the other metrics and their applicability was verified. The feature extraction process for the strain data was straightforward, as after the determination of the critical specimen’s point, the axial strain data were extracted via the DIC technique. The well-established analytical model of Lekhnitskii enhanced the feature performance indicating that mechanics can play an informative role on the

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Fatigue Life Prediction of Composites and Composite Structures

feature selection process. These results can be used in future toward the development of a hybrid model where mechanics can play a key role for the feature extraction process. Considering the importance of delivering a RUL prediction methodology that is generic enough and able to provide reliable real-time estimations, future research should focus mainly on two topics: l

l

Data fusion processes Real time uncertainty quantification

As discussed in Section 20.4.2, data fusion can be performed in three levels. There is not yet an established data fusion method that provides features, which enhance the RUL predictions significantly. Toward that direction, research should be performed on the selection and combination of appropriate SHM technologies, which will produce data that compliments each other and will enable the design of a super-feature that should fulfill the criteria of monotonicity, trendability, and prognosability. The prognostics framework should be flexible enough in order to accommodate any real time uncertainty, that is, unexpected loading event such as impact, which may dramatically change the operation life of the structure. This is important especially for the data-driven algorithms where the training of the algorithm is usually based on forecasted loading conditions. The algorithm should be designed in such a way that can recognize the change in the data histories and incorporate its effect on the analysis of the RUL.

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Index Note: Page numbers followed by f indicate figures, and t indicate tables. A Acoustic emission (AE), 6 Adaptive neuro-fuzzy inference systems (ANFIS), 350–351, 355–357, 371–372 Additive manufacturing (AM), 335 Aging, 1 AM. See Additive manufacturing (AM) American Society for Testing and Materials (ASTM), 2 Analysis of variance (ANOVA), 342 ANFIS. See Adaptive neuro-fuzzy inference system (ANFIS) Angle ply GFRP composite laminates dynamic mechanical analysis, 108 experimental procedure, material and specimens, 103 failure analysis, 127–128 fatigue behavior, 112–127 pure creep behavior, 109–111 quasistatic behavior, 108–109 set-up and instrumentation dynamic mechanical analysis, 104 fatigue experiments, 106–107 instrumentation, 107–108 pure creep experiment, 105 quasistatic experiments, 105 Anisomorphic CFL diagram, 427–428 cases of, 444 formulation, 441–443 procedure for construction, 443 procedure for predicting S-N curves, 443–444 Anisomorphic CLD, 392–393 ANNs. See Artificial neural networks (ANNs) ANOVA. See Analysis of variance (ANOVA) Artificial intelligence (AI) methods artificial neural networks (ANNs), 349 fuzzy logic techniques (FL), 349 genetic algorithms (GAs), 349

genetic programming (GP), 349 Artificial neural networks (ANNs), 349, 353–355, 369–370 Artificial neuron (AN), 353 Autogenous heating, 199 B Bayesian neural networks (BNNs), 713 BHE shear-lag model, 270–272, 271f Brevity and clarity criterion, 253–255 Broutman and Sahu model, 79–82 Budiansky-Hutchinson-Evans shear-lag model, 270–272 “Building block” approach, 639 C Carbon fiber (CF), 335 Carbon fiber-reinforced (CFRP) laminates, 135–136 Carbon fiber-reinforced plastic (CFRP), 653 Carbon nanofibers (CNFs), 143–144 Carbon nanotube (CNT), 135–136 Carboxyl-terminated butadiene acrylonitrile (CTBN), 173–175 CCfatigue, 639 Ceramic-matrix composites (CMCs), 269 Chopped strand mat (CSM), 144 Classical laminate theory (CLT), 540, 684 Poisson’s ratios, 546 shear moduli, 546 stiffness measurement, 543–545 Classic fatigue life prediction methodology assessment of mean stress effect, 391–393 CA fatigue data, 390–391 cycle counting, 389–390 damage summation, 394–395 fatigue failure criterion, 393–394 CLDs. See Constant life diagrams (CLDs) CLT. See Classical laminate theory (CLT) CMCs. See Ceramic-matrix composites (CMCs)

742

Composite Construction Laboratory (CCLab), 103 Composite laminates application to a thick laminate, 260–265 approaches for fatigue of, 240–245 characterization for failure parameters determination of fatigue strengths, 251–253 fatigue failure criteria, 250–251 degradation due to fatigue damage, 255–257 failure criteria and modes, 253–255 progressive damage development and progression, 257–260 Composite materials behavior of, 3–4 damage mechanisms, 99 durability of, 99 energy dissipation, 100 experimental characterization of component testing for research purposes, 6 specimen/component/full-scale testing, 6 specimen testing for engineering purposes, 5 specimen testing for research purposes, 5 fatigue, 3 fatigue nomenclature, 8–9 fatigue test parameters control mode, 7 loading pattern, 7 stress ratio, 7 testing frequency/strain rate, 7 testing temperature, 8 waveform, 8 loading patterns, 100–101 structural changes affects, 100 structures operating in open-air applications, 100–101 Composite materials, fatigue theories for fitting of experimental data, 85–93 prediction results, 93–94 residual strength models Broutman and Sahu, 79–82 Hahn and Kim, 84 Reifsnider and Stinchcomb, 82–83 Schaff and Davidson, 83–84 Compressed natural gas (CNG), 653

Index

Compression-compression (CC), 467 Computational intelligence methods adaptive neuro-fuzzy inference system (ANFIS), 355–357, 371–372 artificial neural networks, 353–355, 369–370 conventional methods comparison, 376–378 gene expression programming algorithm, 361–363, 374–375 genetic programming, 373 GFRP multidirectional laminate, 365–366 modeling examples, 363–375 multidirectional glass/epoxy laminate, 366–367 Computer aided design (CAD), 336, 337f Confidence intervals distance convergence (CIDC), 730–732 Connected anisomorphic CFL diagram, 452, 456 Constant amplitude (CA) fatigue, 388, 495–496 Constant amplitude loading, 10f, 13, 23, 33 Constant fatigue life (CFL) diagram, 428–429 Constant life diagrams (CLDs), 4, 47, 350–351, 389, 412–414, 495, 635 Constant life (CL) lines, 24–25 Convergence (CEm), 730–732 Crack initiation model, validation of application to C_TP_1, 484–488 application to C_TS_1, 477–478 application to C_TS_2, 482–484 application to G_TS_1, 478–482 application to G_TS_2, 488 local stress in matrix, 474–477 Creep-fatigue-loading, 102 profile, 107 Creeprecovery fatigue (CRF), 221 Critical element model, 82–83 “Critical Plane-Critical Component” approach, 640–641 Critical stress ratio, 427, 441–442, 455 Cross-ply CMCs SiC/MAS at 566°C in air atmosphere, 310 SiC/MAS at 1093°C in air atmosphere, 310–315 Cross-validation method, 368 Cumulative distribution function (CDF), 717–718

Index

Cumulative relative accuracy (CRA), 730–732 Cycle counting, 409–410 Cycle-dependent moduli, 557–559 Cyclic creep, 102 Cyclic dissipated energy, 100 Cyclic loadings, 702–703 D Damage tolerant (or fail-safe) design, 12 “Degree of fulfillment”, 357 Digital image correlation (DIC), 2, 714–715 system, 644 technique, 260–262 Digital image correlation technique (DICT), 51 DTU 10- MW reference turbine blade tip deflection, 578–579 fatigue loading, 578 Dynamic mechanical analysis (DMA), 104 E Ecole Polytechnique Federale de Lausanne (EPFL), 103 E-glass fibers with Epon826 epoxy resin (GFRP), 653 Energy method, 169–170 Error backpropagation (EBP), 353–354 Evolutionary algorithms, 358 F Failure analysis, 693 Failure tensor polynomial in fatigue (FTPF), 15–16, 681–682 Fatigue, 1 of composite materials, 3 experiments control mode, 49 grips, 49–50 loading rate, 50 specimen geometry, 50 system stiffness and alignment, 50 Fatigue analysis and life modeling fatigue experiments control mode, 49 grips, 49–50 loading rate, 50

743

specimen geometry, 50 system stiffness and alignment, 50 measurements and sensors, 50–52 S-N diagrams, 57f censoring and run-outs, 59–60 extrapolation of fatigue curves, 60 statistical description of fatigue data, 58–59 S-N formulations adding parameters, 63 S-N curves and CLD, 71–72 S-N curves that take into account the R-value, 65–71 statistical formulations, 64–65 strength-based S-N curve, 63–64, 71 two-parameter S-N curve, 61–63 specimens length and gauge length, 56 manufacturability and batch size, 54 maximum load, 54 planform, 55–56 tabs, 54–55 thickness, 56 width, 57 test frequency creep/time at mean stress effects, 52 frictional heating, 52–53 viscoelastic heating, 53 Fatigue behavior of composites with nanoparticles bending stress, 177–179 cyclic flexural bending fatigue life, 183–184 dispersion and morphology analysis, 184–187 materials specification, 175 setup for flexural bending fatigue, 183–184 specimen preparation, 175–177 static bending strength, 180–182 static testing instruments, 180 test equipment, 180 fatigue life, 112–116 fiber-reinforced composite 3D printing experimental analysis, 340–342 experimental setup, 339 printing equipment, 336–338 specimen preparation, 338–339 statistical analysis, 342–344

744

Fatigue behavior (Continued) hysteresis loops, 116–122 temperature—damage evolution, 122–127 of thick laminates, 245–249 of TP vs. TS laminates autogenous heating under fatigue loading, 218–219 creep-fatigue interaction, 219–225 fiber-dominated fatigue behavior, 212–218 influence of matrix nature, 225–228 matrix-dominated fatigue behavior, 205–212 Fatigue crack prediction crack initiation initiation literature, 466–467 multiscale criterion, 468–473 validation of crack initiation model, 473–488 definition of crack, 465–466 Fatigue damage accumulation existing models, 609–611 under nonstationary loading, 613–614 under stationary loading, 611–613 Fatigue damage accumulation model, 152–153 Fatigue damage hypothesis, 592f Fatigue damage modeling, 695–698 Fatigue damage simulator (FADAS), 519–534 Fatigue failure, 1, 250–251, 393–394, 705t Fatigue life modeling of composite materials, 376–378 adaptive neuro-fuzzy inference system (ANFIS), 355–357, 371–372 artificial neural networks, 353–355 conventional methods comparison, 376–378 gene expression programming algorithm, 361–363, 374–375 genetic programming, 373 GFRP multidirectional laminate, 365 modeling examples, 363–375 Fatigue life modeling/prediction CCfatigue and case studies datasets description, 656–657 multiaxial, 671–675 multidirectional laminate, 657–659

Index

software application, 660–664 variable loading, 665–670 FEMFAT laminate approach experimental work, 643–644 fatigue life prediction, 648–653 fatigue tests, 645–647 FE analysis, 647–648 for laminates, 640–643 multifunctional truck cross member, 653–654 results for quasi-static loading, 644–645 software for, 638–639 Fatigue life prediction classical laminate theory Poisson’s ratios, 546 shear moduli, 546 stiffness measurement, 543–545 under complex irregular loading, 23–32 damage mechanisms and stiffness progresses UD 0 degree, 552–554 UD 45 degree, 552 UD 90 degree, 550–552 evaluation of Nano-NSDM CSM/epoxy composites, 151–155 epoxy resin modified by silica nanoparticles, 146–147 for GFRP with nanoparticles, 147–150 nanoparticles/CSM/polymer hybrid nanocomposites, 160–167 thermoplastic nanocomposites, 155–159 fatigue experiments, 547–549 material properties degradation, 136 micromechanical-energy method energy method, 169–170 evaluation of Nano-EFAT model, 172–173 modeling strategy (Nano-EFAT model), 170–171 tests results, 171–172 micromechanical models, 143–145 Halpin-Tsai micromechanics model, 143–144 Nielsen micromechanics model, 144–145 modeling strategy for nanoparticle composites, 142

Index

Nano-NSDM based on Halpin-Tsai model, 145 based on Nielsen model, 146 normalized fatigue life model, 139–141 normalized stiffness degradation approach, 136–138 predictive method application CLT vs. experiment, 559–562 cycle-dependent moduli, 557–559 Poisson’s ratios, 557–559 predictive models applicability damage mechanisms, 562–563 experimental test procedures, 564 specimen geometry and embedded layers, 563–564 theories, 11–23 diverse fatigue considerations, 21–23 macroscopic failure, 12–16 multiscale modeling, 19–21 strength and stiffness degradation, 16–19 Fatigue life prediction, probabilistic fatigue reliability under non-stationary loading, 622–624 fatigue reliability under stationary loading, 618–620 FORM approach, 621–622 moments matching approach, 620–621 time-dependent fatigue reliability and, 624 Fatigue life prediction, under constant amplitude loading constant fatigue life (CFL) diagram approach, 428–429 extended anisomorphic constant fatigue life (CFL), 451–455 linear constant fatigue life (CFL) inclined Goodman diagram, 432–433 shifted Goodman diagram, 431–432 symmetric and asymmetric Goodman diagrams, 429–431 nonlinear constant fatigue life (CFL) anisomorphic CFL diagram, 441–444 Bell-shaped CFL diagram, 439–441 inclined Gerber diagram, 438 piecewise linear, 434 symmetric and asymmetric Gerber diagrams, 435–436

745

prediction of constant fatigue life (CFL) fiber-dominated fatigue behavior, 445–447 matrix-dominated fatigue behavior, 448–451 Fatigue life prediction, under cyclic complex stress constant life diagrams and S-N curves, 517–519 failure onset conditions, 507–509 fatigue damage simulator (FADAS), 519–534 computational procedure, 525–526 numerical predictions, 526–534 VA cyclic stresses, 523–525 loading-unloading-reloading (L-U-R), 499–502 ply response under quasi-static monotonic loading, 498–499 stiffness degradation post-failure material models, 506–507 pre-failure material models, 504–506 strength degradation due to cyclic loading, 509–517 Fatigue life prediction, under realistic loading conditions classic methodology assessment of mean stress effect, 391–393 CA fatigue data, 390–391 cycle counting, 389–390 damage summation, 394–395 fatigue failure criterion, 393–394 constant life diagrams, 412–414 cycle counting, 409–410 lifetime predictions, 414–417 multidirectional glass/epoxy laminate, 403–409 S-N curves, 410–412 strength degradation models acquiring strength degradation data, 395–397 life prediction using strength degradation, 397 in literature, 395 load sequence effects, 399–401 multiple block loadings, 401 strength degradation vs. classic methodology, 401–403

746

Fatigue lifetime prediction model, 93–94, 93f Fatigue nomenclature, 8–9 Fatigue peak stress, effects of, 283–286 Fatigue Progressive Damage Modeling (FPDM), 264 Fatigue strength, 251–253 Fatigue stress ratio, effects of, 286–288 Fatigue test parameters control mode, 7 loading pattern, 7 stress ratio, 7 testing frequency/strain rate, 7 testing temperature, 8 waveform, 8 Fatigue theories, composite materials fitting of experimental data, 85–93 prediction results, 93–94 residual strength models Broutman and Sahu, 79–82 Hahn and Kim, 84 Reifsnider and Stinchcomb, 82–83 Schaff and Davidson, 83–84 Fiber Bragg grating (FBG) sensors, 34 Fiberglass (FG), 335 Fiber-reinforced additive manufacturing (FRAM), 335 Fiber-reinforced ceramic-matrix composites effects of fatigue peak stress, 283–286 effects of fatigue stress ratio, 286–288 effects of fiber volume fraction, 298–304 effects of matrix crack spacing, 288–298 experimental comparisons cross-ply CMCs, 310–315 unidirectional CMCs, 305–310 fibers failure, 275–276 5D woven CMCs, 321 hysteresis theory, 276–281 interface debonding, 273–275 life prediction method, 281–282 matrix multi-cracking, 272–273 stress analysis, 270–272 2D woven CMCs Nextel 610/Aluminosilicate at 1000°C, 321 SiC/[Si-B4C] at 1200°C in air and in steam atmospheres, 318–320 SiC/SiC at 800°C in air atmosphere, 315 SiC/Si-N-C at 1000°C, 315–318 3D woven CMCs C/SiC at elevated temperature, 321–326

Index

SiC/SiC at elevated temperature, 327–330 Fiber-reinforced composite 3D printing experimental analysis, 340–342 experimental setup, 339 printing equipment, 336–338 specimen preparation, 338–339 statistical analysis, 342–344 Fiber-reinforced polymer (FRP), 101 Fibre failure (FF), 275–276, 506 Finite element (FE), 682 Finite element analysis (FEA), 474, 682 Finite element fatigue (FEMFAT), 639 Finite element method (FEM), 29–30, 642 First ply failure (FPF), 507–508 5D woven CMCs, 321 FL. See Fuzzy logic techniques (FL) Flexural bending fatigue life, 183–184 FOCUS6, 639 FPDM. See Fatigue Progressive Damage Modeling (FPDM) FRAM. See Fiber-reinforced additive manufacturing (FRAM) FTPF. See Failure tensor polynomial in fatigue (FTPF) Fused deposition modeling (FDM), 335–336 Fused filament fabrication (FFF), 335–336 Fuzzy logic (FL) methods, 349 fuzzy sets, 356 linguistic rules, 356 linguistic variables, 355 membership function (MF), 355 G GAs. See Genetic algorithms (GAs) Gene Expression Programming (GEP), 361–363, 374–375 Generalized delta rule, 369 Genetic algorithms (GAs), 349 Genetic algorithms, and OptQuest nonlinear programs (OQNLP), 721 Genetic programming (GP), 349, 373 GENOA, 639 GEP. See Gene Expression Programming (GEP) GFRP. See E-glass fibers with Epon826 epoxy resin (GFRP) Glass-fiber-reinforced plastic (GFRP), 173–175 Glass fiber-reinforced polyester (GFRP), 15

Index

Goodman diagram, 65–66, 66f, 432–433 GP. See Genetic programming (GP) Gradual degradation rules, 694 Graphene nanoplatelets (GPL), 175 Graphite nanoplatelets (GNPs), 175 H Hahn and Kim model, 84 Halpin-Tsai micromechanics model, 143–144 Hashin criteria, 253–255 Hidden layers, 353–354 Hysteresis-based damage parameter, 281 Hysteresis loops, 3–4, 116–122, 117–119f, 280 Hysteresis theory hysteresis-based damage parameter, 281 hysteresis loops, 280 interface complete debonding, 279–280 interface partial debonding, 277–279

747

Linear variable displacement transducer (LVDT), 50 Lin-Lin S-N curve, 667 Lin-log, 363–364, 390 Load-bearing capacity, 246 Load reduction ratio, 247–248 Local maximum principal stress (LMPS), 467 Log-log, 363–364 M

Karhunen-Loeve (KL) expansion method, 615

Markforged Mark Two (MKF), 335 Matrix crack spacing, effects of, 288–298 Matrix multi-cracking, 272–273 Maximum likelihood estimation (MLE) approach, 716 Mean absolute percentage error (MAPE), 730–732 Measure, correlate, and predict (MCP), 699 Mechanical combined loading (MCL), 582 Median absolute percentage error (MdAPE), 730–732 Meta-delamination, 197–198 Miner’s rule, 79–81 “Miner sum”, 418 MLP. See Multilayer perceptron (MLP) Modified fatigue strength ratio, 430 Modified Mann-Kendall (MMK) criterion, 719 Most probable point (MPP), 622 Multilayer perceptron (MLP), 353–354 Multiple R-value CFL diagram, 434 Multiscale Designer (MDS-C), 639 Multiscale modeling, 19–21 Multiwall carbon nanotube (MWCNT), 135–136

L

N

Large field of view (LFoV) scan, 589–591 Last ply failure (LPF), 507–508 Life prediction method, 281–282 Linear constant fatigue life (CFL) inclined Goodman diagram, 432–433 shifted Goodman diagram, 431–432 symmetric and asymmetric Goodman diagrams, 429–431 Linear damage accumulation rule (LDAR), 607–609 Linear elastic fracture mechanics (LEFM), 225

Nano-EFAT model, 170–171 Nielsen micromechanics model, 144–145 Nielsen “no-slip” model, 146–147 Nondestructive fatigue damage methods 3D micro-XCT ex situ, 588–591 TWLI in situ characterization method, 587–588 Nondestructive testing (NDT), 714 Nonhomogenous hidden semi-Markov model (NHHSMM), 713, 715–717 Nonlinear constant fatigue life (CFL) anisomorphic CFL diagram, 441–444

I Initiation zone, 243–244 Institution of Civil Engineers (ICE), 1 Interface debonding, 277–280 Interface shear stress degradation model, 282 Inter-fibre failure (IFF), 506 International Energy Association (IEA), 656–657 Intrinsic fatigue curve (IFC), 607–608 Isolife curves, 65 K

748

Index

Nonlinear constant fatigue life (CFL) (Continued) Bell-shaped CFL diagram, 439–441 inclined Gerber diagram, 438 piecewise linear, 434 symmetric and asymmetric Gerber diagrams, 435–436 Normalized fatigue life model, 139–141 Normalized stiffness degradation approach, 136–138 Normalized stiffness degradation model for nanocomposites (Nano-NSDM) based on Halpin-Tsai model, 145 based on Nielsen model, 146

Prognostics framework case study AE and strain data, 727 AE feature extraction, 724–726 performance metrics, 730–734 RUL estimations, 727–730 strain data feature extraction, 721–723 data fusion process, 720–721 data processing and feature extraction, 718–719 Progressive damage analysis (PDA), 19 Progressive Fatigue Damage Modeling (PFDM), 258 Pure creep behavior, 109–111

O

Q

One-parameter technique, 26–29 Open reading frame (ORF), 363 Optimat Blades database, 85

Quasistatic behavior, 108–109 Quasistatic stress-strain curves, 109f Quasistatic tensile properties, 109t

P

R

Palmgren-Miner rule, 609 Parametric Progressive Damage Model, 257 Particle swarm optimization (PSO), 721 Periodic boundary conditions (PBC), 474 PFDM. See Progressive Fatigue Damage Modeling (PFDM) Poisson’s ratios, 546, 557–559 Polyethylene high density (PEHD), 653 Polymer matrix composite (PMC), 195–196 Precision, mean squared error (MSE), 730–732 Prediction of constant fatigue life (CFL) fiber-dominated fatigue behavior, 445–447 matrix-dominated fatigue behavior, 448–451 Predictive method application cycle-dependent moduli, 557–559 Poisson’s ratios, 557–559 Probabilistic fatigue life prediction fatigue reliability under non-stationary loading, 622–624 fatigue reliability under stationary loading, 618–620 FORM approach, 621–622 moments matching approach, 620–621 time-dependent fatigue reliability and, 624 Probability density function (PDF), 612

Rainflow counting method, 26–29, 29f Rainflow-equivalent-range-mean counting, 33 Reifsnider and Stinchcomb model, 82–83 Remaining useful life (RUL), 711 Representative unit cell (RUC), 474 Residual stiffness, 16–19, 24–25 Residual strength, 5, 12, 15–17, 19, 24–30, 33 models Broutman and Sahu, 79–82 Hahn and Kim, 84 Reifsnider and Stinchcomb, 82–83 Schaff and Davidson, 83–84 Reversed loading, 9 S Safe-life design, 12 Scanning electron microscope (SEM), 160, 548–549 Schaff and Davidson model, 83–84 Sc-N curves, 17–19 SEM. See Scanning electron microscope (SEM) Serviceability limit state, 6 Shear moduli, 546 SLERA. See Strength-Life-Equal-RankAssumption (SLERA)

Index

S-N curve, 2, 5–7, 13, 17–19, 18f, 21, 24–25, 26f, 30–35, 410–412 Softening strips, 196–197 Standard Tessellation (STL), 336 Stiffness-based approach, for fatigue-life prediction classical laminate theory Poisson’s ratios, 546 shear moduli, 546 stiffness measurement, 543–545 damage mechanisms and stiffness progresses UD 0 degree, 552–554 UD 45 degree, 552 UD 90 degree, 550–552 fatigue experiments, 547–549 predictive method application CLT vs. experiment, 559–562 cycle-dependent moduli, 557–559 Poisson’s ratios, 557–559 predictive models applicability damage mechanisms, 562–563 experimental test procedures, 564 specimen geometry and embedded layers, 563–564 Stiffness controlled curves (Sc-N), 17 Stiffness degradation, 600 compression/compression fatigue, 600 tension/compression fatigue, 600–602 Strength degradation models acquiring strength degradation data, 395–397 life prediction using strength degradation, 397 in literature, 395 load sequence effects, 399–401 multiple block loadings, 401 strength degradation vs. classic methodology, 401–403 Strength-Life-Equal-Rank-Assumption (SLERA), 78, 410 Stress analysis, 270–272, 692–693 Structural health monitoring (SHM), 714–715 Sub-critical stress ratio, 456 Sudden degradation rules, 694–695

749

T Takagi-Sugeno ANFIS model, 357f Takagi-Sugeno model, 356–357 ANFIS model, 357f Tension-compression (TC), 467 Tension-tension (TT) fatigue, 467 damage localization, 597–600 gradual stiffness degradation, 594–596 stiffness drop studied using TWLI test setup, 592–594 Thermoplastic (TP)-based composites, 195 Thermosetting (TS)-based composites, 195 Thick laminates, 245–249 Three bundle cross-over point, 596 3D woven CMCs C/SiC at elevated temperature, 321–326 SiC/SiC at elevated temperature, 327–330 ThreeD X-ray computed tomography (XCT), 586–587, 590f TP- and TS-based composites, experimental study experimental set-up, 204–205 materials and specimens description, 203–204 Transilluminated white light imagining (TWLI), 586–587 Transmission electron microscopy (TEM), 160, 176f TWLI. See Transilluminated white light imagining (TWLI) 2D woven CMCs Nextel 610/Aluminosilicate at 1000°C, 321 SiC/[Si-B4C] at 1200°C in air and in steam atmospheres, 318–320 SiC/SiC at 800°C in air atmosphere, 315 SiC/Si-N-C at 1000°C, 315–318 U UEET. See Ultra Efficient Engine Technology (UEET) Ultimate compressive stress (UCS), 33, 392 Ultimate tensile stress (UTS), 33, 108–109, 392 Ultra Efficient Engine Technology (UEET), 269 Uncertainty modeling of external loading, 614–615 of material properties, 615–618

750

V Vacuum-assisted resin transfer molding (VARTM), 403 Variable amplitude (VA) fatigue, 387, 495–496 Variable amplitude loading, 7, 12, 17, 24, 30, 33–34 Vertical displacement transducers (LVDTs), 105 Viscoelastic polymer matrix, 101 W Wear-in Refs, 226 Wear-out model, 13 Weibull model, 281–282 Wind turbine blades deflection of, 575–576 fatigue loading, 573–575 load-carrying composite in fatigue measurement, 582–586 fatigue testing, 580–582 load distribution on, 571–572 loads on wind turbine rotor, 570–571 stresses in blade, 572–573 Wind turbine rotor characterization of, 698–700 fatigue life prediction of fatigue damage criterion, 692–698 loading, 686–688

Index

modeling technique, 684–686 stages of investigation, 682–683 static analysis, 688–692 implementation on fatigue modeling, 700–706 WISPERX spectrum, 85 Wovenply thermoplastic composites fatigue behavior autogenous heating in polymer matrix, 199–200 creep-fatigue interaction, 201–202 influence of reinforcement architecture, 197–198 influence of stress concentration, 202–203 influence of viscous effects, 200–201 TP vs. TS composites, 196–197 X XCT. See X-ray computer tomography technique (XCT) X-ray computer tomography technique (XCT), 588–589 X-ray tomography, 2 Y Ye’s Model, 153–154 Yield stress (YS), 108–109