Durability of Ceramic-Matrix Composites (Woodhead Publishing Series in Composites Science and Engineering) 0081030215, 9780081030219

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Durability of Ceramic-Matrix Composites (Woodhead Publishing Series in Composites Science and Engineering)
 0081030215, 9780081030219

Table of contents :
Cover
Durability of Ceramic-Matrix Composites
Copyright
Contents
Preface
Acknowledgments
1 Introduction and overview of ceramic-matrix composites
1.1 Introduction
1.2 Application of ceramic-matrix composites
1.2.1 Reinforced fibers
1.2.2 Interface phase
1.2.3 Ceramic matrix
1.2.4 Application in aero engines
1.2.5 France
1.2.6 United States
1.2.6.1 High-performance turbine engine technology project
Joint technology demonstration engine
Advanced turbine engine gas generator
Joint turbine advanced gas generator
Joint expendable turbine engine concept demonstrator
1.2.6.2 Ultraefficient engine technology Program
1.2.6.3 Application on the F414 aero engine
1.2.6.4 Federal Aviation Administration continuous lower energy, emissions, and noise program in the United States
1.2.7 Japan
1.2.8 Application in rocket engines
1.2.9 Application in Scramjet Engine
1.2.10 Application in thermal protection systems
1.3 Overview of tensile behavior of ceramic-matrix composites
1.3.1 Experimental observation
1.3.1.1 Unidirectional ceramic-matrix composites
1.3.1.2 Cross-Ply Ceramic-Matrix Composites
1.3.1.3 2D ceramic-matrix composites
1.3.1.4 2.5D ceramic-matrix composites
1.3.1.5 3D ceramic-matrix composites
1.3.2 Theoretical Analysis
1.3.2.1 Initiation matrix cracking
1.3.2.2 Matrix multicracking evolution
1.3.2.3 Fibers Failure
1.3.2.4 Stress–strain curve
1.4 Overview of fatigue behavior of ceramic-matrix composites
1.4.1 Fatigue hysteresis behavior
1.4.1.1 Experimental observation
1.4.1.2 Theoretical analysis
1.4.2 Interface wear behavior
1.4.2.1 Experimental observation
1.4.2.2 Theoretical analysis
1.4.3 Fibers strength degradation
1.4.4 Oxidation embrittlement
1.4.5 Modulus degradation
1.4.6 The effect of loading frequency
1.4.7 The Effect of Stress Ratio
1.5 Overview of lifetime prediction methods of ceramic-matrix composites
1.5.1 S–N curve
1.5.1.1 Unidirectional ceramic-matrix composites
1.5.1.2 Cross-ply ceramic-matrix composites
1.5.1.3 2D ceramic-matrix composites
1.5.1.4 2.5D ceramic-matrix composites
1.5.1.5 3D ceramic-matrix composites
1.5.2 Fatigue life prediction
1.6 Conclusion
References
Further reading
2 Matrix cracking of ceramic–matrix composites
2.1 Introduction
2.2 First-matrix cracking in an oxidation environment at elevated temperature
2.2.1 Stress analysis
2.2.1.1 Downstream stress
2.2.1.2 Upstream stress
2.2.2 Interface debonding
2.2.3 Matrix cracking stress
2.2.4 Results and discussion
2.2.4.1 Effect of fiber-volume fraction on time-dependent fiber–matrix interface debonding and first-matrix cracking stress
2.2.4.2 Effect of fiber–matrix interface debonded energy on time-dependent fiber–matrix interface debonding and first-matri...
2.2.4.3 Effect of fiber–matrix interface shear stress on the time-dependent fiber–matrix interface debonding and first-matr...
2.2.4.4 Effect of oxidation temperature on time-dependent fiber–matrix interface debonding and first-matrix cracking stress
2.2.5 Experimental comparisons
2.3 Matrix multicracking evolution considering fibers poisson contraction
2.3.1 Stress analysis
2.3.2 Interface debonding
2.3.3 Multiple matrix cracking
2.3.4 Results and discussion
2.3.4.1 Effect of fiber–matrix interface frictional coefficient on fiber–matrix interface debonding and matrix multicrackin...
2.3.4.2 Effect of fibers poisson ratio on fiber–matrix interface debonding and matrix multicracking evolution
2.3.4.3 Effect of fiber-volume fraction on fiber–matrix interface debonding and matrix multicracking evolution
2.3.4.4 Effect of applied cycle number on fiber–matrix interface shear stress and matrix axial stress distribution
2.3.5 Experimental comparisons
2.3.5.1 Unidirectional SiC/CAS composite
2.3.5.2 Unidirectional SiC/CAS-II composite
2.3.5.3 Unidirectional SiC/borosilicate composite
2.4 Matrix multicracking evolution considering interface oxidation
2.4.1 Stress analysis
2.4.2 Interface debonding
2.4.3 Multiple matrix cracking
2.4.4 Results and discussion
2.4.4.1 Effect of fiber volume fraction on fiber–matrix interface debonding and matrix cracking evolution
2.4.4.2 Effect of fiber–matrix interface shear stress on interface debonding and matrix cracking evolution
2.4.4.3 Effect of fiber–matrix interface debonded energy on fiber–matrix interface debonding and matrix cracking evolution
2.4.4.4 Effect of oxidation temperature on fiber–matrix interface debonding and matrix cracking evolution
2.4.4.5 Effect of oxidation time on fiber–matrix interface debonding and matrix cracking evolution
2.4.4.6 Comparisons of matrix cracking evolution with and without oxidation
2.4.5 Experimental comparison
2.4.5.1 Unidirectional SiC/CAS composite
2.4.5.2 Unidirectional SiC/borosilicate composite
2.4.5.3 Mini–SiC/SiC composite
2.5 Matrix multicracking evolution of cross-ply ceramic–matrix composites
2.5.1 Stress analysis
2.5.1.1 Undamaged state
2.5.1.2 Matrix cracking mode 1
2.5.1.3 Matrix cracking mode 2
2.5.1.4 Matrix cracking mode 3
2.5.1.5 Matrix cracking mode 5
2.5.2 Energy balance approach
2.5.2.1 Mode 1 cracking evolution
2.5.2.2 Mode 2 cracking evolution
2.5.2.3 Mode 3 cracking evolution
2.5.2.4 Mode 3 cracking occurring between two mode 1 cracks
2.5.2.5 Mode 5 cracking occurring between two mode 1 cracks
2.5.3 Results and discussion
2.5.3.1 Mode 1 cracking evolution
2.5.3.2 Mode 2 cracking evolution
2.5.3.3 Mode 3 cracking evolution
2.5.3.4 New cracking mode 3 and mode 5 occurring between two existing mode 1 transverse cracks
2.6 Conclusion
References
Further reading
3 Tensile strength of ceramic-matrix composites
3.1 Introduction
3.2 Tensile strength under multiple fatigue loading
3.2.1 Stress analysis
3.2.2 Damage models
3.2.2.1 Matrix multicracking evolution
3.2.2.2 Fiber–matrix interface debonding
3.2.2.3 Interface wear
3.2.2.4 Fibers failure
3.2.3 Results and discussion
3.2.3.1 Single matrix cracking
3.2.3.2 Matrix multicracking
3.2.4 Experimental comparisons
3.3 Tensile strength under cyclic loading at elevated temperatures in oxidative environments
3.3.1 Residual strength model
3.3.2 Results and discussion
3.3.2.1 Effect of peak stress on the composite residual strength and fibers fracture
3.3.2.2 Effect of fiber–matrix interface shear stress on composite residual strength and fibers fracture
3.3.2.3 Effect of fiber Weibull modulus on composite residual strength and fibers fracture
3.3.2.4 Effect of fiber strength on composite residual strength and fibers fracture
3.3.2.5 Effect of testing temperature on composite residual strength and fibers fracture
3.3.3 Experimental comparisons
3.3.3.1 SiC/[Si–N–C] composites at room and elevated temperatures
3.3.3.2 SiC/SiC composites at elevated temperatures
3.3.3.3 Nextel 720/Alumina composites at elevated temperatures
3.4 Conclusion
References
Further reading
4 Interface debonding and sliding of ceramic-matrix composites
4.1 Introduction
4.2 Interface debonding and sliding under different loading sequences
4.2.1 Stress analysis
4.2.1.1 Initial loading
4.2.1.2 Unloading
4.2.1.3 Reloading
4.2.2 Interface slip lengths
4.2.2.1 Interface debonding length
4.2.2.2 Interface counter-slip length
4.2.2.3 Interface new-slip length
4.2.3 Results and discussion
4.2.3.1 Effect of loading sequence on the fiber–matrix interface debonding
4.2.3.2 Effect of applied cycle number on fiber–matrix interface debonding
Loading sequence of Case 2
Loading sequence of Case 3
Loading sequence of Case 4
Loading sequence of Case 5
4.2.3.3 Effect of peak stress level on fiber–matrix interface debonding
Loading sequence of Case 2
Loading sequence of Case 3
Loading sequence of Case 4
Loading sequence of Case 5
4.2.3.4 Effect of arbitrary loading sequence on fatigue hysteresis and fiber–matrix interface sliding
4.2.3.5 Effect of fiber volume fraction on fatigue hysteresis and fiber–matrix interface sliding
4.2.3.6 Effect of matrix crack spacing on the fatigue hysteresis loops and fiber–matrix interface sliding
4.2.3.7 Effect of interface properties on the fatigue hysteresis and fiber–matrix interface sliding
4.2.3.8 Effect of fiber–matrix interface wear on the fatigue hysteresis loops and fiber–matrix interface sliding
4.2.4 Experimental comparisons
4.2.4.1 Unidirectional C/SiC composite
First stage fatigue peak stress
Second stage fatigue peak stress
Third stage fatigue peak stress
4.2.4.2 Unidirectional SiC/calcium alumina silicate-II composite
4.2.4.3 Cross-ply C/SiC composite
4.3 Hysteresis dissipated energy under multiple loading sequences
4.3.1 Hysteresis theories
4.3.2 Results and discussion
4.3.2.1 Effect of fiber-volume fraction on multiple loading fiber–matrix interface sliding
4.3.2.2 Effect of matrix crack spacing on multiple loading fiber–matrix interface sliding
4.3.2.3 Effect of low-peak stress level on multiple loading fiber–matrix interface sliding
4.3.2.4 Effect of high-peak stress level on multiple loading fiber–matrix interface sliding
4.3.2.5 Effect of fatigue stress range on multiple loading fiber–matrix interface sliding
4.3.2.6 Comparisons between single and multiple loading stress levels on fiber–matrix interface sliding
4.3.2.7 Comparisons between different loading sequences on fiber–matrix interface sliding
4.3.3 Experimental comparisons
4.3.3.1 C/SiC composite
4.3.3.2 SiC/SiC composite
4.4 Conclusion
References
Further reading
5 Damage evolution of ceramic-matrix composites under cyclic fatigue loading
5.1 Introduction
5.2 Hysteresis-based damage parameters
5.3 Tensile loading–unloading damage evolution
5.3.1 Results and discussion
5.3.1.1 Effect of fiber-volume fraction on hysteresis loops and hysteresis-based damage parameters
5.3.1.2 Effect of matrix cracking space on hysteresis loops and hysteresis-based damage parameters
5.3.1.3 Effect of fiber–matrix interface shear stress on hysteresis loops and hysteresis-based damage parameters
5.3.1.4 Effect of fiber–matrix interface debonded energy on hysteresis loops and hysteresis-based damage parameters
5.3.1.5 Effect of fibers failure on hysteresis loops and hysteresis-based damage parameters
5.3.2 Experimental comparisons
5.3.2.1 Unidirectional C/SiC composite
5.3.2.2 Cross-ply C/SiC composite
5.3.2.3 2D C/SiC composite
5.3.2.4 2.5D C/SiC composite
5.3.2.5 3D braided C/SiC composite
5.3.2.6 3D needled C/SiC composite
5.4 Cyclic fatigue damage evolution
5.4.1 Results and discussion
5.4.1.1 Effect of fiber volume fraction on fiber–matrix interface debonding and hysteresis-based damage parameters
5.4.1.2 Effect of fatigue peak stress on fiber–matrix interface debonding and hysteresis-based damage parameters
5.4.1.3 Effect of fatigue stress ratio on hysteresis-based damage parameters
5.4.1.4 Effect of matrix crack spacing on fiber–matrix interface debonding and hysteresis-based damage parameters
5.4.1.5 Effect of matrix crack mode on fiber–matrix interface debonding and hysteresis-based damage parameter
5.4.1.6 Effect of woven structure on hysteresis-based damage parameters
5.4.2 Experimental comparisons
5.4.2.1 Unidirectional ceramic-matrix composites
SiC/calcium alumina silicate composite at room temperature
SiC/calcium alumina silicate-II composite at room temperature
SiC/1723 composite at room temperature
C/SiC composite at room temperature
C/SiC composite at elevated temperature
5.4.2.2 Cross-ply ceramic-matrix composites
SiC/calcium alumina silicate composite at room temperature
SiC/calcium alumina silicate composite at 700°C in air atmosphere
SiC/calcium alumina silicate composite at 850°C in air atmosphere
C/SiC composite at room temperature
C/SiC composite at 800°C in air atmosphere
SiC/MAS-L composite at 800°C and 1000°C in inert atmosphere
5.4.2.3 2D ceramic-matrix composites
SiC/SiC composite at 600°C, 800°C, and 1000°C in inert condition
SiC/SiC composite at 1000°C in air and in steam atmosphere
SiC/SiC composite at 1200°C in air and steam atmospheres
SiC/SiC composite at 1300°C in air atmosphere
5.4.2.4 3D braided ceramic-matrix composites
5.5 Static fatigue damage evolution
5.5.1 Results and discussion
5.5.1.1 Effect of fatigue peak stress on static fatigue damage evolution
5.5.1.2 Effect of matrix crack spacing on static fatigue damage evolution
5.5.1.3 Effect of fiber volume fraction on static fatigue damage evolution
5.5.1.4 Effect of oxidation temperature on static fatigue damage evolution
5.5.2 Experimental comparisons
5.6 Conclusion
References
Further reading
6 Fatigue life prediction of ceramic-matrix composites based on hysteresis dissipated energy
6.1 Introduction
6.2 Theoretical analysis
6.3 Results and discussions
6.3.1 Effects of fatigue peak stress on fiber–matrix interface debonding, HDE, and HDE-based damage parameters
6.3.2 Effects of fatigue stress ratio on HDE and HDE-based damage parameters
6.3.3 Effects of matrix crack spacing on fiber–matrix interface debonding, HDE, and HDE-based damage parameters
6.3.4 Effects of fiber volume fraction on fatigue life, fiber–matrix interface debonding, HDE, and HDE-based damage parameters
6.4 Experimental comparisons
6.4.1 Unidirectional ceramic-matrix composites
6.4.1.1 SiC/CAS composite
6.4.1.2 SiC/1723 composite
6.4.2 Cross-ply ceramic-matrix composites
6.4.2.1 SiC/MAS at 566°C in air atmosphere
6.4.2.2 SiC/MAS composite at 1093°C in air atmosphere
6.4.3 2D ceramic-matrix composites
6.4.3.1 C/SiC composites at room temperature
6.4.3.2 SiC/SiC composites at room temperature
6.4.3.3 C/SiC composites at elevated temperature
6.4.3.4 SiC/SiC composites at elevated temperatures
6.4.4 2.5D ceramic-matrix composites
6.4.5 3D ceramic-matrix composites
6.4.5.1 C/SiC at room and elevated temperatures
6.4.5.2 SiC/SiC composite at elevated temperature
6.5 Conclusion
References
Further reading
Index
Back Cover

Citation preview

Durability of Ceramic-Matrix Composites

Woodhead Publishing Series in Composites Science and Engineering

Durability of CeramicMatrix Composites

Li Longbiao

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 Copyright © 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. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-08-103021-9 (print) ISBN: 978-0-08-103022-6 (online) For information on all Woodhead Publishing publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Matthew Deans Acquisitions Editor: Gwen Jones Editorial Project Manager: Isabella C. Silva Production Project Manager: Surya Narayanan Jayachandran Cover Designer: Mark Rogers Typeset by MPS Limited, Chennai, India

Contents

Preface Acknowledgments

ix xi

1

1 1 2 2 2 4 4 4 6 8 9 11 12 14 16 25 28 31 35 37 37 38 40 44

Introduction and overview of ceramic-matrix composites 1.1 Introduction 1.2 Application of ceramic-matrix composites 1.2.1 Reinforced fibers 1.2.2 Interface phase 1.2.3 Ceramic matrix 1.2.4 Application in aero engines 1.2.5 France 1.2.6 United States 1.2.7 Japan 1.2.8 Application in rocket engines 1.2.9 Application in Scramjet Engine 1.2.10 Application in thermal protection systems 1.3 Overview of tensile behavior of ceramic-matrix composites 1.3.1 Experimental observation 1.3.2 Theoretical Analysis 1.4 Overview of fatigue behavior of ceramic-matrix composites 1.4.1 Fatigue hysteresis behavior 1.4.2 Interface wear behavior 1.4.3 Fibers strength degradation 1.4.4 Oxidation embrittlement 1.4.5 Modulus degradation 1.4.6 The effect of loading frequency 1.4.7 The Effect of Stress Ratio 1.5 Overview of lifetime prediction methods of ceramic-matrix composites 1.5.1 S N curve 1.5.2 Fatigue life prediction 1.6 Conclusion References Further reading

45 45 57 58 58 73

vi

Contents

2

Matrix cracking of ceramic matrix composites 75 2.1 Introduction 75 2.2 First-matrix cracking in an oxidation environment at elevated temperature 75 2.2.1 Stress analysis 78 2.2.2 Interface debonding 81 2.2.3 Matrix cracking stress 82 2.2.4 Results and discussion 83 2.2.5 Experimental comparisons 86 2.3 Matrix multicracking evolution considering fibers poisson contraction 86 2.3.1 Stress analysis 92 2.3.2 Interface debonding 94 2.3.3 Multiple matrix cracking 95 2.3.4 Results and discussion 96 2.3.5 Experimental comparisons 100 2.4 Matrix multicracking evolution considering interface oxidation 104 2.4.1 Stress analysis 106 2.4.2 Interface debonding 106 2.4.3 Multiple matrix cracking 108 2.4.4 Results and discussion 108 2.4.5 Experimental comparison 115 2.5 Matrix multicracking evolution of cross-ply ceramic matrix composites 121 2.5.1 Stress analysis 121 2.5.2 Energy balance approach 125 2.5.3 Results and discussion 132 2.6 Conclusion 140 References 140 Further reading 143

3

Tensile strength of ceramic-matrix composites 3.1 Introduction 3.2 Tensile strength under multiple fatigue loading 3.2.1 Stress analysis 3.2.2 Damage models 3.2.3 Results and discussion 3.2.4 Experimental comparisons 3.3 Tensile strength under cyclic loading at elevated temperatures in oxidative environments 3.3.1 Residual strength model 3.3.2 Results and discussion 3.3.3 Experimental comparisons 3.4 Conclusion References Further reading

145 145 149 151 152 160 163 166 170 172 180 186 186 191

Contents

vii

4

Interface debonding and sliding of ceramic-matrix composites 4.1 Introduction 4.2 Interface debonding and sliding under different loading sequences 4.2.1 Stress analysis 4.2.2 Interface slip lengths 4.2.3 Results and discussion 4.2.4 Experimental comparisons 4.3 Hysteresis dissipated energy under multiple loading sequences 4.3.1 Hysteresis theories 4.3.2 Results and discussion 4.3.3 Experimental comparisons 4.4 Conclusion References Further reading

193 193 194 196 199 203 221 238 240 242 252 268 270 272

5

Damage evolution of ceramic-matrix composites under cyclic fatigue loading 5.1 Introduction 5.2 Hysteresis-based damage parameters 5.3 Tensile loading unloading damage evolution 5.3.1 Results and discussion 5.3.2 Experimental comparisons 5.4 Cyclic fatigue damage evolution 5.4.1 Results and discussion 5.4.2 Experimental comparisons 5.5 Static fatigue damage evolution 5.5.1 Results and discussion 5.5.2 Experimental comparisons 5.6 Conclusion References Further reading

273 273 276 280 280 293 306 309 319 358 362 368 369 371 374

6

Fatigue life prediction of ceramic-matrix composites based on hysteresis dissipated energy 6.1 Introduction 6.2 Theoretical analysis 6.3 Results and discussions 6.3.1 Effects of fatigue peak stress on fiber matrix interface debonding, HDE, and HDE-based damage parameters 6.3.2 Effects of fatigue stress ratio on HDE and HDE-based damage parameters 6.3.3 Effects of matrix crack spacing on fiber matrix interface debonding, HDE, and HDE-based damage parameters 6.3.4 Effects of fiber volume fraction on fatigue life, fiber matrix interface debonding, HDE, and HDE-based damage parameters

375 375 376 378 379 382 384

391

viii

Contents

6.4

Experimental comparisons 6.4.1 Unidirectional ceramic-matrix composites 6.4.2 Cross-ply ceramic-matrix composites 6.4.3 2D ceramic-matrix composites 6.4.4 2.5D ceramic-matrix composites 6.4.5 3D ceramic-matrix composites 6.5 Conclusion References Further reading Index

398 398 403 425 433 435 446 446 451 453

Preface

Ceramic-matrix composites (CMCs) have the advantages of high-temperature resistance, low density, and low-thermal expansion coefficient, which can significantly reduce engine or system structure quality, and improve the temperature-bearing capacity for components used in high temperatures. CMCs have already been applied in aero, rocket, and scramjet engines, thermal protection system, and many others. Compared to the monolithic ceramic, the mechanical behavior of CMCs has many different characteristics. During the preparation of CMCs, there is microcracking in the matrix. Under cyclic loading, the growth rate of microcracking in the matrix is much slower than that of monolithic ceramic. For monolithic ceramic, the fatigue failure is often caused by single cracking. For CMCs, many cracks appear in the matrix under cyclic loading, which is, however, not the direct cause of fatigue failure. Understanding the failure mechanisms and internal damage evolution represents an important step to ensure the reliability, durability, and safety of CMCs. This book focuses on the durability of fiber-reinforced CMCs at elevated temperatures, especially the damage mechanisms and lifetime, including: 1. Time stress-dependent damage models for the first matrix cracking and multiple matrix cracking of CMCs at elevated temperatures are developed. The relationships between the first matrix cracking stress, multiple matrix cracking evolution, fiber matrix interface debonding and oxidation, oxidation time and temperature, and different matrix cracking modes are established. 2. Cyclic time-dependent strength degradation models at elevated temperatures are developed. The residual strength of different CMCs subjected to cyclic fatigue loading at elevated temperatures are predicted. 3. Cyclic-dependent interface debonding and sliding of CMCs subjected to multiple loading sequences are investigated considering interface wear mechanisms. The relationships between interface debonding and sliding, loading sequence, and applied cycle numbers are established. 4. Hysteresis-based damage evolution models of CMCs subjected to cyclic fatigue loading at elevated temperatures are developed. The relationships between the hysteresis loops, hysteresis dissipated energy (HDE), HDE-based damage parameters, and interface debonding and sliding are established.

x

Preface

5. An energy-based lifetime prediction method of CMCs at elevated temperature is developed. The experimental fatigue lifetime of unidirectional, cross-ply, 2D, 2.5D, and 3D CMCs are predicted for different testing conditions.

I hope this book can help material scientists and engineering designers understand and master the durability mechanisms of CMCs. Li Longbiao October 19, 2019

Acknowledgments

I am grateful to my wife Peng Li and my son Li Shengning for their encouragement. A special thanks to Isabella Conti Silva, Elsevier, for her help with my original manuscript. I am also grateful to the team at Elsevier for their professional assistance.

Introduction and overview of ceramic-matrix composites

1.1

1

Introduction

Ceramic-matrix composites (CMCs) refer to the composites in which reinforced materials are introduced into the ceramic matrix. The reinforced materials are in the dispersed phase and the ceramic matrix is in the continuous phase. The dispersed phase can be continuous fibers, particles, or whiskers. Continuous fiber-reinforced CMCs, by incorporating fibers in ceramic matrices, not only exploit their attractive high-temperature strength, but also reduce the propensity for catastrophic failure (Christin, 2002; Naslain, 2004; Schmidt et al., 2004; DiCarlo and Roode, 2006; Zhang et al., 2006; Watananbe et al., 2016). Compared with superalloy and singlephase ceramics, fiber-reinforced CMCs have the following advantages: G

G

G

Low density (only 1/31/4 of superalloy). They can be used as combustion chamber and turbine parts, and can directly reduce the mass by about 60%; as nozzle flaps, it not only directly reduces the mass, but also helps to balance the total weight of the aircraft. High-temperature resistance. Fiber-reinforced CMCs require less air cooling or no cooling at all. With this material, the cooling air requirement of the combustor chamber or nozzle is significantly reduced, even to zero, so as to improve the combustion efficiency and reduce pollution emissions and noise levels; the cooling structure is greatly simplified or even omitted, thereby reducing the complexity of structural design and further reducing the weight. The structural design is simplified when using this material for turbine components, while the operating temperature is increased. High damage tolerance ability. Compared with single-phase ceramics, the failure strain of fiber-reinforced CMCs is greatly improved. For example, the failure strain of SiC/SiC material developed by chemical vapor infiltration (CVI) is 10 times that of single-phase SiC. This not only facilitates the processing of the components, but also detects degradation in performance before the material is destroyed, thereby improving the reliability of life prediction.

In this chapter the application and mechanical behavior of fiber-reinforced CMCs are introduced. The application of fiber-reinforced CMCs in aero, rocket, and scramjet engines and thermal protection systems (TPSs) are summarized. The damage evolution characteristics under monotonic tensile and cyclic fatigue loading, and the SN curves of unidirectional, cross-ply, 2D, 2.5D, and 3D CMCs are overviewed. The damage mechanisms and models of initial matrix cracking, matrix multicracking evolution, fibers failure, interface wear, fibers strength degradation, and oxidation embrittlement are analyzed. The effects of loading frequency and fatigue stress ratio on the cyclic fatigue behavior are also discussed.

Durability of Ceramic-Matrix Composites. DOI: https://doi.org/10.1016/B978-0-08-103021-9.00001-0 © 2020 Elsevier Ltd. All rights reserved.

2

Durability of Ceramic-Matrix Composites

1.2

Application of ceramic-matrix composites

The mechanical properties of fiber-reinforced CMCs depend on the fibers used, the matrix, and the interface phase and their interactions.

1.2.1 Reinforced fibers As the main bearing part of the composite material, fiber plays an important role in the performance of the product. The influencing factors include fiber type, fiber volume fraction, and the fiber weaving method. For fiber-reinforced CMCs, reinforced fibers can be divided into three types, namely carbon fiber, oxide fiber, and silicon carbide fiber. The basic material properties of carbon, oxide, and silicon carbide fibers are listed in Table 1.1 (Zhang, 2009). Common fiber preforms use a variety of weaving methods such as 2D, 2.5D, and 3D.

1.2.2 Interface phase The interfacial phase connects the reinforcing fibers and the matrix, which determines the bonding strength between the fiber and the matrix and the toughness of fiber-reinforced CMCs. Under loading, the fracture behavior of fiber-reinforced CMCs mainly includes the matrix cracking, crack deflection, fibermatrix interface debonding, fiber fracture, and pullout. Fiber pullout is an important way to release energy, while fibermatrix interface debonding is the premise for the pullout of the fibers from the matrix. When the fibermatrix interface has strong bonding properties, toughness cannot be achieved by the fibers, leading to the brittle fracture of fiber-reinforced CMCs; however, when the fibermatrix interface has weak bonding properties, the matrix cannot transfer the load to the fibers through the interface, and the reinforced effect cannot be realized. The function of the interface phase includes: G

G

G

G

Preventing physical shrinkage and chemical reaction damage to the fibers during preparation. Relieve residual thermal stress at the interface between the fiber and the matrix. Transfer the load between the fiber and the matrix under loading. Improve the interface bonding strength for the energy dissipation mechanisms of interface debonding and fiber pullout, resulting in pseudoplastic behavior.

The interface phase is the key factor affecting the mechanical behavior of fiberreinforced CMCs, and the material used during the interphase should satisfy the following requirements: G

G

G

Low modulus Low-shear strength High-chemical stability

Table 1.1 The basic material properties of carbon fiber, oxide fiber, and SiC fiber. Type

Manufacturer

Trademark

Density (g/cm3)

Diameter (µm)

Tensile strength (GPa)

Tensile modulus (GPa)

Fracture strain (%)

Carbon fiber

Toray

Oxide fiber

3M

SiC fiber

Nippon carbon

T300 T800H T1000 M40 M50 M40J M60J Nextel 610 Nextel 720 Nicalon NL 202 Hi-Nicalon Hi-Nicalon S Tyranno SA Sylramic

1.76 1.80 1.82 1.81 1.91 1.77 1.94 3.75 3.4 2.55

56 5.2 5.3 6.5 6.3 5.2 5.0 1012 12 14

3.53 5.59 7.06 2.77 2.45 4.50 3.80 1.9 2.1 3.0

235 294 294 392 490 392 590 370 260 200

1.5 1.9 2.4 0.6 0.5 1.1 0.7 0.5 0.81 1.1

2.74 3.0 3.0 3.0

14 12 10 10

2.8 2.5 2.6 3.2

270 380 340 380

1.0 0.7 0.7 0.8

UBE Industries Dow Corning

4

Durability of Ceramic-Matrix Composites

There are three types of interface phases: G

G

G

PyC interface phase. The PyC interface phase has a typical layer structure, and matrix cracking can achieve multiple defections during the interface phase. However, at temperatures above 500 C, the PyC interphase starts to oxidize, which decreases the lifetime of the fiber-reinforced CMCs. BN interface phase. The BN interface phase also has a typical layer structure, but with better oxidation resistance. When the thickness of the BN interface phase is higher than 0.25 μm, the fiber pullout occurs for the composite to form. Multilayer interface layer, i.e., (PyCSiC)n. The (PyCSiC)n interface layer integrates the advantages of the PyC and SiC interphases, with the oxidation resistance of the SiC interphase, and multiple cracking deflection behavior of the PyC interphase, which can increase the toughness of the fiber-reinforced CMCs.

1.2.3 Ceramic matrix A ceramic matrix is one of the most important components of composite materials. Its main composition and structure have a significant impact on the comprehensive properties of the materials. On the one hand, the ceramic matrix is exposed to the working environment and needs to withstand the test of temperature, water, and oxygen. On the other hand, the ceramic matrix bears the first load and cracks appear under the external load. The crack propagation mode is an important factor affecting the reliability of composite materials. There are mainly three types of ceramic matrix: G

G

G

Glass-ceramic matrix, which include calcium alumina silicate (i.e., CAS), lithium alumina silicate (i.e., LAS), magnesium alumina silicate (i.e., MAS), and borosilicate, and has poor temperature resistance. Oxide matrix, which include Al2O3, YAG, ZrO2, TiO2, ZrO2TiO2, and ZrO2Al2O3 among others, and has low creep resistance at elevated temperatures. Nonoxide ceramic matrix, which include SiC, Si3N, and SiCBN, among others, and has the advantages of high strength, high hardness, and high-temperature resistance.

The applications of fiber-reinforced CMCs in aero, rocket, and scramjet engines and TPS and braking systems are summarized next.

1.2.4 Application in aero engines Aero engine manufacturers have concentrated their research and development on fiber-reinforced CMCs with oxidation resistance capacity, which include carbon fiberreinforced CMCs, ceramic fiberreinforced CMCs, and whisker-reinforced CMCs such as carbon fiberreinforced silicon carbide (C/SiC), silicon carbide fiberreinforced silicon carbide (SiC/SiC), carbon fiberreinforced silicon nitride (C/Si3N4), carbon fiberreinforced alumina (C/Al2O3), silicon carbidefiber reinforced alumina (SiC/Al2O3), and so forth.

1.2.5 France Since the 1980s, aero-engine designers and manufacturers such as Snecma in France, GE and P&W in the United States, and the engine design and manufacturer

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IHI Company in Japan have investigated and tested the application of hightemperature, fiber-reinforced CMC components in aero gas turbine engines, such as the application on the third generation improved fighter engine F100-PW-229, the fourth generation fighter engines M88-2 and F119, the next generation high bypass ratio commercial engine, and the high thrust-to-weight ratio fighter engine. In the early 1980s, Snecma tried to use CMC nozzle flaps in an M88 aero engine. At that time, although the CMC nozzle flap was much more expensive and had a shorter service life than the high temperature alloy flap, Snecma still carried out a lot of research and experiments in view of the nozzle flap’s great development potential. Snecma developed a SiC/SiC composite (CERASEPR A300) and C/SiC composite (SEPCARBINOXR A262). The SPECARBINOX A262 has a lifetime of 100 hours at intermediate temperatures. The material is characterized by nonbrittleness, failure strength of 250 MPa at room temperature and long service life at temperatures below 973K. In 1996, Snecma successfully applied this material to an M88-2 engine nozzle flap, greatly reducing the quality. In 2002, Snecma fulfilled its goal and began to put it into mass production. At the same time, Snecma tried to apply the fiber-reinforced CMCs to the nozzle internal flap of an M88-2 engine that can withstand high thermal stress. In the 1990s, to solve the problem of short life caused by oxidation damage in the previous generation CMCs, Snecma developed a new generation of fiberreinforced CMCs, CERASEPR A410 and SEPCARBINOXR A500, using selfsealing technology. The methodology of self-sealing technology is to consume partial oxygen into the microcrack in order to prevent oxygen from entering the interface between the fiber and the matrix. The matrix of CERASEPR A410 and SEPCARBINOXR A500 is the same, and is composed of Si, C, and B in a specific order. Its function is to seal the matrix microcracks at a given temperature. The reinforced fibers of CERASEPR A410 and SEPCARBINOXR A500 are Hi-Nicalon and carbon fibers, respectively. The fatigue strength and life of these materials are significantly improved. At present, Snecma has successfully completed the tests of the CERASEPR A410 nozzle flaps in an M88-2E4 engine. Its characteristic light weight and high-temperature resistance are what is required to introduce advanced thrust-vectoring nozzles. In the mid-1990s, Snecma and P&W tried to apply fiber-reinforced CMCs to F100 aero-engine nozzles. First, the specimens were verified by a low-cycle fatigue test. These tests were carried out in a harsher environment (from standard air conditions to 90% steam conditions) than the actual engine operating environment. After determining the feasibility of applying fiber-reinforced CMC nozzle flaps to an F100-PW-229 engine, 8 nozzle flaps were manufactured, including 2 CERASEPR A410 constant thickness nozzle flaps, 2 CERASEPR A410 variable thickness nozzle flaps, 2 SEPCARBINOXR A500 constant thickness nozzle flaps, and 2 SEPCARBINOXR A500 pitching/yaw balancing nozzle flaps. The first six flaps performed the ground acceleration mission test on the F100-PW-229 and F100-PW220 engines in the Sea Level and Altitude Test Chamber of P&W West Palm Beach Sea Level Test Bench and the Arnold Engineering Development Center, respectively, in which 87% tests were completed on the F100-PW-229 engine and 13% tests were completed on the F100-PW-220 engine. In all the tests, the nozzle

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Durability of Ceramic-Matrix Composites

flaps met the 4600 total accelerated cycle life requirements of the metal flaps they replaced, and there was no delamination problem. The test time of two flaps is much longer than the required level, and the longest one reaches 6582 total acceleration cycles. Snecma and P&W are transferring SEPCARBINOXR A500 constant thickness nozzle flaps to field evaluation and are preparing flight tests on F-15E fighter/F100-PW-229 and F-16 fighter/F100-PW-229 engines. In addition, Snecma is also actively developing CERASEP (SiC/SiC) and SEPCARBINOXR (C/SiC) combustor flame tubes.

1.2.6 United States Under the support from high-temperature engine materials technology, highperformance turbine engine technology (IHPTET), ultraefficient engine technology (UEET), and versatile affordable advanced turbine engine projects and programs, aeroengine design and manufacturers in the United States have researched many advanced materials and structures (Liang, 2011). Fiber-reinforced CMCs components are one of the key research areas.

1.2.6.1 High-performance turbine engine technology project To meet the temperature requirements of phase 2 and phase 3 of the IHPTET program, GE, P&W, and Allison have developed and validated many high-temperature CMC components for aero gas turbine engines based on their respective IHPTET program verification engines.

Joint technology demonstration engine The SiC fiber used for the turbine guided vanes and SiC/SiC composite turbine rear frame leading edge parts in phase 3 of IHPTET program were verified on a large turbojet/turbofan engine version of a JTDE. GE and Allison demonstrated the CMC guided low-pressure turbine vanes on the JTDEXTE76/1 engine during phase 2 of the IHPTET program. The guided 3D fiber vanes improved the strength and durability of the engine. The low-density and high-temperature resistance of CMCs reduce significantly the weight of the vanes, and the cooling air requirement.

Advanced turbine engine gas generator For the advanced turbine engine gas generator (ATEGG) demo version of the XTC76/3 for phase 2 of the IHPTET program, GE and Allison obtained material from the NASA EPM (Enabling Propulsion Material) project, and developed and demonstrated the Hi-Nicalon SiC/SiC combustion flame tube with a fiber volume of 40%. On the ATEGG demo of the XTC77/1 for phase 3 of the IHPTET program, GE and Allison developed and demonstrated the CMC hollow high-pressure turbine guide vanes. Compared to the Ni-based superalloy guided vanes, the mass of the CMC vanes was 50% less and the cooling air requirement was reduced by 20%.

Joint turbine advanced gas generator For the joint turbine advanced gas generator (JTAGG) demo version of the XTC97 for phase 3 of the IHPTET program, Honeywell and GE demonstrated the

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CMC high-temperature rise combustor. The CMC combustion chamber obtained a smaller temperature distribution factor at the target oil/gas ratio and achieved good performance.

Joint expendable turbine engine concept demonstrator Fir the joint expendable turbine engine concept (JETEC) demo version of the XTL86 for phase 2 of the IHPTET program, Williams International developed and demonstrated CMC turbine guide vanes, turbine blades, and nozzle. These components constitute the uncooled high-temperature whole composite components that improved the temperature resistance of the engine and reduced the quality of the engine. For the JETEC demo version of phase 3 of the IHPTET program, Allison demonstrated the C/SiC exhaust nozzle, which needs no cooling, thereby improving the performance and lowering the cost.

1.2.6.2 Ultraefficient engine technology Program The key technology of material and structure plays an important role in the UEET program to achieve the goals of shortening takeoff and landing distance by 70%, reducing NOx emissions by 70%, and reducing fuel consumption and cost by 8% 15%. Fiber-reinforced CMCs are the key materials for combustor flame tubes and turbine guide vanes. Laboratory specimens of CMC combustor flame tubes confirmed that they have a stress capacity of 13.78 MPa at 1478K and over 9000 hours of thermal life. The combustion chamber segment was tested as having a life of 200 hours. The development strategy of the UEET program is: G

G

The engine test was carried out under working conditions of the aircraft task cycle to verify the durability of CMC combustion chamber flame tubes at 1478K. To improve the temperature resistance of CMCs and EBCs and develop 1755K and 1922K flame tubes to greatly reduce, or even eliminate, the film cooling, thereby expanding the application range of CMCs.

The temperature resistance of SiC/SiC composites in the EPM program has been significantly improved through the following methods, and research work is being carried out to improve the stress of CMCs during long-life (3001000 hours hot working). G

G

G

G

Improve the fabrication process by reducing or removing the material constituent that causes the creep of CMCs. Improve the heat treating of SYLRAMIC fiber and improve creep resistance. Use the Hi-Nicalon SiC fibers. Reduce the dispersion of composite properties by optimizing the fabrication process.

1.2.6.3 Application on the F414 aero engine In cooperation with Snecma to develop fiber-reinforced CMC nozzle seals for F100-PW-229 aero engines, P&W is using CMC nozzle flaps and seals validated under the IHPTET program to improve the F119 aero engine, which powered the

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Durability of Ceramic-Matrix Composites

world’s most advanced fighter F-22. With the new flaps, the durability of the aero engine is improved significantly while the quality and the cost are reduced. GE has signed a multiyear contract with Goodrich to develop C/SiC nozzle flaps and seals for the higher temperature F414 engine. Goodrich is responsible for providing light-weight and long-life CMCs, and GE is responsible for testing and evaluation. GE has conducted production and flight testing of CMC standard parts. GE also developed and demonstrated the CMC combustion chamber with the support of the TECH56 program. The CMC combustion chamber can increase the working temperature, and possesses long life with few cooling air.

1.2.6.4 Federal Aviation Administration continuous lower energy, emissions, and noise program in the United States The continuous lower energy, emissions, and noise (CLEEN) program is the Federal Aviation Administration’s principal NextGen environmental effort to develop and demonstrate new technologies, procedures, and sustainable alterative jet fuels. Under the CLEEN program, Boeing is teaming with partners ATK/COI-C, Albany Engineered Composites, to develop and demonstrate an acoustic CMC (oxide) exhaust nozzle on a Rolls Royce Trent 1000 engine. The CMC technology will reduce weight and increase the temperature capability of future engine nozzles, reducing aircraft fuel burn by 1% (Petervary and Steyer, 2012).

1.2.7 Japan Japan also recognizes the importance of the application research of fiber-reinforced CMCs in the field of aero gas turbine engines. In the AMG (Advanced Materials Gas-Generator) research and development program, fiber-reinforced CMC combustion chamber flame tubes processed by chemical vapor deposition reached the outlet temperature of 1873K without damage. The blade section of the developed SiC/SiC turbine blisk was exposed to hot gas flow to conduct the rotation test. The working speed reached 30,000 r/min, the blade tip speed reached 386 m/s, and the gas temperature reached 973K. No vibration and no damage were found. The ESPR (Research and Development of Environmentally Compatible Propulsion System for Next-Generation Supersonic Transport) program was launched in 1999, in which the CMC combustion chamber and turbine components were designed and tested. The CMC material was SiC/SiC composite, which adopting 3D weaving, CVI deposition carbon coating on SiC fiber surface, CVI 1 PIP for matrix densification technology, and SiC coating on the component surface by CVI. The CMC turbine outer ring is a SiC/SiC composite. To form the main airflow channel, the outer turbine ring is a disc part arranged around the turbine blade. The turbine outer ring was tested on a high-temperature core engine, and the static strength was 30 times of the load, confirming the strength of the parts. The temperature test results of the CMC turbine outer ring at 1650 C and 15 minutes were

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consistent with the predicted results of the heat transfer analysis. After 20 hours of testing, the CMC turbine outer ring had no cracks, coating stripping, or ablation. IHI Corporation designed and tested a CMC low-pressure turbine guide vane, a SiC/SiC composite, which uses 3D braiding technology. The guide vane leads the swirling airflow to the axial flow, consisting of an airfoil section and edge section. To evaluate the durability of the CMC low-pressure turbine guide vane under hightemperature gas flow, a special testing device was designed for the model turbine guide vane. The test temperature was 1150 C1300 C and the test cycle number was 1000. The results showed that no damage occurred, proving that the CMC turbine guide vane has enough durability (Tamura and Nakamura, 2005). The CMC turbine disk was designed and tested in the AMG program. The material was a SiC/SiC composite with SiTiCO Tyrannoe SiC fiber. The tensile strength of the SiC/SiC composite was 500 MPa at room temperature. The diameter of the turbine disk was 134210 mm, the fiber-to-volume fraction was 32%, and the density was 1.97 g/cm3. Two SiC/SiC turbine disks with a thickness of 22 and 5 mm were fabricated. The strength of the two discs in the rotating test at room temperature was almost the same. The maximum speed of rupture was 32800 rpm (Hisaichi et al., 1999). CMCs have good high-temperature resistance and, therefore, satisfy the hightemperature requirement of aero engines’ hot-section components. They have been successfully applied to the nozzle flaps of the M88-2 engine and the sealing parts of the F100 engine. GE is expected to increase the application of CMC 10-fold over the next 10 years. It will also have a wider application prospect in aero engine turbine components. CMCs do not require air-cooling in hot-section parts, as metal does, therefore, fuel consumption will be significantly reduced and thrust-to-weight ratio will be increased. After successful commercial application of GE’s Leap-X turbine outer ring for commercial engines, there are plans to use CMC materials in the static and rotating components of its next-generation military aircraft engines.

1.2.8 Application in rocket engines The use of C/SiC and SiC/SiC composites to prepare liquid rocket engine (LRE) thrust chambers can reduce engine structural quality, improve engine operating temperature, simplify engine structure design, and greatly improve overall engine performance. Hyper-Therm HTC Inc. and Air Force Research Laboratories used CVI technology to fabricate C/SiC composites LRE thrust chambers. The thrust chamber length is 457 mm, nozzle outlet diameter is 254 mm, and throat diameter is 35 mm. It has passed the hot test under a gas temperature of 2050 C, chamber pressure of 4.1 MPa, and thrust of 1735.2 N, while the throat ablation rate is about 2.54 3 1022 mm/s. Fiber Materials Inc. has prepared a C/SiC composite thrust chamber for solid missile attitude-orbiting rocket engines. The thrust chamber is prepared by the PIP process with a material density of 2.0 g/cm3, throat diameter of 5.08 mm, and a wall thickness of less than 1.5 mm. During the ignition test, the solid propellant flame temperature reached 2038 C, the maximum working pressure reached

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17 MPa, and the average working pressure was 4.6 MPa. After the completion of the 8.11 seconds test run, the diameter change of the thrust chamber throat was only 1.5%. The concepts of “uncooled” and “actively cooled” C/SiC composite thrust chambers were proposed by NASA according to the application requirements and were demonstrated and verified. The ignition test of the uncooled combustor takes LH2/ LO2 as the propellant with the highest pressure 6.9 MPa, and ignition 10 seconds. The steady temperature of the inner wall is more than 1400 C, and the outer wall temperature is about 500 C. The experimental results show that the C/SiC doublewall structure can meet the uncooled requirements proposed by NASA to simplify the cooling problem of the thrust chamber. The ignition test of the SiC/SiC thrust chamber with cooling passage was carried out with LH2/LO2 as the propellant, combustion chamber pressure was 2.7 MPa, and the maximum steady-state temperature of the thrust chamber wall exceeded 2370 C. It passed the hot test for 30 seconds. NASA tested the CMC turbine blisk of a rocket engine turbopump to improve safety and reduce launch costs for the second generation of reusable launch vehicles (RLV). The second-generation RLV aims to increase its safety 100-fold compared with the current RLV, and reduce the cost of launching payloads into low Earth orbit from $10,000 to $1000. Because CMCs can reduce the quality of harmful components, increase the working temperature, do not need to cool down, and enhance the damping capacity, the CMC turbine blisk will play an important role in NASA’s second-generation RLV. The C/SiC nozzle of the third stage LH2/LO2 thrust chamber of Arian 4 was prepared using the CVI process by Snecma. The length of the nozzle is 1016 mm, the outlet diameter is 940 mm, and the total mass is only 25 kg, which is 50 kg less than that of the alloy nozzle of the same volume. In 1989, the C/SiC nozzle successfully completed two high-altitude ignition experiments. The inlet temperature of the nozzle was higher than 1800 C and the working time was 900 seconds. The company has now applied C/SiC composite material to the Arian 5 engine. The European Space Transportation Company, in collaboration with Snecma, carried out the applied research of C/SiC composite LRE and conducted the first ground hot test in 1998. In the ignition test, the chamber pressure is 1 MPa and the maximum working wall temperature of the throat is 1700 C. Under these conditions, the combustor accumulatively worked for 3200 s. In 2003 the improved C/SiC composite combustion chamber worked under a chamber pressure of 1.1 MPa for 5700 seconds. The thrust chambers of altitude and orbit control engines for apogee satellites with different thrust (56000 N) were fabricated using CVI technology. The materials were C/SiC and SiC/SiC. The C/SiC thrust chamber was tested at room pressure of 0.8 MPa and 1.2 MPa. The service life of the chamber was 10002800 seconds and the maximum wall temperature was 14501700 C. The SiC/SiC thrust chamber with 5 N thrust was tested at pressures of 0.8 MPa and 1.0 MPa, respectively. The maximum wall temperature reached 1550 C1600 C and the service life reached 50 hours (Mathieu et al., 1990). Germany’s DASA has devoted major efforts to the development of silicon infiltration reaction sintering technology to prepare C/SiC components and the

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development of winding technology in the fabrication of fiber preforms. The C/SiC composites with nearly complete density can be prepared using this technology with the advantages of high speed, low cost, and net molding. The modulus is 90250 GPa, the tensile strength is 140350 MPa, the thermal expansion coefficient is (3.56.5) 3 1026/K, the thermal conductivity is 5135 W/(m K), and the service temperature is higher than 1700 C. C/SiC thrust chambers for solid rocket engines and LREs were fabricated by silicon infiltration reaction sintering technology. The thrust chamber of 10 N bipropellant satellite engine has no obvious thermal aging and thermal corrosion after 400 thermal cycling tests (cumulative working time 50 hours). DASA also attaches great importance to the development of PIP technology, combined with winding molding technology to prepare a variety of C/SiC components (Beyer et al., 1999). A C/SiC thrust chamber of a 400 N apogee satellite engine was fabricated by DASA. The wall temperature reaches 1550K1850K and the ignition time is 34 minutes. There is no obvious change in structure and shape after ignition.

1.2.9 Application in Scramjet Engine From the initial development of C/C material for Solid Rocket Motor nozzles, Snecma Propulsion Solide (SPS) has developed a family of C/SiC materials to be compatible with the more and more stringent requirements for the propulsion and external structure of future aerospace vehicles. Since 1984 Snecma has proposed its C/SiC NOVOLTEX material for scramjet application. After successful test demonstrations with ONERA with Ramjet composite structure, a new cooperation was started with the UTC group for demonstration of C/SiC use in scramjet. Through the United Technologies Research Center (UTRC) and Snecma initiated Joint Composite Scramjet (JCS) program, the potential of using C/C and C/SiC composite materials in scramjet combustors demonstrated weight savings and increased thermal margins relative to an all-metallic scramjet. G

G

The first applications selected for use of Snecma-produced composite material was for the air-breathing pilots, used for ignition and stabilization in the combustor, and a panel representative of an uncooled wall. These components were jointly designed by the team, produced by Snecma, and tested successfully at Mach 7 flight conditions at the UTRC scramjet test facility in June 2001 (Uhrig and Larrier, 2002). In parallel to the system studies carried within the framework of the JCS program, a USAF-DGA cooperation, the AC3P (Advanced Composite Combustion Chamber Program) funded the development of a C/SiC actively cooled technology as a potential future replacement for the current HySET metallic design. Several panels were manufactured and tested in parallel to a material and structural development study. The final test in April 2003 at the USAF AFRL radiant facility demonstrated a Snecma-manufactured totally leak-free composite heat exchanger under relevant conditions (Bouquet et al., 2003a,b).

Within the frame of the X-33 program, NASA carried activities to develop and assess many innovative technologies required for future RLV. SPS was selected as a suitable light weight, actively cooled, thermostructural composite technology to

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test its applicability to X-33 and RLV Aerospike engines. Within the framework of a French DoD hypersonic program (AC3), CMCs were evaluated for M8 scramjet combustion chamber applications. For that purpose, SPS designed a composite chamber composed of integrated actively kerosene-cooled CMC panels. A cooperative program between USAF and the French DoD was set up to test and characterize the performance of this advanced technology. Snecma with the support of NASA developed a C/SiC composite with an active cooling structure for the inlet of a scramjet engine with the aim of operating at heat flux densities greater than 5 MW/m2. The C/SiC composite prepared by CVI technology was tested in air at 1500 C for 320 minutes and in air at 1700 C for 40 minutes. After fully considering various factors and conducting related experiments, an Nibased alloy was selected as the cooling tube. To solve the thermal mismatch between the alloy and the composite material, the Ni-based alloy tube is inserted into the copper wire braid, then embedded in the groove reserved for the composite material, and finally fixed by the elastic force applied by the metal device. Laboratory tests using water as a coolant have shown that this structure has good heat-transfer characteristics at low-heat flux densities (Bouquet et al., 2003a,b, 2004). The US Air Force Hypersonic Technology Program tested the performance of a variety of composites for application in scramjet inlet leading edge, inlet sidewall, combustor sidewall, and external nozzle in a simulated Mach 8 missile operational environment. The results show that the C/SiC composites with oxidation-resistant coating can withstand 10 minutes of the simulated environmental test with no evidence of erosion and can be, therefore, used as inlet material in a single-use missile and, hopefully, in combustion chambers and nozzles to withstand temperatures up to 1940 C (Dirling Ray, 1998). Refractory Composites, Inc. (RCI) and P&W Hypersonics are performing two parallel development programs under NASA funding for combined cycle engine technology development. The development of an all CMC-based RBCC (rocketbased combined cycle) combustor flow path from inlet to exhaust is the ultimate objective of this work. A C/SiC composite sandwich structure with a cooling component was designed and divided into three layers. The innermost layer facing the high-temperature airflow is a C/SiC composite material, the middle layer is a Ni alloy cooling tube, and the outermost layer is also a C/SiC composite material. The shrinkage ratio of this structure passed the test of the working environment of the simulated scramjet combustion chamber (Paquette et al., 2002).

1.2.10 Application in thermal protection systems The TPS is one of the most critical technologies for developing and securing aerospace vehicles for safe service in extreme environments. The hypersonic vehicles perform long-term, high-speed flight in the atmosphere, which produces high dynamic pressure and aerodynamic heating. The extreme thermal and mechanical loads experienced by the aerospace vehicles place high demands on the TPS and structural design. An effective TPS provides enough protection for the aerospace vehicle structure in the face of severe aerodynamic heating, which helps the main

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structure of the aerospace vehicle to remain within the allowable temperature range. In addition, a low-ablation or zero-ablation TPS needs to maintain an effective aerodynamic layout for flight. With the rapid development of aerospace vehicles, more stringent requirements are imposed on the TPS. The TPS is required to develop from the separation of thermal protection and structure to the integration of thermal protection and structure, and from single ceramic thermal protection to CMCs structures. For the TPSs of aerospace vehicles, it is necessary to transfer mechanical loads at 1900 C in air. CMC is currently the only suitable material (Anderson, 2001). The US X-38 aerospace vehicle adopts thermal protection/structure integrated thermal protection technology. C/SiC is the preferred thermal/structural integrated material because of its high-temperature resistance, low density, and oxidation resistance. A C/SiC heat/structure integration flap has been used on X-38, which is considered as the most successful and advanced application. The German Space Center (DLR) is committed to developing fiber-reinforced CMCs through liquid silicon infiltration technology. Usually this material exhibits excellent thermal shock resistance, high relative mass ratio, and dense matrix (Heidenreich, 2007). Within the framework of the German and Japanese cooperative project Express, DLR has developed a TPS using ceramic tiles, namely Cetex, which is a fiber-reinforced CMC component of C/C-SiC (Alfano, 2010). Cetex connects the stationary points used for ablation and thermal shielding in the recovery capsule. Quality identification tests of the samples have passed the plasma wind tunnel test. In the PWK2 wind tunnel test by Stuttgart University in Germany, the highest temperature sustained by the Cetex sample was 3000K. Although the Japanese spacecraft did not work as expected, the uncontrolled recovery capsule successfully re-entered the atmosphere, resulting in a surface temperature of about 2500K on Cetex, and sustaining no apparent damage on the Cetex surface (Alfano, 2010). German MT Aerospace’s C/SiC composite thermal components have passed the quality identification test in the X-38 project. These components are used in body flaps and wing leading edges of the X-38. DLR provides the C/C-SiC nose cap prepared by liquid silicon infiltration, which is about 740 3 640 3 170 mm3 in size, a shell thickness of about 6 mm, and a mass of about 7 kg. Snecma has also developed a CMC tile TPS that has passed the arc-spraying test with a high temperature of 1500K (Alfano, 2010). DASA uses winding molding and PIP technology to prepare C/SiC thermal protection components (Wulz and Trabandt, 1997; Muu¨hlratzer et al., 1998). The density of the composite is 1.8 g/cm3, Young’s modulus is 6070 GPa, while the tensile strength, compressive strength, and bending strength are 270, 370, and 530 MPa, respectively. The strength at 1500 C is higher than that at room temperature. The residual strength is 80% of the original strength after exposure to air at 1600 C for 30 minutes. The composite has good thermal-shock resistance and can be used for the thermal protection component of reusable aerospace vehicles. In the framework of the European advanced test re-entry testbed expert project led by the European Space Agency, the SiC-coated C/C composite manufactured by the DLR were selected as the aircraft nose caps with the aim to collect data on different physical phenomena in re-entry atmosphere. In the sustained hypersonic flight

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experiment (Shyfe) project funded by the British ministry of defense, the supersonic bearing capacity of the C/SiC spacecraft panel, manufactured by German MT Aerospace Company through CVI technology, was developed (George, 2008). SPS developed a heat shield cover plate made of CMC that separates the mechanical and thermal functions of the material. Parts that perform mechanical functions such as the CMC panels, fasteners, and brackets have good mechanical properties, but do not have good heat-shield properties. Components that perform thermal protection functions such as internal insulation, sealing rings, and heat shields are highly efficient heat insulators, but do not have the necessary carrying capacity. The average surface density of the TPS is 18.08 kg/m2 and the CMC panel is connected to the structure by an adjustable flexible support. This connection allows thermal expansion mismatch between the CMC panel and the structure and has sufficient stiffness to prevent large deformation of the CMC panel and to transfer external load to the structure. In addition, this connection method has the advantage of easy replacement. By performing an arc-heating test on the demo piece (the total arc heating test time is equivalent to 11 re-entry times), the maximum temperature of the panel after the leading edge is 1200 C, while the temperature of the interface between the structure and the support is below 110 C. The Japanese National Aerospace Laboratory used Ube’s Ti-containing polycarbosilane and Si-Ti-C-O fibers as raw materials to fabricate heat-shield panels using PIP technology, and replacing some rigid ceramic tiles of HOPE2X demonstrator. The panels were tested in a simulated re-entry environment. The surface temperature reaches 1310 C1590 C, but the surface catalytic effect is not obvious. After testing, the surface of the sample has no obvious degradation and the mass loss is very small. However, the mass-loss rate increases with the increase of temperature when the temperature exceeds 1450 C, which is caused by the volatilization of Na in the surface sealing glass layer and the decomposition of fiber and matrix in the composite material (Ishikawa and Ogasawara, 2001; Ogasawara et al., 2001).

1.3

Overview of tensile behavior of ceramic-matrix composites

Under tensile loading, the stressstrain curve of fiber-reinforced CMC exhibits obvious nonlinear due to multiple damage mechanisms. The typical tensile stress strain curve of fiber-reinforced CMCs is shown in Fig. 1.1, which can be divided into three regions, that is, the elastic region at low-stress level (region I), nonlinear region at intermediate-stress level (region II), and a second linear region before final fracture (region III). In region I, there is no damage inside of the composite during initial loading and the tensile stressstrain curve is linear. With an increase in applied stress, the first matrix cracking occurs in the matrix-rich region in the form of microcracks, and the stress is defined as σmc. These micro matrix cracks cannot be reflected in the macrotensile stressstrain curve; however, the experimental techniques of acoustic emission (Kim and Pagano, 1991; Morscher, 2014;

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Figure 1.1 The typical tensile stressstrain curve of fiber-reinforced CMCs. CMCs, Ceramic-matrix composites.

Sujidkul et al., 2014; Almansour et al., 2015; Breede et al., 2015; Morscher and Gordon, 2017), microscopic observation (Sørensen and Talreja, 1993), and surface temperature measurement (Holmes and Shuler, 1990) can be used to detect such microcracking. With an increase in applied stress, the tensile stress-strain curve of fiber-reinforced CMCs exhibits nonlinear due to accumulation of matrix cracks, which indicates the beginning of region II. The applied stress corresponding to the beginning of region II is the proportional limit stress. At region II, the number of matrix cracks increases, and are propagated perpendicular to the loading direction. When the matrix cracking propagates to the fibermatrix interface, the matrix cracking will deflect along the interface, and fibermatrix interface debonding occurs. With an increase in applied stress, when the slip region of adjacent matrix crack overlaps with each other, the matrix cracking approaches saturation. Matrix cracking saturation stress is defined as σsat. After the saturation of matrix cracking, region III of tensile stressstrain curve starts. The applied stress is mainly carried by the fibers, and the tangent modulus of the tensile stressstrain curve of dσ/dε is approximately VfEf (i.e., Vf denotes the fiber volume fraction and Ef denotes the fiber elastic modulus) (Curtin, 1991). With increased applied stress, partial fiber fracture, and the broken fibers continue to carry loading through the fibermatrix interface shear stress. When the broken fibers fraction approaches the critical value, the composite fails (Curtin, 1991; Curtin et al., 1998). The matrix initial cracking depends on three factors, that is, matrix fracture toughness, the axial thermal residual stress due to the thermal expansion coefficient mismatch between the fiber and the matrix, and the matrix internal flaw distribution. Due to these factors, matrix cracking cannot approach saturation during single applied stress. The propagation of existing matrix cracking and the appearance of new matrix cracking occur within a certain stress range, which decreases with

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Durability of Ceramic-Matrix Composites

increasing matrix Weibull modulus of m. When the slip regions of the adjacent matrix cracking overlaps, matrix cracking approaches saturation and the stress carried by the matrix will not change anymore. For fiber-reinforced CMCs, a weak interphase is generally introduced between the fiber and the matrix. When the matrix cracking propagates to the fibermatrix interface, the deflection along the fibermatrix interface occurs rather than penetrating through the fibers. After the fibermatrix interface debonding, the fiber slides relative to the matrix in the fibermatrix interface debonded region, and the fibermatrix interface shear stress of τi can be described using the Coulomb friction law (Sørensen et al., 1993). The fibermatrix interface shear stress depends upon three factors, that is, the roughness of fiber surface, the fibermatrix interface radial thermal residual stress and the fibers Poisson contraction effect. The fibermatrix interface shear stress affects the loading transfer between the fiber and the matrix, and then the nonlinear behavior of the tensile stressstrain curve and tensile strength of fiber-reinforced CMCs. When the fibermatrix interface shear stress approaches zero, the fracture of fiber-reinforced CMCs is similar to the fiber bundle fracture process, the tensile strength is affected by the gauge length and decreases as the increase of the gauge length. When the fibermatrix interface shear stress increases, the matrix, after cracking, still carries the load. The load distribution at the plane of matrix cracking obeys the global load sharing (GLS) criterion (Curtin, 1991). The load carried by broken fibers is recovered within the fiber characteristic length through the fibermatrix interface shear stress. When the gauge length of the fiber-reinforced CMCs is greater than the fiber characteristic length, the fracture strength of CMCs will not be affected by the gauge length. When the fibermatrix interface shear stress is much higher, the load distribution at the plane of matrix cracking obeys the local load sharing (LLS) criterion (Xia and Curtin, 2000), the broken fibers cause the stress concentration, the load carried by intact fibers increases, the tensile fracture process of fiber-reinforced CMCs exhibits stochastic behavior and the tensile strength depends on the volume of the gauge length. However, for most fiber-reinforced CMCs, the fibermatrix interface shear stress is low, leading to the nonlinear behavior of the tensile stressstrain curve and fibers pullout at the fracture surface. The tensile behavior of fiber-reinforced CMCs is affected by the loading rate (Spearing et al., 1994). Sørensen and Holmes (1996) investigated the effect of loading rate on the tensile behavior of unidirectional SiC/CAS composites at room temperature. It was found that the proportional limit stress, tensile strength, and failure strain increase with increasing loading rate. At high temperatures, affected by the fiber creep (Zhu et al., 2004a,b) and oxidation (Choi and Gyekenyesi, 2005), the loading rate will also influence the tensile behavior of fiber-reinforced CMCs.

1.3.1 Experimental observation In this section, experimental observations and analyses of damage evolution as well as the stressstrain curves of fiber-reinforced CMCs with different fiber preforms (i.e., unidirectional, cross-ply, 2D, 2.5D, and 3D) under tensile loading are overviewed.

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1.3.1.1 Unidirectional ceramic-matrix composites Prewo (1986) performed tensile experiments on an SiC/LAS composite at room temperature. The tensile stressstrain exhibits nonlinear at a strain of 0.3%, and at the fracture surface of the specimen many fibers pulled out. Marshall and Evans (1985) investigated the damage evolution of an SiC/LAS composite under tensile loading at room temperature using the in situ optical microscopy. The multiple damage mechanisms of matrix cracking and fibers pullout were observed. When the tensile stressstrain curve is deviated, the matrix cracks appear across the sample width and thickness. With an increase in in applied stress, the amount of matrix cracking increases and approaches saturation. After the saturation of matrix cracking, fractured fibers pulled out from the matrix with further matrix cracking. When the applied stress approaches ultimate strength, the fibers pullout causes the ductile fracture of the composite. Pryce and Smith (1992) investigated the tensile and cyclic loadingunloading tensile behavior of unidirectional SiC/calcium aluminosilicate (SiC/CAS) composite at room temperature. When the applied strain approaches 0.08%, matrix cracking first appears and the tensile stressstrain curve is deviated. With an increase in applied stress, the amount of matrix cracking increases. When the strain approaches 0.3%, the matrix cracking approaches saturation and the tensile stressstrain curve is deviated again. When the applied strain approaches 0.78%, the composite tensile strength fractures. Beyerle et al. (1992) related the tensile behavior to constituent properties of unidirectional SiC/CAS composites. When the applied stress approaches 150200 MPa, the tensile stressstrain curve exhibits nonlinear. The polished side surface was observed under an optical microscope and it was found that the first matrix cracking stress occurred around 130150 MPa. Under tensile loading, the modulus changes of composite were measured during partial unloading. The occurrence of matrix cracking caused a decrease in modulus, and after the saturation of matrix cracking, the modulus decreases less. The saturation matrix crack spacing was measured using an optical microscope and the fibermatrix interface shear stress was estimated using the matrix cracking model. The in situ SiC strength inside the composite was also obtained through the fracture mirror test. Daniel et al. (1993) investigated the tensile damage evolution of unidirectional SiC/CAS composites using an in situ optical microscope at room temperature. Under tensile loading, the tensile stressstrain curve is linear at the initial loading, and when the applied stress approaches 100 MPa, matrix cracking appears; however, the tensile stressstrain curve still exhibits linear. When the applied stress approaches 275 MPa, the matrix cracking approaches saturation, and the saturation matrix crack spacing is approximately twice the fiber diameter. Before the saturation of matrix cracking, partial fiber fracture near the matrix cracking plane is in the range of 16 times the fiber diameter length due to the stress concentration at the debonding tip. After the saturation of matrix cracking, the tensile stressstrain curve appears linear again; however, the composite modulus decreases from 125 to 52 GPa.

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Kim and Liaw (2007) investigated the tensile damage evolution of unidirectional SiC/CAS composites using infrared thermography at room temperature. The surface temperature of the sample was increasing obviously at the stage of fibers broken and pullout. The temperature change of the sample during the tensile loading is mainly due to fibers breakage and pullout. Wang et al. (2001) investigated the tensile damage evolution of unidirectional C/SiC composites using an in situ optical microscope at room temperature. The longitudinal matrix cracking appears first along the direction of the fibers due to weak bonding between the fibers and the matrix. With an increase in applied stress, longitudinal cracking propagates and connects with each other, and then penetrates through the entire specimen, leading to the final tensile fracture. The transverse matrix cracking perpendicular to the fiber direction did not appear during tensile loading, mainly due to the weak bonding between the fibers and the matrix. Mah et al. (1985) investigated the tensile behavior of unidirectional SiC/LAS composites at elevated temperatures of 900 C and 1000 C in an oxidative environment. The tensile behavior exhibited degradation under these conditions. When the oxygen content is high, the matrix cracking penetrated through the fibers, leading to the brittle fracture of the composite. When the oxygen content is low, the matrix cracking deflected along the fibermatrix interface, and the fiber bridged the matrix cracking, leading to the toughness fracture of the composite. Jablonski and Bhatt (1990) investigated the tensile behavior of unidirectional silicon carbide fiber-reinforced reaction-bonded silicon nitride matrix composites (SiC/RBSN) composites at room temperature and elevated temperatures of 1300 C and 1500 C in air atmosphere. With increasing testing temperature, the tensile strength and failure strain decrease. At room temperature and elevated temperature of 1300 C in air atmosphere, the matrix cracking appears before tensile failure, and the matrix cracking strain was about 0.1%. At an elevated temperature of 1500 C, the tensile stressstrain curve did not have an obvious deviated point; however, the ductile behavior is better than that at 1300 C. The tensile failure specimen at room and elevated temperatures was observed under the scanning electron microscope (SEM). At room temperature and elevated temperature of 1500 C in air atmosphere, the fibermatrix interface has strong bonding and the matrix cracking penetrated through the fibers; however, at 1300 C in air atmosphere, the fibermatrix interface has weak bonding, the matrix cracking deflects along the fibermatrix interface, and the amount of matrix cracking is much larger. Cao et al. (2001a) investigated the tensile behavior of M40JB-C/SiC and T800-C/ SiC composites at elevated temperatures of 1300 C and 1450 C in inert atmosphere. For the M40JB-C/SiC composite, the tensile stressstrain curve exhibits nonlinear and can be divided into a linear and nonlinear region. The tensile strength is about 374 MPa and tensile modulus is about 137 GPa at 1300 C. However, at 1450 C, the tensile strength decreases to 338 MPa and the tensile modulus decreases to 116 GPa. For the T800-C/SiC composite at 1300 C, the tensile stressstrain curve of T800C/SiC composite is linear and without an obvious nonlinear behavior. The tensile strength is about 392 MPa, which is higher than that of the M40JB-C/SiC composite, and tensile modulus is about 115 GPa, which is lower than that of the M40JB-C/SiC composite.

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For unidirectional fiber-reinforced CMCs, matrix cracking and fibermatrix interface debonding are the major reasons for the nonlinear behavior of tensile stressstrain curve. The fibermatrix interface properties, temperature, and oxidation affect the tensile behavior. When the fibermatrix interface has weak bonding, longitudinal cracking appears along the fiber direction and the tensile stressstrain curve is linear. When the fibermatrix interface has strong bonding, the matrix cracking penetrates through the fibers without the fibermatrix interface debonding and the tensile stressstrain curve is linear. At elevated temperatures in an oxidative environment, the fibermatrix interface oxidation caused the penetration through the fibers and the tensile stressstrain curve appears linear.

1.3.1.2 Cross-Ply Ceramic-Matrix Composites Sbaizero and Evans (1986) investigated the tensile behavior of [0/90]s cross-ply SiC/LAS composites at room temperature. Due to the weak bonding between the 0 and 90 ply, delamination occurred under tensile loading. Zawada et al. (1991) investigated the tensile damage evolution of [0/90]3s crossply SiC/1723 composites at room temperature using an optical microscope. The tensile stressstrain curve appears nonlinear and deviates twice at the applied stress of 55 and 280 MPa, corresponding to the transverse cracking in the 90 ply and matrix cracking in the 0 ply. Wang and Parvizi-Majidi (1992) investigated the tensile damage mechanisms of [03/90/03], [03/902/03], and [03/903/03] cross-ply SiC/CAS composites using an optical microscope at room temperature. Transverse cracking first occurs in the 90 ply, which appears at the interface between the 0 and 90 ply, and then propagates to the 90 ply. With a decrease in thickness of the 90 ply, the initial transverse cracking strain increases, and the propagation rate decreases. For the [03/903/03] cross-ply SiC/CAS composite under tensile loading, when transverse cracking appears in the 90 ply, the cracking quickly penetrates through the thickness of the 90 ply. However, for the [03/90/03] cross-ply SiC/CAS composite under tensile loading, the propagation of transverse cracking in the 90 ply needs higher strain or stress, and when the transverse cracking propagates to the interface between the 0 ply and 90 ply, the cracking propagates through the 0 ply within several times of fiber radius length, and then stops due to the bridged fibers in the 0 ply. The transverse crack spacing decreases with an increase in applied stress and approaches saturation before matrix cracking in the 0 ply. When the applied strain approaches 0.13%, the matrix cracking in the 0 ply occurs in the three-ply forms. The matrix cracking strain in the 0 ply is independent of the 90 ply and its thickness. The initial matrix cracking occurs in the matrix-rich region of the 0 ply, and with an increase in applied strain, the matrix cracking density increases. Most of the matrix cracking propagates and stops at the interface between the 0 and 90 ply; however, some of the matrix cracking propagates to the 90 ply. With continual increasing of applied strain, the fibers in the 0 ply fracture, leading to the failure of the composite. Pryce and Smith (1992) investigated the tensile damage behavior of [0/90]s, [0/90]3s and [0/90]4s cross-ply SiC/CAS composites using an optical microscopic at room temperature. The tensile stressstrain curve of cross-ply SiC/CAS composites

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Durability of Ceramic-Matrix Composites

is nonlinear and deviates twice. When the strain approaches 0.02%0.04%, the tensile stressstrain curve deviates due to the matrix transverse cracking in the 90 ply. With an increase in applied stress, the transverse cracking density in the 90 ply increases, matrix cracking in the 0 ply appears, and the tensile stressstrain curve deviates again. Partial matrix cracking occurs due to the propagation from the transverse cracking in the 90 ply and further matrix cracking appears in the matrix of the 0 ply. The transverse and matrix cracking density increase with applied stress and approaches saturation before final failure. The failure strain is about 0.63%0.65%. The shear-lag model is used to describe the stress distribution in the 0 ply and 90 ply after transverse cracking in the 90 ply. Combing the transverse cracking strength criterion, and assuming the new cracking occurs in the middle of the existing transverse crack spacing, the evolution of transverse cracking in the 90 ply can been predicted. Karandikar and Chou (1993a) investigated the tensile damage behavior of [0/90/ 0/90/0/90/0/90/0], [03/90/03] and [03/903/03] cross-ply SiC/CAS composites using an optical microscope at room temperature. When the strain approaches 0.027% 0.056%, the tensile stressstrain curve deviates for the first time. When the strain approaches 0.15%0.17%, the tensile stressstrain curve deviates again. The transverse cracking in the 90 ply is the initial damage mode. The transverse cracking strain decreases with an increase in the thickness of the 90 ply, which is lowest for the ply form of [03/903/03] and highest for the ply form of [03/90/03]. However, the existence of the 90 ply has less effect on matrix cracking in the 0 ply. The first matrix cracking strain and the saturation matrix cracking density in the 0 ply are much higher than those of 90 ply. The appearance of transverse cracking and matrix cracking decreases the composite modulus and Poisson ratio under tensile loading. The shear-lag model is used to describe the microstress field of the composite after transverse cracking in the 90 ply and matrix cracking in the 0 ply. The composite modulus degradation under tensile loading was predicted by combining the measured transverse and matrix cracking density. Prewo et al. (1989) investigated the tensile behavior of [0/90]4s cross-ply SiC/ LAS-II composites at an elevated temperature of 900 C in air and Ar atmospheres. The composite tensile strength in Ar atmosphere is much higher than that in air atmosphere. Rousseau (1990) investigated the tensile behavior of [0/90]2s cross-ply SiC/CAS composites at room and elevated temperature of 815 C in air atmosphere. The tensile stressstrain curve exhibits obvious nonlinear behavior and deviates twice. However, the composite tensile strength at 815 C in air atmosphere is much lower than that at room temperature. Agins (1993) investigated the tensile damage evolution behavior of [0/90]2s cross-ply SiC/CAS composites at room and elevated temperatures of 700 C and 850 C in air atmosphere. The tensile strength and failure strain decrease with an increase in temperature. At 850 C in air atmosphere, the length of fibers pullout is much larger than that at 700 C, and obvious defects appear on the fiber surface. When the temperature exceeds 800 C, the fibermatrix interface oxidizes, the fibermatrix interface bonding strength decreases, and the fibermatrix interface

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debonds at low-stress level, leading to the increase of fiber loading, decrease of composite tensile strength and failure strain, and increase of fibers pullout length. For the cross-ply fiber-reinforced CMCs under tensile loading, the occurrence of transverse cracking in the 90 ply and matrix cracking in the 0 ply causes the twice deviation on the tensile stressstrain curve. At high temperatures in inert atmosphere, the composite tensile strength is higher than that at room temperature. However, at high temperature in an oxidative environment, the composite tensile strength is lower than that at room temperature due to the fibermatrix interface oxidation.

1.3.1.3 2D ceramic-matrix composites Wang et al. (1991a) investigated the tensile damage behavior of 2D C/SiC composites at room temperature. The damage at the fiber bundle determines the fracture of the composite. The fracture mechanism of the composites is analyzed by observing the uneven fracture surface, and the matrix microcracking is the main cause of the ultimate failure under tensile loading. Wang and Laird (1997) investigated the tensile behavior of 2D C/SiC composites at room temperature. The damage of fiber bundles during fabrication, fluctuation of fiber bundles, and inconsistency of fiber bundles cross-section lead to the lower modulus of the composite and its nonlinear stressstrain behavior. Camus et al. (1996) investigated the tensile damage behavior of 2D C/SiC composites at room temperature. The tensile stressstrain curve exhibits nonlinear due to the multiple energy dissipation mechanisms, including the transverse matrix microcracking, debonding between the fiber bundles and matrix and between the fiber and the matrix inside of fiber bundles, fibers pullout, and the release of thermal residual stress. The mechanical behavior of the 2D C/SiC composite depends on the damage initiation and propagation under tensile loading and the distribution of thermal microcracking and thermal residual stress. Mei et al. (2007) investigated the tensile and cyclic loadingunloading tensile damage evolution of 2D C/SiC composite using the acoustic emission and SEM at room temperature. When the applied stress is lower than 50 MPa, there is no acoustic emission signal and the accumulation acoustic emission energy is nearly zero, and the tensile stressstrain curve exhibits linear. When the applied stress reaches 50 MPa, the acoustic emission signal increases suddenly; and at the applied stress between 50 and 150 MPa, the matrix thermal microcracking caused by the fabrication starts to propagate, and the tangent modulus of the composite decreases with increasing applied stress. When the applied stress exceeds 150 MPa, the tangent modulus of the tensile stressstrain curve increases; however, the acoustic emission signal decreases. When the applied stress reaches 230 MPa, the fibers break and pull out from the matrix, the acoustic emission signal increases greatly, and before the final fracture of the composite, the tangent modulus of the composite decreases again. The composite fails with a large amount of fibers pullout. Jacoben and Brondsted (2001) investigated the tensile damage mechanisms of 2D SiC/SiC composites at room temperature. For SiC/SiC composites, the thermal residual stress is much less than that of C/SiC composites. The tensile stressstrain

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Durability of Ceramic-Matrix Composites

curve appears linear at initial loading. With an increase in applied stress, matrix cracking occurs at the large pores inside of the matrix and then transverse cracking appears at the 90 yarns. With an increase in applied stress, the transverse cracking propagates to the 0 yarns, and at the failure of the composite, the matrix cracking in the 0 yarns does not approach saturation. Calard and Lamon (2002) investigated the tensile damage evolution of 2D SiC/SiC composites at room temperature. Under tensile loading, matrix cracking occurs at the large pores among the fiber bundles and then propagates to the longitudinal and transverse yarns. When the matrix cracking approaches saturation, the fibers start to break and the fiber bundles fracture leads to the final failure of the composite. Morscher et al. (2007) investigated the tensile damage evolution behavior of 2D SiC/SiC composites using acoustic emission at room temperature. The initial matrix cracking stress of 2D SiC/SiC composite increases with increasing fiber volume content along the loading direction. The thermal residual stress is measured through the cyclic loadingunloading testing, and it was found that for the Hi-Nicalon SiC/ SiC composite there is no thermal residual stress. However, for the Sylramic SiC/ SiC composite, thermal residual compressive stress existed in the matrix. The normalized acoustic emission accumulation energy under tensile loading can be used to characterize the matrix multicracking evolution. After tensile failure, the matrix cracking saturation density is measured under an optical microscope. The relationships between the applied stress and the matrix cracking density can be obtained using the measured matrix cracking saturation density and the normalized acoustic emission accumulation energy. Engesser (2004) investigated the tensile behavior of 2D C/SiC composites at an elevated temperature of 550 C in air atmosphere. The elastic modulus under initial loading at an elevated temperature of 550 C is higher than that at room temperature, and the tensile strength and the failure strain at an elevated temperature of 550 C is less than at room temperature. The tensile strength and failure strain decrease with loading rate. When the loading rate decreases, the oxidation time increases at elevated temperature, leading to the oxidation of the fibermatrix interphase and fibers, and then the decrease of the composite tensile strength and failure strain. Sullivan et al. (2007) investigated the tensile stressstrain behavior of 2D C/SiC composites at an elevated temperature of 1100 C in inert atmosphere. The tensile stressstrain curve exhibits nonlinear behavior at initial loading, and the elastic modulus at initial loading, composite tensile strength, and failure strain are much higher than those at room temperature. Lipetzky et al. (1996) investigated the tensile properties of 2D SiC/SiC composites at room temperature and elevated temperatures of 850 C, 1000 C, and 1200 C, respectively. The tensile behavior is largely independent of test temperature below 1000 C. At an elevated temperature of 1200 C, the SiC/SiC composite retains much of its low-temperature stiffness and proportional limit; however, its strength increases. At room temperature, the tensile properties are rate-dependent. When the loading rate increases, the composite Young’s modulus decreases, and the proportional limit and ultimate tensile strength increase.

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Zhu et al. (1999a,b) investigated the tensile stressstrain behavior of 2D SiC/SiC composites at an elevated temperature of 1000 C in Ar atmosphere. The tensile stressstrain curve is linear under initial loading, and the elastic modulus at initial loading, proportional limit stress, composite tensile strength, and failure strain are higher than those at room temperature. Guo and Kagawa (2001) investigated the tensile stressstrain of 2D SiC/SiC composite at 298K, 1200K, 1400K, and 1600K, respectively, in air atmosphere. The tensile stressstrain curves of 2D SiC/SiC composites at these four temperatures exhibit similar behavior, that is, the tensile stressstrain curve is linear under initial loading, and when the applied stress approaches to the critical stress, the tensile stressstrain curve starts to deviate till final fracture. The proportional limit stress decreases with increasing temperature. When the temperature increases to 1200K, the elastic modulus at initial loading decreases a little compared with that at 298K; however, when the temperature exceeds 1200K, the elastic modulus decreases greatly. When the temperature increases to 1200K, the tensile strength is similar with that at 298K; however, when the temperature exceeds 1200K, the tensile strength decreases greatly. The fibers’ in situ strength are obtained using fracture mirror tests. It was found that the fibers’ in situ strength decreases with increasing temperature; however, the fiber Weibull modulus does not change with temperature variation. The tensile stressstrain curves are nonlinear for 2D fiber-reinforced CMCs. The matrix cracking outside and inside yarns and the fibermatrix interface debonding inside the yarns are the main reasons for the nonlinear behavior of the tensile stressstrain curve. Through acoustic emission monitoring, matrix cracking and fibermatrix interface debonding are revealed to be two independent damage processes; however, due to the microstructure and the fibermatrix bonding, the matrix cracking in 2D CMCs may not approach saturation. At elevated temperatures in inert atmosphere, the tensile strength and failure strain are higher than those at room temperature; however, at elevated temperature in air atmosphere, the oxidation of the fibers and interphases have lower tensile strength compared with that at room temperature.

1.3.1.4 2.5D ceramic-matrix composites Dalmaz et al. (1996) investigated the tensile damage evolution of 2.5D C/SiC composites using SEM at room temperature. Under initial loading, there is a large amount of microcracking in the matrix; with an increase in applied stress, this microcracking propagates and connects with new matrix cracking. The microcracking caused by the fabrication and the new matrix cracking caused by the applied stress, lead to the nonlinear behavior of 2.5D C/SiC composites. Ma et al. (2006) investigated the tensile stressstrain behavior of 2.5D C/SiC composites along warp and weft yarns. The tensile stressstrain curves along warp and weft yarns appear obvious nonlinear, and the tensile stressstrain curve can be divided into three regions, that is, the linear region under initial loading, the nonlinear region with increasing applied stress, and the quasi-linear region before the final fracture. The tensile strength and modulus along the warp yarn direction are much

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higher than those along the weft yarn direction. However, the fracture strain along the warp and weft yarns is relatively close to each other. Wang et al. (2008) investigated the tensile damage evolution behavior of 2.5D C/SiC composites using acoustic emission at room temperature. When the applied stress is lower than 20 MPa, the tensile stressstrain curve is linear, there is no occurrence of acoustic emission signal and the composite is without damage. When the stress is higher than 20 MPa, the acoustic emission signal is monitored, and original matrix microcracking propagates; however, the tensile stressstrain curve is still linear. When the applied stress exceeds 50 MPa, the tensile stressstrain curve is nonlinear, the acoustic emission signal increases greatly, bridged matrix cracking appears inside the composite, and debonding occurs when the matrix cracking propagates to the fibermatrix interface. When the applied stress approaches 200 MPa, the acoustic emission signal increases slowly and, after the saturation of the matrix cracking, the tensile stressstrain curve appears linear again till final fracture. The stressstrain curve appears obvious nonlinear for 2.5D fiber-reinforced CMCs. The tensile stressstrain curve can be divided into three regions, that is, the linear elastic region, matrix cracking, the fibermatrix interface debonding nonlinear region, and the secondary linear region after matrix-cracking saturation. Under tensile loading, the fibers broken in the warp yarns lead to the final fracture of the composite.

1.3.1.5 3D ceramic-matrix composites Pan et al. (2005) investigated the damage development of 3D C/SiC composites using acoustic emission at room temperature. The tensile stressstrain curve exhibits nonlinear due to the multiple damage mechanisms of matrix multicracking, fibermatrix interface debonding, and fibers failure. The development of accumulation acoustic emission energy can be divided into three stages, that is, initial damage, critical damage, and fast damage. At the initial damage stage, the damage propagates slowly in the form of primary cracking and weak fibermatrix interface debonding. At the critical damage stage, the damage is mainly caused by the rapid cracking and propagation of the fibermatrix interface in the fiber bundles, and fibers fracture and pull out. At the fast damage stage, the acoustic emission accumulation energy suddenly increases rapidly till final fracture. The accumulation acoustic emission energy can be used to monitor the damage evolution process of 3D C/SiC composites. Luo and Qiao (2003) investigated the tensile behavior of 3D C/SiC composites at room temperature and elevated temperatures of 1100 C and 1500 C under different loading rates (i.e., 0.08, 0.06 and 5.82 mm/min). At room temperature, the tensile strength increases with loading rate; at an elevated temperature of 1500 C, the tensile strength decreases with increasing loading rate; and at an elevated temperature of 1100 C, the tensile strength is independent of loading rate. With an increase in the loading rate at room and elevated temperatures, the fracture strain decreases and the initial elastic modulus increases. Under tensile loading, the damage mechanisms of matrix cracking in the fiber bundles, fibermatrix interface debonding, fibers pullout, and wear between fiber bundles were observed.

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Qiao et al. (2004) investigated the tensile damage evolution behavior of 3D C/SiC composites from room temperature to 1500 C in vacuum atmosphere. The tensile stressstrain curves can be divided into three regions, the modulus of which is dependent on the initial matrix cracking stress and matrix cracking saturation stress. The modulus at the first stage and matrix cracking saturation strain are constant with an increase in temperature. However, the modulus at the second and third stages, matrix first-cracking stress, matrix cracking saturation stress, fracture stress, and matrix first-cracking strain increase with the increase of temperature till 1300 C, and then decrease when the temperature is higher than 1300 C. Mazars et al. (2017) investigated the tensile damage of 3D SiC/SiC composites at room temperature and 1250 C using X-ray microtomography. Image processing and Digital volume correlation residuals are used to identify the damage mechanisms with increasing loads. The matrix cracks first initiate in weft yarns and then develop and proliferate in the weft planes until they coalesce and create a throughthickness failure. The fibers break outside the cracking plane at 25 C, and they break within the main crack plane at 1250 C. For 3D CMCs, damage evolution and damage mechanisms under tensile loading depend on the testing temperature and loading rate. Acoustic emission, electrical resistance, and X-ray microtomography can be used to monitor the damage evolution process. The tensile stressstrain curves can also be divided into three stages, and the first matrix-cracking stress, saturation matrix-cracking stress, and tensile strength depend on the testing temperature.

1.3.2 Theoretical Analysis In this section, the theoretical analysis of initial matrix-cracking, matrix multicracking evolution, fibers failure, and stressstrain curves of fiber-reinforced CMCs under tensile loading are overviewed.

1.3.2.1 Initiation matrix cracking A matrix crack can be classified as long matrix cracking (Aveston et al., 1971; Aveston and Kelly, 1973; Budiansky et al., 1986; Chiang, 2001, 2007) or short matrix cracking (Marshall et al., 1985; Marshall and Cox, 1987; Thouless and Evans, 1988; Cao et al., 1990; Chiang et al., 1993; Danchaivijit and Shetty, 1993; Chiang, 2000) according to the internal defect size. When the matrix internal defect is larger than the characteristic value of co, then the initiation matrix cracking stress is independent on the matrix internal defect. pffiffiffiffiffiffi2=3  1  1=3 Vm rf Ec Em co 5 ξ m 2 Vf 2 τ i E f

(1.1)

where ξm denotes the matrix fracture energy; rf denotes the fiber radius; Vf and Vm denote the fiber and the matrix volume fraction, respectively; Ef, Em, and Ec denote

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the fiber, matrix, and composite elastic modulus, respectively; and τ i denotes the fibermatrix interface shear stress. During the propagation of matrix cracking, the stress field before and after the crack tip remains unchanged, the cracking penetrates through the width and thickness of the specimen, and the matrix cracking propagates in the steadystate condition. However, when the matrix internal defect is less than the characteristic value of co, the initiation matrix cracking stress depends on the matrix interface defect size, and the process of matrix cracking propagation is in the nonsteady state. Kim and Pagano (1991) monitored the first matrix-cracking stress of unidirectional SiC/ CAS, SiC/1723, SiC/BMAS, and SiC/LAS-III composites using an acoustic emission and optical microscope. It was found that the first matrix-cracking stress is far less than the solution ACK model (Aveston et al., 1971), and the microcracking for the first matrix-cracking usually appears in the matrix-rich region. Barsoum et al. (1992) found that the first matrix-cracking stress depends on the maximum fiber nonbridging region caused by the fiber random distribution inside the matrix as the existence of these microcracks do not affect the composite modulus. However, with an increase in applied stress, these microcracks evolve into the short matrix cracking defined by the MCE model (Marshall et al., 1985), and then the long matrix cracking defined by the ACK model (Aveston et al., 1971). σACK cr

6Vf2 Ef Ec2 τ i ξ m 5 rf Vm Em2

!1=3 2 Ec ðαc 2 αm ÞΔΤ

  σMCE 1 a 23=4 cr 5 1 1 8 co σACK cr

(1.2)

(1.3)

where αm and αc denote the matrix and composite thermal expansion coefficient, respectively; ΔT denotes the temperature difference between the fabricated and testing temperature; and a denotes the matrix defect length.

1.3.2.2 Matrix multicracking evolution When the applied stress exceeds the first matrix-cracking stress, new matrix cracking occurs in the existing matrix-crack spacing. There are four approaches to simulate the matrix multicracking evolution: (1) the maximum stress approach (Aveston et al., 1971; Lee and Daniel, 1992), (2) the energy balance approach (Zok and Spearing, 1992; Weitsman and Zhu, 1993), (3) the critical matrix strain energy criterion (Solti et al., 1997; Longbiao et al., 2008), and (4) the stochastic matrix cracking approach (Curtin, 1993; Curtin, 1999; Okabe et al., 1999).

1.3.2.3 Fibers Failure After the saturation of matrix cracking, fibers carry most of the applied stress. With fibers failure, the load is redistributed between the fractured and intact fibers. There are two criteria to determine the load distribution between the intact and fracture

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fibers, that is, GLS (Curtin, 1991) and LLS (Curtin et al., 1998). The GLS criterion assumes that when the fibers fracture, the load is evenly distributed to intact fibers (Thouless and Evans, 1988; Cao and Thouless, 1990; Sutcu, 1989; Schwietert and Steif, 1990; Curtin, 1991; Hild et al., 1994; Xia and Curtin, 2000). The LLS criterion assumes that when the fibers break, the load is distributed to neighboring intact fibers (Dutton et al., 2000; Xia and Curtin, 2000; Okabe et al., 2001; Xia et al., 2002). The GLS criterion neglects the stress concentration caused by the fibers fracture and is suitable for the weak fibermatrix interface bonding condition (i.e., τ i , 50 MPa). Paar et al. (1998) adopted the two-parameter Weibull model to describe fibers strength. The GLS criterion is used to describe the stress distribution after fibers fracture. Combing with the weakest chain model, the fibers failure location and fibers failure volume fraction are analyzed. The predicted tensile stress strain curve coincided with experimental data; however, the predicted tensile strength was higher than the experimental data. Liao and Reifsnider (2000) adopted the Coulomb’s frictional law to describe the fibermatrix interface shear stress in the debonded region and predicted the process of fibers fracture after matrix cracking and fibermatrix interface debonding, composite tensile strength, and fracture toughness based on the OhFinnie model (Oh and Finnie, 1970). Solti et al. (1995a) developed the critical fiber strain energy (CFSE) criterion for simulating the fibers failure in fiber-reinforced CMCs. The fiber characteristic strength in the Weibull model is changed to CFSE, and the fiber Weibull modulus is obtained by fitting the tensile stressstrain curve of the fiber failure section. The model avoids the measurement of the fiber in situ strength and the fiber in situ Weibull modulus; however, the method of obtaining the model parameter by fitting the tensile stressstrain curve needs further verification.

1.3.2.4 Stressstrain curve Pryce and Smith (1992) simulated the tensile stressstrain curve of unidirectional fiber-reinforced CMCs. The simplified shear-lag model is used to describe the microstress field of the damaged composite with matrix cracking and fiber—matrix interface debonding. Combined with the measured matrix cracking density, the tensile stressstrain curve for the stage of matrix cracking and fibermatrix interface debonding is predicted. In the PryceSmith model, the load transfer length is used to determine the fibermatrix interface’s debonded length. Curtin et al. (1998) adopted the matrix stochastic cracking model and fibers failure model to predict the matrix cracking evolution and fibers stochastic failure and predicted the tensile stressstrain curve of unidirectional fiber-reinforced CMC based on the PryceSmith model (Pryce and Smith, 1992). In Curtin’s model, the fibermatrix interface’s debonded length is also determined by the load transfer length, the predicted tensile strength is higher than the experimental data, and the tensile stressstrain curve in the stage of fibers failure has a large difference with experimental results. Solti et al. (1997) adopted the shear-lag model to describe the microstress field of the damaged composite after matrix multicracking and fibermatrix interface debonding. The critical matrix strain energy, the maximum interface shear stress

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criterion, and the CFSE criterion are used to describe the matrix multicracking evolution, fibermatrix interface debonding, and fibers failure. Combined with the shear-lag model, the tensile stressstrain curve of unidirectional fiber-reinforced CMCs is predicted. However, as the parameter in the CFSE model is obtained from fitting the stressstrain curve in the fibers failure stage, the composite tensile strength cannot be predicted. Cheng et al. (2004) predicted the deformation, damage, and failure of unidirectional fiber-reinforced CMCs by combining the Monte Carlo method and FEM, and theoretical results coincided with the tensile stressstrain curve in the stage of matrix cracking, and the predicted tensile strength is higher than the experimental value. The fibermatrix interface debonding criterion adopted by Cheng’s model is simple, and the effect of fibermatrix interface properties on the macrotensile stressstrain curve is not considered. Solti et al. (1995b) analyzed the microstress field of damaged cross-ply fiberreinforced CMCs using the shear-lag model. The critical matrix strain energy, the maximum shear stress, and the CFSE criteria are used to determine the transverse cracking in the 90 ply, matrix cracking in the 0 ply, fibermatrix interface debonding, and fibers failure. By combining the shear-lag model with the damage models, the tensile stressstrain curve of cross-ply fiber-reinforced CMCs is predicted. Xu (2008) investigated the effect of fiber volume content, fibermatrix interface shear stress, and fibermatrix interface bonding strength on the tensile stressstrain curve of cross-ply fiber-reinforced CMCs based on Solti’s model (Solti et al., 1995b). Tao et al. (2009) divided the 2D CMCs into two parts, that is, the orthogonal laying structure and the fiber bundle wave structure. The stressstrain relationship between the orthogonal laying part and the fluctuation part of the fiber bundle is obtained by the micromechanical method and the volume averaging method, respectively. However, the simplified assumption of the model ignores the influence of inner pores in the warp and weft yarns and outside pores on the tensile behavior of the composite, and the effect of matrix cracking and fibermatrix interface debonding in the fluctuation region of warp yarn is not considered. Longbiao et al. (2014a,b, 2015) investigated the tensile behavior of unidirectional and cross-ply C/SiC composites under tensile loading. The matrix cracking, fibermatrix interface debonding and fibers failure were considered using the stochastic cracking model, fracture mechanics approach, and GLS criterion. The damage evolution of the matrix multicracking, fibermatrix interface debonding, fibers failure, and the tensile stressstrain curves of unidirectional and cross-ply C/SiC composites were predicted.

1.4

Overview of fatigue behavior of ceramic-matrix composites

Compared with single ceramics and whisker-reinforced CMCs, the fatigue behavior of fiber-reinforced CMC has many different characteristics. During the fabrication of fiber-reinforced CMCs, the matrix has many microcracks (Wang et al., 1991b,

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2010; Wang and Laird, 1997; Kostopoulos et al., 1997; Dalmaz et al., 1998; Du et al., 2002a,b; Liu, 2003; Sun et al., 2007; Li, 2011, 2018a). Under cyclic loading, the propagation rate of matrix microcracking is slower than that of single ceramics (Holmes and Cho, 1992; Karandikar and Chou, 1992; Pryce and Smith, 1993; Opalski and Mall, 1994; Morscher and Maxwell, 2018). For the single ceramics and whisker-reinforced CMCs, the fatigue failure is caused by single matrix cracking; however, for the fiber-reinforced CMC, there are a large number of matrix cracks in the composite, and matrix cracking is not the direct region for fatigue failure (Opalski, 1992; Mall and Tracy, 1992; Karandikar and Chou, 1993b; Pryce and Smith, 1994; Kim and Liaw, 2005; Maillet et al., 2012). Under cyclic fatigue loading, the mechanical performance (i.e., the strength and modulus) decreases with cyclic loading. At room temperature, matrix cracking, fibermatrix interface debonding, fibermatrix interface sliding, and fibers fracture are the main fatigue damage mechanisms (Wang et al., 2010; Zhu et al., 1996, 1997, 1998, 1999a,b; Mizuno et al., 1996; Dalmaz et al., 1998; Du et al., 2002a,b; Liu, 2003). At elevated temperature, the fibermatrix interface degradation and fibers creep are responsible for fatigue failure (Prewo, 1987; Holmes et al., 1989; Allen et al., 1993; Steiner, 1994; Holmes and Sørensen, 1995; Sun et al., 1996; Yasmin and Bowen, 2004). The microstructure, strength characteristic, and thermal residual stress distribution also change with temperature changes (Reynaud et al., 1994; Reynaud, 1996), which makes the fatigue damage mechanisms of fiber-reinforced CMCs much more difficult to understand. The experimental observation and measurement provide a basis for the study of the fatigue damage mechanism of fiber-reinforced CMCs. At present, the methods used to study fatigue behavior of fiber-reinforced CMCs mainly include: SN curve (Zawada et al., 1990, 1991; Rouby and Reynaud, 1993; Mizuno et al., 1996; Evans, 1997; Zhu et al., 1999a,b; McNulty and Zok, 1999; Haque and Rahman, 2000; Cao et al., 2001b; Zhu and Kagawa, 2001; Sørensen et al., 2002; Du et al., 2002b; Zawada et al., 2003; Liu, 2003; Yasmin and Bowen, 2004; Kim and Liaw, 2005; Singh and Mall, 2018; Ruggles-Wrenn, et al., 2018), residual strength and modulus (Prewo, 1987; Zawada et al., 1991; Holmes and Cho, 1992; Reynaud et al., 1997, 1998; Sørensen et al., 2000), hysteresis loops (Zawada et al., 1991, 2003; Holmes and Cho, 1992; Opalski, 1992; Pryce and Smith, 1993, 1994; Reynaud et al., 1994; ¨ nal, Opalski and Mall, 1994; Evans, 1997; Reynaud, 1996; Mizuno et al., 1996; U 1996a; McNulty and Zok, 1999; Zhu et al., 1999a,b; Sun et al., 2007; Wang et al., 2010), surface matrix cracking density (Zawada et al., 1991; Holmes and Cho, 1992; Karandikar and Chou, 1992, 1993a,b; Pryce and Smith, 1993, 1994; Opalski and Mall, 1994; Lee and Stinchcomb, 1994; Dalmaz et al., 1998; McNulty and Zok, 1999; Vanwijgenhoven et al., 1999; Cao et al., 2001b; Kim and Liaw, 2005), surface temperature rising (Holmes and Cho, 1992; Kim and Liaw, 2005), ultrasonic (Kostopoulos et al., 1999), acoustic emission (Zawada et al., 1990; Haque and Rahman, 2000; Godin et al., 2011; Maillet et al., 2012; Racle et al., 2017; Godin et al., 2018), electrical resistance (Du et al., 2002a, 2002b; Han et al., 2016; Sullivan et al., 2018; Han et al., 2018), digital image correlation (Morscher and Maxwell, 2018), and SEM. Similar to traditional materials, the fatigue life of

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fiber-reinforced CMCs decreases with increasing fatigue peak stress and stress amplitude. At room temperature, the fatigue limit stress is about 60%80% of the tensile strength (for metals, the fatigue limit stress is about 50% tensile strength), which is much higher than the first matrix-cracking stress. In order to cause damage in fiber-reinforced CMCs, the fatigue peak stress should be higher than the first matrix-cracking stress. Under cyclic loading, the modulus decreases and the hysteresis width and residual strain increases. When the particles generated by the matrix wear are mixed in the fibermatrix interface or matrix crack plane, the composite modulus will increase with cyclic loading. For most fiber-reinforced CMCs the matrix has low-fracture toughness and there is no fatigue damage. At room temperature, the degradation of the fibermatrix interface and fibers is the major reason for fatigue damage (Rouby and Reynaud, 1993; Lee and Stinchcomb, 1994; Reynaud, 1996; Mizuno et al., 1996; Zhu et al., 1999a,b; Zhu and Kagawa, 2001). The degradation of the fibermatrix interface is mainly caused by the interface layer fracture and wear, release of thermal residual stress, and temperature rising at the interface. The fibers degradation is caused by the fiber surface defect during fibermatrix interface wear. By the fiber fracture mirror tests, the fibers strength decreases greatly after fatigue loading (McNulty and Zok, 1999). The decrease of the fibermatrix interface causes the increase of the fibermatrix interface debonded length with increasing applied cycles, leading to the increase of residual strain and the decrease of composite modulus. The fiber characteristic length and the stress carried by the fibers increase with decreasing fibermatrix interface shear stress, and the fiber strength decreases due to the interface wear, which increases the failure probability of the fibers. Due to these factors, the fibers continually fracture under cyclic loading, and when the broken fibers fraction approaches the critical value, the composite modulus decreases greatly, and the composite fatigue fractures. At intermediate temperature (400 C800 C) in inert atmosphere, there is no chemical response between the fiber and the matrix, and there is no oxidation and creep. An increase in temperature will affect the radial and axial thermal residual stress at the fibermatrix interface, and then the fibermatrix interface shear stress, fiber, matrix axial stress, and then the fatigue behavior of fiber-reinforced CMCs (Reynaud, 1996), including: G

G

When the radial thermal expansion coefficient of the matrix is less than that of the fiber, when the temperature increases, the radial thermal residual stress at the fibermatrix interface decreases, leading to the decrease of the fibermatrix interface shear stress, and the wear rate of the interface, and also the damage of the composite. However, when the radial thermal expansion coefficient of the matrix is higher than that of the fiber, an increase in temperature increases the fibermatrix interface radial thermal residual stress, leading to the increase of the fibermatrix interface shear stress, and the wear rate of the interface, and also the damage of the composite. When the axial thermal expansion coefficient of the matrix is less than that of the fiber, at room temperature the matrix axial thermal residual stress is compressive stress, and the fiber axial thermal residual stress is tensile stress. With an increase in temperature, the matrix axial thermal residual compressive stress decreases, leading to the decrease of first

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matrix-cracking stress, and when the temperature is high enough, a large amount of cracks appear in the matrix before fatigue loading, which will decrease the fatigue life at elevated temperature. However, when the matrix axial thermal expansion coefficient is higher than that of the fiber, the matrix axial thermal residual stress is tensile stress, and the fiber axial thermal residual stress is compressive stress. With an increase in temperature, the matrix axial thermal residual tensile stress will decrease, leading to the increase of the first matrix-cracking stress, which will increase the fatigue life at elevated temperature.

At high temperatures ( . 800 C) in inert atmosphere, the chemical response occurs at the fiber/matrix interface, and the interphase properties degrade. The fiber creep also occurs when the matrix creep strength is less than that of the fibers. The matrix axial stress will decrease and the fiber axial stress will increase, leading to an increase in the fiber bridge stress and a decrease in fatigue life. However, when the matrix creep strength is higher than that of the fiber, the matrix axial stress will increase, leading to more matrix cracking and fibermatrix interface debonding, ¨ nal, 1996a; Zhu et al., 1997, 1999a,b; and is much more serious fatigue damage. (U Du et al., 2002b; Liu, 2003) At elevated temperatures in air atmosphere, most fiber-reinforced CMCs encounter oxidation embrittlement. Upon first loading to the fatigue peak stress, the matrix cracking appears, oxygen enters inside the composite and oxidizes the fibermatrix interphase or reacts with the matrix to form a new strong fibermatrix interface bonding. The disappearance of the fibermatrix interphase and the presence of a new strong fibermatrix interphase decrease the fatigue life of fiber-reinforced CMCs. (Prewo, 1987; Prewo et al., 1989; Holmes et al., 1989; Rousseau, 1990; Allen et al., 1993; Steiner, 1994; Elahi et al., 1994; Groner, 1994; Holmes and Sørensen, 1995; Sun et al., 1996; Haque and Rahman, 2000; Yasmin and Bowen, 2004; Wu et al., 2006). Fatigue hysteresis, fibermatrix interface wear, fibers strength degradation, oxidation embrittlement, and modulus degradation of fiber-reinforced CMCs under cyclic fatigue loading are overviewed in the next section. The effects of loading frequency and fatigue stress ratio on the fatigue damage are also analyzed.

1.4.1 Fatigue hysteresis behavior Under fatigue loading of fiber-reinforced CMCs, when the fatigue peak stress is higher than the first matrix-cracking stress, upon first loading to the fatigue peak stress, the damage of matrix cracking and fibermatrix interface debonding appear. During the subsequent cyclic loading, the fiber sliding relative to the matrix in the interface debonded region leads to the fatigue stressstrain hysteresis loops of fiber-reinforced CMCs. The shape, location, and area of the fatigue hysteresis loops reveal the fatigue damage extent inside fiber-reinforced CMCs (Fantozzi and Reynaud, 2009, 2014).

1.4.1.1 Experimental observation Marshall and Evans (1985) found the fatigue stressstrain hysteresis loops of unidirectional fiber-reinforced CMCs and attributed the fatigue hysteresis loops to the

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frictional slip between the fiber and the matrix at the interface. Marshall and Oliver (1987) conducted fiber pushin/pushout tests and also found the stressstrain hysteresis loops, and proved interface sliding mechanisms. Minford and Prewo (1986) found the obvious fatigue hysteresis loops of fiberreinforced unidirectional CMCs under tensiontension cyclic fatigue loading at room temperature. Holmes et al. (1989) found the fatigue hysteresis loops and strain ratcheting off unidirectional fiber-reinforced CMCs at elevated temperatures, and the fatigue peak stress of the hysteresis loops was higher than that of first matrixcracking stress. Kotil et al. (1990) investigated the fatigue hysteresis mechanism of unidirectional fiber-reinforced CMCs and attributed to the fatigue hysteresis loops to the sliding process of the pullout and pushin of fracture fibers. Holmes and Cho (1992) analyzed the evolution characteristics of fatigue hysteresis loops of unidirectional fiber-reinforced CMCs at room temperature. At initial cyclic loading, the matrix cracking and fibermatrix interface debonding occur, and fatigue hysteresis modulus decreases quickly. When the cycle exceeds a certain number, the modulus gradually recovers. The recovery of the modulus may be due to the mixed particles at the fibermatrix interface or matrix cracking plane, which increases the fibermatrix interface shear stress. The fatigue hysteresis loops area increases first to the peak value, then decreases, and increases quickly again when approaching final fracture. Zawada et al. (1991) investigated the fatigue hysteresis loops shape and area of cross-ply fiber-reinforced CMCs under tensiontension loading at room temperature. Under initial loading, when the fatigue peak stress exceeds the proportional limit stress, the loading stressstrain curve is nonlinear, and the unloading stress strain curve is linear, and the fatigue hysteresis loop is not closed. At the cycle number of 400,000, the composite fatigue hysteresis modulus and hysteresis loops area decrease obviously; and when the applied cycle reaches 1,000,000, the fatigue hysteresis loops area continually decrease. However, the fatigue hysteresis modulus slowly increases compared with the cycle number of 400,000, due to the mixed particles at the matrix cracking plane or fibermatrix interface. Lynch and Evans (1996) investigated the loadingunloading behavior of three different layups cross-ply fiber-reinforced CMCs (i.e., [0/90], [ 6 45 ], and [ 1 30 / 2 60 ]). Upon unloading and reloading, the stressstrain curves appear as obvious hysteresis loops. Compared with [0/90] cross-ply CMCs, the fatigue hysteresis loops shape of the [ 6 45 ] and [ 1 30 / 2 60 ] composites are very different due to the cracking-closure effect. Reynaud (1996) investigated the tensiontension fatigue behavior of 2D SiC/ SiC and [0/90]s cross-ply SiC/MAS-L composites at elevated temperatures of 600 C, 800 C, and 1000 C in inert atmosphere. For the 2D SiC/SiC composite, the fibermatrix interface radial thermal residual stress is compressive stress, and with increasing temperature, the interface radial thermal residual stress decreases, leading to the decrease of the fibermatrix interface shear stress and the higher fatigue hysteresis dissipated energy than that at room temperature. For the [0/90]s cross-ply SiC/MAS-L composite, the fibermatrix interface radial stress is tensile stress, and with increasing temperature, the fibermatrix interface radial tensile stress

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decreases, leading to the increase of the fibermatrix interface shear stress, and higher fatigue hysteresis dissipated energy than that at room temperature. At elevated temperatures of 800 C1000 C in inert atmosphere, the chemical reaction at the fibermatrix interface leads to the degradation of the fibermatrix interface shear stress. After heat treating for 50 hours at elevated temperature in inert atmosphere, the fibermatrix interface shear stress decreases. Dalmaz et al. (1998) investigated the cyclic fatigue hysteresis loops of 2.5D C/SiC composites at room and elevated temperatures of 600 C in inert atmosphere. Under tensiontension fatigue loading, the fatigue hysteresis loops area decreases with increasing of the applied cycles, due to the degradation of the interface shear stress between the yarns and the matrix. The fatigue hysteresis loops area at an elevated temperature of 600 C is less than that at room temperature due to the matrixcracking closure at elevated temperature. The decrease of the fatigue hysteresis loops area proves the interface wear mechanism. However, at an elevated temperature of 600 C, the fatigue hysteresis modulus decreases first, and when the applied cycle approaches a certain number, the fatigue hysteresis modulus partially recovers due to the matrix-cracking closure and the fiber rotation in the yarns (Dalmaz et al., 1996). Fantozzi and Reynaud (2009) investigated the cyclic tensiontension fatigue behavior of 2.5D SiC/[SiBC] and 2.5D C/[SiBC] composites at an elevated temperature of 1200 C in air atmosphere. For the 2.5D SiC/[SiBC] composite under 50/200 MPa, the fatigue hysteresis dissipated energy decreases with an increase in the applied cycle number due to the fibermatrix interface wear and the degradation of the fibermatrix interface shear stress. For the 2.5D C/[SiBC] composite under static fatigue loading for 144 hours, the fatigue hysteresis loops area decreases obviously, mainly due to the fibermatrix PyC interphase oxidation or the recession of the carbon fibers.

1.4.1.2 Theoretical analysis Kotil et al. (1990) investigated the fatigue hysteresis loops of fiber-reinforced CMCs and found that the fibermatrix interface shear stress affects the fatigue hysteresis loops shape and area. When the fibermatrix interface shear stress is too low or too high, the hysteresis loops are negligible. When the fibermatrix interface shear stress is too low, the fiber can freely slip in the matrix, and the energy dissipation caused by sliding is small. However, when the fibermatrix interface shear stress is too high, the fibermatrix interface debonded length is small, and the energy dissipation caused by sliding is also small. When the fibermatrix interface shear stress is moderate, the fatigue hysteresis dissipated energy can approach to maximum. Cho et al. (1991) divided the fibermatrix interface sliding into partial slip and completely slip and obtained the hysteresis energy dissipated rate for these two conditions of fiber-reinforced CMCs. Pryce and Smith (1993) analyzed the fatigue hysteresis loops when the fibermatrix interface partially debonded by assuming the constant frictional

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interface shear stress. Ahn and Curtin (1997) divided the matrix cracking space into long cracking, medium cracking, and short cracking, analyzed the matrix stochastic cracking on the fatigue hysteresis loops, and compared the results with the PS model (Pryce and Smith, 1993). Solti et al. (2000) adopted the maximum interface shear stress criterion to determine the fibermatrix interface debonded length, unloading interface counter slip length, and reloading new slip length, and predicted the fatigue hysteresis loops of fiber-reinforced CMCs when the fibermatrix interface is chemical bonding. Kirth and Kedward (1995) investigated the cyclic loadingunloading fatigue hysteresis loops models for the fibermatrix interface debonding based on the PS model (Pryce and Smith, 1992). Longbiao et al. (2008) investigated the effect of fibers Poisson contraction on the fatigue hysteresis loops of unidirectional fiber-reinforced CMCs based on Coulomb’s frictional law. The unloading and reloading fiber and matrix axial stress, and the fibermatrix interface shear stress, are solved using the Lame equation. The fracture mechanics approach is adopted to determine the fibermatrix interface debonded length, unloading interface counter slip length and reloading interface new slip length. The unloading and reloading stressstrain relationship are solved to predict the fatigue hysteresis loops. The effects of fibermatrix interface debonded energy and fibermatrix interface frictional coefficient on the fibermatrix interface debonding upon initial loading, the interface counter slip upon unloading and new slip upon reloading are discussed. The predicted results are compared with experimental data and a PS model (Pryce and Smith, 1992). Sørensen et al. (1993) investigated the cyclic loadingunloading stressstrain hysteresis loops of unidirectional fiber-reinforced CMCs using the axisymmetric FEM. The Coulomb’s frictional law is used to describe the fibermatrix interface shear stress. Upon loading, the fibermatrix interface shear stress decreases due to the Poisson contraction, which decreases the elastic modulus; and upon unloading the fibermatrix interface shear stress recovers, leading to the increase of the elastic modulus. Vagaggini et al. (1995) developed fatigue hysteresis loops models based on the HutchinsonJensen fiber pullout model (Hutchison and Jensen, 1990) when the fibermatrix interface is chemical bonding. The fibermatrix interface debonded energy affects the initial fibermatrix interface debonding and relative sliding under the cyclic fatigue loading. When the fibermatrix interface debonded energy is low, the unloading interface counter slip and reloading interface new slip are both independent of the interface debonding. However, when the fibermatrix interface debonded energy is high, the unloading interface counter slip and reloading interface new slip stops at the interface debonding tip, and upon continually unloading or reloading, the slip range remains the same. Hild et al. (1996) developed the fatigue hysteresis loops model of fiberreinforced CMCs using continuum damage mechanics. Their model consists of four state parameters, one experimental observation parameter, and three internal variables. The three internal variables include matrix cracking, interface debonding, and crack opening strain. By determining the damage evolution criterion of these parameters using the loadingunloading testing, it is found that the residual stress

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field has a greater effect on the hysteresis loop width and the inelastic strain evolution, but a smaller effect on the initial unloading elastic modulus. Mei and Cheng (2009) investigated the cyclic loadingunloading hysteresis behavior of 2D, 2.5D, and 3D C/SiC composites, and related the fatigue hysteresis loops shape with the fiber volume fraction along the loading direction. Longbiao et al. (2014a,b) established the cyclic loadingunloading micromechanical model of unidirectional fiber-reinforced CMCs, and analyzed the initial loadingunloadingreloading stress strain relationship and predicted the cyclic loadingunloading stressstrain curve based on the measured matrix cracking evolution curve. Guo et al. (2015) investigated the tensile and shear stressstrain behavior and microstructure characteristics and obtained the material constituent properties based on the hysteresis analysis. Under cyclic shear loading, the stressstrain curve appears as obvious hysteresis behavior, and with increasing unloading peak stress, the hysteresis width and residual strain increase, and the damage inside the composite also increases. Xu (2008) established the fatigue hysteresis loops model of cross-ply fiber-reinforced CMCs considering the effect of transverse and longitudinal matrix cracking, and fibermatrix interface debonding. Yang (2011) established the fatigue hysteresis loops model of 2.5D C/SiC composite considering the effect of microstructure damage on macroproperties based on the modulus reduction method and predicted the fatigue hysteresis loops for different applied cycles. Fang et al. (2016) investigated the tensioncompressive fatigue behavior and failure mechanisms of needled C/SiC composites. With increasing applied cycles, the hysteresis modulus decreases, and the residual strain and hysteresis loops area increase. Longbiao (2016a,b, c,d,e, 2017a,b,c, 2018a) developed the fatigue hysteresis loops model of fiber-reinforced CMCs considering matrix multicracking, fibermatrix interface debonding/wear/oxidation and fibers fracture, predicted the cyclic loadingunloading hysteresis loops for different fatigue peak stress and fatigue hysteresis loops for different cycle number, investigated the effect of multiple-step loading, thermomechanical loading on the fatigue hysteresis loops, and proposed an approach to predict the fiber/matrix interface shear stress of fiber-reinforced CMCs through the fatigue hysteresis loops. They also estimated the fibermatrix interface shear stress of unidirectional, cross-ply, and 2D CMCs for different peak stresses and applied cycles (Li, 2014, 2016f, 2017d), established the relationship between the fatigue hysteresis dissipated energy with surface temperature rising, and developed a hysteresis dissipated energy-based damage parameter to monitor the damage evolution inside of fiber-reinforced CMCs (Li, 2016g,h, 2018b).

1.4.2 Interface wear behavior Under cyclic fatigue loading, the interface wear between the fiber and the matrix decreases the fibermatrix interface shear stress and fibers strength, leading to the degradation of modulus, strength, and lifetime of fiber-reinforced CMCs.

1.4.2.1 Experimental observation Holmes and Cho (1992) characterized the fatigue behavior of fiber-reinforced CMCs through measuring surface temperature. When the fiber slides relative to the

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matrix in the debonded region, the specimen surface temperature increases obviously under cyclic loading. The relationship between the specimen surface temperature and the fibermatrix interface shear stress has been established. The fibermatrix interface shear stress of unidirectional SiC/CAS composite at room temperature was obtained. It was found that the fibermatrix interface shear stress decreases at the initial cyclic loading. Kostopoulos et al. (1997) investigated the tensiontension fatigue damage accumulation of 3D C/SiC composite at room temperature. When the fatigue peak stress is higher than the first matrix-cracking stress upon first loading to the peak stress, multiple matrix cracking occurs in the composite. The matrix cracking approaches a steady state at the initial cyclic loading and remains the same during further cycling. The fibermatrix interface wear decreases the stress concentration at the intersection of the yarns and partially increases the modulus of the composite. At the final stage of cyclic loading, a large amount of fibers failure leads to the final fracture of the composite. The fracture surface of fatigue failure specimens was observed and a large number of fibers pulled out, which proved that the interfacial wear mechanism existed during cyclic loading. Du et al. (2002b) investigated the tensiontension fatigue behavior of 3D C/SiC composites at room temperature and elevated temperatures of 1300 C in vacuum atmosphere. Under high fatigue peak stress, the matrix cracking defects along the fibermatrix interface, and the fibermatrix interface debonds. With increasing cycles, the matrix inside of fiber bundles cracks and the wear between the fiber and the matrix increases. At room temperature, the interface wear caused by the repeated sliding between the fiber and the matrix plays an important role in fatigue failure. However, at 1300 C the effect of the interface wear on the fatigue damage becomes low. Han et al. (2004) investigated the tensiontension cyclic fatigue behavior of 2D and 3D C/SiC composites at elevated temperatures of 1100 C1500 C in vacuum atmosphere. Compared with the 2D C/SiC composite, the fiber pullout length of 3D C/SiC composites at the fracture surface is much longer, indicating the longer fibermatrix interface sliding length.

1.4.2.2 Theoretical analysis Evans (1997) investigated the tensiontension cyclic fatigue behavior of unidirectional SiC/CAS composites at room temperature. Under cyclic fatigue loading, the fatigue hysteresis loops corresponding to different applied cycles are measured, and the fibermatrix interface shear stress is estimated by analyzing the shape of the fatigue hysteresis loops. It was found that the fibermatrix interface shear stress decreases from τ i 5 20 MPa at the first cycle to τ i 5 5 MPa at the 30th applied cycle. The fibermatrix interface wear model was developed to predict the fibermatrix interface shear stress degradation and compared with experimental data. McNulty and Zok (1999) investigated the tensiontension cyclic fatigue behavior of two different fiber-reinforced CMCs of SiC/CAS and SiC/MAS at room temperature. Based on the method developed by Evans (1997), the fibermatrix

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interface shear stress for different applied cycles are obtained. For the SiC/CAS composite, the initial fibermatrix interface shear stress is τ o 5 1012 MPa, and the steady-state fibermatrix interface shear stress is τ s 5 57 MPa. For the SiC/ MAS composite, the initial fibermatrix interface shear stress is τ o 5 2025 MPa, and after cycling 40,000, the fibermatrix interface shear stress decreases to τ s 5 78 MPa. The degradation of fibermatrix interface shear stress is the major reason for the low-cycle fatigue failure. Longbiao (2017d) established the relationship between the fatigue hysteresis dissipated energy and the fibermatrix interface shear stress and analyzed the fibermatrix interface shear stress degradation rate of C/SiC and SiC/SiC composites with different fiber preforms (i.e., unidirectional, cross-ply, 2D, 2.5D, and 3D) at room and elevated temperatures.

1.4.3 Fibers strength degradation Under cyclic fatigue loading, the fibers strength degrades. McNulty and Zok (1999) performed the fiber fracture mirror experiments for pullout fibers at the fracture surface of unidirectional SiC/CAS composites. It was found that after fatigue loading, the fibers strength decreases by approximately 15%. Staehler et al. (2003) performed SEM analyses on the fracture surface of 2D C/SiC composites after fatigue failure. It is found that symmetrical wear caused by yarn relative sliding and wear caused by fibermatrix interface sliding inside the yarn exist on the fiber surface, leading to the fibers strength degradation. Lee and Stinchcomb (1994) investigated the tensiontension fatigue behavior of cross-ply SiC/CAS-II composites at room temperature and developed the fiber strength degradation model. Jiang et al. (2001) obtained the carbon fiber in situ strength of C/SiC composites using the modified fracture mirror method and found that the in situ carbon fiber strength decreases due to defects caused by the fabrication. Longbiao (2016i) investigated the strength degradation of fiber-reinforced CMCs under multiple-step loading, and analyzed the effect of fatigue peak stress and applied cycles on the composite strength degradation. It was found that the fibers strength degradation is the main reason for the degradation of the composite.

1.4.4 Oxidation embrittlement At elevated temperatures in air atmosphere, most fiber-reinforced CMCs exhibit the behavior of oxidation embrittlement (Gowayed et al., 2015). Steiner (1994) investigated the tensiontension fatigue behavior of cross-ply SiC/MAS composite at elevated temperatures of 566 C and 1093 C in air atmosphere. With increasing temperature, the amount of the fibers pullout decreases due to the fibermatrix interface oxidized strong bonding, resulting the brittle fracture of the composite and decrease of the fatigue life. Halbig and Eckel (2000) performed nonstressed oxidation experiments of 2D C/SiC composite and divided the carbon oxidation into two different modes (i.e., reaction control oxidation and diffusion control oxidation) based on the temperature range. The applied stress that increases the matrix opening

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width changes the oxidation mechanism inside the C/SiC composite. Lamouroux et al. (1994) investigated the oxidation mechanism and kinetic of 2D C/SiC composites and developed an oxidation kinetic model of C/SiC composites. RugglesWrenn and Jones (2013) investigated the tensiontension cyclic fatigue behavior of 2D SiC/SiC composites at elevated temperature. It was found that oxidation embrittlement caused the great decrease in fatigue limit stress. Pailler and Lamon (2005) developed the micromechanical fatigueoxidation damage model of fiberreinforced CMCs and predicted the strain evolution with time of 2D SiC/SiC composites under static loading at elevated temperatures in an oxidative environment. Morscher and Cawley (2002) investigated the strength degradation of 2D SiC/SiC composites under static loading at elevated temperatures of 800 C1000 C in an oxidative environment. It was found that the interphase type, the amount of matrix cracking and stress level are the main reasons for affecting the strength degradation. Yin (2001) investigated the oxidation behavior of 3D C/SiC composites in an aero engine gas environment. The mechanical performance of C/SiC composites after oxidation in a gas environment exhibits obvious degradation. Wei (2004) investigated the oxidation behavior of C/SiC composites in air, steam and H2OO2 atmospheres, analyzed the oxidation mechanisms for the different environments, and established the oxidation kinetic model in oxygen environments. Luan (2007) investigated the damage evolution process, damage modes and damage mechanisms of 3D C/SiC composite in an aero engine equivalent simulation environment, wind tunnel simulation environment, and thermal shock simulation environment. They established the stressed-oxidation damage model of 3D C/SiC composite. Liu et al. (2006) investigated the stressed-oxidation behavior of 2D C/SiC composites under the dry and wet oxygen environments as well as fatigue and creep loading. The fatigue life is longer than that of the creep life, and the oxidation of carbon fibers is the main failure mechanism for the creep failure specimen in a dry oxygen environment. However, water vapor aggravates the oxidation of the SiC matrix. Longbiao (2015a, 2017e,f,g,h) investigated the oxidation mechanisms of fiber-reinforced CMCs and analyzed the effect of oxidation time and oxidation temperature on matrix multicracking, tensile strength, damage evolution, and fatigue life of CMCs by combining the damage models and oxidation model.

1.4.5 Modulus degradation Under cyclic fatigue loading, modulus degradation occurs due to multiple fatigue damage mechanisms. Zawada et al. (1990) investigated the tensiontension cyclic fatigue behavior of unidirectional SiC/1723 composites at room temperature. When the fatigue peak stress is lower than the first matrix-cracking stress, the modulus remains unchanged with applied cycles. When the fatigue peak stress is between the first matrix cracking stress and proportional limit stress, the modulus decreases after the initial 10,000 cycles and then recovers slowly. Karandikar and Chou (1993a,b) investigated the tensiontension cyclic fatigue behavior of cross-ply SiC/CAS composites at room temperature. Under cyclic loading, the transverse cracking density and matrix cracking density increase, and the

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saturated matrix cracking density is the same as that under tensile loading at the same applied stress. There is a linear relationship between the matrix cracking density and modulus degradation. Wang and Laird (1997) investigated the tensiontension cyclic fatigue behavior of 2D C/SiC composites at room temperature. Under cyclic loading, matrix cracking in the longitudinal yarns, transverse yarns cracking, fibermatrix interface debonding, fibers fracture, delamination, yarns separation, and fibermatrix interface wear evolve with applied cyclic loading, leading to the degradation of the modulus degradation. Elahi et al. (1994) investigated the tensiontension cyclic fatigue behavior of 2D SiC/SiC composites at room temperature and elevated temperatures of 1000 C in air atmosphere. At room temperature, when the fatigue peak stress is 75% of the tensile strength, under initial cyclic loading, the modulus decreases greatly by 50%, and during the continually cycling, the modulus remains the unchanged. After experiencing 1,540,000 applied cycles, the specimen does not reach failure, and the residual tensile strength and strain do not change due to the fatigue loading. However, the modulus decreases by 42% compared with that under tensile loading. At an elevated temperature of 1000 C in air atmosphere, the fatigue peak stress is 77.4% of the tensile strength, the modulus decreases about 40% during initial cyclic loading, and at the following cyclic loading when the modulus decreases to 50%, the composite fatigue fractures. Ruggles-Wrenn et al. (2011) investigated the tensiontension cyclic fatigue behavior of 2D SiC/SiC composites at an elevated temperatures of 1200 C in air and steam atmospheres. It was found that the modulus decreases by 25% in air atmosphere and 31% in steam atmosphere. Cao et al. (2001b) investigated the flexure fatigue behavior of unidirectional C/SiC composite at room temperature. The evolution of flexure modulus can be divided into three stages,: G

G

G

At the initial cyclic loading, the flexure modulus degradation rate and amplitude are both high. When the flexure modulus decreases by 15%, the modulus changes a little. When the composite fatigue fractures, the modulus decreases suddenly.

Du et al. (2002a) investigated the tensiontension cyclic fatigue behavior of 3D C/SiC composites at room temperature. Under cyclic fatigue loading, the change of Young’s modulus and electric resistance was measured using the resonance method and resistance increment instrument. With increasing applied cycles, the Young’s modulus of the composite showed a significant decrease, slow decrease, and sudden decrease. Most of the reduction in Young’s modulus occurs during the initial 600 cycles. The slow reduction stage accounts for more than 94% of the fatigue life and at this stage, and the Young’s modulus change rate is approximately linear with the logarithm of the cycle number. The electric resistance change rate increases with the increase of the cycle number, and the law of increase can be roughly divided into three stages (i.e., slow increase, step increase, and sharp increase). The electric resistance change rate of the composite reflects the damage extent and damage

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form of the fiber, and can be used as an effective parameter to characterize the fiber damage of the composite. Wang et al. (2010) investigated the tensiontension cyclic fatigue behavior of 2D C/SiC composite at room temperature. At initial 100 applied cycles, the composite modulus decreases rapidly, and the acoustic accumulation energy increases quickly, and the specimen surface temperature increases obviously. However, after 100 applied cycles the composite modulus, acoustic emission accumulation energy, and surface temperature change slowly with cycles. Shi et al. (2015) investigated the tensiontension cyclic fatigue behavior of 3D SiC/SiC composite at elevated temperatures of 1100 C and 1300 C in air atmosphere, and analyzed the effect of temperature, coating, and loading frequency on the fatigue behavior of 3D SiC/SiC composites (i.e., the strain ratcheting and modulus degradation evolution with cycles). It was found that temperature and oxidation are the major reasons for the modulus degradation in 3D SiC/SiC composites. Longbiao et al. (2017) and Longbiao (2017i) investigated the cyclic fatigue behavior of unidirectional SiC/Si3N4 and cross-ply SiC/MAS composites at room and elevated temperatures. At high fatigue peak stress, the modulus decreases rapidly at initial loading, then changes slowly during the following cycles, and decreases greatly before final fracture.

1.4.6 The effect of loading frequency Masuda et al. (1989) conducted fatigue experiments of Si3N4 ceramics at room temperature under different loading frequencies, and found that the loading frequency has no effect on the fatigue life of single ceramics. However, for fiber-reinforced CMCs, the weak bonding between the fiber and the matrix causes energy dissipation during cyclic loading (Tur, 1999). The frictional sliding between the fiber and the matrix during loading and unloading releases the energy dissipation in a form of heat, leading to the increase of specimen surface temperature. The increase of surface temperature was also observed in fiber-reinforced polymer matrix composites; however, the viscoplasticity of the polymer matrix was the main reason for the increase of surface temperature (Dan-Jumbo et al., 1989). Holmes and Shuler (1990) and Holmes and Cho (1992) observed the temperature rising of unidirectional SiC/CAS-II and 2D C/SiC composites under cyclic tensiontension loading. For the 2D C/SiC composite, when the fatigue peak stress is σmax 5 250 MPa, the loading frequency is f 5 85 Hz, and the stress ratio is R 5 0.04, the specimen surface temperature increases by 32K. Kotch and Grathwohl (1992) performed the tensiontension cyclic fatigue experiments at the high loading frequency of f 5 100 Hz and found that the surface temperature increases greatly. Chawla (1997) performed tensiontension cyclic fatigue experiments of cross-ply [0/90] and [ 6 45] SiC/Si3N4 composites at the loading frequency of f 5 100 Hz, and found that ply layup affects the surface temperature rising at high loading frequency. For the fiber-reinforced CMCs unloading cyclic fatigue loading, the loading frequency affects the sliding rate between the fiber and the matrix, and then the fibermatrix interface wear extent and frictional heating behavior (Sørensen and Holmes, 1995;

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Hou et al., 2007). Many researchers investigated the effect of loading frequency on the fatigue life and frictional heating behavior of fiber-reinforced CMCs. Thomas et al. (1993) found that when the loading frequency is in the range of f 5 0.310 Hz, the fatigue life and fatigue hysteresis loops are not affected by the loading frequency. Holmes et al. (1994) investigated the effect of loading frequency on the tensiontension cyclic fatigue behavior of unidirectional SiC/CAS composites at room temperature. The loading frequencies are f 5 25, 75, 150, and 350 Hz, the fatigue peak stress is σmax 5 180240 MPa, the fatigue valley stress is σmin 5 10 MPa, and the fatigue limit cycle is defined as 5 3 106. The fatigue life decreases with increasing of the loading frequency. For all these loading frequencies, the fatigue limit stress is far lower than the proportional limit stress. Under cyclic fatigue loading, the surface temperature rise is obvious. When the fatigue peak stress is σmax 5 240 MPa and the loading frequency is f 5 350 Hz, the surface temperature increases up to 160K. At high-loading frequency, the internal temperature increases greatly, leading to the decrease of the fibermatrix interface radial thermal residual stress and then the fibermatrix interface shear stress. These factors result in fatigue life decrease of SiC/CAS composites at high loading frequency. Shuler et al. (1993) investigated the effect of loading frequency on the tensiontension cyclic fatigue behavior of 2D C/SiC composites at room temperature. The loading frequencies are f 5 1, 10 and 50 Hz, the fatigue stress ratio is R 5 0.1, the fatigue limit cycle is defined as 106, and the fatigue peak stress is σmax 5 310405 MPa. At the same fatigue peak stress, the fatigue life decreases with an increase in loading frequency. The residual strength after fatigue loading is higher than the original tensile strength. When the loading frequencies are f 5 1 and 10 Hz, the fatigue limit stress is σmax 5 335 MPa, and about 80% of the tensile strength. However, when the loading frequency is f 5 50 Hz, the fatigue limit stress is lower than 310 MPa. Under high-loading frequency, the internal temperature increases obviously, which results in low-fatigue life. The residual strength increases by about 20% after cycling 106 at the fatigue peak stress of σmax 5 335 MPa and loading frequencies of f 5 1 and 10 Hz, mainly due to the decrease of the stress concentration at the fiber bundles; intersection after cyclic loading. Chawla et al. (1996) investigated the effect of loading frequency on the tensiontension cyclic fatigue behavior of 2D SiC/SiC composites with different interphase thicknesses. The loading frequencies are f 5 100 and 350 Hz, the fatigue peak stresses are σmax 5 120 and 150 MPa, and the interphase thicknesses are 0.33 and 1.1 μm. Under the same fatigue peak stress, the fatigue life with the thick interphase is much higher than that with the thin interphase. The temperature rise of a thin interphase is higher than that with a thick interphase. A thick interphase protects the composite from wear damage at high loading frequency and decreases the surface temperature and improves fatigue performance. Staehler et al. (2003) investigated the effect of loading frequency on the tensiontension cyclic fatigue behavior of 2D C/SiC composites at room temperature. The fatigue loading frequencies are f 5 4, 10 and 375 Hz, the fatigue stress ratio is R 5 0.05, and the fatigue limit cycle is defined as 107. Under high fatigue peak

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stress, the fatigue life is independent of the loading frequency. However, under low fatigue peak stress, the fatigue life at the loading frequencies of f 5 4 and 40 Hz is much higher than that under the high-loading frequency of f 5 375 Hz. When the loading frequencies are f 5 4 and 40 Hz, the fatigue limit stress is 375 MPa; and when the loading frequency is f 5 375 Hz, the fatigue limit stress is 350 MPa. Under cyclic fatigue loading, the relative sliding between the fiber and the matrix results in the specimen’s surface temperature rising, and the loading frequency and fatigue peak stress affects the extent of temperature rising. At the loading frequency of f 5 375 Hz, rapid rising temperature results in the oxidation of carbon fibers, which is the main reason for the fatigue life reduction. Mall and Engesser (2006) investigated the effect of loading frequency on the tensiontension cyclic fatigue behavior of 2D C/SiC composites at an elevated temperature of 550 C in air atmosphere. The fatigue stress ratio is R 5 0.05, the fatigue loading frequencies are f 5 0.1, 10 and 375 Hz, and the fatigue peak stress is σmax 5 105500 MPa. At the same fatigue peak stress, the fatigue life increases with the loading frequency, which is different from that at room temperature. When the fatigue peak stress is low, the fatigue life for the loading frequencies of f 5 0.1 and 10 Hz is almost the same; however, when the loading frequency is f 5 375 Hz, the fatigue lifetime is much longer than that at the loading frequencies of f 5 0.1 and 10 Hz. At elevated temperatures, the oxidation of carbon fibers is the main reason for the fatigue life difference of the composites between room temperature and elevated temperature. When the loading frequency is low, the oxidation results in fatigue failure; however, at high loading frequency, the internal temperature rises greatly to close the matrix cracking, or self-healing of the matrix cracking, when the SiC matrix reacts with the oxygen, which stops the further oxidation of carbon fibers and increases the fatigue life. Mall and Ahn (2008) investigated the effect of loading frequency on the tensiontension cyclic fatigue behavior of 2D Nextel 720/alumina composites at room temperature. The fatigue stress ratio is R 5 0.05, the fatigue loading frequencies are f 5 1, 100 and 900 Hz, and the corresponding fatigue limit cycles are 105, 107, and 108. When the loading frequency increases from f 5 1 to f 5 900 Hz, the fatigue life increases with the increase of the loading frequency. When the fatigue peak stress is σmax 5 120 MPa, the fatigue life is 600, 6000, and 108 for the loading frequency of f 5 1, 100, and 900 Hz, respectively. When the fatigue peak stress exceeds 130 MPa, the fatigue life at the loading frequency of f 5 1 and 100 Hz is almost the same. At f 5 900 Hz, the surface temperature increases to 75 C; however, for the loading frequencies of f 5 1 and 100 Hz, the surface temperature only increases by 5 C. At the high loading frequency of f 5 900 Hz, the internal temperature rising results in the rebonding between the fiber and the matrix, which increases the fatigue life. Ruggles-Wrenn et al. (2008) investigated the effect of loading frequency on the tensiontension cyclic fatigue behavior of 2D Nextel 720/alumina composites at an elevated temperature of 1200 C in air and in steam atmospheres. The fatigue stress ratio is R 5 0.05, the fatigue peak stress is σmax 5 100170 MPa at an elevated temperature of 1200 C in air atmosphere and loading frequencies of f 5 0.1 and

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1 Hz, σmax 5 75170 MPa at 1200 C in steam atmosphere, and the loading frequencies of f 5 0.1, 1 and 10 Hz. In air atmosphere, the loading frequencies of f 5 0.1 and 1 Hz have little effect on fatigue life. The fatigue limit stress is 170 MPa, which is about 88% of the tensile strength. The residual strength after 105 cycles does not decrease; however, the modulus decreases by 30%. In steam atmosphere, the loading frequency affects the fatigue life. When the loading frequency is f 5 10 Hz, the fatigue limit stress is 150 MPa, and is about 78% of the tensile strength. The residual strength after 106 cycles decreases by 4%, and the modulus decreases by 7%. When the loading frequency is f 5 1 Hz, the fatigue limit stress is 125 MPa, and is about 69% of the tensile strength. The residual strength decreases by 12%, and the modulus decreases by 20%. When the loading frequency is f 5 0.1 Hz, the fatigue limit stress is lower than 75 MPa. The specimen fracture surface was observed under SEM, and it was found that there exists multiple fiber pullout for the loading frequency of f 5 10 Hz, and for f 5 0.1 Hz there is nearly no fibers pullout. Liu (2003) investigated the effect of loading frequency on the fatigue life of 3D C/SiC composites at an elevated temperature of 1500 C in vacuum atmosphere. The loading frequencies are f 5 20 and 60 Hz and the fatigue limit cycle is defined as 106. The fatigue life decreases with decreasing loading frequency. When the loading frequency is f 5 20 Hz, the fatigue limit stress is 230240 MPa; and when the loading frequency is f 5 60 Hz, the fatigue limit stress is 240250 MPa. The creep damage at 1500 C is the main reason for the decreased fatigue life at lowloading frequency. When the fibermatrix interface is strong bonding, the effect of loading frequency on the fatigue life of fiber-reinforced CMCs is weakened. Chawla et al. (1998) investigated the high-frequency fatigue behavior of 2D SiC/SiCON composites with strong fibermatrix interface bonding. The loading frequency is f 5 100 Hz and the fatigue limit cycle is defined as 107. The fatigue limit stress is 200 MPa and is about 80% of the tensile strength. Compared with the fatigue experimental data of SiC/SiCON composites at the loading frequency of f 5 1 Hz (LaraCurzio et al., 1995), the loading frequency has little effect on the fatigue life of SiC/SiCON composites. Compared to fiber-reinforced CMCs with weak fibermatrix interface bonding, the surface temperature shows an obvious decrease. Due to the strong interface bonding for the SiC/SiCON composite, the fibermatrix interface debonded length is short, which weakens the influence of interface wear on the fatigue life and reduces the frictional heating inside the composite and temperature rising at the specimen surface, resulting in high fatigue performance. Vanswijgenhovena et al. (1998) investigated the effect of loading frequency on the cyclic fatigue behavior of 2D SiC/SiCON composites. The loading frequency changes from f 5 0.01 to 350 Hz. The fatigue peak stresses are σmax 5 58, 83, and 92 MPa and the fatigue valley stress is σmin 5 10 MPa. The loading frequency has no effect on fatigue limit stress. At the same fatigue peak stress, the failure cycle increases with increasing loading frequency, and the failure time decreases with increasing loading frequency. When the fatigue peak stress is σmax 5 58 MPa and the loading frequency is f 5 100 Hz, the specimen experienced 107 cycles without

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fatigue failure. When the loading frequency is f 5 350 Hz, the specimen cycled 5 3 107 without fatigue failure, and when the loading frequency is f 5 0.83 Hz, the specimen cycled 106 without fatigue failure. The fatigue limit stress is independent of loading frequency and is lower than the proportional limit stress. The residual strength of the surviving specimen decreased by 46%. When the fatigue peak stress is σmax 5 83 and 92 MPa, the failure cycle increases with loading frequency; however, the failure time decreases with increasing loading frequency (i.e., the failure time decreases from 36 hours at f 5 0.1 Hz to 1 hours at f 5 350 Hz when σmax 5 92 MPa).

1.4.7 The Effect of Stress Ratio Allen and Bowen (1993) investigated the effect of stress ratio on the flexure fatigue behavior of unidirectional SiC/CAS composites at room temperature. The loading frequency is f 5 50 Hz, the stress ratio is R 5 0.1 and 0.5, and the fatigue limit cycle is defined as 106. The fatigue limit stress decreases with increasing fatigue stress ratio. Holmes (1999) investigated the effect of stress ratio on the tensiontension cyclic fatigue behavior of unidirectional SiC/CAS composites at room temperature. The loading frequency is f 5 200 Hz, the stress ratio is R 5 0.05 and 0.5, and the fatigue limit cycle is 108. The fatigue life increases with increasing stress ratio, and the fatigue limit stress decreases with increasing stress ratio. When the stress ratio is R 5 0.05, the fatigue limit stress is 212 MPa, and when the stress ratio is R 5 0.5, the fatigue limit stress is 240 MPa. At the same fatigue peak stress, the matrix cracking density increases with increasing stress ratio. When the fatigue peak stress is σmax 5 240 MPa, the amount of matrix cracking at stress ratio R 5 0.5 is much higher than that at R 5 0.05. However, the specimen failed at R 5 0.05 and cycled for 106 at R 5 0.5, indicating that the matrix cracking cannot be used to determine fatigue failure. At the same fatigue peak stress, the temperature rise at R 5 0.05 is much higher than that at R 5 0.5. For the low-stress ratio, the stress amplitude and the sliding range between the fiber and the matrix are high, leading to the increase of the surface temperature. Holmes (1991) investigated the effect of stress ratio on the fatigue life of unidirectional SiC/Si3N4 composites at an elevated temperature of 1200 C in air atmosphere. The loading frequency is f 5 10 Hz and the stress ratios are R 5 0.1, 0.3, and 0.5. When the fatigue peak stress is lower than the proportional limit stress, the stress ratio has little effect on fatigue life, and the damage is mainly caused by creep. However, when the fatigue peak stress is higher than the proportional limit stress, the fatigue life increases with increasing stress ratio, and the damage is mainly caused by cyclic loading. Yasmin and Bowen (2004) investigated the effect of stress ratio on the flexure fatigue behavior of cross-ply [0/90]4s SiC/CAS-II composites at an elevated temperature of 800 C in air atmosphere. The fatigue loading frequency is f 5 10 Hz, the stress ratio is R 5 0.1, 0.5, and 0.8, and the fatigue limit cycle is defined as 106. At the same fatigue peak stress, the fatigue life increases with increasing stress ratio. When the fatigue stress ratio is R 5 0.1, the specimen cycled for 5060, and when

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the fatigue stress ratio is R 5 0.5 and 0.8, the specimen cycled for 106 without fatigue failure. Moschelle (1994) investigated the effect of stress ratio on the fatigue behavior of 2D SiC/SiC composites at room temperature. The loading frequency is f 5 10 Hz, the stress ratio is R 5 0.1 and 21, and the fatigue limit cycle is defined as 106. The fatigue life at R 5 21 is less than that at R 5 0.1. At the same fatigue peak stress, compared with tensiontension fatigue, the tensioncompression fatigue caused much more damage in SiC/SiC composites. Kalluri et al. (2006) investigated the effect of stress ratio on the fatigue behavior of 2D SiC/SiC composites at elevated temperatures of 1038 C and 1204 C in air atmosphere. The loading frequency is f 5 20 Hz, the stress ratio is R 5 0.05 and 0.5, and the fatigue peak stress is σmax 5 179 MPa. It was found that the fatigue life at R 5 0.5 is much higher than that at R 5 0.05. Zhu et al. (2004a,b) investigated the effect of stress ratio on the low-cycle fatigue behavior of 3D SiC/[SiTiCO] composites at room temperature. The fatigue loading frequency is f 5 0.02 Hz and the stress ratio is R 5 0.1 and 21. It was found that the stress ratio had little effect on the fatigue life. Under tensioncompressive loading, the fatigue failure is mainly caused by tension damage. Liu (2003) investigated the effect of stress ratio on the fatigue life of 3D C/SiC composites at an elevated temperature of 1500 C in vacuum atmosphere. The fatigue loading frequency is f 5 60 Hz, and the fatigue stress ratio is R 5 0.1 and 0.5. When the fatigue stress ratio is R 5 0.5, the fatigue limit stress is 230240 MPa. However, when the fatigue stress ratio is R 5 0.1, the fatigue limit stress is 240250 MPa. The fatigue life decreases with increasing stress ratio for C/SiC composites at 1500 C.

1.5

Overview of lifetime prediction methods of ceramic-matrix composites

Under cyclic fatigue loading, the lifetime of fiber-reinforced CMCs depends on the fiber preforms (i.e., unidirectional, cross-ply, 2D, 2.5D, and 3D) and the testing conditions. The fatigue life of fiber-reinforced CMCs with different fiber preforms and their lifetime prediction methods are discussed in this section.

1.5.1 SN curve The fatigue life SN curves of unidirectional, cross-ply, 2D, 2.5D, and 3D fiberreinforced CMCs at room and elevated temperatures are overviewed in this section.

1.5.1.1 Unidirectional ceramic-matrix composites Cao et al. (2001b) investigated the flexure fatigue behavior of unidirectional C/SiC composites at room temperature and obtained their flexure fatigue life. When the

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fatigue limit cycle is defined as 5,000,000, the fatigue limit stress is approximately 379 MPa, which is 69% of the flexure strength. Minford and Prewo (1986) investigated the tensiontension fatigue behavior of unidirectional SiC/LAS composites at room temperature. Based on the fibermatrix interface bonding condition, the SiC/LAS composite is divided into strong fibermatrix interface bonding of SiC/LAS-I and wear fibermatrix interface bonding of SiC/LAS-II. For the SiC/LAS-I composite, the tensile stressstrain curve is linear, the tensile strength is about 261 MPa, the fatigue loading frequency is f 5 10 Hz, the fatigue peak stresses are σmax 5 138, 172, and 207 MPa, the valley stress is σmin 5 20.7 MPa, and the fatigue limit cycle is defined as 100,000. When the fatigue peak stress is σmax 5 138 and 172 MPa, the specimen survives after 100,000 cycles, the residual tensile strength is the same as the original tensile strength, and the modulus remains the same under cyclic loading. However, when the fatigue peak stress is σmax 5 207 MPa, the specimen cycled for 2160 and fatigue failed. For the SiC/LAS-II composite, the tensile stressstrain curve is nonlinear, and the proportional limit stress and tensile strength are 270 and 520 MPa, respectively. The fatigue loading frequency is f 5 5 Hz, the fatigue stress ratio is R 5 0.1, the fatigue peak stresses are σmax 5 225, 275, 310, and 335 MPa, and the fatigue limit cycle is defined as 100,000. When the fatigue peak stress is lower than the proportional limit stress, the modulus remains unchanged during cyclic loading, the specimen survives after experiencing 100,000 cycles, and the residual tensile strength is higher than the original tensile strength. When the fatigue peak stress is higher than the proportional limit stress, the modulus decreases with increasing cycles and composite failure occurs before 100,000 cycles. Zawada et al. (1990) investigated the tensiontension fatigue behavior of unidirectional SiC/1723 composites at room temperature. The fatigue loading frequency is f 5 10 Hz, the fatigue stress ratio is R 5 0.1, and the fatigue limit cycle is defined as 1,000,000. The fatigue limit stress is far higher than the first matrix-cracking stress and the proportional limit stress (i.e., σmax , σmc), and reaches about 65% of the tensile strength. When the fatigue peak stress is less than the first matrixcracking stress, the composite modulus remains unchanged. However, when the fatigue peak stress is between the first matrix-cracking stress and the proportional limit stress (i.e., σmc ,σmax , σpls), the composite modulus decreases at the initial 10,000 applied cycles and then recovers gradually. Evans (1997) investigated the tensiontension fatigue behavior of unidirectional SiC/CAS composites at room temperature. The fatigue loading frequency is f 5 10 Hz, the stress ratio is R 5 0.05, the fatigue peak stress is σmax 5 280 MPa, and the fatigue limit cycle is defined as 40,000. It was found that the fatigue limit stress of SiC/CAS composites is about 70% of the tensile strength. McNulty and Zok (1999) investigated the low-cycle tensiontension fatigue behavior of unidirectional SiC/CAS and SiC/MAS composites at room temperature. The fatigue loading frequency is f 5 0.5 Hz and the fatigue peak stress is σmax 5 280 MPa for SiC/CAS composites, and σmax 5 369 MPa for SiC/MAS composites. The fatigue limit cycle is defined as 40,000. At room temperature, the fatigue limit stress of SiC/CAS composites is about 65% of the tensile strength; and for SiC/ MAS composites, the fatigue limit stress is about 80% of the tensile strength.

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Sørensen et al. (2000) investigated the high-cycle tensiontension fatigue behavior of unidirectional SiC/CAS-II composites at room temperature. The fatigue loading frequency is f 5 200 Hz and the fatigue peak stress is σmax 5 240 MPa. After experiencing 108 applied cycles, the residual strength is about 85% of the tensile strength. However, after cycling for 105, the strength remains the same. The composite strength decreases greatly before the final fracture. Therefore the decline in strength occurs before failure, rather than gradually decaying with the cycle. There is a special region in the center of the high-frequency fatigue failure sample in which there appears to be no fibers pullout, but there is a large amount of fibers pullout around that region. Under high-frequency fatigue loading, the internal temperature rising leads to the chemical reaction at the fibermatrix interface, resulting in an increase of the fibermatrix interface bond strength, which is the main reason for the high frequency and high-cyclic fatigue failure of SiC/CAS-II composites. Sørensen et al. (2002) investigated the high-frequency tensiontension fatigue behavior of unidirectional SiC/CAS-II composites at room temperature. The fatigue loading frequency is f 5 200 and 500 Hz, the fatigue peak stress is σmax 5 160280 MPa, the fatigue valley stress is σmin 5 10 MPa, and the fatigue limit cycle is 108. The fatigue limit stress is 212 MPa at the loading frequency of f 5 200 Hz, which is lower than the proportional limit stress of σpls 5 380 MPa, and is about 42% of the tensile strength. When the applied cycle number increases to 108, the surface temperature and fatigue hysteresis modulus still change with applied cycle, indicating the continual damage of the composite. Prewo (1987) investigated the flexure fatigue behavior of unidirectional SiC/ LAS-II composites at room temperature and elevated temperatures of 600 C and 900 C in air atmosphere. The fatigue loading frequency is f 5 5 Hz and the fatigue limit cycle is defined as 100,000. At room temperature, the fatigue limit stress is higher than the proportional limit stress (i.e., σpls 5 500 MPa), and the residual flexure strength (710813 MPa) is higher than the original flexure strength (σuts 5 750 MPa). At 600 C in air atmosphere, the fatigue limit stress and the residual flexure strength in both approach the proportional limit stress (σpls 5 350 MPa), indicating the material performance degradation. At 900 C in air atmosphere, the fatigue limit stress is lower than the proportional limit stress (σpls 5 300 MPa); however, the residual strength approaches the original flexure strength (σuts 5 550 MPa). At 600 C in air atmosphere, the oxidation embrittlement is the main reason for the fatigue performance degradation of SiC/LAS-II composites. However, at 900 C in air atmosphere, the fibermatrix interface forms the oxidation layer at initial cyclic loading, which stops further oxidation of the interface layer, leading to the better fatigue performance than that achieved at 600 C. Allen et al. (1993) investigated the flexure fatigue behavior of unidirectional SiC/CAS composites at room temperature and elevated temperatures of 800 C and 1000 C in air atmosphere. At room temperature, the fatigue loading frequency is f 5 50 Hz, the fatigue stress ratio is R 5 0.1 and 0.5, and the fatigue limit cycle is defined as 106. At elevated temperatures of 800 C and 1000 C in air atmosphere, the fatigue loading frequency is f 5 10 Hz and the stress ratio is R 5 0.1. At room temperature under the same fatigue peak stress, the cyclic fatigue life is less than that of static fatigue, and the fatigue life decreases with increasing peak stress.

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Upon first loading to the peak stress (i.e., the peak stress is higher than the first matrix-cracking stress), matrix cracking appears at the surface of the specimen. During the following cyclic fatigue loading, the matrix cracking propagates along the direction perpendicular to the fiber. The fibermatrix interface wear decreases the fiber bridge traction stress, leading to the propagation of the matrix cracking with increasing applied cycle. At 800 C in air atmosphere under the same fatigue peak stress, the cyclic fatigue life is less than that of static fatigue. The fiber pullout length at the fracture surface is less than that at room temperature. At 1000 C in air atmosphere under the same peak stress, the cyclic fatigue life is the same as that under static fatigue. The oxidation at elevated temperatures results in the increase of fibermatrix interface bonding strength, leading to fatigue failure at elevated temperatures. Holmes et al. (1989) investigated the tensiontension fatigue behavior of unidirectional SiC/Si3N4 composite at elevated temperature of 1000 C in air atmosphere. The fatigue loading frequency is f 5 10 Hz, the fatigue stress ratio is R 5 0.1, and the fatigue limit cycle is defined as 2 3 106. When the fatigue peak stress is lower than the proportional limit stress, the specimen cycled without fatigue failure. When the fatigue peak stress is higher than the proportional limit stress, the specimen failed before approaching the fatigue limit cycle. When the fatigue peak stress is σmax 5 220 MPa, the failure cycle is 270,000750,000; and when the fatigue peak stress is σmax 5 280 MPa, the failure cycle is 400033,000. Under cyclic loading, the composite modulus decreases, and the obvious stressstrain hysteresis loops and the strain ratcheting appear. Holmes and Sørensen (1995) investigated the tensiontension fatigue behavior of unidirectional SiC/Si3N4 composites at elevated temperature of 1200 C in air atmosphere. The fatigue loading frequency is f 5 10 Hz, the fatigue peak stress is σmax 5 180, 200, and 240 MPa, and the valley stress is σmin 5 10 MPa. When the fatigue peak stress is σmax 5 180 MPa, which is lower than the proportional limit stress of σpls 5 196 MPa, no matrix cracking occurred under cyclic loading, and the stressstrain curve is linear without hysteresis behavior; however, strain ratcheting appears. When the fatigue peak stress is σmax 5 200 and 240 MPa, which is higher than the proportional limit stress, the fatigue hysteresis width increases with peak stress. Under low fatigue peak stress, which is lower than the proportional limit stress, the composite damage is caused mainly due to creep. However, when the fatigue peak stress is higher than the proportional limit stress, the composite damage is caused by fatigue mechanisms.

1.5.1.2 Cross-ply ceramic-matrix composites Zawada et al. (1991) investigated the tensiontension fatigue behavior of cross-ply [0/90]3s SiC/1723 composites at room temperature. The loading frequency is f 5 10 Hz, the stress ratio is R 5 0.1, and the fatigue limit cycle is defined as 106. At room temperature, the fatigue limit stress is 70% of the tensile strength and is higher than the proportional limit stress of 0 ply. Under low-fatigue peak stress, the modulus reduction is small; under high-fatigue peak stress, the modulus greatly

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decreases till final fracture. Under moderate fatigue peak stress, the modulus decreases greatly at first, and then recovers slowly, due to the mixed particles at the matrix cracking plane or fibermatrix interface. After experiencing 106 cycles, the residual strength and modulus decrease with increasing fatigue peak stress. Lee and Stinchcomb (1994) investigated the tensiontension fatigue behavior of cross-ply [0/90]4s SiC/CAS-II composites at room temperature. The loading frequency is f 5 10 Hz, the stress ratio is R 5 0.1, the fatigue peak stress is 65%, and 85% of the tensile strength. The modulus degradation can be divided into three stages, namely, the initial fast decay stage (10%15% fatigue life), intermediate slow descent stage (75%80% fatigue life), and sharp descent stage (10%15% fatigue life) before failure. When the fatigue peak stress is higher than the proportional limit stress, the fiber debonding and transverse cracking in the 90 ply is the major reason for initial modulus decay. When the transverse cracking approaches saturation, the stress redistributes between the 90 and 0 ply, and the matrix cracking and the fibermatrix interface debonding occur in the 0 ply, leading to intermediate slow degradation. Before final fracture, the fiber break and the modulus degradation greatly increase. Kim and Liaw (2005) investigated the high cycle fatigue behavior of cross-ply [0/90]4s SiC/CAS composites at room temperature. The fatigue loading frequency is f 5 20 Hz, the stress ratio is R 5 0.1, and the fatigue peak stress is 65%95% tensile strength (σuts 5 180 MPa). The fatigue life decreases with increasing peak stress, the fatigue limit stress is 100 MPa, and is 55% of the tensile strength. The relationship between the fatigue microdamage mechanisms and rising temperature has been established, and the temperature increase can be divided into three stages: G

G

G

At initial cyclic stage, the matrix cracking occurs without fibermatrix interface debonding, and the temperature increase at the specimen surface is not obvious. At intermediate cyclic stage, the matrix cracking propagates with fibermatrix interface debonding, and the surface temperature gradually increases. Before final fracture, the surface temperature increases rapidly as the broken fibers pull out from the matrix.

Opalski (1992) investigated the tensiontension, tensioncompression, and compressioncompression fatigue behavior of cross-ply [0/90]2s SiC/CAS composites at room temperature. The fatigue loading frequency is f 5 10 Hz and the fatigue limit cycle is 106. The fatigue limit stress under tensiontension fatigue loading is about 50% tensile strength. Under compressioncompression fatigue loading with the peak stress of σmax 5 140 and 210 MPa, the specimens do not reach fatigue failure. Under tensioncompression loading with the tensile fatigue peak stress of σmax 5 140 MPa, the fatigue life decreases with increasing compressive stress. Longitudinal cracking appears between the transverse cracking in the 90 ply, which connects the transverse cracking, resulting in much more serious damage. Mall and Tracy (1992) investigated the tensiontension fatigue behavior of crossply [0/45/90]s SiC/CAS composites at room temperature. The fatigue loading frequency is f 5 10 Hz, the fatigue stress ratio is R 5 0.1, and the fatigue limit cycle is

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defined as 106. The fatigue limit stress is about 67% of tensile strength, which corresponds to the stress, without causing damage in the 0 ply. Vanwijgenhoven et al. (1999) investigated the tensiontension fatigue behavior of [0]16, [0]12, [02/902]s, [0/90]3s, [452/-452]2s, and [02/452/-452/902]s SiC/BMAS composites at room temperature. The loading frequency is f 5 3 Hz, the stress ratio is R 5 0.1, and the fatigue limit cycle is defined as 106. For the unidirectional composite, the fatigue limit stress is 375 MPa, which is about 60% of the tensile strength; for [0/90] composite, the fatigue limit stress is 175 MPa, which is about 45% of the tensile strength; for [452/ 2 452]2s composite, the fatigue limit stress is 70 MPa, which is about 70% of the tensile strength; and for the [02/452/ 2 452/902]s composite, the fatigue limit stress is 110 MPa, which is about 65% of the tensile strength. The damage under cyclic loading is the same as that under tensile loading; however, fatigue damage is caused under lower stress levels; for the unidirectional, [0/90] and [02/452/ 2 452/902]s composites, the fatigue life is controlled by the 0 ply, and the increasing of matrix cracking density, decrease of the fibermatrix interface shear stress, and the decay of fibers strength all result in the decrease of residual strength. However, for the [452/ 2 452]2s composite, the fatigue life is more affected by delamination. Prewo et al. (1989) investigated the tensiontension fatigue behavior of crossply [0/90]4s-SiC/LAS-III composites at elevated temperature of 900 C in air and in Ar atmosphere. The fatigue loading frequency is f 5 7.210 Hz, the stress ratio is R 5 0.1, and the fatigue limit cycle is defined as 105. The fatigue limit stress at 900 C in air atmosphere is approximately 90 MPa, which approaches to the proportional limit stress, and is higher than the fatigue limit stress in Ar atmosphere. The residual strength at 900 C in air atmosphere is about 85 MPa, which is far lower than the tensile strength (i.e., σuts 5 150 MPa). The residual strength in Ar approaches the tensile strength (i.e., σuts 5 200 MPa). Rousseau (1990) investigated the tensiontension fatigue behavior of cross-ply [0/90]2s-SiC/CAS composites at room temperature and elevated temperature of 815 C in air atmosphere. At room temperature, there are two proportional limit stresses in tensile stressstrain curve (i.e., σpls,l 5 35 MPa and σpls,h 5 105 MPa). At 815 C in air atmosphere, there are also two proportional limit stresses in the tensile stressstrain curve (i.e., σpls,l 5 59 MPa and σpls,h 5 104 MPa). The fatigue loading frequency is f 5 0.3 Hz and the fatigue stress ratio is R 5 00.15. At room temperature, when the fatigue peak stress is σmax 5 135 MPa (i.e., σmax . σpls, 5 h 5 105 MPa), the specimen cycled 2 3 10 without fatigue failure. At elevated tem perature of 815 C in air atmosphere, when the fatigue peak stress is σmax 5 70 MPa (i.e., σmax , σpls,h 5 104 MPa), the specimen cycled for 2000 and then fatigue failed. The fracture surface of the failure specimen at elevated temperature was observed and there was no fibers pullout at the fracture surface. The weak fibermatrix interface was oxidized into strong interface bonding. Oxidation at elevated temperatures is the main reason for the decrease of the fatigue life. Steiner (1994) investigated the tensiontension fatigue behavior of cross-ply [0/90]4s-SiC/MAS composites at elevated temperatures of 566 C and 1093 C in air atmosphere. The fatigue loading frequency is f 5 1 and 10 Hz. The damage

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mechanism of SiC/MAS composite is independent of temperature and loading frequency; however, the modulus degradation rate and fatigue damage extent are affected by the temperature, and the fatigue life decreases with increasing temperature due to the fiber and interphase oxidation at elevated temperature. The fracture surface of the fatigue failure specimen was observed under the SEM. With an increase in the testing temperature, the region of the fibers pullout decreases, due to the oxidized fibermatrix interface strong bonding. Yasmin and Bowen (2004) investigated the flexure fatigue behavior of cross-ply [0/90]4s-SiC/CAS-II composites at room temperature and elevated temperature of 800 C in air atmosphere. The fatigue loading frequency is f 5 10 Hz, the stress ratio is R 5 0.1, and the fatigue limit cycle is defined as 106. At room temperature, the fatigue limit stress is higher than the proportional limit stress and is about 40% of the flexure strength. At 800 C, the fatigue limit stress is the first matrix-cracking stress, which is reduced by 50%, compared with that at room temperature, due to the fibermatrix interface oxidation. At room temperature, after experiencing 106 cycles, the residual strength is about 85% of the original strength, mainly due to the degradation of the fibermatrix interface shear stress. At 800 C in air atmosphere, the residual strength after experiencing 106 cycles is the same as the original strength. The increase of the fatigue performance at elevated temperature is mainly due to the transfer from the weak fibermatrix interface bonding to oxidize strong fibermatrix interface bonding. The fibermatrix interface strong bonding increases the fatigue performance.

1.5.1.3 2D ceramic-matrix composites Wang and Laird (1997) investigated the tensiontension fatigue behavior of 2D C/SiC composites at room temperature. The fatigue loading frequency is f 5 1 Hz, the stress ratio is R 5 0, the fatigue peak stress is σmax 5 320380 MPa, and the fatigue limit cycle is defined as 106. The fatigue limit stress is about 80% of the tensile strength. Under cyclic fatigue loading, seven damage modes were observed; matrix cracking, weft yarns cracking, fibermatrix interface debonding, fibers break, delamination, separation of the yarns, and matrix wear. The evolution of these seven damage modes degrades the modulus. Under cyclic loading, the high-loading rate and repeated loading lead to serious damage of delamination and yarns separation. Sun et al. (2007) investigated the tensiontension fatigue behavior of 2D C/SiC composites. The loading frequency is f 5 10 Hz, the stress ratio is R 5 0.1, the fatigue peak stress is 80%90% of the tensile strength (i.e., σuts 5 268 MPa), and the fatigue limit cycle is defined as 5 3 105. At room temperature, the fatigue limit stress is about 80%B85% of the tensile strength, and when the fatigue peak stress exceeds 88% of the tensile strength, fatigue failure quickly occurs. Hou et al. (2005) investigated the tensiontension fatigue behavior of notched 2D C/SiC composites at room and elevated temperature in vacuum atmosphere. The loading frequency is f 5 60 Hz, the stress ratio is R 5 0.1, and the fatigue limit cycle is 106. The fatigue limit stress is about 80%90% tensile strength at the same temperature. At initial cyclic fatigue loading, the damage near the notch propagates

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quickly with a large amount of transverse matrix cracking; with an increase in applied cycles, the damage grows slowly with more damage modes. Rouby and Reynaud (1993) investigated the tensiontension fatigue behavior of 2D SiC/SiC composites at room temperature. The fatigue loading frequency is f 5 1 Hz, the stress ratio is R 5 0, the fatigue peak stress is σmax 5 50175 MPa, and the fatigue limit cycle is defined as 106. At room temperature, the fatigue limit stress is 135 MPa, which is about 75% of the tensile strength. According to the fatigue peak stress, the fatigue damage can be divided into three conditions: G

G

G

When the fatigue peak stress is higher than the tensile strength, upon first loading to the fatigue peak stress, the composite fatigue fractures. When the fatigue peak stress is between the fatigue limit stress (i.e., 75% of the tensile strength) and the tensile strength, the fatigue failure occurs at a certain number of applied cycles (i.e., N 5 510,000). When the fatigue peak stress is lower than the fatigue limit stress, the specimen survived within 106 cycles; however, when the fatigue peak stress is higher than the first matrixcracking stress, the shape of the fatigue hysteresis loops evolves with the cycles.

¨ nal (1996a,b) investigated the tensiontension fatigue behavior of 2D SiC/SiC U composites at room temperature and elevated temperature of 1300 C in N2 atmosphere. The fatigue loading frequency is f 5 0.5 Hz and the stress ratio is R 5 0.1. The creep affects the fatigue damage with fibers brittle fracture, and there was few fibers pullout at the fracture surface. Zhu et al. (1997, 1998, 1999a,b) investigated the tensiontension fatigue behavior of 2D SiC/SiC composites at room temperature and elevated temperature of 1000 C in Ar atmosphere. At room temperature, the fatigue loading frequency is f 5 10 Hz; and at elevated temperature, the loading frequency is f 5 20 Hz. The stress ratio is R 5 0.1 and the fatigue limit cycle is defined as 107. At room temperature, the fatigue limit stress is 160 MPa, which is higher than the proportional limit stress (i.e., σpls 5 80 MPa), and is about 70%80% of the tensile strength. At 1000 C in Ar atmosphere, the fatigue life SN curve can be divided into three regions: G

G

G

Low cyclic region (i.e., N , 104). There is no obvious difference in fatigue life between room temperature and high temperature. When the fatigue peak stress is less than 180 MPa, the fatigue life at 1000 C in Ar atmosphere decreases quickly; however, there is no fatigue life decrease at room temperature. Fatigue limit region. When the fatigue limit cycle is defined as 107, the fatigue limit stress at elevated temperature is 75 MPa, which is far below the proportional limit stress (i.e., σpls 5 100 MPa), and is about 30% of the tensile strength at elevated temperature.

Zhu (2006) investigated the effect of oxidation on fatigue behavior of 2D SiC/ SiC composites. After oxidation for 100 hours at 600 C in air atmosphere, the fibermatrix interphase decreases, leading to the decrease of fatigue life by 13%. After oxidation for 100 hours at 800 C in air atmosphere, the fibermatrix interface forms strong SiO2 interphase, resulting in shorter fatigue life. Groner (1994) investigated the tensiontension fatigue behavior of 2D SiC/SiC composites with/without hole at elevated temperature of 1100 C in air atmosphere. For the SiC/SiC composite without hole, the fatigue limit stress at 1100 C is about

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105 MPa, which is about 45% of the tensile strength. However, for the SiC/SiC composites with hole, the fatigue limit stress is about 95 MPa, which is about 42% of the tensile strength. The SN curves for the specimen with/without hole are similar. The hole has little effect on the fatigue life of 2D SiC/SiC composites at elevated temperature. Lee et al. (1998) investigated the tensiontension fatigue behavior of 2D SiC/ SiNC composites at room temperature and elevated temperature of 1000 C in air atmosphere. The fatigue loading frequency is f 5 1 Hz, the stress ratio is R 5 0.05, and the fatigue limit cycle is defined as 106. At room temperature, the fatigue limit stress is 160 MPa, which is higher than the proportional limit stress (σpls 5 85 MPa), and is about 75% tensile strength (σuts 5 197 MPa). When the fatigue peak stress is σmax 5 175 MPa, the fatigue hysteresis loops were measured, and no strain ratcheting appeared. At 1000 C, the fatigue limit stress is 110 MPa, which is 51% of tensile strength (σuts 5 214 MPa). The hysteresis loops area at elevated temperature is much larger than that at room temperature, mainly due to the decrease of thermal residual stress and fibers oxidation. Haque and Rahman (2000) investigated the tensiontension fatigue behavior of 2D SiC/SiNC composites at room temperature and elevated temperatures of 700 C and 1000 C in air atmosphere. The fatigue loading frequency is f 5 1 Hz, the fatigue stress ratio is R 5 0.1, and the fatigue limit cycle is defined as 106. The fatigue life decreases with increasing temperature. At room temperature, the fatigue limit stress is about 80% of the tensile strength; at an elevated temperature of 700 C in air atmosphere, the fatigue limit stress is about 50% of the tensile strength; and at an elevated temperature of 1000 C in air atmosphere, the fatigue limit stress is about 20% of the tensile strength. At room temperature, the fatigue life of SiC/SiNC composites can be divided into three regions: G

G

G

At high fatigue peak stress, the fibers failure caused fatigue fracture. At low fatigue peak stress, the fibermatrix interface wear decreases the fiber bridge traction stress, resulting in matrix cracking propagation, delamination, and fibers fracture, and then fatigue fracture. The fatigue limit region.

Steel et al. (2001) investigated the tensiontension fatigue behavior of 2D Nextel 720/alumina composites at room temperature and elevated temperature of 1200 C in air atmosphere. The fatigue loading frequency is f 5 1 Hz, the fatigue stress ratio is R 5 0.05, and the fatigue limit cycle is defined as 105. At room temperature, the fatigue limit stress is 102 MPa, which is about 70% of the tensile strength (σuts 5 144 MPa); and at an elevated temperature of 1200 C in air atmosphere, the fatigue limit stress is 122 MPa, which is about 87% tensile strength (σuts 5 140 MPa). The fatigue damage mechanism at elevated temperature is similar with that at room temperature; however, the fiber creep affects the fatigue behavior. Zawada et al. (2003) investigated the tensiontension cyclic fatigue behavior of 2D Nextel 610/Al2O3SiO2 composites at room temperature and elevated temperature of 1000 C in air atmosphere. The fatigue loading frequency is f 5 1 Hz, the stress ratio is R 5 0.05, and the fatigue limit cycle is 105. At room temperature, the

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fatigue limit stress is 170 MPa and about 85% of the tensile strength; and at elevated temperature of 1000 C in air atmosphere, the fatigue limit stress is 150 MPa and about 85% of the tensile strength. At elevated temperature, when the fatigue peak stress is σmax 5 100150 MPa, the modulus decreases 5%10% during the initial 1000 cycles, and then the modulus remains constant with increasing cycles till the fatigue limit cycles, indicating no obvious accumulation damage. After experiencing 106 cycles at elevated temperature, and then cooled down to room temperature, the residual strength at room temperature is the same with the original strength. The hysteresis loops were measured during cycling, and the hysteresis loops area is low, only 35 kJ/m3. Ruggles-Wrenn et al. (2012) investigated the tensiontension cyclic fatigue behavior of 2D SiC/[SiCB4C] composites at an elevated temperature of 1200 C in air and in steam atmospheres. When the loading frequency is f 5 0.1 Hz, the fatigue life in steam atmosphere is greatly reduced compared with that in air atmosphere. However, when the loading frequency is f 5 1 Hz, the steam atmosphere has little effect on the fatigue life. Lanser and Ruggles-Wrenn (2016) investigated the tensioncompressive cyclic fatigue behavior of 2D Nextel 720/alumina composites at an elevated temperature of 1200 C in air and in steam atmospheres. In air atmosphere, the fatigue limit stress is about 40% of the tensile strength; and in steam atmosphere, the fatigue limit stress is about 35% of the tensile strength. Ruggles-Wrenn and Lee (2016) investigated the tensiontension cyclic fatigue behavior of 2D SiC/SiC composites at an elevated temperature of 1300 C in air and in steam atmospheres. In air atmosphere, the fatigue limit stress is about 28% of the tensile strength; however, in steam atmosphere, the fatigue limit stress is about 19% of the tensile strength. Sabelkin et al. (2016) investigated the tensiontension fatigue behavior of 2D SiC/SiC composites at an elevated temperature of 1205 C in combustion chamber environment and in air atmosphere. Compared with the testing conditions in air, the fatigue life in the combustion chamber environment decreases nearly 100 times, mainly due to the oxidation embrittlement at elevated temperature. Cheng et al. (2010) investigated the tensiontension cyclic fatigue behavior of 2D C/SiC composites at an elevated temperature of 1300 C in oxidative environment. It was found that the fatigue limit stress decreases greatly for the testing condition with high oxygen content. Dong et al. (2016) investigated the tensiontension cyclic fatigue behavior of 2D C/(SiCSiBC)m composites at elevated temperatures of 3001200 C in air atmosphere. When the temperature is lower than 400 C, the fatigue damage of 2D C/(SiCSiBC)m is mainly caused by stress and temperature; and when the temperature reaches 550 C, the temperature, stress, and oxidation affect the fatigue damage, and the oxidation mechanism is affected by the temperature. Longbiao (2017j,k,l) investigated the tensiontension cyclic fatigue behavior of 2D SiC/SiC and cross-ply SiC/MAS composites at elevated temperature in air atmosphere. The fatigue life is affected by the fibermatrix interface wear and interphase oxidation and decreases greatly compared with that at room temperature.

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1.5.1.4 2.5D ceramic-matrix composites Dalmaz et al. (1998) investigated the tensiontension cyclic fatigue behavior of 2.5D C/SiC composites at room and elevated temperatures in inert atmosphere. The fatigue loading frequency is f 5 1 and 10 Hz. Due to the thermal expansion coefficient mismatch between the fiber and the matrix, cracking occurs inside of the matrix and yarns. The thermal residual tensile stress exists between the yarns and the interphase layer, leading to weak interface bonding. The fatigue behavior of 2.5D C/SiC composites depends on the interface shear strength between the fiber and the matrix. When the temperature is below 1000 C, the repeated sliding between the yarnsyarns and yarnsmatrix results in the recession of the fibermatrix interface. The fatigue damage depends on the cyclic number, not the cyclic time. However, with increasing temperature, the thermal residual stress between the fiber and the matrix decreases, which increases the elastic modulus and fatigue life. When the temperature is above 1000 C, creep affects the fatigue behavior, and the fatigue performance degrades again.

1.5.1.5 3D ceramic-matrix composites Kostopoulos et al. (1997) investigated the tensiontension cyclic fatigue behavior of uncoated/coated 3D C/SiC composites at room temperature. The fatigue loading frequency is f 5 10 Hz, the stress ratio is R 5 0.1, and the fatigue limit cycle is 106. Under fatigue loading, the fatigue limit stress for the uncoated and coated C/SiC composites is nearly the same, which approaches to 70% of tensile strength. The tensile test was conducted after fatigue loading and it was found that the tensile stressstrain curve is linear, the residual strength decreases by 18%, and the modulus decreases by 18%. The resonance frequency and damping coefficient were measured under cyclic loading to monitor the fatigue damage accumulation (Kostopoulos et al., 1999). When the fatigue peak stress is high, the damping coefficient increases with cycles till fatigue failure, which indicates the energy dissipation inside of composite. Under cyclic loading, when the fatigue peak stress is higher than the matrix strength, upon first loading to the peak stress, a large amount of matrix cracking appears together with fibermatrix interface debonding. The matrix cracking approaches to the steady state during initial cyclic loading and remains constant during the following cycles. The matrix cracking is the premise for the fatigue failure of 3D C/SiC composites; however, it is not the direct reason. The interface wear between the fiber and the matrix inside of yarns, and between yarns and yarns are the main reasons for fatigue failure. When cycling to the steady-state stage, the interface wear reduces the stress concentration at the intersection of the yarns, resulting the slow recovery of the modulus. At the final stage of the cyclic loading, the local fibers failure causes the fatigue fracture. The fracture surface of the failure specimen was observed. The fibers pullout confirms the fibermatrix interface wear mechanism during cyclic loading.

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Butkus et al. (1990) investigated the tensiontension cyclic fatigue behavior of 3D SiC/SiC composites at room temperature. The loading frequency is f 5 10 Hz, the stress ratio is R 5 0.1, the fatigue peak stress is σmax 5 100250 MPa, and the fatigue limit cycle is 106. The fatigue limit stress at room temperature is 215 MPa and about 90% of the tensile strength. When the fatigue peak stress is lower than 215 MPa, the modulus decreases during the first 10 applied cycles and then remains unchanged with increasing cycles. Du et al. (2002b) investigated the tensiontension cyclic fatigue behavior of 3D C/SiC composites at room temperature and elevated temperature of 1300 C in vacuum atmosphere. The fatigue loading frequency is f 5 60 Ha, the stress ratio is R 5 0.1, and the fatigue limit cycle is defined as 106. The fatigue limit stress at 1300 C is 285 MPa, and about 94% of the tensile strength The fatigue limit stress at room temperature is 235 MPa, and about 85% of the tensile strength. The fiber pullout length at 1300 C is much longer than that at room temperature. The fatigue damage mainly originates from the braided intersection of the fiber bundles. With the increase of fatigue cycles, the damage of the matrix around the fiber bundles increases. Liu (2003) investigated the tensiontension cyclic fatigue behavior of 3D C/SiC composites at room temperature and elevated temperatures of 1100 C, 1300 C, and 1500 C in vacuum atmosphere. The fatigue life SN curve can be divided into three regions, namely, short life, long life, and infinite life. The thermal residual stress caused by the thermal expansion coefficient mismatch changes with the temperature. At room temperature, the fibermatrix interface radial thermal residual stress is tensile stress; however, above the fabrication temperature, the fibermatrix interface radial thermal residual stress is compressive stress. The fiber pullout length at the fracture surface is different (i.e., the longest at room temperature, and the shortest at 1500 C). The fatigue cracks originate mainly from the fiber weaving intersections, and the mode of matrix crack propagation mainly depends on the fibermatrix interface layer and the internal flaw inside of the matrix. Zhang et al. (2009) investigated the fatigue damage evolution of 3D SiC/SiC composites at 1300 C in oxidative environment and analyzed the fatigue failure mechanism. Luo et al. (2016) investigated the tensiontension fatigue behavior of 3D SiC/SiC composites at 1300 C in air atmosphere and revealed the fatigue damage mechanisms at elevated temperature of matrix initial cracking and propagation, fibermatrix interface debonding, and fibers, pullout, and bridging. Under highfatigue peak stress (i.e., the fatigue peak stress is higher than the proportional limit stress), the fiber plays the key role on the fatigue damage evolution and fatigue life; under low fatigue peak stress (i.e., the fatigue peak stress is lower than the proportional limit stress), the matrix plays the key role on the fatigue damage evolution and fatigue life. When the fatigue peak stress is between the fatigue limit stress and proportional limit stress, the fiber and matrix both affect the fatigue damage evolution. Han et al. (2004) investigated the cyclic fatigue behavior of 2D and 3D C/SiC composites at elevated temperatures of 1100 C1500 C in vacuum atmosphere. Compared with 2D C/SiC composites, the tensile strength and the proportional limit

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stress are much higher for 3D C/SiC composites. The fiber pullout length is much longer, indicating a longer sliding distance at the fibermatrix interface. Ren (2004) investigated the tensiontension cyclic fatigue and fatigue/creep behavior of 3D C/SiC composites in Ar atmosphere, dry oxygen, and wet oxygen atmospheres. The elastic modulus decreases with applied cycles. Under fatigue/ creep loading at 1300 C, the damage mechanisms include matrix cracking and fiber and fiber bundles fracture and pullout. In Ar atmosphere, the main damage mechanism includes the fibers fracture and pullout; however, in dry and wet oxygen atmospheres, besides the abovementioned damage mechanisms, fibers oxidation also occurs. Wu et al. (2006) investigated the tensiontension cyclic fatigue behavior of 3D C/SiC composites at 1300 C in H2OO2Ar atmosphere. The loading frequency is f 5 1 Hz and the stress ratio is R 5 0.1. Compared with tensile loading, the fibermatrix interface debonding, sliding and fibers pullout under cyclic loading are much more obvious, with more fibers pullout and longer pullout length. Under tensile loading, the composite failure depends on the fibers strength distribution and fibermatrix interface properties. Under cyclic loading, the fatigue peak stress is lower than the first matrix-cracking stress, the braiding angle decreases with increasing cycles, which reduces the fibermatrix interface bonding strength, resulting in matrix cracking and propagating. When matrix cracking appears, the oxygen and water vapor enter the internal of composite through cracks, leading to oxidation of the fibermatrix PyC interphase and SiC matrix. When the opening of matrix cracking is small, the oxygen that enters the composite is less, the PyC interphase is partially oxidized, and the SiC matrix is much less oxidized. When the opening of matrix cracking is wide, more oxygen penetrates into the material, which makes the PyC interphase completely oxidized and the silicon carbide matrix seriously oxidized.

1.5.2 Fatigue life prediction Under cyclic fatigue loading of fiber-reinforced CMCs, the fibermatrix interface wear decreases the fibermatrix interface shear stress and fibers strength, leading to the increasing of fibers fracture with cycles. When the residual strength of the composite approaches to the fatigue peak stress, the composite fatigue failure (Reynaud 1996). Evans (1997) predicted the fatigue life SN curve of unidirectional SiC/CAS composite based on the Reynaud model (Reynaud 1996). However, the effect of fiber strength degradation and fibers oxidation at elevated temperature on the fatigue life is not considered. Case et al. (1998) developed an approach to predict the fatigue life of fiber-reinforced CMC based on the residual strength and critical element model; however, the relationship between the microdamage parameter and the macroresidual strength is not established. Sujidkul et al. (2014) relates the damage mechanisms of matrix cracking and fibers fracture with electric resistance change and developed an approach to monitor the damage evolution of fiber-reinforced CMCs using the electric resistance; however, the use of electric resistance method is limited by the environment and temperature. Gyekenyesi et al.

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(2005) developed a macrofatigue life prediction model of NASA Life based on the residual strength. However, the microstructure characteristics and the effect of environment are not considered in the fatigue life model. Min et al. (2014) obtained the microstress field of the fiber, matrix, and fibermatrix interface by combining the FEM stress analysis with the 3-phase micromechanics model and predicted the fatigue life of 2D C/SiC composites at room temperature. Xu (2008) considered the damage mechanisms of fibermatrix interface wear and the degradation of fibermatrix interface shear stress, and combined the fibers fracture model with the interface wear model, predicted the fatigue life of unidirectional and cross-ply fiber-reinforced CMCs at room temperature. Lu et al. (2014) investigated the tensiontension fatigue behavior of 2D C/SiC composites at room temperature and predicted the fatigue life using the nominal stress method. Luan et al. (2004) investigated the stress corrosion mechanism and fatigue life prediction of 2D C/SiC composites at elevated temperature. Longbiao (2015b,c) developed an approach to predict the fatigue life of fiber-reinforced CMCs based on the fatigue hysteresis dissipated energy, and considered the fibermatrix interface wear mechanism, and predicted the fatigue damage evolution and fatigue life of unidirectional fiberreinforced CMCs.

1.6

Conclusion

In this chapter the mechanical behavior and damage characteristics of fiberreinforced CMCs with different fiber preforms (i.e., unidirectional, cross-ply, 2D, 2.5D, and 3D) have been overviewed. The experimental observation and theoretical models for initial matrix cracking, matrix multicracking evolution, fibers failure, tensile stressstrain curves, fatigue hysteresis loops, interface wear, fibers strength degradation, oxidation embrittlement, and modulus degradation were discussed. The damage evolution and fatigue lifetime of fiber-reinforced CMCs depend on the fiber preforms and testing conditions, and are also affected by the loading frequency and fatigue stress ratio.

References Agins, D.M., 1993. Static Fracture Behavior of a Ceramic Matrix Composite at Elevated Temperatures (M.S. thesis). Air Force Institute of Technology. Ahn, B.K., Curtin, W.A., 1997. Strain and hysteresis by stochastic matrix cracking in ceramic matrix composites. J. Mech. Phys. Solids 45, 177209. Alfano, D., 2010. Spectroscopic Properties of Carbon Fiber Reinforced Silicon Carbide Composites for Aerospace Applications, Properties and Applications of Silicon Carbide. Italian Aerospace Research Centre, Italy. Allen, R.F., Bowen, P., 1993. Fatigue and fracture of a SiC/CAS continuous fiber reinforced glass ceramic matrix composite at ambient and elevated temperatures. Ceram. Eng. Sci. Process 14, 265272.

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Further reading Longbiao, L., 2018c. Damage monitoring and life prediction of cross-ply SiC/CAS ceramicmatrix composites at room and elevated temperatures under cyclic loading. J. Aerospace Eng 31, 04017084. Papenburg, U., Walter, S., Selzer, M., Beyer, S., Laube, H., Langel, G., 1997. Advanced ceramic matrix composites (CMC’s) for space propulsion systems. In: The 33rd Joint Propulsion Conference and Exhibit, Joint Propulsion Conferences.

Matrix cracking of ceramicmatrix composites

2.1

2

Introduction

Ceramicmatrix composites (CMCs) possess excellent mechanical performance, especially at elevated temperatures, and have already been applied on hot-section components in aero engines, such as combustion chamber wall tiles and turbine vanes and blades (Christin, 2002; Naslain, 2004; Schmidt et al., 2004; DiCarlo and Roode, 2006; Zhang et al., 2006; Stephen, 2010; Watananbe et al., 2016). The matrix failure strain is less than that of the fiber, and matrix cracking is the first main damage mechanism for fiber-reinforced CMCs. Thermomechanical loading, thermal shock, and complex loading can lead to multiple matrix cracks (Cox and Marshall, 1996; Sevener et al., 2017). These matrix cracks degrade the mechanical performance of CMCs and form paths for the ingress of the environment, oxidizing the interphase, fibers, or even matrix at elevated temperatures, leading to the final fracture of CMCs (Filipuzzi et al., 1994; Lamouroux et al., 1994; Verrilli et al., 2004; Halbig et al., 2008; Li, 2015a,b, 2017, 2018). The fiber preforms, interface properties, testing temperature, and environment affect the matrix cracking density or the opening of matrix cracking (Smith et al., 2008; Simon et al., 2017; Gowayed et al., 2015). It is necessary to develop models of matrix cracking or matrix multiple cracking of fiber-reinforced CMCs especially when considering oxidation at elevated temperatures (Parthasarathy et al., 2018). In this chapter, first-matrix cracking and multiple-matrix cracking of fiberreinforced CMCs are investigated using the energy balance approach. The damage mechanisms of interface debonding and oxidation are considered. The relationships between the first-matrix cracking stress (FMCS), matrix multicracking evolution, interface debonding and oxidation, oxidation time and temperature, and different matrix cracking modes are established. The FMCS and matrix multicracking evolution for different fiber-volume fraction, interface properties, and oxidation temperature and time are analyzed. The FMCS and matrix multicracking evolution of different CMCs are then predicted.

2.2

First-matrix cracking in an oxidation environment at elevated temperature

During the application of CMCs, the composites suffer oxidation damage at elevated temperatures (Santhosh et al., 2013). The mechanical behavior of fiberreinforced CMCs is remarkably different at the stress above or below the FMCS. When the applied stress is below FMCS, CMCs exhibit linear elastic behavior Durability of Ceramic-Matrix Composites. DOI: https://doi.org/10.1016/B978-0-08-103021-9.00002-2 © 2020 Elsevier Ltd. All rights reserved.

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without damage inside the composite; however, when the applied stress is above FMCS, matrix cracking and interface debonding between the fiber and the matrix leads to the nonlinear behavior of CMCs. Jablonski and Bhatt (1990) investigated the tensile behavior of unidirectional silicon carbide fiber-reinforced reactionbonded silicon nitride matrix composites at elevated temperatures of 1300 C and 1500 C in air atmosphere. At 1300 C in air atmosphere, matrix cracking appears before tensile failure and the matrix cracking strain was about 0.1%. At 1500 C, the tensile stressstrain curve did not have obvious deviated point; however, the ductile behavior is better than that at 1300 C. At 1500 C in air atmosphere, the fibermatrix interface is strong bonding, and matrix cracking penetrated through the fibers. However, at 1300 C in air atmosphere, the fibermatrix interface is weak bonding, the matrix cracking deflects along the fibermatrix interface, and the amount of matrix cracking is much larger. Cao et al. (2001) investigated the tensile behavior of M40JB-C/SiC and T800-C/SiC composites at elevated temperatures of 1300 C and 1450 C in inert atmosphere. For the M40JB-C/SiC composite, the tensile strength is about 374 MPa and tensile modulus is about 137 GPa at 1300 C; however, at 1450 C, the tensile strength decreases to about 338 MPa and the tensile modulus decreases to about 116 GPa. For the T800-C/SiC composite at 1300 C, the tensile strength is about 392 MPa, which is higher than that of M40JBC/SiC composite, and tensile modulus is about 115 GPa, which is lower than that of M40JB-C/SiC composite. Agins (1993) investigated the tensile behavior of crossply [0/90]2s-SiC/CAS composites at elevated temperatures of 700 C and 850 C in air atmosphere. At 850 C in air atmosphere, the length of fibers pullout is much larger than that at 700 C, and the fibers surface has obvious defects. When the temperature exceeds 800 C, the fibermatrix interface oxidizes and its bonding strength decreases. The fibermatrix interface debonds at a low-stress level, leading to the increase of fiber loading, the decrease of composite tensile strength and failure strain, and the increase of fibers pullout length. Engesser (2004) investigated the tensile behavior of 2D C/SiC composites at an elevated temperature of 550 C in air atmosphere. The elastic modulus under initial loading at 550 C is higher than that at room temperature, and the tensile strength and the failure strain is less than that at room temperature. The tensile strength and failure strain decrease with decreasing loading rate. When the loading rate decreases, the oxidation time increases at elevated temperatures, leading to the oxidation of the fibermatrix interphase and fibers, and then a decrease in the composite tensile strength and failure strain. Lipetzky et al. (1996) investigated the tensile behavior of 2D SiC/SiC composites at elevated temperatures of 850 C, 1000 C, and 1200 C. The tensile behavior is largely independent of test temperatures below 1000 C. At 1200 C, the material retains much of its low-temperature stiffness and proportional limit; however, the strength increases. Zhu et al. (1999) investigated the tensile behavior of 2D SiC/SiC composites at an elevated temperature of 1000 C in inert atmosphere. The tensile stressstrain curve is linear under initial loading, and the elastic modulus at initial loading, proportional limit stress, composite tensile strength, and failure strain are higher than those at room temperature. Guo and Kagawa (2001) investigated the tensile behavior of 2D SiC/SiC composite at 298K, 1200K, 1400K,

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77

and 1600K in air atmosphere. The tensile stressstrain curves of 2D SiC/SiC composite at 850 C, 1000 C, and 1200 C exhibit similar behavior, that is, the tensile stressstrain curve is linear under initial loading, and when the applied stress approaches critical stress, the tensile stressstrain curve starts to deviate till final fracture. The proportional limit stress decreases with increasing temperature. When the temperature increases to 1200K, the elastic modulus at initial loading decreases a little compared to that at 298K; however, when the temperature exceeds 1200K, the elastic modulus decreases greatly. When the temperature increases to 1200K, the tensile strength is similar to that at 298K; however, when the temperature exceeds 1200K, the tensile strength decreases greatly. Fibers in situ strengths are obtained using fracture mirror tests. It was found that fibers in situ strength decreases with increasing temperature; however, the fiber Weibull modulus does not change with increasing temperature. Luo and Qiao (2003) investigated the tensile behavior of 3D C/SiC composites at elevated temperatures of 1100 C and 1500 C under different loading rates (i.e., 0.06 and 5.82 mm/min). At 1100 C, the tensile strength is independent of loading rate. With increasing loading rate at elevated temperatures, the fracture strain decreases, and the initial elastic modulus increases. Under tensile loading, the damage mechanisms of matrix cracking in the fiber bundles, fibermatrix interface debonding, fibers pullout, and wear between fiber bundles were observed. Qiao et al. (2004) investigated the tensile behavior of 3D C/SiC composites from room temperature to 1500 C in vacuum atmosphere. The tensile stressstrain curve can be divided into three stages, the modulus of which are dependent on the initial matrix cracking stress and matrix cracking saturation stress. The modulus at the first stage and matrix cracking saturation stress are constant with increasing temperature; however, the modulus at the second and third stages, matrix first cracking stress, and matrix cracking saturation stress, fracture stress and matrix first cracking strain increase with increasing temperature till 1300 C, and then decreases when the temperature is higher than 1300 C. Mazars et al. (2017) investigated the tensile damage of 3D SiC/SiC composites at room temperature and 1250 C using X-ray microtomography. Image processing and digital volume correlation residuals were used to identify the damage mechanisms with increasing loads. The matrix cracks first initiate in weft yarns, develop and proliferate in the weft planes until they coalesce, and then create a through-thickness failure. The fibers break outside the cracking plane at 25 C and they break within the main crack plane at 1250 C. For the application of CMCs at elevated temperatures, matrix cracking leads to the mechanical performance degradation caused by the oxidation of the interphase, fibers, or matrix. The FMCS is a key design parameter for hot-section CMC components. During the long-term application at elevated temperatures, the FMCS may degrade with operation time or cycle, due to the oxidation through microcracking caused by thermal expansion coefficient mismatch, which affects the durability and reliability of CMC components. (Lamouroux et al., 1994; Casas and MartinezEsnaola, 2003; Halbig et al., 2008; Murthy et al., 2008). The energy balance approach (Aveston et al., 1971; Budiansky et al., 1986; Rajan and Zok, 2014) and fracture mechanics approach (Marshall et al., 1985; McCartney, 1987) are two main methods to determine the FMCS of fiber-reinforced CMCs.

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In the next section, the FMCS of fiber-reinforced CMCs is predicted using the energy balance approach. The interface oxidation damage mechanism at elevated temperature is considered in the stress analysis and interface debonding. The theoretical relationships among the FMCS, interface debonding, and oxidation temperature and time are established. The FMCS for different fiber-volume fractions, interface properties, and oxidation time and oxidation temperature are analyzed. The FMCS versus the oxidation time curves of C/SiC composites with different interface bonding properties at 700 C are then predicted.

2.2.1 Stress analysis The thermal expansion coefficient mismatch between carbon fiber and the silicon carbide matrix causes microcracks in the SiC matrix during fabrication processing. These microcracks serve as paths for the ingress of oxidative atmosphere into CMCs, as shown in Fig. 2.1. The interface oxidation length of ζ can be determined using the equation (Casas and Martinez-Esnaola, 2003): h  ϕ ti ζ 5 ϕ1 1 2 exp 2 2 b where b is a delay factor (Casas and Martinez-Esnaola, 2003).

Figure 2.1 A schematic of crack-tip, interface debonding and oxidation. Source: Reproduced from Li, L., 2017. Modeling matrix cracking of fiber-reinforced ceramic-matrix composites under oxidation environment at elevated temperature. Theor. Appl. Fract. Mech. 87, 110119.

(2.1)

Matrix cracking of ceramicmatrix composites

23

ϕ1 5 7:021 3 10

  8231 3 exp T

  17; 090 ϕ2 5 227:1 3 exp 2 T

79

(2.2a)

(2.2b)

2.2.1.1 Downstream stress Under tensile loading, the CMCs can be divided into three regions: (1) region I, the downstream region; (2) region II, the intermediate region; and (3) region III, the upstream region, as shown in Fig. 2.1. A unit cell in region I is extracted from the composite, as shown in Fig. 2.2. The length of the unit cell is half of the matrix crack spacing of lc/2, and the interface oxidation length and interface debonded length are ζ and ld, respectively. At the interface oxidation region, the interface shear stress τ f is low; and in the interface slip region, the interface shear stress τ i is high. In region I, the force equilibrium equation of the fiber in the interface debonded region can be determined using the equation: dσf ðxÞ 2τ i ðxÞ 52 dx rf

(2.3)

Figure 2.2 A schematic of shear-lag model considering interface oxidation and debonding. Source: Reproduced from Li, L., 2017. Modeling matrix cracking of fiber-reinforced ceramicmatrix composites under oxidation environment at elevated temperature. Theor. Appl. Fract. Mech. 87, 110119.

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The boundary conditions of the fiber and matrix axial stress at the matrix crack plane can be determined using the equations: σ Vf

(2.4)

σ m ð x 5 0Þ 5 0

(2.5)

σ f ð x 5 0Þ 5

The relationship between the fiber and matrix axial stress and the applied stress can be determined using the equation: Vf σf ðxÞ 1 Vm σm ðxÞ 5 σ

(2.6)

Solving Eqs. (2.3)(2.6), the axial stress of the fiber and the matrix in the interface oxidation and debonded region of region I can be determined using the equations:

σD f ðxÞ 5

σD m ðxÞ 5

8 σ 2τ f > > 2 x; > > rf < Vf

xAð0; ζ Þ

σ 2τ f 2τ i > > > 2 ζ2 ðx 2 ζ Þ; > : Vf rf rf 8 Vf τ f > > x; 2 > > < V m rf

(2.7) xAðζ; ld Þ

xAð0; ζ Þ

Vf τ f Vf τ i > > > 2 ζ 12 ðx 2 ζ Þ; > : V m rf Vm r f

(2.8) xAðζ; ld Þ

The fiber and matrix have the same strain in the interface bonded region of region I. εf 5 εm 5 εc 5

σ Ec

(2.9)

The axial stress of the fiber and the matrix in the interface bonded region of region I can be determined using the equations: σD f 5

Ef σ Ec

(2.10)

σD m5

Em σ Ec

(2.11)

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81

2.2.1.2 Upstream stress The fiber and matrix axial stress in region III can be determined using the equations: σU f 5

Ef σ Ec

(2.12)

σU m5

Em σ Ec

(2.13)

2.2.2 Interface debonding When matrix cracking deflects along the fibermatrix interface, interface debonding occurs (Gao et al., 1988; Hsueh, 1996; Sun and Singh, 1998). The fracture mechanics interface debonding criterion is used to determine the fibermatrix interface debonding, (Gao et al., 1988) with the equation: ξd 5 2

F @wf ð0Þ 1 2 4πrf @ld 2

ð ld 0

τi

@vðxÞ dx @ld

(2.14)

where ξd denotes the interface debonded energy; F(5πrf2σ/Vf) denotes the fiber load at the matrix cracking plane; wf (0) denotes the fiber axial displacement at the matrix cracking plane; and v(z) denotes the relative displacement between the fiber and the matrix. The fiber and matrix axial displacement can be determined using the equations: ðx σf wf ðxÞ 5 dz E N f ð ld  σ σ τi τf  2 5 dx 1 ðx 2 ld Þ 1 ðld 2ζ Þ2 2 ζ 2 2ζld 1 x2 Vf Ef r f Ef r f Ef N Ec (2.15) ðx

σm dx E N m ð ld  σ Vf τ i Vf τ f  2 5 dx 2 ðld 2ζ Þ2 1 ζ 2 2ζld 1 x2 r f Vm E m r f Vm E m N Ec

wm ðxÞ 5

(2.16)

The relative displacement can be determined using the equation:   vðzÞ 5 wf ðxÞ 2 wm ðxÞ  σ Ec τ i Ec τ f  2 5 ðld 2 xÞ 2 ðld 2ζ Þ2 1 ζ 2 2ζld 1 x2 Vf E f r f V m Ef Em r f Vm Ef Em (2.17)

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Substituting wf(x 5 0) and v(x) into Eq. (2.14) leads to the form of the equation: Ec τ 2i τiσ 2Ec τ f τ i ðld 2ζ Þ2 2 ðld 2 ζ Þ 1 ζ ðld 2 ζ Þ Vf Ef r f Vm E f Em r f Vm Ef Em 2

Ec τ 2f τf σ r f V m Em σ 2 ζ1 1 ζ 2 2 ξd 5 0 Vf Ef Ef Ec 4Vf2 r f Vm Ef Em

(2.18)

Solving Eq. (2.18), the interface debonded length can be determined using the equation: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   τf r f Vm Em σ r f Vm Em Ef ld 5 1 2 2 ξd ζ1 τi 2 Vf Ec τ i Ec τ 2i

(2.19)

2.2.3 Matrix cracking stress The energy relationship to evaluate the FMCS can be determined using the equation (Budiansky et al., 1986): 0 1 2 3 ð ld ð R  2 V   V 1 r τ ð z Þ @ f i A2πrdrdz 4 f σU 2σD 1 m σU 2σD 2 5dz1 f f 2πR2 Gm 2ld rf r Em m m 2N Ef 0 1 4Vf ld A (2.20) 5Vm ξm 1 @ ξd rf

1 2

ðN

in which ξ m is the matrix fracture energy; and Gm is the matrix shear modulus. Substituting the fiber and matrix stresses of Eqs. (2.7), (2.8), (2.10)(2.13) and the interface debonded length of Eq. (2.19) into Eq. (2.20), the energy balance equation leads to the equation: η1 σ 2 1 η2 σ 1 η3 5 0

(2.21)

where η1 5

Vm Em ld V f Ef E c

η2 5 2

2τ i τf ðld 2ζ Þ2 1 ζ ð2ld 2 ζ Þ rf E f τi

(2.22a)

(2.22b)

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83

0 1 0 1  "  2 # 4 @ Vf Ec A τ i 2 4V E f c τf 3 3 A τ f τ i ζ ðld 2 ζ Þ η3 5 ðld 2ζ Þ 1 τ i ζ 1 @ rf 3 Vm Ef Em Vm Ef Em rf2 2 0 1 3 0 1 τf 4Vf ld A (2.22c) ξd 3 4ld 2 @1 2 Aζ 5 2 Vm ξ m 2 @ τi rf

2.2.4 Results and discussion The time-dependence of FMCS, fibermatrix interface debonded length, and interface oxidation ratio versus the oxidation time curves of C/SiC composites for different fiber-volume fractions, fibermatrix interface shear stress, interface debonding energy, and oxidation temperature are analyzed in this section.

2.2.4.1 Effect of fiber-volume fraction on time-dependent fibermatrix interface debonding and first-matrix cracking stress The time-dependence of FMCS, fibermatrix interface debonded length of ld/rf and the interface oxidation ratio of ζ/ld versus the oxidation time curves of C/SiC composites at elevated temperature of 800 C for different fiber-volume fractions of Vf 5 30% and 35% are shown in Fig. 2.3. When the fiber-volume fraction increases from Vf 5 30% to 35%, the FMCS increases at the same oxidation time, the interface debonded length decreases at the same oxidation time, and the interface oxidation ratio increases at the same oxidation time.

2.2.4.2 Effect of fibermatrix interface debonded energy on time-dependent fibermatrix interface debonding and first-matrix cracking stress The time-dependence of FMCS, fibermatrix interface debonded length of ld/rf and the interface oxidation ratio of ζ/ld versus the oxidation time curves of C/SiC composites at 800 C for different fibermatrix interface debonded energy of ξd 5 0.5 and 1.0 J/m2 are shown in Fig. 2.4. When the fibermatrix interface debonded energy increases from ξ d 5 0.51.0 J/m2, the fibermatrix interface debonding length decreases, and the interface oxidation ratio increases, leading to the increase of the FMCS at the same oxidation time.

2.2.4.3 Effect of fibermatrix interface shear stress on the timedependent fibermatrix interface debonding and firstmatrix cracking stress The time-dependence of FMCS, fibermatrix interface debonded length of ld/rf and the fibermatrix interface oxidation ratio of ζ/ld versus the oxidation time curves

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Durability of Ceramic-Matrix Composites

Figure 2.3 (A) The FMCS versus the oxidation time curves; (B) the fibermatrix interface debonded length of ld/rf versus the oxidation time curves; and (C) the fibermatrix interface oxidation ratio of ζ/ld versus the oxidation time curves of C/SiC for different fiber-volume fractions of Vf 5 30% and 35%. FMCS, First-matrix cracking stress.

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85

Figure 2.4 (A) The FMCS versus the oxidation time curves; (B) the fibermatrix interface debonded length of ld/rf versus the oxidation time curves; and (C) the fibermatrix interface oxidation ratio of ζ/ld versus the oxidation time curves of C/SiC for different interface debonded energies of ξ d 5 0.5 and 1.0 J/m2. FMCS, First-matrix cracking stress.

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Durability of Ceramic-Matrix Composites

of C/SiC composites at 800 C for different fibermatrix interface shear stress in the slip region of τ i 5 10 and 15 MPa are shown in Fig. 2.5. When the fibermatrix interface shear stress in the slip region increases from τ i 5 10 to 15 MPa, the fibermatrix interface debonding length decreases, and the fibermatrix interface oxidation ratio increases, leading to the increase of FMCS. The time-dependence of FMCS, fibermatrix interface debonded length of ld/rf and interface oxidation ratio of ζ/ld versus the oxidation time curves of C/SiC composites at 800 C for different fibermatrix interface shear stress in the oxidation region of τ f 5 1 and 3 MPa are shown in Fig. 2.6. When the fibermatrix interface shear stress in the oxidation region increases, the fibermatrix interface debonding length decreases, and the interface oxidation ratio increases, leading to the increase of FMCS.

2.2.4.4 Effect of oxidation temperature on time-dependent fibermatrix interface debonding and first-matrix cracking stress The time-dependence of FMCS, fibermatrix interface debonded length of ld/rf and the interface oxidation ratio of ζ/ld versus oxidation time curves of C/SiC composites for different oxidation temperature of Tem 5 600 C and 800 C are shown in Fig. 2.7. When the oxidation temperature increases from Tem 5 600 C to 800 C, the fibermatrix interface debonded length and the interface oxidation ratio increase, leading to the decrease of FMCS at the same oxidation time.

2.2.5 Experimental comparisons The FMCS of C/SiC composites after unstresses oxidation at elevated temperature of 700 C are predicted for different interface properties (i.e., the strong interface bonding and the weak interface bonding). For strong interface bonding, FMCS decreases with increasing oxidation time, that is, from 37 MPa without oxidation to 20 MPa with 6 hours oxidation. For weak interface bonding, FMCS decreases with increasing oxidation time, from 27 MPa without oxidation to 13 MPa with 6 hours oxidation. The experimental and predicted FMCS versus the oxidation time curves of C/SiC at 700 C are shown in Fig. 2.8. Strong fibermatrix interface bonding can improve the oxidation resistance of C/SiC at elevated temperatures.

2.3

Matrix multicracking evolution considering fibers poisson contraction

Matrix multicracking affects the mechanical behavior of fiber-reinforced CMCs. For unidirectional SiC/CAS composites under tensile loading at room temperature, when the applied strain approaches 0.08%, matrix cracking appears and the tensile stressstrain curve is deviated. With an increase in applied stress, the amount of matrix cracking increases, and when the strain approaches to 0.3%, the tensile stressstrain curve is deviated again and the matrix cracking approaches saturation

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87

Figure 2.5 (A) The FMCS versus the oxidation time curves; (B) the fibermatrix interface debonded length of ld/rf versus the oxidation time curves; and (C) the fibermatrix interface oxidation ratio of ζ/ld versus the oxidation time curves of C/SiC for different fibermatrix interface shear stresses in the slip region of τ i 5 10 and 15 MPa. FMCS, First-matrix cracking stress.

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Durability of Ceramic-Matrix Composites

Figure 2.6 (A) The FMCS versus the oxidation time curves; (B) the fibermatrix interface debonded length of ld/rf versus the oxidation time curves; and (C) the fibermatrix interface oxidation ratio of ζ/ld versus the oxidation time curves of C/SiC for different fibermatrix interface shear stresses in the oxidation region of τ f 5 1 and 3 MPa. FMCS, First-matrix cracking stress.

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89

Figure 2.7 (A) The FMCS versus the oxidation time curves; (B) the fibermatrix interface debonded length versus the oxidation time curves; and (C) the fibermatrix interface oxidation ratio versus the oxidation time curves of C/SiC for different oxidation temperatures of Tem 5 600 C and 800 C. FMCS, First-matrix cracking stress.

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Durability of Ceramic-Matrix Composites

Figure 2.8 The FMCS versus the oxidation time curves of C/SiC composite at 700 C. FMCS, First-matrix cracking stress.

(Pryce and Smith, 1992). For unidirectional C/SiC composites under tensile loading at room temperature, the longitudinal matrix cracking appears first along the direction of the fibers due to the weak interface bonding between the fiber and the matrix. With increasing applied stress, the longitudinal cracking propagates and connects with each other, and then penetrates through the entire specimen, leading to final tensile fracture. The transverse matrix cracking perpendicular to the fiber direction did not appear during tensile loading, mainly due to the weak interface bonding between the fiber and the matrix (Wang et al., 2001). For the cross-ply SiC/CAS composite under tensile loading at room temperature, the transverse cracking firstly occurs in the 90-degree ply, which appears at the interface between the 0 and 90-degree ply, and then propagates to the 90-degree ply (Wang and Parvizi-Majidi, 1992). With decreasing thickness of 90-degree ply, the first transverse cracking strain increases, and the propagation rate decreases. For [03/903/03]SiC/CAS composites under tensile loading, when transverse cracking appears at the 90-degree ply, the cracking quickly penetrates through the thickness of the 90degree ply. However, for [03/90/03] SiC/CAS composites under tensile loading, the propagation of transverse cracking in the 90-degree ply needs increasing strain or stress, and when the transverse cracking propagates to the interface between the 0- and 90-degree ply, the cracking propagates through the 0-degree ply within several times of the fiber radius length, and then stops due to the bridged fibers in the 0-degree ply. The transverse crack spacing decreases with increasing applied

Matrix cracking of ceramicmatrix composites

91

stress, and approaches saturation before matrix cracking in the 0-degree ply. When the strain approaches 0.13%, matrix cracking in the 0-degree ply [03/90/03] SiC/ CAS occurs for the three ply forms mentioned above. The matrix cracking strain in the 0-degree ply is independent of the 90-degree ply and its thickness. The initiation matrix cracking occurs in the matrix-rich region in the 0-degree ply, and with increasing strain, matrix cracking density and length increase. Most of the matrix cracking propagates and stops at the interface between the 0- and 90-degree plys; however, some of the matrix cracking propagates to the 90-degree ply. With continually increasing strain, the fibers in the 0-degree ply fracture, leading to the failure of the composite. For 2D C/SiC composites under tensile loading at room temperature, when the applied stress is lower than 50 MPa there is no acoustic emission signal, the accumulation acoustic emission energy is nearly zero, and the tensile stressstrain curve exhibits linear. When the applied stress reaches 50 MPa, the acoustic emission signal increases suddenly. At applied stress between 50 and 150 MPa, the matrix thermal microcracking caused by the fabrication starts to propagate, and the tangent modulus of the composite decreases with increasing applied stress. When the applied stress exceeds 150 MPa, the tangent modulus of the tensile stressstrain curve increases with applied stress; however, the acoustic emission signal decreases. When the applied stress reaches 230 MPa, the fibers break and pullout from the matrix, the acoustic emission signal increases greatly, and, before the final fracture of the composite, the tangent modulus of the composite decreases again. The composite fails with a large amount of fibers pullout (Mei et al., 2007). For the 2D SiC/SiC composite under tensile loading at room temperature, the FMCS increases with the fiber volume content along the loading direction. The normalized acoustic emission accumulation energy under tensile loading can be used to characterize the matrix multicracking evolution. After tensile failure, the matrix cracking saturation density is measured under optical microscope. The relationships between the applied stress and the matrix cracking density can be obtained using the measured matrix cracking saturation density and the normalized acoustic emission accumulation energy (Morscher et al., 2007). For the 2.5D C/SiC composite under tensile loading at room temperature, when the applied stress is lower than 20 MPa, the tensile stressstrain curve is linear, there is no occurrence of acoustic emission signal, and the composite is without damage. When the stress is higher than 20 MPa, the acoustic emission signal is monitored, and original matrix microcracking propagates; however, the tensile stressstrain curve is still linear. When the applied stress exceeds 50 MPa, the tensile stressstrain curve is nonlinear, the acoustic emission signal increases greatly, bridged matrix cracking appears inside of the composite, and debonding occurs when the matrix cracking propagates to the fibermatrix interface. When the applied stress approaches 200 MPa, the acoustic emission signal increases slowly, and after the saturation of the matrix cracking, the tensile stressstrain curve appears linear again till final fracture (Wang et al., 2008). For the 3D C/SiC composite under tensile loading at room temperature, the tensile stressstrain curve exhibits nonlinear due to multiple damage mechanisms of the matrix multicracking, fibermatrix interface debonding, and fibers failure. The development of accumulation acoustic emission energy can be divided into

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Durability of Ceramic-Matrix Composites

three stages, namely, initial damage stage, critical damage stage, and rapid damage stage. At the initial damage stage, the damage propagates slowly in the form of primary cracking and weak fibermatrix interface debonding. At the critical damage stage, the damage is mainly caused by the rapid cracking and propagation of the fibermatrix interface in the fiber bundles, and fibers fracture and pullout. At the fast damage stage, the acoustic emission accumulation energy suddenly increases rapidly till final fracture (Pailler and Lamon, 2005). In the next section, the multiple matrix cracking of fiber-reinforced CMCs is investigated considering fiber Poisson contraction. The Coulomb frictional law and the shear-lag model is combined to perform the stress analysis in the interface slip region and interface bonding region. The fracture mechanics approach is adopted to determine the fibermatrix interface debonding length. The evolution of matrix multicracking and the fibermatrix interface debonded length versus the applied stress for different interface frictional coefficient, fiber volume fraction, interface debonded energy, and applied cycle number are analyzed. The matrix multiple cracking evolution of different CMCs are then predicted.

2.3.1 Stress analysis A unit cell is used to perform the stress analysis in the interface debonded region and interface bonded region, which is extracted from the ceramic composite as shown in Fig. 2.2. The fiber and matrix axial stress and the fibermatrix interface shear stress can be determined using the following equations for the interface debonded region and interface bonded region (Chiang, 2007). 8σ   αvf >  ðσ 2 σÞ eλx 2 1 ; xA½0; ld  2  > > Vf αvf 1 γvm > f

σ αv ð σ 2 σ Þ > 4 5 2ððρðx2ld ÞÞ=rf Þ 1 σfo ; >  f  λld > : Vf 2 Vf αvf 1 γvm e 2 1 2 σfo e

2

3 l c xA4ld ; 5 2 (2.23)

8   αvf >   ðσ 2 σÞ eλx 2 1 ; xA½0; ld  > > > Vm αvf 1 γvm > > > > > 82 9 3 > > > < σ = >  λl  αv ð σ 2 σ Þ < σ f 4 2   e d 2 1 2 σfo 5e2ððρðx2ld ÞÞ=rf Þ 1 σfo ; 2 γ σ m ð x Þ 5 Vm : Vf ; Vf αvf 1 γvm > > > > 2 3 > > > > > lc > > > xA4ld ; 5 (2.24) > : 2

Matrix cracking of ceramicmatrix composites

93

8 r αv λ > f f  ðσ 2 σÞeλx ; xA½0; ld  > > > 2Vf αvf 1 γvm > < 2 3 τ i ðxÞ 5 >   > ρ σ αv ð σ 2 σ Þ f > λld 4 5 2ððρðx2ld ÞÞ=rf Þ ; > > : 2 Vf 2 Vf αvf 1 γvm  e 2 1 2 σfo e

2

3 l c xA4ld ; 5 2 (2.25)

where vf and vm denote the fiber and matrix Poisson ratio, respectively; α 5 Em/Ef, Em and Ef denote the matrix and fiber elastic modulus, respectively; γ 5 Vf/Vm; and ρ denotes the shear-lag parameter.   2Gm α 1 γ 2 2k αvf 1 γvm   ρ 5 Em ln R=rf 2

(2.26)

where Gm denotes the matrix shear modulus. ln

    2lnVf 1 Vm 3 2 Vf R 52 rf 4Vm2

1 2 2vm k   σ; σfo 5 Vm α 1 γ 2 2k αvf 1 γvm

(2.27) " # 1 γ ð1 2 2vm kÞ   σ 12 σmo 5 Vm α 1 γ 2 2k αvf 1 γvm (2.28)

σ5

  Vf q i 1 1 vm 1 2γ 1 α 1 2 vf ; αvf

λ5

2μk ; rf

k5

αvf 1 γvm   1 1 vm 1 2γ 1 α 1 2 vf (2.29)

where μ denotes the interface frictional coefficient; qi denotes the interface thermal residual clamping stress.   Ef Em αf 2 αm ΔΤ   qi 5 Ef ð 1 1 v m Þ 1 Em 1 2 v f

(2.30)

where ΔΤ denotes the temperature difference between the fabricated temperature T0 and testing temperature T1 (ΔΤ 5 T1 2 T0).

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Durability of Ceramic-Matrix Composites

2.3.2 Interface debonding Interface debonding is determined using the fracture mechanics approach as shown in Eq. (2.14) (Gao et al., 1988). The fiber and the matrix axial displacements can be determined using the equations: 8 2 3 1 2 2vf k < σ αvf ðσ 2 σÞ 4eλld 2 eλx  2 ðld 2 xÞ5 wf ðxÞ 5 2 ðl 2 xÞ 2  :Vf d Ef λ Vf αvf 1 γvm 9     ð1 2 2vm kÞ lc =2 2 ld σ = k0 lc =2 2 x σ  

1 2 Ef Vm α 1 γ 2 2k αvf 1 γvm ;

(2.31)

8 2 3 1 2 2vm k < αvf ðσ 2 σÞ 4eλld 2 eλx   2 ðld 2 xÞ5 wm ðxÞ 5 2 Em :Vm αvf 1 γvm λ 9      α 1 2 2vf k lc =2 2 ld σ = k0 lc =2 2 x σ   2 1 Ef Vm α 1 γ 2 2k αvf 1 γvm ;

(2.32)

The relative displacement between the fiber and matrix can be determined using the equation: 2 3 λld λx   e 2 e vðxÞ 5 wf ðxÞ 2 wm ðxÞ 5 2 αvf ðσ 2 σÞ4 2 ðld 2 xÞ5 λ 2 3 1 2 2v k 1 2 2v k 1 2 2vf k m f  1  5 1 ðld 2 xÞσ 34 Vf Ef Vm Em αvf 1 γvm Vf Ef αvf 1 γvm

(2.33)

Substituting wf(x 5 0) and v(x) into Eq. (2.14), leads to the equation: "

 αvf ð11β Þðσ 2σÞ  λld   e 21 2σ αvf 1γvm

#2 5

4Vf2 Ef ð1 1 β Þ   ξd rf 1 2 2vf k

(2.34)

Solving Eq. (2.34) leads to the equation: ld 5

 

αvf 1 γvm ðσ 2 σ0 Þ 1 ln 1 1 αvf ð1 1 β Þðσ 2 σÞ λ

(2.35)

Matrix cracking of ceramicmatrix composites

95

where γ ð1 2 2vm kÞ  ; σ0 5 β5  α 1 2 2vf k

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4Vf2 Ef ð1 1 β Þ   ξd rf 1 2 2vf k

(2.36)

where σ0 denotes interface initial debonded stress. When ld 5 lc/2, the interface complete debonded stress can be determined using the equation:     αvf ð1 1 β Þ eL=2λ 2 1 σ 1 αvf 1 γvm σ0     σb 5 αvf ð1 1 β Þ eL=2λ 2 1 1 αvf 1 γvm

(2.37)

2.3.3 Multiple matrix cracking The critical matrix strain energy (CMSE) criterion is used to determine matrix multicracking evolution (Solti et al., 1995). The matrix strain energy can be determined using the equation: Um 5

1 2Em

ð Am

ð lc 0

σ2m ðxÞdxdAm

(2.38)

where Am denotes the cross-section area of the matrix in the unit cell. Substituting the matrix axial stress in Eq. (2.24) into Eq. (2.38), the matrix strain energy for the interface partial debonding can be determined using the equation:  α2 v2f Am ðσ 2σÞ2 e2λld 2 4eλld 1 2λld 1 3 Am 2  1 Um 5 σmo lc =2 2 ld  2 2 2λ E m Vm Em αvf 1γvm

(2.39)

When the interface completely debonds, the matrix strain energy can be determined using the equation:   α2 v2f Am ðσ 2σÞ2 eλlc 2 4eλlc =2 1 λlc 1 3 L 5 Um σ; L; Ld 5  2 2 2λ Vm2 Em αvf 1γvm

(2.40)

The CMSE can be determined using the equation: 1 σ2 Ucrm 5 kAm l0 mocr 2 Em

(2.41)

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Durability of Ceramic-Matrix Composites

where k is the CMSE parameter; and l0 is the initial matrix crack spacing and σmocr can be determined using the equation: " # 1 γ ð1 2 2vm kÞ   σcr 12 σmocr 5 Vm α 1 γ 2 2k αvf 1 γvm

(2.42)

where 6Vf2 Ef Ec2 τ i ξ m σcr 5 rf Vm Em2

!1=3 2 Ec ðαc 2 αm ÞΔΤ

(2.43)

Multiple matrix cracking can be determined using the energy balance relationship defined in the equation: Um ðσ . σcr ; lc ; ld Þ 5 Ucrm ðσcr ; l0 Þ

(2.44)

2.3.4 Results and discussion The evolution of matrix multiple cracking and fibermatrix interface debonding versus applied stress of SiC/borosilicate composites are analyzed for different interface frictional coefficient, fibers Poisson ratio, fiber volume fraction, interface debonded energy, and applied cycles.

2.3.4.1 Effect of fibermatrix interface frictional coefficient on fibermatrix interface debonding and matrix multicracking evolution The evolution of matrix multicracking and fibermatrix interface debonding versus the applied stress curves for different fibermatrix interface frictional coefficient of μ 5 0.5 and 1.0 are shown in Fig. 2.9. When the fibermatrix interface frictional coefficient increases from μ 5 0.5 to 1.0, the fibermatrix interface debonded length decreases at the same applied stress, leading to the increase of the matrix cracking density.

2.3.4.2 Effect of fibers poisson ratio on fibermatrix interface debonding and matrix multicracking evolution The evolution of matrix multicracking and fibermatrix interface debonding versus the applied stress curves for different fiber Poisson ratios of υf 5 0.2 and 0.3 are shown in Fig. 2.10. When the fiber Poisson ratio increases, the fibermatrix interface debonded length increases, leading to the decrease of the matrix cracking density at the same applied stress.

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97

Figure 2.9 (A) The matrix cracking density versus the applied stress curves; and (B) the fibermatrix interface debonded length versus the applied stress curve for different interface frictional coefficient of μ 5 0.1 and 1.0. FMCS, First-matrix cracking stress.

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Figure 2.10 (A) The matrix cracking density versus the applied stress curves; and (B) the fibermatrix interface debonded length versus the applied stress curves for different fibers Poisson ratios of υf 5 0.2 and 0.3.

2.3.4.3 Effect of fiber-volume fraction on fibermatrix interface debonding and matrix multicracking evolution The evolution of matrix multicracking and fibermatrix interface debonding versus the applied stress curves for different fiber volume fraction of Vf 5 30% and 35%

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99

Figure 2.11 (A) The matrix cracking density versus the applied stress curves; and (B) the fibermatrix interface debonded length versus the applied stress curves for different fiber volume fraction of Vf 5 30% and 35%.

are shown in Fig. 2.11. When the fiber volume fraction increases, the interface debonded length decreases at the same applied stress, leading to the increase of matrix cracking density.

2.3.4.4 Effect of applied cycle number on fibermatrix interface shear stress and matrix axial stress distribution Under cyclic fatigue loading, the fibermatrix interface shear stress of fiberreinforced CMCs degrades with increasing applied cycles due to interface wear

100

Durability of Ceramic-Matrix Composites

(Cho et al., 1991; Holmes and Cho, 1992; Evans et al., 1995; Vagaggini et al., 1995; Fantozzi and Reynaud, 2009; Rouby and Louet, 2002). For SiC/CAS-II composites, the fibermatrix interface shear stress decreases from over 20 MPa to about 5 MPa during first 25,000 cycles (Holmes and Cho, 1992). The relationship of the fibermatrix interface shear stress versus the applied cycles can be determined using the equation: (Evans et al., 1995)  η τ i ðN Þ 5 τ io 1 1 2 exp2ωN ðτ i min 2 τ io Þ

(2.45)

where τ io denotes the initial fibermatrix interface shear stress, that is, τ i(N) at N 5 1, before cyclic fatigue loading; τ imin denotes the steady-state fibermatrix interface shear stress during cycling; and ω and η are empirical constants. The evolution of the fibermatrix interface shear stress versus the applied cycles curve is shown in Fig. 2.12A. The fibermatrix interface shear stress decreases from τ i 5 30 MPa at the applied cycle number of N 5 1 to τ i 5 10 MPa at the applied cycle number of N 5 45. Under cyclic fatigue loading, the stress carried by the matrix decreases with the decreasing fibermatrix interface shear stress, as shown in Fig. 2.12B. Based on the CMSE criterion, that is, when the matrix strain energy approaches the critical value, matrix cracking and matrix cracking density remain the same with increasing applied cycles.

2.3.5 Experimental comparisons The evolution of the matrix multiple cracking and fibermatrix interface debonding of different fiber-reinforced CMCs are predicted considering constant fibermatrix interface shear stress, and the interface shear stress is described using the Coulomb frictional law.

2.3.5.1 Unidirectional SiC/CAS composite The matrix multiple cracking density and the fibermatrix interface debonded length versus the applied stress curves of unidirectional SiC/CAS composites at room temperature are shown in Fig. 2.13. The FMCS is about 100 MPa and the matrix cracking saturation stress is in the range of 270280 MPa. Considering fibers Poisson contraction, the interface debonded length decreases, and the matrix cracking density decreases at the same applied stress, in contrast to constant interface shear stress. The matrix multiple cracking evolution considering fiber Poisson contraction agrees with experimental data.

2.3.5.2 Unidirectional SiC/CAS-II composite The matrix multiple cracking density and the fibermatrix interface debonded length versus the applied stress curves of unidirectional SiC/CAS-II composites at room temperature are shown in Fig. 2.14. The FMCS is about 120 MPa and the

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101

Figure 2.12 (A) The fibermatrix interface shear stress versus applied cycles curves; and (B) the matrix axial stress distribution along the fiber axial direction for different numbers of applied cycles.

matrix cracking saturation stress is about 240 MPa. Considering fiber Poisson contraction, the interface debonded length decreases, and the matrix cracking density decreases at the same applied stress, in contrast to constant interface shear stress. The matrix multiple cracking evolution considering fiber Poisson contraction agrees with experimental data.

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Figure 2.13 (A) The matrix cracking density versus the applied stress curves; and (B) the fibermatrix interface debonded length versus the applied stress curves for unidirectional SiC/CAS composite.

2.3.5.3 Unidirectional SiC/borosilicate composite The matrix multiple cracking density and the fibermatrix interface debonded length versus the applied stress curves of unidirectional SiC/borosilicate composites at room temperature are shown in Fig. 2.15. The FMCS is about σmc 5 150 MPa and the matrix cracking saturation stress is about σsat 5 390400 MPa. When considering

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103

Figure 2.14 (A) The matrix cracking density versus the applied stress curve; and (B) the fibermatrix interface debonded length versus the applied stress curve for unidirectional SiC/CAS-II composite.

fiber Poisson contraction, the interface debonded length decreases, and the matrix cracking density decreases at the same applied stress, in contrast to constant interface shear stress. The matrix multiple cracking evolution considering fibers Poisson contraction agrees with experimental data.

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Figure 2.15 (A) The matrix cracking density versus the applied stress; and (B) the fibermatrix interface debonded length versus the applied stress curve for unidirectional SiC/borosilicate composite.

2.4

Matrix multicracking evolution considering interface oxidation

At elevated temperatures in air atmosphere, most fiber-reinforced CMCs experience oxidation embrittlement (Gowayed et al., 2015). Steiner (1994) investigated the tensiontension fatigue behavior of cross-ply SiC/MAS composites at

Matrix cracking of ceramicmatrix composites

105

elevated temperatures of 566 C and 1093 C in air atmosphere. With increasing temperature, the amount of the fibers pullout decreases due to the fibermatrix interface oxidized strong bonding, resulting in the brittle fracture of the composite and decrease of fatigue life. Halbig and Eckel (2000) performed nonstressed oxidation experiments on 2D C/SiC composites and divided carbon oxidation into two different modes, that is, reaction control oxidation and diffusion control oxidation, based on temperature range. The applied stress that increases the matrix opening width, changes the oxidation mechanism inside the C/SiC composite. Lamouroux et al. (1994) investigated the oxidation mechanism and kinetics of 2D C/SiC composites and developed the oxidation kinetic model of C/SiC composite. Ruggles-Wrenn and Jones (2013) investigated the tensiontension fatigue behavior of 2D SiC/SiC composites at elevated temperatures. It was found that the oxidation embrittlement caused the great decrease in fatigue limit stress. Pailler and Lamon (2005) developed the micromechanical fatigue/oxidation damage model of fiber-reinforced CMCs and predicted the strain evolution with time of 2D SiC/SiC composites under static loading at elevated temperatures in an oxidative environment. Morscher and Cawley (2002) investigated the strength degradation of 2D SiC/SiC composites under static loading at elevated temperatures of 8001000 C in an oxidative environment. It was found that the interphase type, the amount of matrix cracking and the stress level are the main reasons affecting the strength degradation. Yin (2001) investigated the oxidation behavior of 3D C/SiC composites in the aero-engine gas environment. The mechanical performance of C/SiC composites after oxidation in the gas environment exhibits obvious degradation. Wei (2004) investigated the oxidation behavior of C/SiC composites in air, steam, and H2OO2 atmospheres, analyzed the oxidation mechanisms for different environments, and established the oxidation kinetic model in an oxygen environment. Luan (2007) investigated the damage evolution process, damage modes and damage mechanisms of 3D C/SiC composites in an aero-engine, wind-tunnel simulation environment, and thermal-shock simulation environments, and established the stressed-oxidation damage model of 3D C/SiC composites. Liu et al. (2006) investigated the stressed-oxidation behavior of C/SiC composite under the dry and wet oxygen environment as well as fatigue and creep loading. The fatigue life is longer than that of the creep life, and the oxidation of carbon fibers is the main failure mechanism for the creep failure specimen in a dry oxygen environment. However, water vapor aggravates the oxidation of the SiC matrix. In this section, the evolution of matrix multiple cracking, fibermatrix interface debonding, and interface oxidation of different fiber-reinforced CMCs are investigated. The theoretical relationship between the matrix cracking density, fibermatrix interface debonding ratio, fibermatrix interface oxidation ratio, and applied stress are established. The matrix cracking and fibermatrix interface debonding versus applied stress curves for different fiber volume fraction, fibermatrix interface shear stress, interface debonded energy, and oxidation temperature and time are discussed. The matrix multiple cracking evolution of different fiber-reinforced CMCs are then predicted considering interface oxidation.

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2.4.1 Stress analysis The fiber and matrix axial stress distribution in the interface oxidation region and interface slip region can be determined using the equations: 8 σ 2τ f > > 2 x; xAð0; ζ Þ > < Vf rf σ f ðxÞ 5 σ 2τ f 2τ i > > > : Vf 2 rf ζ 2 rf ðx 2 ζ Þ; 8 Vf τ f > > x; xAð0; ζ Þ 2 > < V m rf σm ðxÞ 5 Vf τ f Vf τ i > > > : 2 Vm rf ζ 1 2 Vm rf ðx 2 ζ Þ;

xAðζ; ld Þ

xAðζ; ld Þ

(2.46)

(2.47)

In the interface bonded region, the fiber and matrix axial stress and the fibermatrix interface shear stress can be determined using the equations:

  Vm 2τ f 2τ i x 2 ld σf ðxÞ 5 σfo 1 σmo 2 ζ2 ðld 2 ζ Þ exp 2ρ (2.48) Vf rf rf rf

  Vf τ f Vf τ i x 2 ld ζ 12 ðld 2 ζ Þ 2 σmo exp 2ρ σm ðxÞ 5 σmo 1 2 Vm r f Vm rf rf

(2.49)

  ρ Vm 2τ f 2τ i x 2 ld σmo 2 ζ2 ðld 2 ζ Þ exp 2ρ 2 Vf rf rf rf

(2.50)

τ i ðxÞ 5

2.4.2 Interface debonding The fibermatrix interface debonding length considering the interface oxidation can be determined using Eq. (2.14) (Gao et al., 1988). The axial displacements of the fiber and matrix can be determined using the equations: wf ðxÞ 5

ð lc =2 z

σf ðxÞ dx Ef

0 1  σ τf  τ σ l i fo @ c 2 ld A ðld 2 xÞ 2 2ζld 2 ζ 2 2 x2 2 ðld 2ζ Þ2 1 5 Vf Ef r f Ef rf E f Ef 2 0 13 2 32 r f 4 Vm 2τ f 2τ i lc =2 2 ld A5 σmo 2 ζ2 ðld 2 ζ Þ541 2 exp@ 2ρ 1 ρEf Vf rf rf rf (2.51)

Matrix cracking of ceramicmatrix composites

wm ðxÞ 5

ð lc =2 z

107

σ m ðxÞ dx Em

0 1  Vf τ f  V τ σ l f i mo c @ 2 ld A 5 2ζld 2 ζ 2 2 x2 1 ðld 2ζ Þ2 1 r f Vm Em r f Vm Em Em 2 0 13 2 32 rf 4 Vf τ f Vf τ i l =2 2 l c d A5 2 σmo 2 2 ζ 22 ðld 2 ζ Þ541 2 exp@ 2ρ ρEm r f Vm r f Vm rf (2.52) The relative displacement between the fiber and the matrix can be determined using the equation:   vðxÞ 5 wf ðxÞ 2 wm ðxÞ 5

 σ Ec τ f  Ec τ i ðld 2 xÞ 2 2ζld 2 ζ 2 2 x2 2 ðld 2ζ Þ2 Vf E f r f V m Em Ef rf Vm Em Ef

0 13 2 32 r f Ec 4 τf τi l =2 2 l c d A5 1 σmo 2 2 ζ 2 2 ðld 2 ζ Þ541 2 exp@ 2ρ ρVm Em Ef rf rf rf (2.53) Substituting wf (z 5 0) and v(z) into Eq. (2.14), leads to the equation: Ec τ 2i Ec τ 2i τiσ 2Ec τ f τ i ðld 2ζ Þ2 1 ðld 2 ζ Þ 2 ðld 2 ζ Þ 1 ζ ðld 2 ζ Þ Vf E f r f Vm E m Ef ρVm Em Ef r f Vm Em E f 2

Ec τ 2f rf τ i σ Ec τ f τ i τf σ r f Vm Em σ 2 1 ζ2 1 ζ2 ζ1 2 ξd 5 0 2ρVf Ef Vf Ef r f Vm Em E f ρVm Em Ef 4Vf2 Ef Ec (2.54)

Solving Eq. (2.53), the interface debonded length can be determined using the equation: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   s    τf r f Vm Em σ 1 r f 2 r f Vm E m Ef ld 5 1 2 2 1 ξd 2 ζ1 ρ τi 2 V f Ec τ i 2ρ Ec τ 2i

(2.55)

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Durability of Ceramic-Matrix Composites

2.4.3 Multiple matrix cracking Substituting the matrix axial stresses in Eqs. (2.47) and (2.49) into Eq. (2.38), the matrix strain energy when the interface is partially debonding can be determined using the equation: 8 Am 2 x; xA½0; ζ  > > > Vf rf > > > > > < σ 2 2τ f ζ 2 2τ i ðx 2 ζ Þ; xA½ζ; l  d σf ðxÞ 5 Vf rf rf > 2 3 1 2 3 0 > > > > > V τ τ x 2 l l m f i d c > A; xA4ld ; 5 4 5 @ > > : σfo 1 Vf σmo 2 2 rf ζ 2 2 rf ðld 2 ζ Þ exp 2ρ rf 2 (4.1) 8 V τ f f > 2 x;xA½0;ζ  > > > V m rf > > > > > Vf τ f Vf τ i < 2 ζ 12 ðx2ζ Þ;xA½ζ;ld  Vm r f Vm r f σm ðxÞ5 > 2 3 1 2 3 0 > > > > > V τ V τ x2l l f f f i dA c > 4 5 @ > ; xA4ld ; 5 > : σmo 2 σmo 22 Vm rf ζ 22 Vm rf ðld 2ζ Þ exp 2ρ rf 2 (4.2) 8 τ f ;xA½0;ζ  > > > > > < τ i ;xA½ζ;ld  2 3 1 2 3 0 τ i ðxÞ5 > ρ V τ 2τ x2l l dA c > > 4 m σmo 22 f ζ 2 i ðld 2ζ Þ5exp@2ρ ; xA4ld ; 5 > > : 2 Vf rf rf rf 2

(4.3)

Interface debonding and sliding of ceramic-matrix composites

197

where ρ denotes the shear-lag model parameter. (Budiansky et al., 1986) ρ2 5

4Ec Gm V m Em E f ϕ

(4.4)

and   2lnVf 1 Vm 3 2 Vf ϕ52 2Vm2 σfo 5

  Ef σ 1 Ef αc 2 αf ΔΤ Ec

σmo 5

Em σ 1 Em ðαc 2 αm ÞΔΤ Ec

(4.5)

(4.6)

(4.7)

4.2.1.2 Unloading Upon unloading, the counter slip occurs in the interface debonded region. At the unloading transition stress of σtr_u, the interface counter-slip length approaches the interface wear length ζ, i.e., y(σ . σtr_u) , ζ. When σ , σtr_u, the interface counter-slip length y is larger than the interface wear length ζ, that is, y (σ , σtr_u) . ζ. When σ . σtr_u, the fiber axial stress distribution upon unloading can be determined using the equation: 8 σ 2τ f > > σf ðxÞ5 1 x;xA½0;y > > V rf f > > > > > σ 2τ f > > ð2y2xÞ;xA½y;ζ  σf ðxÞ5 1 > > > V rf f < σ 2τ f 2τ i > σf ðxÞ5 1 ð2y2ζ Þ2 ðx2ζ Þ;xA½ζ;ld  > > V r rf > f f > > 2 3 1 2 3 0 > > > > > V 2τ 2τ x2l l m f i dA c > > ; xA4ld ; 5 ð2y2ζ Þ2 ðld 2ζ Þ5exp@2ρ > σf ðxÞ5σfo 1 4 σmo 1 : Vf rf rf rf 2 (4.8) When σ , σtr_u, the fiber axial stress distribution upon unloading can be determined using the equation:

198

Durability of Ceramic-Matrix Composites

8 σ 2τ f > > > σf ðxÞ5 V 1 r x;xA½0;ζ  > f f > > > > > σ 2τ 2τ i f > > σf ðxÞ5 1 ζ1 ðx2ζ Þ;xA½ζ;y > > > Vf rf rf < σ 2τ f 2τ i > σf ðxÞ5 1 ζ1 ð2y2ζ 2xÞ;xA½y;ld  > > Vf rf rf > > > 2 3 2 3 > >   > > > V 2τ 2τ x2l l m f i d c > 4 5 > ; xA4ld ; 5 > : σf ðxÞ5σfo 1 Vf σmo 1 rf ζ 1 rf ð2y2ζ 2ld Þ exp 2ρ rf 2 (4.9)

4.2.1.3 Reloading Upon reloading, new slip occurs in the counter-slip region. At the reloading transition stress of σtr_r, the interface new-slip length approaches the interface wear length ζ (i.e., z(σ , σtr_r) , ζ). When σ . σtr_r, the reloading interface new-slip length is larger than the interface wear length ζ (i.e., z (σ . σtr_u) . ζ). When σ , σtr_r, the fiber axial stress distribution upon reloading can be determined using the equation: 8 σ 2τ f > > σf ðxÞ5 2 x; xA½0;z > > V rf > f > > > > σ 2τ f > > σf ðxÞ5 2 ð2z 2xÞ;xA½z; ζ  > > > V rf f > > > > > < σ ðxÞ5 σ 2 2τ f ð2z 2ζ Þ1 2τ i ðx2 ζ Þ; xA½ζ; y f Vf rf rf > > > σ 2τ f 2τ i > > > σf ðxÞ5 2 ð2z 2ζ Þ1 ð2y 2ζ 2 xÞ; xA½y;ld  > > V r rf > f f > > 2 3 2 3 > >   > > V 2τ 2τ x2 l l > m f i d c > > ð2z2 ζ Þ1 ð2y2 ζ 2ld Þ5exp 2ρ σ ðxÞ5 σfo14 σmo 2 ; xA4ld ; 5 > : f Vf rf rf rf 2 (4.10) When σ . σtr_r, the fiber axial stress distribution upon reloading can be determined using the equation:

Interface debonding and sliding of ceramic-matrix composites

199

8 σ 2τ f > > σf ðxÞ5 2 x;xA½0;ζ  > > Vf rf > > > > > σ 2τ f 2τ i > > > > σf ðxÞ5 V 2 r ζ 2 r ðx2ζ Þ;xA½ζ;z > f f f > > > > > σ 2τ 2τ < σ ðxÞ5 2 f ζ 2 i ð2z2ζ 2xÞ;xA½z;y f Vf rf rf > > > σ 2τ 2τ > f i > > σf ðxÞ5 2 ζ 2 ð2z2ζ 22y1xÞ;xA½y;ld  > > V r r > f f f > > 2 3 2 3 > >   > > V 2τ 2τ x2l l > m f i d c > > ζ 2 ð2z2ζ 22y1ld Þ5exp 2ρ σ ðxÞ5σfo 1 4 σmo 2 ; xA4ld ; 5 > : f Vf rf rf rf 2 (4.11)

4.2.2 Interface slip lengths 4.2.2.1 Interface debonding length The fracture mechanics approach is used to determine the interface debonded length. (Gao et al., 1988) F @wf ð0Þ 1 2 ξd 5 4πrf @ld 2

ð lc =2 0

τ i ðxÞ

@vðxÞ dx @ld

(4.12)

where ξ d denotes the interface debonded energy; F(5πrf2σ/Vf) denotes the fiber load at the matrix crack plane; wf (0) denotes the fiber axial displacement at the matrix crack plane; and v(x) denotes the relative displacement between the fiber and the matrix. The axial displacement of fiber and matrix can be described using the equations: wf ðxÞ 5

ð lc =2 x

σf ðxÞ dx Ef

   σ τf  τi σfo lc 2 2 2 5 2 ld ðld 2 xÞ 2 2ζld 2 ζ 2 x 2 ðld 2ζ Þ 1 Vf Ef r f Ef rf E f Ef 2 2 32 3   r f 4 Vm 2τ f 2τ i l =2 2 l c d 5 1 σmo 2 ζ2 ðld 2 ζ Þ541 2 exp 2ρ ρEf Vf rf rf rf (4.13)

200

Durability of Ceramic-Matrix Composites

wm ðxÞ 5

ð lc =2 x

σ m ðxÞ dx Em

   Vf τ f  Vf τ i σmo lc 2 2 2 5 2 ld 2ζld 2 ζ 2 x 1 ðld 2ζ Þ 1 r f Vm Em r f Vm Em Em 2 2 32 3   rf 4 Vf τ f Vf τ i l =2 2 l c d 5 2 σmo 2 2 ζ 22 ðld 2 ζ Þ541 2 exp 2ρ ρEm r f Vm rf V m rf (4.14) The relative displacement is given by the equation:   vðxÞ 5 wf ðxÞ 2 wm ðxÞ  σ Ec τ f  Ec τ i 5 ðld 2 xÞ 2 2ζld 2 ζ 2 2 x2 2 ðld 2ζ Þ2 Vf E f r f V m Em Ef rf Vm Em Ef 3 2 32   r f Ec 4 τf τi l =2 2 l c d 5 1 σmo 2 2 ζ 2 2 ðld 2 ζ Þ541 2 exp 2ρ ρVm Em Ef rf rf rf (4.15) Substituting wf (x 5 0) and v(x) into the interface debonding criterion, leads to the form of the equation: Ec τ 2i Ec τ 2i τiσ 2Ec τ f τ i ðld 2ζ Þ2 1 ðld 2 ζ Þ 2 ðld 2 ζ Þ 1 ζ ðld 2 ζ Þ Vf E f r f Vm E m Ef ρVm Em Ef r f Vm Em E f 2

Ec τ 2f rf τ i σ Ec τ f τ i τf σ r f Vm Em σ 2 1 ζ2 1 ζ2 ζ1 2 ξd 5 0 2ρVf Ef Vf Ef r f Vm Em E f ρVm Em Ef 4Vf2 Ef Ec (4.16)

Solving Eq. (4.16), the interface debonded length is given by the equation:   sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    2 τf r f Vm Em σ 1 rf r f Vm E m Ef 2 1 ξd ld 5 1 2 2 ζ1 ρ τi 2 V f Ec τ i 2ρ Ec τ 2i

(4.17)

4.2.2.2 Interface counter-slip length Upon unloading, the axial displacements of the fiber and the matrix can be described using the equations:

Interface debonding and sliding of ceramic-matrix composites

201

ð lc =2

 2τ f σf ðxÞ σ τf  2 dx 5 ðl d 2 xÞ 2 ζ 1 x2 2 2ζy 1 ζ ðl d 2 yÞ E V E r E r f Ef f f f f f x   2τ i τi σfo lc 2 2 1 2 ld ðy2ζ Þ 2 ð2y2ζ 2ld Þ 1 r f Ef rf E f Ef 2 2 32 3   r f 4 Vm 2τ f 2τ i l =2 2 l c d 5 1 σmo 1 ζ1 ð2y 2 ζ 2 ld Þ541 2 exp 2ρ ρEf Vf rf rf rf

wf ðxÞ 5

(4.18) ð lc =2

 2Vf τ i σm ðxÞ Vf τ f  2 2 dx52 ζ 1x 22ζld 2 ðy2ζ Þ2 E r V E r f V m Em m f m m x   Vf τ i σmo lc 2 2ld 1 ð2y2ζ2ld Þ 1 r f V m Em Em 2 3 2 32   rf 4 Vf τ f Vf τ i lc =22ld 5 2 σmo 12 ζ 12 ð2y2ζ 2ld Þ5412exp 2ρ ρEm Vm r f Vm rf rf

wm ðxÞ5

(4.19) The relative displacement is given by the equation:    σ τ f Ec  2 ðld 2xÞ2 x 1ζ 2 22ζld vðxÞ5 wf ðxÞ2wm ðxÞ 5 Vf Ef r f Vm Em Ef 2Ec τ i τ i Ec ðy2ζ Þ2 2 ð2y2ζ2ld Þ2 r f V m Em Ef r f V m Em Ef 3 2 32   r f 4 Vm 2τ f 2τ i l =22l c d 5 1 σmo 1 ζ1 ð2y2ζ 2ld Þ5412exp 2ρ ρEf Vf rf rf rf 2 32 3   rf 4 Vf τ f Vf τ i lc =22ld 5 1 σmo 12 ζ 12 ð2y2ζ 2ld Þ5412exp 2ρ ρEm Vm rf Vm r f rf 1

(4.20) Substituting wf (x 5 0) and v(x) into Eq. (4.12), leads to the form of the equation: r f Vm Em 2 τf σ τiσ rf τ i σ σ 1 ζ1 ð2y 2 ζ 2 ld Þ 2 Vf Ef Vf E f 2ρVf Ef 4Vf2 Ef Ec 1

Ec τ 2f 2Ec τ f τ i Ec τ f τ i ζ2 1 ζ ð2y 2 ζ 2 ld Þ 2 ζ r f V m Em Ef r f V m Em E f ρVm Em Ef

1

Ec τ 2i Ec τ 2i ð2y2ζ 2ld Þ2 2 ð2y 2 ζ 2 ld Þ 2 ξd 5 0 r f V m Em Ef ρVm Em Ef

(4.21)

202

Durability of Ceramic-Matrix Composites

Solving Eq. (4.21), the unloading interface counter-slip length is given by the equation: 8 2 39   sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    2 = 1< τf r V E σ 1 r r V E E f m m f f m m f ld ðσmax Þ1 12 2 1 ξd 5 y5 2 ζ 24 2 ; 2: τi 2 Vf Ec τ i ρ 2ρ Ec τ i (4.22)

4.2.2.3 Interface new-slip length Upon reloading, the axial displacement of the fiber and the matrix can be described using the equations: ð lc =2

σf ðxÞ σ τf  2 2 τf dx5 ðld 2xÞ2 2z 2x 1 ð2z2ζ Þ2 E V E r E r E f f f f f f f x 0 1 2τ f 2τ i τ σ l i fo c 2 ð2z2ζ Þðld 2ζ Þ1 ðy2ζ Þ2 2 ð2y2ζ2ld Þ2 1 @ 2ld A rf E f rf E f r f Ef Ef 2 0 13 2 32 r f 4 Vm 2τ f 2τ i l =22l c d A5 1 σmo 2 ð2z2ζ Þ1 ð2y2ζ 2ld Þ5412exp@2ρ ρEf Vf rf rf rf

wf ðxÞ5

(4.23) wm ðxÞ5

ð lc =2 x

σm ðxÞ dx Em

Vf τ f  2 2  Vf τ f 2Vf τ f 2z 2x 2 ð2z2ζ Þ2 1 ð2z2ζ Þðld 2ζ Þ r f V m Em r f Vm Em r f Vm Em 0 1 2Vf τ i V τ σ l f i mo c @ 2ld A 2 ðy2ζ Þ2 1 ð2y2ζ2ld Þ2 1 r f V m Em rf Vm Em Em 2 0 13 2 32 rf 4 Vf τ f Vf τ i lc =22ld A5 2 σmo 22 ð2z2ζ Þ12 ð2y2ζ 2ld Þ5412exp@2ρ ρEm Vm r f Vm r f rf

5

(4.24)

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203

The relative displacement is given by the equation:   vðxÞ5 wf ðxÞ2wm ðxÞ σ Ec τ f  2 2  Ec τ f 5 ðld 2xÞ2 2z 2x 1 ð2z2ζ Þ2 Vf Ef rf Vm Em Ef r f Vm Em Ef 2Ec τ f 2Ec τ i Ec τ i ð2z2ζ Þðld 2ζ Þ1 ðy2ζ Þ2 2 ð2y2ζ2ld Þ2 r f Vm Em E f r f Vm Em Ef r f V m Em Ef 0 13 2 32 rf 4Vm 2τ f 2τ i l =22l c d A5 1 σmo 2 ð2z2ζ Þ1 ð2y2ζ 2ld Þ5412exp@2ρ ρEf Vf rf rf rf 0 13 2 32 rf 4 Vf τ f Vf τ i l =22l c d A5 1 σmo 22 ð2z2ζ Þ12 ð2y2ζ 2ld Þ5412exp@2ρ ρEm V m rf Vm r f rf 2

(4.25) Substituting wf(x 5 0) and v(x) into Eq. (4.12), leads to the form of the equation: r f Vm E m σ 2 τf σ τiσ rf τ i σ 2 ð2z 2 ζ Þ 1 ð2y 2 ζ 2 ld Þ 2 Vf E f Vf Ef 2ρVf Ef 4Vf2 Ef Ec 1

Ec τ 2f 2Ec τ f τ i Ec τ f τ i ð2z2ζ Þ2 2 ð2z 2 ζ Þð2y 2 ζ 2 ld Þ 1 ð2z 2 ζ Þ r f Vm Em Ef r f V m Em Ef ρVm Em Ef

1

Ec τ 2i Ec τ 2i ð2y2ζ 2ld Þ2 2 ð2y 2 ζ 2 ld Þ 2 ξd 5 0 r f Vm Em Ef ρVm Em Ef (4.26)

Solving Eq. (4.26), the reloading interface new-slip length is given by the equation: " "     1 τf r f Vm E m σ 1 2 ζ2 yðσmin Þ 2 ld 1 1 2 2 ρ τi 2 V f Ec τ i sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ##)  2 rf rf Vm Em Ef 1 ξd 2 2ρ Ec τ 2i

τi z5 τf

(

(4.27)

4.2.3 Results and discussion Composite strain can be determined using the equation: 2 εc 5 Ef l c

ð lc =2

  σf ðxÞdx 2 αc 2 αf ΔΤ

(4.28)

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The unloading stressstrain relationship can be determined using the equations: σ 2ld 2τ f  4τ f 2τ i 2 ð2y2ζ Þ2 22y2 1 ð2y2ζ Þðld 2ζ Þ2 ðld 2ζ Þ2 Vf Ef lc rf Ef lc r f Ef l c r f Ef l c 2 32 3   2rf 4Vm 2τ f 2τ i lc =22ld 5 1 σmo 1 ð2y2ζ Þ2 ðld 2ζ Þ5412exp 2ρ ρEf lc Vf rf rf rf     2σfo lc 2ld 2 αc 2αf ΔΤ; y,ζ 1 Ef lc 2

εc unloading 5

(4.29) 2σld 2τ f 2 4τ f 4τ i 2τ i 1 ζ 1 ζ ðld 2ζ Þ1 ðy2ζ Þ2 2 ð2y2ζ2ld Þ2 Vf Ef lc rf Ef lc rf Ef lc rf Ef lc rf Ef lc 3 2 32   2rf 4Vm 2τ f 2τ i l =22l c d 5 1 σmo 1 ζ1 ð2y2ζ 2ld Þ5412exp 2ρ ρEf lc Vf rf rf rf     2σfo lc 1 2ld 2 αc 2αf ΔΤ; y.ζ Ef lc 2

εc unloading 5

(4.30) The reloading stressstrain relationship can be determined using the equations: 2σ 4τ f 2 2τ f 4τ f ld 2 z 1 ð2z2ζ Þ2 2 ð2z2ζ Þðld 2ζ Þ Vf Ef l c rf Ef lc r f Ef l c rf Ef lc 2 32 3   2rf 4Vm 2τ f 2τ i lc =22ld 5 σmo 2 ð2z2ζ Þ1 ð2y2ζ 2ld Þ5412exp 2ρ 1 ρEf lc Vf rf rf rf     4τ i 2τ i 2σfo lc 2ld 2 αc 2αf ΔΤ; z,ζ ðy2ζ Þ2 2 ð2y2ζ2ld Þ2 1 1 rf Ef lc r f Ef l c Ef l c 2

εc reloading 5

(4.31) εc reloading 5

2σ 2τ f 2 4τ f 4τ i ld 1 ζ 2 ζld 2 ðz2ζ Þ2 Vf Ef l c rf Ef lc rf Ef lc rf Ef lc

0

1

4τ i 2τ i 2σfo @lc 2ld A ð2z2ζ2yÞ2 2 ð2z2ζ22y1ld Þ2 1 rf Ef lc rf Ef lc Ef lc 2 0 13 2 32 2rf 4Vm 2τ f 2τ i l =22l c d A5 1 σmo 2 ζ 2 ð2z2ζ 22y1ld Þ5412exp@2ρ ρEf lc Vf rf rf rf   2 αc 2αf ΔΤ; z.ζ

1

(4.32)

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205

Holmes and Cho (1992) characterized the fatigue behavior of fiber-reinforced CMCs through measuring the surface temperature. When the fiber slides relative to the matrix in the debonded region, the specimen surface temperature increases under cyclic loading. The relationship between the specimen surface temperature and the fibermatrix interface shear stress has been established to obtain the fibermatrix interface shear stress of unidirectional SiC/CAS composites at room temperature, and it was found that the fibermatrix interface shear stress decreases at the initial cyclic loading. Du et al. (2002) investigated the tensiontension cyclic fatigue behavior of 3D C/SiC composites at room temperature and an elevated temperature of 1300 C in vacuum atmosphere. Under high-fatigue peak stress, the matrix cracking deflects along the fibermatrix interface, and the fibermatrix interface debonds. With increasing cycles, the matrix inside of fiber bundles cracks, and the wear between the fiber and the matrix increases. At room temperature, the interface wear caused by the repeated sliding between the fiber and the matrix plays an important role in fatigue failure; however, at an elevated temperature of 1300 C, the effect of the interface wear on the fatigue damage becomes low. Han et al. (2004) investigated the tensiontension cyclic fatigue behavior of 2D and 3D C/SiC composites at elevated temperatures in the range of 1100 C1500 C in vacuum atmosphere. Compared to the 2D C/SiC composites, the fibers pullout length of 3D C/SiC composites at the fracture surface is much longer, indicating the longer fibermatrix interface sliding length. Evans et al. (1995) investigated the tensiontension cyclic fatigue behavior of unidirectional SiC/CAS composites at room temperature. Under cyclic fatigue loading, the fatigue hysteresis loops corresponding to different applied cycles are measured, and the fibermatrix interface shear stress is estimated through analyzing the shape of the fatigue hysteresis loops. It was found that the fibermatrix interface shear stress decreases from τ i 5 20 MPa at the first cycle to τ i 5 5 MPa at the 30th applied cycle. The fibermatrix interface wear model was developed to predict the fibermatrix interface shear stress degradation and then compared with experimental data. The evolution of the fibermatrix interface shear stress versus the applied cycles can be determined using the equation (Evans et al., 1995):    τ i ðN Þ 5 τ io 1 1 2 exp 2ωN λ ðτ i min 2 τ io Þ

(4.33)

where τ io denotes the initial interface shear stress, i.e., τ i(N) at N 5 1, before fatigue loading; τ imin denotes the steady-state interface shear stress during cycling; and ω and λ are empirical constants.

4.2.3.1 Effect of loading sequence on the fibermatrix interface debonding The fibermatrix interface debonding ratio of 2ld/lc versus the applied cycles curves for five different loading sequences are shown in Fig. 4.2. The five loading sequences are given as: 1. Case 1, σmax 5 180 MPa and lc 5 310 μm; or σmax 5 200 MPa and lc 5 220 μm. 2. Case 2, σmax1 5 180 MPa, lc 5 310 μm and N1 5 5000; and σmax2 5 200 MPa and lc 5 220 μm.

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Figure 4.2 (A) The fibermatrix interface debonded ratio of 2ld/lc versus the applied cycle number curves for the loading sequences of Cases 1, 2, and 4; (B) the fibermatrix interface debonded ratio of 2ld/lc versus the applied cycle number curves for the loading sequences of Cases 1, 3, and 5; (C) the fibermatrix interface debonded ratio of 2ld/lc versus the applied cycle number curves for the loading sequences of Cases 2 and 3; and (D) the fibermatrix interface debonded ratio of 2ld/lc versus the applied cycle number curves for the loading sequences of Cases 4 and 5. 3. Case 3, σmax1 5 200 MPa, lc 5 220 μm and N1 5 5000; and σmax2 5 180 MPa and lc 5 220 μm. 4. Case 4, σmax1 5 180 MPa, lc 5 310 μm and N1 5 1000; and σmax2 5 200 MPa, lc 5 220 μm and N2 5 1000. 5. Case 5, σmax1 5 200 MPa, lc 5 220 μm and N1 5 1000; and σmax2 5 180 MPa, lc 5 220 μm and N2 5 1000.

The fibermatrix interface debonding ratio of 2ld/lc versus the applied cycles curves for the loading sequences of Cases 1, 2, and 4 are shown in Fig. 4.2A.

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207

Figure 4.2 (Continued).

The fatigue peak stress of Case 1 is 180 MPa; for Case 2 and 4, the fatigue peak stress is 180 and 200 MPa with different interval cycles (i.e., N 5 5000 for Case 2, and N 5 1000 for Case 4). The fibermatrix interface debonding ratio of 2ld/lc is the lowest for the loading sequence of Case 1, and the highest for the loading sequence of Case 2. The fatigue peak stress and cycle interval affect the fibermatrix interface debonding extent inside the fiber-reinforced CMCs. The fibermatrix interface debonding ratio of 2ld/lc versus the applied cycles curves for the loading sequences of Case 1, 3, and 5 are shown in Fig. 4.2B. The fatigue peak stress for the loading sequence of Case 1 is 200 MPa; for the loading sequences of Cases 3 and 5, the fatigue peak stress is 200 and 180 MPa with different interval cycles (i.e., N 5 5000 for the loading sequence of Case 3 and N 5 1000 for the loading sequence of Case 5). The fibermatrix interface debonding ratio of 2ld/lc is the highest for the loading sequence of Case 1, and the lowest for the

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loading sequence of Case 5. The fatigue peak stress and applied cycle interval affect the extent of the fibermatrix interface debonding inside the fiber-reinforced CMCs. The fibermatrix interface debonding ratio of 2ld/lc versus the applied cycles curves for the loading sequences of Cases 2 and 3 are shown in Fig. 4.2C. The fatigue peak stress for the loading sequence of Case 2 is 180 and 200 MPa with the cycle interval of N 5 5000. The fatigue peak stress for the loading sequence of Case 3 is 200 and 180 MPa with the applied cycle interval of N 5 5000. At the initial N 5 5000 cycle, the fibermatrix interface debonding ratio for the loading sequence of Case 3 is much higher than that of the loading sequence of Case 2. However, at the following 5000 cycles, the fibermatrix interface debonding ratio of 2ld/lc for the loading sequence of Case 2 is much higher than that of the loading sequence of Case 3. The fibermatrix interface debonding ratio of 2ld/lc versus the applied cycles curves for the loading sequences of Cases 4 and 5 are shown in Fig. 4.2D. The fatigue peak stress for the loading sequence of Case 4 is 180 and 200 MPa with the cycle interval of N 5 1000. The fatigue peak stress for the loading sequence of Case 5 is 200 and 180 MPa with the applied cycle interval of N 5 1000. The fibermatrix interface debonding ratio of 2ld/lc increases with increasing fatigue peak stress and applied cycles.

4.2.3.2 Effect of applied cycle number on fibermatrix interface debonding Loading sequence of Case 2 The fibermatrix interface debonding ratio of 2ld/lc versus the applied cycles for the loading sequence of Case 2 with different cycle intervals are shown in Fig. 4.3. With increasing applied cycle interval from N1 5 1000 to N1 5 9000, the fibermatrix interface debonding ratio of 2ld/lc at the applied cycle number for N 5 10,000 cycle decreases. When the cycle interval is N1 5 1000, the fibermatrix interface debonding ratio is 2ld/lc 5 0.97 at the cycle number for N 5 10,000. When the cycle interval is N1 5 3000, the fibermatrix interface debonding ratio is 2ld/ lc 5 0.96 at the cycle number for N 5 10,000. When the cycle interval is N1 5 5000, the fibermatrix interface debonding ratio is 2ld/lc 5 0.94 at the cycle number for N 5 10,000. When the cycle interval is N1 5 7000, the fibermatrix interface debonding ratio is 2ld/lc 5 0.9 at the cycle number for N 5 10,000. When the cycle interval is N1 5 9000, the fibermatrix interface debonding ratio is 2ld/lc 5 0.86 at the cycle number for N 5 10,000.

Loading sequence of Case 3 The fibermatrix interface debonding ratio of 2ld/lc versus the applied cycles curves for the loading sequence of Case 3 with different cycle intervals are shown in Fig. 4.4. The fibermatrix interface debonding ratio of 2ld/lc increases with applied cycles, and decreases rapidly at the cycle interval of N1, and then increases slowly with applied cycles. When the cycle interval is N1 5 1000, the fibermatrix

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209

Figure 4.3 The fibermatrix interface debonded ratio of 2ld/lc versus the applied cycle number curves of the loading sequence of Case 2 with different cycle intervals for N1 5 1000, 3000, 5000, 7000, and 9000.

Figure 4.4 The fibermatrix interface debonded ratio of 2ld/lc versus the applied cycle number curves of the loading sequence of Case 3 with different cycle intervals for N1 5 1000, 3000, 5000, 7000, and 9000.

interface debonding ratio is 2ld/lc 5 0.83 at the cycle number for N 5 10,000; when the cycle interval is N1 5 3000, the fibermatrix interface debonding ratio is 2ld/ lc 5 0.82 at the cycle number for N 5 10,000; when the cycle interval is N1 5 5000, the fibermatrix interface debonding ratio is 2ld/lc 5 0.81 at the cycle number for N 5 10,000. When the cycle interval is N1 5 7000, the fibermatrix interface debonding ratio is 2ld/lc 5 0.83 at the cycle number for N 5 10,000. When the cycle

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Figure 4.5 The fibermatrix interface debonded ratio of 2ld/lc versus the applied cycle number curves of the loading sequence of Case 4 with different cycle intervals of N1 5 1000 and N2 5 2000, and N1 5 2000 and N2 5 1000.

interval is N1 5 9000, the fibermatrix interface debonding ratio is 2ld/lc 5 0.84 at the cycle number for N 5 10,000.

Loading sequence of Case 4 The fibermatrix interface debonding ratio of 2ld/lc versus the applied cycles curves for the loading sequence of Case 4 with different cycle intervals of N1 5 1000 and N2 5 2000, and N1 5 2000 and N2 5 1000 are shown in Fig. 4.5. For the cycle interval of N1 5 1000 and N2 5 2000, the fibermatrix interface debonded ratio of 2ld/lc at the fatigue peak stress of σmax2 5 200 MPa is larger than that for the cycle interval of N1 5 2000 and N2 5 1000. When the applied cycle number is N 5 3000, the fibermatrix interface debonding ratio is 2ld/lc 5 0.48 for the cycle interval of N1 5 1000 and N2 5 2000, and 2ld/lc 5 0.46 for the cycle interval of N1 5 2000 and N2 5 1000. When the applied cycle number is N 5 6000, the fibermatrix interface debonding ratio is 2ld/lc 5 0.84 for the cycle interval of N1 5 1000 and N2 5 2000, and 2ld/lc 5 0.83 for the cycle interval of N1 5 2000 and N2 5 1000. When the applied cycle number is N 5 9000, the fibermatrix interface debonding ratio is 2ld/lc 5 0.94 for the cycle interval of N1 5 1000 and N2 5 2000, and 2ld/lc 5 0.92 for the cycle interval of N1 5 2000 and N2 5 1000.

Loading sequence of Case 5 The fibermatrix interface debonding ratio of 2ld/lc versus the applied cycles curves for the loading sequence of Case 5 with different cycle intervals of N1 5 1000 and N2 5 2000, and N1 5 2000 and N2 5 1000 are shown in Fig. 4.6. For the cycle interval of N1 5 1000 and N2 5 2000, the fibermatrix interface debonded ratio of 2ld/lc at the fatigue peak stress of σmax2 5 200 MPa is less than that for the cycle interval of N1 5 2000 and N2 5 1000.

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Figure 4.6 The fibermatrix interface debonded ratio of 2ld/lc versus the applied cycle number curves of the loading sequence of Case 5 with different cycle intervals of N1 5 1000 and N2 5 2000, and N1 5 2000 and N2 5 1000.

4.2.3.3 Effect of peak stress level on fibermatrix interface debonding Loading sequence of Case 2 The fibermatrix interface debonding ratio of 2ld/lc versus the applied cycles curves for the loading sequence of Case 2 with different low-fatigue peak stresses of σmax1 5 140, 160, 180, and 200 MPa and the same high-fatigue peak stress of σmax2 5 240 MPa are shown in Fig. 4.7A. With increasing low-fatigue peak stress of σmax1, the fibermatrix interface debonding ratio of 2ld/lc increases, and the applied cycles for completely interface debonding decreases. When the low-fatigue peak stress is σmax1 5 140 MPa, the applied cycle number for the fibermatrix interface complete debonding of 2ld/lc 5 1 is N 5 8862. When the low-fatigue peak stress is σmax1 5 160 MPa, the applied cycle number for the fibermatrix interface complete debonding of 2ld/lc 5 1 is N 5 8372. When the low-fatigue peak stress is σmax1 5 180 MPa, the applied cycle number for the fibermatrix interface complete debonding of 2ld/lc 5 1 is N 5 7708. When the low-fatigue peak stress is σmax1 5 200 MPa, the applied cycle number for the fibermatrix interface complete debonding of 2ld/lc 5 1 is N 5 6830. The fibermatrix interface debonding ratio of 2ld/lc versus the applied cycles curves for the loading sequence of Case 2 with different high-fatigue peak stress of σmax2 5 140, 180, 220, and 260 MPa and the same low-fatigue peak stress of σmax1 5 120 MPa are shown in Fig. 4.7B. With increasing high-fatigue peak stress, the fibermatrix interface debonding ratio increases. When the applied cycle number is N 5 10,000, the fibermatrix interface debonding ratio is 2ld/lc 5 0.54, 0.75, 0.97, and 1.0 for the fatigue peak stress of σmax2 5 140, 180, 220, and 260 MPa, respectively.

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Figure 4.7 The fibermatrix interface debonded ratio of 2ld/lc versus the applied cycle number curves corresponding to (A) Case 2: σmax2 5 240 MPa and σmax1 5 140, 160, 180, and 200 MPa with N1 5 5000; and (B) Case 2: σmax1 5 120 MPa and σmax2 5 140, 180, 220, and 260 MPa with N1 5 5000.

Loading sequence of Case 3 The fibermatrix interface debonding ratio of 2ld/lc versus the applied cycles curves for the loading sequence of Case 3 with different high-fatigue peak stresses of σmax1 5 140, 160, 180, and 200 MPa and the same low-fatigue peak stress of σmax2 5 120 MPa are shown in Fig. 4.8A. With increasing high-fatigue peak stress, the fibermatrix interface debonding ratio of 2ld/lc increases. At the applied cycle number of N 5 3000, the fibermatrix interface debonding ratio is 2ld/lc 5 0.42, 0.49, 0.56, and 0.63 at the high-fatigue peak stresses of σmax1 5 140, 160, 180, and 200 MPa, respectively.

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Figure 4.8 The fibermatrix interface debonded ratio of 2ld/lc versus the applied cycle number curve corresponding to (A) Case 3: σmax1 5 140, 160, 180, and 200 MPa and σmax2 5 120 MPa with the cycle interval of N1 5 3000; and (B) Case 3: σmax1 5 200 MPa and σmax2 5 120, 140, 160, and 180 MPa with the cycle interval of N1 5 3000.

The fibermatrix interface debonding ratio of 2ld/lc versus the applied cycles curves for the loading sequence of Case 3 with different low-fatigue peak stress of σmax2 5 120, 140, 160, and 180 MPa and the same high-fatigue peak stress of σmax1 5 200 MPa are shown in Fig. 4.8B. With increasing low-fatigue peak stress, the fibermatrix interface debonding ratio of 2ld/lc increases, and the applied cycle for the interface completely debonding decreases. At the applied cycle number of N 5 10,000, the fibermatrix interface debonding ratio is 2ld/lc 5 0.66, 0.78, 0.92, and 1.0 at the fatigue peak stresses of σmax2 5 120, 140, 160, and 180 MPa, respectively.

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Figure 4.9 The fibermatrix interface debonded ratio of 2ld/lc versus the applied cycle number curves corresponding to (A) Case 4: σmax1 5 140, 160, and 180 MPa and σmax2 5 200 MPa with the cycle interval of N1 5 N2 5 1000; and (B) Case 4: σmax1 5 140 MPa and σmax2 5 160, 180, and 200 MPa with the cycle interval of N1 5 N2 5 1000.

Loading sequence of Case 4 The fibermatrix interface debonding ratio of 2ld/lc versus the applied cycles curves for the loading sequence of Case 4 with different low-fatigue peak stresses of σmax1 5 140, 160, and 180 MPa and the same high-fatigue peak stress of σmax2 5 200 MPa are shown in Fig. 4.9A. With increasing of the low-fatigue peak stress, the fibermatrix interface debonding ratio of 2ld/lc increases, and the applied cycle for the interface completely debonding decreases. The fibermatrix interface debonding ratio of 2ld/lc versus the applied cycles curves for the loading sequence of Case 4 with different high-fatigue peak stresses

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215

of σmax2 5 160, 180, and 200 MPa and the same low-fatigue peak stress of σmax1 5 140 MPa are shown in Fig. 4.9B. With increasing high-fatigue peak stress, the fibermatrix interface debonding ratio of 2ld/lc increases, and the applied cycle for the interface completely debonding decreases. At the applied cycle number of N 5 10,000, the fibermatrix interface debonding ratio is 2ld/lc 5 0.67, 0.8, and 0.93 at the fatigue peak stresses of σmax2 5 160, 180, and 200 MPa, respectively.

Loading sequence of Case 5 The fibermatrix interface debonding ratio of 2ld/lc versus the applied cycles curves for the loading sequence of Case 5 with different high-fatigue peak stresses of σmax1 5 160, 180, and 200 MPa and the same low-fatigue peak stress of σmax2 5 140 MPa are shown in Fig. 4.10A. With increasing high-fatigue peak stress, the fibermatrix interface debonding ratio of 2ld/lc increases. At the applied cycle number of N 5 9000, the fibermatrix interface debonding ratio is 2ld/lc 5 0.66, 0.79, and 0.92 at the high-fatigue peak stresses of σmax1 5 160, 180, and 200 MPa, respectively. The fibermatrix interface debonding ratio versus applied cycles curves for Case 5 with different low-fatigue peak stresses of σmax2 5 140, 160, and 180 MPa and the same high-fatigue peak stress of σmax1 5 200 MPa are shown in Fig. 4.10B. With increasing low-fatigue peak stress, the fibermatrix interface debonding ratio increases. At the applied cycle number of N 5 10,000, the fibermatrix interface debonding ratio is 2ld/lc 5 0.53, 0.66, and 0.8 at the low-fatigue peak stresses of σmax2 5 140, 160, and 180 MPa, respectively.

4.2.3.4 Effect of arbitrary loading sequence on fatigue hysteresis and fibermatrix interface sliding The arbitrary loading sequence affects the fatigue hysteresis loops and the fibermatrix interface sliding of fiber-reinforced CMCs. The loading sequence is given by Eq. (4.34): σmax1 ð180 MPaÞ ! σmax2 ð300 MPaÞ ! σmax3 ð240 MPaÞ ! σmax4 ð200 MPaÞ ! σmax5 ð320 MPaÞ ! σmax6 ð180 MPaÞ

(4.34)

The fatigue hysteresis loops and the fibermatrix interface sliding for the loading sequence in Eq. (4.34) are shown in Figs. 4.114.15. Considering the fibermatrix interface wear mechanisms, the low fibermatrix interface shear stress in the wear region affects the interface slip length and the fatigue hysteresis loops. When the loading fatigue peak stress is σmax2 5 300 MPa, the fatigue hysteresis dissipated energy (HDE) without considering the fibermatrix interface wear is U 5 40 kPa and the fibermatrix interface sliding ratio is 2y/lc 5 2z/lc 5 38%. The fatigue HDE considering the fibermatrix interface wear is U 5 55 kPa and the fibermatrix interface sliding ratio is 2y/lc 5 2z/lc 5 60%.

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Figure 4.10 The fibermatrix interface debonded ratio of 2ld/lc versus the applied cycle number curve corresponding to (A) Case 5: σmax1 5 160, 180, and 200 MPa and σmax2 5 140 MPa with the cycle interval of N1 5 N2 5 1000; and (B) Case 5: σmax1 5 200 MPa and σmax2 5 140, 160, and 180 MPa with the cycle interval of N1 5 N2 5 1000.

When the loading fatigue peak stress is σmax3 5 240 MPa, the fatigue HDE without considering the fibermatrix interface wear is U 5 20 kPa and the fibermatrix interface sliding ratio is 2y/lc 5 2z/lc 5 30%. The fatigue HDE considering the fibermatrix interface wear is U 5 32 kPa and the fibermatrix interface sliding ratio is 2y/lc 5 2z/lc 5 52%. When the loading fatigue peak stress is σmax4 5 240 MPa, the fatigue HDE without considering the fibermatrix interface wear is U 5 12 kPa and the fibermatrix interface sliding ratio is 2y/lc 5 2z/lc 5 26%. The fatigue HDE considering the

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217

Figure 4.11 (A) The fatigue hysteresis loops; and (B) the fibermatrix interface slip ratio of 2y(z)/lc versus the applied stress at the loading fatigue peak stress of σmax2 5 300 MPa.

fibermatrix interface wear is U 5 21 kPa and the fibermatrix interface sliding ratio is 2y/lc 5 2z/lc 5 47%. When the loading fatigue peak stress is σmax5 5 320 MPa, the fatigue HDE without considering the fibermatrix interface wear is U 5 49 kPa and the fibermatrix interface sliding ratio is 2y/lc 5 2z/lc 5 41%. The fatigue HDE considering the fibermatrix interface wear is U 5 65 kPa and the fibermatrix interface sliding ratio is 2y/lc 5 2z/lc 5 62%. When the loading fatigue peak stress is σmax6 5 180 MPa, the fatigue HDE without considering the fibermatrix interface wear is U 5 9 kPa and the fibermatrix interface sliding ratio is 2y/lc 5 2z/lc 5 23%. The fatigue HDE considering the fibermatrix interface wear is U 5 16 kPa and the fibermatrix interface sliding ratio is 2y/lc 5 2z/lc 5 44%.

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Figure 4.12 (A) The fatigue hysteresis loops; and (B) the fibermatrix interface slip ratio of 2y(z)/lc versus the applied stress at the loading fatigue peak stress of σmax3 5 240 MPa.

4.2.3.5 Effect of fiber volume fraction on fatigue hysteresis and fibermatrix interface sliding The fatigue hysteresis loops and the fibermatrix interface sliding ratio of 2y(z)/lc for different fiber volume fractions of Vf 5 35% and 40% are shown in Fig. 4.16. When the fiber volume fraction increases, the fibermatrix interface debonding and sliding range decreases, leading to decreases of peak and valley strain, and the hysteresis area; however, there is an increase of hysteresis modulus.

4.2.3.6 Effect of matrix crack spacing on the fatigue hysteresis loops and fibermatrix interface sliding The fatigue hysteresis loops and the fibermatrix interface sliding ratio of 2y(z)/lc for different matrix crack spacing are shown in Fig. 4.17. When the matrix crack

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219

Figure 4.13 (A) The fatigue hysteresis loops; and (B) the fibermatrix interface slip ratio of 2y(z)/lc versus the applied stress at the loading fatigue peak stress of σmax4 5 200 MPa.

spacing increases, the fibermatrix interface sliding ratio of 2y(z)/lc decreases, leading to the decrease of the fatigue peak and valley strain, and the fatigue hysteresis area; however, there is an increase of fatigue hysteresis modulus.

4.2.3.7 Effect of interface properties on the fatigue hysteresis and fibermatrix interface sliding The fatigue hysteresis loops and the fibermatrix interface sliding for different fibermatrix interface shear stresses of τ i 5 15 and 20 MPa are shown in Fig. 4.18. When the fibermatrix interface shear stress increases, the fibermatrix interface sliding ratio decreases, leading to the decrease of fatigue peak and valley strain, and the fatigue hysteresis area; however, there is an increase of the fatigue hysteresis modulus.

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Figure 4.14 (A) The fatigue hysteresis loops; and (B) the fibermatrix interface slip ratio of 2y(z)/lc versus the applied stress at the loading fatigue peak stress of σmax5 5 320 MPa.

The fatigue hysteresis loops and the fibermatrix interface sliding ratio of 2y(z)/lc for different fibermatrix interface debonded energies of ξd 5 0.1 and 0.2 J/m2 are shown in Fig. 4.19. When the fibermatrix interface debonded energy increases, the fibermatrix interface sliding ratio decreases, leading to the decrease of the fatigue peak and valley strain, and the fatigue hysteresis area; however, there is an increase of fatigue hysteresis modulus.

4.2.3.8 Effect of fibermatrix interface wear on the fatigue hysteresis loops and fibermatrix interface sliding The fatigue hysteresis loops and fibermatrix interface sliding ratio of 2y(z)/lc for different fibermatrix interface shear stresses in the wear region of τ f 5 5 and

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Figure 4.15 (A) The fatigue hysteresis loops; and (B) the fibermatrix interface slip ratio of 2y(z)/lc versus the applied stress at the loading fatigue peak stress of σmax6 5 180 MPa.

10 MPa are shown in Fig. 4.20. When the fibermatrix interface shear stress in the wear region increases, the peak and valley strain decreases, and the fatigue hysteresis area decreases; however, the fatigue hysteresis modulus increases due to the decrease of the fibermatrix interface sliding ratio.

4.2.4 Experimental comparisons The experimental fatigue hysteresis loops and fibermatrix interface sliding of unidirectional and cross-ply C/SiC composites subjected to multiple fatigue loadings are predicted in this section.

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Figure 4.16 (A) The fatigue hysteresis loops; (B) the fibermatrix interface slip ratio of 2y (z)/lc versus the applied stress for different fiber volume fractions.

4.2.4.1 Unidirectional C/SiC composite The fatigue hysteresis loops and the fibermatrix interface sliding of unidirectional C/SiC composites under different loading sequences and applied cycles are analyzed.

First stage fatigue peak stress The fatigue hysteresis loops at the first stage fatigue peak stress of σmax1 5 260 MPa are shown in Fig. 4.21, corresponding to the applied cycle number of N 5 1000, 3000, 5000, 10,000, 30,000, and 50,000. The fatigue hysteresis loops under the first stage fatigue peak stress of σmax1 5 260 MPa correspond to the fibermatrix interface slip Case II with the partially interface debonding and partially relative sliding between the fiber and the matrix. When the applied cycle

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Figure 4.17 (A) The fatigue hysteresis loops; and (B) the fibermatrix interface slip ratio of 2y(z)/lc versus applied stress for different matrix crack spacing.

number increases, the fatigue hysteresis modulus decreases, the fatigue HDE increases, and the fatigue peak and residual strains also increase. The fibermatrix interface sliding ratio of y(z)/ld versus the unloading and reloading stresses are shown in Fig. 4.22, corresponding to the applied cycle number of N 5 1000, 3000, 5000, 10,000, 30,000, and 50,000. With increasing applied cycles, the fibermatrix interface sliding ratio increases from y/ld 5 35% at the applied cycle number of N 5 1000 to y/ld 5 44% at the applied cycle number of N 5 50,000.

Second stage fatigue peak stress The fatigue hysteresis loops at the second stage fatigue peak stress of σmax2 5 280 MPa are shown in Fig. 4.23, corresponding to the applied cycle

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Figure 4.18 (A) The fatigue hysteresis loops; and (B) the fibermatrix interface slip ratio of 2y(z)/lc versus the applied stress for different fibermatrix interface shear stress of τ i 5 15 and 20 MPa.

number of N 5 1000, 5000, 10,000, 30,000, and 50,000. The fatigue hysteresis loops under the second stage fatigue peak stress of σmax2 5 280 MPa correspond to the fibermatrix interface slip Case II with the partially interface debonding and relative sliding. When the applied cycle number increases, the fatigue hysteresis modulus decreases, the fatigue HDE increases, and the fatigue peak and residual strains also increase. The fibermatrix interface sliding ratio of y(z)/ld versus the unloading and reloading stresses are shown in Fig. 4.24, corresponding to the applied cycle number of N 5 1000, 5000, 10,000, 30,000, and 50,000. With increasing of the applied cycles, the fibermatrix interface sliding ratio increases from y(z)/ld 5 45% at the applied cycle number of N 5 1000 to y(z)/ld 5 51% at the applied cycle number of N 5 50,000.

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Figure 4.19 (A) The fatigue hysteresis loops; and (B) the fibermatrix interface slip ratio of 2y(z)/lc versus applied stress for different fibermatrix interface debonded energy of ξ d 5 0.1 and 0.2 J/m2.

Third stage fatigue peak stress The fatigue hysteresis loops at the third stage fatigue peak stress of σmax3 5 300 MPa are shown in Fig. 4.25, corresponding to the applied cycle number of N 5 500, 700, and 1000. The fatigue hysteresis loops under the third stage fatigue peak stress of σmax3 5 300 MPa correspond to the fibermatrix interface slip Case II with partial interface debonding and relative sliding. When the applied cycle number increases, the fatigue hysteresis modulus decreases, the fatigue HDE increases, and the fatigue peak and residual strains also increase. The fibermatrix interface sliding ratio of y(z)/ld versus the unloading and reloading stresses are shown in Fig. 4.26, corresponding to the applied cycle number of N 5 500, 700, and 1000. With increasing of the applied cycles, the

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Figure 4.20 (A) The fatigue hysteresis loops; and (B) the fibermatrix interface slip ratio of 2y(z)/lc versus the applied stress for different fibermatrix interface shear stress in the wear region of τ f 5 5 and 10 MPa.

fibermatrix interface sliding ratio increases from y(z)/ld 5 55% at the applied cycle number of N 5 500 to y(z)/ld 5 59% at the applied cycle number of N 5 1000.

4.2.4.2 Unidirectional SiC/calcium alumina silicate-II composite Holmes and Cho (1992) investigated the multiple loading fatigue behavior of unidirectional SiC/CAS-II composites at room temperature with a loading frequency of 25 Hz. The multiple loading fatigue hysteresis loops and the fibermatrix interface sliding ratio of y(z)/ld versus the unloading and reloading stress curves are shown in Fig. 4.27.

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Figure 4.21 The fatigue hysteresis loops of unidirectional C/SiC composite under the first stage fatigue peak stress of σmax1 5 260 MPa at the applied cycle number of (A) N 5 1000; (B) N 5 3000; (C) N 5 5000; (D) N 5 10,000; (E) N 5 30,000; and (F) N 5 50,000.

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Figure 4.21 (Continued).

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Figure 4.22 The fibermatrix interface sliding ratio of y(z)/ld of unidirectional C/SiC composites under the first stage fatigue peak stress of σmax1 5 260 MPa at the applied cycle number of (A) N 5 1000; (B) N 5 3000; (C) N 5 5000; (D) N 5 10,000; (E) N 5 30,000; and (F) N 5 50,000.

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Figure 4.22 (Continued).

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Interface debonding and sliding of ceramic-matrix composites

Figure 4.23 The fatigue hysteresis loops of unidirectional C/SiC composites under the second stage fatigue peak stress of σmax2 5 280 MPa at the applied cycle number of (A) N 5 1000; (B) N 5 5000; (C) N 5 10,000; (D) N 5 30,000; and (E) N 5 50,000.

231

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Figure 4.23 (Continued).

Under the first stage fatigue peak stress of σmax1 5 160 MPa, the fatigue hysteresis loops correspond to the interface slip Case II with the partial interface debonding and relative sliding, and the fibermatrix interface sliding ratio is y(z)/ ld 5 60%. After experiencing 25,000 cycles and at the second stage fatigue peak stress of σmax2 5 180 MPa, the fatigue hysteresis loops correspond to the interface slip Case IV with complete interface debonding and relative sliding, the fibermatrix interface sliding ratio is y(z)/ld 5 100%, the unloading transition stress for interface complete sliding is σtr_fu 5 18.5 MPa, and the reloading transition stress for interface complete sliding is σtr_fr 5 171.5 MPa.

Figure 4.24 The fibermatrix interface sliding ratio of y(z)/ld of unidirectional C/SiC composites under the second stage fatigue peak stress of σmax2 5 280 MPa at the applied cycle number of (A) N 5 1000; (B) N 5 5000; (C) N 5 10,000; (D) N 5 30,000; and (E) N 5 50,000.

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Figure 4.24 (Continued).

After experiencing 25,000 cycles and at the third stage fatigue peak stress of σmax2 5 200 MPa, the fatigue hysteresis loops correspond to the interface slip Case IV with complete interface debonding and relative sliding, the fibermatrix interface sliding ratio is y(z)/ld 5 100%, the unloading transition stress for the interface complete sliding is σtr_fu 5 86 MPa, and the reloading transition stress for interface complete sliding is σtr_fr 5 124 MPa. After experiencing 25,000 cycles and at the fourth stage fatigue peak stress of σmax4 5 220 MPa, the fatigue hysteresis loops correspond to the interface slip Case IV with complete interface debonding and relative sliding, the fibermatrix interface sliding ratio is y(z)/ld 5 100%, the unloading transition stress for the interface complete sliding is σtr_fu 5 136 MPa, and the reloading transition stress for interface complete sliding is σtr_fr 5 94 MPa.

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Figure 4.25 The fatigue hysteresis loops of unidirectional C/SiC composites under the third stage fatigue peak stress of σmax3 5 300 MPa at the applied cycle number of (A) N 5 500; (B) N 5 700; and (C) N 5 1000.

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Figure 4.26 The fibermatrix interface sliding ratio of y(z)/ld of unidirectional C/SiC composites under the third stage fatigue peak stress of σmax3 5 300 MPa at the applied cycle number of (A) N 5 500; (B) N 5 700; and (C) N 5 1000.

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Figure 4.27 (A) The experimental and predicted fatigue hysteresis loops; and (B) the fibermatrix interface slip ratio of y(z)/ld versus the applied stress of unidirectional SiC/ CAS-II composites. CAS, Calcium alumina silicate.

After experiencing 25,000 cycles and at the fourth stage fatigue peak stress of σmax5 5 240 MPa, the fatigue hysteresis loops correspond to the interface slip Case IV with complete interface debonding and relative sliding, the fibermatrix interface sliding ratio is y(z)/ld 5 100%, the unloading transition stress for the interface complete sliding is σtr_fu 5 136.5 MPa, and the reloading transition stress for the interface complete sliding is σtr_fr 5 113.5 MPa.

4.2.4.3 Cross-ply C/SiC composite The cyclic fatigue hysteresis loops of cross-ply C/SiC composites under multiple fatigue loading of σmax1 5 85 MPa and σmax2 5 90 MPa at room temperature are

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Figure 4.28 The fatigue hysteresis loops at different fatigue peak stresses of (A) σmax1 5 85 MPa; and (B) σmax2 5 90 MPa.

shown in Fig. 4.28. Under the first stage fatigue loading of σmax1 5 85 MPa, the fatigue hysteresis loops correspond to the interface slip Case 2 with partial interface debonding and frictional sliding. Under the second stage fatigue loading of σmax2 5 90 MPa, the fatigue hysteresis loops correspond to the interface slip Case 2 with partial interface debonding and frictional sliding.

4.3

Hysteresis dissipated energy under multiple loading sequences

Under fatigue loading of fiber-reinforced CMCs, when the fatigue peak stress is higher than the FMCS, upon first loading to the fatigue peak stress, matrix cracking

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239

and fibermatrix interface debonding appear. During the subsequent cyclic loading, the fiber sliding is relative to the matrix in the debonded region, leading to fatigue stressstrain hysteresis loops of fiber-reinforced CMCs. The shape, location, and area of the fatigue hysteresis loops reveal the fatigue damage inside the composite (Fantozzi and Reynaud, 2009, 2014). Zawada et al. (1991) investigated the fatigue hysteresis loops shape and area of cross-ply fiber-reinforced CMCs under tensiontension loading at room temperature. Under initial loading, when the fatigue peak stress exceeds the proportional limit stress, the loading stressstrain curve is nonlinear, the unloading stressstrain curve is linear, and the fatigue hysteresis loop is not closed. At cycle number 400,000, the composite fatigue hysteresis modulus and hysteresis loops area decrease obviously. When the applied cycle reaches 1,000,000, the fatigue hysteresis loops area continually decreases; however, the fatigue hysteresis modulus slowly increases compared with the cycle number of 400,000, due to the mixed particles at the matrix cracking plane or fibermatrix interface. Lynch and Evans (1996) investigated the loading/unloading behavior of three different layups cross-ply fiber-reinforced CMCs, namely, [0/90], [ 6 45 degrees], and [ 1 30 degrees/60 degrees]. Upon unloading and reloading, the stressstrain curves appear obvious hysteresis loops. Compared with [0/90] crossply CMCs, the fatigue hysteresis loops shape of [ 6 45 degrees] and [ 1 30 degrees/ 2 60 degrees] composites are very different due to the cracking closure effect. Reynaud (1996) investigated the tensiontension fatigue behavior of 2D SiC/SiC and [0/90]s-SiC/MAS-L composites at elevated temperatures of 600 C, 800 C, and 1000 C in inert atmosphere. For the 2D SiC/SiC composite, the fibermatrix interface radial thermal residual stress is compressive stress, and with increasing temperature the interface radial thermal residual stress decreases, leading to the decrease of the fibermatrix interface shear stress and higher fatigue HDE than that at room temperature. For the [0/90]s cross-ply SiC/MAS-L composite, the fibermatrix interface radial stress is tensile stress, and with increasing temperature the fibermatrix interface radial tensile stress decreases, leading to the increase of the fibermatrix interface shear stress, and higher fatigue HDE than that at room temperature. At elevated temperatures of 800 C1000 C in inert atmosphere, the chemical reaction at the fibermatrix interface leads to the degradation of fibermatrix interface shear stress. After heat treating for 50 hours at elevated temperatures in inert atmosphere, the fibermatrix interface shear stress decreases. Dalmaz et al. (1998) investigated the fatigue hysteresis loops of 2.5D C/SiC composites at room and elevated temperature of 600 C in inert atmosphere. Under tensiontension fatigue loading, the fatigue hysteresis loops area decreases with increasing applied cycles at room and elevated temperature due to the degradation of the interface shear stress between the yarns and the matrix. The fatigue hysteresis loops area at 600 C is less than that at room temperature, due to matrix cracking closure at elevated temperature. The decrease of the fatigue hysteresis loops area confirms the interface wear mechanism. However, at 600 C, the fatigue hysteresis modulus decreases at first, and when the applied cycle approaches a certain number, the hysteresis modulus partially recovers due to matrix cracking closure and the fiber rotation in the yarns (Dalmaz et al., 1996). Longbiao (2014) established the cyclic loadingunloading micromechanical model of unidirectional fiber-reinforced

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CMCs and analyzed the initial loading-unloading-reloading stressstrain relationships and predicted the cyclic loadingunloading stressstrain curve based on the measured matrix cracking evolution curve. Under multiple loading sequences, the fatigue HDE can be used to reflect the interface sliding state inside of CMCs. In this section, the fatigue HDE and fibermatrix interface slip corresponding to single or multiple loading stress levels and different loading sequences are investigated.

4.3.1 Hysteresis theories Under multiple loading sequences, the interface shear stress in the wear and sliding region affects interface debonding and sliding. The interface debonded length at σmax1 is defined as ζ, and the new interface debonded length at σmax2 is defined as ld. The fatigue peak stress for the interface partially debonding and completely sliding is determined using the equation: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# " Vf Ec τ i Vm Em Ef ρ2 11 114 ξ 2 σmin σt 5 2 ρVm Em rf Ec τ 2i d

(4.35)

The fatigue peak stress for the interface completely debonding is determined using the equation: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# "   Vf Ec τ i lc τ f 2ζ Vm Em Ef ρ2 11ρ 2ρ 12 1 114 ξ σb 5 ρVm Em rf τ i rf rf Ec τ 2i d

(4.36)

The fatigue peak stress for the interface completely debonding and sliding can be determined using the equation: σp 5 2

Vf Ec l c Vf Ec ζ τi 2 4 Vm Em r f V m Em r f

 12

 τf τ i 1 σmin τi

(4.37)

Based on the interface debonding and sliding state, the fatigue hysteresis loops of CMCs under multiple loading sequences can be divided into four different cases. 1. Case I, the fibermatrix interface partially debonding and the fiber completely sliding in the interface debonded region. 2. Case II, the fibermatrix interface partially debonding and the fiber partially sliding in the interface debonded region. 3. Case III, the fibermatrix interface completely debonding and the fiber partially sliding in the interface debonded region. 4. Case IV, the fibermatrix interface completely debonding and the fiber completely sliding in the interface debonded region.

The fiber axial stress distributions for Cases I, II, III, and IV upon unloading and reloading are shown in Figs. 4.294.32.

Interface debonding and sliding of ceramic-matrix composites

Figure 4.29 The interface slip Case I for (A) unloading; and (B) reloading.

Figure 4.30 The interface slip Case II for (A) unloading; and (B) reloading.

Figure 4.31 The interface slip Case III for (A) unloading; and (B) reloading.

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Figure 4.32 The interface slip Case IV for (A) unloading; and (B) reloading.

Under cyclic fatigue loading, the stressstrain hysteresis loops area is the energy lost during the corresponding cycle, which is defined using the equation: U5

ð σmax σmin



εc

unloading ðσÞ 2 εc reloading ðσÞ



(4.38)

The fatigue hysteresis loops, fatigue HDE, fibermatrix interface debonding, and sliding of unidirectional SiC/CAS-II composites under the fatigue loading sequence of σmax1 5 100 MPa and σmax2 5 140 MPa are shown in Fig. 4.33. With decreasing fibermatrix interface shear stress in the sliding region, the fatigue hysteresis loops correspond to the interface slip Cases I, II, III, and IV. The fatigue HDE increases to the peak value first and then decreases, the interface debonding changes from partial to complete debonding, and the fibermatrix interface sliding changes from completely sliding to partially sliding and then completely sliding.

4.3.2 Results and discussion Under multiple loading sequences, the effects of fiber volume fraction, matrix cracking, peak stress, and loading sequence on the fatigue damage evolution of SiC/CAS-II composites are analyzed in this section.

4.3.2.1 Effect of fiber-volume fraction on multiple loading fibermatrix interface sliding The fatigue HDE, hysteresis modulus, fibermatrix interface debonding, and sliding ratio versus the fibermatrix interface shear stress curves under multiple loading sequence of σmax1 5 100 MPa and σmax2 5 180 MPa for different fiber volume fractions are shown in Fig. 4.34. When the fiber volume fraction increases, the fatigue HDE decreases with the fibermatrix interface partially debonding, and

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Figure 4.33 (A) The fatigue hysteresis loops for different interface slip cases; (B) the fatigue HDE versus the fibermatrix interface shear stress curve; (C) the fibermatrix interface debonding ratio of 2ld/lc versus the fibermatrix interface shear stress curve; and (D) the fibermatrix interface sliding ratio of y/ld versus the fibermatrix interface shear stress curve under multiple loading sequence of σmax1 5 100 MPa and σmax2 5 140 MPa. HDE, Hysteresis dissipated energy.

when the fibermatrix interface completely debonds, the fatigue HDE approaches to the same value with high or low fiber volume fraction. The fatigue hysteresis modulus increases at the same fibermatrix interface shear stress; the fibermatrix interface debonding ratio decreases when the interface is partially debonding, the fibermatrix interface shear stress for the interface completely debonding decreases. The fibermatrix interface sliding ratio increases when the interface is partially debonding, and decreases when the interface completely debonds.

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Figure 4.33 (Continued).

4.3.2.2 Effect of matrix crack spacing on multiple loading fibermatrix interface sliding The fatigue HDE, fatigue hysteresis modulus, fibermatrix interface debonding, and sliding ratio versus the fibermatrix interface shear stress curves under multiple loading sequence of σmax1 5 100 MPa and σmax2 5 180 MPa for different matrix crack spacing are shown in Fig. 4.35. When the matrix crack spacing increases, the fatigue HDE decreases with the fibermatrix interface partially debonding and increases when the fibermatrix interface completely debonds, the peak fatigue HDE value increases, and the fatigue HDE versus the fibermatrix interface shear stress curve moves to the left side. The fatigue hysteresis modulus increases at the same fibermatrix interface shear stress. The fibermatrix interface debonding ratio decreases when the interface is partially debonding, and the fibermatrix

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Figure 4.34 (A) The fatigue HDE versus the fibermatrix interface shear stress curve; (B) the fatigue hysteresis modulus versus the fibermatrix interface shear stress curve; (C) the fibermatrix interface debonded ratio versus the fibermatrix interface shear stress curve; and (D) the fibermatrix interface sliding ratio versus the interface shear stress curve under multiple loading sequence of σmax1 5 100 MPa and σmax2 5 180 MPa. HDE, Hysteresis dissipated energy.

interface shear stress corresponding to complete debonding decreases. The interface sliding ratio is the same with high or low matrix crack spacing; however, when the interface completely debonds, the interface sliding ratio decreases.

4.3.2.3 Effect of low-peak stress level on multiple loading fibermatrix interface sliding The fatigue HDE, fatigue hysteresis modulus, fibermatrix interface debonding, and sliding ratio versus the fibermatrix interface shear stress curves

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Figure 4.34 (Continued).

under multiple loading sequence of σmax2 5 180 MPa for different low-fatigue peak stresses of σmax1 5 100 and 140 MPa are shown in Fig. 4.36. When the low-fatigue peak stress increases, the fatigue HDE increases with the fibermatrix interface partially debonding, and decreases when the fibermatrix interface completely debonds, the peak value of fatigue HDE decreases, and the fatigue HDE versus the fibermatrix interface shear stress curve moves to the right side. The fatigue hysteresis modulus decreases at the same fibermatrix interface shear stress. The fibermatrix interface debonding ratio increases when the interface is partially debonding and the fibermatrix interface shear stress corresponding to the interface completely debonding increases. The interface sliding ratio increases when the interface partially or completely debonds.

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Figure 4.35 (A) The fatigue HDE versus the fibermatrix interface shear stress curve; (B) the fatigue hysteresis modulus versus the fibermatrix interface shear stress curve; (C) the fibermatrix interface debonded ratio versus the interface shear stress curve; and (D) the fibermatrix interface sliding ratio versus the interface shear stress curve under multiple loading sequence of σmax1 5 100 MPa and σmax2 5 180 MPa.

4.3.2.4 Effect of high-peak stress level on multiple loading fibermatrix interface sliding The fatigue HDE, fatigue hysteresis modulus, fibermatrix interface debonding, and sliding ratio versus the fibermatrix interface shear stress curves under multiple loading sequence of σmax1 5 120 MPa for different high-fatigue peak stresses of σmax2 5 140 and 180 MPa are shown in Fig. 4.37. When the high-fatigue peak stress increases, the fatigue HDE increases with the fibermatrix interface partially or completely debonding. However, when the fatigue hysteresis loops correspond to the interface slip Case IV, the fatigue HDE for low- or high-peak stress approaches

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Figure 4.35 (Continued).

the same value, and the peak fatigue HDE value increases. The fatigue hysteresis modulus decreases at the same interface shear stress. The fibermatrix interface debonding ratio decreases with the fibermatrix interface partially debonding, and the fibermatrix interface shear stress for the interface completely debonding increases. The fibermatrix interface sliding decreases when the interface is partially debonding and increases when the interface completely debonds.

4.3.2.5 Effect of fatigue stress range on multiple loading fibermatrix interface sliding The fatigue HDE, fatigue hysteresis modulus, fibermatrix interface debonding, and sliding ratio versus the fibermatrix interface shear stress curves under multiple loading sequence of σmax1 5 100 MPa and σmax2 5 180 MPa for different

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Figure 4.36 (A) The fatigue HDE versus the fibermatrix interface shear stress curve; (B) the fatigue hysteresis modulus versus the fibermatrix interface shear stress curve; (C) the fibermatrix interface debonded ratio versus the fibermatrix interface shear stress curve; and (D) the fibermatrix interface sliding ratio versus the interface shear stress curve under multiple loading sequence of σmax2 5 180 MPa for different low-fatigue peak stresses of σmax1 5 100 and 140 MPa. HDE, Hysteresis dissipated energy.

fatigue stress ranges are shown in Fig. 4.38. When the fatigue stress increases, the fatigue HDE increases with the fibermatrix interface partially or completely debonding. The fatigue hysteresis modulus decreases at the same fibermatrix interface shear stress. The fibermatrix interface debonding ratio depends on the fatigue peak stress and is independent on the fatigue stress range. The fibermatrix interface sliding ratio increases with the fibermatrix interface partially or completely debonding, and the fibermatrix interface shear stress for complete sliding increases.

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Figure 4.36 (Continued)

4.3.2.6 Comparisons between single and multiple loading stress levels on fibermatrix interface sliding The fatigue HDE, fibermatrix interface debonding, and sliding ratio versus the fibermatrix interface shear stress curves under single fatigue peak stresses of σmax 5 180 and 240 MPa and multiple fatigue peak stresses of σmax1 5 180 MPa and σmax2 5 240 MPa are shown in Fig. 4.39. Under multiple fatigue peak stresses of σmax1 5 180 MPa and σmax2 5 240 MPa, the peak fatigue HDE value is the highest, and the fatigue HDE is the same with that under single fatigue peak stress of σmax 5 240 MPa when the fibermatrix interface is partially debonding. The fibermatrix interface debonding ratio is the highest at the same fibermatrix interface shear stress, and the fibermatrix interface shear stress for the interface completely debonding is also the highest compared with single fatigue peak stresses of σmax 5 180 and 240 MPa. The interface sliding ratio is higher compared to that

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Figure 4.37 (A) The fatigue HDE versus the fibermatrix interface shear stress curve; (B) the fatigue hysteresis modulus versus the fibermatrix interface shear stress curve; (C) the fibermatrix interface debonded ratio versus the fibermatrix interface shear stress curve; and (D) the fibermatrix interface sliding ratio versus the fibermatrix interface shear stress curve under a multiple loading sequence of σmax1 5 120 MPa for different high-peak stresses of σmax2 5 140 and 180 MPa. HDE, Hysteresis dissipated energy.

at the same fatigue peak stress of σmax 5 240 MPa, lower than that at the same fatigue peak stress of σmax 5 180 MPa, and the interface shear stress corresponding to complete sliding is the highest.

4.3.2.7 Comparisons between different loading sequences on fibermatrix interface sliding The fatigue HDE, fatigue hysteresis modulus, fibermatrix interface debonding, and sliding ratio versus the fibermatrix interface shear stress curves for different loading sequences are shown in Fig. 4.40. Compared with the lowhigh peak stress loading

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Figure 4.37 (Continued)

sequence (i.e., σmax1 , σmax2), the fatigue HDE decreases when the fibermatrix interface is partially or completely debonding under the highlow peak stress loading sequence (i.e., σmax1 . σmax2). The fibermatrix interface debonding ratio increases when the fibermatrix interface is partially debonding and the fibermatrix interface shear stress corresponding to the fibermatrix interface completely debonding increases under highlow peak stress loading sequence (i.e., σmax1 . σmax2). The fibermatrix interface sliding ratio decreases when the fibermatrix interface is partially or completely debonding and the interface shear stress corresponding to complete sliding decreases under highlow peak stress loading sequence (i.e., σmax1 . σmax2).

4.3.3 Experimental comparisons For different fiber preforms in CMCs, an effective coefficient of the fiber volume content along the loading direction (ECFL) is defined using the equation:

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Figure 4.38 (A) The fatigue HDE versus the fibermatrix interface shear stress curve; (B) the fatigue hysteresis modulus versus the fibermatrix interface shear stress curve; (C) the fibermatrix interface debonded ratio versus the fibermatrix interface shear stress curve; and (D) the fibermatrix interface sliding ratio versus the interface shear stress curve under multiple loading sequence of σmax1 5 100 MPa and σmax2 5 180 MPa for different stress ranges. HDE, Hysteresis dissipated energy.

φ5

Vf

axial

Vf

(4.39)

where Vf and Vf_axial denote the total fiber volume fraction in the composite and the effective fiber volume fraction in the cyclic loading direction. For 2D, 2.5D, 3D, and needled CMCs, the values of φ are 0.5, 0.75, 0.93, and 0.375, respectively. The fatigue hysteresis loops and the fibermatrix interface sliding of C/SiC and SiC/ SiC composites with different fiber preforms under multiple loading sequences are predicted in this section.

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Figure 4.38 (Continued).

4.3.3.1 C/SiC composite Mei and Cheng (2009) investigated the cyclic fatigue behavior of needled, 2D, 2.5D, and 3D C/SiC composites under multiple fatigue loading sequences. For the needled C/SiC composite, the fatigue hysteresis loops and the fibermatrix interface sliding ratio versus the unloading and reloading stress curves under the fatigue peak stresses of σmax1 5 95 MPa, σmax2 5 125 MPa, and σmax3 5 155 MPa are shown in Fig. 4.41. The fibermatrix interface sliding ratio increases from y(z)/ld 5 65% at a loading fatigue peak stress of σmax1 5 95 MPa to y(z)/ld 5 86% at a loading fatigue peak stress of σmax3 5 155 MPa, as shown in Fig. 4.42. For the 2D C/SiC composite, the fatigue hysteresis loops and the fibermatrix interface sliding ratio versus the unloading and reloading applied

Figure 4.39 (A) The fatigue HDE versus the fibermatrix interface shear stress curve; (B) the fibermatrix interface debonded ratio versus the fibermatrix interface shear stress curve; and (C) the fibermatrix interface sliding ratio versus the fibermatrix interface shear stress curve under single fatigue peak stresses of σmax 5 180 and 240 MPa and multiple fatigue peak stresses of σmax1 5 180 MPa and σmax2 5 240 MPa. HDE, Hysteresis dissipated energy.

Figure 4.40 (A) The fatigue HDE versus the fibermatrix interface shear stress curve; (B) the fibermatrix interface debonded ratio versus the fibermatrix interface shear stress curve; and (C) the fibermatrix interface sliding ratio versus the interface shear stress curve for different loading sequences. HDE, Hysteresis dissipated energy.

Figure 4.41 The fatigue hysteresis loops of needled C/SiC composite under different loading fatigue peak stresses of (A) σmax1 5 95 MPa; (B) σmax2 5 125 MPa; and (C) σmax3 5 155 MPa.

Figure 4.42 The fibermatrix interface sliding ratio of needled C/SiC composite under different fatigue peak stresses of (A) σmax1 5 95 MPa; (B) σmax2 5 125 MPa; and (C) σmax3 5 155 MPa.

Figure 4.43 The fatigue hysteresis loops of 2D C/SiC composite under different fatigue peak stresses of (A) σmax1 5 180 MPa; (B) σmax2 5 200 MPa; and (C) σmax3 5 220 MPa.

Figure 4.44 The fibermatrix interface sliding ratio of 2D C/SiC composite under different fatigue peak stresses of (A) σmax1 5 180 MPa; (B) σmax2 5 200 MPa; and (C) σmax3 5 220 MPa.

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stress curves under multiple fatigue peak stresses of σmax1 5 180 MPa, σmax2 5 200 MPa, and σmax3 5 220 MPa are shown in Fig. 4.43. The fatigue hysteresis loops correspond to the interface slip Case IV, and the fibermatrix interface sliding ratio is y(z)/ld 5 1 for different fatigue peak stresses, as shown in Fig. 4.44. For the 2.5D C/SiC composite, the fatigue hysteresis loops and the fibermatrix interface sliding ratio versus the unloading and reloading applied stress curves under multiple fatigue peak stresses of σmax1 5 95 MPa and σmax2 5 150 MPa are shown in Fig. 4.45. The fatigue hysteresis loops correspond to the interface slip Case II, and the fibermatrix interface sliding ratio is y(z)/ld 5 88% and

Figure 4.45 The fatigue hysteresis loops of 2.5D C/SiC composite under different fatigue peak stresses of (A) σmax1 5 95 MPa; and (B) σmax2 5 150 MPa.

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Figure 4.46 The fibermatrix interface sliding ratio of 2.5D C/SiC composite under different fatigue peak stresses of (A) σmax1 5 95 MPa; and (B) σmax2 5 150 MPa.

y(z)/ld 5 70% for the fatigue peak stress of σmax1 5 95 MPa and σmax2 5 150 MPa, respectively, as shown in Fig. 4.46. For the 3D C/SiC composite, the fatigue hysteresis loops and the fibermatrix interface sliding ratio versus the unloading and reloading applied stress curves under the multiple fatigue peak stresses of σmax1 5 265 MPa, σmax2 5 285 MPa, and σmax3 5 300 MPa are shown in Fig. 4.47. The fatigue hysteresis loops correspond to the interface slip Case II, and the fibermatrix interface sliding ratio increases from y(z)/ld 5 63% at the loading fatigue peak stress of σmax1 5 265 MPa to y(z)/ld 5 64% at the loading fatigue peak stress of σmax3 5 300 MPa, as shown in Fig. 4.48.

Interface debonding and sliding of ceramic-matrix composites

Figure 4.47 The fatigue hysteresis loops of 3D C/SiC composite under different fatigue peak stresses of (A) σmax1 5 265 MPa; (B) σmax2 5 285 MPa; and (C) σmax3 5 300 MPa.

263

Figure 4.48 The fibermatrix interface sliding ratio of 3D C/SiC composite under different fatigue peak stresses of (A) σmax1 5 265 MPa; (B) σmax2 5 285 MPa; and (C) σmax3 5 300 MPa.

Interface debonding and sliding of ceramic-matrix composites

Figure 4.49 The fatigue hysteresis loops of 2D SiC/SiC composite under different fatigue peak stresses of (A) σmax1 5 180 MPa; (B) σmax2 5 200 MPa; and (C) σmax3 5 220 MPa.

265

Figure 4.50 The fibermatrix interface sliding ratio of 2D SiC/SiC composite under different fatigue peak stresses of (A) σmax1 5 180 MPa; (B) σmax2 5 200 MPa; and (C) σmax3 5 220 MPa.

Interface debonding and sliding of ceramic-matrix composites

267

4.3.3.2 SiC/SiC composite McNulty and Zok (1999) investigated the cyclic fatigue behavior of 2D SiC/SiC composites under multiple fatigue loading sequences. The fatigue hysteresis loops under multiple fatigue peak stresses of σmax1 5 180 MPa, σmax2 5 200 MPa, and σmax3 5 220 MPa are shown in Fig. 4.49. The fatigue hysteresis loops correspond to the interface slip Case II, and the fibermatrix interface sliding ratio are y(z)/ ld 5 87%, 83%, and 80% corresponding to the different fatigue peak stresses of σmax1 5 180 MPa, σmax2 5 200 MPa, and σmax3 5 220 MPa, respectively, as shown in Fig. 4.50.

Figure 4.51 The fatigue hysteresis loops of 2D SiC/SiC composite under different fatigue peak stresses of (A) σmax1 5 175 MPa; (B) σmax2 5 190 MPa; (C) σmax3 5 205 MPa; and (D) σmax4 5 215 MPa.

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Figure 4.51 (Continued).

The fatigue hysteresis loops under different fatigue peak stresses of σmax1 5 175 MPa, σmax2 5 190 MPa, σmax3 5 205 MPa, and σmax4 5 215 MPa are shown in Fig. 4.51. The fibermatrix interface sliding ratio are y(z)/ld 5 82%, 80%, 76%, and 75% for the fatigue peak stresses of σmax1 5 175 MPa, σmax2 5 190 MPa, σmax3 5 205 MPa, and σmax4 5 215 MPa, respectively, as shown in Fig. 4.52.

4.4

Conclusion

The fibermatrix interface debonding and sliding of fiber-reinforced CMCs have been investigated in this chapter by considering interface wear mechanisms and

Interface debonding and sliding of ceramic-matrix composites

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Figure 4.52 The fibermatrix interface sliding ratio of 2D SiC/SiC composite under different fatigue peak stresses of (A) σmax1 5 175 MPa; (B) σmax2 5 190 MPa; (C) σmax3 5 205 MPa; and (D) σmax4 5 215 MPa.

different loading sequences. The fracture mechanics approach was adopted to determine the fibermatrix interface debonding and sliding length. The theoretical relationships between the fibermatrix interface debonding and sliding, loading sequences, peak stress level, applied cycle number, material properties, and damage states have been established. The evolution of fatigue hysteresis loops, fatigue hysteresis area, and hysteresis modulus versus the fibermatrix interface shear stress have been analyzed, which related with the fibermatrix interface debonding and sliding damage states. The experimental fibermatrix interface debonding and sliding and the fatigue hysteresis loops of CMCs with different fiber preforms have been predicted for different loading sequences.

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Figure 4.52 (Continued).

References Budiansky, B., Hutchinson, J.W., Evans, A.G., 1986. Matrix fracture in fiber-reinforced ceramics. J. Mech. Phys. Solids 34, 167189. Cho, C.D., Holmes, J.W., Barber, J.R., 1991. Estimation of interfacial shear in ceramic composites from frictional heating measurements. J. Am. Ceram. Soc. 74, 28022808. Curtin, W.A., 1991. Theory of mechanical properties of ceramic matrix composites. J. Am. Ceram. Soc. 74, 28372845. Dalmaz, A., Reynaud, P., Rouby, D., Fantozzi, G., 1996. Damage propagation in carbon/silicon carbide composites during tensile tests under SEM. J. Mater. Sci. 31, 42134219. Dalmaz, A., Reynaud, P., Rouby, D., Fantozzi, G., Abbe, F., 1998. Mechanical behavior and damage development during cyclic fatigue at high-temperature of a 2.5D carbon/SiC composite. Compos. Sci. Technol. 58, 693699.

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Du, S., Qiao, S., Ji, G., Han, D., 2002. Tension-tension fatigue behaviour of 3D-C/SiC composite at room temperature and 1300 C. J. Mater. Eng. 9, 2225. Evans, A.G., Zok, F.W., McMeeking, R.M., 1995. Fatigue of ceramic matrix composites. Acta Metall. Mater. 43, 859875. Fang, G., Gao, X., Song, Y., 2016. Tension-compression fatigue behaviour and failure mechanisms of needled C/SiC composite. J. Mater. Eng. 44, 7882. Fantozzi, G., Reynaud, P., 2009. Mechanical hysteresis in ceramic matrix composites. Mater. Sci. Eng. A 521522, 1823. Fantozzi, G., Reynaud, P., 2014. Mechanical behavior of SiC fiber-reinforced ceramic matrix composites. Compr. Hard Mater. 2, 345366. Gao, Y.C., Mai, Y., Cotterell, B., 1988. Fracture of fiber-reinforced materials. J. Appl. Math. Phys. 39, 550572. Guo, H., Jia, P., Wang, B., Jiao, G., Zeng, Z., 2015. Study on constituent properties of a 2DSiC/SiC composite by hysteresis measurements. Chin. J. Theor. Appl. Mech. 47, 260269. Han, D., Qiao, S.R., Li, M., Hou, J.T., Wu, X.J., 2004. Comparison of fatigue and creep behavior between 2D and 3D-C/SiC composites. Acta Metall. Sin. (Engl. Lett.) 17, 569574. Holmes, J.W., Cho, C.D., 1992. Experimental observation of frictional heating in fiberreinforced ceramics. J. Am. Ceram. Soc. 75, 929938. Hutchinson, J.W., Jensen, H.M., 1990. Models of fiber debonding and pullout in brittle composites with friction. Mech. Mater. 9, 139163. Kostopoulos, V., Vellios, L., Pappas, Y.Z., 1997. Fatigue behavior of 3-d SiC/SiC composites. J. Mater. Sci. 32, 215220. Longbiao, L., 2013a. Fatigue hysteresis behavior of cross-ply C/SiC ceramic matrix composites at room and elevated temperatures. Mater. Sci. Eng. A 586, 160170. Longbiao, L., 2013b. Modeling hysteresis behavior of cross-ply C/SiC ceramic matrix composites. Compos., B 53, 3645. Longbiao, L., 2014. Modeling fatigue hysteresis behavior of unidirectional C/SiC ceramicmatrix composite. Compos., B 66, 466474. Longbiao, L., 2015a. Synergistic effect of arbitrary loading sequence and interface wear on the fatigue hysteresis loops of carbon fiber-reinforced ceramic-matrix composites. Eng. Fract. Mech. 146, 6788. Longbiao, L., 2015b. Modeling the effect of interface wear on fatigue hysteresis behavior of carbon fiber-reinforced ceramic-matrix composites. Appl. Compos. Mater. 22, 887920. Longbiao, L., 2016a. Synergistic effect of interface wear and loading sequence on interface debonding and relative slipping of fiber-reinforced ceramic-matrix composites. Theor. Appl. Fract. Mech. 86, 171186. Longbiao, L., 2016b. Interfacial debonding and slipping of carbon fiber-reinforced ceramicmatrix composites subjected to different fatigue loading sequences. J. Aerosp. Eng. 29, 04016029-1. Longbiao, L., 2017a. Modeling of fatigue hysteresis behaviour in carbon fiber-reinforced ceramic-matrix composites under multiple loading stress levels. J. Compos. Mater. 51, 971983. Longbiao, L., 2017b. Interfacial debonding and slipping of carbon fiber-reinforced ceramicmatrix composites under two-stage cyclic loading. Compos. Interfaces 24, 417445. Longbiao, L., 2018. A hysteresis energy dissipation based model for multiple loading damage in continuous fiber-reinforced ceramic-matrix composites. Compos., B . Available from: https://doi.org/10.1016/j.compositesb.2018.11.012.

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Longbiao, L., Song, Y.D., 2010. An approach to estimate interface shear stress of ceramic matrix composites from hysteresis loops. Appl. Compos. Mater. 17, 309328. Longbiao, L., Song, Y.D., 2011. Influnce of fiber failure on fatigue hysteresis loops of ceramic matrix composites. J. Reinforced Plast. Compos. 30, 1225. Longbiao, L., Song, Y.D., Sun, Y.C., 2013. Estimate interface shear stress of unidirectional C/SiC ceramic matrix composites from hysteresis loops. Appl. Compos. Mater. 20, 693707. Longbiao, L., Song, Y.D., Sun, Z.G., 2009a. Influence of interface de-bonding on the fatigue hysteresis loops of ceramic matrix composites. Chin. J. Solids Mech. 30, 814. Longbiao, L., Song, Y.D., Sun, Z.G., 2009b. Effect of fiber Poisson contraction on fatigue hysteresis loops of ceramic matrix composites. J. Nanjing Univ. Aeronautics Astronautics 41, 181186. Lynch, C.S., Evans, A.G., 1996. Effects of off-axis loading on the tensile behavior of a ceramic matrix composite. J. Am. Ceram. Soc. 79, 31133123. McNulty, J.C., Zok, F.W., 1999. Low-cycle fatigue of Nicalon-fiber-reinforced ceramic composites. Compos. Sci. Technol. 59, 15971607. Mei, H., Cheng, L., 2009. Comparison of the mechanical hysteresis of carbon/ceramic-matrix composites with different fiber preforms. Carbon 47, 10341042. Pryce, A.W., Smith, P.A., 1993. Matrix cracking in unidirectional ceramic matrix composites under quasi-static and cyclic loading. Acta Metall. Mater. 41, 12691281. Reynaud, P., 1996. Cyclic fatigue of ceramic-matrix composites at ambient and elevated temperatures. Compos. Sci. Technol. 56, 809814. Shi, Z.F., Zhou, L.M., 2002. Interfacial damage in fiber-reinforced composites subjected to tension fatigue loading. Fatigue Fract. Eng. Mater. Struct. 25, 445457. Solti, J.P., Mall, S., Robertson, D.D., 1995. Modeling damage in unidirectional ceramicmatrix composites. Compos. Sci. Technol. 54, 5566. Sørensen, B.F., Talreja, R., Sorensen, O.T., 1993. Micromechanical analysis of damage mechanisms in ceramic matrix composites during mechanical and thermal loading. Composites 24, 124140. Vagaggini, E., Domergue, J.M., Evans, A.G., 1995. Relationships between hysteresis measurements and the constituent properties of ceramic matrix composites: I, Theory. J. Am. Ceram. Soc. 78, 27092720. Xia, Z., Curtin, W.A., 2000. Tough-to-brittle transitions in ceramic-matrix composites with increasing interfacial shear stress. Acta Meter. 48, 48794892. Zawada, L.P., Butkus, L.M., Hartman, G.A., 1991. Tensile and fatigue behavior of silicon carbide fiber-reinforced aluminosilicate glass. J. Am. Ceram. Soc. 74, 28512858.

Further reading Sun, Y.J., Singh, R.N., 1998. The generation of multiple matrix cracking and fiber-matrix interfacial debonding in a glass composite. Acta Mater. 46, 16571667.

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

5.1

5

Introduction

When ceramic-matrix composites (CMCs) are subjected to cyclic fatigue loading, there are multiple fatigue damage mechanisms, including matrix cracking, and fibermatrix interface debonding and sliding, leading to the opening and closure of matrix cracks during each cycle (Evans et al., 1995; Rouby and Reynaud, 1993). Upon unloading and reloading, the stressstrain hysteresis loops appear due to the damage mechanisms of repeated sliding at the debonded interface (Fantozzi and Reynaud, 2009; Longbiao, 2011, 2013a,b, 2014, 2015a,b, 2016a,b,c,d,e, 2017a,b). The evolution of fatigue hysteresis loops depends on the material properties, peak stress, and internal damage. For unidirectional SiC/calcium alumina silicate (CAS)II composites at room temperature (Holmes and Cho 1992), with increasing applied cycles, the hysteresis modulus first decreases and then approaches the constant value before decreasing rapidly at fatigue fracture. The hysteresis area first increases and then decreases, and increases rapidly at fatigue fracture. For cross-ply C/SiC composites at room temperature and 800 C (Longbiao, 2013a,b), the fatigue hysteresis modulus and area decrease with applied cycles. For 2D SiC/SiC composites at 600 C, 800 C, and 1000 C (Reynaud, 1996), the hysteresis area increases with applied cycles and testing temperature. For the 2.5D SiC/[Si 2 B 2 C] at 1200 C (Fantozzi and Reynaud, 2009), the hysteresis area decreases with applied cycles. The degradation of fatigue hysteresis modulus or hysteresis area reflect the internal matrix cracking and interface sliding damage. In the theoretical research area, Kotil et al. (1990) investigated the fatigue hysteresis loops of fiber-reinforced CMCs and found that the fibermatrix interface shear stress affects the fatigue hysteresis loops shape and area. When the fibermatrix interface shear stress is too low or too high, the hysteresis loops are negligible. When the fibermatrix interface shear stress is too low, the fiber can freely slip in the matrix, and the energy dissipation caused by sliding is small. However, when the fibermatrix interface shear stress is too high, the fibermatrix interface debonded length is small, and the energy dissipation caused by sliding is also small. When the fibermatrix interface shear stress is moderate, the fatigue hysteresis dissipated energy (HDE) can approach to the maximum. Cho et al. (1991) divided the fibermatrix interface sliding into partial and complete slipping and obtained the hysteresis energy dissipated rate for these two conditions. Pryce and Smith (1993) analyzed the fatigue hysteresis loops when the fibermatrix Durability of Ceramic-Matrix Composites. DOI: https://doi.org/10.1016/B978-0-08-103021-9.00005-8 © 2020 Elsevier Ltd. All rights reserved.

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interface is partially debonding by assuming the constant frictional interface shear stress. Ahn and Curtin (1997) divided matrix cracking into long-crack, mediumcrack, and short-crack spacing, analyzed the matrix stochastic cracking on the fatigue hysteresis loops, and compared the results with the PS model (Pryce and Smith, 1993). Solti et al. (2000) adopted the maximum interface shear stress criterion to determine the fibermatrix interface debonded length, unloading interface counter slip length, and reloading new-slip length, and predicted the fatigue hysteresis loops of fiber-reinforced CMCs when the fibermatrix interface is chemically bonded. Keith and Kedward (1995) investigated the cyclic loadingunloading hysteresis loops models for the fibermatrix interface debonding based on the PS model (Pryce and Smith, 1993). Longbiao et al. (2008) investigated the effect of fibers Poisson contraction on the fatigue hysteresis loops of unidirectional fiberreinforced CMC based on the Coulomb’s frictional law. The unloading and reloading fiber and matrix axial stress and the fibermatrix interface shear stress are solved using the Lame equation. The fracture mechanics approach is adopted to determine the fibermatrix interface debonded length, unloading interface counter slip length, and reloading interface new slip length. The unloading and reloading stressstrain relationship are solved to predict the hysteresis loops. The effects of fibermatrix interface debonded energy and fibermatrix interface frictional coefficient on the fibermatrix interface debonding upon initial loading, the interface counter slip upon unloading and new slip upon reloading are discussed in the next section. The predicted results are compared with experimental data and the PS model (Pryce and Smith, 1993). Sørensen et al. (1993) investigated the cyclic loadingunloading stressstrain hysteresis loops of unidirectional fiber-reinforced CMCs using axisymmetric FEM. The Coulomb’s law is used to describe the fibermatrix interface shear stress. The change of the fibermatrix interface shear stress along the interface debonded length is analyzed. Upon loading, the fibermatrix interface shear stress decreases due to Poisson contraction, which decreases the elastic modulus. Upon unloading the fibermatrix interface shear stress recovers, leading to the increase of the elastic modulus. Vagaggini et al. (1995) developed the fatigue hysteresis loops models based on the HutchinsonJensen fiber pullout model (Hutchinson and Jensen, 1990) when the fibermatrix interface is chemically bonded. It was found that the fibermatrix interface debonded energy affects the initial fibermatrix interface debonding and relative sliding under fatigue loading. When the fibermatrix interface debonded energy is low, the unloading interface counter slip and reloading interface new slip are independent of interface debonding. However, when the fibermatrix interface debonded energy is high, the unloading interface counter slip and reloading interface new slip stops at the interface debonding tip, and upon continually unloading or reloading, the slip range remains the same. Hild et al. (1996) developed the fatigue hysteresis loops model of the fiber-reinforced CMCs using continuum damage mechanics. The model consists of four state parameters, one experimental observation parameter, and three internal variables. The three internal variables include matrix cracking, interface debonding, and crack-opening strain. By determining the damage evolution criterion of these parameters using the

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

275

loadingunloading testing, it is found that the residual stress field has a greater effect on the hysteresis loop width and the inelastic strain evolution, but a smaller effect on the initial unloading elastic modulus. Mei and Cheng (2009) investigated the cyclic loadingunloading hysteresis behavior of 2D, 2.5D, and 3D C/SiC composites, and related the hysteresis loops shape with the fiber volume fraction along the loading direction. Longbiao et al. (2014) established the cyclic loadingunloading micromechanical model of unidirectional fiber-reinforced CMCs, and analyzed the initial loadingunloadingreloading stressstrain relationship and predicted the cyclic loadingunloading stressstrain curve based on the measured matrix cracking evolution curve. Guo et al. (2015) investigated the tensile and shear stressstrain behavior and microstructure characteristic and obtained the material constituent properties based on the hysteresis analysis. Under cyclic shear loading, the stressstrain curve shows obvious hysteresis behavior, and with increasing unloading peak stress, the hysteresis width and residual strain increase, as well as the damage inside of the composite, also increase. Xu (2008) established the fatigue hysteresis loops model of cross-ply fiber-reinforced CMCs considering the effect of transverse and longitudinal matrix cracking, fibermatrix interface debonding. Yang (2011) established the fatigue hysteresis loops model of 2.5D C/SiC composites considering the effect of microstructure damage on macroproperties based on the modulus reduction method and predicted the fatigue hysteresis loops for different applied cycles. Fang et al. (2016) investigated the tensioncompressive fatigue behavior and failure mechanisms of needled C/SiC composites. With increasing cycles, the hysteresis modulus decreases, the residual strain, and hysteresis loops area increase. Longbiao (2016a,b,c,d,e, 2017a,b,c, 2018) developed a fatigue hysteresis loops model of fiber-reinforced CMCs considering matrix multicracking, fibermatrix interface debonding/wear/oxidation, and fibers fracture, predicted the cyclic loadingunloading hysteresis loops for different peak stress and fatigue hysteresis loops for different cycle numbers, investigated the effect of multiple-step loading, thermomechanical loading on the fatigue hysteresis loops, proposed an approach to predict the fibermatrix interface shear stress of fiber-reinforced CMCs through the fatigue hysteresis loops, estimated the fibermatrix interface shear stress of unidirectional, cross-ply, and 2D CMCs for different peak stresses and applied cycles, established the relationship between the fatigue HDE with surface temperature rising, and developed a HDE-based damage parameter, to monitor the damage evolution inside of fiber-reinforced CMCs. Under cyclic fatigue loading, the fatigue hysteresis area or HDE can be used an effect tool to monitor the internal damage evolution of CMCs. Upon unloading and reloading, the sliding between the fiber and the matrix in the debonded region is the main reason for the hysteresis loops. The theoretical relationships between the hysteresis loops, HDE, HDE-based damage parameters, and interface debonding or sliding are established. The effects of fiber volume content, fatigue peak stress, fatigue stress ratio, and matrix crack spacing on the HDE and HDE-based damage parameter versus applied cycles are discussed in the next section. The experimental fatigue HDE and HDE-based damage parameters of unidirectional, cross-ply, 2D, 2.5D, and 3D CMCs are then predicted.

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5.2

Durability of Ceramic-Matrix Composites

Hysteresis-based damage parameters

When the fatigue peak stress is higher than the initial matrix cracking stress, hysteresis loops appear upon unloading and reloading due to the sliding fatigue damage mechanisms of CMCs. The debonding and sliding range between the fiber and the matrix in the debonded region affect the hysteresis shape, hysteresis location, and hysteresis area. Marshall and Evans (1985) found the fatigue stressstrain hysteresis loops of unidirectional fiber-reinforced CMCs and attributed the hysteresis loops to the frictional slip between the fiber and the matrix at the interface. Marshall and Oliver (1987) conducted the fiber push-in/push-out tests, and also found the stressstrain hysteresis loops, and confirmed certain interface sliding mechanisms. Minford and Prewo (1986) performed the tensiontension fatigue experiments of unidirectional fiber-reinforced CMCs at room temperature and found obvious fatigue hysteresis loops. Holmes et al. (1989) found fatigue hysteresis loops and strain ratcheting of unidirectional fiber-reinforced CMCs at elevated temperatures, and found that the fatigue peak stress of hysteresis loops is higher than that of first matrix-cracking stress (FMCS). Kotil et al. (1990) investigated the fatigue hysteresis mechanism of unidirectional fiber-reinforced CMCs and attributed to the fatigue hysteresis loops to the sliding process of pull-out and push-in of fractured fibers. Holmes and Cho (1992) investigated the fatigue hysteresis loops characteristic of unidirectional fiber-reinforced CMCs at room temperature. At initial cyclic loading, matrix cracking and fibermatrix interface debonding occur, and fatigue hysteresis modulus decreases quickly. When the cycle exceeds a certain number, the modulus gradually recovers. The recovery of the modulus may be due to the mixed particles at the fibermatrix interface or matrix cracking plane, which increases the fibermatrix interface shear stress. The fatigue hysteresis loops area increases first to the peak value, then decreases, and increases quickly when approaching final fracture. Zawada et al. (1991) investigated the fatigue hysteresis loops shape and area of cross-ply fiber-reinforced CMCs under tensiontension loading at room temperature. Under initial loading, when the fatigue peak stress exceeds the proportional limit stress, the loading stressstrain curve is nonlinear, the unloading stressstrain curve is linear, and the fatigue hysteresis loop is not closed. At cycle number of 400,000, the composite fatigue hysteresis modulus and hysteresis loops area decrease obviously. When the applied cycle reaches 1,000,000, the fatigue hysteresis loops area continually decreases; however, the fatigue hysteresis modulus slowly increases compared with the cycle number of 400,000 due to the mixed particles at the matrix cracking plane or fibermatrix interface. Lynch and Evans (1996) investigated the loadingunloading behavior of three different layups crossply fiber-reinforced CMC, namely, [0/90], [ 6 45 ], and [ 1 30 / 2 60 ]. Upon unloading and reloading, the stressstrain curves appear as obvious hysteresis loops. Compared with [0/90] cross-ply CMCs, the fatigue hysteresis loops shape of [ 6 45 ] and [ 1 30 / 2 60 ] composites are vastly different due to the cracking closure effect. Reynaud (1996) investigated the tensiontension fatigue behavior of 2D SiC/SiC, and [0/90]s-SiC/MAS-L composites at elevated temperatures of 600 C,

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

277

800 C, and 1000 C in inert atmosphere. For 2D SiC/SiC composites, the fibermatrix interface radial thermal residual stress is compressive stress, and with increasing temperature, the interface radial thermal residual stress decreases, leading to the decrease of the fibermatrix interface shear stress and the higher fatigue HDE than that at room temperature. For the [0/90]s-SiC/MAS-L composite, the fibermatrix interface radial stress is tensile stress, and with increasing temperature, the fibermatrix interface radial tensile stress decreases, leading to the increase of the fibermatrix interface shear stress, and the higher fatigue HDE than that at room temperature. At elevated temperature of 800 C1000 C in inert atmosphere, the chemical reaction at the fibermatrix interface leads to the degradation of the fibermatrix interface shear stress. After heat treating for 50 hours at elevated temperature in inert atmosphere, the fibermatrix interface shear stress decreases. Dalmaz et al. (1999) investigated the fatigue hysteresis loops of 2.5D C/SiC composite at room and elevated temperature of 600 C in inert atmosphere. Under tensiontension fatigue loading, the fatigue hysteresis loops area decreases with increasing applied cycles at room and elevated temperature due to the degradation of the interface shear stress between the yarns and the matrix. The fatigue hysteresis loops area at 600 C is less than that at room temperature due to the matrix cracking closure at elevated temperature. The decrease of the fatigue hysteresis loops area proves the interface wear mechanism. However, at 600 C, the fatigue hysteresis modulus decreases at first, and when the applied cycle approaches a certain number, the hysteresis modulus partially recovers due to the matrix cracking closure and the fiber rotation in the yarns (Dalmaz et al., 1996). Fantozzi and Reynaud (2009) investigated the tensiontension fatigue behavior of 2.5D SiC/ [SiBC] and 2.5D C/[SiBC] composites at elevated temperature of 1200 C in air atmosphere. For the 2.5D SiC/[SiBC] composite under 250/200 MPa, the fatigue HDE decreases with increasing cycle number due to the fibermatrix interface wear and the degradation of the fibermatrix interface shear stress. For the 2.5D C/[SiBC] composite under static fatigue loading for 144 hours, the fatigue hysteresis loops area decreases obviously, mainly due to the fibermatrix PyC interphase oxidation or the recession of the carbon fibers. Based on the analysis of fibermatrix interface debonding and sliding, the fatigue hysteresis loops can be divided into four different cases: 1. Case I, the interface is partially debonding and the fiber completely sliding relative matrix in the interface debonded region. 2. Case II, the interface is partially debonding and the fiber partially sliding relative matrix in the interface debonded region. 3. Case III, the interface is completely debonded and the fiber partially sliding relative matrix in the interface debonded region. 4. Case IV, the interface is completely debonded and the fiber completely sliding relative matrix in the interface debonded region.

to to to to

The unloading and reloading stressstrain relationships of CMCs for the interface partially debonding can be determined using the equations:

278

Durability of Ceramic-Matrix Composites

εc

pu

 TU 1 Δσ τ i y2 τ i ð2y 2 ld Þð2y 1 ld 2 lc Þ  14 2 2 αc 2 αf ΔΤ Ef rf l c Ef r f l c Ef

5

(5.1) εc

pr

5

TR 1 Δσ τ i z2 τ i ðy22zÞ2 24 14 Ef Ef rf lc E f rf l c

 τ i ðld 2 2y 1 2zÞðld 1 2y 2 2z 2 lc Þ  12 2 αc 2 αf ΔΤ rf l c Ef

(5.2)

where TU and TR denote the intact fibers stress upon unloading and reloading, respectively; Δσ denotes the extra applied stress carried by intact fibers from gross-slip of adjacent failure fibers; and y and z denote the interface counter-slip length and newslip length, respectively. Equation (5.3) is used to determine the interface counter slip length; and Eq. (5.4) is used to determine the interface new slip length. (    1 r f Vm Em 1 y 5 TU 2 ld 2 2 ρ 2 Ec τ i vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi )    u 2 rf2 Vf Vm Ef Em TU u rf σ r f Vm Em Ef t 2 ξd TU 2 1 2ρ 2 Vf 4Ec2 τ i 2 Ec τ i 2 (    1 r f V m Em 1 z 5y2 TR 2 ld 2 2 ρ 2 Ec τ i vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi )    u 2 rf2 Vf Vm Ef Em TR u r σ r f V m Em Ef 2 t 2ρf 2 2 ξ T 1 R d Vf 4Ec2 τ i 2 Ec τ i 2

(5.3)

(5.4)

The unloading and reloading stressstrain relationships of CMCs for the interface completely debonding can be determined using the equations:  2   TU 1 Δσ τ i y2 τ i 2y2lc =2 εc fu 5 14 22 2 αc 2 αf ΔΤ Ef Ef rf lc Ef rf l c εc

fr

TR 1 Δσ τ i z2 τ i ðy22zÞ2 24 14 Ef E f rf l c Ef rf lc  2   τ i lc =222y12z 22 2 αc 2 αf ΔΤ Ef rf l c

(5.5)

5

(5.6)

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

279

where y5

  rf Ef ðT 2 TU Þ 2 ðσmax 2 σÞ 4τ i Ec

z 5 yðσmin Þ 2

(5.7)

  rf Ef ðT 2 TR Þ 2 ðσmax 2 σÞ 4τ i Ec

(5.8)

where T denotes the intact fibers stress. The fatigue hysteresis area or HDE can be determined using the equation: ð σmax

U5

σmin



εc

unloading ðσÞ 2 εc reloading ðσÞ



(5.9)

where εc_unload and εc_reload denote unloading and reloading strain, respectively. Substituting unloading and reloading strains of Eqs. (5.1), (5.2), (5.5), and (5.6) corresponding to interface partially and completely debonding into Eq. (5.9), the fatigue HDE U can be obtained. The HDE-based damage parameter Φ is defined using the equation: Φ5

Un 2 Uinitial Ue

(5.10)

in which Un denotes the HDE at the Nth cycle; Uinitial denotes the HDE at the second cycle; and Ue denotes the elastic strain energy. Eq. (5.11) is used to determine the elastic strain energy. Ue 5

1 ðσmax 2 σmin Þðεmax 2 εmin Þ 2

(5.11)

The fatigue hysteresis width Δε is defined using the equation;   σmin 1 σmax Δε 5 εc unloading 2 εc 2

 reloading

σmin 1 σmax 2

 (5.12)

The fatigue hysteresis modulus E is defined using the equation: E5

σmax 2 σmin εc ðσmax Þ 2 εc ðσmin Þ

(5.13)

280

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Durability of Ceramic-Matrix Composites

Tensile loadingunloading damage evolution

The tensile loadingunloading hysteresis loops of unidirectional, cross-ply, 2D, 2.5D, and 3D C/SiC composites are predicted in this section. An effective coefficient of the fiber volume fraction along the loading direction (ECFL) is introduced to describe the fiber preforms and can be determined with the equation: φ5

Vf

axial

Vf

(5.14)

where Vf and Vf_axial denote the total fiber volume fraction in the composites and the effective fiber volume fraction in the cyclic loading direction. The values of parameter φ corresponding to unidirectional, cross-ply, 2D, 2.5D, 3D, and needled C/SiC composites are 1, 0.5, 0.5, 0.75, 0.93, and 0.375, respectively.

5.3.1 Results and discussion The ceramic composite system of SiC/CAS is as a case study. The effects of fiber volume fraction, matrix cracking density, fibermatrix interface shear stress, fibermatrix interface debonded energy, and fibers failure on the hysteresis loops, hysteresis area, hysteresis width, and hysteresis modulus are analyzed.

5.3.1.1 Effect of fiber-volume fraction on hysteresis loops and hysteresis-based damage parameters The fatigue hysteresis loops for different fiber-volume fractions of Vf 5 30% and 40% are shown in Fig. 5.1 corresponding to different fatigue peak stresses of σmax 5 200, 250, 300, 350, and 400 MPa. The fatigue hysteresis-based damage parameters of HDE, hysteresis width, and hysteresis modulus for different fiber volume fractions and fatigue peak stresses are provided in Table 5.1. When the fatigue peak stress is σmax 5 200 MPa, the fatigue HDE decreases from U 5 22.3 kPa at the fiber volume fraction of Vf 5 30% to U 5 5.8 kPa at the fiber volume fraction of Vf 5 40%; the fatigue hysteresis width decreases from Δε 5 0.01679% at the fiber volume fraction of Vf 5 30% to Δε 5 0.00479% at the fiber volume fraction of Vf 5 40%; and the fatigue hysteresis modulus increases from E 5 99.8 GPa at the fiber volume fraction of Vf 5 30% to E 5 121.5 GPa at the fiber volume fraction of Vf 5 40%. When the fatigue peak stress is σmax 5 250 MPa, the fatigue HDE decreases from U 5 43.6 kPa at the fiber volume fraction of Vf 5 30% to U 5 15.5 kPa at the fiber volume fraction of Vf 5 40%; the fatigue hysteresis width decreases from Δε 5 0.02623% at the fiber volume fraction of Vf 5 30% to Δε 5 0.00936% at the fiber volume fraction of Vf 5 40%; and the fatigue hysteresis modulus increases from E 5 95.8 GPa at the fiber volume fraction of Vf 5 30% to E 5 118.5 GPa at the fiber volume fraction of Vf 5 40%.

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

Figure 5.1 The fatigue hysteresis loops for different fiber volume fractions of Vf 5 30% and 40% at the fatigue peak stresses of (A) σmax 5 200 MPa; (B) σmax 5 250 MPa; (C) σmax 5 300 MPa; (D) σmax 5 350 MPa; and (E) σmax 5 400 MPa.

281

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Durability of Ceramic-Matrix Composites

Figure 5.1 (Continued)

Table 5.1 The fatigue hysteresis-based damage parameters for different fiber-volume fractions. Fatigue peak stress (MPa)

HDE (kPa) Vf 5 30%

200 250 300 350 400

22.3 43.6 75.3 119.6 178.6

HDE, Hysteresis dissipated energy.

Hysteresis width (%)

Hysteresis modulus (GPa)

Vf 5 40%

Vf 5 30%

Vf 5 40%

Vf 5 30%

Vf 5 40%

5.8 15.5 26.5 42.1 62.9

0.01679 0.02623 0.03788 0.05142 0.06716

0.00479 0.00936 0.0133 0.01811 0.02365

99.8 95.8 92.1 88.7 85.5

121.5 118.5 116.5 114.5 112.6

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

283

When the fatigue peak stress is σmax 5 300 MPa, the fatigue HDE decreases from U 5 75.3 kPa at the fiber volume fraction of Vf 5 30% to U 5 26.5 kPa at the fiber volume fraction of Vf 5 40%; the fatigue hysteresis width decreases from Δε 5 0.03788% at the fiber volume fraction of Vf 5 30% to Δε 5 0.0133% at the fiber volume fraction of Vf 5 40%; and the fatigue hysteresis modulus increases from E 5 92.1 GPa at the fiber volume fraction of Vf 5 30% to E 5 116.5 GPa at the fiber volume fraction of Vf 5 40%. When the fatigue peak stress is σmax 5 350 MPa, the fatigue HDE decreases from U 5 119.6 kPa at the fiber volume fraction of Vf 5 30% to U 5 42.1 kPa at the fiber volume fraction of Vf 5 40%; the fatigue hysteresis width decreases from Δε 5 0.05142% at the fiber volume fraction of Vf 5 30% to Δε 5 0.01811% at the fiber volume fraction of Vf 5 40%; and the fatigue hysteresis modulus increases from E 5 88.7 GPa at the fiber volume fraction of Vf 5 30% to E 5 114.5 GPa at the fiber volume fraction of Vf 5 40%. When the fatigue peak stress is σmax 5 400 MPa, the fatigue HDE decreases from U 5 178.6 kPa at the fiber volume fraction of Vf 5 30% to U 5 62.9 kPa at the fiber volume fraction of Vf 5 40%; the fatigue hysteresis width decreases from Δε 5 0.06716% at the fiber volume fraction of Vf 5 30% to Δε 5 0.02365% at the fiber volume fraction of Vf 5 40%; and the fatigue hysteresis modulus increases from E 5 85.5 GPa at the fiber volume fraction of Vf 5 30% to E 5 112.6 GPa at the fiber volume fraction of Vf 5 40%. When the fiber volume fraction increases, the fibermatrix interface debonded length and sliding range decreases; the fatigue HDE and fatigue hysteresis width decrease, the hysteresis modulus increases. At the same fiber volume fraction, when the fatigue peak stress increases, the fibermatrix interface debonding length and sliding range increase; and the fatigue HDE and hysteresis width increase, and the fatigue hysteresis modulus decreases.

5.3.1.2 Effect of matrix cracking space on hysteresis loops and hysteresis-based damage parameters The fatigue hysteresis loops for different matrix cracking space are shown in Fig. 5.2 corresponding to different fatigue peak stresses of σmax 5 200, 250, 300, 350, and 400 MPa. The fatigue hysteresis-based damage parameters of HDE, hysteresis width, and hysteresis modulus for different matrix crack spacing and fatigue peak stresses are shown in Table 5.2. When the fatigue peak stress is σmax 5 200 MPa, the fatigue HDE decreases from U 5 8.3 kPa at the matrix cracking space of lc 5 420 μm to U 5 4.8 kPa at the matrix cracking space of lc 5 725 μm; the fatigue hysteresis width decreases from Δε 5 0.0063% at the matrix cracking space of lc 5 420 μm to Δε 5 0.00368% at the matrix cracking space of lc 5 725 μm; and the fatigue hysteresis modulus increases from E 5 115.1 GPa at the matrix cracking space of lc 5 420 μm to E 5 118.6 GPa at the matrix cracking space of lc 5 725 μm. When the fatigue peak stress is σmax 5 250 MPa, the fatigue HDE decreases from U 5 34.2 kPa at the matrix cracking space of lc 5 198 μm to U 5 20 kPa at the

284

Durability of Ceramic-Matrix Composites

Figure 5.2 The fatigue hysteresis loops for different matrix cracking space at the fatigue peak stresses of (A) σmax 5 200 MPa; (B) σmax 5 250 MPa; (C) σmax 5 300 MPa; (D) σmax 5 350 MPa; and (E) σmax 5 400 MPa.

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

285

Figure 5.2 (Continued)

Table 5.2 The fatigue hysteresis-based damage parameters for different matrix crack spacing. Fatigue peak stress (MPa)

Matrix cracking space (µm)

HDE (kPa)

Hysteresis width (%)

Hysteresis modulus (GPa)

200

420 725 198 339 147 252 140 240 140 240

8.3 4.8 34.2 20 80.3 46.7 132.9 77.5 198.4 115.7

0.0063 0.00368 0.02062 0.01203 0.04025 0.02341 0.05714 0.03333 0.07462 0.04353

115.1 118.6 102.9 110.7 93 103.8 88.2 100.3 84.7 97.6

250 300 350 400

HDE, Hysteresis dissipated energy.

286

Durability of Ceramic-Matrix Composites

matrix cracking space of lc 5 339 μm; the fatigue hysteresis width decreases from Δε 5 0.02062% at the matrix cracking space of lc 5 198 μm to Δε 5 0.01203% at the matrix cracking space of lc 5 339 μm; and the fatigue hysteresis modulus increases from E 5 102.9 GPa at the matrix cracking space of lc 5 198 μm to E 5 110.7 GPa at the matrix cracking space of lc 5 339 μm. When the fatigue peak stress is σmax 5 300 MPa, the fatigue HDE decreases from U 5 80.3 kPa at the matrix cracking space of lc 5 147 μm to U 5 46.7 kPa at the matrix cracking space of lc 5 252 μm; the fatigue hysteresis width decreases from Δε 5 0.04025% at the matrix cracking space of lc 5 147 μm to Δε 5 0.02341% at the matrix cracking space of lc 5 252 μm; and the fatigue hysteresis modulus increases from E 5 93 GPa at the matrix cracking space of lc 5 147 μm to E 5 103.8 GPa at the matrix cracking space of lc 5 252 μm. When the fatigue peak stress is σmax 5 350 MPa, the fatigue HDE decreases from U 5 132.9 kPa at the matrix cracking space of lc 5 140 μm to U 5 77.5 kPa at the matrix cracking space of lc 5 240 μm; the fatigue hysteresis width decreases from Δε 5 0.05714% at the matrix cracking space of lc 5 140 μm to Δε 5 0.03333% at the matrix cracking space of lc 5 240 μm; and the fatigue hysteresis modulus increases from E 5 88.2 GPa at the matrix cracking space of lc 5 140 μm to E 5 100.3 GPa at the matrix cracking space of lc 5 240 μm. When the fatigue peak stress is σmax 5 400 MPa, the fatigue HDE decreases from U 5 198.4 kPa at the matrix cracking space of lc 5 140 μm to U 5 115.7 kPa at the matrix cracking space of lc 5 240 μm; the fatigue hysteresis width decreases from Δε 5 0.07462% at the matrix cracking space of lc 5 140 μm to Δε 5 0.04353% at the matrix cracking space of lc 5 240 μm; and the fatigue hysteresis modulus increases from E 5 84.7 GPa at the matrix cracking space of lc 5 140 μm to E 5 97.6 GPa at the matrix cracking space of lc 5 240 μm. When the matrix crack spacing increases, the fibermatrix interface debonding and sliding extent decrease; the fatigue HDE and hysteresis width decrease, and the fatigue hysteresis modulus increases.

5.3.1.3 Effect of fibermatrix interface shear stress on hysteresis loops and hysteresis-based damage parameters The fatigue hysteresis loops for different fibermatrix interface shear stress are shown in Fig. 5.3 corresponding to different fatigue peak stresses of σmax 5 200, 250, 300, 350, and 400 MPa. The fatigue hysteresis-based damage parameters of HDE, hysteresis width and hysteresis modulus for different fibermatrix interface shear stress and fatigue peak stresses are shown in Table 5.3. When the fatigue peak stress is σmax 5 200 MPa, the fatigue HDE decreases from U 5 10.1 kPa at the interface shear stress of τ i 5 10 MPa to U 5 5.5 kPa at the interface shear stress of τ i 5 20 MPa; the fatigue hysteresis width decreases from Δε 5 0.00812% at the interface shear stress of τ i 5 10 MPa to Δε 5 0.00459% at the interface shear stress of τ i 5 20 MPa; and the fatigue hysteresis modulus increases from E 5 115.8 GPa at the interface shear stress of τ i 5 10 MPa to E 5 122.1 GPa at the interface shear stress of τ i 5 20 MPa.

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

287

Figure 5.3 The fatigue hysteresis loops for different fibermatrix interface shear stress at the fatigue peak stresses of (A) σmax 5 200 MPa; (B) σmax 5 250 MPa; (C) σmax 5 300 MPa; (D) σmax 5 350 MPa; and (E) σmax 5 400 MPa.

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Durability of Ceramic-Matrix Composites

Figure 5.3 (Continued)

When the fatigue peak stress is σmax 5 250 MPa, the fatigue HDE decreases from U 5 27.3 kPa at the interface shear stress of τ i 5 10 MPa to U 5 14.2 kPa at the interface shear stress of τ i 5 20 MPa; the fatigue hysteresis width decreases from Δε 5 0.01656% at the interface shear stress of τ i 5 10 MPa to Δε 5 0.00858% at the interface shear stress of τ i 5 20 MPa; and the fatigue hysteresis modulus increases from E 5 110.8 GPa at the interface shear stress of τ i 5 10 MPa to E 5 119.4 GPa at the interface shear stress of τ i 5 20 MPa. When the fatigue peak stress is σmax 5 300 MPa, the fatigue HDE decreases from U 5 47.7 kPa at the interface shear stress of τ i 5 10 MPa to U 5 23.8 kPa at the interface shear stress of τ i 5 20 MPa; the fatigue hysteresis width decreases from Δε 5 0.02395% at the interface shear stress of τ i 5 10 MPa to Δε 5 0.01197% at the interface shear stress of τ i 5 20 MPa; and the fatigue hysteresis modulus increases from E 5 107.6 GPa at the interface shear stress of τ i 5 10 MPa to E 5 117.7 GPa at the interface shear stress of τ i 5 20 MPa.

Table 5.3 The fatigue hysteresis-based damage parameter for different fibermatrix interface shear stress. Fatigue peak stress (MPa)

200 250 300 350 400 HDE, Hysteresis dissipated energy.

HDE (kPa)

Hysteresis width (%)

Hysteresis modulus (GPa)

τ i 5 10 MPa

τ i 5 20 MPa

τ i 5 10 MPa

τ i 5 20 MPa

τ i 5 10 MPa

τ i 5 20 MPa

10.1 27.3 47.7 75.8 113.2

5.5 14.2 23.8 37.9 56.6

0.00812 0.01656 0.02395 0.0326 0.04257

0.00459 0.00858 0.01197 0.01629 0.02128

115.8 110.8 107.6 104.6 101.8

122.1 119.4 117.7 115.9 114.2

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Durability of Ceramic-Matrix Composites

When the fatigue peak stress is σmax 5 350 MPa, the fatigue HDE decreases from U 5 75.8 kPa at the interface shear stress of τ i 5 10 MPa to U 5 37.9 kPa at the interface shear stress of τ i 5 20 MPa; the fatigue hysteresis width decreases from Δε 5 0.0326% at the interface shear stress of τ i 5 10 MPa to Δε 5 0.01629% at the interface shear stress of τ i 5 20 MPa; and the fatigue hysteresis modulus increases from E 5 104.6 GPa at the interface shear stress of τ i 5 10 MPa to E 5 115.9 GPa at the interface shear stress of τ i 5 20 MPa. When the fatigue peak stress is σmax 5 400 MPa, the fatigue HDE decreases from U 5 113.2 kPa at the interface shear stress of τ i 5 10 MPa to U 5 56.6 kPa at the interface shear stress of τ i 5 20 MPa; the fatigue hysteresis width decreases from Δε 5 0.04257% at the interface shear stress of τ i 5 10 MPa to Δε 5 0.02128% at the interface shear stress of τ i 5 20 MPa; and the fatigue hysteresis modulus increases from E 5 101.8 GPa at the interface shear stress of τ i 5 10 MPa to E 5 114.2 GPa at the interface shear stress of τ i 5 20 MPa. When the fibermatrix interface shear stress increases, the fibermatrix interface debonded length, unloading interface counter slip length, and new-slip length upon reloading all decrease, leading to the decrease of the fatigue HDE and hysteresis width and the increase of fatigue hysteresis modulus.

5.3.1.4 Effect of fibermatrix interface debonded energy on hysteresis loops and hysteresis-based damage parameters The fatigue hysteresis loops for different fibermatrix interface debonded energy are shown in Fig. 5.4 corresponding to different fatigue peak stresses of σmax 5 200, 250, 300, and 350 MPa. The fatigue hysteresis-based damage parameters of HDE, hysteresis width and hysteresis modulus for different fibermatrix interface debonded energy, and fatigue peak stresses are provided in Table 5.4. When the fatigue peak stress is σmax 5 200 MPa, the fatigue HDE decreases from U 5 9.4 kPa at the interface debonded energy of ξ d 5 0.1 J/m2 to U 5 0 kPa at the interface debonded energy of ξd 5 1 J/m2; the fatigue hysteresis width decreases from Δε 5 0.0071% at the interface debonded energy of ξd 5 0.1 J/m2 to Δε 5 0 at the interface debonded energy of ξd 5 1 J/m2; and the fatigue hysteresis modulus increases from E 5 119 GPa at the interface debonded energy of ξd 5 0.1 J/m2 to E 5 126.9 GPa at interface debonded energy of ξd 5 1 J/m2. When the fatigue peak stress is σmax 5 250 MPa, the fatigue HDE decreases from U 5 18.4 kPa at the interface debonded energy of ξ d 5 0.1 J/m2 to U 5 8.2 kPa at the interface debonded energy of ξd 5 1 J/m2; the fatigue hysteresis width decreases from Δε 5 0.01108% at the interface debonded energy of ξ d 5 0.1 J/m2 to Δε 5 0.00466% at the interface debonded energy of ξd 5 1 J/m2; and the fatigue hysteresis modulus increases from E 5 116.5 GPa at the interface debonded energy of ξd 5 0.1 J/m2 to E 5 119.9 GPa at interface debonded energy of ξ d 5 1 J/m2. When the fatigue peak stress is σmax 5 300 MPa, the fatigue HDE decreases from U 5 31.8 kPa at the interface debonded energy of ξ d 5 0.1 J/m2 to U 5 26.1 kPa at the interface debonded energy of ξd 5 1 J/m2; the fatigue hysteresis width decreases from Δε 5 0.01597% at the interface debonded energy of ξ d 5 0.1 J/m2 to Δε 5 0.01379%

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

291

Figure 5.4 The fatigue hysteresis loops for different fibermatrix interface debonded energies at the fatigue peak stresses of (A) σmax 5 200 MPa; (B) σmax 5 250 MPa; (C) σmax 5 300 MPa; and (D) σmax 5 350 MPa.

at the interface debonded energy of ξ d 5 1 J/m2; and the fatigue hysteresis modulus increases from E 5 114.2 GPa at the interface debonded energy of ξd 5 0.1 J/m2 to E 5 115.1 GPa at interface debonded energy of ξd 5 1 J/m2. When the fatigue peak stress is σmax 5 350 MPa, the fatigue HDE decreases from U 5 50.5 kPa at the interface debonded energy of ξ d 5 0.1 J/m2 to U 5 49.9 kPa at the interface debonded energy of ξd 5 1 J/m2; the fatigue hysteresis width decreases from Δε 5 0.02173% at the interface debonded energy of ξd 5 0.1 J/m2 to Δε 5 0.02159% at the interface debonded energy of ξd 5 1 J/m2; and the fatigue hysteresis modulus remains unchanged. When the fibermatrix interface debonded energy increases, the interface debonded length and interface slip length decrease when the interface partially

292

Durability of Ceramic-Matrix Composites

Figure 5.4 (Continued)

Table 5.4 The fatigue hysteresis-based damage parameter for different fibermatrix interface debonded energies. Fatigue peak stress (MPa)

200 250 300 350

HDE (kPa)

Hysteresis width (%)

Hysteresis modulus (GPa)

ξd 5 0.1 J/m2

ξd 5 1 J/m2

ξd 5 0.1 J/m2

ξd 5 1 J/m2

ξd 5 0.1 J/m2

ξd 5 1 J/m2

9.4 18.4 31.8 50.5

0 8.2 26.1 49.9

0.0071 0.01108 0.01597 0.02173

0 0.00466 0.01379 0.02159

119 116.5 114.2 111.9

126.9 119.9 115.1 111.9

HDE, Hysteresis dissipated energy.

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

293

debonding; however, when the interface is completely debonded, the interface debonded energy has no effect on the interface debonding and sliding length. The fatigue HDE and hysteresis width decreases with increasing interface debonded energy when the interface is partially debonding, and the fatigue hysteresis modulus increases with the interface debonded energy when the interface is partially debonding. When the interface is completely debonded, the fatigue HDE, hysteresis width and hysteresis modulus with different interface debonded energies approach the same value.

5.3.1.5 Effect of fibers failure on hysteresis loops and hysteresisbased damage parameters The fatigue hysteresis loops with/without considering fibers failure are shown in Fig. 5.5 corresponding to different fatigue peak stresses of σmax 5 300, 350, and 400 MPa. The fatigue hysteresis-based damage parameters of HDE, hysteresis width and hysteresis modulus with/without considering fibers failure for different fatigue peak stresses are provided in Table 5.5. When the fatigue peak stress is σmax 5 300 MPa, the fatigue HDE increases from U 5 31.8 kPa without fibers failure to U 5 34.8 kPa when the fibers failure probability is 3.6%; the fatigue hysteresis width increases from Δε 5 0.01596% without fibers failure to Δε 5 0.01746% when the fibers failure probability is 3.6%; and the fatigue hysteresis modulus decreases from E 5 114.1 GPa without fibers failure to E 5 113.7 GPa when the fibers failure probability is 3.6%. When the fatigue peak stress is σmax 5 350 MPa, the fatigue HDE increases from U 5 50.5 kPa without fibers failure to U 5 61.4 kPa when the fibers failure probability is 6.2%; the fatigue hysteresis width increases from Δε 5 0.02173% without fibers failure to Δε 5 0.02638% when the fibers failure probability is 6.2%; and the fatigue hysteresis modulus decreases from E 5 113.3 GPa without fibers failure to E 5 111.2 GPa when the fibers failure probability is 6.2%. When the fatigue peak stress is σmax 5 400 MPa, the fatigue HDE increases from U 5 75.4 kPa without fibers failure to U 5 115.6 kPa when the fibers failure probability is 12%; the fatigue hysteresis width increases from Δε 5 0.02838% without fibers failure to Δε 5 0.04343% when the fibers failure probability is 12%; and the fatigue hysteresis modulus decreases from E 5 109.7 GPa without fibers failure to E 5 107.2 GPa when the fibers failure probability is 12%. When the fibers failure occurs under cyclic tensile loading, the stress carried by intact fibers increases due to the fibers broken and sliding, leading to the increase of the fibermatrix interface debonding and sliding length. The fatigue HDE and fatigue hysteresis width increase, and the fatigue hysteresis modulus decreases when the fibers broken occurs inside of fiber-reinforced CMCs.

5.3.2 Experimental comparisons The fatigue hysteresis loops and fatigue damage of unidirectional, cross-ply, 2D, 2.5D, and 3D fiber-reinforced CMCs are predicted.

294

Durability of Ceramic-Matrix Composites

Figure 5.5 The fatigue hysteresis loops with/without considering fibers failure at the fatigue peak stresses of (A) σmax 5 300 MPa; (B) σmax 5 350 MPa; and (C) σmax 5 400 MPa.

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

295

5.3.2.1 Unidirectional C/SiC composite The cyclic tensile hysteresis loops of unidirectional C/SiC composites at room temperature are shown in Fig. 5.6. The composite tensile strength is 265 MPa and the failure strain is 0.27%. The experimental and theoretical hysteresis loops corresponding to different peak stresses are shown in Fig. 5.7. When the peak stress increases, the fibermatrix interface debonding length increases, and the interface counter-slip and new-slip length increases. The experimental and predicted fatigue HDE and hysteresis modulus are given in Table 5.6. The experimental fatigue HDE values are U 5 2.2, 2.4, 2.8, 4.8, 5.9, 9.8, 13.5, 18, 23.5, and 30 kPa at the fatigue peak stresses of σmax 5 80, 100, 120, 140, 160, 180, 200, 220, 240, and 260 MPa. The theoretical fatigue HDE values are U 5 2, 2.2, 2.8, 4.6, 6.2, 10, 13.7, 18.2, 24, Table 5.5 The fatigue hysteresis-based damage parameter with/without considering fibers failure. Fatigue peak stress (MPa)

Fibers failure probability (%)

HDE (kPa)

Hysteresis width (%)

Hysteresis modulus (GPa)

300

 3.6  6.2  12

31.8 34.8 50.5 61.4 75.4 115.6

0.01596 0.01746 0.02173 0.02638 0.02838 0.04343

114.1 113.7 113.3 111.2 109.7 107.2

350 400

HDE, Hysteresis dissipated energy.

Figure 5.6 The cyclic tensile hysteresis loops of unidirectional C/SiC composites at room temperature. Source: Reproduced from Longbiao, L., 2016a. Hysteresis loops of carbon fiber-reinforced ceramic-matrix composites with different fiber preforms. Ceram. Int. 42, 1653516551.

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Durability of Ceramic-Matrix Composites

Figure 5.7 The fatigue hysteresis loops of unidirectional C/SiC composite corresponding to (A) σmax 5 80 MPa; (B) σmax 5 100 MPa; (C) σmax 5 120 MPa; (D) σmax 5 140 MPa; (E) σmax 5 160 MPa; (F) σmax 5 180 MPa; (G) σmax 5 200 MPa; (H) σmax 5 220 MPa; (I) σmax 5 240 MPa; and (J) σmax 5 260 MPa.

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

Figure 5.7 (Continued)

297

298

Durability of Ceramic-Matrix Composites

Figure 5.7 (Continued)

and 32 kPa at the fatigue peak stresses of σmax 5 80, 100, 120, 140, 160, 180, 200, 220, 240, and 260 MPa. The experimental fatigue hysteresis modulus are E 5 122, 121, 119, 118, 116, 115, 114, 112.6, 111.3, and 110 GPa at the fatigue peak stresses of σmax 5 80, 100, 120, 140, 160, 180, 200, 220, 240, and 260 MPa. The theoretical fatigue hysteresis modulus are E 5 122.6, 121.1, 119.6, 118.2, 116.8, 115.3, 114, 112.6, 111.3, and 110 GPa at the fatigue peak stresses of σmax 5 80, 100, 120, 140, 160, 180, 200, 220, 240, and 260 MPa.

5.3.2.2 Cross-ply C/SiC composite The experimental and theoretical fatigue hysteresis loops of cross-ply C/SiC composites corresponding to different peak stresses of σmax 5 60, 80, 100, and 120 MPa are shown in Fig. 5.8. The composite tensile strength is 124 MPa and the failure strain is 0.24%. The experimental and theoretical fatigue HDE and fatigue

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

299

Figure 5.7 (Continued)

hysteresis modulus are given in Table 5.7. The experimental fatigue HDE are U 5 3.7, 8, 11.9, and 15.2 kPa at the fatigue peak stresses of σmax 5 60, 80, 100, and 120 MPa; and the theoretical fatigue HDE are U 5 1.6, 7.9, 15.6, and 20 kPa at the fatigue peak stresses of σmax 5 60, 80, 100, and 120 MPa. The experimental fatigue hysteresis modulus are E 5 72.4, 68.8, 58.3, and 52.9 GPa at the fatigue peak stresses of σmax 5 60, 80, 100, and 120 MPa. The theoretical fatigue hysteresis modulus are E 5 74, 70, 59, and 54 GPa at the fatigue peak stresses of σmax 5 60, 80, 100, and 120 MPa.

5.3.2.3 2D C/SiC composite Cheng (2010) investigated the cyclic tensile hysteresis behavior of 2D C/SiC composites at room temperature. The composite strength is 215 MPa and the failure strain is 0.73%. The experimental and theoretical fatigue hysteresis loops

300

Durability of Ceramic-Matrix Composites

Table 5.6 The fatigue hysteresis-based damage parameter of unidirectional c/sic composite. Fatigue peak stress (MPa) 80 100 120 140 160 180 200 220 240 260

HDE (kPa)

Hysteresis modulus (GPa)

Experimental

Predicted

Experimental

Predicted

2.2 2.4 2.8 4.8 5.9 9.8 13.5 18 23.5 30

2 2.2 2.8 4.6 6.2 10 13.7 18.2 24 32

122 121 119 118 116 115 114 112.6 111.3 110

122.6 121.1 119.6 118.2 116.8 115.3 114 112.6 111.3 110

HDE, Hysteresis dissipated energy.

Figure 5.8 The experimental and predicted cyclic tensile fatigue hysteresis loops of crossply C/SiC composites at room temperature. Table 5.7 The fatigue hysteresis-based damage parameter of cross-ply C/SiC composite. Fatigue peak stress (MPa) 60 80 100 120 HDE, Hysteresis dissipated energy.

HDE (kPa)

Hysteresis modulus (GPa)

Experimental

Predicted

Experimental

Predicted

3.7 8 11.9 15.2

1.6 7.9 15.6 20

72.4 68.8 58.3 52.9

74 70 59 54

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

301

corresponding to different peak stresses of σmax 5 75, 90, 120, 135, 150, 165, 180, 195, and 210 MPa are shown in Fig. 5.9. The experimental and theoretical fatigue HDE and fatigue hysteresis modulus are given in Table 5.8. The experimental fatigue HDE are U 5 2.6, 3.1, 8.3, 11.7, 18.5, and 60.9 kPa at the fatigue peak stresses of σmax 5 75, 90, 120, 135, and 210 MPa. The theoretical fatigue HDE are U 5 0.9, 2.5, 10.5, 10.6, 22.7, and 60.9 kPa at the fatigue peak stresses of σmax 5 75, 90, 120, 135, and 210 MPa. The experimental fatigue hysteresis modulus are E 5 87.5, 81.6, 71.9, 66.9, 61.7, and 42.8 GPa at the fatigue peak stresses of σmax 5 75, 90, 120, 135, and 210 MPa. The theoretical fatigue hysteresis modulus are E 5 88, 83, 74, 69, 63, and 43 GPa at the fatigue peak stresses of σmax 5 75, 90, 120, 135, and 210 MPa.

5.3.2.4 2.5D C/SiC composite Wang et al. (2008) investigated the cyclic tensile hysteresis behavior of 2.5D C/SiC composites at room temperature. The composite strength is 308 MPa and the failure strain is 0.68%. The experimental and theoretical fatigue hysteresis loops corresponding to different peak stresses of σmax 5 115, 155, 195, 230, and 265 MPa are shown in Fig. 5.10. The experimental and theoretical fatigue HDE and fatigue hysteresis modulus are given in Table 5.9. The experimental fatigue HDE are U 5 5.8, 12.9, 30.4, 43.7, and 61.1 kPa at the fatigue peak stresses of σmax 5 115, 155, 195, 230, and 265 MPa. The theoretical fatigue HDE are U 5 3.6, 15.3, 58.2, 68.8, and 83.8 kPa at the fatigue peak stresses of σmax 5 115, 155, 195, 230, and 265 MPa. The experimental fatigue hysteresis modulus is E 5 110.6, 93.3, 73.2, 66.7, and 64.2 GPa at the fatigue peak stresses of σmax 5 115, 155, 195, 230, and 265 MPa. The theoretical fatigue hysteresis modulus are E 5 111, 94, 74, 67 and 66 GPa at the fatigue peak stresses of σmax 5 115, 155, 195, 230, and 265 MPa.

5.3.2.5 3D braided C/SiC composite Mei and Cheng (2009) investigated the cyclic tensile hysteresis behavior of 3D braided C/SiC composites at room temperature. The composite strength is σUTS 5 308 MPa and the failure strain is 0.59%. The experimental and theoretical fatigue hysteresis loops corresponding to different peak stresses of σmax 5 245, 260, 275, and 300 MPa are shown in Fig. 5.11. The experimental and theoretical fatigue HDE and fatigue hysteresis modulus are given in Table 5.10. The experimental fatigue HDE are U 5 35.1, 47.2, 63.3, and 76.2 kPa at the fatigue peak stresses of σmax 5 245, 260, 275, and 300 MPa. The theoretical fatigue HDE are U 5 56.4, 89.8, 120, and 128 kPa at the fatigue peak stresses of σmax 5 245, 260, 275, and 300 MPa. The experimental fatigue hysteresis modulus are E 5 70.5, 66.1, 62.6, and 59.1 GPa at the fatigue peak stresses of σmax 5 245, 260, 275, and 300 MPa. The theoretical fatigue hysteresis modulus are E 5 71, 67, 64, and 61 GPa at the fatigue peak stresses of σmax 5 245, 260, 275, and 300 MPa.

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Figure 5.9 The fatigue hysteresis loops of 2D C/SiC composites corresponding to (A) σmax 5 75 MPa; (B) σmax 5 90 MPa; (C) σmax 5 120 MPa; (D) σmax 5 135 MPa; (E) σmax 5 150 MPa; (F) σmax 5 165 MPa; (G) σmax 5 180 MPa; (H) σmax 5 195 MPa; and (I) σmax 5 210 MPa.

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

Figure 5.9 (Continued)

303

304

Figure 5.9 (Continued)

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305

Table 5.8 The fatigue hysteresis-based damage parameter of 2D C/SiC composite. Fatigue peak stress (MPa) 75 90 120 135 150 210

HDE (kPa)

Hysteresis modulus (GPa)

Experimental

Predicted

Experimental

Predicted

2.6 3.1 8.3 11.7 18.5 60.9

0.9 2.5 10.5 10.6 22.7 60.9

87.5 81.6 71.9 66.9 61.7 42.8

88 83 74 69 63 43

HDE, Hysteresis dissipated energy.

Figure 5.10 The experimental and predicted cyclic loadingunloading hysteresis loops of 2.5D C/SiC composites at room temperature.

5.3.2.6 3D needled C/SiC composite Xie et al. (2016) investigated the cyclic tensile hysteresis behavior of 3D needled C/SiC composite at room temperature. The composite strength is σUTS 5 92.6 MPa and the failure strain is 0.48%. The experimental and theoretical fatigue hysteresis loops corresponding to different peak stresses of σmax 5 45, 55, 65, 80, and 90 MPa are shown in Fig. 5.12. The experimental and theoretical fatigue HDE and fatigue hysteresis modulus are given in Table 5.11. The experimental fatigue HDE are U 5 1.6, 2.9, 4.4, 6.2, and 10.6 kPa at the fatigue peak stresses of σmax 5 45, 55, 65, 80, and 90 MPa. The theoretical fatigue HDE are U 5 4.7, 9.8, 18.9, 35.4, and 53.5 kPa at the fatigue peak stresses of σmax 5 45, 55, 65, 80, and 90 MPa. The experimental fatigue hysteresis modulus are E 5 34.1, 31.6, 29.1, 27.4, and 25.4 GPa at the fatigue peak stresses of σmax 5 45, 55, 65, 80, and 90 MPa. The

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Table 5.9 The fatigue hysteresis-based damage parameter of 2.5D C/SiC composite. Fatigue peak stress (MPa) 115 155 195 230 265

HDE (kPa)

Hysteresis modulus (GPa)

Experimental

Predicted

Experimental

Predicted

5.8 12.9 30.4 43.7 61.1

3.6 15.3 58.2 68.8 83.8

110.6 93.3 73.2 66.7 64.2

111 94 74 67 66

HDE, Hysteresis dissipated energy.

Figure 5.11 The experimental and predicted cyclic tensile hysteresis loops of 3D braided C/SiC composites at room temperature.

theoretical fatigue hysteresis modulus are E 5 35, 32, 29, 28, and 25.7 GPa at the fatigue peak stresses of σmax 5 45, 55, 65, 80, and 90 MPa.

5.4

Cyclic fatigue damage evolution

Under cyclic fatigue loading of CMCs, the interface shear stress decreases with increasing applied cycles due to the damage mechanism of interface wear at room temperature (Holmes and Cho, 1992; Rouby and Reynaud, 1993; Evans et al., 1995; Reynaud, 1996; Fantozzi and Reynaud, 2009) and interface oxidation at elevated temperatures (Mall and Engesser, 2006; Longbiao, 2013a,b). The temperature increase can be used an effective tool to monitor the interface wear process under cyclic fatigue loading of CMCs (Holmes and Shuler, 1990; Cho et al., 1991;

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307

Table 5.10 The fatigue hysteresis-based damage parameter of 3D braided C/SiC composite. Fatigue peak stress (MPa) 245 260 275 300

HDE (kPa)

Hysteresis modulus (GPa)

Experimental

Predicted

Experimental

Predicted

35.1 47.2 63.3 76.2

56.4 89.8 120 128

70.5 66.1 62.6 59.1

71 67 64 61

HDE, Hysteresis dissipated energy.

Figure 5.12 The experimental and predicted cyclic tensile hysteresis loops of 3D needled C/SiC composites at room temperature.

Table 5.11 The fatigue hysteresis-based damage parameter of 3D needled C/SiC composite. Fatigue peak stress (MPa) 45 55 65 80 90

HDE (kPa)

Hysteresis modulus (GPa)

Experimental

Predicted

Experimental

Predicted

1.6 2.9 4.4 6.2 10.6

4.7 9.8 18.9 35.4 53.5

34.1 31.6 29.1 27.4 25.4

35 32 29 28 25.7

HDE, Hysteresis dissipated energy.

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Holmes et al., 1994; Kim and Liaw, 2005; Liu et al., 2008). Holmes and Cho (1992) characterized the fatigue behavior of fiber-reinforced CMCs through measuring surface temperature. When the fiber sliding relative to the matrix in the debonded region, the specimen surface temperature increases obviously under cyclic loading. The relationship between the specimen surface temperature and the fibermatrix interface shear stress has been established and obtained the fibermatrix interface shear stress of unidirectional SiC/CAS composites at room temperature, and it was found that the fibermatrix interface shear stress decreases at the initial cyclic loading. Kostopoulos et al. (1997) investigated the tensiontension fatigue damage accumulation of 3D C/SiC composites at room temperature. When the fatigue peak stress is higher than the FMCS, upon first loading to the peak stress, multiple matrix cracking occurs in the composite. The matrix cracking approaches to the steady at the initial cyclic loading and remains the same during the further cycling. The fibermatrix interface wear decreases the stress concentration at the intersection of the yarns and partially increases the modulus of the composite. At the final stage of cyclic loading, a large amount of fibers failure leads to the final fracture of the composite. The fracture surface of fatigue failure specimens was observed and a large number of fibers were pulled out, which confirmed that the interfacial wear mechanism existed during cyclic loading. Du et al. (2002) investigated the tensiontension fatigue behavior of 3D C/SiC composites at room temperature and elevated temperature of 1300 C in vacuum atmosphere. Under high fatigue peak stress, the matrix cracking defects along the fibermatrix interface, and the fibermatrix interface debonds. With increasing cycles, the matrix inside of fiber bundles cracks, and the wear between the fiber and the matrix increases. At room temperature, the interface wear caused by the repeated sliding between the fiber and the matrix plays an important role in fatigue failure; however, at 1300 C, the effect of the interface wear on the fatigue damage becomes low. Han et al. (2004) investigated the tensiontension fatigue behavior of 2D and 3D C/SiC composites at elevated temperatures in the range of 1100 C1500 C in vacuum atmosphere. Compared with 2D C/SiC composites, the fiber pullout length of the 3D C/SiC composite at the fracture surface is much longer, indicating the longer fibermatrix interface sliding length. Evans et al. (1995) investigated the tensiontension fatigue behavior of unidirectional SiC/ CAS composites at room temperature. Under cyclic fatigue loading, the fatigue hysteresis loops corresponding to different applied cycles are measured, and the fibermatrix interface shear stress is estimated by analyzing the shape of the fatigue hysteresis loops. It was found that the fibermatrix interface shear stress decreases from τ i 5 20 MPa at the first cycle to τ i 5 5 MPa at the 30th applied cycle. The fibermatrix interface wear model was developed to predict the fibermatrix interface shear stress degradation and compared with experimental data. McNulty and Zok (1999) investigated the tensiontension fatigue behavior of SiC/CAS and SiC/MAS composites at room temperature. Based on the method developed by Evans et al. (1995), the fibermatrix interface shear stress for different applied cycles are obtained. For the SiC/CAS composite, the initial fibermatrix interface shear stress is τ o 5 1012 MPa, and the steady-state

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309

fibermatrix interface shear stress is τ s 5 57 MPa. For the SiC/MAS composite, the initial fibermatrix interface shear stress is τ o 5 2025 MPa, and after experiencing 40,000 cycles, the fibermatrix interface shear stress decreases to τ s 5 78 MPa. The degradation of fibermatrix interface shear stress is the major reason for the low-cycle fatigue failure. Longbiao (2017d) established the relationship between the fatigue HDE and the fibermatrix interface shear stress and analyzed the fibermatrix interface shear stress degradation rate of C/SiC and SiC/ SiC composites with different fiber preforms (i.e., unidirectional, cross-ply, 2D, 2.5D, and 3D) at room and elevated temperatures. Evans et al. (1995) developed an approach to evaluate the fibermatrix interface shear stress through hysteresis loops analysis using the Vagaggini’s models (Vagaggini et al., 1995). The evolution of the interface shear stress versus applied cycles can be determined using the equation (Evans et al., 1995):    τ i ðN Þ 5 τ io 1 1 2 exp 2ωN λ ðτ i min 2 τ io Þ

(5.15)

where τ io denotes the initial fibermatrix interface shear stress, that is, τ i(N) at N 5 1, before fatigue loading; τ imin denotes the steady-state fibermatrix interface shear stress during cycling; and ω and η are empirical constants.

5.4.1 Results and discussion The effects of fiber volume content, fatigue peak stress, matrix crack spacing, multiple matrix cracking modes, and woven structures on the fatigue damage evolution of SiC/CAS composites are investigated in this section.

5.4.1.1 Effect of fiber volume fraction on fibermatrix interface debonding and hysteresis-based damage parameters The fibermatrix interface debonding ratio, hysteresis modulus, HDE, and HDEbased damage parameter versus applied cycles for different fibers volume fraction are shown in Fig. 5.13. When the applied cycle number increases, the fibermatrix interface debonding ratio 2ld/lc increases first and then approaches a constant of 1.0, corresponding to the interface partially debonding and completely debonding. The fatigue hysteresis modulus first decreases slowly, then decreases rapidly, and finally remains constant. The fatigue HDE and HDE-based damage parameter increases first to the peak value, then decreases, and then remains constant. When the fiber volume faction increases, the fibermatrix interface debonding ratio decreases when the interface partially debonds, and the applied cycles for the interface completely debonding increases. The fatigue hysteresis modulus increases at the same applied cycle, the HDE decreases at the same applied cycle, and the HDE-based damage parameter decreases when the interface is partially debonding and increases when the interface is completely debonded.

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Figure 5.13 (A) The fibermatrix interface debonded ratio versus applied cycle curves; (B) the fatigue hysteresis modulus versus applied cycle number curves; (C) the fatigue HDE versus applied cycle number curves; and (D) the fatigue HDE-based damage parameters versus applied cycle number curves for different fiber volume fraction. HDE, Hysteresis dissipated energy.

5.4.1.2 Effect of fatigue peak stress on fibermatrix interface debonding and hysteresis-based damage parameters The fibermatrix interface debonding ratio, fatigue hysteresis modulus, HDE, and HDE-based damage parameters versus applied cycles for different fatigue peak stress curves are shown in Fig. 5.14. When the fatigue peak stress increases, the fibermatrix interface debonding ratio increases when the interface partially debonding, and the applied cycles for interface completely debonding decreases.

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

311

Figure 5.13 (Continued)

The fatigue hysteresis modulus decreases due to the large interface debonding and sliding range. The HDE increases corresponding to the interface partially and completely debonding, and the HDE-based damage parameter increases when the interface is partially debonding and decreases when the interface is completely debonded.

5.4.1.3 Effect of fatigue stress ratio on hysteresis-based damage parameters The fatigue HDE and HDE-based damage parameter versus the applied cycle curves for different fatigue stress ratios are shown in Fig. 5.15. When the fatigue stress ratio increases, the fatigue HDE decreases when the interface partially or completely debonding; and the fatigue HDE-based damage parameter decreases

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Figure 5.14 (A) The fibermatrix interface debonded ratio versus applied cycle curves; (B) the fatigue hysteresis modulus versus the applied cycle number curves; (C) the fatigue HDE versus the applied cycle number curves; and (D) the fatigue HDE-based damage parameter versus the applied cycle number curve for different fatigue peak stresses. HDE, Hysteresis dissipated energy.

when the interface is partially debonding and increases when the interface completely debonds.

5.4.1.4 Effect of matrix crack spacing on fibermatrix interface debonding and hysteresis-based damage parameters The fibermatrix interface debonding ratio, fatigue hysteresis modulus, HDE, and HDE-based damage parameters versus applied cycle curves for different matrix crack

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

313

Figure 5.14 (Continued)

spacing are shown in Fig. 5.16. When the matrix crack spacing increases, the fibermatrix interface debonding ratio decreases when the interface is partially debonding, and the applied cycle number for the interface completely debonding increases. The fatigue hysteresis modulus increases when the interface is partially or completely debonding. The fatigue HDE and HDE-based damage parameters decrease when the interface is partially debonding and increase when the interface is completely debonded.

5.4.1.5 Effect of matrix crack mode on fibermatrix interface debonding and hysteresis-based damage parameter Under cyclic fatigue loading, there are five different matrix cracking modes in 2D CMCs, as shown in Fig. 2.26. The matrix cracking modes 3 and 5 involve the

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Figure 5.15 (A) The fatigue HDE versus cycle number curves; and (B) the fatigue HDEbased damage parameters versus cycle number curves for different fatigue stress ratio. HDE, Hysteresis dissipated energy.

interface debonding and matrix cracking in the longitudinal ply. The composite unloading and reloading strain is related with the strain of matrix cracking modes 3 and 5, as shown in the equations: εc

unload

5 ηε3

unload

εc

reload

5 ηε3

reload

1 ð1 2 ηÞε5

1 ð1 2 ηÞε5

unload

reload

(5.16) (5.17)

in which ε3_unload and ε3_reload denote the unloading and reloading strain of the matrix cracking mode 3, respectively; ε5_unload and ε5_reload denote the unloading

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315

Figure 5.16 (A) The fibermatrix interface debonding ratio versus applied cycle curves; (B) the fatigue hysteresis modulus versus cycle number curves; (C) the fatigue HDE versus cycle number curves; and (D) the fatigue HDE-based damage parameters versus cycle number curves for different matrix crack spacing. HDE, Hysteresis dissipated energy.

and reloading strain of the matrix cracking mode 5, respectively; and η denotes the composite damage parameter. The fatigue HDE for 2D CMCs can be determined using the equation: U 5 ηU3 1 ð1 2 ηÞU5

(5.18)

in which U3 and U5 denote the fatigue HDE of matrix cracking mode 3 and mode 5, respectively.

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Figure 5.16 (Continued)

The fatigue HDE-based damage parameter Φ for 2D CMCs can be determined using the equation: Φ 5 ηΦ3 1 ð1 2 ηÞΦ5

(5.19)

in which Φ3 and Φ5 denote the fatigue HDE-based damage parameter of matrix cracking mode 3 and mode 5, respectively. The fatigue hysteresis modulus for 2D CMCs can be determined using the equation: E 5 ηE3 1 ð1 2 ηÞE5

(5.20)

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317

Figure 5.17 (A) The fibermatrix interface debonding ratio versus applied cycle curves; (B) the fatigue hysteresis modulus versus cycle number curves; (C) the fatigue HDE versus applied cycle number curves; and (D) the fatigue HDE-based damage parameters versus applied cycle number curves for different matrix cracking modes. HDE, Hysteresis dissipated energy.

where E3 and E5 denote the fatigue hysteresis modulus of matrix cracking mode 3 and mode 5, respectively. The fibermatrix interface debonding ratio, fatigue hysteresis modulus, HDE, and HDE-based damage parameters versus applied cycles are shown in Fig. 5.17. The fibermatrix interface debonding ratio of cracking mode 3 is higher than that of mode 5 when the interface is partially debonding. The fatigue hysteresis modulus of cracking mode 5 is much higher than that of cracking mode 3 when the interface is partially or completely debonded. The fatigue HDE and HDE-based damage

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Figure 5.17 (Continued)

parameters of cracking mode 3 is much higher than those of cracking mode 5 when the interface is partially or completely debonding.

5.4.1.6 Effect of woven structure on hysteresis-based damage parameters The fatigue HDE versus the interface shear stress curves using 1D fatigue hysteresis loops models, and fatigue hysteresis loops models considering multiple matrix cracking modes are shown in Fig. 5.18. The fatigue HDE of cracking mode 3 is larger than that of cracking mode 5 at the same interface shear stress. When the damage parameter is η 5 0.4, the fatigue HDE versus the interface shear stress curve approaches to that using an equivalent 1D fatigue hysteresis loops model.

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

319

Figure 5.18 The fatigue HDE versus the interface shear stress curves of woven CMCs using an equivalent 1D hysteresis loops model, and cross-ply hysteresis loops model for cracking mode 3, cracking mode 5, and composite with different damage parameter η. CMCs, Ceramic-matrix composites; HDE, hysteresis dissipated energy. Source: Reproduced from Longbiao, L., 2016b. Damage development in fiber-reinforced ceramic-matrix composites under cyclic fatigue loading using hysteresis loops at room and elevated temperatures. Int. J. Fract. 199, 3958.

5.4.2 Experimental comparisons The fibermatrix interface shear stress, fatigue hysteresis modulus, fatigue HDE, and HDE-based damage parameters as functions of applied cycle number are analyzed for CMCs with different fiber preforms.

5.4.2.1 Unidirectional ceramic-matrix composites The fibermatrix interface shear stress, fatigue hysteresis modulus, fatigue HDE, and HDE-based damage parameters versus applied cycles of unidirectional SiC/CAS, SiC/CAS-II, SiC/1723, and C/SiC composites at room temperature, and C/SiC composites at an elevated temperature of 800 C in air atmosphere are analyzed in this section.

SiC/calcium alumina silicate composite at room temperature Evans et al. (1995) investigated the tensiontension cyclic fatigue behavior of unidirectional SiC/CAS composites at room temperature. The fatigue peak stress is σmax 5 280 MPa and the loading frequency is f 5 10 Hz. The fibermatrix interface shear stress, fatigue hysteresis loops, fatigue HDE, and HDE-based damage parameters versus applied cycles are shown in Fig. 5.19. The fibermatrix interface shear stress decreases from τ i 5 22 MPa at initial cycle to the steady-state value of τ i 5 5 MPa at 109th cycle. Due to the decrease of the fibermatrix interface shear stress, the fibermatrix interface debonding and sliding length increases, leading to

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Figure 5.19 (A) the fatigue HDE versus the applied cycle number curves; (B) the fatigue HDE-based damage parameters versus the applied cycle curves; and (C) the fibermatrix interface shear stress versus the applied cycle curves of unidirectional SiC/CAS composites. CAS, Calcium alumina silicate; HDE, hysteresis dissipated energy.

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

321

the evolution of the fatigue hysteresis loops. The fatigue hysteresis loops at the cycle number N 5 1 and 5 correspond to the interface slip Case II. The fatigue hysteresis loops at the cycle number N 5 109 correspond to the interface slip Case IV. The fatigue HDE and HDE-based damage parameters increase first with applied cycles to the peak value, and then decreases to the steady-state value, corresponding to the steady-state interface shear stress.

SiC/calcium alumina silicate-II composite at room temperature Holmes and Cho (1992) investigated the tensiontension cyclic fatigue behavior of unidirectional SiC/CAS-II composites at room temperature. The fatigue peak stress is σmax 5 180 MPa and the loading frequency is f 5 25 Hz. The fatigue HDE versus the interface shear stress curve is shown in Fig. 5.20A. With decreasing fibermatrix interface shear stress, the fatigue HDE increases first to the peak value and then decreases corresponding to the interface slip Cases II, I, III, and IV. The fibermatrix interface shear stress corresponding to different applied cycles can be obtained using the theoretical fatigue HDE versus the interface shear stress curve and the experimental HDE values. The fibermatrix interface shear stress decreases from τ i 5 25 MPa at the applied cycle number N 5 1 to τ i 5 7 MPa at the applied cycle number N 5 3200. The fatigue HDE and HDE-based damage parameters increase with applied cycles and then approach steady-state values corresponding to the interface partially and completely debonding, as shown in Figs. 5.20B and C. The degradation of the fibermatrix interface shear stress leads to the decrease of the fatigue hysteresis modulus with applied cycles, as shown in Fig. 5.20D.

SiC/1723 composite at room temperature Zawada et al. (1991) investigated the tensiontension fatigue behavior of unidirectional SiC/1723 composite at room temperature. The fatigue peak stress is σmax 5 500 MPa and the fatigue stress ratio is R 5 0.1. The fibermatrix interface shear stress decreases from τ i 5 32 MPa at the applied cycle number N 5 1 to τ i 5 27 MPa at the applied cycle number N 5 10,011. Due to the decrease of the fibermatrix interface shear stress, the fibermatrix interface debonding and sliding length increases with applied cycles. The experimental and theoretical fatigue hysteresis loops corresponding to different applied cycles are shown in Fig. 5.21A. The fatigue hysteresis loops at the applied cycle number N 5 10, 20, and 143 correspond to the interface slip Case II. The fatigue hysteresis loops at the applied cycle number N 5 1010 and 10,011 correspond to the interface slip Case III. The fatigue HDE and HDE-based damage parameters first increase with applied cycles, and then approach the steady-state values, corresponding to the interface partially and completely debonding, as shown in Figs. 5.21B and C.

C/SiC composite at room temperature The tensiontension fatigue hysteresis loops of C/SiC composites under fatigue peak stresses of σmax 5 140, 180, and 240 MPa are shown in Figs. 5.22, 5.23, and 5.24. When the fatigue peak stress is σmax 5 140 MPa, the fatigue hysteresis modulus and fatigue HDE decrease with applied cycles. The fatigue hysteresis modulus decreases from E 5 178 GPa at the applied cycle number N 5 1 to E 5 156 GPa at

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Figure 5.20 (A) The fatigue HDE versus the fibermatrix interface shear stress curve; (B) the fatigue HDE versus the applied cycle number curve; (C) the fatigue HDE-based damage parameter versus the applied cycle number curve; and (D) the fatigue hysteresis modulus versus applied cycle number curve of unidirectional SiC/CAS-II composites. CAS, Calcium alumina silicate; HDE, Hysteresis dissipated energy.

the applied cycle number N 5 1,106,240. The fatigue HDE degrades from U 5 18 kPa at the applied cycle number N 5 1 to U 5 7.7 kPa at the applied cycle number N 5 15,155. The theoretical fatigue HDE versus the fibermatrix interface shear stress curve is shown in Fig. 5.25A. Using the experimental HDE value and the theoretical HDE versus the fibermatrix interface shear stress curve, the fibermatrix interface shear stress for different applied cycles can be obtained, as shown in Fig. 5.25B. The fibermatrix interface shear stress decreases from τ i 5 7.8 MPa at the applied cycle number N 5 1 to τ i 5 0.27 MPa at the applied cycle

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

323

Figure 5.20 (Continued)

number N 5 1,005,541. With decreasing fibermatrix interface shear stress, the fibermatrix interface debonding and sliding length increases with applied cycles, and the interface slip condition of the fatigue hysteresis loops transfers from the Case II to Case IV. The fatigue HDE and HDE-based damage parameters versus the applied cycle curves are shown in Fig. 5.25C and D. The fatigue HDE and HDE-based damage parameters increase first to the peak value, and then approach the steady-state value, corresponding to the interface partially and completely debonding. When the fatigue peak stress is σmax 5 180 MPa, the fatigue hysteresis modulus decreases from E 5 175 GPa at the cycle number N 5 1 to E 5 150 GPa at the cycle number N 5 1,071,870. The fatigue HDE decreases from U 5 32 kPa at the cycle number N 5 1 to U 5 5.4 kPa at the cycle number N 5 1,039,332. The theoretical fatigue HDE versus the fibermatrix interface shear stress curve is shown in

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Figure 5.21 (A) The experimental and theoretical fatigue hysteresis loops corresponding to different applied cycles; (B) the fatigue HDE versus the applied cycles; and (C) the fatigue HDE-based damage parameters versus the applied cycles of unidirectional SiC/1723 composites. HDE, Hysteresis dissipated energy.

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

Figure 5.22 The fatigue hysteresis loops of unidirectional C/SiC composites under the fatigue peak stress of σmax 5 140 MPa corresponding to (A) N 5 1; (B) N 5 15,155; (C) N 5 58,804; (D) N 5 234,783; (E) N 5 665,129; and (F) N 5 816,908.

325

326

Figure 5.22 (Continued)

Durability of Ceramic-Matrix Composites

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

Figure 5.23 The fatigue hysteresis loops of unidirectional C/SiC composites under the fatigue peak stress of σmax 5 180 MPa corresponding to (A) N 5 100; (B) N 5 78,932; (C) N 5 131,833; (D) N 5 399,271; (E) N 5 747,151; and (F) N 5 1,039,332.

327

328

Figure 5.23 (Continued)

Durability of Ceramic-Matrix Composites

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

Figure 5.24 The fatigue hysteresis loops of unidirectional C/SiC composites under the fatigue peak stress of σmax 5 240 MPa corresponding to (A) N 5 1; (B) N 5 100; (C) N 5 1000; (D) N 5 10,000; (E) N 5 100,000; and (F) N 5 1,000,000.

329

330

Figure 5.24 (Continued)

Durability of Ceramic-Matrix Composites

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

331

Figure 5.25 (A) The fatigue HDE versus the fibermatrix interface shear stress curve; (B) the fibermatrix interface shear stress versus applied cycles curve; (C) the fatigue HDE versus cycle number curve; and (D) the fatigue HDE-based damage parameters versus applied cycle number curve of unidirectional C/SiC composite at room temperature under different fatigue peak stresses. HDE, Hysteresis dissipated energy.

Fig. 5.25A. Using the experimental HDE value and the theoretical HDE versus the fibermatrix interface shear stress curve, the fibermatrix interface shear stress for different applied cycles can be obtained, as shown in Fig. 5.25B. The fibermatrix interface shear stress decreases from τ i 5 8.3 MPa at the cycle number N 5 1 to τ i 5 0.28 MPa at the cycle number N 5 1,039,332. With decreasing fibermatrix interface shear stress, the fibermatrix interface debonding and sliding length increases with applied cycles, the interface slip condition of the fatigue hysteresis loops transfers from Case II to Case IV. The fatigue HDE and HDE-based damage

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Durability of Ceramic-Matrix Composites

Figure 5.25 (Continued)

parameters versus the applied cycle curves are shown in Fig. 5.25C and D. The fatigue HDE and HDE-based damage parameters increase first to the peak value, and then approach the steady-state value, corresponding to the interface partially and completely debonding. When the fatigue peak stress is σmax 5 240 MPa, the fatigue hysteresis modulus degrades from E 5 169 GPa to E 5 128 GPa after the first 10 cycles. The fatigue HDE degrades from U 5 56 kPa at the cycle number N 5 1 to U 5 8 kPa at the cycle number N 5 1,000,000. The theoretical fatigue HDE versus the interface shear stress curve is shown in Fig. 5.25A. Using the experimental HDE value and the theoretical HDE value versus the fibermatrix interface shear stress curve, the fibermatrix interface shear stress for different applied cycles can be obtained, as shown in Fig. 5.25B. The fibermatrix interface shear stress decreases from τ i 5 8 MPa at the cycle number N 5 1 to τ i 5 0.3 MPa at the cycle number

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N 5 1,000,000. With decreasing fibermatrix interface shear stress, the fibermatrix interface debonding and sliding length increases with applied cycles, and the interface slip condition of the fatigue hysteresis loops transfers from Case II to Case IV. The fatigue HDE and HDE-based damage parameters versus the applied cycle curves are shown in Fig. 5.25C and D. The fatigue HDE and HDE-based damage parameters increase first to the peak value, and then approach the steadystate value, corresponding to the interface partially and completely debonding.

C/SiC composite at elevated temperature The tensiontension fatigue hysteresis loops of C/SiC composites under fatigue peak stresses of σmax 5 180 and 250 MPa are shown in Figs. 5.26 and 5.27. When the fatigue peak stress is σmax 5 180 MPa, the fatigue hysteresis modulus versus the applied cycles curve can be divided into three regions, that is, the initial rapid degradation region, the intermediate slow degradation region, and the final rapid degradation region. The fatigue HDE value degrades from U 5 30 kPa at the applied cycle number N 5 1 to U 5 5.8 kPa at the applied cycle number N 5 87,000. The theoretical fatigue HDE value versus the fibermatrix interface shear stress curve is shown in Fig. 5.28A. Using the experimental HDE value and the theoretical HDE value versus the fibermatrix interface shear stress curve, the fibermatrix interface shear stress for different applied cycles can be obtained, as shown in Fig. 5.28B. The fibermatrix interface shear stress decreases from τ i 5 8.4 MPa at the applied cycle number N 5 1 to τ i 5 0.29 MPa at the applied cycle number N 5 87,000. With decreasing fibermatrix interface shear stress, the fibermatrix interface debonding and sliding length increases with applied cycles, and the interface slip condition of the fatigue hysteresis loops transfers from Case II to Case IV. The fatigue HDE and HDE-based damage parameters versus the applied cycle curves are shown in Fig. 5.28C and D. The fatigue HDE and HDE-based damage parameters increase first to the peak value, and then approach the steadystate value, corresponding to the interface partially and completely debonding. When the fatigue peak stress is σmax 5 250 MPa, the fatigue hysteresis modulus decreases from E 5 210 GPa at the applied cycle number N 5 1 to E 5 83 GPa at the applied cycle number N 5 24541. The fatigue HDE degrades from U 5 62 kPa at the applied cycle number N 5 1 to U 5 7.2 kPa at the applied cycle number N 5 24,000. The theoretical fatigue HDE versus the fibermatrix interface shear stress curve is shown in Fig. 5.28A. Using the experimental fatigue HDE value and the theoretical HDE versus the fibermatrix interface shear stress curve, the fibermatrix interface shear stress for different applied cycles can be obtained, as shown in Fig. 5.28B. The fibermatrix interface shear stress decreases from τ i 5 7.9 MPa at the applied cycle number N 5 1 to τ i 5 0.28 MPa at the applied cycle number N 5 24,000. With decreasing fibermatrix interface shear stress, the fibermatrix interface debonding and sliding length increases with applied cycles, and the interface slip condition of the fatigue hysteresis loops transfers from Case II to Case IV. The fatigue HDE and HDE-based damage parameters versus the applied cycle curves are shown in Fig. 5.28C and D. The fatigue HDE and HDE-based

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Figure 5.26 The fatigue hysteresis loops of unidirectional C/SiC composites under the fatigue peak stress of σmax 5 180 MPa corresponding to (A) N 5 1; (B) N 5 10; (C) N 5 8000; (D) N 5 21,000; (E) N 5 65,000; and (F) N 5 85,000.

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

Figure 5.26 (Continued)

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Figure 5.27 The fatigue hysteresis loops of unidirectional C/SiC composites under the fatigue peak stress of σmax 5 250 MPa corresponding to (A) N 5 10; (B) N 5 1000; (C) N 5 5000; (D) N 5 10,000; (E) N 5 15,000; and (F) N 5 24,000.

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

Figure 5.27 (Continued)

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Figure 5.28 (A) The fatigue HDE versus the fibermatrix interface shear stress curve; (B) the fibermatrix interface shear stress versus applied cycles curve; (C) the fatigue HDE versus cycle number curve; and (D) the fatigue HDE-based damage parameter versus applied cycle number curve of unidirectional C/SiC composites at elevated temperature under different fatigue peak stresses. HDE, Hysteresis dissipated energy.

damage parameters increase first to the peak value, and then approach the steadystate value, corresponding to the interface partially and completely debonding.

5.4.2.2 Cross-ply ceramic-matrix composites SiC/calcium alumina silicate composite at room temperature Opalski and Mall (1994) investigated the cyclic tensiontension fatigue behavior of cross-ply SiC/CAS composites at room temperature with the loading frequency

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Figure 5.28 (Continued)

of f 5 10 Hz. When the fatigue peak stress is σmax 5 180 MPa, the theoretical fatigue HDE versus the fibermatrix interface shear stress curve is shown in Fig. 5.29A. The fatigue HDE cross-ply composite and corresponding matrix cracking modes 3 and 5 increase with decreasing interface shear stress to the peak value and then decrease. Using the theoretical fatigue HDE value versus the fibermatrix interface shear stress curve and the experimental HDE values for different applied cycles, the fibermatrix interface shear stress for different applied cycles can be obtained. The interface shear stress decreases from τ i 5 19 MPa at the applied cycle number N 5 1 to τ i 5 12 MPa at the applied cycle number N 5 5000, as shown in Fig. 5.29B. With decreasing fibermatrix interface shear stress, the fatigue hysteresis evolves with applied cycles, corresponding to interface slip Cases II, III, and IV, as shown in Fig. 5.29C. The fatigue hysteresis modulus versus the applied cycle

Figure 5.29 (A) The fatigue HDE versus the interface shear stress curve; (B) the interface shear stress versus the applied cycle number curve; (C) the fatigue hysteresis loops; (D) the fatigue hysteresis modulus versus applied cycle number curve; (E) the fatigue HDE versus applied cycle number curve; and (F) the fatigue HDE-based damage parameter versus cycle number curve of cross-ply SiC/CAS composites under the fatigue peak stress of σmax 5 180 MPa at room temperature. CAS, Calcium alumina silicate; HDE, hysteresis dissipated energy.

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

Figure 5.29 (Continued)

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curves of cross-ply composites and cracking modes 3 and 5 are shown in Fig. 5.29D. The composite hysteresis modulus decreases from E 5 55 GPa at the applied cycle number N 5 1 to E 5 48 GPa at the applied cycle number N 5 5000. The experimental and theoretical fatigue HDE and HDE-based damage parameters of cross-ply composites and matrix cracking modes 3 and 5 versus applied cycle number curves are shown in Fig. 5.29E and F.

SiC/calcium alumina silicate composite at 700 C in air atmosphere Opalski and Mall (1994) investigated the cyclic tensiontension fatigue behavior of cross-ply SiC/CAS composites at an elevated temperature of 700 C. The fatigue HDE, hysteresis modulus, and peak strain versus applied cycles for different applied cycle curves and peak stresses are shown in Fig. 5.30. The fatigue HDE decreases with applied cycles for different fatigue peak stresses of σmax 5 110, 83, and 69 MPa, corresponding to the interface slip Case IV. When the fatigue peak stress increases, the fatigue HDE and the degradation rate with increasing applied cycles increase due to multiple matrix cracking modes and interface damage. The fatigue hysteresis modulus decreases with applied cycles for different fatigue peak stresses of σmax 5 110, 83, and 69 MPa, and the degradation rate depends on the fatigue peak stress due to the matrix cracking extent inside the cross-ply composite. The fatigue peak strain increases with applied cycles for different fatigue peak stresses of σmax 5 69 and 110 MPa, while the increasing rate also depends on the peak stress level.

SiC/calcium alumina silicate composite at 850 C in air atmosphere Opalski and Mall (1994) investigated the cyclic tensiontension fatigue behavior of cross-ply SiC/CAS composites at an elevated temperature of 850 C. The fatigue HDE, fatigue hysteresis modulus, and peak strain versus applied cycle curves of cross-ply SiC/CAS composites at an elevated temperature of 850 C are shown in Fig. 5.31. The evolution of the fatigue HDE versus applied cycle curves depends on the fatigue peak stress. When the fatigue peak stress is σmax 5 55 MPa, the fatigue HDE increases with applied cycles, corresponding to the interface slip Case II. When the fatigue peak stress is σmax 5 52 MPa, the fatigue HDE increases to the peak value and then decreases, corresponding to interface slip Cases III and IV. The fatigue hysteresis modulus decreases with applied cycles for different fatigue peak stresses of σmax 5 52 and 55 MPa, and the degradation rate increases with peak stress due to the stress-dependent damage of matrix cracking and interface debonding. The fatigue peak strain increases with applied cycles and the increasing rate increases with peak stress due to high interface debonding and sliding range at high peak stress.

C/SiC composite at room temperature The experimental fatigue HDE value versus applied cycles curves and fatigue HDE value versus the fibermatrix interface shear stress curves of C/SiC composites for different peak stresses at room temperature are shown in Fig. 5.32. The fatigue HDE decreases with applied cycles at the peak stresses of σmax 5 87 and 105 MPa, and the degradation rate of HDE increases with peak stress, corresponding to the

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Figure 5.30 (A) The fatigue HDE versus applied cycle curves; (B) the normalized fatigue hysteresis modulus versus applied cycles; and (C) the peak strain versus applied cycles of cross-ply SiC/CAS composites at an elevated temperature of 700 C. CAS, Calcium alumina silicate; HDE, hysteresis dissipated energy.

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Figure 5.31 (A) The fatigue HDE versus applied cycle curves; (B) the normalized fatigue hysteresis modulus versus applied cycles; and (C) the peak strain versus applied cycles of cross-ply SiC/CAS composites at an elevated temperature of 850 C. CAS, Calcium alumina silicate; HDE, hysteresis dissipated energy.

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Figure 5.32 (A) The experimental fatigue HDE versus the applied cycles; and (B) the theoretical fatigue HDE versus the fibermatrix interface shear stress of C/SiC composites at room temperature. HDE, Hysteresis dissipated energy.

interface slip Case IV due to the stress-dependent interface debonding and sliding range. Using the theoretical HDE versus applied cycle curve and experimental HDE value, the fibermatrix interface shear stress for different applied cycles and peak stresses can be obtained. When the fatigue peak stress is σmax 5 87 MPa, the fibermatrix interface shear stress degrades from τ i 5 6.2 MPa at the applied cycle number N 5 1 to τ i 5 2 MPa at the applied cycle number N 5 100. When the fatigue peak stress is σmax 5 105 MPa, the fibermatrix interface shear stress degrades from τ i 5 7.3 MPa at the applied cycle number N 5 1 to τ i 5 2.1 MPa at the applied cycle number N 5 100. The degradation rate of the fibermatrix interface shear stress increases with peak stress due to the stress-dependent interface wear rate.

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C/SiC composite at 800 C in air atmosphere The experimental fatigue HDE value versus the applied cycle curves and fatigue HDE value versus the fibermatrix interface shear stress curves of C/SiC composites for different peak stresses at 800 C are shown in Fig. 5.33. When the fatigue peak stresses are σmax 5 97.5 and 105 MPa, the experimental fatigue HDE decreases with the applied cycles, and the HDE and HDE degradation rate both increase with the fatigue peak stress. Using the theoretical fatigue HDE value versus the fibermatrix interface shear stress curves and experimental fatigue HDE values, the fibermatrix interface shear stress for different applied cycles and peak stresses can be obtained. When the fatigue peak stress is σmax 5 97.5 MPa, the

Figure 5.33 (A) The experimental fatigue HDE versus the applied cycles; (B) the theoretical fatigue HDE versus the fibermatrix interface shear stress of C/SiC composites at an elevated temperature of 800 C. HDE, Hysteresis dissipated energy.

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fibermatrix interface shear stress degrades from τ i 5 6.2 MPa at the applied cycle number N 5 1 to τ i 5 0.5 MPa at the applied cycle number N 5 12,000. When the fatigue peak stress is σmax 5 105 MPa, the fibermatrix interface shear stress degraded from τ i 5 5.5 MPa at the applied cycle number N 5 1 to τ i 5 0.8 MPa at the applied cycle number N 5 100. The degradation rate of the fibermatrix interface shear stress with applied cycles increases with the fatigue peak stress level.

SiC/MAS-L composite at 800 C and 1000 C in inert atmosphere Reynaud (1996) investigated cyclic tensiontension fatigue behavior of cross-ply SiC/MAS-L composites at 800 C and 1000 C. The experimental fatigue HDE value versus applied cycles and theoretical fatigue HDE value versus the interface shear stress curves are shown in Fig. 5.34. The fatigue HDE decreases with the applied cycles and increases with increasing temperature at the same applied cycle numbers and fatigue peak stress of σmax 5 110 MPa, corresponding to interface slip Case IV. Using the theoretical HDE value versus the fibermatrix interface shear stress curve and the experimental fatigue HDE values, the fibermatrix interface shear stress for different applied cycles and temperatures can be obtained. At 800 C, the fibermatrix interface shear stress degrades from τ i 5 1.3 MPa at the applied cycle number N 5 5 to τ i 5 0.7 MPa at the applied cycle number N 5 34,162. At 1000 C, the fibermatrix interface shear stress degrades from τ i 5 2.7 MPa at the applied cycle number N 5 6 to τ i 5 0.9 MPa at the applied cycle number N 5 133,925.

5.4.2.3 2D ceramic-matrix composites SiC/SiC composite at 600 C, 800 C, and 1000 C in inert condition Reynaud (1996) investigated the cyclic tensiontension fatigue behavior of 2D SiC/SiC composites at elevated temperatures of 600 C, 800 C, and 1000 C. The experimental fatigue HDE value versus applied cycle curves for different testing temperatures of 600 C, 800 C, and 1000 C under the fatigue peak stress of σmax 5 130 MPa are shown in Fig. 5.35A. The fatigue HDE value increases with applied cycles, corresponding to interface slip Case II, and at the same applied cycle, the HDE value increases with temperature. Using the theoretical HDE value versus applied cycle curves and experimental HDE values, the fibermatrix interface shear stress for different applied cycles and temperatures can be obtained, as shown in Fig. 5.35B. The experimental and predicted fibermatrix interface shear stress versus the applied cycle number curves at elevated temperatures of 600 C, 800 C, and 1000 C are shown in Fig. 5.36A. At an elevated temperature of 600 C, the fibermatrix interface shear stress decreases from τ i 5 35 MPa at the applied cycle number N 5 1 to τ i 5 20 MPa at the applied cycle number N 5 333,507. At an elevated temperature of 800 C, the fibermatrix interface shear stress decreases from τ i 5 22 MPa at the applied cycle number N 5 1 to τ i 5 12.5 MPa at the applied cycle number N 5 97894. At an elevated temperature of 1000 C, the fibermatrix interface shear stress decreases from τ i 5 18 MPa at the applied cycle number N 5 1 to τ i 5 8.5 MPa at the applied cycle number N 5 117,055.

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Figure 5.34 (A) The experimental HDE value versus applied cycles; (B) the theoretical HDE value versus the interface shear stress of cross-ply SiC/MAS-L composites under the fatigue peak stress of σmax 5 110 MPa at 800 C and 1000 C. HDE, Hysteresis dissipated energy.

The experimental and predicted fatigue HDE value versus the applied cycle number curves at elevated temperatures of 600 C, 800 C, and 1000 C are shown in Fig. 5.36B. The predicted HDE value agreed with experimental data, and increases with applied cycles. At an elevated temperature of 600 C, the fatigue HDE increases from U 5 5.5 kPa at the applied cycle number N 5 1 to U 5 9.6 kPa at the applied cycle number N 5 400,000. At an elevated temperature of 800 C, the fatigue HDE increases from U 5 9.2 kPa at the applied cycle number N 5 1 to the peak value of U 5 15.4 kPa at the applied cycle number N 5 122,364. At an elevated temperature of 1000 C, the fatigue HDE value first increases with the applied cycle number from U 5 10.7 kPa at the applied cycle number N 5 1 to the peak

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Figure 5.35 (A) The fatigue HDE value versus applied cycle number curves; and (B) the theoretical fatigue HDE value versus cycle number curve of 2D SiC/SiC composites under the fatigue peak stress of σmax 5 130 MPa at elevated temperature. HDE, Hysteresis dissipated energy.

value of U 5 22.6 kPa at the applied cycle number N 5 375,365, and then remains constant till cycle number N 5 400,000.

SiC/SiC composite at 1000 C in air and in steam atmosphere Michael (2010) investigated the cyclic tensiontension fatigue behavior of 2D SiC/ SiC composites at 1000 C with the loading frequency of f 5 1 Hz and a stress ratio of R 5 0.1. The experimental fatigue HDE versus applied cycle curves and the theoretical fatigue HDE versus the applied cycle number curves of 2D SiC/SiC composites

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Figure 5.36 (A) The experimental and theoretical fibermatrix interface shear stress versus the applied cycle number curves; and (B) the experimental and theoretical fatigue HDE value versus the applied cycle number curves of 2D SiC/SiC composites under the fatigue peak stress of σmax 5 130 MPa at elevated temperatures. HDE, Hysteresis dissipated energy.

under different fatigue peak stresses at an elevated temperature of 1000 C in air and steam atmospheres are shown in Fig. 5.37. When the fatigue peak stress is σmax 5 80 MPa in air atmosphere, the experimental fatigue HDE value increases with applied cycles, corresponding to interface slip Case II. Using the theoretical HDE value versus the fibermatrix interface shear stress curve and the experimental HDE values, the fibermatrix interface shear stress for different applied cycles can be obtained. When the fatigue peak stress is σmax 5 60 MPa in steam atmosphere, the experimental fatigue HDE increases with the applied cycles,

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Figure 5.37 (A) The experimental fatigue HDE value versus the applied cycle curves; and (B) the theoretical fatigue HDE value versus the fibermatrix interface shear stress curves of 2D SiC/SiC composites under different fatigue peak stresses at 1000 C in air and steam atmospheres. HDE, Hysteresis dissipated energy.

corresponding to interface slip Case II. Using the theoretical HDE value versus the fibermatrix interface shear stress curve and the experimental fatigue HDE values, the fibermatrix interface shear stress for different applied cycles can be obtained. When the fatigue peak stress is σmax 5 100 MPa in steam atmosphere, the experimental fatigue HDE value increases with applied cycles, corresponding to interface slip Case II. Using the theoretical fatigue HDE value versus the fibermatrix interface shear stress curve and the experimental HDE values, the fibermatrix interface shear stress for different applied cycles can be obtained.

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Figure 5.38 (A) The experimental and theoretical fibermatrix interface shear stress versus the applied cycle number curves; and (B) the experimental and theoretical fatigue HDE values versus the applied cycle number curves of 2D SiC/SiC composites under different fatigue peak stresses at 1000 C in air and steam atmospheres. HDE, Hysteresis dissipated energy.

The experimental and theoretical fibermatrix interface shear stress and the fatigue HDE value versus the applied cycle curves are shown in Fig. 5.38. When the fatigue peak stress is σmax 5 80 MPa in air atmosphere, the fibermatrix interface shear stress decreases from τ i 5 15 MPa at the applied cycle number N 5 2 to τ i 5 10 MPa at the applied cycle number N 5 30,000. The predicted fatigue HDE value versus applied cycles curve agree with experimental data (i.e., from U 5 4.7 kPa at the applied cycle number N 5 1 to U 5 7.2 kPa at the applied cycle

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number N 5 40,000). When the fatigue peak stress is σmax 5 60 MPa in steam atmosphere, the fibermatrix interface shear stress decreases from τ i 5 15 MPa at the applied cycle number N 5 2 to τ i 5 3 MPa at the applied cycle number N 5 190,000. The predicted fatigue HDE versus the applied cycles curve agree with experimental data (i.e., from U 5 1.5 kPa at the applied cycle number N 5 1 to U 5 5.9 kPa at the applied cycle number N 5 20,000). When the fatigue peak stress is σmax 5 100 MPa in steam atmosphere, the fibermatrix interface shear stress decreases from τ i 5 15 MPa at the applied cycle number N 5 2 to τ i 5 8 MPa at the applied cycle number N 5 10,000. The predicted fatigue HDE versus the applied cycles curve agree with experimental data (i.e., from U 5 9.1 kPa at the applied cycle number N 5 1 to U 5 16.8 kPa at the applied cycle number N 5 20,000).

SiC/SiC composite at 1200 C in air and steam atmospheres Jacob (2010) investigated the cyclic tensiontension fatigue behavior of 2D SiC/ SiC composites at 1200 C with the loading frequency of f 5 0.1 Hz and a stress ratio of R 5 0.05. The experimental fatigue HDE value versus applied cycle curves and the theoretical fatigue HDE value versus the applied cycle number curves of 2D SiC/SiC composites at 1200 C in air and steam atmospheres are shown in Fig. 5.39. When the fatigue peak stress is σmax 5 140 MPa in air atmosphere, the experimental fatigue HDE increases with the applied cycles, corresponding to interface slip Case II. Using the theoretical fatigue HDE value versus the fibermatrix interface shear stress curve and the experimental fatigue HDE values, the fibermatrix interface shear stress for different applied cycles can be obtained. When the fatigue peak stress is σmax 5 140 MPa in steam atmosphere, the experimental fatigue HDE increases with the applied cycles, corresponding to interface slip Case II. Using the theoretical fatigue HDE value versus the fibermatrix interface shear stress curve and the experimental HDE values, the fibermatrix interface shear stress for different applied cycles can be obtained. The experimental and theoretical fibermatrix interface shear stress and the fatigue HDE versus the applied cycle curves of 2D SiC/SiC composites at 1200 C in air and steam atmospheres are shown in Fig. 5.40. When the fatigue peak stress is σmax 5 140 MPa in air atmosphere, the fibermatrix interface shear stress decreases from τ i 5 15 MPa at the applied cycle number N 5 1000 to τ i 5 3 MPa at the applied cycle number N 5 30,000. The predicted fatigue HDE value versus the applied cycles curve agree with experimental data (i.e., from U 5 3.9 kPa at the applied cycle number N 5 1 to U 5 26 kPa at the applied cycle number N 5 40,000). When the fatigue peak stress is σmax 5 140 MPa in steam atmosphere, the fibermatrix interface shear stress decreases from τ i 5 17 MPa at the applied cycle number N 5 100 to τ i 5 3.2 MPa at the applied cycle number N 5 10,000. The predicted fatigue HDE value versus the applied cycles curve agree with experimental data (i.e., from U 5 4.4 kPa at the applied cycle number N 5 1 to the peak value of U 5 26.2 kPa at the applied cycle number N 5 19,875, and then remains to be constant till N 5 20,000).

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Figure 5.39 (A) The experimental fatigue HDE value versus the applied cycle curves; and (B) the theoretical fatigue HDE value versus the fibermatrix interface shear stress curves of 2D SiC/SiC composites at 1200 C in air and steam atmospheres. HDE, Hysteresis dissipated energy.

SiC/SiC composite at 1300 C in air atmosphere Zhu et al. (1998) investigated cyclic tensiontension fatigue behavior of 2D SiC/ SiC composites at 1300 C in air atmosphere with the loading frequency of f 5 20 Hz and a stress ratio of R 5 0.1. The experimental fatigue HDE value versus applied cycle curves and the theoretical fatigue HDE value versus the applied cycle number curves of 2D SiC/SiC composites at 1300 C in air atmosphere under different fatigue peak stresses are shown in Fig. 5.41. When the fatigue peak stress is σmax 5 90 MPa, the experimental fatigue HDE increases with applied cycles, corresponding to interface slip Case II.

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Figure 5.40 (A) The experimental and theoretical fibermatrix interface shear stress versus the applied cycle number curves; and (B) the experimental and theoretical fatigue HDE values versus the applied cycle number curves of 2D SiC/SiC composites at 1200 C in air and steam atmospheres. HDE, Hysteresis dissipated energy.

Using the theoretical fatigue HDE value versus the fibermatrix interface shear stress curve and the experimental fatigue HDE values, the fibermatrix interface shear stress for different applied cycles can be obtained. When the fatigue peak stress is σmax 5 120 MPa, the experimental fatigue HDE increases with applied cycles, corresponding to interface slip Case II. Using the theoretical fatigue HDE value versus the fibermatrix interface shear stress curve and the experimental fatigue HDE values, the fibermatrix interface shear stress for different applied cycles can be obtained.

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Figure 5.41 (A) The experimental fatigue HDE value versus the applied cycle curves; and (B) the theoretical fatigue HDE values versus the fibermatrix interface shear stress curves of 2D SiC/SiC composites at 1300 C in air atmosphere. HDE, Hysteresis dissipated energy.

The experimental and theoretical fibermatrix interface shear stress and the fatigue HDE value versus the applied cycle curves of 2D SiC/SiC composites at 1300 C in air atmosphere are shown in Fig. 5.42. When the fatigue peak stress is σmax 5 90 MPa, the fibermatrix interface shear stress decreases from τ i 5 12 MPa at the applied cycle number N 5 6000 to τ i 5 3 MPa at the applied cycle number N 5 2,800,000. The predicted fatigue HDE value versus the applied cycle curves agree with experimental data (i.e., from U 5 1.6 kPa at the

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Figure 5.42 (A) The experimental and theoretical fibermatrix interface shear stress versus the applied cycle number curves; and (B) the experimental and theoretical fatigue HDE values versus the applied cycle number curves of 2D SiC/SiC composites at 1300 C in air atmosphere. HDE, Hysteresis dissipated energy.

applied cycle number N 5 1 to U 5 7.9 kPa at the applied cycle number N 5 3,000,000). When the fatigue peak stress is σmax 5 120 MPa, the fibermatrix interface shear stress decreases from τ i 5 18 MPa at the applied cycle number N 5 100 to τ i 5 3.7 MPa at the applied cycle number N 5 36,000. The predicted fatigue HDE versus the applied cycle curves agree with experimental data (i.e., from U 5 3.6 kPa at the applied cycle number N 5 1 to U 5 15.4 kPa at the applied cycle number N 5 40,000).

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5.4.2.4 3D braided ceramic-matrix composites Shi et al. (2015) investigated the cyclic tensiontension fatigue behavior of 3D SiC/SiC composites at 1300 C with the loading frequency of f 5 1 Hz and a stress ratio of R 5 0.1. The experimental and theoretical fatigue HDE values versus the applied cycle curves, the theoretical fatigue HDE value versus the fibermatrix interface shear stress curve, and the experimental and theoretical fibermatrix interface shear stress versus the applied cycle number curves of 3D SiC/SiC composites at 1300 C in air atmosphere are shown in Fig. 5.43. When the fatigue peak stress is σmax 5 100 MPa, the experimental fatigue HDE value increases with applied cycles, corresponding to interface slip Case II. Using the theoretical fatigue HDE value versus the fibermatrix interface shear stress curve and the experimental fatigue HDE values, the fibermatrix interface shear stress for different applied cycles can be obtained. The fibermatrix interface shear stress decreases from τ i 5 11.6 MPa at the applied cycle number N 5 10 to τ i 5 2.5 MPa at the applied cycle number N 5 400. The predicted fatigue HDE value versus the applied cycles curve agree with experimental data (i.e., from U 5 6.7 kPa at the applied cycle number N 5 1 to U 5 20 kPa at the applied cycle number N 5 400).

5.5

Static fatigue damage evolution

Under static fatigue loading, interface oxidation affects the internal damage evolution of fiber-reinforced CMCs. The effect of oxidation on the hysteresis loops is investigated using the oxidation region propagating model. The timedependent oxidation region is controlled by the interface frictional slip and diffusion of oxygen through multiple matrix cracking. Upon unloading and reloading, the interface slip and counter slip occur in the interface oxidation, slip, and debonded regions with different interface shear stresses. The theoretical relationship between the static fatigue hysteresis loops, hysteresis area, interface debonding and sliding, and oxidation time are established in this section. The influences of peak stress, matrix cracking, fiber volume fraction, and oxidation temperature on the damage evolution of CMCs under static fatigue loading are analyzed. The static fatigue hysteresis loops of C/[SiBC] composites at 1200 C are predicted. The static fatigue hysteresis loops for different oxidation times, the HDE value versus the oxidation time curve, the interface debonding ratio 2ld/lc versus the oxidation time curve, the interface oxidation ratio ξ/ld versus the oxidation time curve, and the interface sliding ratio y/ld curve of C/SiC composites at 800 C and σmax 5 180 MPa are shown in Fig. 5.44. The static fatigue hysteresis loops evolve with increasing oxidation time, corresponding to the interface slip Cases II, III, and IV. The fatigue HDE value increases first with oxidation time to the peak value, then decreases to the steady-state value. The interface

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Figure 5.43 (A) The experimental and theoretical fatigue HDE values versus the applied cycle curves; (B) the theoretical fatigue HDE versus the fibermatrix interface shear stress curve; and (C) the experimental and theoretical fibermatrix interface shear stress versus the applied cycle number curves of 3D SiC/SiC composites at 1300 C in air atmosphere. HDE, Hysteresis dissipated energy.

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Figure 5.44 (A) The static fatigue hysteresis loops; (B) the fatigue HDE value versus the oxidation time curve; (C) the fibermatrix interface debonding ratio of 2ld/lc versus the oxidation time curve; (D) the fibermatrix interface oxidation ratio of ξ/ld versus the oxidation time curve; and (E) the fibermatrix interface slip ratio of y/ld versus the oxidation time curve of C/SiC composites at 800 C. HDE, Hysteresis dissipated energy.

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361

Figure 5.44 (Continued)

debonded length increases with oxidation time under constant static fatigue peak stress due to the interface oxidation at the debonded region. An increase in the oxidation region and low interface shear stress in the oxidation region propagate the interface debonded length, the interface debonding ratio increases with oxidation time, and approaches to 1.0 when the interface completely debonds. The interface oxidation ratio increases with oxidation time, and when the interface oxidation region occupies the entire interface debonded region, the interface oxidation ratio approaches 1.0. The interface counter slip length is also affected by the interface oxidation region, and increases with oxidation time. When the interface counter-slip length approaches the interface debonded length, the hysteresis loops correspond to interface slip Case IV.

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Figure 5.45 (A) The static fatigue HDE value versus the oxidation time curve; (B) the fibermatrix interface debonded ratio of 2ld/lc versus the oxidation time curve; (C) the fibermatrix interface oxidation ratio of ξ/ld versus the oxidation time curve; and (D) the fibermatrix interface counter-slip ratio of y/ld versus the oxidation time curve at different fatigue peak stresses of σmax 5 180 and 200 MPa. HDE, Hysteresis dissipated energy.

5.5.1 Results and discussion The effects of static peak stress, matrix crack spacing, fiber volume fraction, and oxidation temperature on fatigue HDE values, interface debonding, oxidation, and slip ratios are analyzed in this section.

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Figure 5.45 (Continued)

5.5.1.1 Effect of fatigue peak stress on static fatigue damage evolution The static fatigue HDE value, interface debonding, oxidation, and slip ratio versus the oxidation time curves at different peak stresses are shown in Fig. 5.45. When the static peak stress increases, the static fatigue HDE value increases at the same oxidation time, corresponding to the different interface slip cases (i.e., Cases II, III, and IV). The interface debonding ratio increases, and the oxidation time for approaching complete debonding decreases. The interface oxidation ratio decreases when the interface partially debonds; however, when the interface completely debonds, the interface oxidation ratio for different peak stresses approach the same

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Figure 5.46 (A) The static fatigue HDE value versus the oxidation time curve; (B) the fibermatrix interface debonded ratio of 2ld/lc versus the oxidation time curve; (C) the fibermatrix interface oxidation ratio of ξ/ld versus the oxidation time curve; and (D) the fibermatrix interface counter-slip ratio of y/ld versus the oxidation time curves for different matrix crack spacings of lc 5 200 and 240 μm. HDE, Hysteresis dissipated energy.

value. The interface counter slip ratio increases when the interface is partially debonding, and decreases when the interface is completely debonded.

5.5.1.2 Effect of matrix crack spacing on static fatigue damage evolution The static fatigue HDE value, interface debonding, oxidation, and slip ratio versus the oxidation time curves at different matrix crack spacings are shown in Fig. 5.46.

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

365

Figure 5.46 (Continued)

When the matrix crack spacing increases, the fatigue HDE decreases, and the oxidation time of static fatigue hysteresis loops corresponding to interface slip Case IV increases. The interface debonding ratio decreases, and the oxidation time for complete debonding increases. The interface oxidation ratio decreases when the interface completely debonds, and the interface slip ratio decreases when the interface is completely debonded.

5.5.1.3 Effect of fiber volume fraction on static fatigue damage evolution The static fatigue HDE value, interface debonding, oxidation, and slip ratio versus the oxidation time curves at different fiber volume fractions are shown in Fig. 5.47.

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Durability of Ceramic-Matrix Composites

Figure 5.47 (A) The static fatigue HDE value versus the oxidation time curve; (B) the fibermatrix interface debonded ratio of 2ld/lc versus the oxidation time curve; (C) the fibermatrix interface oxidation ratio of ξ/ld versus the oxidation time curve; and (D) the fibermatrix interface counter-slip ratio of y/ld versus the oxidation time curves for different fiber volume fractions of Vf 5 35% and 40%. HDE, Hysteresis dissipated energy.

When the fiber volume fraction increases, the static fatigue HDE value decreases corresponding to the interface slip Cases II, III, and IV. The interface debonding ratio decreases, and the oxidation time corresponding to the interface completely debonding increases The interface oxidation ratio increases when the interface is partially debonding and approaches the same value when the interface is completely debonded. The interface slip ratio increases when the interface is partially debonding and decreases when the interface completely debonds.

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

367

Figure 5.47 (Continued)

5.5.1.4 Effect of oxidation temperature on static fatigue damage evolution The static fatigue HDE value, interface debonding, oxidation, and slip ratio versus the oxidation time curves at different oxidation temperatures are shown in Fig. 5.48. When the oxidation temperature increases, the fatigue HDE evolves rapidly with oxidation time due to the increasing rate of the interface oxidation region propagation. The interface debonding ratio increases rapidly with oxidation time, and the interface debonding ratio is much higher when the interface is partially debonding. The interface oxidation ratio increases at the same oxidation time, and the oxidation time corresponding to the interface completely oxidation decreases. The interface counter slip ratio increases when the interface partially or completely debonds.

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Durability of Ceramic-Matrix Composites

Figure 5.48 (A) The static fatigue HDE value versus the oxidation time curve; (B) the fibermatrix interface debonded ratio of 2ld/lc versus the oxidation time curve; (C) the fibermatrix interface oxidation ratio of ξ/ld versus the oxidation time curve; and (D) the fibermatrix interface counter-slip ratio of y/ld versus the oxidation time curves at 800 C and 900 C. HDE, Hysteresis dissipated energy.

5.5.2 Experimental comparisons Fantozzi and Reynaud (2009) investigated the static fatigue behavior of C/[Si 2 B 2 C] composites at 1200 C under σmax 5 170 MPa. The static fatigue hysteresis loops for different oxidation time curves and the static fatigue HDE value versus the oxidation time curves are shown in Fig. 5.49. The static fatigue hysteresis loops evolve with oxidation time, corresponding to interface slip Cases II, III,

Damage evolution of ceramic-matrix composites under cyclic fatigue loading

369

Figure 5.48 (Continued)

and IV. The static fatigue HDE value increases with oxidation time, first to the peak value and then decreases with oxidation time. The predicted static fatigue hysteresis loops and fatigue HDE value versus the oxidation time curve agree with experimental data, as shown in Fig. 5.49.

5.6

Conclusion

The damage evolution of fiber-reinforced CMCs under cyclic fatigue or static fatigue loading at room and elevated temperatures have been investigated in this

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Durability of Ceramic-Matrix Composites

Figure 5.49 (A) The static fatigue hysteresis loops for different oxidation time curves; and (B) the static fatigue HDE value versus the oxidation time curve of C/[Si 2 B 2 C] composites at 1200 C. HDE, Hysteresis dissipated energy. Source: Reproduced from Longbiao, L. 2016e. Modeling the effect of oxidation on hysteresis loops of carbon fiber-reinforced ceramic-matrix composites under static fatigue at elevated temperature. J. Eur. Ceram. Soc. 36, 465480.

chapter using the fatigue hysteresis-based damage parameter, that is, HDE, HDEbased damage parameter, hysteresis modulus and peak strain. The effects of fiber volume fraction, matrix cracking density, interface shear stress and debonded energy, and fibers failure on the fatigue damage evolution of CMCs have been discussed. The experimental HDE and HDE-based damage parameters of different CMCs under cyclic loadingunloading tensile, cyclic fatigue loading, and static fatigue loading have been predicted.

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References Ahn, B.K., Curtin, W.A., 1997. Strain and hysteresis by stochastic matrix cracking in ceramic matrix composites. J. Mech. Phys. Solids 45, 177209. Cheng, Q., 2010. Simulation and Assessment on the Low Velocity Impact Properties of Plain Woven C/SiC Composites (Ph.D. thesis). Northwestern Polytechnical University, Xi’an, China. Cho, C., Holmes, J.W., Barber, J.R., 1991. Estimation of interfacial shear in ceramic composites from frictional heating measurements. J. Am. Ceram. Soc. 74, 28022808. Dalmaz, A., Reynaud, P., Rouby, D., Fantozzi, G., 1996. Damage propagation in carbon/silicon carbide composites during tensile tests under SEM. J. Mater. Sci. 31, 42134219. Dalmaz, A., Reynaud, P., Rouby, D., Fantozzi, G., Abbe, F., Bourgeon, M., 1999. Cyclic fatigue behavior at room temperature and at high temperature under inert atmosphere of a C/SiC multilayer composites. Key Eng. Mater. 164, 325328. Du, S., Qiao, S., Ji, G., Han, D., 2002. Tension-tension fatigue behaviour of 3D-C/SiC composite at room temperature and 1300 C. J. Mater. Eng. 9, 2225. Evans, A.G., Zok, F.W., McMeeking, R.M., 1995. Fatigue of ceramic matrix composites. Acta Metall. Mater. 43, 859875. Fang, G., Gao, X., Song, Y., 2016. Tension-compression fatigue behaviour and failure mechanisms of needled C/SiC composite. J. Mater. Eng. 44, 7882. Fantozzi, G., Reynaud, P., 2009. Mechanical hysteresis in ceramic matrix composites. Mater. Sci. Eng. A 521522, 1823. Guo, H., Jia, P., Wang, B., Jiao, G., Zeng, Z., 2015. Study on constituent properties of a 2DSiC/SiC composite by hysteresis measurements. Chin. J. Theor. Appl. Mech. 47, 260269. Han, D., Qiao, S.R., Li, M., Hou, J.T., Wu, X.J., 2004. Comparison of fatigue and creep behavior between 2D and 3D-C/SiC composites. Acta Metall. Sin. (Engl. Lett.) 17, 569574. Hild, F., Burr, A., Leckie, F.A., 1996. Matrix cracking and debonding of ceramic-matrix composites. Int. J. Solids Struct. 33, 12091220. Holmes, J.W., Shuler, S.F., 1990. Temperature rise during fatigue of fiber-reinforced ceramics. J. Mater. Sci. Lett. 9, 1290. Holmes, J.W., Cho, C.D., 1992. Experimental observations of frictional heating in fiberreinforced ceramics. J. Am. Ceram. Soc. 75, 929938. Holmes, J.W., Kotil, T., Founds, W.T., 1989. High temperature fatigue of SiC-fiberreinforced Si3N4 ceramic composites. Proceedings of Symposium on High-Temperature Composites. Techomics, Basel, Switzerland, and Lancaster, UK. Holmes, J.W., Wu, X., Sorensen, B.F., 1994. Frequency dependence of fatigue life and internal heating of a fiber-reinforced/ceramic-matrix composite. J. Am. Ceram. Soc. 77, 32843286. Hutchinson, J.W., Jensen, H.M., 1990. Models of fiber debonding and pullout in brittle composites with friction. Mech. Mater. 9, 139163. Jacob, D., 2010. Fatigue behavior of an advanced SiC/SiC composite with an oxidation inhibited matrix at 1200 C in air and in steam. In: AFIT/GEA/ENY/10-M07. Keith, W.P., Kedward, K.T., 1995. The stress-strain behavior of a porous unidirectional ceramic matrix composite. Composites 26, 163174. Kim, J., Liaw, P.K., 2005. Characterization of fatigue damage modes in nicalon/calcium aluminosilicate composites. J. Eng. Mater. Technol. 127, 815.

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Kotil, T., Holmes, J.W., Comninou, M., 1990. Origin of hysteresis observed during fatigue of ceramic-matrix composites. J. Am. Ceram. Soc. 73, 18791883. Kostopoulos, V., Vellios, L., Pappas, Y.Z., 1997. Fatigue behavior of 3-d SiC/SiC composites. J. Mater. Sci. 32, 215220. Longbiao, L., 2011. Fatigue Damage Models and Life Prediction of Long-Fiber-Reinforced Ceramic-Matrix Composites (Ph.D. thesis). Nanjing University of Aeronautics and Astronautics, Nanjing, China. Longbiao, L., 2013a. Fatigue hysteresis behavior of cross-ply C/SiC ceramic matrix composites at room and elevated temperatures. Mater. Sci. Eng. A 586, 160170. Longbiao, L., 2013b. Estimate interface shear stress of unidirectional C/SiC ceramic matrix composites from hysteresis loops. Appl. Compos. Mater. 20, 693707. Longbiao, L., 2014. Modeling fatigue hysteresis behavior of unidirectional C/SiC ceramicmatrix composites. Compos., B 66, 466474. Longbiao, L., 2015a. A hysteresis dissipated energy-based parameter for damage monitoring of carbon fiber-reinforced ceramic-matrix composites under fatigue loading. Mater. Sci. Eng. A 634, 188201. Longbiao, L., 2015b. Damage monitoring of unidirectional C/SiC ceramic-matrix composite under cyclic fatigue loading using a hysteresis loss energy-based damage parameter at room and elevated temperatures. Appl. Compos. Mater. 23, 357374. Longbiao, L., 2016a. Hysteresis loops of carbon fiber-reinforced ceramic-matrix composites with different fiber preforms. Ceram. Int. 42, 1653516551. Longbiao, L., 2016b. Damage development in fiber-reinforced ceramic-matrix composites under cyclic fatigue loading using hysteresis loops at room and elevated temperatures. Int. J. Fract. 199, 3958. Longbiao, L., 2016c. Fatigue hysteresis of carbon fiber-reinforced ceramic-matrix composites at room and elevated temperatures. Appl. Compos. Mater. 23, 127. Longbiao, L., 2016d. Effects of temperature, oxidation and fiber preforms on interface shear stress degradation in fiber-reinforced ceramic-matrix composites. Mater. Sci. Eng. A 674, 588603. Longbiao, L., 2016e. Modeling the effect of oxidation on hysteresis loops of carbon fiberreinforced ceramic-matrix composites under static fatigue at elevated temperature. J. Eur. Ceram. Soc. 36, 465480. Longbiao, L., 2017a. Synergistic effects of temperature, oxidation and stress level on fatigue damage evolution and lifetime prediction of cross-ply SiC/CAS ceramicmatrix composites through hysteresis-based parameters. J. Mater. Eng. Perform. 26, 56815693. Longbiao, L., 2017b. Fatigue hysteresis behaviour of cross-ply C/SiC ceramic matrix composites at room and elevated temperatures. Mater. Sci. Eng. A 586, 160170. Longbiao, L., 2017c. Modeling thermomechanical fatigue hysteresis loops of long-fiberreinforced ceramic-matrix composites under out-of-phase cyclic loading condition. Int. J. Fatigue 105, 3442. Longbiao, L., 2017d. Comparisons of interface shear stress degradation rate between C/SiC and SiC/SiC ceramic-matrix composites under cyclic fatigue loading at room and elevated temperatures. Compos. Interfaces 24, 171202. Longbiao, L., 2018. Hysteresis loops of fiber-reinforced ceramic-matrix composites under inphase/out-of-phase thermomechanical and isothermal cyclic loading. Compos. Interfaces 25, 855882. Longbiao, L., Song, Y., Sun, Z., 2008. Influence of fiber Poisson contraction on matrix cracking development of ceramic matrix composites. J. Aerosp. Power 23, 21962201.

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Longbiao, L., Song, Y., Sun, Y., 2014. Modeling tensile behavior of unidirectional C/SiC ceramic matrix composites. Mech. Compos. Mater. 49, 659672. Liu, C.D., Cheng, L.F., Luan, X.G., Lin, B., Zhou, J., 2008. Damage evolution and real-time non-destructive evaluation of 2D carbon-fiber/SiC-matrix composites under fatigue loading. Mater. Lett. 62, 39223924. Lynch, C.S., Evans, A.G., 1996. Effects of off-axis loading on the tensile behavior of a ceramic matrix composite. J. Am. Ceram. Soc. 79, 31133123. Mall, S., Engesser, J.M., 2006. Effects of frequency on fatigue behavior of CVI C/SiC at elevated temperature. Compos. Sci. Technol. 66, 863874. Marshall, D.B., Evans, A.G., 1985. Failure mechanisms in ceramic-fiber/ceramic-matrix composites. J. Am. Ceram. Soc. 68, 225231. Marshall, D.B., Oliver, W.C., 1987. Measurement of interfacial mechanical properties in fiber-reinforced ceramic composites. J. Am. Ceram. Soc. 70, 542548. McNulty, J.C., Zok, F.W., 1999. Low cycle fatigue of Nicalon-fiber-reinforced ceramic composites. Compos. Sci. Technol. 59, 15971607. Mei, H., Cheng, L.F., 2009. Comparison of the mechanical hysteresis of carbon/ceramic matrix composites with different fiber preforms. Carbon 47, 10341042. Michael, K., 2010. Fatigue behavior of a SiC/SiC composite at 1000 C in air and steam. In: AFIT/GAE/ENY/10-D01. Minford, E., Prewo, K.M., 1986. Fatigue of silicon carbide reinforced lithium-aluminosilicate glass-ceramics. Tailoring Multiphase and Composite Ceramics. Plenum Publishing Corporation, New York, ISBN: 978-1-4613-2233-7. Opalski, F.A., Mall, S., 1994. Tension-compression fatigue behavior of a silicon carbide calcium-aluminosilicate ceramic matrix composites. J. Reinforced Plast. Compos. 13, 420438. Pryce, A.W., Smith, P.A., 1993. Matrix cracking in unidirectional ceramic matrix composites under quasi-static and cyclic loading. Acta Metall. Mater. 41, 12691281. Reynaud, P., 1996. Cyclic fatigue of ceramic-matrix composites at ambient and elevated temperatures. Compos. Sci. Technol. 56, 809814. Rouby, D., Reynaud, P., 1993. Fatigue behavior related to interface modification during load cycling in ceramic-matrix fiber composites. Compos. Sci. Technol. 48, 109118. Shi, D.Q., Jing, X., Yang, X.G., 2015. 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, 3844. Solti, J.P., Robertson, D.D., Mall, S., 2000. Estimation of interfacial properties from hysteresis energy loss in unidirectional ceramic matrix composites. Adv. Compos. Mater. 9, 161173. Sørensen, B.F., Talreja, R., Sorensen, O.T., 1993. Micromechanical analysis of damage mechanisms in ceramic matrix composites during mechanical and thermal loading. Composites 24, 124140. Vagaggini, E., Domergue, J.M., Evans, A.G., 1995. Relationships between hysteresis measurements and the constituent properties of ceramic matrix composites: I, Theory. J. Am. Ceram. Soc. 78, 27092720. Wang, Y.Q., Zhang, L.T., Cheng, L.F., Ma, J.Q., Zhang, W.H., 2008. Tensile performance and damage evolution of a 2.5D C/SiC composite characterized by acoustic emission. Appl. Compos. Mater. 15, 183188. Xie, J.B., Fang, G.D., Chen, Z., Liang, J., 2016. Modeling of nonlinear mechanical behavior for 3D needled C/C-SiC composites under tensile load. Appl. Compos. Mater. 23, 783797.

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Xu, R., 2008. Modeling Hysteresis Loops and Life Prediction in Cross-Ply Ceramic Matrix Composites (Master thesis). Nanjing University of Aeronautics and Astronautics, Nanjing, China. Yang, F., 2011. Research on Fatigue Behaviour of 2.5D Woven Ceramic Matrix Composites (Master thesis). Nanjing University of Aeronautics and Astronautics, Nanjing, China. Zawada, L.P., Butkus, L.M., Hartman, G.A., 1991. Tensile and fatigue behavior of silicon carbide fiber-reinforced aluminosilicate glass. J. Am. Ceram. Soc. 74, 28512858. Zhu, S.J., Mizuno, M., Nagano, Y., Cao, J.W., Kagawa, Y., Kaya, H., 1998. Creep and fatigue behavior in an enhanced SiC/SiC composite at high temperature. J. Am. Ceram. Soc. 81, 22692277.

Further reading Budiansky, B., Hutchinson, J.W., Evans, A.G., 1986. Matrix fracture in fiber-reinforced ceramics. J. Mech. Phys. Solids 34, 167189. Curtin, W.A., 1993. Multiple matrix cracking in brittle matrix composites. Acta Metall. Mater. 41, 13691377. Kuo, W.S., Chou, T.W., 1995. Multiple cracking of unidirectional and cross-ply ceramic matrix composites. J. Am. Ceram. Soc. 78, 745755. Lamon, J., 2001. A micromechanics-based approach to the mechanical behavior of brittlematrix composites. Compos. Sci. Technol. 61, 22592272. Morscher, G.N., Singh, M., Kiser, J.D., Freedman, M., Bhatt, R., 2007. Modeling stressdependent matrix cracking and stress-strain behavior in 2D woven SiC fiber reinforced CVI SiC composites. Compos. Sci. Technol. 67, 10091017.

Fatigue life prediction of ceramicmatrix composites based on hysteresis dissipated energy

6.1

6

Introduction

When fiber-reinforced ceramic-matrix composites (CMCs) are subjected to cyclic fatigue loading, multiple fatigue damage mechanisms such as matrix cracking, fibermatrix interface debonding and fibers fracture occur upon first loading to the fatigue peak stress (Gowayed et al., 2015a,b). When the applied cycle number increases, the fibermatrix interface wear occurs in the debonded region, leading to the degradation of the fibermatrix interface shear stress. The repeated sliding between the fiber and the matrix reduces the load transfer capacity between the fiber and the matrix, and the fibers strength (Longbiao, 2013, 2014, 2015a,b,c, 2016a,b,c, 2017a). At elevated temperature in oxidative environment, the oxygen enters inside the material through matrix cracks, leading to the oxidation of the fibermatrix interface and the fibers. When the oxidation time increases, the oxidation region propagates along the fibermatrix interface or the matrix cracking. The fibermatrix interface shear stress and the fibers strength in the oxidation region degrade. Under cyclic fatigue loading, the broken fibers fraction increases as does the degradation of the fibermatrix interface shear stress and the fibers strength, caused by the damage mechanisms of fibermatrix interface wear or interface oxidation. With increasing applied cycles, the composite modulus and strength decrease when the fibers gradually fracture. The composite fatigue fractures when the broken fibers fraction approaches the critical value (Longbiao, 2015a). Under cyclic fatigue loading of fiber-reinforced CMCs, the fibermatrix interface wear decreases the fibermatrix interface shear stress and fibers strength, leading to an increase in fibers fracture with applied cycles. When the residual strength of the composite approaches fatigue peak stress, the composite fatigue failure occurs (Reynaud, 1996). Evans et al. (1995) predicted the fatigue life SN curve of unidirectional SiC/CAS composites based on the Reynaud model (Reynaud, 1996). However, the effect of fiber strength degradation and fibers oxidation at elevated temperatures on the fatigue life is not considered. Case et al. (1998) developed an approach to predict the fatigue life of fiber-reinforced CMCs based on the residual strength and critical element model; however, the relationship between the microdamage parameter and the macroresidual strength is not established. Sujidkul et al. (2014) relates the damage mechanisms of matrix cracking and fibers fracture with electric resistance change and developed an approach to monitor the damage evolution of fiber-reinforced CMCs using electric resistance; however, the use of the Durability of Ceramic-Matrix Composites. DOI: https://doi.org/10.1016/B978-0-08-103021-9.00006-X © 2020 Elsevier Ltd. All rights reserved.

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electric resistance method is limited by the environment and temperature. Gyekenyesi et al. (2005) developed a macrofatigue life prediction model of NASA Life based on residual strength. However, the microstructure characteristic and the effect of the environment are not considered in the fatigue life model. Min et al. (2014) obtained the microstress field of fiber, matrix, and fibermatrix interface by combining FEM stress analysis with the three-phase micromechanics model and predicted the fatigue life of 2D C/SiC composites at room temperature. Xu (2008) considered the damage mechanisms of fibermatrix interface wear and the degradation of fibermatrix interface shear stress, and combined the fibers fracture model with the interface wear model to predict the fatigue life of unidirectional and cross-ply fiber-reinforced CMCs at room temperature. Lu et al. (2014) investigated the tensiontension fatigue behavior of 2D C/SiC composites at room temperature and predicted the fatigue life using the nominal stress method. Luan et al. (2004) investigated the stress corrosion mechanism and fatigue life prediction of 2D C/SiC composites at elevated temperature. The multiple fatigue damage mechanisms of matrix cracking, fibermatrix interface debonding, sliding, wear, and oxidation, and the fibers failure can be monitored through fatigue hysteresis dissipated energy (HDE). Under cyclic fatigue loading, the matrix cracking density, fibermatrix interface debonded length, fibermatrix interface shear stress degradation, and fibers failure affect the area of the fatigue hysteresis loops. Based on the fatigue hysteresis loops models considering multiple fatigue damage mechanisms of matrix cracking, fibermatrix interface debonding, wear, and oxidation, and fibers fracture, the fatigue HDE and fatigue HDE-based damage parameters can be obtained and used to monitor damage development and predict the fatigue lifetime of fiberreinforced CMCs. Longbiao (2015b,c) developed an approach to predict the fatigue life of fiber-reinforced CMCs based on the fatigue HDE, considered the fibermatrix interface wear mechanism, and predicted the fatigue damage evolution and fatigue life of unidirectional fiber-reinforced CMCs. In this chapter, fatigue life predictions of fiber-reinforced CMCs based on fatigue HDE is investigated. The theoretical relationships between the fatigue hysteresis loops, multiple fatigue damage mechanisms of matrix cracking, fibermatrix interface debonding and fibers fracture, HDE, and HDE-based damage parameters are established. The effects of fatigue peak stress, fatigue stress ratio, matrix crack spacing, and fiber volume fraction on the fibermatrix interface debonding and sliding, HDE, and HDE-based damage parameters are analyzed. The experimental fatigue life SN curves of unidirectional, cross-ply, 2D, 2.5D, and 3D fiber-reinforced CMCs at room and elevated temperatures are predicted.

6.2

Theoretical analysis

When fiber-reinforced CMCs are subjected to cyclic fatigue loading, the fatigue hysteresis loops occur as the fiber sliding relative to the matrix in the fibermatrix interface debonding region. The fatigue hysteresis loops area is the energy

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dissipation for each cycle and can reflect the damage evolution of fiber-reinforced CMCs. The fatigue HDE can be determined using the equation: ð σmax

U5

σmin



 εunloading ðσÞ 2 εreloading ðσÞ dσ

(6.1)

where εunloading denotes the composite unloading strain; and εreloading denotes the composite reloading strain. The HDE-based damage parameter can be determined using the equation: Φ5

Un 2 Uinitial Ue

(6.2)

where Un denotes the HDE at the Nth applied cycle; Uinitial denotes the HDE at the first applied cycle; and Ue denotes the elastic strain energy, which is determined using the equation: Ue 5

1 ðσmax 2 σmin Þðεmax 2 εmin Þ 2

(6.3)

When the fiber-reinforced CMCs are subjected to the cyclic fatigue loading between valley and peak stresses, the relative sliding between the fiber and the matrix degrades the fibermatrix interface shear stress and the fibers strength, leading to the gradual fracture of the fibers (Rouby and Reynaud, 1993; Evans et al., 1995; Solti et al., 1997; Fantozzi et al., 2001; Mall and Engesser, 2006; Gowayed et al., 2015a,b; Longbiao, 2013). The global load sharing criteria combined with the two parameter Weibull model are adopted in the present analysis to determine the stress distribution between intact and broken fibers, as well as the broken fibers fraction under cyclic fatigue loading. The relationship between the applied stress and intact and broken fibers stress can be determined using the equations (Curtin, 2000):   σ 5 T 1 2 Pf 1 hTb iPf Vf

(6.4)

where Pf is the fiber broken fraction; and hTbi denotes the broken fibers stress.   mf11    T σ 0 mf τ i Pf 5 1 2 exp 2 σk σ0 ðN Þ τ i ðN Þ

(6.5)

2 3 (  mf 11  mf τ ð N Þ τ i i 5 σ 0 ðN Þ σk 3 σ0σð0N Þ 1 2 exp4 2 σTk σ0 T τi τ i ðN Þ 8 9 mf <  mf 11  = T τ i σk 2 exp 2 σTk σ k ðN Þ : Pf τ i ðN Þ ;

T hTb i 5 Pf

 mf 11 

mf

(6.6)

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where mf denotes the Weibull modulus of fiber; σk denotes the characteristic strength of the fiber; σ0 denotes the fiber reference strength at a length l0; σ0(N) denotes the fiber strength at the Nth cycle; and τ i(N) denotes the interface shear stress at the Nth cycle. Combining Eqs. (6.4)(6.6), the relationship between the applied stress, intact fibers stress, interface shear stress, fiber strength, and fiber Weibull modulus is determined using the equation: ( "   #)   σ mf 11 σ ðN Þmf τ ðN Þ σ T mf 11 σ0 mf τ i k 0 i 5T 1 2 exp 2 Vf σ0 τi σk T σ 0 ðN Þ τ i ðN Þ (6.7) With increasing applied cycles, the fiber strength decreases due to the interface wear, and the degradation of fiber strength can be determined using the equation (Lee and Stinchcomb, 1994): σ0 ðN Þ 5 σ0 ½1 2 p1 ðlog N Þp2 

(6.8)

where p1 and p2 are model parameters. The interface wear can also degrade the fibermatrix interface shear stress, and the relationship between the applied cycles and the fibermatrix interface shear stress can be determined using the equation (Evans et al., 1995):

τ io 2 τ i ðN Þ 5 1 2 exp 2ωN λ τ io 2 τ imin

(6.9)

where τ io denotes the fibermatrix interface shear stress at the first applied cycle; τ imin denotes the fibermatrix interface shear stress at steady-state; and ω and λ are model parameters. Combining Eqs. (6.7)(6.9), the intact fiber stress T can be determined for different fatigue peak stress and applied cycles. Substituting Eqs. (6.8) and (6.9) and intact fiber stress T into Eq. (6.5), the fiber failure probability corresponding to different applied cycle number can be determined. When the fraction of broken fibers approaches the critical value, the composite fatigue fractures.

6.3

Results and discussions

When fiber-reinforced CMCs are subjected to cyclic fatigue loading, the fibers gradually fracture, which affects the HDE and HDE-based damage parameters. The effects of fatigue peak stress, fatigue stress ratio, matrix crack spacing, and fiber volume fraction on the fibermatrix interface debonding and sliding, HDE and HDE-based damage parameters of SiC/CAS composites are discussed in this section.

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6.3.1 Effects of fatigue peak stress on fibermatrix interface debonding, HDE, and HDE-based damage parameters The fatigue life SN data corresponding to different peak stresses of σmax 5 210, 220, 240, 300, and 320 MPa is shown in Fig. 6.1, in which the corresponding failure cycles are 20,370, 4920, 3428, 2405, and 2171, respectively. When the fatigue peak stress decreases, the corresponding failure cycle increases. The relationships between the fibermatrix interface debonding, fibermatrix interface shear stress, fatigue peak stress, and applied cycles are shown in Fig. 6.2. The fibermatrix interface debonded ratio (2ld/lc) versus the fibermatrix interface shear stress curves at σmax 5 210 and 320 MPa are shown in Fig. 6.2A. The fibermatrix interface debonded ratio versus the interface shear stress curve can be divided into two regions, namely, the interface partial debonding (2ld/lc , 1) region and complete debonding (2ld/lc 5 1) region. Under constant fatigue peak stress of σmax 5 210 or 320 MPa, the fibermatrix interface debonded ratio of 2ld/lc increases with decreasing fibermatrix interface shear stress. When the fatigue peak stress is σmax 5 210 MPa, the fibermatrix interface debonded ratio is 2ld/ lc 5 0.74 at τ i 5 25 MPa. When the fatigue peak stress is σmax 5 320 MPa, the fibermatrix interface debonded ratio is 2ld/lc 5 0.46 at τ i 5 25 MPa. When the fatigue peak stress increases, the fibermatrix interface shear stress corresponding to the complete debonding (i.e., 2ld/lc 5 1) increases. When the fatigue peak stress is σmax 5 210 MPa, the fibermatrix interface shear stress for complete debonding

Figure 6.1 The fatigue life SN data of SiC/CAS composites. Source: Reproduced from Longbiao, L., 2015b. A hysteresis dissipated energy-based damage parameter for life prediction of carbon fiber-reinforced ceramic-matrix composites under fatigue loading. Composites, B 82, 108128.

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Figure 6.2 (A) The fibermatrix interface debonded ratio of 2ld/lc versus the fibermatrix interface shear stress curve; and (B) the fibermatrix interface debonded ratio of 2ld/lc versus the applied cycle number curves at different fatigue peak stresses.

is τ i 5 12 MPa. When the fatigue peak stress is σmax 5 320 MPa, the fibermatrix interface shear stress for complete debonding is τ i 5 19 MPa. The fibermatrix interface debonding ratio of (2ld/lc) versus the applied cycles curves for different fatigue peak stresses of σmax 5 210 and 320 MPa are shown in Fig. 6.2B. When the cycle number increases, the fibermatrix interface debonding ratio increases due to the decrease of the fibermatrix interface shear stress caused by the interface wear. The fibermatrix interface debonding ratio versus the

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applied cycle number curve can be divided into two regions, namely, the interface partial debonding (2ld/lc , 1) region and complete debonding (2ld/lc 5 1) region. When the fibermatrix interface partially debonds, the fibermatrix interface debonding ratio increases with fatigue peak stress at the same applied cycle. When the cycle number is N 5 10, the fibermatrix interface debonding ratio is 2ld/ lc 5 0.36 at σmax 5 320 MPa. When the fatigue peak stress is σmax 5 210 MPa, the fibermatrix interface debonding ratio is 2ld/lc 5 0.23 at cycle number N 5 10. When the fatigue peak stress increases, the applied cycle number for interface complete debonding decreases. When the fatigue peak stress is σmax 5 320 MPa, the cycle number is N 5 1194 for fibermatrix interface complete debonding. When the fatigue peak stress decreases to σmax 5 210 MPa, the cycle number is N 5 1639 for the fibermatrix interface complete debonding. The relationships between the fatigue HDE, HDE-based damage parameter, fatigue peak stress, and applied cycles are shown in Fig. 6.3. The fatigue HDE versus applied cycle number curves for different fatigue peak stresses of σmax 5 210, 240, and 320 MPa are shown in Fig. 6.3A. At high fatigue peak stress of σmax 5 320 MPa, the fatigue HDE increases with applied cycles till fatigue fracture. At intermediate fatigue peak stress of σmax 5 240 MPa, the fatigue HDE increases with applied cycles to the peak value and then decreases with applied cycles till fatigue fracture. At low fatigue peak stress of σmax 5 210 MPa, the fatigue HDE increases with applied cycles to the peak value, then decreases, and increases again with applied cycles till fatigue fracture. The peak HDE increases with peak stress, and the failure applied cycle number decreases with peak stress. The HDE-based damage parameter Φ versus applied cycle number curves for different fatigue peak stresses of σmax 5 210, 240, and 320 MPa are shown in Fig. 6.3B. At high fatigue peak stress of σmax 5 320 MPa, the HDE-based damage parameter Φ increases with applied cycles till fatigue fracture. At intermediate peak stress of σmax 5 240 MPa, the HDE-based damage parameter Φ increases with applied cycles to the peak value and then decreases with applied cycles till fatigue fracture. At low peak stress of σmax 5 210 MPa, the HDE-based damage parameter Φ increases with applied cycles to the peak value, then decreases, and increases again with applied cycles till fatigue fracture. The evolution of fatigue HDE and HDE-based damage parameter can be divided into three cases: 1. Case I, at high peak stress level, the HDE and HDE-based damage parameter increase with applied cycles till fatigue fracture. 2. Case II, at intermediate peak stress level, the HDE and HDE-based damage parameter increase with applied cycles to the peak value and then decrease with applied cycles till fatigue fracture. 3. Case III, at low peak stress level, the HDE and HDE-based damage parameter increase with applied cycles to the peak value, then decrease, and increase again with applied cycles till fatigue fracture.

The damage mechanisms of fibermatrix interface debonding, wear, and fiber failure affect the evolution characteristics of HDE and HDE-based damage parameters with applied cycles.

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Figure 6.3 (A) The fatigue HDE versus applied cycle number curve; and (B) the HDE-based damage parameter versus applied cycle number curve at different peak stresses of σmax 5 210, 240, and 320 MPa. HDE, Hysteresis dissipated energy.

6.3.2 Effects of fatigue stress ratio on HDE and HDE-based damage parameters The relationships between the fatigue peak stress, fatigue stress ratio, HDE, and applied cycles are shown in Fig. 6.4. When the fatigue peak stress is σmax 5 320 MPa, the fatigue HDE increases till final fatigue fracture with applied cycles; and when the fatigue stress ratio increases, the fatigue HDE decreases at the same applied cycle. When the fatigue peak stress is σmax 5 240 MPa, the fatigue

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Figure 6.4 The fatigue HDE versus applied cycle number curves for different stress ratio at fatigue peak stresses of (A) σmax 5 320 MPa; (B) σmax 5 240 MPa; and (C) σmax 5 210 MPa. HDE, Hysteresis dissipated energy.

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HDE increases first and then decreases with applied cycles; and when the fatigue stress ratio increases, the fatigue HDE decreases at the same applied cycle. When the fatigue peak stress is σmax 5 210 MPa, the fatigue HDE increases to the peak value first, then decreases, and increases again with applied cycles till fatigue fracture; and with increasing fatigue stress ratio, the fatigue HDE decreases at the same applied cycle. The relationships between fatigue peak stress, fatigue stress ratio, HDE-based damage parameter, and applied cycles are shown in Fig. 6.5. When the fatigue peak stress is σmax 5 300 MPa, the HDE-based damage parameter increases till final fatigue fracture with applied cycles; and when the fatigue stress ratio increases, the HDE-based damage parameter decreases at the same applied cycle. When the fatigue peak stress is σmax 5 240 MPa, the HDE-based damage parameter increases first and then decreases with applied cycles; and when the fatigue stress ratio increases, the HDE-based damage parameter decreases when the interface is partially debonding (i.e., 2ld/lc , 1), and increases when the interface completely debonds (i.e., 2ld/lc 5 1). When the fatigue peak stress is σmax 5 210 MPa, the HDE-based damage parameter increases to the peak value first, then decreases, and increases again with applied cycles till fatigue fracture With increasing fatigue stress ratio, the HDE-based damage parameter decreases when the interface is partially debonding (i.e., 2ld/lc , 1), and increases when the interface completely debonds (i.e., 2ld/lc 5 1).

6.3.3 Effects of matrix crack spacing on fibermatrix interface debonding, HDE, and HDE-based damage parameters The relationships between the fibermatrix interface debonding ratio of 2ld/lc, interface shear stress, matrix crack spacing, and fatigue peak stress are shown in Fig. 6.6. Under constant fatigue peak stress, with increasing matrix crack spacing, the fibermatrix interface debonding ratio decreases at the same interface shear stress when the interface is partially debonding, and the interface shear stress corresponds to the complete debonding decrease. When the fatigue peak stress is σmax 5 320 MPa, the fibermatrix interface shear stress is τ i 5 19 MPa for the fibermatrix complete debonding at lc 5 140 μm. When the matrix crack spacing is lc 5 200 μm, the fibermatrix interface shear stress for interface complete debonding is τ i 5 14 MPa; and when the matrix crack spacing is lc 5 240 μm, the fibermatrix interface shear stress for the interface complete debonding is τ i 5 12 MPa. When the fatigue peak stress is σmax 5 240 MPa, the fibermatrix interface shear stress is τ i 5 14 MPa for the fibermatrix interface complete debonding at lc 5 140 μm. When the matrix crack spacing is lc 5 200 μm, the fibermatrix interface shear stress for the interface complete debonding is τ i 5 10 MPa. When the matrix crack spacing is lc 5 240 μm, the fibermatrix interface shear stress for the interface complete debonding is τ i 5 8 MPa. When the fatigue peak stress is σmax 5 210 MPa, the fibermatrix interface shear stress is τ i 5 12 MPa for the fibermatrix interface complete debonding at lc 5 140 μm.

Fatigue life prediction of ceramic-matrix composites based on hysteresis dissipated energy

Figure 6.5 The HDE-based damage parameter versus applied cycle number curves for different stress ratio at fatigue peak stresses of (A) σmax 5 320 MPa; (B) σmax 5 240 MPa; and (C) σmax 5 210 MPa. HDE, Hysteresis dissipated energy.

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Figure 6.6 The fibermatrix interface debonded ratio versus the interface shear stress curves for different matrix crack spacing of lc 5 140, 200, and 240 μm at fatigue peak stresses of (A) σmax 5 320 MPa; (B) σmax 5 240 MPa; and (C) σmax 5 210 MPa.

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When the matrix crack spacing is lc 5 200 μm, the fibermatrix interface shear stress for the interface complete debonding is τ i 5 8 MPa; and when the matrix crack spacing is lc 5 240 μm, the fibermatrix interface shear stress for the interface complete debonding is τ i 5 7 MPa. The relationships between the fibermatrix interface debonding ratio of 2ld/lc, applied cycle number, matrix crack spacing, and fatigue peak stress are shown in Fig. 6.7. Under constant fatigue peak stress, with increasing matrix crack spacing, the fibermatrix interface debonding ratio decreases at the same applied cycle number when the interface partial debonding, and the applied cycle number corresponding to the complete interface debonding increases. When the fatigue peak stress is σmax 5 320 MPa, the critical applied cycle number is N 5 1194 for the fibermatrix interface complete debonding at lc 5 140 μm. When the matrix crack spacing is lc 5 200 μm, the critical applied cycle number for the interface complete debonding is N 5 1499; and when the matrix crack spacing is lc 5 240 μm, the critical applied cycle number for the interface complete debonding is N 5 1645. When the fatigue peak stress is σmax 5 240 MPa, the critical applied cycle number is N 5 1509 for the fibermatrix interface complete debonding at lc 5 140 μm. When the matrix crack spacing is lc 5 200 μm, the critical applied cycle number for the interface complete debonding is N 5 1825. When the matrix crack spacing is lc 5 240 μm, the critical applied cycle number for the interface complete debonding is N 5 1983. When the fatigue peak stress is σmax 5 210 MPa, the critical applied cycle number is N 5 1639 for the fibermatrix interface complete debonding at lc 5 140 μm. When the matrix crack spacing is lc 5 200 μm, the critical applied cycle number for the interface complete debonding is N 5 1957. When the matrix crack spacing is lc 5 240 μm, the critical applied cycle number for the interface complete debonding is N 5 2118. The relationships between the fatigue HDE, applied cycle number, matrix crack spacing, and fatigue peak stress are shown in Fig. 6.8. Under high fatigue peak stress of σmax 5 320 MPa, the fatigue HDE increases with applied cycles till fatigue fracture. With increasing matrix crack spacing, the fatigue HDE decreases. Under intermediate fatigue peak stress of σmax 5 240 MPa, the fatigue HDE increases first and then decreases with applied cycles. With increasing matrix crack spacing, the fatigue HDE decreases when the interface partially debonds (i.e., 2ld/lc , 1), and increases when the interface completely debonds (i.e., 2ld/lc 5 1). Under low fatigue peak stress of σmax 5 210 MPa, the fatigue HDE increases first, then decreases, and increases again with applied cycles. With increasing matrix crack spacing, the fatigue HDE decreases when the interface is partially debonding (i.e., 2ld/lc , 1), and increases when the interface completely debonds (i.e., 2ld/lc 5 1). The relationships between the HDE-based damage parameter, applied cycle number, matrix crack spacing, and fatigue peak stress are shown in Fig. 6.9. Under high fatigue peak stress of σmax 5 320 MPa, the HDE-based damage parameter increases with applied cycles till fatigue fracture. With increasing matrix crack spacing, the HDE-based damage parameter decreases. Under intermediate fatigue peak stress of σmax 5 240 MPa, the HDE-based damage parameter increases first and then decreases with applied cycles. With increasing matrix crack spacing, the

Figure 6.7 The fibermatrix interface debonded ratio versus the applied cycle curves for different matrix crack spacing of lc 5 140, 200, and 240 μm at the fatigue peak stresses of (A) σmax 5 320 MPa; (B) σmax 5 240 MPa; and (C) σmax 5 210 MPa.

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Figure 6.8 The fatigue HDE versus the applied cycle number curves for different matrix cracking space of lc 5 140, 200, and 240 μm at the fatigue peak stresses of (A) σmax 5 320 MPa; (B) σmax 5 240 MPa; and (C) σmax 5 210 MPa. HDE, Hysteresis dissipated energy.

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Figure 6.9 The HDE-based damage parameters versus the applied cycle number curves for different matrix cracking spaces of lc 5 140, 200, and 240 μm at the fatigue peak stresses of (A) σmax 5 320 MPa; (B) σmax 5 240 MPa; and (C) σmax 5 210 MPa. HDE, Hysteresis dissipated energy.

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HDE-based damage parameter decreases when the interface is partially debonding (i.e., 2ld/lc , 1), and increases when the interface completely debonds (i.e., 2ld/ lc 5 1). Under low fatigue peak stress of σmax 5 210 MPa, the HDE-based damage parameter increases first, then decreases, and increases again with applied cycles. With increasing matrix crack spacing, the HDE-based damage parameter decreases when the interface is partially debonding (i.e., 2ld/lc , 1), and increases when the interface completely debonds (i.e., 2ld/lc 5 1). When the matrix crack spacing increases, the fibermatrix interface debonding and sliding range decreases when the interface debonds partially. The interface shear stress for complete debonding decreases, the applied cycles for interface complete debonding increases, and the HDE and HDE-based damage parameter also decrease.

6.3.4 Effects of fiber volume fraction on fatigue life, fibermatrix interface debonding, HDE, and HDE-based damage parameters The fatigue life SN data of different fiber volume fractions are shown in Fig. 6.10, in which the fatigue life increases with fiber volume fraction. When the fiber volume is Vf 5 35%, the fatigue failure cycles are 1584, 1840, 2670, 2889, and 3428 for the fatigue peak stresses of σmax 5 320, 300, 240, 220, and 210 MPa. When the fiber volume is Vf 5 40%, the fatigue failure cycles are 2171, 2405, 3428, 4920, and 20,370 for the fatigue peak stresses of σmax 5 320, 300, 240, 220, and

Figure 6.10 The fatigue life SN data for different fiber volume fractions. Source: Reproduced from Longbiao, L., 2015b. A hysteresis dissipated energy-based damage parameter for life prediction of carbon fiber-reinforced ceramic-matrix composites under fatigue loading. Composites, B 82, 108128.

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210 MPa. When the fiber volume is Vf 5 45%, the fatigue failure cycles are 2569, 2744, 9923, 291,295, and 1,000,000 for the fatigue peak stresses of σmax 5 320, 300, 240, 220, and 210 MPa. The relationships between the fibermatrix interface debonding ratio, interface shear stress, fatigue peak stress, and fiber volume fraction are shown in Fig. 6.11. Under the same fatigue peak stress, with increasing fiber volume fraction, the fibermatrix interface debonding ratio 2ld/lc decreases at the same interface shear stress when the interface is partially debonding. The interface shear stress corresponding to the interface complete debonding decreases. When the fatigue peak stress is σmax 5 320 MPa, the fibermatrix interface shear stress for the interface complete debonding is τ i 5 25 MPa at the fiber volume of Vf 5 35%. When the fiber volume is Vf 5 40%, the fibermatrix interface shear stress for the interface complete debonding is τ i 5 19 MPa. When the fiber volume is Vf 5 45%, the fibermatrix interface shear stress for the interface complete debonding is τ i 5 15 MPa. When the fatigue peak stress is σmax 5 240 MPa, the fibermatrix interface shear stress for the interface complete debonding is τ i 5 18 MPa at the fiber volume of Vf 5 35%. When the fiber volume is Vf 5 40%, the fibermatrix interface shear stress for the interface complete debonding is τ i 5 14 MPa. When the fiber volume is Vf 5 45%, the fibermatrix interface shear stress for the interface complete debonding is τ i 5 11 MPa. When the fatigue peak stress is σmax 5 210 MPa, the fibermatrix interface shear stress for the interface complete debonding is τ i 5 15 MPa at the fiber volume of Vf 5 35%; when the fiber volume is Vf 5 40%, the fibermatrix interface shear stress for the interface complete debonding is τ i 5 12 MPa; and when the fiber volume is Vf 5 45%, the fibermatrix interface shear stress for the interface complete debonding is τ i 5 9 MPa. The relationships between the fibermatrix interface debonding ratio, applied cycle number, fatigue peak stress, and fiber volume fraction are shown in Fig. 6.12. Under the same fatigue peak stress, when the fiber volume fraction increases, the fibermatrix interface debonding ratio of 2ld/lc decreases at the same applied cycle when the interface is partially debonding; and the applied cycle number corresponding to the interface complete debonding increases. When the fatigue peak stress is σmax 5 320 MPa, the critical applied cycle number for the interface complete debonding is N 5 916 at the fiber volume of Vf 5 35%. When the fiber volume is Vf 5 40%, the critical applied cycle number for the interface complete debonding is N 5 1194; and when the fiber volume is Vf 5 45%, the critical applied cycle number for the interface complete debonding is N 5 1437. When the fatigue peak stress is σmax 5 240 MPa, the critical applied cycle number for the interface complete debonding is N 5 1265 at the fiber volume of Vf 5 35%. When the fiber volume is Vf 5 40%, the critical applied cycle number for the interface complete debonding is N 5 1509; and when the fiber volume is Vf 5 45%, the critical applied cycle number for the interface complete debonding is N 5 1734. When the fatigue peak stress is σmax 5 210 MPa, the critical applied cycle number for the interface complete debonding is N 5 1402 at the fiber volume of Vf 5 35%. When the fiber volume is Vf 5 40%, the critical applied cycle number for the interface complete debonding is

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Figure 6.11 The fibermatrix interface debonded ratio versus the interface shear stress curves for different fiber volume fractions at (A) σmax 5 320 MPa; (B) σmax 5 240 MPa; (C) σmax 5 210 MPa.

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Figure 6.12 The fibermatrix interface debonded ratio versus the applied cycle number curve for different fiber volume fractions of Vf 5 35%, 40%, and 45% at the fatigue peak stresses of (A) σmax 5 320 MPa; (B) σmax 5 240 MPa; and (C) σmax 5 210 MPa.

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N 5 1639; and when the fiber volume is Vf 5 45%, the critical applied cycle number for the interface complete debonding is N 5 1861. The relationships between the fatigue HDE, applied cycle number, fatigue peak stress, and fiber volume fraction are shown in Fig. 6.13. Under high fatigue peak stress of σmax 5 320 MPa, when the fiber volumes are Vf 5 35%, 40%, and 45%, the fatigue HDE increases with applied cycles till final fatigue fracture. When the fiber volume increases, the fatigue HDE decreases at the same applied cycle. Under intermediate fatigue peak stress of σmax 5 240 MPa, when the fiber volumes are Vf 5 35% and 40%, the fatigue HDE increases to the peak value, and then decreases till fatigue fracture with increasing applied cycles. When the fiber volume is Vf 5 45%, the fatigue HDE increases to the peak value, then decreases, and increases again till fatigue fracture with increasing applied cycles. When the fiber volume increases, the fatigue HDE decreases at the same applied cycle when the interface partially or completely debonds. Under low fatigue peak stress of σmax 5 210 MPa, when the fiber volume is Vf 5 35%, the fatigue HDE increases to the peak value and then decreases till fatigue fracture with increasing applied cycles. When the fiber volumes are Vf 5 40% and 45%, the fatigue HDE increases to the peak value, then decreases, and increases again with applied cycles. When the fiber volume increases, the fatigue HDE decreases at the same applied cycle when the interface partially or completely debonds. The relationships between the HDE-based damage parameter, applied cycle number, fatigue peak stress, and fiber volume fraction are shown in Fig. 6.14. Under high fatigue peak stress of σmax 5 320 MPa, when the fiber volume is Vf 5 35%, the HDE-based damage parameter increases with applied cycles till final fatigue fracture. When the fiber volume increases, the HDE-based damage parameter decreases at the same applied cycle. Under intermediate fatigue peak stress of σmax 5 240 MPa, when the fiber volumes are Vf 5 35% and 40%, the HDE-based damage parameter increases to the peak value, and then decreases till fatigue fracture with increasing applied cycles. When the fiber volume is Vf 5 45%, the HDEbased damage parameter increases to the peak value, then decreases, and increases again till fatigue fracture with increasing applied cycles. When the fiber volume increases, the HDE-based damage parameter decreases at the same applied cycle when the interface partially or completely debonds. Under low fatigue peak stress of σmax 5 210 MPa, when the fiber volume is Vf 5 35%, the HDE-based damage parameter increases to the peak value and then decreases till fatigue fracture with increasing applied cycles. When the fiber volumes are Vf 5 40% and 45%, the HDE-based damage parameter increases to the peak value, then decreases, and increases again with applied cycles. When the fiber volume increases, the HDEbased damage parameter decreases at the same applied cycle when the interface partially or completely debonds. When the fiber volume increases, the fibermatrix interface debonding and sliding range decreases. The fibermatrix interface debonding ratio decreases when the interface partial debonding, the fatigue HDE and HDE-based damage parameter decrease when the interface partial or complete debonding, and the fatigue life increases.

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Figure 6.13 The fatigue HDE versus the applied cycle number curves for different fiber volume fractions of Vf 5 35%, 40%, and 45% at the fatigue peak stresses of (A) σmax 5 320 MPa; (B) σmax 5 240 MPa; and (C) σmax 5 210 MPa. HDE, Hysteresis dissipated energy.

Fatigue life prediction of ceramic-matrix composites based on hysteresis dissipated energy

Figure 6.14 The HDE-based damage parameter versus applied cycle number curves for different fiber volume fractions of Vf 5 35%, 40%, and 45% at the fatigue peak stresses of (A) σmax 5 320 MPa; (B) σmax 5 240 MPa; and (C) σmax 5 210 MPa. HDE, Hysteresis dissipated energy.

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Experimental comparisons

The experimental fatigue HDE and HDE-based damage parameter, the broken fibers fraction, and fatigue life SN curves of unidirectional, cross-ply, 2D, 2.5D, and 3D fiber-reinforced CMCs are predicted in this section.

6.4.1 Unidirectional ceramic-matrix composites Cao et al. (2001) investigated the flexure fatigue behavior of unidirectional C/SiC composite at room temperature and obtained its flexure fatigue life. When the fatigue limit cycle is defined as 5,000,000, the fatigue limit stress is approximately 379 MPa, which is 69% of the flexure strength. Minford and Prewo (1986) investigated the tensiontension cyclic fatigue behavior of unidirectional SiC/LAS composites at room temperature. Based on the fibermatrix interface bonding condition, the SiC/LAS composite is divided into the strong fibermatrix interface bonding of SiC/LAS-I and weak fibermatrix interface bonding of SiC/LAS-II. For the SiC/LAS-I composite, the tensile stress strain curve is linear, the tensile strength is about σUTS 5 261 MPa, the fatigue loading frequency is f 5 10 Hz, and the fatigue peak stresses are σmax 5 138 MPa, 172 MPa, and 207 MPa. The valley stress is σmin 5 20.7 MPa and the fatigue limit cycle is defined as 100,000. When the fatigue peak stress is σmax 5 138 MPa and 172 MPa, the specimen survives after 100,000 cycles, and the residual tensile strength is the same as the original tensile strength. The modulus remains the same under cyclic loading; however, when the fatigue peak stress is σmax 5 207 MPa, the specimen cycled 2160 and fatigue failed. For the SiC/LAS-II composite, the tensile stressstrain curve is nonlinear, and the proportional limit stress and the tensile strength are 270 and 520 MPa, respectively. The fatigue loading frequency is f 5 5 Hz, the fatigue stress ratio is R 5 0.1, and the fatigue peak stresses are σmax 5 225, 275, 310, and 335 MPa. The fatigue limit cycle is defined as 100,000. When the fatigue peak stress is lower than the proportional limit stress, the modulus remains unchanged during cyclic loading, and the specimen survives after experiencing 100,000 cycles. The residual tensile strength is higher than the original tensile strength. When the fatigue peak stress is higher than the proportional limit stress, the modulus decreases with increasing cycles and there is composite failure before reaching 100,000 cycles. Zawada et al. (1990) investigated the tensiontension cyclic fatigue behavior of unidirectional SiC/1723 composites at room temperature. The fatigue loading frequency is f 5 10 Hz, the fatigue stress ratio is R 5 0.1, and the fatigue limit cycle is defined as 1,000,000. The fatigue limit stress is far higher than the first matrix cracking stress (FMCS) and the proportional limit stress (i.e., σmax , σmc), and reaches about 65% of the tensile strength. When the fatigue peak stress is less than the FMCS, the composite modulus remains unchanged. However, when the fatigue peak stress is between the FMCS and the proportional limit stress

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(i.e., σmc ,σmax , σpls), the composite modulus decreases at the initial 10,000 applied cycles and then recovers gradually. Evans et al. (1995) investigated the tensiontension cyclic fatigue behavior of unidirectional SiC/CAS composite at room temperature. The fatigue loading frequency is f 5 10 Hz, and the stress ratio is R 5 0.05, and the fatigue peak stress is σmax 5 280 MPa and the fatigue limit cycle is defined as 40,000. It was found that the fatigue limit stress of SiC/CAS composite is about 70% of the tensile strength. McNulty and Zok (1999) investigated the low-cycle tensiontension fatigue behavior of unidirectional SiC/CAS and SiC/MAS composites at room temperature. The fatigue loading frequency is f 5 0.5 Hz, the fatigue peak stress is σmax 5 280 MPa for the SiC/CAS composite, and σmax 5 369 MPa for the SiC/ MAS composite. The fatigue limit cycle is defined as 40,000. At room temperature, the fatigue limit stress of the SiC/CAS composite is about 65% of the tensile strength; and for the SiC/MAS composite, the fatigue limit stress is about 80% of the tensile strength. Sørensen et al. (2000) investigated the high-cycle tensiontension fatigue behavior of unidirectional SiC/CAS-II composites at room temperature. The fatigue loading frequency is f 5 200 Hz and the fatigue peak stress is σmax 5 240 MPa. After experiencing 108 applied cycles, the residual strength is about 85% of the tensile strength; however, after cycling for 105, the strength remains the same. The composite strength decreases greatly before the final fracture. Therefore the decline in strength occurs before failure, rather than gradually decaying with increasing cycles. There is a special region in the center of the high-frequency fatigue failure sample in which there appears no fibers pullout, but there is a large amount of fibers pullout around this region. Under high-frequency fatigue loading, the internal temperature increase leads to a chemical reaction at the fibermatrix interface, resulting in an increase of the fibermatrix interface bond strength, which is the main reason for the high-frequency and high-cyclic fatigue failure of SiC/CAS-II composites. Sørensen et al. (2002) investigated the high-frequency tensiontension fatigue behavior of unidirectional SiC/CAS-II composites at room temperature. The fatigue loading frequency is f 5 200 and 500 Hz, the fatigue peak stress is σmax 5 160280 MPa, the fatigue valley stress is σmin 5 10 MPa, and the fatigue limit cycle is 108. The fatigue limit stress is 212 MPa at the loading frequency of f 5 200 Hz, which is lower than the proportional limit stress of σpls 5 380 MPa, and is about 42% of the tensile strength. When cycling for 108, the surface temperature and fatigue hysteresis modulus change with applied cycles, indicating the continual damage of the composite. Prewo (1987) investigated the flexure fatigue behavior of unidirectional SiC/ LAS-II composites at room temperature and elevated temperatures of 600 C and 900 C in air atmosphere. The fatigue loading frequency is f 5 5 Hz, and the fatigue limit cycle is defined as 100,000. At room temperature, the fatigue limit stress is higher than the proportional limit stress (i.e., σpls 5 500 MPa), and the residual flexure strength (710813 MPa) is higher than the original flexure strength (σUTS 5 750 MPa). At 600 C in air atmosphere, the fatigue limit stress and the

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residual flexure strength both approach the proportional limit stress (σpls 5 350 MPa), indicating the material performance degradation. At 900 C in air atmosphere, the fatigue limit stress is lower than the proportional limit stress (σpls 5 300 MPa); however, the residual strength approaches the original flexure strength (σUTS 5 550 MPa). At 600 C in air atmosphere, oxidation embrittlement is the main reason for the fatigue performance degradation of SiC/LAS-II composites; however, at 900 C in air atmosphere, the fibermatrix interface forms the oxidation layer at initial cyclic loading, which stops further oxidation of the interface layer, leading to the better fatigue performance than that at 600 C. Allen et al. (1993) investigated the flexure fatigue behavior of unidirectional SiC/CAS composites at room temperature and elevated temperatures of 800 C and 1000 C in air atmosphere. At room temperature, the fatigue loading frequency is f 5 50 Hz, the fatigue stress ratio is R 5 0.1 and 0.5, and the fatigue limit cycle is defined as 106. At elevated temperatures of 800 C and 1000 C in air atmosphere, the fatigue loading frequency is f 5 10 Hz and the stress ratio is R 5 0.1. At room temperature under the same fatigue peak stress, the cyclic fatigue life is less than that of static fatigue, and the fatigue life decreases with increasing peak stress. Upon first loading to the peak stress (i.e., the peak stress is higher than the FMCS), the matrix cracking appears at the surface of the specimen. During the following cyclic fatigue loading, the matrix cracking propagates along the direction perpendicular to the fiber. The fibermatrix interface wear decreases the fiber bridge traction stress, leading to the propagation of the matrix cracking with increasing cycles. At 800 C in air atmosphere under the same fatigue peak stress, the cyclic fatigue life is less than that of static fatigue. The fiber pullout length at the fracture surface is less than that at room temperature. At 1000 C in air atmosphere under the same peak stress, the cyclic fatigue life is the same as that under static fatigue. Oxidation at elevated temperature results in the increase of fibermatrix interface bond strength, leading to fatigue failure at elevated temperatures. Holmes et al. (1989) investigated the tensiontension fatigue behavior of unidirectional SiC/Si3N4 composites at elevated temperature of 1000 C in air atmosphere. The fatigue loading frequency is f 5 10 Hz, the fatigue stress ratio is R 5 0.1, and the fatigue limit cycle is defined as 2 3 106. When the fatigue peak stress is lower than the proportional limit stress, the specimen cycled without fatigue failure. When the fatigue peak stress is higher than the proportional limit stress, the specimen failed before approaching the fatigue limit cycle. When the fatigue peak stress is σmax 5 220 MPa, the failure cycle is in the range of 270,000750,000. When the fatigue peak stress is σmax 5 280 MPa, the failure cycle is in the range of 400033,000. Under cyclic loading, the composite modulus decreases, and obvious stressstrain hysteresis loops and strain ratcheting appear. Holmes and Sørensen (1995) investigated the tensiontension fatigue behavior of unidirectional SiC/Si3N4 composites at elevated temperature of 1200 C in air atmosphere. The fatigue loading frequency is f 5 10 Hz, the fatigue peak stress is σmax 5 180, 200, and 240 MPa, and the valley stress is σmin 5 10 MPa. When the fatigue peak stress is σmax 5 180 MPa, which is lower than the proportional limit stress of σpls 5 196 MPa, there is no matrix cracking under cyclic loading, and the

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stressstrain curve is linear without hysteresis behavior; however, strain ratcheting appears. When the fatigue peak stress is σmax 5 200 and 240 MPa, which is higher than the proportional limit stress, the fatigue hysteresis width increases with peak stress. Under low fatigue peak stress, which is lower than the proportional limit stress, the composite damage is caused mainly due to creep; however, when the fatigue peak stress is higher than the proportional limit stress, composite damage is caused by fatigue mechanisms.

6.4.1.1 SiC/CAS composite The tensiontension fatigue experiments were conducted at room temperature with the fatigue stress ratio of R 5 0.05 and the loading frequency of f 5 10 Hz (Evans et al., 1995). The fatigue life SN curves, fatigue HDE, and HDE-based damage parameter versus applied cycles curves and the fibermatrix interface shear stress degradation with cycle number curves of unidirectional SiC/CAS composites are shown in Fig. 6.15. The experimental and theoretical predicted fatigue life SN curves of unidirectional SiC/CAS composites are shown in Fig. 6.15A. The fatigue life SN curve can be divided into two regions, that is, low-cyclic fatigue region at high fatigue peak stress due to the degradation of the fibermatrix interface shear stress and high-cyclic fatigue region at low fatigue peak stress due to degradation of fiber strength. The fatigue limit stress of SiC/CAS composites at room temperature approaches 62% of the tensile strength. The fatigue HDE versus applied cycle number curves at high, intermediate, and low fatigue peak stresses of σmax 5 420, 380, and 340 MPa are shown in Fig. 6.15B. Under high fatigue peak stress of σmax 5 420 MPa, the fatigue HDE increases with applied cycles till fatigue fracture. Under intermediate fatigue peak stress of σmax 5 380 MPa, the fatigue HDE increases with applied cycles till fatigue fracture. Under low fatigue peak stress of σmax 5 340 MPa, the fatigue HDE first increases to the peak value, then decreases, and increases again with applied cycles till fatigue fracture. The fatigue HDE-based damage parameter versus applied cycle number curves at high, intermediate, and low fatigue peak stresses of σmax 5 420, 380, and 340 MPa are shown in Fig. 6.15C. Under high fatigue peak stress of σmax 5 420 MPa, the fatigue HDE-based damage parameter increases with applied cycles till fatigue fracture. Under intermediate fatigue peak stress of σmax 5 380 MPa, the HDE-based damage parameter increases first to the peak value, then decreases, and increases again with applied cycles till fatigue fracture. Under low fatigue peak stress of σmax 5 340 MPa, the HDE-based damage parameter first increases to the peak value, then decreases, and increases again with applied cycles till fatigue fracture. The evolution of the fatigue HDE and HDE versus applied cycles is due to the degradation of the fibermatrix interface shear stress caused by the wear at the fibermatrix interface, as shown in Fig. 6.15D.

Figure 6.15 (A) The fatigue life SN curves; (B) the fatigue HDE versus applied cycles curves; (C) the HDE-based damage parameter versus applied cycles; and (D) the fibermatrix interface shear stress versus applied cycles curve of unidirectional SiC/CAS composites at room temperature. HDE, Hysteresis dissipated energy.

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6.4.1.2 SiC/1723 composite The tensiontension fatigue experiments of unidirectional SiC/1723 composites were conducted at room temperature with the fatigue stress ratio of R 5 0.1 and the loading frequency of f 5 10 Hz (Zawada et al., 1991). The fatigue life SN curves, fatigue HDE, and HDE-based damage parameter versus applied cycles curves and the fibermatrix interface shear stress degradation with cycle number curves of unidirectional SiC/1723 composites are shown in Fig. 6.16. The experimental and theoretical predicted fatigue life SN curves of unidirectional SiC/1723 composites are shown in Fig. 6.16A. The fatigue life SN curve can be divided into two regions, that is, low-cyclic fatigue region at high fatigue peak stress due to the degradation of the fibermatrix interface shear stress and high-cyclic fatigue region at low fatigue peak stress due to degradation of fiber strength. The fatigue limit stress of SiC/CAS composites at room temperature approaches 70% of the tensile strength. The fatigue HDE versus applied cycle number curves at different fatigue peak stresses of σmax 5 650, 600, and 550 MPa are shown in Fig. 6.16B, in which the fatigue HDE increases slowly first, and then rapidly with applied cycles due to the damage mechanisms of degradation of the fibermatrix interface shear stress and fibers failure. The HDE-based damage parameter versus applied cycle number curves at different fatigue peak stresses of σmax 5 650, 600, and 550 MPa are shown in Fig. 6.16C. The HDE-based damage parameter increases with cycle number and can be divided into two regions, that is, the slow increase region due to the degradation of fibermatrix interface shear stress, and the rapid increase region due to fibers failure. The degradation of the fibermatrix interface shear stress versus applied cycles curves are shown in Fig. 6.16D, in which the fibermatrix interface shear stress approaches the steady-state value rapidly.

6.4.2 Cross-ply ceramic-matrix composites Zawada et al. (1991) investigated the tensiontension fatigue behavior of cross-ply [0/90]3s SiC/1723 composites at room temperature. The loading frequency is f 5 10 Hz, the stress ratio is R 5 0.1, and the fatigue limit cycle is defined as 106. At room temperature, the fatigue limit stress is 70% of the tensile strength and is higher than the proportional limit stress of 0 degree ply. Under low fatigue peak stress, the modulus reduction is small. Under high fatigue peak stress, the modulus greatly decreases till final fracture. Under moderate fatigue peak stress, the modulus decreases greatly at first and then recovers slowly, due to the mixed particles at the matrix cracking plane or fibermatrix interface. After experiencing 106 cycles, the residual strength and modulus decrease with increasing fatigue peak stress. Lee and Stinchcomb (1994) investigated the tensiontension fatigue behavior of cross-ply [0/90]4s SiC/CAS-II composites at room temperature. The loading frequency is f 5 10 Hz, the stress ratio is R 5 0.1, and the fatigue peak stress is 65%

Figure 6.16 (A) The fatigue life SN curves; (B) the fatigue HDE versus applied cycles curves; (C) the HDE-based damage parameter versus applied cycles; and (D) the fibermatrix interface shear stress versus applied cycles curve of unidirectional SiC/1723 composites at room temperature. HDE, Hysteresis dissipated energy.

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and 85% of the tensile strength. The modulus degradation can be divided into three stages, that is, initial fast decay stage (10%15% fatigue life), intermediate slow descent stage (75%80% fatigue life), and sharp descent stage (10%15% fatigue life) before failure. When the fatigue peak stress is higher than the proportional limit stress, the fiber debonding and transverse cracking in the 90 degrees ply is the major reason for initial modulus decay. When the transverse cracking approaches saturation, stress redistribution between the 90 and 0 degrees ply, and matrix cracking and fibermatrix interface debonding occur in the 0 degree ply, leading to the intermediate slow degradation; and before final fracture, the fibers break and the modulus decreases greatly. Kim and Liaw (2005) investigated the high-cycle fatigue behavior of cross-ply [0/90]4s SiC/CAS composites at room temperature. The fatigue loading frequency is f 5 20 Hz, the stress ratio is R 5 0.1, and the fatigue peak stress is in the range of 65%95% tensile strength (σUTS 5 180 MPa). The fatigue life decreases with increasing peak stress, the fatigue limit stress is 100 MPa, and is 55% of the tensile strength. The relationship between the fatigue microdamage mechanisms and temperature increase has been established. The temperature increase can be divided into three stages: G

G

G

At initial cyclic stage, the matrix cracking occurs without fibermatrix interface debonding and the temperature increase at the specimen surface is not obvious. At intermediate cyclic stage, the matrix cracking propagates with fibermatrix interface debonding and the surface temperature gradually increases. Before final fracture, the surface temperature increases rapidly as the broken fibers pull out from the matrix.

Opalski (1992) investigated the tensiontension, tensioncompression, and compressioncompression fatigue behavior of cross-ply [0/90]2s SiC/CAS composites at room temperature. The fatigue loading frequency is f 5 10 Hz and the fatigue limit cycle is 106. The fatigue limit stress under tensiontension fatigue loading is about 50% tensile strength. Under compressioncompression fatigue loading with peak stress of σmax 5 140 and 210 MPa, the specimens do not reach fatigue failure. Under tensioncompression loading with the tensile fatigue peak stress of σmax 5 140 MPa, the fatigue life decreases with increasing compressive stress. The longitudinal cracking appears between the transverse cracking in the 90 degrees ply that connects the transverse cracking, resulting in much more serious damage. Mall and Tracy (1992) investigated the tensiontension fatigue behavior of [0/45/90]s SiC/CAS composites at room temperature. The fatigue loading frequency is f 5 10 Hz, the fatigue stress ratio is R 5 0.1, and the fatigue limit cycle is defined as 106. The fatigue limit stress is about 67% tensile strength, which corresponds to the stress without causing damage in 0 degree ply. Vanwijgenhoven et al. (1999) investigated the tensiontension fatigue behavior of [0]16, [0]12, [02/902]s, [0/90]3s, [452/ 2 452]2s, and [02/452/ 2 452/902]s SiC/ BMAS composites at room temperature. The loading frequency is f 5 3 Hz, the stress ratio is R 5 0.1, and the fatigue limit cycle is defined as 106. For the

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unidirectional composite, the fatigue limit stress is 375 MPa, which is about 60% of the tensile strength. For [0/90] composite, the fatigue limit stress is 175 MPa, which is about 45% of the tensile strength. For [452/ 2 452]2s composite, the fatigue limit stress is 70 MPa, which is about 70% of the tensile strength. For the [02/452/ 2 452/ 902]s composite, the fatigue limit stress is 110 MPa, which is about 65% of the tensile strength. The damage under cyclic loading is the same as that under tensile loading; however, the fatigue damage is caused under lower stress levels. For the unidirectional, [0/90] and [02/452/ 2 452/902]s composite, the fatigue life is controlled by the 0 degree ply, and an increase in matrix cracking density, decrease in fibermatrix interface shear stress, and the decay of fibers strength all result in the decrease of residual strength. However, for the [452/ 2 452]2s composite, the fatigue life is much more affected by delamination. Prewo et al. (1989) investigated the tensiontension fatigue behavior of crossply [0/90]4s SiC/LAS-III composites at elevated temperature of 900 C in air and Ar atmosphere. The fatigue loading frequency is f 5 7.210 Hz, the stress ratio is R 5 0.1, and the fatigue limit cycle is defined as 105. The fatigue limit stress at 900 C in air atmosphere is approximately 90 MPa, which approaches the proportional limit stress and is higher than the fatigue limit stress in Ar atmosphere. The residual strength at 900 C in air atmosphere is about 85 MPa, which is far lower than the tensile strength (i.e., σUTS 5 150 MPa). The residual strength in Ar approaches the tensile strength (i.e., σUTS 5 200 MPa). Rousseau (1990) investigated the tensiontension fatigue behavior of [0/90]2s cross-ply SiC/CAS composites at room temperature and elevated temperature of 815 C in air atmosphere. At room temperature, there are two proportional limit stresses in tensile stressstrain curve, that is, σpls,l 5 35 MPa and σpls,h 5 105 MPa. At 815 C in air atmosphere, there are also two proportional limit stress in the tensile stressstrain curve, that is, σpls,l 5 59 MPa and σpls,h 5 104 MPa. The fatigue loading frequency is f 5 0.3 Hz and the fatigue stress ratio is R 5 00.15. At room temperature, when the fatigue peak stress is σmax 5 135 MPa (i.e., σmax . σpls, 5 h 5 105 MPa), the specimen cycled 2 3 10 without fatigue failure. At elevated tem perature of 815 C in air atmosphere, when the fatigue peak stress is σmax 5 70 MPa (i.e., σmax , σpls,h 5 104 MPa), the specimen cycled for 2000 and then fatigue failed. The fracture surface of the failure specimen at elevated temperature was observed and there was no fibers pullout at the fracture surface. The weak fibermatrix interface was oxidized into strong interface bonding. The oxidation at elevated temperature is the main reason for the decrease of the fatigue life at elevated temperature. Steiner (1994) investigated the tensiontension fatigue behavior of cross-ply [0/90]4s SiC/MAS composites at elevated temperatures of 566 C and 1093 C in air atmosphere. The fatigue loading frequency is f 5 1 and 10 Hz. The damage mechanism of SiC/MAS composite is independent of temperature and loading frequency; however, the modulus degradation rate and fatigue damage extent are affected by the temperature, and the fatigue life decreases with increasing temperature, due to the fiber and interphase oxidation at elevated temperature. The fracture surface of the fatigue failure specimen was observed under scanning electron microscope.

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With increasing testing temperature, the region of the fibers pullout decreases due to the oxidized fibermatrix interface strong bonding. Yasmin and Bowen (2004) investigated the flexure fatigue behavior of [0/90]4s cross-ply SiC/CAS-II composites at room temperature and elevated temperature of 800 C in air atmosphere. The fatigue loading frequency is f 5 10 Hz, the stress ratio is R 5 0.1, and the fatigue limit cycle is defined as 106. At room temperature, the fatigue limit stress is higher than the proportional limit stress and is about 40% of the flexure strength. At 800 C, the fatigue limit stress is the FMCS, which is reduced by 50% compared to that at room temperature due to the fibermatrix interface oxidation. At room temperature, after experiencing 106 cycles, the residual strength is about 85% of the original strength, mainly due to the degradation of the fibermatrix interface shear stress. At 800 C in air atmosphere, the residual strength after experiencing 106 cycles is the same as the original strength. The increase of the fatigue performance at elevated temperature is mainly due to the transfer from the weak fibermatrix interface bonding to oxidized strong fibermatrix interface bonding. The fibermatrix interface strong bond increases the fatigue performance. The evolution of fatigue HDE, fibermatrix interface shear stress, and broken fibers fraction versus applied cycles, and the fatigue life SN curves of cross-ply SiC/MAS composites at elevated temperatures in air atmosphere are analyzed in this section.

6.4.2.1 SiC/MAS at 566 C in air atmosphere The fatigue life SN curve of cross-ply SiC/MAS composites at 566 C in air atmosphere is shown in Fig. 6.17, in which the loading frequency is f 5 1 Hz. The composite tensile strength is about σUTS 5 292 MPa, and the fatigue limit stress is about 60 MPa when the maximum cycle number is defined as 100,000. The fatigue HDE at fatigue peak stresses of σmax 5 103, 120, and 137 MPa decrease with applied cycles, and the fatigue HDE increases with fatigue peak stress at the same applied cycle. The fibermatrix interface shear stress decreases with applied cycles, and the broken fibers fraction increases with cycle number. The fatigue life of cross-ply SiC/MAS composites at 566 C in air atmosphere is greatly reduced due to the oxidation of the interphase and fibers. When the fatigue peak stress is σmax 5 137 MPa, the fatigue HDE, fibermatrix interface shear stress and broken fibers fraction versus applied cycles curves are shown in Fig. 6.18. Under cyclic fatigue loading, the experimental fatigue HDE decreases from U 5 5.4 kPa at the applied cycle number N 5 4 to U 5 4.4 kPa at the applied cycle number N 5 230. The theoretical fatigue HDE decreases from U 5 5.8 kPa at the applied cycle number N 5 1 to U 5 4.3 kPa at the applied cycle number N 5 1000. The fibermatrix interface shear stress decreases from τ i 5 1.2 MPa at cycle number N 5 4 to τ i 5 1 MPa at cycle number N 5 230. The broken fibers fraction increases from 3% at cycle number N 5 1 to 39.9% at cycle number N 5 1273.

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Figure 6.17 The fatigue life SN curves of cross-ply SiC/MAS composites at the loading frequency of f 5 1 and 10 Hz at elevated temperature of 566 C in air atmosphere.

When the fatigue peak stress is σmax 5 120 MPa, the fatigue HDE, fibermatrix interface shear stress and broken fibers fraction versus applied cycles curves are shown in Fig. 6.19. Under cyclic fatigue loading, the experimental fatigue HDE decreases from U 5 4.5 kPa at cycle number of N 5 3 to U 5 3.2 kPa at cycle number of N 5 105; and the theoretical fatigue HDE decreases from U 5 4.9 kPa at cycle number of N 5 1 to U 5 2.8 kPa at the applied cycle number of N 5 1000. The fibermatrix interface shear stress decreases from τ i 5 1.2 MPa at cycle number of N 5 3 to τ i 5 0.8 MPa at cycle number of N 5 105. The broken fibers fraction increases from 1.7% at cycle number of N 5 1 to 39.9% at cycle number of N 5 5433. When the fatigue peak stress is σmax 5 103 MPa, the fatigue HDE, fibermatrix interface shear stress, and broken fibers fraction versus applied cycles curves are shown in Fig. 6.20. Under cyclic fatigue loading, the experimental fatigue HDE decreases from U 5 2.7 kPa at cycle number N 5 4 to U 5 2.4 kPa at cycle number N 5 920. The theoretical fatigue HDE decreases from U 5 3.0 kPa at cycle number N 5 1 to U 5 2.4 kPa at cycle number N 5 1000. The fibermatrix interface shear stress decreases from τ i 5 1 MPa at cycle number N 5 4 to τ i 5 0.8 MPa at cycle number N 5 920. The broken fibers fraction increases from 0.9% at cycle number N 5 1 to 34% at cycle number N 5 19,931. When the loading frequency is f 5 10 Hz, the fatigue life SN curve of crossply SiC/MAS composites at elevated temperature of 566 C in air atmosphere is shown in Fig. 6.17. The composite tensile strength is about σUTS 5 292 MPa, and the fatigue limit stress is about 90 MPa when the maximum cycle number is defined as 1,000,000. The fatigue HDE and fibermatrix interface shear stress degrades

Figure 6.18 (A) The fatigue HDE versus applied cycles curves; (B) the fibermatrix interface shear stress versus applied cycles curves; and (C) the broken fibers fraction versus applied cycles curve of cross-ply SiC/MAS composites at loading frequency of f 5 1 Hz and fatigue peak stress of σmax 5 137 MPa. HDE, Hysteresis dissipated energy.

Figure 6.19 (A) The fatigue HDE versus applied cycles curves; (B) the fibermatrix interface shear stress versus applied cycles curves; and (C) the broken fibers fraction versus applied cycles curve of cross-ply SiC/MAS composites at loading frequency of f 5 1 Hz and fatigue peak stress of σmax 5 120 MPa. HDE, Hysteresis dissipated energy.

Figure 6.20 (A) The fatigue HDE versus applied cycles curves; (B) the fibermatrix interface shear stress versus applied cycles curves; and (C) the broken fibers fraction versus applied cycles curve of cross-ply SiC/MAS composites at loading frequency of f 5 1 Hz and fatigue peak stress of σmax 5 103 MPa. HDE, Hysteresis dissipated energy.

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with applied cycles, and the broken fibers fraction increases with applied cycles. The fatigue limit stress at the loading frequency of f 5 10 Hz is higher than that at the loading frequency of f 5 1 Hz. When the fatigue peak stress is σmax 5 137 MPa, the fatigue HDE, fibermatrix interface shear stress, and broken fibers fraction versus applied cycles curves are shown in Fig. 6.21. Under cyclic fatigue loading, the experimental fatigue HDE decreases from U 5 6.5 kPa at cycle number N 5 2 to U 5 3.6 kPa at cycle number N 5 7730. The theoretical fatigue HDE decreases from U 5 7.2 kPa at cycle number N 5 1 to U 5 3.4 kPa at the applied cycle number N 5 8000. The fibermatrix interface shear stress decreases from τ i 5 1.5 MPa at cycle number N 5 2 to τ i 5 0.8 MPa at cycle number N 5 7730. The broken fibers fraction increases from 4% at cycle number N 5 1 to 39% at cycle number N 5 4403. When the fatigue peak stress is σmax 5 120 MPa, the fatigue HDE, fibermatrix interface shear stress, and broken fibers fraction versus applied cycles curves are shown in Fig. 6.22. Under cyclic fatigue loading, the experimental fatigue HDE decreases from U 5 2.5 kPa at cycle number N 5 6 to U 5 1.3 kPa at the applied cycle number N 5 6149. The theoretical fatigue HDE decreases from U 5 3.8 kPa at cycle number N 5 1 to U 5 1.1 kPa at cycle number N 5 8000. The fibermatrix interface shear stress decreases from τ i 5 0.7 MPa at cycle number N 5 6 to τ i 5 0.25 MPa at cycle number N 5 6150. The broken fibers fraction increases from 2% at cycle number N 5 1 to 40% at cycle number N 5 42,455. When the fatigue peak stress is σmax 5 103 MPa, the fatigue HDE, fibermatrix interface shear stress, and broken fibers fraction versus applied cycles curves are shown in Fig. 6.23. Under cyclic fatigue loading, the experimental fatigue HDE decreases from U 5 1.3 kPa at cycle number N 5 2 to U 5 0.7 kPa at cycle number N 5 570. The theoretical fatigue HDE decreases from U 5 1.5 kPa at cycle number N 5 1 to U 5 0.7 kPa at cycle number N 5 1000. The fibermatrix interface shear stress decreases from τ i 5 0.5 MPa at cycle number N 5 2 to τ i 5 0.28 MPa at cycle number N 5 570. The broken fibers fraction increases from 1% at cycle number N 5 1 to 40% at cycle number N 5 400,081.

6.4.2.2 SiC/MAS composite at 1093 C in air atmosphere The fatigue life SN curve of cross-ply SiC/MAS composites at 1093 C in air atmosphere is shown in Fig. 6.24, in which the loading frequency is f 5 1 Hz. The composite tensile strength is about σUTS 5 209 MPa, and when the fatigue peak stress is σmax 5 59 MPa the corresponding failure cycle number is N 5 507,348. Under high fatigue peak stresses of σmax 5 103, 120, and 137 MPa, the fatigue HDE increases to the peak value and then decreases with applied cycles. Under low fatigue peak stress of σmax 5 96 MPa, the fatigue HDE decreases with applied cycles. The fibermatrix interface shear stress decreases with applied cycles, and the broken fibers fraction increases with applied cycles. When the loading frequency is f 5 1 Hz, the fatigue limit stress of cross-ply SiC/MAS composites at 1093 C is higher than that at 566 C.

Figure 6.21 (A) The fatigue HDE versus applied cycles curves; (B) the fibermatrix interface shear stress versus applied cycles curves; and (C) the broken fibers fraction versus applied cycles curve of cross-ply SiC/MAS composites at loading frequency of f 5 10 Hz and fatigue peak stress of σmax 5 137 MPa. HDE, Hysteresis dissipated energy.

Figure 6.22 (A) The fatigue HDE versus applied cycles curves; (B) the fibermatrix interface shear stress versus applied cycles curves; and (C) the broken fibers fraction versus applied cycles curve of cross-ply SiC/MAS composites at loading frequency of f 5 10 Hz and fatigue peak stress of σmax 5 120 MPa. HDE, Hysteresis dissipated energy.

Figure 6.23 (A) The fatigue HDE versus applied cycles curves; (B) the fibermatrix interface shear stress versus applied cycles curves; and (C) the broken fibers fraction versus applied cycles curve of cross-ply SiC/MAS composites at loading frequency of f 5 10 Hz and fatigue peak stress of σmax 5 103 MPa. HDE, Hysteresis dissipated energy.

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Figure 6.24 The fatigue life SN curves of cross-ply SiC/MAS composites at loading frequencies of f 5 1 and 10 Hz and 1093 C in air atmosphere.

When the fatigue peak stress is σmax 5 137 MPa, the fatigue HDE, fibermatrix interface shear stress, and broken fibers fraction versus applied cycles curves are shown in Fig. 6.25. Under cyclic fatigue loading, the experimental fatigue HDE decreases from U 5 43.8 kPa at cycle number N 5 4 to U 5 32.8 kPa at cycle number N 5 25. The theoretical fatigue HDE increases from U 5 40.6 kPa at cycle number N 5 1 to U 5 43.4 kPa at cycle number N 5 5, and then decreases to U 5 28 kPa at cycle number N 5 100. The fibermatrix interface shear stress decreases from τ i 5 9.8 MPa at cycle number N 5 4 to τ i 5 5.6 MPa at cycle number N 5 25. The broken fibers fraction increases from 8% at cycle number N 5 1 to 38% at cycle number N 5 38. When the fatigue peak stress is σmax 5 120 MPa, the fatigue HDE, fibermatrix interface shear stress, and broken fibers fraction versus applied cycles curves are shown in Fig. 6.26. Under cyclic fatigue loading, the experimental fatigue HDE decreases from U 5 34.9 kPa at cycle number N 5 3 to U 5 22.5 kPa at cycle number N 5 75. The theoretical fatigue HDE increases from U 5 33.6 kPa at cycle number N 5 1 to U 5 35.1 kPa at cycle number N 5 3, and then decreases to U 5 23 kPa at cycle number N 5 100. The fibermatrix interface shear stress decreases from τ i 5 8.2 MPa at cycle number N 5 3 to τ i 5 3.6 MPa at cycle number N 5 75. The broken fibers fraction increases from 4% at cycle number N 5 1 to 40% at cycle number N 5 371. When the fatigue peak stress is σmax 5 103 MPa, the fatigue HDE, fibermatrix interface shear stress, and broken fibers fraction versus applied cycles curves are shown in Fig. 6.27. Under cyclic fatigue loading, the experimental fatigue HDE decreases from U 5 25.5 kPa at cycle number N 5 4 to U 5 6.5 kPa at cycle number N 5 10,608. The theoretical fatigue HDE increases from U 5 22 kPa at cycle

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Figure 6.25 (A) The fatigue HDE versus applied cycles curves; (B) the fibermatrix interface shear stress versus applied cycles curves; and (C) the broken fibers fraction versus applied cycles curve of cross-ply SiC/MAS composites at loading frequency of f 5 1 Hz and fatigue peak stress of σmax 5 137 MPa. HDE, Hysteresis dissipated energy.

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Figure 6.26 (A) The fatigue HDE versus applied cycles curves; (B) the fibermatrix interface shear stress versus applied cycles curves; and (C) the broken fibers fraction versus applied cycles curve of cross-ply SiC/MAS composites at loading frequency of f 5 1 Hz and fatigue peak stress of σmax 5 120 MPa. HDE, Hysteresis dissipated energy.

Figure 6.27 (A) The fatigue HDE versus applied cycles curves; (B) the fibermatrix interface shear stress versus applied cycles curves; and (C) the broken fibers fraction versus applied cycles curve of cross-ply SiC/MAS composites at loading frequency of f 5 1 Hz and fatigue peak stress of σmax 5 103 MPa. HDE, Hysteresis dissipated energy.

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number N 5 1 to U 5 26 kPa at cycle number N 5 3, and then decreases to U 5 7.4 kPa at cycle number of N 5 20,000. The fibermatrix interface shear stress decreases from τ i 5 7.6 MPa at cycle number N 5 4 to τ i 5 1.1 MPa at cycle number N 5 10,608. The broken fibers fraction increases from 2% at cycle number N 5 1 to 39% at cycle number N 5 31,053. When the fatigue peak stress is σmax 5 96 MPa, the fatigue HDE, fibermatrix interface shear stress, and broken fibers fraction versus applied cycles curves are shown in Fig. 6.28. Under cyclic fatigue loading, the experimental fatigue HDE decreases from U 5 16 kPa at cycle number N 5 3 to U 5 4.4 kPa at cycle number N 5 23,067. The theoretical fatigue HDE decreases from U 5 25.1 kPa at cycle number N 5 1 to U 5 4.3 kPa at the number N 5 30,000. The fibermatrix interface shear stress decreases from τ i 5 6.2 MPa at cycle number N 5 3 to τ i 5 1.0 MPa at cycle number N 5 23,067. The broken fibers fraction increases from 1% at cycle number N 5 1 to 18% at cycle number N 5 204,364. When the loading frequency is f 5 10 Hz, the fatigue life SN curve of crossply SiC/MAS composites at 1093 C in air atmosphere is shown in Fig. 6.24. The composite tensile strength is about 209 MPa, and when the fatigue peak stress is σmax 5 59 MPa, the corresponding failure cycle number is N 5 58,7694. The fatigue HDE at the fatigue peak stresses of σmax 5 103, 96, and 86 MPa decreases with applied cycles. The fibermatrix interface shear stress decreases with applied cycles. The broken fibers fraction increases with applied cycles for different fatigue peak stresses of σmax 5 103, 96, and 86 MPa. The fatigue limit stress at loading frequency of f 5 10 Hz is higher than that at loading frequency of f 5 1 Hz. When the fatigue peak stress is σmax 5 103 MPa, the fatigue HDE, fibermatrix interface shear stress, and broken fibers fraction versus applied cycles curves are shown in Fig. 6.29. Under cyclic fatigue loading, the experimental fatigue HDE decreases from U 5 13 kPa at cycle number N 5 6 to U 5 3.1 kPa at cycle number N 5 94,044. The theoretical fatigue HDE decreases from U 5 16.6 kPa at cycle number N 5 1 to U 5 3.8 kPa at cycle number N 5 100,000. The fibermatrix interface shear stress decreases from τ i 5 2.4 MPa at cycle number N 5 6 to τ i 5 0.6 MPa at cycle number N 5 94,044. The broken fibers fraction increases from 12% at cycle number N 5 1 to 40% at cycle number N 5 67. When the fatigue peak stress is σmax 5 96 MPa, the fatigue HDE, fibermatrix interface shear stress, and broken fibers fraction versus applied cycles curves are shown in Fig. 6.30. Under cyclic fatigue loading, the experimental fatigue HDE decreases from U 5 8 kPa at cycle number N 5 6 to U 5 3.6 kPa at cycle number N 5 28,400. The theoretical fatigue HDE decreases from U 5 13.4 kPa at cycle number N 5 1 to U 5 4.2 kPa at cycle number N 5 30,000. The fibermatrix interface shear stress decreases from τ i 5 2 MPa at cycle number N 5 6 to τ i 5 0.8 MPa at cycle number N 5 28,400. The broken fibers fraction increases from 3% at cycle number N 5 1 to 37% at cycle number N 5 104,127. When the fatigue peak stress is σmax 5 86 MPa, the fatigue HDE, fibermatrix interface shear stress, and broken fibers fraction versus applied cycles curves are shown in Fig. 6.31. Under cyclic fatigue loading, the experimental fatigue HDE decreases from U 5 6 kPa at cycle number N 5 7 to U 5 1.6 kPa at cycle number

Figure 6.28 (A) The fatigue HDE versus applied cycles curves; (B) the fibermatrix interface shear stress versus applied cycles curves; and (C) the broken fibers fraction versus applied cycles curve of cross-ply SiC/MAS composites at loading frequency of f 5 1 Hz and fatigue peak stress of σmax 5 96 MPa. HDE, Hysteresis dissipated energy.

Figure 6.29 (A) The fatigue HDE versus applied cycles curves; (B) the fibermatrix interface shear stress versus applied cycles curves; and (C) the broken fibers fraction versus applied cycles curve of cross-ply SiC/MAS composites at loading frequency of f 5 10 Hz and fatigue peak stress of σmax 5 103 MPa. HDE, Hysteresis dissipated energy.

Figure 6.30 (A) The fatigue HDE versus applied cycles curves; (B) the fibermatrix interface shear stress versus applied cycles curves; and (C) the broken fibers fraction versus applied cycles curve of cross-ply SiC/MAS composites at loading frequency of f 5 10 Hz and fatigue peak stress of σmax 5 96 MPa. HDE, Hysteresis dissipated energy.

Figure 6.31 (A) The fatigue HDE versus applied cycles curves; (B) the fibermatrix interface shear stress versus applied cycles curves; and (C) the broken fibers fraction versus applied cycles curve of cross-ply SiC/MAS composites at loading frequency of f 5 10 Hz and fatigue peak stress of σmax 5 86 MPa. HDE, Hysteresis dissipated energy.

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N 5 1,063,330. The theoretical fatigue HDE decreases from U 5 8.5 kPa at cycle number N 5 1 to U 5 2 kPa at cycle number N 5 1,100,000. The fibermatrix interface shear stress decreases from τ i 5 2 MPa at cycle number N 5 6 to τ i 5 0.4 MPa at cycle number N 5 1,063,330. The broken fibers fraction increases from 2.5% at cycle number N 5 1 to 25% at cycle number N 5 194,793.

6.4.3 2D ceramic-matrix composites There are many researchers performing experimental investigations on the fatigue behavior of 2D CMCs. Wang and Laird (1997) investigated the tensiontension cyclic fatigue behavior of 2D C/SiC composites at room temperature. The fatigue loading frequency is f 5 1 Hz, the stress ratio is R 5 0, the fatigue peak stress is σmax 5 320380 MPa, and the fatigue limit cycle is defined as 106. The fatigue limit stress is about 80% of the tensile strength. Under cyclic fatigue loading, seven damage modes were observed, namely, matrix cracking, weft yarns cracking, fibermatrix interface debonding, fibers break, delamination, separation of the yarns, and matrix wear. The evolution of these seven damage modes degrades the modulus. Under cyclic loading, the high loading rate and repeated loading lead to the serious damage of delamination and yarns separation. Sun et al. (2007) investigated the tensiontension cyclic fatigue behavior of 2D C/SiC composites. The loading frequency is f 5 10 Hz, the stress ratio is R 5 0.1, the fatigue peak stress is 80%90% of the tensile strength (i.e., σUTS 5 268 MPa), and the fatigue limit cycle is defined as 5 3 105. At room temperature, the fatigue limit stress is about 80%85% of the tensile strength, and when the fatigue peak stress exceeds 88% of the tensile strength, fatigue failure quickly occurs. Hou et al. (2005) investigated the tensiontension cyclic fatigue behavior of notched 2D C/SiC composites at room and elevated temperatures in vacuum atmosphere. The loading frequency is f 5 60 Hz, the stress ratio is R 5 0.1, and the fatigue limit cycle is defined as 106. The fatigue limit stress is about 80%90% of the tensile strength at the same temperature. At initial cyclic fatigue loading, the damage near the notch propagates quickly with a large amount of transverse matrix cracking. With increasing applied cycles, the damage grows slowly with more damage modes. Rouby and Reynaud (1993) investigated the tensiontension cyclic fatigue behavior of 2D SiC/SiC composites at room temperature. The fatigue loading frequency is f 5 1 Hz, the stress ratio is R 5 0, the fatigue peak stress is σmax 5 50175 MPa, and the fatigue limit cycle is defined as 106. At room temperature, the fatigue limit stress is 135 MPa, which is about 75% of the tensile strength. According to the fatigue peak stress, the fatigue damage can be divided into three conditions: G

G

G

When the fatigue peak stress is higher than the tensile strength upon first loading to the fatigue peak stress, the composite fatigue fractures. When the fatigue peak stress is between the fatigue limit stress (i.e., 75% of the tensile strength) and the tensile strength, the fatigue failure occurs after a certain number of applied cycles (i.e., N 5 510,000). When the fatigue peak stress is lower than the fatigue limit stress, the specimen survived 106 cycles; however, when the fatigue peak stress is higher than the FMCS, the shape of the fatigue hysteresis loops evolves with cycles.

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¨ nal (1996a,b) investigated the tensiontension cyclic fatigue behavior of 2D U SiC/SiC composites at room temperature and elevated temperature of 1300 C in N2 atmosphere. The fatigue loading frequency is f 5 0.5 Hz and the stress ratio is R 5 0.1. The creep affects the fatigue damage with fibers brittle fracture, and there was few fibers pullout at the fracture surface. Zhu et al. (1997, 1998, 1999) investigated the tensiontension cyclic fatigue behavior of 2D SiC/SiC composites at room temperature and an elevated temperature of 1000 C in Ar atmosphere. At room temperature, the fatigue loading frequency is f 5 10 Hz; and at elevated temperature, the loading frequency is f 5 20 Hz. The stress ratio is R 5 0.1 and the fatigue limit cycle is defined as 107. At room temperature, the fatigue limit stress is 160 MPa, which is higher than the proportional limit stress (i.e., σpls 5 80 MPa), and is about 70%80% of the tensile strength. At an elevated temperature of 1000 C in Ar atmosphere, the fatigue life SN curve can be divided into three regions: G

G

G

Low-cyclic region, i.e., N , 104. There is no obvious difference in fatigue life between room temperature and high temperature. When the fatigue peak stress is less than 180 MPa, the fatigue life at 1000 C in Ar atmosphere decreases quickly; however, there is no fatigue life decrease at room temperature. Fatigue limit region. When the fatigue limit cycle is defined as 107, the fatigue limit stress at elevated temperature is 75 MPa, which is far below the proportional limit stress (i.e., σpls 5 100 MPa), and is about 30% of the tensile strength at elevated temperature.

Zhu (2006) investigated the effect of oxidation on fatigue behavior of 2D SiC/ SiC composites. After oxidation for 100 hours at 600 C in air atmosphere, the fibermatrix interface decreases, leading to the decrease of fatigue life by 13%. After oxidation for 100 hours at 800 C in air atmosphere, the fibermatrix interface forms strong SiO2 interphase, resulting in shorter fatigue life. Groner (1994) investigated the tensiontension cyclic fatigue behavior of 2D SiC/SiC composites with/ without a hole at an elevated temperature of 1100 C in air atmosphere. For the SiC/ SiC composite without a hole, the fatigue limit stress at 1100 C is about 105 MPa, which is about 45% of the tensile strength; however, for the SiC/SiC composite with a hole, the fatigue limit stress is about 95 MPa, which is about 42% of the tensile strength. The SN curves for the specimen with/without a hole are similar. The hole has little effect on the fatigue life of 2D SiC/SiC composites at elevated temperature. Lee et al. (1998) investigated the tensiontension cyclic fatigue behavior of 2D SiC/SiNC composites at room temperature and an elevated temperature of 1000 C in air atmosphere. The fatigue loading frequency is f 5 1 Hz, the stress ratio is R 5 0.05, and the fatigue limit cycle is defined as 106. At room temperature, the fatigue limit stress is 160 MPa, which is higher than the proportional limit stress (σpls 5 85 MPa), and is about 75% tensile strength (σUTS 5 197 MPa). When the fatigue peak stress is σmax 5 175 MPa, the fatigue hysteresis loops were measured and no strain ratcheting appeared. At 1000 C, the fatigue limit stress is 110 MPa, which is 51% tensile strength (σUTS 5 214 MPa). The fatigue hysteresis loops area at elevated temperature is much larger than that at room temperature, mainly due to the decrease of thermal residual stress and fibers oxidation. Haque and Rahman (2000) investigated the tensiontension cyclic fatigue behavior of 2D SiC/SiNC composites at room temperature and elevated temperatures of 700 C and 1000 C in

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air atmosphere. The fatigue loading frequency is f 5 1 Hz, the fatigue stress ratio is R 5 0.1, and the fatigue limit cycle is defined as 106. The fatigue life decreases with increasing testing temperature. At room temperature, the fatigue limit stress is about 80% of the tensile strength; at 700 C in air atmosphere, the fatigue limit stress is about 50% of the tensile strength; and at 1000 C in air atmosphere, the fatigue limit stress is about 20% of the tensile strength. At room temperature, the fatigue life of SiC/SiNC composites can be divided into three regions: G

G

G

At high fatigue peak stress, the fibers failure caused fatigue fracture. At low fatigue peak stress, the fibermatrix interface wear decreases the fiber bridge traction stress, resulting in matrix cracking propagation, delamination, and fibers fracture, and then fatigue fracture. The fatigue limit region.

Steel et al. (2001) investigated the tensiontension cyclic fatigue behavior of 2D Nextel 720/alumina composites at room temperature and an elevated temperature of 1200 C in air atmosphere. The fatigue loading frequency is f 5 1 Hz, the fatigue stress ratio is R 5 0.05, and the fatigue limit cycle is defined as 105. At room temperature, the fatigue limit stress is 102 MPa, which is about 70% of the tensile strength (σUTS 5 144 MPa). At 1200 C in air atmosphere, the fatigue limit stress is 122 MPa, which is about 87% tensile strength (σUTS 5 140 MPa). The fatigue damage mechanism at elevated temperature is similar with that at room temperature; however, fiber creep affects the fatigue behavior. Zawada et al. (2003) investigated the tensiontension fatigue behavior of 2D Nextel 610/Al2O3SiO2 composites at room temperature and elevated temperature of 1000 C in air atmosphere. The fatigue loading frequency is f 5 1 Hz, the stress ratio is R 5 0.05, and the fatigue limit cycle is 105. At room temperature, the fatigue limit stress is 170 MPa and about 85% of the tensile strength; and at 1000 C in air atmosphere, the fatigue limit stress is 150 MPa and about 85% of the tensile strength. At elevated temperature, when the fatigue peak stress is σmax 5 100150 MPa, the modulus decreases by 5%10% during the initial 1000 cycles, and then the modulus remains constant with increasing cycles till the fatigue limit cycles, indicating no obvious accumulation damage. After experiencing 106 cycles at elevated temperature, and then cooled down to room temperature, the residual strength at room temperature is the same as the original strength. The hysteresis loops were measured during cycling, and the hysteresis loops area is low, only 35 kPa. Ruggles-Wrenn et al. (2012) investigated the tensiontension cyclic fatigue behavior of 2D SiC/[SiC-B4C] composites at an elevated temperature of 1200 C in air and steam atmospheres. When the loading frequency is f 5 0.1 Hz, the fatigue life in steam atmosphere greatly reduced compared with that in air atmosphere; however, when the loading frequency is f 5 1 Hz, the steam atmosphere has little effect on the fatigue life. Lanser and Ruggles-Wrenn (2016) investigated the tensioncompressive fatigue behavior of 2D Nextel 720/alumina composites at an elevated temperature of 1200 C in air and steam atmosphere. In air atmosphere, the fatigue limit stress is about 40% of the tensile strength; and in steam atmosphere, the fatigue limit stress is about 35% of the tensile strength. Ruggles-Wrenn and Lee (2016) investigated the tension tension fatigue behavior of 2D SiC/SiC composites at an elevated temperature of

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1300 C in air and steam atmospheres. In air atmosphere, the fatigue limit stress is about 28% of the tensile strength; however, in steam atmosphere, the fatigue limit stress is about 19% of the tensile strength. Sabelkin et al. (2016) investigated the tensiontension cyclic fatigue behavior of 2D SiC/SiC composites at an elevated temperature of 1205 C in a combustion chamber environment and in air atmosphere. Compared with the testing condition in air, the fatigue life in combustion chamber environment decreases nearly 100 times, mainly due to the oxidation embrittlement at elevated temperature. Cheng et al. (2010) investigated the tensiontension cyclic fatigue behavior of 2D C/SiC composites at an elevated temperature of 1300 C in oxidative environment. It was found that the fatigue limit stress decreases greatly for the testing condition with high oxygen content. Dong et al. (2016) investigated the tensiontension cyclic fatigue behavior of 2D C/[SiCSiBC]m composites at elevated temperatures in the range of 300 C1200 C in air atmosphere. When the temperature is lower than 400 C, the fatigue damage of 2D C/[SiCSiBC]m composites is mainly caused by stress and temperature; and when the temperature reaches 550 C, the temperature, stress, and oxidation affect the fatigue damage, and the oxidation mechanism is affected by the temperature. Longbiao (2017b,c,d) investigated the tensiontension cyclic fatigue behavior of 2D SiC/SiC and cross-ply SiC/MAS composites at elevated temperatures in air atmosphere. The fatigue life is affected by the fibermatrix interface wear and interface oxidation and decreases greatly compared to that at room temperature.

6.4.3.1 C/SiC composites at room temperature The fatigue life SN curves of 2D C/SiC composites at room temperature are shown in Fig. 6.32. Shuler et al. (1993) investigated the tensiontension cyclic

Figure 6.32 The fatigue life SN curves of 2D C/SiC composites at room temperature.

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fatigue behavior of 2D C/SiC composites at room temperature. The composite tensile strength is σUTS 5 420 MPa, and the fatigue limit stress approaches 83% of the tensile strength. The broken fibers fraction versus applied cycles curves at the fatigue peak stresses of 80%, 83%, 86%, 89%, 91%, and 96% of the tensile strength are shown in Fig. 6.33A. Lu et al. (2014) investigated the tensiontension cyclic fatigue behavior of 2D C/SiC composites at room temperature. The composite tensile strength is σUTS 5 264 MPa, and the fatigue limit stress approaches to 80% of the tensile strength. The broken fibers fraction versus applied cycle number curves at the fatigue peak stresses of 85%, 86%, 87%, 88%, 89%, and 90% of the tensile strength are shown in Fig. 6.33B. Min et al. (2014) investigated the tensiontension cyclic fatigue behavior of 2D C/SiC composites at room temperature. The composite tensile strength is σUTS 5 500 MPa, and the fatigue limit stress approaches 78% of the tensile strength. The broken fibers fraction versus applied cycle number curves at the fatigue peak stresses of 77%, 82%, 86%, and 92% of the tensile strength are shown in Fig. 6.33C.

6.4.3.2 SiC/SiC composites at room temperature The fatigue life SN curves of 2D SiC/SiC composites at room temperature are shown in Fig. 6.34. Reynaud (1996) investigated the tensiontension cyclic fatigue behavior of 2D SiC/SiC composites at room temperature. The composite tensile strength is σUTS 5 170 MPa, and the fatigue limit stress approaches 75% of the tensile strength. The broken fibers fraction versus applied cycles curves for different fatigue peak stresses of 95%, 90%, and 85% of the tensile strength are shown in Fig. 6.35A. Mizuno et al. (1996) investigated the tensiontension fatigue behavior of 2D SiC/SiC composites at room temperature. The fatigue limit stress approaches 85% of the tensile strength. The broken fibers fraction versus applied cycles curves for different fatigue peak stresses of 95%, 90%, and 85% of the tensile strength are shown in Fig. 6.35B.

6.4.3.3 C/SiC composites at elevated temperature The fatigue life SN curves of 2D C/SiC composites at elevated temperature are shown in Fig. 6.36. Mall and Engesser (2006) investigated the tensiontension cyclic fatigue behavior of 2D C/SiC composites at an elevated temperature of 550 C in air atmosphere. The composite tensile strength is σUTS 5 487 MPa, and the fatigue limit stress approaches 30% of the tensile strength. The broken fibers fraction versus applied cycles curves for different fatigue peak stresses of 72%, 56%, 36%, and 22% of the tensile strength are shown in Fig. 6.37A. Cheng et al. (2010) investigated the tensiontension fatigue behavior of 2D C/SiC composites at an elevated temperature of 1300 C in an oxidative environment. The composite tensile strength is σUTS 5 300 MPa, and the fatigue limit stress approaches 27% of the tensile strength. The broken fibers fraction versus applied cycles curves for different fatigue peak stresses of 80%, 70%, 60%, and 50% of the tensile strength are shown in Fig. 6.37B.

Figure 6.33 The broken fibers fraction versus applied cycles curves of 2D C/SiC composites. (A) The broken fiber fraction versus applied cycles curves at the fatigue peak stresses of 80%, 83%, 86%, 89%, 91% and 96% of the tensile strength; (B) the broken fiber fraction versus applied cycle number curves at the fatigue peak stresses of 85%, 86%, 87%, 88%, 89% and 90% of the tensile strength; and (C) the broken fiber fraction versus applied cycle number curves at the fatigue peak stresses of 77%, 82%, 86% and 92% of the tensile strength.

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Figure 6.34 The fatigue life SN curves of 2D SiC/SiC composites at room temperature.

6.4.3.4 SiC/SiC composites at elevated temperatures The fatigue life SN curves of 2D SiC/SiC composites at elevated temperature are shown in Fig. 6.38. Mall (2005) investigated the tensiontension cyclic fatigue behavior of 2D SiC/SiC composites at an elevated temperature of 750 C. The composite tensile strength is σUTS 5 345 MPa. Under 0% (dry) moisture atmosphere, the fatigue limit stress approaches 67% of the tensile strength. The broken fibers fraction versus applied cycles curves for the fatigue peak stresses of 70%, 80%, and 90% of the tensile strength are shown in Fig. 6.39A. Under 60% moisture content atmosphere, the fatigue limit stress approaches 49% of the tensile strength. The broken fibers fraction versus applied cycles curves for the fatigue peak stresses of 70%, 80%, and 90% of the tensile strength are shown in Fig. 6.39B. Mizuno et al. (1996) investigated the tensiontension cyclic fatigue behavior of 2D SiC/SiC composites at an elevated temperature of 1000 C in Ar atmosphere. The composite tensile strength is σUTS 5 251 MPa, and the fatigue limit stress approaches 30% of the tensile strength. The broken fibers fraction versus the applied cycles curves for the fatigue peak stresses of 60%, 70%, and 80% of the tensile strength are shown in Fig. 6.39C. Michael (2010) investigated the tensiontension cyclic fatigue behavior of 2D SiC/SiC composites at an elevated temperature of 1000 C. The composite tensile strength is σUTS 5 114 MPa. In air atmosphere, the fatigue limit stress approaches 20% of the tensile strength. The broken fibers fraction versus the applied cycles curves for the fatigue peak stresses of 70%, 80%, and 90% of the tensile strength are shown in Fig. 6.39D. In steam atmosphere, the fatigue limit stress approaches 12% of the tensile strength. The broken fibers fraction versus the applied cycles

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Figure 6.35 The broken fibers fraction versus applied cycles curves of 2D SiC/SiC composites at room temperature. (A) The broken fiber fraction versus applied cycles curves for different fatigue peak stresses of 95%, 90% and 85% of the tensile strength; and (B) the broken fiber fraction versus applied cycles curves for different fatigue peak stresses of 95%, 90% and 85% of the tensile strength.

curves for the fatigue peak stresses of 85%, 90%, and 95% of the tensile strength are shown in Fig. 6.39E. Jacob (2010) investigated the tensiontension cyclic fatigue behavior of 2D SiC/SiC composites at an elevated temperature of 1200 C. The composite tensile strength is σUTS 5 306 MPa. In air atmosphere, the fatigue limit stress approaches 18% of the tensile strength. The broken fibers fraction versus applied cycles curves for different fatigue peak stresses of 70%, 80%, and 90% of the tensile strength are

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Figure 6.36 The fatigue life SN curves of 2D C/SiC composites at elevated temperature.

shown in Fig. 6.39F. In steam atmosphere, the fatigue limit stress approaches 10% of the tensile strength. The broken fibers fraction versus the applied cycles curves for different fatigue peak stresses of 60%, 70%, and 80% of the tensile strength are shown in Fig. 6.39G. Ruggles-Wrenn and Sharma (2011) investigated the tensiontension cyclic fatigue behavior of 2D SiC/SiC composites at an elevated temperature of 1300 C. The composite tensile strength is σUTS 5 241 MPa. In air atmosphere, the fatigue limit stress approaches 42% of the tensile strength. The broken fibers fraction versus the applied cycles curves for different fatigue peak stresses of 70%, 80%, and 90% of the tensile strength are shown in Fig. 6.39H. In steam atmosphere, the fatigue limit stress approaches 40% of the tensile strength. The broken fibers fraction versus the applied cycles curves for different fatigue peak stresses of 70%, 80%, and 90% of the tensile strength are shown in Fig. 6.39I.

6.4.4 2.5D ceramic-matrix composites Dalmaz et al. (1998) investigated the tensiontension cyclic fatigue behavior of 2.5D C/SiC composites at room and elevated temperatures in inert atmosphere. The fatigue loading frequency is f 5 1 and 10 Hz. Due to the thermal expansion coefficient mismatch between the fiber and the matrix, cracking occurs inside the matrix and yarns. The thermal residual tensile stress exists between the yarns and the interface layer, leading to weak interface bonding. The fatigue behavior of 2.5D C/SiC composites depends on the interface shear strength between the fiber and the matrix. When the temperature is below 1000 C, the repeated sliding between the yarns/yarns and yarns/matrix results in the recession of the fibermatrix interface. The fatigue damage depends on the cyclic number, not the cyclic time. However,

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Figure 6.37 The broken fibers fraction versus applied cycles curves of 2D C/SiC composites at elevated temperature. (A) The broken fiber fraction versus applied cycles curves for different fatigue peak stresses of 72%, 56%, 36% and 22% of the tensile strength; and (B) the broken fibers fraction versus applied cycles curves for different fatigue peak stresses of 80%, 70%, 60% and 50% of the tensile strength.

with increasing temperature, the thermal residual stress between the fiber and the matrix decreases, which increases the elastic modulus and fatigue life. When the temperature is above 1000 C, creep affects the fatigue behavior and the fatigue performance degrades again.

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Figure 6.38 The fatigue life SN curves of 2D SiC/SiC composites at elevated temperatures.

The fatigue life SN curves and the broken fibers fraction versus applied cycles of 2.5D C/SiC composites at room and elevated temperatures are shown in Figs. 6.40 and 6.41. Zhang et al. (2013) investigated the tensiontension fatigue behavior of 2.5D C/SiC composites at room temperature. The composite tensile strength is σUTS 5 212 MPa, and the fatigue limit stress to 85% of the tensile strength. The broken fibers fraction versus applied cycles curves for different fatigue peak stresses of 85%, 89%, 94%, 96%, and 99% of the tensile strength are shown in Fig. 6.41A. Yang (2011) investigated the tensiontension cyclic fatigue behavior of 2.5D C/ SiC composites at an elevated temperature of 800 C in air atmosphere. The composite tensile strength is σUTS 5 280 MPa, and the fatigue limit stress approaches 28% of the tensile strength. The broken fibers fraction versus the applied cycles curves for different fatigue peak stresses of 50%, 60%, 70%, and 80% of the tensile strength are shown in Fig. 6.41B.

6.4.5 3D ceramic-matrix composites Many researchers performed investigations on the fatigue behavior of 3D CMCs. Kostopoulos et al. (1997) investigated the tensiontension cyclic fatigue behavior of uncoated/coated 3D C/SiC composites at room temperature. The fatigue loading frequency is f 5 10 Hz, the stress ratio is R 5 0.1, and the fatigue limit cycle is 106. Under cyclic fatigue loading, the fatigue limit stress for the uncoated and coated C/ SiC composites is nearly the same, which approaches 70% of tensile strength. The tensile test was conducted after fatigue loading, and it was found that the tensile

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Figure 6.39 The broken fibers fraction versus the applied cycles curves of 2D SiC/SiC composites at elevated temperatures. (A) The broken fiber fraction versus applied cycles curves for the fatigue peak stresses of 70%, 80% and 90% of the tensile strength; (B) the broken fiber fraction versus applied cycles curves for the fatigue peak stresses of 70%, 80% and 90% of the tensile strength; (C) the broken fiber fraction versus the applied cycles curves for the fatigue peak stresses of 60%, 70% and 80% of the tensile strength; (D) the broken fiber fraction versus the applied cycles curves for the fatigue peak stresses of 70%, 80% and 90% of the tensile strength; (E) the broken fiber fraction versus the applied cycles curves for the fatigue peak stresses of 85%, 90% and 95% of the tensile strength; (F) the broken fiber fraction versus applied cycles curves for different fatigue peak stresses of 70%, 80% and 90% of the tensile strength; (G) the broken fiber fraction versus the applied cycles curves for different fatigue peak stresses of 60%, 70% and 80% of the tensile strength; (H) the broken fiber fraction versus the applied cycles curves for different fatigue peak stresses of 70%, 80% and 90% of the tensile strength; and (I) the broken fiber fraction versus the applied cycles curves for different fatigue peak stresses of 70%, 80% and 90% of the tensile strength.

Fatigue life prediction of ceramic-matrix composites based on hysteresis dissipated energy

Figure 6.39 (Continued).

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Figure 6.39 (Continued).

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Figure 6.39 (Continued).

Figure 6.40 The fatigue life SN curves of 2.5D C/SiC composites at room temperature and elevated temperature.

stressstrain curve is linear, the residual strength decreases by 18%, and the modulus decreases by 18%. The resonance frequency and damping coefficient were measured under cyclic loading to monitor the fatigue damage accumulation (Kostopoulos et al., 1999). When the fatigue peak stress is high, the damping coefficient increases with applied cycles till fatigue failure, which indicates the energy dissipation inside the composite. Under cyclic loading, when the fatigue peak stress is higher than the matrix strength, upon first loading to the peak stress there appears a large amount of matrix cracking with fibermatrix interface debonding.

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Figure 6.41 The broken fibers fraction versus the applied cycles curves of 2.5D C/SiC composites at room and elevated temperature. (A) The broken fiber fraction versus applied cycles curves for different fatigue peak stresses of 85%, 89%, 94%, 96%, and 99% of the tensile strength; and (B) the broken fiber fraction versus the applied cycles curves for different fatigue peak stresses of 50%, 60%, 70% and 80% of the tensile strength.

The matrix cracking approaches the steady-state during initial cyclic loading and remains constant during the following cycles. The matrix cracking is the premise for the fatigue failure of 3D C/SiC composites; however, it is not the direct reason. The interface wear between the fiber and the matrix inside of yarns, and between yarns and yarns are the main reasons for fatigue failure. When cycling to the steadystate stage, the interface wear reduces the stress concentration at the intersection of the yarns, resulting in the slow recovery of the modulus. At the final stage of the cyclic loading, the local fibers failure causes the fatigue fracture. The fracture surface of the failure specimen was observed. The fibers pullout confirmed the fibermatrix interface wear mechanism during cyclic loading. Butkus et al. (1990)

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investigated the tensiontension cyclic fatigue behavior of 3D SiC/SiC composites at room temperature. The loading frequency is f 5 10 Hz, the stress ratio is R 5 0.1, the fatigue peak stress is σmax 5 100250 MPa, and the fatigue limit cycle is 106. The fatigue limit stress at room temperature is 215 MPa, and about 90% of the tensile strength. When the fatigue peak stress is lower than 215 MPa, the modulus decreases during first 10 applied cycles and then remains unchanged with increasing cycles. Du et al. (2002) investigated the tensiontension cyclic fatigue behavior of 3D C/SiC composites at room temperature and an elevated temperature of 1300 C in vacuum atmosphere. The fatigue loading frequency is f 5 60 Ha, the stress ratio is R 5 0.1, and the fatigue limit cycle is defined as 106. The fatigue limit stress at 1300 C is 285 MPa, and about 94% of the tensile strength. The fatigue limit stress at room temperature is 235 MPa, and about 85% of the tensile strength. The fiber pullout length at 1300 C is much longer than that at room temperature. The fatigue damage mainly originates from the braided intersection of the fiber bundles. With the increase of fatigue cycles, the damage of the matrix around the fiber bundles increases. Liu (2003) investigated the tensiontension cyclic fatigue behavior of 3D C/SiC composites at room temperature and elevated temperatures of 1100 C, 1300 C, and 1500 C in vacuum atmosphere. The fatigue life SN curve can be divided into three regions, that is, short life, long life, and infinite life regions. The thermal residual stress caused by the thermal expansion coefficient mismatch changes with increasing temperature. At room temperature, the fibermatrix interface radial thermal residual stress is tensile stress; however, above the fabrication temperature, the fibermatrix interface radial thermal residual stress is compressive stress. The fiber pullout length at the fracture surface is obvious different, that is, the longest at room temperature and the shortest at 1500 C. The fatigue cracks originate mainly from the fiber weaving intersections, and the mode of matrix crack propagation mainly depends on the fibermatrix interface layer and the internal flaw inside of the matrix. Zhang et al. (2009) investigated the fatigue damage evolution of 3D SiC/SiC composites at 1300 C in an oxidative environment and analyzed the fatigue failure mechanism. Luo et al. (2016) investigated the tensiontension fatigue behavior of 3D SiC/SiC composites at 1300 C in air atmosphere and revealed the fatigue damage mechanisms at elevated temperature of matrix initial cracking and propagation, fibermatrix interface debonding, fibers fracture, and fibers pullout and bridging. Under high fatigue peak stress, that is, the fatigue peak stress is higher than the proportional limit stress, the fiber plays the key role on the fatigue damage evolution and fatigue life. Under low fatigue peak stress, that is, the fatigue peak stress is lower than the proportional limit stress, the matrix plays the key role on the fatigue damage evolution and fatigue life. When the fatigue peak stress is between the fatigue limit stress and proportional limit stress, the fiber and matrix both affect the fatigue damage evolution. Han et al. (2004) investigated the fatigue behavior of 2D and 3D C/SiC composites at elevated temperatures in the range of 1100 C1500 C in vacuum atmosphere. Compared with 2D C/SiC composites, the tensile strength and the proportional limit stress is much higher for 3D C/SiC composites. The fiber pullout length is much longer, indicating a longer sliding distance at the fibermatrix interface. Ren (2004) investigated the tensiontension cyclic fatigue and fatigue/creep behavior of 3D C/SiC composites in Ar, dry oxygen, and wet

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oxygen atmospheres. The elastic modulus decreases with applied cycles. Under fatigue/creep loading at 1300 C, the damage mechanisms include matrix cracking and fiber and fiber bundles fracture and pullout. In Ar atmosphere, the main damage mechanism includes the fibers fracture and pullout; however, in dry and wet oxygen, besides the abovementioned damage mechanisms, fibers oxidation also occurs. Wu et al. (2006) investigated the tensiontension cyclic fatigue behavior of 3D C/SiC composites at an elevated temperature of 1300 C in H2OO2Ar atmosphere. The loading frequency is f 5 1 Hz and the stress ratio is R 5 0.1. Compared with tensile loading, the fibermatrix interface debonding, sliding, and fibers pullout under cyclic loading are much more obvious, with more fibers pullout and longer pullout length. Under tensile loading, the composite failure depends on the fibers strength distribution and fibermatrix interface properties. Under cyclic loading, the fatigue peak stress is lower than the FMCS, the braiding angle decreases with cycles, which reduces the fibermatrix interface bonding strength, resulting in matrix cracking and propagating. When matrix cracking appears, the oxygen and water vapor entered the internal of composite through cracks, leading to the oxidation of fibermatrix PyC interphase and SiC matrix. When the opening of matrix cracking is small, the oxygen entering the composite is less, the PyC interface is partially oxidized, and the SiC matrix is much less oxidized. When the opening of matrix cracking is wide, more oxygen penetrates into the material, which makes the PyC interface completely oxidized and the silicon carbide matrix seriously oxidized.

6.4.5.1 C/SiC at room and elevated temperatures The fatigue life SN curves and broken fibers fraction versus applied cycles curves of 3D C/SiC composites at room and elevated temperatures are shown in Figs. 6.42 and 6.43. Du et al. (2002) investigated the tensiontension fatigue behavior of 3D

Figure 6.42 The fatigue life SN curves of 3D C/SiC composites at room and elevated temperatures.

Figure 6.43 The broken fibers fraction versus applied cycles curves of 3D C/SiC composites at room and elevated temperatures. (A) The broken fiber fraction versus applied cycles curves for different fatigue peak stresses of 87%, 89%, 90% and 94% of the tensile strength; (B) the broken fiber fraction versus applied cycles curves for the fatigue peak stresses of 88% and 97% of the tensile strength; (C) the broken fiber fraction versus applied cycles curves for the fatigue peak stresses of 83%, 93%, 98% and 99% of the tensile strength; and (D) the broken fiber fraction versus applied cycles curves for different fatigue peak stresses of 90%, 92%, 95%, 96% and 98% of the tensile strength.

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C/SiC composites at room temperature. The composite tensile strength is σUTS 5 276 MPa, and the fatigue limit stress approaches to 85% of the tensile strength. The broken fibers fraction versus applied cycles curves for different fatigue peak stresses of 87%, 89%, 90%, and 94% of the tensile strength are shown in Fig. 6.43A. Du et al. (2002) investigated the tensiontension cyclic fatigue behavior of 3D C/SiC composites at elevated temperatures of 1100 C and 1300 C in vacuum atmosphere. At 1100 C, the composite tensile strength is σUTS 5 360 MPa, and the fatigue limit stress approaches to 95% of the tensile strength. The broken fibers fraction versus applied cycles curves for the fatigue peak stresses of 88% and 97% of the tensile strength are shown in Fig. 6.43B. At 1300 C, the composite tensile strength is σUTS 5 304 MPa, and the fatigue limit stress approaches to 93% of the tensile strength. The broken fibers fraction versus applied cycles curves for the fatigue peak stresses of 83%, 93%, 98%, and 99% of the tensile strength are shown in Fig. 6.43C. Du and Qiao (2011) investigated the tensiontension cyclic fatigue behavior of 3D C/SiC composites at an elevated temperature of 1500 C in vacuum atmosphere. The composite tensile strength is σUTS 5 261 MPa, and the fatigue limit stress approaches 90% of the tensile strength. The broken fibers fraction versus applied cycles curves for different fatigue peak stresses of 90%, 92%, 95%, 96%, and 98% of the tensile strength are shown in Fig. 6.43D.

6.4.5.2 SiC/SiC composite at elevated temperature The fatigue life SN curves and broken fibers fraction versus applied cycles curves of 3D SiC/SiC composite at elevated temperatures are shown in Figs. 6.44 and 6.45. Shi et al. (2015) investigated the tensiontension cyclic fatigue

Figure 6.44 The fatigue life SN curves of 3D SiC/SiC composites at elevated temperatures.

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Figure 6.45 The broken fibers fraction versus applied cycles curves of 3D SiC/SiC composites at elevated temperatures. (A) The broken fiber fraction versus applied cycles curves for different fatigue peak stresses of 36%, 46% and 60% of the tensile strength; and (B) the broken fiber fraction versus the applied cycles curves for different fatigue peak stresses of 26%, 43%, 55%, 59% and 68% of the tensile strength.

behavior of 3D SiC/SiC composites at elevated temperatures of 1100 C and 1300 C in air atmosphere. At an elevated temperature of 1100 C, the composite tensile strength is σUTS 5 220 MPa, and the fatigue limit stress approaches to 27% of the tensile strength. The broken fibers fraction versus applied cycles curves for different fatigue peak stresses of 36%, 46%, and 60% of the tensile strength are shown in Fig. 6.45A. At an elevated temperature of 1300 C, the composite tensile strength is σUTS 5 198 MPa, and the fatigue limit stress approaches to 22% of the tensile strength. The broken fibers fraction versus the

446

Durability of Ceramic-Matrix Composites

applied cycles curves for different fatigue peak stresses of 26%, 43%, 55%, 59%, and 68% of the tensile strength are shown in Fig. 6.45B.

6.5

Conclusion

When CMCs are subjected to cyclic fatigue loading, there exist multiple fatigue damage mechanisms, including matrix cracking, fibermatrix interface debonding and sliding, interface wear, and fibers fracture. These fatigue damage mechanisms affect the evolution of fatigue HDE and HDE-based damage parameter. In this chapter, the theoretical relationships between the fatigue HDE, HDE-based damage parameter, fibermatrix interface debonding ratio, fibers broken fraction, and fatigue life have been established. The effects of fatigue peak stress, fatigue stress ratio, matrix crack spacing, and fibers volume fraction on the evolution of fatigue HDE, HDE-based damage parameter, and fibermatrix interface debonding and sliding have been discussed. The experimental damage evolution and lifetime prediction of unidirectional SiC/CAS and SiC/1723 composites at room temperature, cross-ply SiC/MAS composites at 566 C and 1093 C, 2D C/SiC composites at room temperature, 550 C and 1300 C, 2D SiC/SiC composites at room temperature, 750 C, 1000 C, 1200 C, and 1300 C, 2.5D C/SiC composites at room temperature and 800 C, 3D C/SiC composites at room temperature and 1300 C and 1500 C, and 3D SiC/SiC composites at room temperature and 1100 C and 1300 C have been predicted.

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Further reading Shi, J., 2001. Tensile fatigue and life prediction of a SiC/SiC composite. In: Proceeding of ASME Turbo Expo 2001. June 47, 2001, New Orleans, LA.

Index

Note: Page numbers followed by “f” and “t” refer to figures and tables, respectively. 0-9, and Symbols 2D ceramic-matrix composites, 21 23 cyclic fatigue damage evaluation of, 347 357 SiC/SiC composite, at 600 C, 800 C and 1000 C in inert condition, 347 349, 349f, 350f SiC/SiC composite, at 1000 C in air and in steam atmosphere, 349 353, 351f, 352f SiC/SiC composite, at 1200 C in air and in steam atmospheres, 353, 354f, 355f SiC/SiC composite, at 1300 C in air atmosphere, 354 357, 356f, 357f fatigue life prediction of, 425 433 C/SiC composites, at elevated temperature, 429 430, 433f, 434f C/SiC composites, at room temperature, 428 429, 428f, 430f SiC/SiC composites, at elevated temperatures, 431 433, 435f, 439f SiC/SiC composites, at room temperature, 429, 431f, 432f S N curves of, 51 54 2D C/SiC composites matrix multicracking evolution, 104 105 tensile loading/unloading damage evolution, 299 301, 304f, 305t 2.5D ceramic-matrix composites, 23 24 fatigue life prediction of, 433 435, 439f, 440f S N curves of, 55 2.5D C/SiC composites, tensile loading/ unloading damage evolution of, 301, 305f, 306t 3D braided CMCs, cyclic fatigue damage evaluation of, 358, 359f

3D braided C/SiC composites, tensile loading/unloading damage evolution of, 301 304, 306f, 307t 3D ceramic-matrix composites, 24 25 fatigue life prediction of, 435 446 C/SiC composites, at room and elevated temperatures, 442 444, 442f, 443f SiC/SiC composite, at elevated temperature, 444 446, 444f, 445f S N curves of, 55 57 3D needled C/SiC composites, tensile loading/unloading damage evolution of, 305 306, 307f, 307t A ACK model, 26 Actively cooled C/SiC composite thrust chambers, 10 Advanced Materials Gas-Generator (AMG), 8 9 Advanced turbine engine gas generator (ATEGG), 6 Aero engines, CMC applications in, 4 F414 aero engine, 7 8 Air Force Research Laboratories, 9 Allison, 6 AMG. See Advanced Materials GasGenerator (AMG) Applied cycle number effect, on fiber matrix interface debonding, 208 210, 209f, 210f, 211f Arbitrary loading sequence effect, on fatigue hysteresis and fiber matrix interface sliding, 215 217, 217f, 218f, 219f, 220f, 221f ATEGG. See Advanced turbine engine gas generator (ATEGG)

454

B Boeing, 8 C Carbon fiber, basic material properties of, 3t Ceramic matrix, 4 Ceramic-matrix composites (CMCs), 1 applications of, 2 14 aero engines, 4 ceramic matrix, 4 France, 4 6 interface phase, 2 4 Japan, 8 9 reinforced fibers, 2 rocket engines, 9 11 scramjet engine, 11 12 thermal protection systems, 12 14 United States. See United States, CMC applications in damage evolution under cyclic fatigue loading, 273 fatigue behavior of. See Fatigue behavior of ceramic-matrix composites lifetime prediction methods of. See Lifetime prediction methods of ceramic-matrix composites tensile behavior of. See Tensile behavior of ceramic-matrix composites tensile strength of, 145 CERASEPR A300, 5 CERASEPR A410, 5 6 CFSE. See Critical fiber strain energy (CFSE) criterion Chemical vapor infiltration (CVI), 1, 10 CLEEN. See Continuous lower energy, emissions, and noise (CLEEN) program CMCs. See Ceramic-matrix composites (CMCs) Continuous lower energy, emissions, and noise (CLEEN) program, 8 Critical fiber strain energy (CFSE) criterion, 26 28 Critical matrix strain energy (CMSE) criterion, 95 96, 108 Cross-ply ceramic-matrix composites, 19 21 cyclic fatigue damage evaluation of, 338 347 C/SiC composite, at 800 oC in air atmosphere, 346 347, %346f

Index

C/SiC composite, at room temperature, 342 345 SiC/CAS composite, at 700 oC in air % atmosphere, 342, 343f SiC/CAS composite, at 850 oC in air % atmosphere, 342, 344f SiC/CAS composite, at room temperature, 338 342, 341f SiC/MAS-L composite, at 800 oC and 1000 oC in inert atmosphere,%347, 348f % fatigue life prediction of, 403 425 SiC/MAS composite, at 566 oC in air % 409f, atmosphere, 407 412, 408f, 410f, 411f, 413f, 414f, 415f SiC/MAS composite, at 1093 oC in air atmosphere, 412 425, 416f,% 417f, 418f, 419f, 421f, 422f, 423f, 424f matrix multicracking evolution of, 121 140, 122f energy balance approach, 125 132 mode 1 cracking evolution, 132, 133f mode 2 cracking evolution, 132 134, 134f, 135f mode 3 cracking evolution, 135, 136f, 137f mode 3 cracking occurring between two mode 1 transverse cracks, 135 140, 137f, 138f, 139f mode 5 cracking occurring between two mode 1 transverse cracks, 135 140, 137f, 138f, 139f stress analysis, 121 124 S N curves of, 48 51 Cross-ply C/SiC composites interface slip lengths, 237 238, 238f tensile loading/unloading damage evolution, 298 299, 300f, 300t C/SiC composites at 800 C in air atmosphere, 346 347, 346f at elevated temperature fatigue life prediction, 429 430, 433f, 434f, 442 444, 442f, 443f unidirectional, 333 338, 335f, 337f, 339f hysteresis dissipated energy under multiple loading sequences, 254 266, 259f, 260f, 261f, 262f, 263f, 264f

Index

at room temperature cross-ply, 342 345 fatigue life prediction, 428 429, 428f, 430f, 442 444, 442f, 443f unidirectional, 321 333, 326f, 330f, 332f CVI. See Chemical vapor infiltration (CVI) Cyclic fatigue loading CMC damage evolution under, 273 at elevated temperatures in oxidative environments, 166 185 fiber strength effect, on composite residual strength and fibers fracture, 174, 178f fiber Weibull modulus effect, on composite residual strength and fibers fracture, 174, 177f fiber matrix interface shear stress effect, on composite residual strength and fibers fracture, 174, 175f, 176f Nextel 720/alumina composites, at elevated temperatures, 184 185, 184f, 185f peak stress effect on composite residual strength and fibers fracture, 172 173, 173f residual strength model, 170 172 SiC/SiC composites, at elevated temperatures, 182 183 SiC/[Si N C] composites, at room and elevated temperatures, 180 181, 180f, 181f testing temperature effect, on composite residual strength and fibers fracture, 174 179, 179f fiber matrix interface shear stress, 99 100, 101f interface debonding and sliding, 195 196, 195f matrix axial stress distribution, 99 100, 101f D Damage evolution of CMC, under cyclic fatigue loading, 273 2D CMCs, 347 357 SiC/SiC composite, at 600 oC, 800 oC % % and 1000 oC in inert condition, % 347 349, 349f, 350f

455

SiC/SiC composite, at 1000 oC in air % and in steam atmosphere, 349 353, 351f, 352f SiC/SiC composite, at 1200 oC in air and in steam atmospheres,%353, 354f, 355f SiC/SiC composite, at 1300 oC in air % 357f atmosphere, 354 357, 356f, 3D braided CMCs, 358, 359f cross-ply CMCs, 338 347 C/SiC composite, at 800 oC in air atmosphere, 346 347, %346f C/SiC composite, at room temperature, 342 345 SiC/CAS composite, at 700 oC in air % atmosphere, 342, 343f SiC/CAS composite, at 850 oC in air % atmosphere, 342, 344f SiC/CAS composite, at room temperature, 338 342, 341f SiC/MAS-L composite, at 800 oC and 1000 oC in inert atmosphere,%347, 348f % stress effect, on fiber/matrix fatigue peak interface debonding and hysteresisbased damage parameters, 310 311, 313f fatigue stress ratio effect, on hysteresisbased damage parameter, 311 312, 314f fiber volume fraction effect, on fiber/ matrix interface debonding and hysteresis-based damage parameters, 309, 311f hysteresis-based damage parameters, 276 279 matrix crack mode effect, on fiber/matrix interface debonding and hysteresisbased damage parameter, 313 318, 318f matrix crack spacing effect, on fiber/ matrix interface debonding and hysteresis-based damage parameters, 312 313, 316f static fatigue damage evolution, 358 369, 361f experimental comparisons, 368 369, 370f fatigue peak stress, effect of, 363 364, 363f

456

Damage evolution of CMC, under cyclic fatigue loading (Continued) matrix crack spacing, effect of, 364 365, 365f oxidation temperature, effect of, 367, 369f volume fraction, effect, of, 365 366, 367f tensile loading/unloading damage evolution, 280 306 2D C/SiC composites, 299 301, 304f, 305t 2.5D C/SiC composites, 301, 305f, 306t 3D braided C/SiC composites, 301 304, 306f, 307t 3D needled C/SiC composites, 305 306, 307f, 307t cross-ply C/SiC composites, 298 299, 300f, 300t fiber volume fraction effect, on hysteresis loops and hysteresis-based damage parameter, 280 283, 282f, 282t fiber/matrix interface debonded energy effect, on hysteresis loops and hysteresis-based damage parameter, 290 293, 292f, 292t fiber/matrix interface shear stress effect, on hysteresis loops and hysteresisbased damage parameter, 286 290, 288f, 289t fibers failure effect, on hysteresis loops and hysteresis-based damage parameter, 293, 294f, 295t matrix cracking space effect, on hysteresis loops and hysteresis-based damage parameter, 283 286, 285f, 285t unidirectional C/SiC composites, 295 298, 295f, 299f, 300t unidirectional CMCs, 319 338 C/SiC composites, at elevated temperature, 333 338, 335f, 337f, 339f C/SiC composites, at room temperature, 321 333, 326f, 330f, 332f SiC/1723 composite, at room temperature, 321, 324f SiC/CAS composite, at room temperature, 319 321, 320f

Index

SiC/CAS-II composite, at room temperature, 321, 323f woven structure effect, on hysteresisbased damage parameter, 318, 319f Damage models, 152 160 fiber matrix interface debonding, 153 155, 161f fibers failure, 158 160 interface wear, 156 158, 157f matrix multicracking evolution, 153, 154f DASA, 10 11, 13 DLR. See German Space Center (DLR) Downstream stress, 79 80, 79f E Energy balance approach, 125 132 mode 1 cracking evolution, 125 127, 126f mode 2 cracking evolution, 127, 128f mode 3 cracking evolution, 127 128, 129f mode 3 cracking occurring between two mode 1 cracks, 129 131, 130f mode 5 cracking occurring between two mode 1 cracks, 131 132, 131f ESPR (Research and Development of Environmentally Compatible Propulsion System for NextGeneration Supersonic Transport) program, 8 European Space Agency, 13 14 European Space Transportation Company, 10 F F414 aero engine, CMC applications in, 7 8 Fatigue behavior of ceramic-matrix composites, 28 45 fibers strength degradation, 37 hysteresis behavior, 31 35 experimental observation, 31 33 theoretical analysis, 33 35 interface wear behavior, 35 37 experimental observation, 35 36 theoretical analysis, 36 37 loading frequency effect, 40 44 modulus degradation, 38 40 oxidation embrittlement, 37 38 stress ratio effect, 44 45

Index

Fatigue hysteresis loops arbitrary loading sequence, effect of, 215 217, 217f, 218f, 219f, 220f, 221f fiber/matrix interface debonded energy, effect of, 290 293, 292f, 292t fiber/matrix interface shear stress, effect of, 286 290, 288f, 289t fibers failure, effect, of, 293, 294f, 295t fiber volume fraction, effect of, 218, 222f, 280 283, 282f, 282t interface properties, effect of, 219 220 interface wear, effect of, 220 221, 226f matrix cracking space, effect of, 283 286, 285f, 285t matrix crack spacing, effect of, 218 219, 223f Fatigue life prediction, of CMC based on HDE, 375 experimental comparisons, 398 446 2D ceramic-matrix composites, 425 433 2.5D ceramic-matrix composites, 433 435 3D ceramic-matrix composites, 435 446 cross-ply ceramic-matrix composites, 403 425 unidirectional ceramic-matrix composites, 398 403 results, 378 397 fatigue peak stress, effect of, 379 381, 379f, 380f, 382f fatigue stress ratio, effect of, 382 384, 383f fiber volume fraction, effect of, 391 397, 391f, 394f, 396f, 397f matrix crack spacing, effect of, 384 391, 386f, 388f, 389f, 390f theoretical analysis, 376 378 Fatigue lifetime prediction, 57 58 Fatigue peak stress effect of fiber matrix interface debonding, HDE, and HDE-based damage parameters, 310 311, 313f, 379 381, 379f, 380f, 382f static fatigue damage evolution, 363 364, 363f

457

in unidirectional C/SiC composites first stage, 222 223, 228f, 230f second stage, 223 224, 232f, 234f third stage, 225 226, 235f, 236f Fatigue stress range effect, on multiple loading fiber matrix interface sliding, 248 249, 254f Fatigue stress ratio effect, on hysteresisbased damage parameter, 311 312, 314f Federal Aviation Administration CLEEN program, 8 FEM. See Finite element method (FEM) Fiber Materials Inc., 9 10 Fiber matrix interface debonded energy, effect of first-matrix cracking stress, 83, 85f matrix multicracking evolution and interface oxidation, 109, 113f time-dependent fiber matrix interface debonding, 83, 85f Fiber matrix interface debonding, 2, 14 16, 19, 23 24, 26 29, 31 32, 34 35, 39, 49, 51, 55 57, 193 damage models, 153 155, 161f under different loading sequences, 194 238, 195f interface slip lengths. See Interface slip lengths stress analysis, 196 199 fatigue peak stress, effect of, 310 311, 313f, 379 381, 379f, 380f, 382f fiber volume fraction, effect of, 309, 311f, 391 397, 391f, 394f, 396f, 397f first-matrix cracking stress, 81 82 fiber matrix interface debonded energy, effect of, 83, 85f fiber matrix interface shear stress, effect of, 83 86, 87f, 88f fiber volume fraction, effect of, 83, 84f oxidation temperature, effect of, 86, 89f hysteresis dissipated energy under multiple loading sequences, 238 268 C/SiC composite, 254 266, 259f, 260f, 261f, 262f, 263f, 264f comparisons between different loading sequences effect, on fiber matrix interface sliding, 251 252, 256f

458

Fiber matrix interface debonding (Continued) comparisons between single and multiple loading stress level effect, on fiber matrix interface sliding, 250 251, 255f fatigue stress range effect, on multiple loading fiber matrix interface sliding, 248 249, 254f fiber volume fraction effect, on multiple loading fiber matrix interface sliding, 242 243, 246f high-peak stress level effect, on multiple loading fiber matrix interface sliding, 247 248, 252f hysteresis theories, 240 242, 241f, 242f, 244f low-peak stress level effect, on multiple loading fiber matrix interface sliding, 245 246, 250f matrix crack spacing effect, on multiple loading fiber matrix interface sliding, 244 245, 248f SiC/SiC composite, 265f, 266f, 267 268, 268f, 270f length, 199 200 matrix crack mode, effect of, 313 318, 318f matrix crack spacing, effect of, 312 313, 316f, 384 391, 386f, 388f, 389f, 390f matrix multicracking and fibers Poisson contraction, 94 95 fiber matrix interface frictional coefficient, effect of, 96, 97f matrix multicracking evolution and interface oxidation, 106 107 fiber matrix interface debonded energy, effect of, 109, 113f fiber matrix interface shear stress, effect of, 109, 111f, 112f fiber volume fraction, effect of, 109, 110f oxidation temperature, effect of, 109 115, 114f oxidation time, effect of, 115, 116f time-dependent fiber matrix interface debonded energy, effect of, 83, 85f

Index

fiber matrix interface shear stress, effect of, 83 86, 87f, 88f fibers Poisson ratio, effect of, 96 97, 98f fiber volume fraction, effect of, 83, 84f, 98 99, 99f oxidation temperature, effect of, 86, 89f, 109 115, 114f Fiber matrix interface frictional coefficient, effect of fiber matrix interface debonding, 96, 97f matrix multicracking evolution, 96, 97f fiber matrix interface shear stress, effect of composite residual strength and fibers fracture, 174, 175f, 176f cyclic fatigue loading, effect of, 99 100, 101f first-matrix cracking stress, 83 86, 87f, 88f fiber matrix interface debonding, 109, 111f, 112f matrix cracking evolution, 109, 111f, 112f time-dependent fiber matrix interface debonding, 83 86, 87f, 88f Fiber-reinforced CMCs, 2, 7 See also individual entries advantages of, 1 tensile behavior of, 16 Fibers failure, 26 27 damage models, 158 160 Fibers Poisson contraction, matrix multicracking evolution and, 86 103 experimental comparisons, 100 103 unidirectional SiC/borosilicate composite, 102 103, 104f unidirectional SiC/CAS composite, 100, 102f unidirectional SiC/CAS-II composite, 100 101, 103f interface debonding, 94 95 multiple matrix cracking, 95 96 results cyclic fatigue loading, 99 100, 101f fiber matrix interface frictional coefficient, effect of, 96, 97f fibers Poisson ratio, effect of, 96 97, 98f fiber volume fraction, effect of, 98 99, 99f stress analysis, 92 93

Index

Fibers Poisson ratio, effect of Fiber matrix interface debonding, 96 97, 98f matrix multicracking evolution, 96 97, 98f Fiber strength degradation, 37 effect on composite residual strength and fibers fracture, 174, 178f Fiber volume fraction, effect of fatigue hysteresis, 218, 222f first-matrix cracking stress, 83, 84f fiber matrix interface debonding, 98 99, 99f, 109, 110f, 309, 311f fiber matrix interface sliding, 218, 222f hysteresis-based damage parameters, 309, 311f matrix cracking evolution, 109, 110f matrix multicracking evolution, 98 99, 99f multiple loading fiber matrix interface sliding, 242 243, 246f time-dependent fiber matrix interface debonding, 83, 84f Fiber Weibull modulus effect, on composite residual strength and fibers fracture, 174, 177f Finite element method (FEM), 28, 149 150 axisymmetric, 34 First-matrix cracking stress (FMCS), 75 fiber matrix interface debonded energy, effect of, 83, 85f fiber matrix interface shear stress, effect of, 83 86, 87f, 88f fiber volume fraction, effect of, 83, 84f interface debonding, 81 82 matrix cracking stress, 82 83 in oxidation environment at elevated temperature, 75 86 interface debonding, 81 82 stress analysis, 78 81, 78f oxidation temperature, effect of, 86, 89f FMCS. See First-matrix cracking stress (FMCS) France, CMC applications in, 4 6 G GE, 4 9 German Space Center (DLR), 13 14 Global load sharing (GLS) criterion, 16, 26 28, 149 150, 158

459

GLS. See Global load sharing (GLS) criterion Goodrich, 8 H HDE. See Hysteresis dissipated energy (HDE) HDE-based damage parameters fatigue peak stress, effect of, 379 381, 379f, 380f, 382f fatigue stress ratio, effect of, 382 384, 383f fiber volume fraction, effect of, 391 397, 391f, 394f, 396f, 397f matrix crack spacing, effect of, 384 391, 386f, 388f, 389f, 390f High-peak stress level effect, on multiple loading fiber matrix interface sliding, 247 248, 252f High-performance turbine engine technology (IHPTET), 6 7 advanced turbine engine gas generator, 6 joint expendable turbine engine concept demonstrator, 7 joint technology demonstration engine, 6 joint turbine advanced gas generator, 6 7 Honeywell, 6 7 Hutchinson Jensen fiber pullout model, 34, 273 275 Hyper-Therm HTC Inc., 9 Hysteresis-based damage parameters, of cyclic fatigue damage evaluation, 276 279 fatigue peak stress, effect of, 310 311, 313f fatigue stress ratio, effect of, 311 312, 314f fiber/matrix interface debonded energy, effect of, 290 293, 292f, 292t fiber/matrix interface shear stress, effect of, 286 290, 288f, 289t fibers failure, effect, of, 293, 294f, 295t fiber volume fraction, effect of, 280 283, 282f, 282t, 309, 311f matrix cracking space, effect of, 283 286, 285f, 285t matrix crack mode, effect of, 313 318, 318f matrix crack spacing, effect of, 312 313, 316f woven structure, effect of, 318, 319f

460

Hysteresis dissipated energy (HDE), 215 217, 275 fatigue life prediction of CMC based on, 375 under multiple loading sequences, 238 268 comparisons between different loading sequences effect, on fiber matrix interface sliding, 251 252, 256f comparisons between single and multiple loading stress level effect, on fiber matrix interface sliding, 250 251, 255f fatigue stress range effect, on multiple loading fiber matrix interface sliding, 248 249, 254f fiber volume fraction effect, on multiple loading fiber matrix interface sliding, 242 243, 246f high-peak stress level effect, on multiple loading fiber matrix interface sliding, 247 248, 252f hysteresis theories, 240 242, 241f, 242f, 244f low-peak stress level effect, on multiple loading fiber matrix interface sliding, 245 246, 250f matrix crack spacing effect, on multiple loading fiber matrix interface sliding, 244 245, 248f Hysteresis theories, 240 242, 241f, 242f, 244f I IHI Corporation, 9 IHPTET. See High-performance turbine engine technology (IHPTET) Initiation matrix cracking, 25 26 Interface counter-slip length, 200 202 Interface debonding. See Fiber matrix interface debonding Interface new-slip length, 202 203 Interface oxidation, matrix multicracking evolution and, 104 120 experimental comparison, 115 120 mini-SiC/SiC composite, 118 120, 120f unidirectional SiC/borosilicate composite, 115 117

Index

unidirectional SiC/CAS composite, 115, 118f interface debonding, 106 107 multiple matrix cracking, 108 results, 108 115 comparisons of matrix cracking evolution with and without oxidation, 115, 117f fiber matrix interface debonded energy, effect of, 109, 113f fiber matrix interface shear stress, effect of, 109, 111f, 112f fiber volume fraction, effect of, 109, 110f oxidation temperature, effect of, 109 115, 114f oxidation time, effect of, 115, 116f stress analysis, 106 Interface phase of ceramic-matrix composites, 2 4 Interface properties effect, on fatigue hysteresis and fiber matrix interface sliding, 219 220 Interface slip lengths, 199 203 experimental comparisons, 221 238 cross-ply C/SiC composites, 237 238, 238f unidirectional C/SiC composite, 222 226, 228f, 230f, 232f, 234f, 235f, 236f unidirectional SiC/calcium alumina silicate-II composites, 226 237, 237f results applied cycle number effect, on fiber matrix interface debonding, 208 210, 209f, 210f, 211f arbitrary loading sequence effect, on fatigue hysteresis and fiber matrix interface sliding, 215 217, 217f, 218f, 219f, 220f, 221f fiber volume fraction effect, on fatigue hysteresis and fiber matrix interface sliding, 218, 222f interface properties effect, on fatigue hysteresis and fiber matrix interface sliding, 219 220 interface wear effect, on fatigue hysteresis loops and fiber matrix interface sliding, 220 221, 226f

Index

loading sequence effect, on fiber matrix interface debonding, 205 208, 207f matrix crack spacing effect, on fatigue hysteresis loops and fiber matrix interface sliding, 218 219, 223f peak stress level effect, on fiber matrix interface debonding, 211 215, 212f, 213f, 214f, 216f Interface wear behavior, 35 37 experimental observation, 35 36 theoretical analysis, 36 37 effect of damage models, 156 158, 157f fatigue hysteresis loops and fiber matrix interface sliding, 220 221, 226f J Japan, CMC applications in, 8 9 Japanese National Aerospace Laboratory, 14 JCS. See Joint Composite Scramjet (JCS) program JETEC. See Joint expendable turbine engine concept (JETEC) demonstrator Joint Composite Scramjet (JCS) program, 11 Joint expendable turbine engine concept (JETEC) demonstrator, 7 Joint technology demonstration engine (JTDE), 6 Joint turbine advanced gas generator (JTAGG), 6 7 JTAGG. See Joint turbine advanced gas generator (JTAGG) JTDE. See Joint technology demonstration engine (JTDE) L Lifetime prediction methods of ceramicmatrix composites, 45 58 fatigue, 57 58 S N curves, 45 57 2D ceramic-matrix composites, 51 54 2.5D ceramic-matrix composites, 55 3D ceramic-matrix composites, 55 57 cross-ply ceramic-matrix composites, 48 51 unidirectional ceramic-matrix composites, 45 48

461

LLS. See Local load sharing (LLS) criterion Loading frequency effect, on CMC fatigue behavior, 40 44 Loading sequence effect, on fiber matrix interface debonding, 205 208, 207f Local load sharing (LLS) criterion, 16, 26 27, 158 Low-peak stress level effect, on multiple loading fiber matrix interface sliding, 245 246, 250f M M40JB-C/SiC composites, tensile behavior of, 18 Matrix axial stress distribution, cyclic fatigue loading effect on, 99 100, 101f Matrix cracking mode 1, 123, 132, 133f energy balance approach, 125 127, 126f mode 3 cracking occurring between two mode 1 cracks, 129 131, 130f mode 5 cracking occurring between two mode 1 cracks, 131 132, 131f Matrix cracking mode 2, 123, 132 134, 134f, 135f energy balance approach, 127, 128f Matrix cracking mode 3, 123 124, 135, 136f, 137f occurring between two mode 1 cracks, 129 131, 130f, 135 140, 137f, 138f, 139f energy balance approach, 127 128, 129f Matrix cracking mode 5, 124 occurring between two mode 1 cracks, 135 140, 137f, 138f, 139f energy balance approach, 131 132, 131f Matrix cracking of ceramic-matrix composites, 75 cross-ply ceramic-matrix composites, matrix multicracking evolution of, 121 140, 122f energy balance approach, 125 132 stress analysis, 121 124 and fibers Poisson contraction, 86 103 cyclic fatigue loading, 99 100, 101f fiber matrix interface frictional coefficient, effect of, 96, 97f

462

Matrix cracking of ceramic-matrix composites (Continued) fibers Poisson ratio, effect of, 96 97, 98f fiber volume fraction, effect of, 98 99, 99f interface debonding, 94 95 multiple matrix cracking, 95 96 stress analysis, 92 93 unidirectional SiC/borosilicate composite, 102 103, 104f unidirectional SiC/CAS composite, 100, 102f unidirectional SiC/CAS-II composite, 100 101, 103f first-matrix cracking in oxidation environment at elevated temperature, 75 86 experimental comparisons, 86, 90f fiber matrix interface debonded energy, effect of, 83, 85f fiber matrix interface shear stress, effect of, 83 86, 87f, 88f fiber volume fraction, effect of, 83, 84f interface debonding, 81 82 matrix cracking stress, 82 83 oxidation temperature, effect of, 86, 89f stress analysis, 78 81, 78f and interface oxidation, 104 120 comparisons of matrix cracking evolution with and without oxidation, 115, 117f fiber matrix interface debonded energy, effect of, 109, 113f fiber matrix interface shear stress, effect of, 109, 111f, 112f fiber volume fraction, effect of, 109, 110f interface debonding, 106 107 mini-SiC/SiC composite, 118 120, 120f multiple matrix cracking, 108 oxidation temperature, effect of, 109 115, 114f oxidation time, effect of, 115, 116f stress analysis, 106 unidirectional SiC/borosilicate composite, 115 117 unidirectional SiC/CAS composite, 115, 118f

Index

Matrix cracking stress, 82 83 Matrix crack mode effect, on fiber/matrix interface debonding and hysteresisbased damage parameter, 313 318, 318f Matrix crack spacing, effect of fatigue hysteresis loops and fiber matrix interface sliding, 218 219, 223f fiber/matrix interface debonding and hysteresis-based damage parameters, 312 313, 316f multiple loading fiber matrix interface sliding, 244 245, 248f static fatigue damage evolution, 364 365, 365f Matrix multicracking, 161 163, 165f, 166f, 167f. See also Matrix multicracking evolution Matrix multicracking evolution, 26. See also matrix multicracking damage models, 153, 154f and fibers Poisson contraction. See Matrix multicracking evolution and fibers Poisson contraction and interface oxidation. See Matrix multicracking evolution and interface oxidation Matrix multicracking evolution and fibers Poisson contraction, 86 103 experimental comparisons, 100 103 unidirectional SiC/borosilicate composite, 102 103, 104f unidirectional SiC/CAS composite, 100, 102f unidirectional SiC/CAS-II composite, 100 101, 103f interface debonding, 94 95 multiple matrix cracking, 95 96 results cyclic fatigue loading, 99 100, 101f fiber matrix interface frictional coefficient, effect of, 96, 97f fibers Poisson ratio, effect of, 96 97, 98f fiber volume fraction, effect of, 98 99, 99f stress analysis, 92 93 Matrix multicracking evolution and interface oxidation, 104 120

Index

experimental comparison, 115 120 mini-SiC/SiC composite, 118 120, 120f unidirectional SiC/borosilicate composite, 115 117 unidirectional SiC/CAS composite, 115, 118f interface debonding, 106 107 multiple matrix cracking, 108 results, 108 115 comparisons of matrix cracking evolution with and without oxidation, 115, 117f fiber matrix interface debonded energy, effect of, 109, 113f fiber matrix interface shear stress, effect of, 109, 111f, 112f fiber volume fraction, effect of, 109, 110f oxidation temperature, effect of, 109 115, 114f oxidation time, effect of, 115, 116f stress analysis, 106 MCE model, 26 Modulus degradation, 38 40 Monte Carlo method, 28, 149 150 MT Aerospace Company, 13 14 Multiple fatigue loading, tensile strength under, 149 165 damage models, 152 160 fiber matrix interface debonding, 153 155, 161f fibers failure, 158 160 interface wear, 156 158, 157f matrix multicracking evolution, 153, 154f experimental comparisons, 163 165, 168f, 169f results, 160 163 matrix multicracking, 161 163, 165f, 166f, 167f single matrix cracking, 160 161, 162f, 163f, 164f stress analysis, 151 152 Multiple loading sequences, hysteresis dissipated energy under, 238 268 experimental comparisons, 252 268 C/SiC composite, 254 266, 259f, 260f, 261f, 262f, 263f, 264f SiC/SiC composite, 265f, 266f, 267 268, 268f, 270f

463

hysteresis theories, 240 242, 241f, 242f, 244f results, 242 252 comparisons between different loading sequences effect, on fiber matrix interface sliding, 251 252, 256f comparisons between single and multiple loading stress level effect, on fiber matrix interface sliding, 250 251, 255f fatigue stress range effect, on multiple loading fiber matrix interface sliding, 248 249, 254f fiber volume fraction effect, on multiple loading fiber matrix interface sliding, 242 243, 246f high-peak stress level effect, on multiple loading fiber matrix interface sliding, 247 248, 252f low-peak stress level effect, on multiple loading fiber matrix interface sliding, 245 246, 250f matrix crack spacing effect, on multiple loading fiber matrix interface sliding, 244 245, 248f N Nextel 720/alumina composites, at elevated temperatures, 184 185, 184f, 185f O Oxidation embrittlement, 37 38 Oxidation temperature, effect of first-matrix cracking stress, 86, 89f oxidation temperature, effect of, 109 115, 114f static fatigue damage evolution, 367, 369f time-dependent fiber matrix interface debonding, 86, 89f Oxidation time, effect of fiber matrix interface debonding, 115, 116f matrix cracking evolution, 115, 116f Oxide fiber, basic material properties of, 3t P Peak stress level effect, on fiber matrix interface debonding, 211 215, 212f, 213f, 214f, 216f P&W Hypersonics, 4 8, 12 PWK2 wind tunnel test, 13

464

R RCI. See Refractory Composites, Inc. (RCI) Refractory Composites, Inc. (RCI), 12 Reusable launch vehicles (RLV), 10 RLV. See Reusable launch vehicles (RLV) Rocket engines, CMC applications in, 9 11 S Scramjet engine, CMC applications in, 11 12 SEPCARBINOXR A500, 5 6 SiC/1723 composite fatigue life prediction of, 403, 404f at room temperature, 321, 324f SiC/CAS composite at 700 oC in air atmosphere, 342, 343f % in air atmosphere, 342, 344f at 850 oC fatigue %life prediction of, 401 402, 402f at room temperature unidirectional, 319 321, 320f cross-ply, 338 342, 341f SiC/CAS-II composite, at room temperature, 321, 323f SiC/MAS composite, fatigue life prediction of at 566 oC in air atmosphere, 407 412, % 409f, 410f, 411f, 413f, 414f, 408f, 415f at 1093 oC in air atmosphere, 412 425, % 417f, 418f, 419f, 421f, 422f, 416f, 423f, 424f SiC/MAS-L composite, at 800 oC and 1000 oC in inert atmosphere, %347, 348f SiC/SiC% composite at 600 oC, 800 oC and 1000 oC in inert % % % 350f condition, 347 349, 349f, at 1000 oC in air and in steam % atmosphere, 349 353, 351f, 352f at 1200 oC in air and in steam % atmospheres, 353, 354f, 355f at 1300 oC in air atmosphere, 354 357, % 357f 356f, at elevated temperatures, 182 183 fatigue life prediction of at room temperature, 429, 431f, 432f at elevated temperatures, 431 433, 435f, 439f, 444 446, 444f, 445f hysteresis dissipated energy under multiple loading sequences, 265f, 266f, 267 268, 268f, 270f

Index

SiC fiber, basic material properties of, 3t SiC/[Si N C] composites, at room and elevated temperatures, 180 181, 180f, 181f Single matrix cracking, 160 161, 162f, 163f, 164f Sliding of ceramic-matrix composites, 193 under different loading sequences, 194 238, 195f interface slip lengths. See Interface slip lengths stress analysis, 196 199 Snecma, 4 8 Joint Composite Scramjet program, 11 Snecma Propulsion Solide (SPS), 11 12, 14 SPECARBINOX A262, 5 SPS. See Snecma Propulsion Solide (SPS) Static fatigue damage evolution, 358 369, 361f experimental comparisons, 368 369, 370f fatigue peak stress, effect of, 363 364, 363f matrix crack spacing, effect of, 364 365, 365f oxidation temperature, effect of, 367, 369f volume fraction, effect, of, 365 366, 367f Stress analysis cross-ply ceramic-matrix composites, matrix multicracking evolution of, 121 124, 122f matrix cracking mode 1, 123 matrix cracking mode 2, 123 matrix cracking mode 3, 123 124 matrix cracking mode 5, 124 undamaged state, 122 first-matrix cracking stress, 78 81, 78f downstream stress, 79 80, 79f upstream stress, 81 interface debonding and sliding, 196 199 initial loading, 196 197 loading, 197 198 reloading, 198 199 matrix multicracking and fibers Poisson contraction, 92 93 matrix multicracking evolution and interface oxidation, 106 multiple fatigue loading, 151 152 Stress ratio effect, on CMC fatigue behavior, 44 45 Stress strain curve, 27 28

Index

T T800-C/SiC composites, tensile behavior of, 18 TECH56 program, 8 Tensile behavior of ceramic-matrix composites, 14 28 experimental observation, 16 25 2D ceramic-matrix composites, 21 23 2.5D ceramic-matrix composites, 23 24 3D ceramic-matrix composites, 24 25 cross-ply ceramic-matrix composites, 19 21 unidirectional ceramic-matrix composites, 17 19 theoretical analysis, 25 28 fibers failure, 26 27 initiation matrix cracking, 25 26 matrix multicracking evolution, 26 stress strain curve, 27 28 Tensile loading/unloading damage evolution, 280 306 experimental comparisons, 293 306 2D C/SiC composites, 299 301, 304f, 305t 2.5D C/SiC composites, 301, 305f, 306t 3D braided C/SiC composites, 301 304, 306f, 307t 3D needled C/SiC composites, 305 306, 307f, 307t cross-ply C/SiC composites, 298 299, 300f, 300t unidirectional C/SiC composites, 295 298, 295f, 299f, 300t results, 280 293 fiber/matrix interface debonded energy effect, on hysteresis loops and hysteresis-based damage parameter, 290 293, 292f, 292t fiber/matrix interface shear stress effect, on hysteresis loops and hysteresisbased damage parameter, 286 290, 288f, 289t fibers failure effect, on hysteresis loops and hysteresis-based damage parameter, 293, 294f, 295t fiber volume fraction effect, on hysteresis loops and hysteresis-based damage parameter, 280 283, 282f, 282t

465

matrix cracking space effect, on hysteresis loops and hysteresis-based damage parameter, 283 286, 285f, 285t Tensile strength of ceramic-matrix composites, 145 cyclic loading at elevated temperatures in oxidative environments, 166 185 experimental comparisons, 180 185, 180f, 181f, 184f, 185f residual strength model, 170 172 results, 172 179, 173f, 175f, 176f, 177f, 178f, 179f multiple fatigue loading, 149 165 damage models, 152 160, 154f, 157f, 161f experimental comparisons, 163 165, 168f, 169f results, 160 163, 162f, 163f, 164f, 165f, 166f, 167f stress analysis, 151 152 Tensile stress, 14 15, 15f Testing temperature effect, on composite residual strength and fibers fracture, 174 179, 179f Thermal protection systems (TPS), 1, 12 14 Time-dependent fiber matrix interface debonding fiber matrix interface debonded energy, effect of, 83, 85f fiber matrix interface shear stress, effect of, 83 86, 87f, 88f fibers Poisson ratio, effect of, 96 97, 98f fiber volume fraction, effect of, 83, 84f, 98 99, 99f oxidation temperature, effect of, 86, 89f, 109 115, 114f TPS. See Thermal protection systems (TPS) U UEET. See Ultraefficient engine technology (UEET) program Ultraefficient engine technology (UEET) program, 7 Uncooled C/SiC composite thrust chambers, 10 Undamaged state of composites, 122 Unidirectional ceramic-matrix composites, 17 19

466

Unidirectional ceramic-matrix composites (Continued) cyclic fatigue damage evaluation of, 319 338 C/SiC composites, at elevated temperature, 333 338, 335f, 337f, 339f C/SiC composites, at room temperature, 321 333, 326f, 330f, 332f SiC/1723 composite, at room temperature, 321, 324f SiC/CAS composite, at room temperature, 319 321, 320f SiC/CAS-II composite, at room temperature, 321, 323f fatigue life prediction of, 398 403 SiC/1723 composite, 403, 404f SiC/CAS composite, 401 402, 402f S N curves of, 45 48 Unidirectional C/SiC composites at elevated temperature, 333 338, 335f, 337f, 339f first stage fatigue peak stress, 222 223, 228f, 230f interface slip lengths, 222 226 at room temperature, 321 333, 326f, 330f second stage fatigue peak stress, 223 224, 232f, 234f tensile loading/unloading damage evolution, 295 298, 295f, 299f, 300t third stage fatigue peak stress, 225 226, 235f, 236f Unidirectional SiC/borosilicate composite matrix multicracking evolution and fibers Poisson contraction, 102 103, 104f matrix multicracking evolution and interface oxidation, 115 117 Unidirectional SiC/CAS composite matrix multicracking evolution and fibers Poisson contraction, 100, 102f

Index

matrix multicracking evolution and interface oxidation, 115, 118f Unidirectional SiC/CAS-II composite interface slip lengths, 226 237, 237f matrix multicracking evolution and fibers Poisson contraction, 100 101, 103f United States, CMC applications in, 6 8 CLEEN program, 8 F414 aero engine, 7 8 high-performance turbine engine technology, 6 7 joint technology demonstration engine, 6 advanced turbine engine gas generator, 6 joint turbine advanced gas generator, 6 7 joint expendable turbine engine concept demonstrator, 7 ultraefficient engine technology program, 7 United Technologies Research Center (UTRC), 11 Upstream stress, 81 US Air Force Hypersonic Technology Program, 12 UTRC. See United Technologies Research Center (UTRC) V Volume fraction effect, on static fatigue damage evolution, 365 366, 367f W Williams International, 7 Woven structure effect, on hysteresis-based damage parameter, 318, 319f X X-33 program, 11 12 X-38 aerospace vehicles, 13