Fracture Analysis of Layered Beams With an Elastically Coupled Behavior and Hygrothermal Stresses: Application to Metal-to-Composite Adhesive Joints (Springer Theses) 9783031176203, 9783031176210, 3031176200

Table of contents :
Supervisor’s Foreword
Preface
Parts of this thesis have been published in the following articles:
Acknowledgments
Contents
Nomenclature
Latin Symbols
Greek Symbols
Abbreviations and Acronyms
Subscripts, Superscripts, and Conventions
Other
1 Introduction
1.1 Background and Motivation
1.1.1 General Background
1.1.2 Motivation
1.2 General Aim and Specific Objectives of the Book
1.2.1 General Aim of the Thesis
1.2.2 Specific Objectives of the Book
1.2.3 The Book at a Glance
1.3 Organization and Structure of the Book
1.3.1 Organization of the Book
1.3.2 Structure of the Book
1.4 Contribution of the Thesis
1.5 Basic Concepts and Definitions
1.5.1 Layered, Generally Layered, Multilayered, Laminated
1.5.2 Crack, Delamination, and Disbonding
1.5.3 Definitions for Beams
1.5.4 Data Reduction Scheme (DRS)
References
2 The Effect of Residual Hygrothermal Stresses on the Energy Release Rate and Mode Mixity of Interfacial Cracks in Beams with Bending–Extension Coupling
2.1 Introduction
2.1.1 State of the Art
2.1.2 Present Work
2.2 Problem Description and Analysis Approach
2.2.1 Definition of the Scientific Problem
2.2.2 The Proposed Analytical Model
2.3 Mathematical Formulation of the Problem
2.3.1 Kinematic Assumptions
2.3.2 Constitutive Laws
2.3.3 Conditions of Static Equilibrium
2.3.4 Conditions of Displacement Continuity
2.4 Solution to the Problem
2.4.1 Derivation of the Governing Equation
2.4.2 Solution to the Governing Equation
2.4.3 Internal Forces and Moments
2.4.4 Crack-Tip Forces
2.4.5 Energy Release Rate (ERR) and Mode Mixity (MM)
2.5 Extraction of Closed-Form Equations
2.5.1 Reduction to Typical Test Configurations
2.5.2 Effect of Contact Between the Two Sublaminates
2.5.3 Reduction to Previous Equations in the Literature
2.6 Validation Through Finite Element Analysis (FEA)
2.6.1 Application: A Typical Glass Aluminum Reinforced Epoxy (GLARE)
2.6.2 Finite Element Analyses (FEAs)
2.7 Results
2.7.1 Example 1: Double Cantilever Beam (DCB) Test
2.7.2 Example 2: End-Notched Flexure (ENF) Test
2.8 Discussion
2.9 Conclusions
References
3 Fracture Toughness of Metal-to-Composite Adhesive Joints with Bending–Extension Coupling and Residual Thermal Stresses
3.1 Introduction
3.1.1 State of the Art
3.1.2 Present Work
3.2 Description of the Technical and Scientific Problem
3.2.1 The Metal-to-Composite Adhesive Joint Under Consideration
3.2.2 Challenges in the Design and Data Reduction of Fracture Tests for Dissimilar Adhesive Joints
3.3 Design and Data Reduction of Fracture Tests
3.3.1 Design of the Fracture Toughness Tests
3.3.2 Data Reduction Approach
3.4 Experimental Methods
3.4.1 Materials and Geometry
3.4.2 Fracture Toughness Experiments
3.5 Finite Element Analyses (FEAs)
3.5.1 Two-Dimensional (2D) Finite Element Analyses (FEAs)
3.5.2 Three-Dimensional (3D) Finite Element Analyses (FEAs)
3.5.3 Cohesive Zone Modeling (CZM)
3.6 Results
3.6.1 The Effect of Residual Thermal Stresses (RTS) on the Total Energy Release Rate (ERR) and Mode Mixity (MM)
3.6.2 Crack Propagation Behavior
3.6.3 Data Reduction Using Various Analytical Expressions
3.7 Discussion
3.8 Conclusions
References
4 Interfacial Fracture Toughness of a Titanium-to-CFRP Adhesive Joint
4.1 Introduction
4.1.1 State of the Art
4.1.2 Present Work
4.2 The Titanium-to-CFRP Joint Under Consideration
4.2.1 Configuration
4.2.2 Materials
4.2.3 The Proposed Manufacturing Options (MOs)
4.3 Experimentation
4.3.1 Mechanical Experiments
4.3.2 Fractographic Investigation
4.3.3 Experimental Data Reduction
4.4 Results
4.4.1 Double Cantilever Beam (DCB) Experiments
4.4.2 End-Notched Flexure (ENF) Experiments
4.5 Discussion
4.6 Conclusions
References
5 Energy Release Rate and Mode Partitioning of Moment-Loaded Fracture Tests on Layered Beams with Bending–Extension Coupling and Hygrothermal Stresses
5.1 Introduction
5.1.1 State of the Art
5.1.2 Present Work
5.2 Existing Analytical Expressions
5.3 The Proposed Data Reduction Scheme (DRS)
5.3.1 Problem Description and Approach of the Analysis
5.3.2 Calculation of the Crack-Tip Forces
5.3.3 Calculation of the Energy Release Rate (ERR) and Mode Mixity (MM)
5.3.4 Special Cases and Refinement to Pre-existing Equations
5.4 Numerical Validation
5.4.1 Case Studies
5.4.2 Finite Element Analysis (FEA)
5.5 Experimental Methods
5.5.1 Double Cantilever Beam Test Fixture with Uneven Bending Moments (DCB-UBM Test Fixture)
5.5.2 The Titanium-to-CFRP Adhesive Joint
5.5.3 Execution of the Experiments
5.6 Results
5.6.1 Validation of the Data Reduction Scheme (DRS)
5.6.2 Application in Actual Experiments
5.7 Discussion
5.8 Conclusions
References
6 Closed-Form Solution for Interfacially Cracked Layered Beams with Bending–Extension Coupling and Hygrothermal Stresses
6.1 Introduction
6.1.1 State of the Art
6.1.2 Present Work
6.2 Problem Description and Assumptions
6.2.1 Motivation
6.2.2 Definition of the Problem
6.2.3 Assumptions
6.3 Mathematical Formulation of the Problem
6.3.1 Kinematic Assumptions
6.3.2 Constitutive Laws
6.3.3 Conditions of Static Equilibrium
6.3.4 Conditions of Displacement Continuity
6.3.5 Boundary Conditions
6.4 Extraction of Closed-Form Expressions
6.4.1 Solution Strategy
6.4.2 Solution Along the Uncracked Region
6.4.3 Solution Along the Cracked Region
6.4.4 Complete Solution
6.5 Refinements and a Suggested Improved Solution
6.5.1 Reduction to Unified Test Configurations
6.5.2 Refinement for a Clamped Crack Tip
6.5.3 Refinements for Common Material Systems
6.5.4 The Balanced Case
6.5.5 On Improving the Equations for the Contact Force
6.6 Application
6.6.1 Application Examples
6.6.2 Application Results
6.7 Discussion
6.8 Conclusions
References
7 Conclusion
7.1 General Summary and Conclusions
7.1.1 Literature Review
7.1.2 Derivation and Implementation of a New Analytical Solution
7.1.3 Numerical Validation of the Analytical Model
7.1.4 Interfacial Cracks Inside Layered Materials
7.1.5 Mode-Mixity (MM) Response
7.1.6 The Effect of Residual Thermal Stresses (RTS)
7.1.7 The Titanium-to-CFRP Adhesive Joint
7.1.8 Some Additional Novelty Points
7.2 Strengths of the Proposed Analytical Model and Expressions
7.3 Ongoing Work and Suggestions for Future Work
7.3.1 The Titanium-to-CFRP Adhesive Joint
7.3.2 Contact Problem
7.3.3 Compliance
7.3.4 Determination of the Energy Release Rate (ERR)
7.3.5 Boundary Conditions
7.3.6 Three-Dimensional (3D) Data Reduction Scheme (DRS)
7.3.7 Further Refinement of the Analytical Model
7.3.8 Further Development of the Rigid Joint Model
References
Appendix A First-Order Shear Deformation Theory Considering an Elastically Coupled Behavior and Residual Hygrothermal Stresses
A.1 Stiffness of a Single Layer
A.2 Constitutive Behavior of a Single Layer
A.3 Equivalent Stiffness of a Layered Plate
A.4 Equivalent Stiffness of a Layered Beam
A.4.1 Plane-Strain Conditions
A.4.2 Plane-Stress Conditions
A.4.3 Homogeneous Beam
A.5 Relations Between Stiffness and Compliance Coefficients
References
Appendix B Fracture Mode Partitioning Adopting the Crack Closure Integral Approach
B.1 Crack Closure Integral
B.2 Fracture Mode Partitioning
B.3 Discussion and Concluding Remarks
References
Appendix C Virtual Crack Closure Technique
References
Appendix D Aircraft Application of the Titanium-to-CFRP Adhesive Joint
D.1 The Technological Problem
D.2 Effect of Laminar Flow on the Design of Aerodynamic Surfaces of Aircrafts
D.3 Laminar Flow Design Approaches
D.3.1 The Natural Laminar Flow (NLF) and Laminar Flow Control (LFC) Approaches
D.3.2 The Hybrid Laminar Flow Control (HLFC) Approach
D.4 Hybrid Laminar Flow Control (HLFC) Application of the Present Joint
References
Appendix E Manufacturing the Titanium-to-CFRP Adhesive Joint
E.1 Production of the Joint
E.1.1 Manufacturing Option (MO) 1
E.1.2 Manufacturing Option (MO) 2
E.1.3 Manufacturing Option (MO) 3
E.1.4 Manufacturing Option (MO) 4
E.2 Preparation of the Test Specimens
References
Appendix F Analytical Approaches for Fracture Mode Decoupling
F.1 Bimaterial Joints
F.1.1 Design Criterion 1: Equal Strains
F.1.2 Design Criterion 2: Equal Stiffnesses
F.1.3 Comparison
F.2 More Complex Cases and a Concluding Remark
References
Appendix G Related to the Work in Chap. 6
G.1 Basic Equations of the Analytical Model
G.2 Strain Measures
G.2.1 Uncracked Region
G.2.2 Cracked Region
G.3 Refinement for a Clamped Crack Tip
G.4 Stiffness and Thermal Coefficients of the Sublaminates
Reference
About the Author
Index

Citation preview

Springer Theses Recognizing Outstanding Ph.D. Research

Panayiotis Tsokanas

Fracture Analysis of Layered Beams With an Elastically Coupled Behavior and Hygrothermal Stresses Application to Metal-to-Composite Adhesive Joints

Springer Theses Recognizing Outstanding Ph.D. Research

Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists.

Theses may be nominated for publication in this series by heads of department at internationally leading universities or institutes and should fulfill all of the following criteria • They must be written in good English. • The topic should fall within the confines of Chemistry, Physics, Earth Sciences, Engineering and related interdisciplinary fields such as Materials, Nanoscience, Chemical Engineering, Complex Systems and Biophysics. • The work reported in the thesis must represent a significant scientific advance. • If the thesis includes previously published material, permission to reproduce this must be gained from the respective copyright holder (a maximum 30% of the thesis should be a verbatim reproduction from the author’s previous publications). • They must have been examined and passed during the 12 months prior to nomination. • Each thesis should include a foreword by the supervisor outlining the significance of its content. • The theses should have a clearly defined structure including an introduction accessible to new PhD students and scientists not expert in the relevant field. Indexed by zbMATH.

Panayiotis Tsokanas

Fracture Analysis of Layered Beams With an Elastically Coupled Behavior and Hygrothermal Stresses Application to Metal-to-Composite Adhesive Joints Doctoral Thesis accepted by the University of Patras, Patras, Greece

Author Dr. Panayiotis Tsokanas Applied Mechanics Laboratory Department of Mechanical Engineering and Aeronautics University of Patras Patras, Greece

Supervisor Prof. Theodoros Loutas Applied Mechanics Laboratory Department of Mechanical Engineering and Aeronautics University of Patras Patras, Greece

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-3-031-17620-3 ISBN 978-3-031-17621-0 (eBook) https://doi.org/10.1007/978-3-031-17621-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

We are a reshuffling of nothing. Dimitri Nanopoulos

To my beloved parents, my father Efstathios, and my mother Constantine, for their devotion.

Supervisor’s Foreword

It is my great pleasure to preface this book, which is a renewed reprint of Dr. Panayiotis Tsokanas’s Ph.D. thesis. The book is included in Springer Theses series that “brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences.” The thesis was conducted at the Applied Mechanics Laboratory, which belongs to the Department of Mechanical Engineering and Aeronautics of the University of Patras, Greece, under my supervision. Dr. Tsokanas started his doctoral studies in late 2016 with a three-year research scholarship from the Hellenic Foundation for Research and Innovation. His research area fell within the intersection of fracture mechanics, solid mechanics, and engineering materials (e.g., composites, adhesive joints). After a critical evaluation of the state of the art, Dr. Tsokanas developed his thesis on the topic of fracture analysis of layered materials with an elastically coupled response and residual hygrothermal stresses. He successfully completed his Ph.D. with an oral defense of his thesis on August 2021, and the examination committee assessed his work as excellent. The thesis is structured as an assemblage of nine journal papers: six published in high-quality journals and three more to be submitted for publication. Moreover, the work has been presented at various international, well-recognized conferences (e.g., European Conference on Composite Materials). In my opinion, what distinguishes Dr. Tsokanas’s work is that it identified and established a significant gap in the literature; I will briefly highlight this gap now. Both in industry and in academic research, there has been an increased interest in material systems consisting of several dissimilar layers (e.g., dissimilar adhesive joints, multidirectional composites, fiber metal laminates). Let us imagine a beam-like structure made of such a layered material. Such a structure typically exhibits two certain peculiarities: an elastically coupled behavior and residual hygrothermal stresses (e.g., if cured/bonded in elevated temperature). Given that the research community is generally interested in characterizing the interfacial fracture of laboratory specimens with a beam-like geometry, the question that follows is whether there is a solid theoretical framework that considers both these peculiarities and can properly analyze the fracture toughness of this structure. After carefully investigating the literature, the thesis came up with a negative answer. Thus, the starting point of the thesis was ix

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Supervisor’s Foreword

the following: the fracture mechanics of engineering materials with an elastically coupled response and residual (hygro)thermal stresses has not been satisfactorily studied. Dr. Tsokanas seized this research opportunity and worked enthusiastically. He delved into the scientific areas of mechanics of composites, theoretical and experimental fracture mechanics, adhesive bonding of dissimilar materials, and material characterization. In a nutshell, the thesis developed (i.e., formulated, validated through finite element modeling, and implemented in actual and virtual experiments) an analytical framework and a plethora of closed-form formulae for the calculation of the fracture toughness (i.e., energy release rate and associated mode mixity) and investigation of fracture behavior of such non-conventional beams. In parallel, the thesis investigated—in the framework of the CS2 TICOAJO project—the fracture behavior of a titanium-to-composite adhesive joint of interest to the aerospace industry. Dr. Tsokanas published his first two articles in the Engineering Fracture Mechanics and Composites Part B: Engineering journals, laying the foundation for much further research that is still developing by him and his collaborators. The present book contains a mathematical formulation that is clear and elegant, while the proposed closed-form expressions are perfectly organized and properly explained. The book includes only essential results in exemplary illustrations, diagrams, and a few tables. Moreover, the background theory is briefly presented with accuracy within several chapters and sometimes in the appendices. The most impressive characteristic of this book is its general utility, given that the proposed closed-form expressions can cover many different cases of engineering materials and loading configurations. Lastly, the writing style of the book is attractive. Through his outstanding thesis, Dr. Tsokanas has left a significant mark on the field. The outputs of the thesis open new avenues of research and future developments; currently, Dr. Tsokanas, a couple of my Ph.D. students, and some of my undergraduate students are all working on concepts inspired by Dr. Tsokanas’s work. I am particularly pleased with this dissertation that, in my belief, will serve as a valuable guide for all interested students, researchers, and engineers working on the general area of fracture analysis of layered material systems. Moreover, I hope that the insights and novel ideas of the book will be useful to further develop methods for the analysis and data reduction of fracture toughness tests on non-conventional specimens with possible bending–extension coupling and residual hygrothermal stresses. I am proud that Panayiotis is the first Ph.D. graduate of the University of Patras to receive the prestigious Springer Theses Award that recognizes outstanding Ph.D. theses worldwide. It has been a great pleasure working with him for the last five years. Patras, Greece May 2022

Prof. Theodoros Loutas

Preface

The present monograph is a renewed reproduction of my Ph.D. thesis, which was submitted to the University of Patras, Greece, in partial fulfillment of the degree of Doctor of Philosophy. The presented work was carried out in the Applied Mechanics Laboratory, which belongs to the Department of Mechanical Engineering and Aeronautics of the University of Patras. The work was undertaken from January 2017 to December 2020 and was supervised by Prof. Theodoros Loutas, Associate Professor at the University of Patras. The thesis was defended by me in front of a seven-member examination committee and was approved unanimously with an excellent grade. In this book, I chose to include only the chapters of my thesis that are already published as peer-reviewed journal papers (the publication list follows). In contrast, three chapters of my thesis will be included in future journal publications and are excluded from this book. These removed chapters are considered largely independent works, so the coherence and understanding of the book are not prevented by removing them. The reader has free access to the complete version of my Ph.D. thesis through this link: http://hdl.handle.net/10442/hedi/50501. I edited this monograph during my visiting stay at the Department of Civil and Industrial Engineering of the University of Pisa, Italy. I want to thank Prof. Paolo S. Valvo, Mr. Paolo Fisicaro, and the rest members of the Mechanics of Advanced Materials and Structures group for their hospitality and support, as well as for our interesting discussions. I apologize in advance for any errata the reader may find while reading this book. For possible comments that occur to the reader on the topics examined in the book, for supplementary material, or for any other reason, the reader can contact me at this email address: [email protected]. Pisa, Italy May 2022

Panayiotis Tsokanas

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Parts of this thesis have been published in the following articles:

1. Tsokanas, P., & Loutas, T. (2019). Hygrothermal effect on the strain energy release rates and mode mixity of asymmetric delaminations in generally layered beams. Engineering Fracture Mechanics, 214, 390–409. https://doi.org/10.1016/ j.engfracmech.2019.03.006 2. Tsokanas, P., Loutas, T., Kotsinis, G., Kostopoulos, V., van den Brink, W. M., & Martin de la Escalera, F. (2020). On the fracture toughness of metal-composite adhesive joints with bending–extension coupling and residual thermal stresses effect. Composites Part B: Engineering, 185, 107694. https://doi.org/10.1016/j. compositesb.2019.107694 3. Loutas, T., Tsokanas, P., Kostopoulos, V., Nijhuis, P., & van den Brink, W. M. (2021). Mode I fracture toughness of asymmetric metal-composite adhesive joints. Materials Today: Proceedings, 34(1), 250–259. https://doi.org/10.1016/ j.matpr.2020.03.075 4. Tsokanas, P., Loutas, T., & Nijhuis, P. (2020). Interfacial fracture toughness assessment of a new titanium-CFRP adhesive joint: an experimental comparative study. Metals, 10(5), 699. https://doi.org/10.3390/met10050699 5. Tsokanas, P., Loutas, T., Kotsinis, G., van den Brink, W. M., & Nijhuis, P. (2021). Strain energy release rate and mode partitioning of moment-loaded elastically coupled laminated beams with hygrothermal stresses. Composite Structures, 259, 113237. https://doi.org/10.1016/j.compstruct.2020.113237 6. Tsokanas, P., & Loutas, T. (2022). Closed-form solution for interfacially cracked layered beams with bending–extension coupling and hygrothermal stresses. European Journal of Mechanics—A/Solids, 96, 104658. https://doi.org/10.1016/ j.euromechsol.2022.104658

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Acknowledgments

I would like to kindly acknowledge all those who have contributed to the progress and successful completion of the present thesis. First and foremost, I would like to sincerely thank the academic supervisor of the thesis, Prof. Theodoros Loutas, for the incisive supervision of my work. Prof. Loutas was always present to effectively coordinate, supervise, and evaluate every step of the work presented in this thesis, as well as to advise me whenever I needed it. I have spent countless hours together with Prof. Loutas, both in the laboratory and outside working hours in several study rooms in the city of Patras, on business trips and walks, discussing all of the topics being developed in the thesis. I owe him for the many hours he has devoted—often on weekends—not only to critically evaluate the literature to help me build the theoretical framework for the present thesis but also to work alongside me on some of the equations presented in this thesis. For all the time he has dedicated, for his high-quality mentorship, and for the advice he has provided me with on a professional and personal level, I am indebted to him. Secondly, I would like to deeply thank the second member of my advisor committee, Prof. Vassilis Kostopoulos, for the overall supervision of the project, the valuable pieces of advice, and the insightful feedback he has given me. Above all, I warmly thank Prof. Kostopoulos because, as my first academic mentor when I was still a third-year undergraduate student, he showed great confidence in me and helped me develop my enthusiasm for academic studies. In addition, I warmly thank the third member of the advisory committee, Prof. Dimitrios Zarouchas (Delft University of Technology, the Netherlands), for his comments on the work and our pleasant collaboration during the course of the TICOAJO research project. I would also like to thank several of my colleagues who have contributed significantly to various aspects of the thesis. First, many thanks go to my TICOAJO partners from the Royal Netherlands Aerospace Centre, the Netherlands, especially Mr. Wouter M. van den Brink and Mr. Peter Nijhuis, who manufactured the test specimens and assisted with the execution of part of the experiments. In addition, I thank my colleague and friend, Mr. Georgios Kotsinis (University of Patras, Greece), for our significant collaboration in the finite element analyses. Moreover, the assistance of my colleagues from the Applied Mechanics Laboratory, University xv

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Acknowledgments

of Patras, Greece—especially Mr. Dimitrios Pegkos and Dr. George Sotiriadis— during the execution of most of the experiments is gratefully acknowledged. I also thank the TICOAJO partners from the Structural Integrity and Composites research group, Delft University of Technology, the Netherlands—especially Dr. Wandong Wang, Prof. Johannes A. Poulis, Prof. Sofia Teixeira de Freitas, and Prof. Dimitrios Zarouchas—for performing the novel surface pre-treatment of both titanium and CFRP. Lastly, I would like to thank the following people: Dr. Yasser Essa and Dr. Federico Martin de la Escalera (Aernnova, Spain) for our enjoyable collaboration in the framework of the TICOAJO project; Mr. Marcelo Müller (Royal Netherlands Aerospace Centre, the Netherlands) for assisting with the execution of the DCB-UBM experiments; Mr. Lucas Adamos (University of Patras, Greece) for our useful discussions and excellent collaboration in one of the papers of his work; Mr. Georgios Galanopoulos (University of Patras, Greece) for assisting with the digital image correlation analysis; and Ms. Anatoli Mitrou (University of Potro, Portugal) for kindly proofreading a large part of the manuscript and providing me with useful suggestions for improving the language. The remaining members of the examination committee (Prof. Theodore P. Philippidis [University of Patras, Greece], Dr. Anastasios P. Vassilopoulos [École Polytechnique Fédérale de Lausanne, Switzerland], Prof. Konstantinos I. Tserpes [University of Patras, Greece], and Prof. Konstantinos Anyfantis [National Technical University of Athens, Greece]) are acknowledged for agreeing to be on the committee. I also sincerely thank them for reading my thesis and providing me with helpful corrections and comments. I especially thank Prof. Philippidis, who carefully read the entire manuscript and provided me with several corrections and interesting suggestions for improvements. In addition, I express my sincere thanks to the members of the Composite Construction Laboratory, Switzerland, for their excellent hospitality during my stay in Lausanne. Furthermore, I acknowledge the anonymous reviewers of the journals to which I submitted various parts of the thesis for their constructive criticism, comments, and recommendations. Their reviews significantly improved my work. Since Prof. Paolo S. Valvo (University of Pisa, Italy) is one of the leading scientists in Europe on the subjects related to my thesis, I took the initiative to kindly ask him to read my thesis and give me feedback on it. For his willingness to read the entire final manuscript and provide me with many interesting comments and suggestions for improvement, I would like to sincerely thank him. His comments helped me to significantly improve the scientific accuracy of the thesis. It would be an oversight not to appreciate the financial support for the present thesis provided by the Clean Sky 2 TICOAJO project, the Hellenic Foundation for Research and Innovation, and the European Cooperation in Science and Technology (COST action CERTBOND). At this point, I would like to thank the laboratory members (and good friends of mine) Athanasios K., Gregory P., and Dimitrios R., as well as my relatively new but very close friend Konstantinos P., and, of course, Konstantinos Y., for making the four years of my Ph.D. studies more enjoyable and interesting. I also thank the Ph.D.

Acknowledgments

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student and member of Prof. Loutas’s group, Dimitrios M., for the fruitful time we have spent working together all these years. For the considerable patience she showed during all those hours I was in front of my computer, I cannot fail to thank my girlfriend, Myrsini. Last but not least, I would like to sincerely thank all the members of my family: my beloved father Efstathios and my beloved mother Constantine, as well as my brothers Charis and Elias and my sister Venia. Thank you for providing me—along with other members of my wider family and friendship circle—with significant emotional support during the years of my Ph.D. studies. Thank you all. Patras, Greece August 2021

Panayiotis Tsokanas

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 General Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 General Aim and Specific Objectives of the Book . . . . . . . . . . . . . . . 1.2.1 General Aim of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Specific Objectives of the Book . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 The Book at a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Organization and Structure of the Book . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Organization of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Structure of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Contribution of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Basic Concepts and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Layered, Generally Layered, Multilayered, Laminated . . . . . 1.5.2 Crack, Delamination, and Disbonding . . . . . . . . . . . . . . . . . . . 1.5.3 Definitions for Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.4 Data Reduction Scheme (DRS) . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 7 7 7 8 9 9 9 10 11 12 12 12 12 14 14

2 The Effect of Residual Hygrothermal Stresses on the Energy Release Rate and Mode Mixity of Interfacial Cracks in Beams with Bending–Extension Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Present Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Problem Description and Analysis Approach . . . . . . . . . . . . . . . . . . . 2.2.1 Definition of the Scientific Problem . . . . . . . . . . . . . . . . . . . . . 2.2.2 The Proposed Analytical Model . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Mathematical Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . 2.3.1 Kinematic Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Constitutive Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 18 19 21 21 21 25 25 26

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2.3.3 Conditions of Static Equilibrium . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Conditions of Displacement Continuity . . . . . . . . . . . . . . . . . 2.4 Solution to the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Derivation of the Governing Equation . . . . . . . . . . . . . . . . . . . 2.4.2 Solution to the Governing Equation . . . . . . . . . . . . . . . . . . . . . 2.4.3 Internal Forces and Moments . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Crack-Tip Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5 Energy Release Rate (ERR) and Mode Mixity (MM) . . . . . . 2.5 Extraction of Closed-Form Equations . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Reduction to Typical Test Configurations . . . . . . . . . . . . . . . . 2.5.2 Effect of Contact Between the Two Sublaminates . . . . . . . . . 2.5.3 Reduction to Previous Equations in the Literature . . . . . . . . . 2.6 Validation Through Finite Element Analysis (FEA) . . . . . . . . . . . . . . 2.6.1 Application: A Typical Glass Aluminum Reinforced Epoxy (GLARE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Finite Element Analyses (FEAs) . . . . . . . . . . . . . . . . . . . . . . . 2.7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Example 1: Double Cantilever Beam (DCB) Test . . . . . . . . . 2.7.2 Example 2: End-Notched Flexure (ENF) Test . . . . . . . . . . . . 2.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Fracture Toughness of Metal-to-Composite Adhesive Joints with Bending–Extension Coupling and Residual Thermal Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Present Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Description of the Technical and Scientific Problem . . . . . . . . . . . . . 3.2.1 The Metal-to-Composite Adhesive Joint Under Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Challenges in the Design and Data Reduction of Fracture Tests for Dissimilar Adhesive Joints . . . . . . . . . . 3.3 Design and Data Reduction of Fracture Tests . . . . . . . . . . . . . . . . . . . 3.3.1 Design of the Fracture Toughness Tests . . . . . . . . . . . . . . . . . 3.3.2 Data Reduction Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Materials and Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Fracture Toughness Experiments . . . . . . . . . . . . . . . . . . . . . . . 3.5 Finite Element Analyses (FEAs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Two-Dimensional (2D) Finite Element Analyses (FEAs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 29 31 31 33 34 35 37 38 39 44 48 48 48 50 55 55 56 57 59 61

65 65 66 69 71 71 72 72 73 76 79 79 80 82 83

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3.5.2 Three-Dimensional (3D) Finite Element Analyses (FEAs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Cohesive Zone Modeling (CZM) . . . . . . . . . . . . . . . . . . . . . . . 3.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 The Effect of Residual Thermal Stresses (RTS) on the Total Energy Release Rate (ERR) and Mode Mixity (MM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Crack Propagation Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Data Reduction Using Various Analytical Expressions . . . . . 3.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Interfacial Fracture Toughness of a Titanium-to-CFRP Adhesive Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Present Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Titanium-to-CFRP Joint Under Consideration . . . . . . . . . . . . . . 4.2.1 Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 The Proposed Manufacturing Options (MOs) . . . . . . . . . . . . 4.3 Experimentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Mechanical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Fractographic Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Experimental Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Double Cantilever Beam (DCB) Experiments . . . . . . . . . . . . 4.4.2 End-Notched Flexure (ENF) Experiments . . . . . . . . . . . . . . . 4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Energy Release Rate and Mode Partitioning of Moment-Loaded Fracture Tests on Layered Beams with Bending–Extension Coupling and Hygrothermal Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Present Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Existing Analytical Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Proposed Data Reduction Scheme (DRS) . . . . . . . . . . . . . . . . . . 5.3.1 Problem Description and Approach of the Analysis . . . . . . . 5.3.2 Calculation of the Crack-Tip Forces . . . . . . . . . . . . . . . . . . . . 5.3.3 Calculation of the Energy Release Rate (ERR) and Mode Mixity (MM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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85 87 88

88 90 92 95 95 97 101 101 102 102 104 104 104 105 108 108 110 117 118 118 121 123 124 126

129 129 130 132 133 136 136 138 139

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5.3.4 Special Cases and Refinement to Pre-existing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Numerical Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Finite Element Analysis (FEA) . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Double Cantilever Beam Test Fixture with Uneven Bending Moments (DCB-UBM Test Fixture) . . . . . . . . . . . . 5.5.2 The Titanium-to-CFRP Adhesive Joint . . . . . . . . . . . . . . . . . . 5.5.3 Execution of the Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Validation of the Data Reduction Scheme (DRS) . . . . . . . . . 5.6.2 Application in Actual Experiments . . . . . . . . . . . . . . . . . . . . . 5.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Closed-Form Solution for Interfacially Cracked Layered Beams with Bending–Extension Coupling and Hygrothermal Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Present Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Problem Description and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Definition of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Mathematical Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . 6.3.1 Kinematic Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Constitutive Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Conditions of Static Equilibrium . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Conditions of Displacement Continuity . . . . . . . . . . . . . . . . . 6.3.5 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Extraction of Closed-Form Expressions . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Solution Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Solution Along the Uncracked Region . . . . . . . . . . . . . . . . . . 6.4.3 Solution Along the Cracked Region . . . . . . . . . . . . . . . . . . . . . 6.4.4 Complete Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Refinements and a Suggested Improved Solution . . . . . . . . . . . . . . . . 6.5.1 Reduction to Unified Test Configurations . . . . . . . . . . . . . . . . 6.5.2 Refinement for a Clamped Crack Tip . . . . . . . . . . . . . . . . . . . 6.5.3 Refinements for Common Material Systems . . . . . . . . . . . . . 6.5.4 The Balanced Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.5 On Improving the Equations for the Contact Force . . . . . . . .

140 144 144 146 148 148 149 150 151 151 154 155 156 158

161 161 162 164 166 166 167 169 170 170 170 171 174 174 175 175 177 182 183 186 186 187 187 189 189

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6.6 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Application Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Application Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

190 190 192 200 201 202

7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 General Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Derivation and Implementation of a New Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Numerical Validation of the Analytical Model . . . . . . . . . . . . 7.1.4 Interfacial Cracks Inside Layered Materials . . . . . . . . . . . . . . 7.1.5 Mode-Mixity (MM) Response . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.6 The Effect of Residual Thermal Stresses (RTS) . . . . . . . . . . . 7.1.7 The Titanium-to-CFRP Adhesive Joint . . . . . . . . . . . . . . . . . . 7.1.8 Some Additional Novelty Points . . . . . . . . . . . . . . . . . . . . . . . 7.2 Strengths of the Proposed Analytical Model and Expressions . . . . . 7.3 Ongoing Work and Suggestions for Future Work . . . . . . . . . . . . . . . . 7.3.1 The Titanium-to-CFRP Adhesive Joint . . . . . . . . . . . . . . . . . . 7.3.2 Contact Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Compliance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Determination of the Energy Release Rate (ERR) . . . . . . . . . 7.3.5 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.6 Three-Dimensional (3D) Data Reduction Scheme (DRS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.7 Further Refinement of the Analytical Model . . . . . . . . . . . . . 7.3.8 Further Development of the Rigid Joint Model . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205 205 205 206 206 208 208 208 209 209 210 211 211 211 212 212 212 213 213 213 214

Appendix A: First-Order Shear Deformation Theory Considering an Elastically Coupled Behavior and Residual Hygrothermal Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Appendix B: Fracture Mode Partitioning Adopting the Crack Closure Integral Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Appendix C: Virtual Crack Closure Technique . . . . . . . . . . . . . . . . . . . . . . . 233 Appendix D: Aircraft Application of the Titanium-to-CFRP Adhesive Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Appendix E: Manufacturing the Titanium-to-CFRP Adhesive Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Appendix F: Analytical Approaches for Fracture Mode Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

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Appendix G: Related to the Work in Chap. 6 . . . . . . . . . . . . . . . . . . . . . . . . . 255 About the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

Nomenclature

Latin Symbols a ai a20 , a0 , aM , aN a0 a0 Ai b bi Bi c ci c0 C Ci di dx Di E Ea, Eb, Ec Ei E xi

Crack length of the beam or specimen, (mm) Extensional compliance of sublaminate i, (mm/N) Constants used in the governing equation (Eqs. 2.29 and 6.14), (–) Initial crack length of the beam or specimen, (mm) Propagated crack length after the first step of the DCB experiment (Chap. 3 only), (mm) Extensional stiffness of sublaminate i, (N/mm) Width of the beam or specimen, (mm) BEC compliance of sublaminate i, (1/N) BEC stiffness of sublaminate i, (N) Lever length of the loading apparatus for the MMB test (Chap. 2 only), (mm) Shear compliance of sublaminate i, (mm/N) Coefficient in the exponential equation giving the N1 (x) (Eq. 2.36) and the N1(u) (x) (expression No. 1 in Table 6.1), (–) Compliance, (mm2 /N) Shear stiffness of sublaminate i, (N/mm) Bending compliance of sublaminate i, (1/N mm) Infinitesimal portion along beam length, (mm) Bending stiffness of sublaminate i, (N mm) Young’s modulus of an isotropic material, (GPa) Young’s modulus of sub-beams a, b, and c (Table 5.1 and Eq. 5.5 only), (GPa) Young’s modulus of sublaminate i (if made of an isotropic material), (GPa) Effective Young’s modulus along x-axis of sublaminate i (Chap. 3 only), (GPa) xxv

xxvi

E 11 , E 22 , E 33 G G GC GC,ini GI , GII , GIII GIC , GIIC GIC,ini , GIIC,ini G 12 , G 23 , G 13 h h hi h int h low , h up

i Ii J k  le 1 , 2 L Lm Ma , Ms Mc Mi (x) (u) M(c) i (x), Mi (x) Mi0 MTH i MiC (x) MI , MII

Nomenclature

Young’s modulus along 1-, 2-, and 3-axis, (GPa) Shear modulus of an isotropic material, (GPa) (Total) ERR, (N/mm) Critical ERR, (N/mm) Critical ERR for crack initiation, (N/mm) Mode I, mode II, and mode III contributions to the ERR, (N/mm) Critical mode I and mode II ERR, (N/mm) Critical mode I and mode II ERR for crack initiation, (N/mm) Shear modulus in 12-, 23-, and 13-plane, (GPa) Total thickness of the beam or specimen, (mm) Auxiliary symbol for the thickness; h = h/2 (Chap. 5 only), (mm) Thickness of sublaminate i, (mm) Thickness of the interfacial layer between the two sublaminates, (mm) Vertical distance between the neutral axis and the bottom surface of sublaminate 1 and between the top surface and the neutral axis of sublaminate 1 (Chap. 3 only), (mm) Sublaminate index, (–) Moment of inertia, (mm3 ) J -integral, (N/mm) Stiffness of the traction-versus-separation law (Chap. 3 only), (N/mm3 ) Length of the uncracked region of the beam, (mm) Length of the finite element in the crack propagation direction (Chap. 3 only), (mm) Moment arms (Chap. 5 only), (mm) Active length of the test configuration, (mm) Total length of the beam model, (mm) Antisymmetric and symmetric bending moments in the DCB-UBM test, (N) Self-equilibrium bending moment at the crack tip, (N) Internal bending moment, (N) Internal bending moment along the cracked and uncracked region (Chap. 6 only), (N) Internal bending moment at the crack tip, from the left side, (N) Hygrothermal moment resultant, (N) Steady-state value of the internal moment Mi (x), (N) Bending moments in a pure mode I (MI ) and pure mode II (MII ) moment-loaded test, (N)

Nomenclature

MT (x) M1 , M2 n Nc Ne Ni (x) Ni(c) (x), Ni(u) (x) Ni0 NiTH NiC (x) NT N1 , N2 P Pc Pc,ini cl Pcont , Pcont Pe Pe,ini PI , PII PI,ini , PII,ini PMMB P1 , P2 Qc Qi (x) (u) Q(c) i (x), Qi (x) QiC Qi0 QT Tcur u i (x) u cl i (x)

xxvii

Total resultant bending moment acting on the middle plane of sublaminate 2, (N) Unequal bending moments applied, (N) Safety factor (Eqs. 3.1 and 3.2 only), (–) Self-equilibrium axial force at the crack tip, (N/mm) Number of elements in the cohesive zone (Chap. 3 only), (–) Internal axial force, (N/mm) Internal axial force along the cracked and uncracked region (Chap. 6 only), (N/mm) Internal axial force at the crack tip, from the left side, (N/mm) Hygrothermal force resultant, (N/mm) Steady-state value of the internal force Ni (x), (N/mm) Total resultant axial force acting on the middle plane of sublaminate 2, (N/mm) Unequal axial forces applied, (N/mm) Force couple in the DCB-UBM test, (N/mm) Applied load in the ENF and SLB tests, (N/mm) Crack initiation load in the ENF test, (N/mm) Contact force between the upper and lower cracked arms based on the clamped and semi-rigid models, (N/mm) Applied load in the DCB test, (N/mm) Crack initiation load in the DCB test, (N/mm) Applied load in a pure-mode DCB and ENF test, (N/mm) Crack initiation load in a pure-mode DCB and ENF test, (N/mm) Applied load in the MMB test, (N/mm) Unequal vertical forces applied, (N/mm) Self-equilibrium transverse force at the crack tip, (N/mm) Internal transverse shear force, (N/mm) Internal transverse shear force along the cracked and uncracked region (Chap. 6 only), (N/mm) Steady-state value of the internal force Qi (x), (N/mm) Internal transverse shear force at the crack tip, from the left side, (N/mm) Total resultant transverse shear force acting on the middle plane of sublaminate 2, (N/mm) Curing temperature, (°C) Axial displacement of the middle plane of sublaminate i, (mm) Axial displacement of the middle plane of sublaminate i based on the clamped crack-tip model (Chap. 6 only), (mm)

xxviii

Nomenclature

(u) u (c) i (x), u i (x)

u 1 (x), u¯ 2 (x) U Ui (x, z i ) V1 , V2 , V3 , V4 wi (x) wicl (x) wi(c) (x), wi(u) (x) w 1 (x), w ¯ 2 (x) Wi (x, z i ) x xi , z i X(x), X(c) (x), X(u) (x)

Axial displacement of the middle plane of sublaminate i along the cracked and uncracked region (Chap. 6 only), (mm) Axial displacement of the bottom plane of sublaminate 1 and the top plane of sublaminate 2, (mm) Total potential energy available for crack propagation (Chap. 1 only), (N mm) Displacement of sublaminate i in x-direction, (mm) Auxiliary constants introduced to simplify the analytical expressions (Chap. 6 only), (–) Transverse displacement of the middle plane of sublaminate i, (mm) Transverse displacement of the middle plane of sublaminate i based on the clamped crack-tip model (Chap. 6 only), (mm) Transverse displacement of the middle plane of sublaminate i along the cracked and uncracked region (Chap. 6 only), (mm) Transverse displacement of the bottom plane of sublaminate 1 and the top plane of sublaminate 2, (mm) Displacement of sublaminate i in z i -direction, (mm) Abscissa indicating the horizontal distance between the crack-tip cross-section and arbitrary cross-section, (mm) Local coordinates, located in the middle plane of sublaminate i, (mm) Generic quantity (an auxiliary definition) (Chap. 6 only), (–)

Greek Symbols αMi αNi α11 , α22 β0 γi (x) γi(c) (x), γi(u) (x) δc δe δI , δII

Curvature of sublaminate i due to RHTS, (1/mm) Axial strain of sublaminate i due to RHTS, (–) CTE along 1- and 2-axis, (·10–6 /°C) Unbalance parameter; β0 = −ξ , (1/N) Shear strain of sublaminate i, (–) Shear strain of sublaminate i along the cracked and uncracked region (Chap. 6 only), (–) Load-point displacement in the ENF test, (mm) Load-point displacement in the DCB test, (mm) Load-point displacement in a pure-mode DCB and ENF test, (N/mm)

Nomenclature

xxix

δI , δII Δ ΔH ΔT Δu(x) Δw(x) Δw(c) (x) εi (x) εi(c) (x), εi(u) (x) η κi (x) κi(c) (x), κi(u) (x) λ ν12 , ν21 ξ ξI , ξII σ (x) σyTi. , σyAl. τ (x) τI0 , τII0 ϕi (x) ϕicl (x) ϕi(c) (x), ϕi(u) (x) ψI , ψII ψIC,ini , ψIIC,ini

Flexibility coefficients, (mm/N) Constant used in the governing equation (Eqs. 2.29 and 6.14), (–) Percentage of moisture exchange, (%) Temperature difference (between the operational and stress-free temperatures), (°C or K) Relative axial displacement of the interfacial plane, (mm) Relative transverse displacement of the interfacial plane, (mm) Relative transverse displacement of the interfacial plane along the cracked region (Chap. 6 only), (mm) Axial strain of sublaminate i, (–) Axial strain of sublaminate i along the cracked and uncracked region (Chap. 6 only), (–) Parameter introduced to simplify the analytical expressions, (mm/N) Curvature of sublaminate i, (1/mm) Curvature of sublaminate i along the cracked and uncracked region (Chap. 6 only), (1/mm) Decay rate, (1/mm) Major and minor Poisson’s ratio, (–) Parameter introduced to simplify the analytical expressions, (1/N) Auxiliary parameters (Eqs. 3.1 and 3.2 only), (–) Interfacial normal stress, (N/mm2 ) Yield stress of the titanium and aluminum (Chap. 3 only), (MPa) Interfacial shear stress, (N/mm2 ) Maximum stress of the traction-versus-separation law for the DCB and ENF test (Chap. 3 only), (GPa) Cross-sectional rotation of the middle plane of sublaminate i, (rad) Cross-sectional rotation of the middle plane of sublaminate i based on the clamped crack-tip model (Chap. 6 only), (rad) Cross-sectional rotation of the middle plane of sublaminate i along the cracked and uncracked region (Chap. 6 only), (rad) MMR during the ENF (ψI ) and DCB or DCB-UBM (ψII ) test, (%) MMR during the ENF (ψIC,ini ) and DCB or DCB-UBM (ψIIC,ini ) test at the crack initiation point, (%)

Abbreviations and Acronyms BEC CCM CFRP CLPT

Bending–extension coupling Compliance calibration method Carbon fiber-reinforced polymer Classical laminated plate theory

xxx

CLT CPT CTE CTE CZM DCB DCB-UBM DRS ENF ERR FEA FML FSDT GFRP GLARE HLFC LEFM LFC MD MM MMB MMR MO NLF PEKK PTFE RHTS RTM RTS SIF SLB TICOAJO UD VARTM VCCT

Nomenclature

Classical lamination theory Classical plate theory Coefficient of thermal expansion Crack-tip element Cohesive zone model(ing) Double cantilever beam Double cantilever beam (test) with uneven bending moments Data reduction scheme End-notched flexure Energy release rate Finite element analysis Fiber metal laminate First-order shear deformation theory Glass fiber-reinforced polymer Glass laminate aluminum reinforced epoxy Hybrid laminar flow control Linear elastic fracture mechanics Laminar flow control Multidirectional Mode mixity Mixed-mode bending Mode-mixity ratio Manufacturing option Natural laminar flow Polyetherketoneketone Polytetrafluoroethylene Residual hygrothermal stresses Resin transfer molding Residual thermal stresses Stress intensity factor Single-leg bending TItanium COmposite Adhesive JOints Unidirectional Vacuum-assisted resin transfer molding Virtual crack closure technique

Subscripts, Superscripts, and Conventions Sublaminate index (unless otherwise stated), where i ∈ {1, 2} Fracture mode (unless otherwise stated): mode I, mode II, and mode III Region of the beam: cracked and uncracked (Chap. 6 only)

Nomenclature

xxxi

Quantity based on the clamped crack-tip model Bottom surface of sublaminate 1 and top surface of sublaminate 2

Other 1D, 2D, 3D

One-, two-, and three-dimensional

Chapter 1

Introduction

1.1 Background and Motivation This section starts by presenting the general background of the thesis (Sect. 1.1.1). This includes a brief reference to five central topics: energy release rate (ERR) and fracture toughness, structural theories, interfacial models, fracture toughness characterization, and layered materials of interest. Next, in Sect. 1.1.2, the motivation for the present thesis is presented.

1.1.1 General Background 1.1.1.1

Energy Release Rate (ERR) and Fracture Toughness

The mechanical analysis of a crack inside a material is most commonly conducted in the context of fracture mechanics. In particular, under linear elastic fracture mechanics (LEFM) conditions, the most common parameter to predict the initiation and propagation of a crack is probably the ERR,1 G [2, pp. 115–119]—see also the book by Friedrich [3] for its application in composites. This essential parameter 1

In the literature, many authors use the term “strain energy release rate” (commonly abbreviated as SERR)—even ASTM standards do so. Nevertheless, it would be more appropriate to use the term “energy release rate” (ERR) without “strain.” In fact, G is defined as the derivative of the total potential energy (i.e., strain energy plus applied load potential energy). The latter vanishes for tests conducted at fixed grips (as ASTM standards suggest). However, if external loads are applied (e.g., bending moments, as in the DCB-UBM test), then, strictly speaking, it is incorrect to use the term SERR. Thus, we prefer to use the term ERR throughout the thesis. For more information on this term, the reader is referred to the fundamental works by Broek [2, pp. 115–119] and Anderson [1, pp. 41–46].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 P. Tsokanas, Fracture Analysis of Layered Beams With an Elastically Coupled Behavior and Hygrothermal Stresses, Springer Theses, https://doi.org/10.1007/978-3-031-17621-0_1

1

2

1 Introduction

expresses the rate at which energy is released as a material undergoes fracture. The ERR is defined mathematically as the decrease in total potential energy per unit area of new surface created by the propagation of the crack: G=−

∂U , ∂a

(1.1)

where U is the total potential energy available for crack propagation, and a is the crack area (or crack length for a two-dimensional [2D] problem). The failure criterion for the ERR states that a crack will start to grow when the available ERR, G, becomes greater than/equal to a critical value, GC (G ≥ GC ). We now introduce the compliance, C, of a generally shaped elastic body loaded by a point force. This magnitude is given as C=

δ , P

(1.2)

where δ is the displacement of the force-application point, and P is the intensity of the force. Using this definition, the ERR can be obtained by differentiating the compliance with respect to the crack length, according to the well-known Irwin–Kies relation [5]: G=

P 2 dC . 2 da

(1.3)

Depending on the structural model adopted (see Sects. 1.1.1.2 and 1.1.1.3), various analytical expressions for compliance have been obtained in the literature. Correspondingly, multiple expressions for the ERR can be deduced through Eq. (1.3). We will return to this in the course of the book. On the one hand, theoretical (i.e., analytical) methods and numerical (e.g., finite element analysis [FEA]) models are constantly being developed by researchers to compute the expected values of G in “classical” test configurations (e.g., double cantilever beam [DCB], end-notched flexure [ENF], mixed-mode bending [MMB]) or in real-world structural components (e.g., a stiffened panel for aerospace application). The ERR of a linearly elastic layered beam can be calculated via many alternative theoretical methods: the compliance-based method (i.e., application of Eq. 1.3), using energetic theorems—e.g., Irwin’s [4] crack closure integral, the virtual crack closure technique (VCCT) [6], or the J -integral approach [12], to list a few. Some of the existing methods are discussed in Chap. 2. On the other hand, experimental techniques are used to assess the critical ERR, GC , of specific material systems, which is also called the (interfacial) fracture toughness. Analyzing such experiments is complicated because, as is known, the crack usually propagates under a mix of the three basic fracture modes (mixed-mode condition): mode I or opening, mode II or sliding, and mode III or tearing. In such cases, the fracture toughness, GC , depends on the mode mixity (MM), which stands as a measure of

1.1 Background and Motivation

crack

3

crack front

sublaminate 1

sublaminate 2 Fig. 1.1 A generally layered beam with an asymmetric crack

the relative amounts of these three fracture modes. The (total) ERR, G, is the (algebraic)2 sum of three modal contributions: GI , GII , and GIII . An appropriate theoretical model should be able to predict these three contributions. This topic is also discussed in Chap. 2.

1.1.1.2

Structural Theories

We now consider an elastic beam (or plate) that is affected by a crack; see, for example, the beam in Fig. 1.1. Imagine that the plane defined by the crack (crack plane) ideally splits the beam into two sub-beams: sublaminates 1 and 2. The structural behaviors of the sub-beams and of the entire beam are typically modeled through structural theories. These theories can be grouped into two main categories: beam theories and plate theories. Here, we briefly present the fundamental structural theories, to which we will frequently refer throughout the book. The most elementary structural theory used for the mechanical analysis of beams is the Euler–Bernoulli beam theory or, simply, Euler beam theory (also commonly referred to as simple beam theory or classical beam theory). This theory is built on the assumption that, after deformation, the plane cross-sections of the beam remain plane and perpendicular to the middle plane.3 In other words, this theory completely neglects shear deformation, albeit it exists in many real-world problems. The so-called classical laminated beam theory is an extension of Euler beam theory to account for laminated beams (Euler’s assumption holds). To consider shear deformation at the first order, Timoshenko [13] proposed a theory known as Timoshenko

2

We add the term algebraic here with the intention of being consistent with some mode partitioning methods that may furnish negative values for the modal contributions. Nevertheless, several doubts could be raised about the physical meaningfulness of such approaches (see [19]). 3 In the literature, the middle plane (or mid-thickness plane) is sometimes referred to as the centerline.

4

1 Introduction

beam theory. After deformation, this theory allows for relative rotations between the plane cross-sections of the beam and its middle plane. Euler and Timoshenko beam theories correspond to Kirchhoff–Love and Mindlin– Reissner plate theories, respectively. Kirchhoff–Love plate theory belongs to the family of so-called classical plate theories (CPTs). In these theories, it is assumed that, after deformation, the plane cross-sections of the plate remain plane and perpendicular to the midsurface. Again, this is the result of neglecting transverse shear deformations. As with beams, classical laminated plate theories (CLPTs) represent an extension of CPTs for laminated plates (Kirchhoff’s assumption holds). To consider shear deformations through the thickness of a plate, Reissner [10, 11] and Mindlin [8] proposed a well-known theory belonging to the family of so-called first-order shear deformation theories (FSDTs). Their theory is commonly known as the Mindlin– Reissner theory, and it assumes the displacement field as linear variations of the displacements of the middle plane. This theory is intended to be applied to thick plates4 in which Kirchhoff’s hypothesis is relaxed: after deformation, the normal to the midsurface remains straight but not perpendicular to the midsurface. The theory was later extended to account for laminated plates; the resulting theory is sometimes called the “first-order shear deformation laminated plate theory” or, more simply, “FSDT for laminated plates.” The higher-order shear deformation theories for both beams and plates assume that, after deformation, the plane cross-sections may not remain plane. This assumption comes closer to reality, enabling better agreement between the structural models and three-dimensional (3D) elastic analyses. Since the pioneering work by Reddy [9], several modified higher-order theories have been developed.

1.1.1.3

Interface Models

We now return to the general problem at hand: we consider a crack within a beam structure made of several dissimilar material layers (see Fig. 1.1). As explained above, the crack splits the beam into two sub-beams, the modeling of which typically follows some of the above structural theories. Besides adopting an appropriate structural theory for the sub-beams, the interface between them must also be suitably modeled. To this end, a wide range of models— which are referred to in the relevant literature as interface models, interface joint models, or, sometimes, models of displacement continuity—have been proposed hitherto. These models are characterized by growing complexity, ranging from rigidjoint models to elastic-interface models and cohesive zone models (CZMs). 4

Mindlin–Reissner theory is used for plates with a thickness of the order of one-tenth of the planar dimensions. In contrast, Kirchhoff–Love theory applies to thinner plates. However, we do not overlook the fact that this distinction is valid for homogeneous and isotropic plates. For laminated or anisotropic plates, the appropriateness of the theories adopted should be based not only on their geometry but also on their elastic properties. For instance, a composite plate exhibiting very low shear modulus, G zx , may not be accurately modeled by the Kirchhoff–Love theory, even if it is thin.

1.1 Background and Motivation

5

Choosing an appropriate interface model is critical for the accurate prediction of the structural response—in terms of, for example, displacements, internal forces and moments, and interfacial stresses—of the beam element at hand. Additionally, it is essential for the correct determination of the fracture toughness and the crack propagation threshold. This is because the interface model determines the stress field near the crack tip. For example, a rigid-joint model represents the stress field using concentrated forces and moments, whereas an elastic-interface model represents it as a distribution of normal and shear stresses. The stress field, in turn, determines the values of the fracture toughness parameters.

1.1.1.4

Interfacial Fracture Characterization

An interfacial crack is a major failure mode for a wide range of layered materials. For example, separation between the laminae of a (fiber-reinforced) composite laminate, commonly known as delamination or interlaminar fracture (Fig. 1.2), is a major lifelimiting factor for the material. Interfacial cracks are also observed in several other layered materials, such as sandwich panels, bonded joints, and multilayered ceramics. An interfacial crack may have several causes: small defects during manufacturing and low-energy impacts are two examples. The crack propagates under static or dynamic loads due to peak interfacial stresses, and it may lead to the large-scale failure of the structural component. As already mentioned, crack growth simultaneously involves three modes: opening, sliding, and tearing. To assess fracture toughness in each of these modes, specific laboratory tests have been developed, some of which are now standardized. For example, the DCB test is the standard test for pure-mode I conditions, and the ENF is used for pure mode II. For mixed-mode I/II fractures, the MMB test has been standardized by ASTM for unidirectional (UD) laminates. Assuming linear behavior, the MMB test can be considered as the superposition of the DCB and ENF tests. Under these conditions, and assuming linear behavior, it is still possible to partition the distinct contributions of the two modes following the procedures illustrated in the standards. crack

Fig. 1.2 Delamination and interfacial disbonding delamination

interfacial disbonding

composite laminate

adhesive joint

6

1.1.1.5

1 Introduction

Layered Materials

Layered materials—i.e., materials that consist of two or more dissimilar layers— have begun to attract increasing interest over the last decades and to gain uses in various industries. Within this general group labeled layered materials, we include various different material systems in this thesis, such as multidirectional (MD) fiberreinforced plastics, dissimilar adhesive joints, and fiber metal laminates (FMLs). Steel-to-concrete assemblages, laminated wood, and sandwich composites are further examples of multilayered systems. The widespread application of layered materials is hindered by the requirement for more complex analytical tools than for conventional materials (e.g., metals, UD composites). As a matter of fact, Fig. 1.3 shows some representative examples of material systems in which the existing analytical models used to interpret the standard tests cannot be applied. In addition, there is a lack of standard testing procedures for the fracture toughness of such materials. Anisotropy and non-homogeneity, as well as the multiple potential planes of crack propagation, result in the possible simultaneous co-existence of all fracture modes. In this case, mode partitioning is far from easy. With specimens featuring asymmetries, standardized procedures for mode partitioning are not applicable. Although symmetric when intact, a laminate that features a crack propagating out of the middle plane of the specimen can also be characterized as nonsymmetric (see Fig. 1.3d). In the latter case, the two sublaminates into which the crack divides the laminate may be nonsymmetric, and they may show an elastically coupled response. Moreover, many layered material systems are subjected to aggressive environments. For instance, MD FRPs and sandwich composites are currently being pursued for defense applications, where the material will have to survive extreme environments characterized by very high and very low temperatures and humidities. In summary, the basic peculiarity of layered materials is that, in the presence of a crack, an asymmetric configuration is often created (see Fig. 1.3). In addition, for multimaterial connections, additional phenomena come into play, such as an elastically coupled behavior and the possible induction of hygrothermal stresses. All these factors make the mechanical analysis of cracks inside such materials a challenging yet significant task. Fig. 1.3 An asymmetric crack in a a homogeneous beam and b a bimaterial joint, c a crack in the interface between the core and skin of a sandwich, and d an asymmetric crack in a symmetric laminate

a

b

c

d

1.2 General Aim and Specific Objectives of the Book

7

1.1.2 Motivation Following the above description of the general background, we present the motivation for the present work. As stated in the previous section, an ever-increasing interest can be noted among both the academic and industrial community for the use of generally layered structures (e.g., dissimilar adhesive joints, MD composites, FMLs). These structures are intended to be used in a range of applications in several industries. The interfacial fracture toughness is an essential property of these structures. A significant scientific effort has been made to investigate this property theoretically, experimentally, and numerically. Various aspects of the problem, such as crack-root rotations, large thickness, and deformability of the crack tip, have been examined. In the following chapter, we provide a concise review of the relevant literature. The starting point of the present work was as follows. We want to characterize the interfacial fracture toughness of a beam-like structure, such as the one shown in Fig. 1.1. Such a structure exhibits certain “peculiarities” (see the previous section) in contrast to most beam structures studied to date in the literature. Essentially, these peculiarities can be categorized as follows: • Elastically coupled behavior • Hygrothermal stresses The question that follows is whether there is a solid theoretical framework to effectively analyze the fracture toughness of a hybrid structure (such as the one defined above), considering both peculiarities. After carefully investigating the literature— specific sections in chapters that follow present the relevant literature—, we came up with a negative answer. This was the starting point of the present thesis.

1.2 General Aim and Specific Objectives of the Book In this section, we explain how the general motivation for the research described above took on a specific form through the aim and objectives that the thesis intended to achieve.

1.2.1 General Aim of the Thesis The general aim of the present thesis consists of a central goal and a parallel goal, as follows: • The central goal is to investigate the interfacial fracture toughness and mode partitioning of beam-type structures featuring two specific “peculiarities”: an elastically coupled behavior and hygrothermal stresses. The beam under consideration also involves the following effects: material and thickness dissimilarity, shear

8

1 Introduction

deformability, and crack-root rotations. To achieve our central goal, we developed (i.e., formulated, validated, and implemented) an analytical framework for determining the fracture toughness and performing mode partitioning of such “peculiar” beams. • The parallel goal is to study the fracture behavior and fracture toughness performance of a metal-to-composite adhesive joint that is of great interest to the aerospace industry. This study is part of a larger research project—this is the Clean Sky 2 TItanium COmposite Adhesive JOints (TICOAJO) project—, which explored a titanium-to-carbon fiber-reinforced polymer (CFRP) joint to be implemented in the hybrid laminar flow control (HLFC) system of the wing of future aircraft. The thesis presents the first demonstration of the applicability of the analytical model proposed by the same thesis (see central goal) in actual experiments that come from a genuine need in the aerospace industry. This is how the general and the parallel goal of the thesis are connected.

1.2.2 Specific Objectives of the Book The above general aim is divided into the following specific objectives: • Collection, evaluation, and organization of the existing literature (Sects. i.1.1, with i = 2, 3, ..., 6) • Formulation of an analytical framework for the fracture analysis of beam structures with bending–extension coupling (BEC) and residual hygrothermal stresses (RHTS) (Chaps. 2, 5, and 6) • Validation of the accuracy of the analytical expressions developed through comparisons with FEAs (Chaps. 2, 3, and 5) • Implementation of the analytical framework in the (theoretical) fracture analysis of various material systems (e.g., FMLs, dissimilar adhesive joints, MD composites) and testing configurations (e.g., DCB, ENF, double cantilever beam [test] with uneven bending moments [DCB-UBM]) (Chaps. 2, 3, 5, and 6) • Application of the analytical expressions as data reduction schemes (DRSs) in different laboratory tests (namely, DCB, ENF, and DCB-UBM); in addition, comparison with pre-existing data-reduction equations (Chaps. 3–5) • Experimental and numerical investigation of the fracture behavior of a novel metal-to-composite adhesive joint of interest for the aerospace community (Chaps. 3–5) • Analytical and numerical determination of the fracture toughness of the joint (Chaps. 3–5)

1.3 Organization and Structure of the Book

9

1.2.3 The Book at a Glance The present monograph deals with the issue of fracture analysis of generally layered beam-like structures with BEC and hygrothermal stresses. In particular, it develops (i.e., formulates, validates, and implements) an analytical framework for the calculation of the fracture toughness of such non-conventional beams. In parallel, it investigates the fracture behavior of a metal-to-composite adhesive joint of interest to the aerospace industry. The general problem concerns a beam structure that may feature several “peculiarities”: it may consist of multiple layers of dissimilar materials; it may have asymmetries in terms of layer thicknesses; it may feature an elastically coupled behavior (in particular, BEC); it may be loaded by arbitrary mechanical loads (i.e., concentrated forces and bending moments); and it may contain RHTS. To tackle this problem, we build a generic analytical model that determines the fracture toughness (i.e., ERR and MM) of beams with all the peculiarities just mentioned. Classical theories in the discipline of mechanics (e.g., beam theory, mechanics of composite materials, energetic methods) and important tools in the field of fracture mechanics (e.g., crack-tip element [CTE], crack closure integral, J -integral) are employed while developing this novel framework. With reference to the state of the art, the proposed analyses extend the level of knowledge and enable us to study a variety of, for example, new hybrid material systems, geometries, and testing setups. All new analytical expressions were validated through FEAs, employing mainly the VCCT and secondarily the cohesive zone modeling (CZM). The new expressions are compared with existing, simpler ones, highlighting the usefulness of the new expressions. A plethora of analytical, numerical, and experimental case studies of the fracture toughness of several material systems (e.g., FMLs, metal-to-composite adhesive joints, MD composite laminates) and test configurations (e.g., DCB, ENF, DCBUBM) have been carried out. As a parallel project, the thesis investigated the technological problem of the fracture analysis of a titanium-to-CFRP adhesive joint to be applied in the HLFC system of future aircraft. Extensive experimental, analytical, and numerical methods were combined to understand the fracture behavior and extract the fracture toughness of this joint under various conditions.

1.3 Organization and Structure of the Book 1.3.1 Organization of the Book The present book is organized into seven chapters. The organization follows the book-by-publication approach, and, hence, each chapter is quite independent of the others.

10

1 Introduction

Chapters 2–6 comprise the main body of the book, and each of them has been published in international refereed journals; the relevant publications are listed in the Front Matter and in each specific chapter. In addition, each of them starts with a short literature review on the specific subject studied in that chapter. We decided to organize the literature review in this way, rather than having a central review at the beginning of the book, due to the largely independent nature of the topics discussed in different chapters. In addition, the methods used in Chaps. 2–6 are outlined in the respective chapter. This style provides the reader with a more logical flow of ideas throughout the work. Moreover, Chaps. 2–6 each contain independent sections for results, discussion, and conclusions. It was our aim for each of the main chapters to retain its independence from the others to enable any chapter to be read without the need to frequently visit the others to collect missing information. However, this undoubtedly necessitates some duplication of information between the main chapters. The book ends with Chap. 7, which presents the general summary and conclusions of the overall work. In this way, the various topics covered are combined into a whole.

1.3.2 Structure of the Book The rest of the book is structured into six chapters. In Chap. 2 [14], the analytical model proposed by the thesis is presented and validated. This model is created first in a general framework before being refined to provide closed-form expressions for various test configurations of interest. Validation of the analytical model is performed through FEA. An application study on an FML is conducted to highlight the effect of residual thermal stresses (RTS) on the ERR and MM. In Chap. 3 [16], an engineering approach is introduced to tackle problems related to the analysis and characterization of fracture toughness tests on non-standard beams (like dissimilar adhesive joints). The study shows the effect of thermal stresses on fracture toughness. It also shows the error introduced by analyzing our data with pre-existing, simpler DRSs. Thus, the chapter concludes by describing the utility of the proposed model regarding correct data reduction. In Chap. 4 [7, 18], an experimental characterization of a metal-to-composite adhesive joint—the one introduced in the previous chapter—with BEC and RTS is performed for the first time. In this way, the chapter demonstrates a way for using the proposed model for experimental data reduction of such hybrid beam-like structures. In Chap. 5 [17], the analytical model is extended to cover the case of the DCBUBM test. We solve the problem for the new boundary conditions that the specific new test dictates. The new analytical expressions are validated by comparison with numerical predictions. Furthermore, actual experiments are performed on the metalto-composite joint and post-processed by the new analytical equations. In Chap. 6 [15], a theoretical mechanical analysis is performed based on the formulation from Chap. 2. This analysis calculates various mechanical magnitudes of interest. For all of them, closed-form analytical expressions are derived, including,

1.4 Contribution of the Thesis

11

again, the effects of BEC and RHTS. The mechanical model is believed to be useful for future derivations of magnitudes of interest to fracture mechanics. In Chap. 7, which is the last chapter of the book, the major conclusions of the entire work are provided, along with recommendations for future research. The chapter starts by summarizing the main findings of the thesis and continues by setting out the primary contributions of the work to scientific knowledge. Lastly, some key aspects of the ongoing work are presented, along with suggestions for future research.

1.4 Contribution of the Thesis The contribution to knowledge of the present thesis can be outlined as follows: • To the best of our knowledge, this work proposes for the first time to simultaneously consider two critical effects in the theoretical fracture analysis of beams: BEC and residual (hygrothermal) stresses. The thesis shows that both of these effects, either individually or in synergy, can be quite significant for the ERR. Various characteristic examples of material systems and loading conditions are presented to support our findings. It is our contention that this idea can (and should) be used to increase the accuracy of many analytical models of the literature—some of them are reviewed in Chaps. 2, 5, and 6—that are used in the analysis of non-conventional material systems. • An analytical framework for the determination of the ERR and MM of beamtype structures with BEC and RHTS is developed (i.e., formulated, validated, and implemented). In other words, the thesis proceeds to a first embodiment of the idea mentioned in the previous bullet-point. A handful of new closed-form equations are generated, which are characterized by great generality since they can cover a range of fracture toughness tests (e.g., DCB, ENF, MMB, single-leg bending [SLB], DCB-UBM) and material systems (e.g., FMLs, dissimilar adhesive joints, multilayered composites). • The fracture behavior and fracture toughness of a titanium-to-CFRP adhesive joint to be used in the aerospace sector is characterized experimentally using various laboratory tests. Several MOs are evaluated, while the experimental results are supported by FEAs using both the VCCT and the CZM. In addition, data reduction is performed using the analytical expressions proposed in the thesis. • Analytical modeling is proposed for the theoretical problem of interfacial cracks inside layered materials with BEC and RHTS. The semi-rigid interface joint model is employed in our analysis. A multitude of equations are generated regarding mechanical quantities and are explicitly reported within the book. It is the strong opinion of us that these analytical expressions prepare the ground for various future extensions, as indicated in Chaps. 6 and 7. The above list contains the central contributions of the thesis in a general way. The specific contributions of Chaps. 2–6 are clearly stated in the sections of the respective chapters titled “Present work.” Thus, it is unnecessary to repeat them here.

12

1 Introduction

1.5 Basic Concepts and Definitions This section clarifies some concepts and terms that are used very frequently in the book. Most of them have been defined in the literature in more than one way, and this may be confusing for the reader. In every case, the definitions given below may not be “universally accepted,” but they are certainly the ones we follow in this work.

1.5.1 Layered, Generally Layered, Multilayered, Laminated At various points in the book, the reader encounters the terms “layered,” “generally layered,” “multilayered,” and “laminated,” which describe either a material or a structural element. In the context of the present work, all these terms refer to the same thing, and we use them all to simply avoid the monotonous repetition of a single term. Specifically, they refer to a composite material (or structure) that consists of two or more layers of different isotropic materials or differently oriented anisotropic materials bonded together. Based on this definition, a UD composite of eight identical layers is not a layered material because its (orthotropic) layers are not oriented in different directions.

1.5.2 Crack, Delamination, and Disbonding A crack inside a beam is defined as an “ideal” discontinuity that is straight and through-the-width. The straight plane defined by the presence of the crack is referred to as the crack plane. If the crack lies in the interface between two different layers of a layered material, then it is called an interfacial crack. In this case, which is the basic case in the thesis, the terms “crack plane” and “interfacial plane” coincide. An interfacial crack may represent either delamination or interfacial disbonding (Fig. 1.2), depending on the specific problem at hand. Delamination is a crack between two laminae of a laminated composite material, while interfacial disbonding refers to a crack between two adherents of a bonded joint.

1.5.3 Definitions for Beams We assume a beam featuring a crack, as shown in Fig. 1.1. The crack splits the beam into two sub-beams that are called sublaminates. The following definitions for this beam are used in the thesis: • Generic beam: this is the general case. The beam may feature any peculiarity, such as material and thickness asymmetries, BEC, and RHTS.

1.5 Basic Concepts and Definitions

13

Table 1.1 Basic definitions for asymmetric beams with an interfacial crack in terms of elastic properties and thicknesses Definition

Conditions

Generic beam

–

Similar beam

a1 = a2 , b1 = b2 = 0, c1 = c2 , d1 = d2 , and h 1 = h 2

Dissimilar beam

If it is not similar

Symmetric beam

a1 = a2 , b1 = −b2 , c1 = c2 , d1 = d2 , h 1 = h 2 , and αN1 = αN2 = αM1 = αM2 = 0

Asymmetric beam

If it is not symmetric

Elastically uncoupled sublaminate

bi = 0

Elastically coupled sublaminate

bi = 0

• Similar beam: this is the simplest case. The two sublaminates are identical and constitute a homogeneous (or isotropic) beam. • Dissimilar beam: any beam that is not similar • Symmetric beam: every beam that obeys the so-called symmetry conditions5 • Asymmetric beam: any beam that is not symmetric. In simpler words, we first clarify that the term asymmetric is expressed with respect to the crack plane. Thus, an asymmetric beam is one where the two sublaminates feature an asymmetry with respect to the crack plane in terms of either material or thickness. The terms dissimilar beam and asymmetric beam, in the way they have been defined here, coincide. • Bilayer or bimaterial beam: a beam consisting of two material layers only, with the crack lying in their interface. We also assume that both layers constitute a homogeneous (or isotropic) beam. • Homogeneous beam: any beam characterized by a set of properties that are independent of position. A homogeneous beam does not feature BEC. • Homogenized beam: any layered beam that, despite having an arbitrary stacking sequence, after undergoing a homogenization process, it can be described by a set of properties as is typically done for homogeneous beams Table 1.1 expresses the above definitions through conditions for the elastic constants, hygrothermal coefficients, and thicknesses of the beam.

These conditions are the following: a1 = a2 , b1 = −b2 , c1 = c2 , d1 = d2 , h 1 = h 2 , and αNi = αMi = 0. The notation is explained in the Nomenclature section and is not repeated here.

5

14

1 Introduction

1.5.4 Data Reduction Scheme (DRS) For fracture toughness characterization, we normally define as DRS a closed-form analytical expression for the ERR(s).6 This equation considers the geometric and material properties of the specimen at hand, the applied loading and boundary conditions, and, in many tests, the crack length. We will now specify the concept of a closed-form expression. A mathematical expression is defined as such7 if it contains a finite number of only constants, variables, explicit functions (e.g., exponent, logarithm, trigonometric functions, inverse hyperbolic functions), and certain operations (i.e., –, +, ×, and ÷), but no, for example, limit, differentiation, or integration.8 Thus, closed-form expressions are considered a particular case of the class of analytical expressions, and they are exact, as opposed to numerical expressions, which are always approximate.

References 1. Anderson TL (1994) Fracture mechanics: fundamentals and applications, 2nd edn. CRC Press 2. Broek D (1982) Elementary engineering fracture mechanics. Springer. https://doi.org/10.1007/ 978-94-009-4333-9 3. Friedrich K (1989) Application of fracture mechanics to composite materials. Elsevier 4. Irwin GR (1958) Fracture. In: Flugge S (ed) Handbuch der physik, vol VI. Springer, pp 551–590 5. Irwin GR, Kies JA (1954) Critical energy rate analysis of fracture strength. Welding J Res Supplement 33:193–198 6. Krueger R (2004) Virtual crack closure technique: history, approach, and applications. Appl Mech Rev 57(2):109–143. https://doi.org/10.1115/1.1595677 7. Loutas T, Tsokanas P, Kostopoulos V, Nijhuis P, van den Brink WM (2021) Mode I fracture toughness of asymmetric metal-composite adhesive joints. Mater Today Proc 34(1):250–259. https://doi.org/10.1016/j.matpr.2020.03.075 8. Mindlin RD (1951) Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J Appl Mech 18(1):31–38. https://doi.org/10.1115/1.4010217 9. Reddy JN (1984) A simple higher-order theory for laminated composite plates. J Appl Mech 51(4):745–752. https://doi.org/10.1115/1.3167719 10. Reissner E (1944) On the theory of bending of elastic plates. J Math Phys 23(1–4):184–191. https://doi.org/10.1002/sapm1944231184 11. Reissner E (1945) The effect of transverse shear deformation on the bending of elastic plates. J Appl Mech 12(2):A69–A77. https://doi.org/10.1115/1.4009435 12. Rice JR (1968) A path independent integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech 35(2):379–386. https://doi.org/10.1115/1.3601206 6

The term “data reduction equation” may be more appropriate in the present thesis than the term “DRS.” This is because this thesis is interested in developing equations that could be used for data-reduction purposes. However, we chose to use the term “DRS,” as it prevails in the relevant literature. 7 This definition is the one we adopt in the present work, although different authors or sources may use alternative definitions. 8 From an engineering point of view, a wider range of equations, such as Fourier series (used, for instance, in the analysis of beams and shells), are considered to be closed-form expressions.

References

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13. Timoshenko SP (1955) Strength of materials. Volume 1: elementary theory and problems, 3rd edn. D. Van Norstrand 14. Tsokanas P, Loutas T (2019) Hygrothermal effect on the strain energy release rates and mode mixity of asymmetric delaminations in generally layered beams. Eng Fract Mech 214:390–409. https://doi.org/10.1016/j.engfracmech.2019.03.006 15. Tsokanas P, Loutas T (2022) Closed-form solution for interfacially cracked layered beams with bending–extension coupling and hygrothermal stresses. Eur J Mech A Solids 96:104658. https://doi.org/10.1016/j.euromechsol.2022.104658 16. Tsokanas P, Loutas T, Kotsinis G, Kostopoulos V, van den Brink WM, Martin de la Escalera F (2020) On the fracture toughness of metal-composite adhesive joints with bending–extension coupling and residual thermal stresses effect. Compos B Eng 185:107694. https://doi.org/10. 1016/j.compositesb.2019.107694 17. Tsokanas P, Loutas T, Kotsinis G, van den Brink WM, Nijhuis P (2021) Strain energy release rate and mode partitioning of moment-loaded elastically coupled laminated beams with hygrothermal stresses. Compos Struct 259:113237. https://doi.org/10.1016/j.compstruct. 2020.113237 18. Tsokanas P, Loutas T, Nijhuis P (2020) Interfacial fracture toughness assessment of a new titanium-CFRP adhesive joint: an experimental comparative study. Metals 10(5):699. https:// doi.org/10.3390/met10050699 19. Valvo PS (2020) A physically consistent virtual crack closure technique accounting for contact and interpenetration. Procedia Struct Integr 28:2350–2369. https://doi.org/10.1016/j.prostr. 2020.11.083

Chapter 2

The Effect of Residual Hygrothermal Stresses on the Energy Release Rate and Mode Mixity of Interfacial Cracks in Beams with Bending–Extension Coupling

Key Points The key points of this chapter are as follows: • Fracture mode partitioning for generally layered beams, including hygrothermal effects • Derivation of analytical expressions for the ERR and MM of various test configurations • Determination of the influence of RTS on the fracture toughness of FMLs • Analytical modeling of the contact problem, which is inherent to the ENF test

2.1 Introduction As stated in Chap. 1, generally layered materials and adhesive joints consisting of materials with different mechanical or physical properties are being used increasingly often in many high-performance structural applications in several industries (e.g., aerospace, automotive, wind energy). Typical examples are metal-to-composite adhesive joints [1] and FMLs [47]. The joining process of such materials usually involves co-curing or secondary bonding. Both methods involve heating at elevated temperatures, at least in high-performance applications. Manufacturing at high temperatures results in RTS along the interface(s). A similar effect can be seen in cases where humidity is exchanged between the structure and its environment during its service life, giving rise to hygroscopic stresses.1 1

An illustrative example of temperature and moisture exchange comes from the aerospace industry. During a typical flight, an aircraft is subjected to a wide spectrum of temperatures and relative humidities. While the aircraft is on the runway, temperatures can rise up to 60 °C with 100% relative humidity. During take-off, there is significant temperature and moisture release. While cruising at an

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 P. Tsokanas, Fracture Analysis of Layered Beams With an Elastically Coupled Behavior and Hygrothermal Stresses, Springer Theses, https://doi.org/10.1007/978-3-031-17621-0_2

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2 The Effect of Residual Hygrothermal Stresses on the Energy Release …

Defects such as delaminations and disbondings are critical to the structural integrity of laminated composite structures and adhesive joints, compromising the overall mechanical performance of the component. As a result, several experimental, analytical, and numerical studies have been devoted to these defects over the last few decades. However, this has been done mainly for simple cases, such as symmetric and balanced composites and similar adhesive joints. In contrast, literature concerning dissimilar materials is much more limited, while research has largely been devoted to the characterization and performance of FMLs. Moreover, delamination or interfacial disbonding phenomena in the presence of hygrothermal stresses need special attention, possibly calling for new DRSs.

2.1.1 State of the Art As discussed in the previous chapter, to assess delamination or interfacial disbonding failure, researchers have employed several approaches based on fracture mechanics. In these approaches, the ERR [9, 51] or stress intensity factor (SIF) [18, 36, 38] are the critical properties that are evaluated. In addition, for problems related to mixed-mode fractures in generally layered beams, extensive studies have focused on determining the total ERR and performing mode partitioning [8, 24, 33, 37, 45, 48–50]. The subsequent section focuses on theoretical methods investigating the effects of the RHTS on the ERR and mode partitioning.

2.1.1.1

Effect of Residual Hygrothermal Stresses (RHTS) on Fracture Toughness

RHTS are not considered in the works cited above and are ignored in most experimental delamination or interfacial disbonding studies in the relevant literature. Nevertheless, these stresses typically exist in multilayered structures due to the mismatch between the coefficients of thermal expansion (CTEs) and hygroscopic expansion of different layers. Thus, it could be crucial to incorporate the effect of RHTS when evaluating the fracture toughness of generally layered specimens subjected to temperature or moisture changes from the manufacturing temperature or moisture. For instance, even laminates manufactured at room temperature can experience thermal stresses when the service temperatures are very low (e.g., at cruising altitude in aircraft or in cryogenic applications). Temperature changes induce bending moments and deformations at the interfaces of fracture beam specimens with dissimilar adherents. Consequently, a non-zero ERR

altitude of some 37,000 ft, the temperature is usually –55 °C, and the relative humidity is normally lower than 30%. Conversely, during landing and parking, the temperature increases again, and moisture uptake is also observed.

2.1 Introduction

19

exists in the coupon without the application of any mechanical loads. Some of the most relevant studies that have addressed this problem will now be reviewed. First, Jiao et al. [20] described the effect of RTS on the evaluation of the interfacial fracture toughness of asymmetric DCB specimens. Although their study incorporated the thermal misfit between different adherents, it ignored the vital thermal misfit between the adhesive and the adherents. A general method to calculate the global ERR accounting for RTS was also presented in Nairn’s [25] pioneering work. Nairn [26] and Guo et al. [17] incorporated the effect of an adhesive layer in evaluating the adhesive fracture toughness of a DCB specimen. They formulated the ERR, which includes RTS. In addition, ENF and MMB tests were performed to investigate the effects of load conditions on interfacial fracture toughness. However, there is a specific limitation in Nairn’s [27] expressions, as the transverse shear effect is not considered. Nairn [27] provided a correction method to account for the transverse shear by modifying the crack length. Furthermore, this method calculated the ERR in heterogeneous laminates with different thermal expansion properties. Nairn’s method was also used by Yokozeki et al. [54] to calculate the total ERR in bimaterial DCB, ENF, and MMB specimens. The semi-rigid interface joint model, originally proposed by Wang and Qiao [50], was used to calculate the mode I and II ERRs in laminates with RTS [52] and for the fracture analysis of sandwich structures by Yokozeki [53] and Yokozeki et al. [55]. In recent years, little work has been done in this area. In terms of RTS, Zhang and Wang [56] undertook a fracture analysis using the principle of superposition and the three interface joint models mentioned above (rigid, semi-rigid, and flexible). They developed analytical expressions for interface delamination in layered structures considering both transverse shear effects and residual stresses. Meanwhile, Qiao and Liu [30] calculated the total ERR of bimaterial fracture beam specimens while considering the effect of the RHTS. They used the three interface joint models and showed that the flexible joint model produced the best predictions for all coupons. Lastly, the recent theoretical work by Toftegaard and Sørensen [40] increases our understanding of how the effect of RTS could be introduced into analytical expressions for fracture toughness.

2.1.2 Present Work With the aid of Irwin’s [19] crack closure integral method, new equations for mode I, mode II, and total ERRs are derived here for an asymmetric interfacial crack between two generally layered sublaminates. Both sublaminates are assumed to feature BEC and to be stressed by RHTS. In addition, the interface between the two sublaminates is supposed to be semi-rigid (rotationally flexible). The analytical framework proposed in the present work enriches the analytical model proposed by Yokozeki [52], who solved a similar problem without considering the effect of BEC. In addition, 2D FEA is used to validate the proposed analytical formulation for a typical case of an FML.

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2 The Effect of Residual Hygrothermal Stresses on the Energy Release …

The critical contributions of the analytical framework presented in this chapter are outlined below: • New analytical expressions are provided for the mode I and mode II contributions to the ERR of an interfacial crack between two generally layered sublaminates. The effects of shear deformations, crack-root rotations, BEC, and RHTS are considered. Mode partitioning is achieved using a global partitioning approach. • The constitutive equations and displacement continuity conditions for a semirigid joint model are revisited. The governing differential equation is solved from the beginning, leading to new generalized expressions that could treat generally layered sublaminates with BEC and RHTS. • The expressions of GI and GII are obtained for a general, arbitrarily loaded configuration and are then reduced for some of the most typical test configurations of paramount interest in experimental fracture mechanics (MMB, DCB, ENF, and SLB configurations). • The effects of RTS on the total ERR and MM are quantified in detail and discussed through an example of a glass laminate aluminum reinforced epoxy (GLARE) with an asymmetric crack that is tested with the DCB and ENF configurations. • The contact problem during an asymmetric ENF test is investigated. The effect of contact forces between the sublaminates, which is crucial, is analytically modeled where possible, extending Nairn’s [27] analytical framework for Timoshenko beams. Limitations are highlighted and discussed. The rest of the chapter is organized as follows. Section 2.2 describes the scientific problem at hand and introduces the basic features of the analytical model proposed. Next, in Sect. 2.3, the mathematical formulation of the problem is presented. This formulation consists of four sets of equations: the kinematic assumptions, the constitutive laws, the conditions of static equilibrium, and the conditions of displacement continuity. Following this, in Sect. 2.4, the solution to the mathematical problem is presented. This section starts by deriving the governing differential equation of the problem and continues by solving this equation. Next, the analytical expressions for the internal forces and moments and the crack-tip forces are obtained. The section ends by providing the expressions for the total ERR and its mode I and mode II contributions. In Sect. 2.5, the general expressions obtained for the ERRs are reduced for four conventional test configurations: the MMB, DCB, ENF, and SLB tests. This section also deals with the contact problem appearing between the two sublaminates and refines the proposed analytical expressions to align with previous expressions in the literature. In Sect. 2.6, the FEA methods used to validate the proposed analytical expressions are presented. Next, Sect. 2.7 offers the total ERR and mode-mixity ratio (MMR) as the applied load increases in the DCB and ENF test configurations, both with and without considering the effect of the RTS. Section 2.8 discusses particular points of interest, and Sect. 2.9 presents the conclusions drawn from the work in this chapter. The present chapter is an extended version of our journal paper in [41]. Compared to this paper, the chapter provides more information on the underlying theory,

2.2 Problem Description and Analysis Approach

21

more details on the analytical formulation, and additional suggestions for future research.

2.2 Problem Description and Analysis Approach This section introduces the proposed analytical modeling. Firstly, Sect. 2.2.1 describes the specific problem under investigation. Next, Sect. 2.2.2 presents the analytical model by highlighting its central features and underlying assumptions.

2.2.1 Definition of the Scientific Problem Figure 2.1 presents a schematic of the general problem addressed in the present chapter. As shown in the figure, we consider a beam-type structure with a rectangular cross-section with a width b, total thickness h, and a generic stacking sequence (i.e., nonsymmetric and unbalanced).2 Before applying external loads, the beam features a straight, through-the-width crack that is arbitrarily located along the thickness of the beam. In addition, RHTS may be developed3 while the beam is loaded with concentrated mechanical loads (i.e., forces and moments). We initially make the significant assumption that the geometry, physical and elastic properties, applied loads, and restraints of our problem are such that the beam can be modeled as a 2D planar beam. Following this assumption, the beam can be made of several layers of different materials and thicknesses (or, of course, of fiber-reinforced layers with different fiber orientations). Furthermore, the present model allows arbitrary stacking sequences—including those with BEC, provided that the beam and the two sub-beams resulting from the presence of the crack are not affected by out-of-plane effects, such as out-of-plane shear and torsion.4

2.2.2 The Proposed Analytical Model Under the assumptions presented above, this section turns to the corresponding 2D5 problem under investigation, which is depicted schematically in Fig. 2.2. 2

Typical examples of such structures include FMLs, dissimilar adhesive joints, and, of course, composite laminates consisting of layers of different materials or different fiber orientations. 3 Usually, such stresses are induced after a manufacturing process at a high temperature or in cases where the difference between the service temperature and manufacturing temperature is high. 4 For example, many symmetric and nonsymmetric cross-ply and angle-ply composite laminates can be analyzed by our model. 5 Common analytical expressions and DRSs (see, for example, those cited in Sect. 2.1.1) are based on 2D theories, which assume a straight crack front and a uniform ERR distribution along the width

22

concentrated mechanical loads

2 The Effect of Residual Hygrothermal Stresses on the Energy Release …

crack: delamination or interfacial disbonding crack front

beam element

support

generic stacking sequence

Fig. 2.1 A generally layered beam with an asymmetric crack loaded by arbitrary forces and moments. RHTS may also be developed

As shown in the figure, the model features a cantilever beam (with its right-hand end clamped and its left-hand end free) made of an arbitrarily layered material. This material is obtained by laminating a sequence of different layers of materials that, in the present work, are assumed to be linearly elastic and orthotropic. An interfacial crack with a length equal to a exists at the left-hand end of the beam. It is essential to highlight that the term crack can describe either delamination or interfacial disbonding, depending on whether the beam structure under consideration is, for example, a (multilayered) composite or an adhesive joint between two similar or dissimilar materials, respectively. As shown in Fig. 2.2, the crack splits the beam into two, hereafter referred to as sublaminate6 1 (the upper sublaminate) and sublaminate 2 (the lower sublaminate).7 These two sublaminates have rectangular cross-sections and can be arbitrarily layered as the total beam. In this way, both sublaminates (can) feature BEC. We also assume that the two sublaminates may have dissimilar thicknesses, as well as the layers that make up each sublaminate.

of the beam. Even though this assumption is logical for b