Crack initiation in composites at micro and meso scales. Development and applications of finite fracture mechanics

Universidad de Sevilla Grupo de Elasticidad y Resistencia de Materiales DOCTORATE IN ENGINEERING Crack initiation in composites at micro and meso scales Development and applications of finite fracture mechanics Doctoral Dissertation of: Israel Garc´ıa Garc´ıa Advisor: Prof. Vladislav Mantiˇc 2014 Acknowledgements Doctoral study is essentially a time to learn the principles and methods for conducting research. The process of learning implies generally the collaboration and interaction with other people. Thus, to be fair, this thesis as well as the whole learning can be considered the result of the work of all these people who taught, assisted or helped me, whether consciously or not, during this time. The following lines are dedicated to recognize their contribution. First, I would like to thank my doctoral advisor Dr. Vladislav Mantiˇc for his continuous support, guidance and endless motivation. I would like also to acknowledge his efforts and time inversed in inspiring discussions and ideas which are the source of the most relevant results reflected in this thesis. In addition, I am greatly indebted by his meticoulous care in reviewing the manuscripts of the articles associated to this thesis. I also wish to extend my thanks to all my colleagues of the Group of Elasticity and Strength of Materials (GERM) for providing an inspiring working and confortable atmosphere which helped me to develop the work reflected here. A special mention is deserved by Dr. Federico Par´ıs, head of the GERM, for promoting this environment. I would like also to thank him along with Dr. Antonio Bl´azquez for the interesting discussions and suggestions about the problem of transverse cracking in cross-ply laminates. During these years, I have also had the opportunity of taking advantage of the experience and computational codes generated by the previous thesis developed in the group. Thus, I would like to express my sincere gratitude to Dr. Enrique Graciani and Antonio Bl´azquez for their BEM codes and Dr. Luis T´avara for his parallel implementation of the Graciani’s BEM code. Thanks are also due to Dr. Elena Correa for her support with the problem of fiber-matrix debonding and to Dr. Alberto Barroso for assist me with his experience in multimaterial corners. I would like to extend my sincere appreciation to Antonio Ca˜nas, Jos´e Ram´on Rodr´ıguez, Isabel del Pr´ıncipe and Miguel Mu˜noz who help and assist me in the laboratory. I am particularly indebted to Jes´us Justo and Anna¨elle Simon for their collaboration to carry out the experiments to evaluate the size effect in cross-ply laminates. Next, I would like to thank to my colleagues in the L3 office for help me in those small things that, in spite of being not visible here, have been essential for the adequate development of my doctoral studies. This gratitude is extendable to my first office mates Luis T´avara, Jos´e Reinoso, Daniane Vicentini, Afonso i ii Leite, Christos Panagiotopoulous and Jes´us Justo on the one hand and the new ones on the other hand: Rafael Le´on, Francisco Moreno, Daniel Garc´ıa and Juan Manuel Gonz´alez. During these years I had the honor of collaborating with Dr. Dominique Leguillon from the Universit´e Pierre et Marie Curie (Paris 6). This collaboration was particularly intense during my stay in Paris of 3 months but also during the visits of Dr. Leguillon to Seville. I would like to express my gratitude to him for the interesting discussions and his insightful comments. A entire chapter of this thesis is a result of this collaboration. In addition, I had the opportunity of collaborating with Dr. Anthony R. Ingraffea from the Cornell Fracture Group (CFG) at Cornell University during my stay of 4 months in Ithaca, NY. I appreciated his kindness and his interesting suggestions. I would like also to express my gratitude to all my colleagues within the (CFG), in particular to Dr. Bruce J. Carter who assisted me with the use of the software FRANC3D NG. I had also the privilege of performing a joint research with Dr. Marco Paggi from the Politecnico di Torino (now at IMT Lucca). I am greatly indebted for performing the FEM modeling of the fiber-matrix debonding and provide me with results to compare the models developed here. His motivation and comments during our joint research is acknowledged. I would like to thank the european tax-paying public as this thesis was supported mainly by the Spanish Ministry of Education by a “Beca de Formaci´on de Profesorado Universitario” (F.P.U. 2009/3968) and the European Social Fund. The partial support by the Spanish Ministry of Science and Innovation (Project MAT2009-14022), the Spanish Ministry of Economy and Competitiveness (Project MAT2012-37387), and the Government of the Autonomous Community of Andalusia (Project TEP-04051) is also acknowledged. I want to thank to my friends, cousins and uncles for their warm friendship, care and enthousiasm during this years. A special thanks to Pablo C´ardenas for sharing so many moments which are indispensable in spite of not being reflected directly in this thesis. I would like particularly to thank to Dr. Miguel Fosas for our discussions about the sense of the doctoral studies, the analyses about the research and the academia and for being a permanent support since I met him. My sincere thanks to my parents Tom´as and Mar´ıa del Carmen and my sister Carla for their patience, care, support and encouragement. In particular, I would like to express my gratitude to my parents for educating me on the passion for knowledge and limitless curiosity. This thesis is undoubtedly a consequence of this education. Finally, my very special thanks to Miriam for sharing my moments of frustration and joyful laughter and for taking care of me more than I myself. Contents Contents iii 1 Introduction 1 1.1 Compositematerials ......................... 1 1.2 Failure in composites . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Objectives............................... 7 1.4 Outlineofthethesis ......................... 9 2 From damage to fracture: crack initiation 11 2.1 Crack initiation from the classical fracture mechanics perspective 11 2.2 Do cracks initiate or exist before loading? . . . . . . . . . . . . . 15 2.3 Predicting crack initiation . . . . . . . . . . . . . . . . . . . . . . 19 2.3.1 Damage mechanics . . . . . . . . . . . . . . . . . . . . . . 19 2.3.2 Atomistic modeling . . . . . . . . . . . . . . . . . . . . . . 19 2.3.3 Cohesive zone models . . . . . . . . . . . . . . . . . . . . 20 2.3.4 Finite fracture mechanics . . . . . . . . . . . . . . . . . . 25 2.3.4.1 Fixed-critical-length criteria . . . . . . . . . . . 26 Stress-based criteria . . . . . . . . . . . . . . . . . 27 Pointwise stress criterion . . . . . . . . . . . . 27 Averaged stress criterion . . . . . . . . . . . . 28 Energy-based criteria . . . . . . . . . . . . . . . . . 29 Incremental energy criterion . . . . . . . . . . 29 Inherent flaw model . . . . . . . . . . . . . . . 29 2.3.4.2 Variable-critical-length criteria . . . . . . . . . . 30 JScriterion ...................... 30 Coupled stress and energy criterion . . . . . . . . . 30 3 The coupled criterion of the finite fracture mechanics 33 3.1 Background.............................. 34 3.2 Leguillon’s hypothesis . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3 Stresscriterion ............................ 40 3.4 Energycriterion ........................... 46 3.4.1 Released energy . . . . . . . . . . . . . . . . . . . . . . . . 48 iii iv Contents 3.4.1.1 Method based on integrating the energy release rate......................... 49 3.4.1.2 Method based on the external-work increment . 50 3.4.1.3 Incremental crack closure technique . . . . . . . 51 3.4.2 Dissipated energy . . . . . . . . . . . . . . . . . . . . . . . 53 3.4.3 General form of the energy criterion . . . . . . . . . . . . 57 3.5 Coupledcriterion........................... 58 3.5.1 General form of the coupled criterion . . . . . . . . . . . . 58 3.5.2 Failure behavior predicted by the coupled criterion . . . . 60 4 Failure initiation in long-fiber reinfonced composites under transverse loads 63 4.1 Symmetry on the crack initiation . . . . . . . . . . . . . . . . . . 65 4.1.1 Stress criterion . . . . . . . . . . . . . . . . . . . . . . . . 67 4.1.2 Energy criterion . . . . . . . . . . . . . . . . . . . . . . . 68 4.1.2.1 Released energy . . . . . . . . . . . . . . . . . . 69 4.1.2.2 Dissipated energy . . . . . . . . . . . . . . . . . 71 4.1.2.3 Final expression of the energy criterion . . . . . 73 4.1.3 Coupled criterion . . . . . . . . . . . . . . . . . . . . . . . 74 4.1.4 Analysis and discussion of results . . . . . . . . . . . . . . 77 4.1.4.1 Difference in critical loads for symmetric and asymmetric post-failure configurations . . . . . . . . . 77 4.1.4.2 Stress-strain curves predicted by the coupled stress and energy criterion of the FFM . . . . . . . . . 81 4.1.4.3 Discussion of previous experimental results . . . 84 4.2 Comparison of the size effect predicted by the coupled criterion with the predictions of a cohesive zone model . . . . . . . . . . . 85 4.2.1 CZM: formulation and predictions . . . . . . . . . . . . . 86 4.2.2 FFM coupled criterion: formulation and predictions . . . 92 4.2.3 Fiber-size effect predictions by CZM and FFM . . . . . . 96 4.2.3.1 Asymptotic behavior for CZM and its physical interpretation ................... 96 4.2.3.2 Asymptotic behavior for FFM and its physical interpretation . . . . . . . . . . . . . . . . . . . 101 4.2.4 Limits of validity of CZM and FFM methods . . . . . . . 103 4.3 Crack initiation under biaxial transverse loads . . . . . . . . . . . 104 4.3.1 Stresses in a single inclusion under a remote biaxial transverseload........................... 105 4.3.2 The solution for a crack at the interface of a single inclusion under a remote biaxial transverse load . . . . . . . . . . . 108 4.3.3 Interface crack onset at a single inclusion under a remote biaxial transverse load . . . . . . . . . . . . . . . . . . . . 112 4.3.3.1 Stress criterion . . . . . . . . . . . . . . . . . . . 112 4.3.3.2 Incremental energy criterion . . . . . . . . . . . 113 4.3.3.3 Coupled stress and energy criterion . . . . . . . 117 4.3.3.4 Post-crack-onset evolution . . . . . . . . . . . . 125 Contents v 4.3.3.5 Applicability of the open model of interface cracks ........................... 125 4.3.4 Size effect of the inclusion radius aon the crack onset and its variations with the load biaxiality . . . . . . . . . . . . 126 4.3.5 Experimental procedure for the measurement of the brittleness number γ, interface tensile strength σcand fracture toughness G1c ......................... 129 4.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 131 Appendices to Chapter 4 135 4.A Dimensional analysis of the energy release rate of a debond at the fiber-matrix interface . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.B Domain of existence of the angle θ0where the normal traction vanishes................................ 136 4.C Functions χand pof the expression of Toya for stresses along the interface with a partial debond . . . . . . . . . . . . . . . . . . . 137 4.D Expression of the dimensionless ERR ˆ G.............. 137 4.E Asymptotic analysis of ˆ Gand Gfor a vanishing critical semiangle ∆θc.................................. 137 4.F Partitioning of ˆ Ginto two components ˆ GIand ˆ GII ........ 138 4.G Expressions for computational implementation . . . . . . . . . . 139 4.H Proof of inequality dG dθd≤dGc dθdfor θd= ∆θE c= ∆θE min ....... 139 5 Failure initiation at spherical-particle reinforced composites 141 5.1 A theoretical model based on the finite fracture mechanics . . . . 142 5.1.1 Stress criterion . . . . . . . . . . . . . . . . . . . . . . . . 144 5.1.2 Energy criterion . . . . . . . . . . . . . . . . . . . . . . . 147 5.1.3 Coupled criterion . . . . . . . . . . . . . . . . . . . . . . . 153 5.2 Results and comparison with experiments . . . . . . . . . . . . . 154 5.3 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . 158 6 Transverse cracking in cross-ply laminates 159 6.1 2D generalized-plane-strain model of transverse crack initiation . 162 6.1.1 Stress criterion . . . . . . . . . . . . . . . . . . . . . . . . 162 6.1.2 Energy criterion . . . . . . . . . . . . . . . . . . . . . . . 165 6.1.3 Coupled stress and energy criterion . . . . . . . . . . . . . 170 6.1.4 Post crack-onset evolution . . . . . . . . . . . . . . . . . . 175 6.1.5 Effect of the laminate geometry . . . . . . . . . . . . . . . 176 6.1.5.1 Influence of the inner-ply thickness for a fixed t0/t90 -sizeeffect ................. 176 6.1.5.2 Influence of the geometric parameter t0/t90 . . . 178 6.1.6 Comparison with experimental evidences from the literature179 6.1.6.1 Experimental evidences for the size effect in the critical strain . . . . . . . . . . . . . . . . . . . . 180 6.1.6.2 Experimental evidences for the arrest length of cracks........................ 183 vi Contents 6.1.7 An experimental procedure for the measurement of the transverse fracture toughness . . . . . . . . . . . . . . . . 184 6.2 3D model of transverse crack initiation . . . . . . . . . . . . . . . 185 6.2.1 Crack geometry and locations studied . . . . . . . . . . . 186 6.2.2 Stresses at the inner ply before crack initiation according toa3Danalysis ....................... 188 6.2.3 Stress criterion for the 3D model . . . . . . . . . . . . . . 191 6.2.4 Energy released at the onset of a long-shallow crack . . . 193 6.2.5 Energy criterion for the 3D model . . . . . . . . . . . . . 203 6.2.6 Coupled criterion . . . . . . . . . . . . . . . . . . . . . . . 205 6.2.7 Results and discussion . . . . . . . . . . . . . . . . . . . . 206 6.3 2D generalized-plane-strain model considering the influence of residualthermalstresses.......................... 210 6.3.1 Stress criterion considering residual thermal stresses . . . 210 6.3.2 Energy criterion considering residual thermal stresses . . 213 6.3.3 Coupled criterion for the thermomechanical finite fracture mechanicsmodel....................... 217 6.4 Experimental validation through a new set of experiments with geometrically similar laminates . . . . . . . . . . . . . . . . . . . 220 6.4.1 Specimen fabrication . . . . . . . . . . . . . . . . . . . . . 221 6.4.2 Testing procedure . . . . . . . . . . . . . . . . . . . . . . 226 6.4.3 Results and discussion . . . . . . . . . . . . . . . . . . . . 232 6.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . 237 Appendices to Chapter 6 239 6.A An expression for E22/˜ E22 ...................... 239 6.B An expression for ε0,∆T yy ....................... 240 6.C An expression for k(90),∆T...................... 241 6.D Dimensional analysis of the energy release rate assuming generalizedplanestrain ........................... 243 6.E Proof of that the function ghas a minimum strictly between the location of the maximum of ˆ Gand the interface . . . . . . . . . . 244 6.F Proof of that ∆ˆamin,g is a lower bound for the arrest length . . . 246 7 Crack initiation at a weak surface in a V-notch 247 7.1 The singular stress field at a v-notch . . . . . . . . . . . . . . . . 249 7.2 Stresscriterion ............................ 252 7.3 Energycriterion ........................... 255 7.3.1 Released energy . . . . . . . . . . . . . . . . . . . . . . . . 256 7.3.2 Dissipated energy . . . . . . . . . . . . . . . . . . . . . . . 256 7.3.3 Incremental energy criterion. Final expression . . . . . . . 258 7.4 Coupledcriterion........................... 258 7.5 Analysis of a weak surface at α= 0◦in a V-notch . . . . . . . . . 260 7.5.1 Crack initiation at the weak surface . . . . . . . . . . . . 261 7.5.2 Crack initiation in the bulk at the optimum angle . . . . 262 7.6 Predictions of the critical load at failure in an Arcan test . . . . 263 Contents vii 7.6.1 Crack initiation at the weak surface . . . . . . . . . . . . 265 7.6.2 Crack initiation in the bulk at the optimum angle . . . . 266 7.6.3 Global prediction of failure load . . . . . . . . . . . . . . 268 7.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . 269 8 Conclusions and future developments 273 8.1 Conluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 273 8.2 Future developments . . . . . . . . . . . . . . . . . . . . . . . . . 277 Bibliography 283 61. Introduction have motivated a huge effort on studying the failure in composites during the last decades. In spite of having motivated an intense research activity during decades, failure in composites has not been characterized enough yet. The reason is mainly the high complexity of the mechanisms governing the global failure. These mechanisms are associated to the microstructure at different scales and even to the interaction between them. This is a direct consequence of the defining feature of composites: the presence of several constituent materials which are combined on different scales. As a consequence, the presence of interfaces between dissimilar materials and a high variability in the elastic, strength and fracture properties corresponding to the different constituent materials can be found at different scales. This affects strongly the failure mechanisms at these scales inducing a high complexity to these mechanisms seen from a macro scale perspective. These failure mechanisms are particularly complex in laminates composed by plies with different orientations, which are intensely employed in the majority of applications described in previous section. A large number of failure criteria based on very different assumptions have been proposed for these laminates. The coexistence of several failure criteria has motivated the organization of world wide exercises to evaluate the accuracy of the predictions given by the criteria proposed by comparing with a set of experiments, see Soden et al. (1998); Hinton et al. (2002); Kaddour and Hinton (2013). Results showed a great margin for accuracy improvement. In spite of the large number of failure criteria proposed, according to the estimation by Sun et al. (1996), the most used criteria in the 1990s were the “maximum strain” and the “maximum stress” criteria, being used by about 52% of composite designers. This estimation also shows a great diversity in the failure criteria employed. The maximum stress or strain criteria are based on assuming a set of critical values for the stress or the strain, respectively, corresponding to certain relevant orientations. The failure is predicted when the stress or the strain along one of these directions reaches its corresponding critical value. Note that the most used criteria are based on the simple assumption that a finite number of different failure mechanisms exists, which are activated exclusively by a certain level of either strain or stress along a certain direction. Therefore, they neglect the existence of other failure mechanisms and even the interaction between the mechanisms considered. On the contrary, the majority of the other failure mechanisms predict an interaction between the failure mechanisms. Thus, the majority use of the simplest one evidences a low level of trust of the designer community in the prediction given by the rest of failure criteria, in particular, in the form that the failure mechanisms are predicted to interact. Previous conclusions are based on estimations published in 1996 which could have changed given the fast and continuous evolution of the discipline. According to Par´ıs (2001), a comprehensive understanding of the physical basis which governs the failure in composites is necessary in order to improve and generate new physical-based failure criteria. These criteria have to take into account the inherent complexity of the micro and meso structure of these types of material systems. This has motivated the study of the mechanisms of crack 1.3. Objectives 7 and damage propagation at these scales. An enormous progress has been made. In fact, a lot of results can be found in the literature characterizing the crack propagation at different scales. A revision of the previous works dealing with the problems studied in this thesis can be found in the corresponding chapters. In contrast with the number of works dealing with the crack propagation at these scales, the crack initiation at micro and meso scales is still an open matter, which has been relatively under-developed. The main reason for this is the classical inability of the linear elastic fracture mechanics to predict the initiation of a crack in an intact material or interface. Some works have been developed in order to study the initiation by computational methods as cohesive zone models requiring large computational resources. Although crack initiation is a controversial concept as discussed in Chapter 2, adequately interpreted it can be considered as the first step in a wide variety of failure mechanisms. Therefore, its prediction is very relevant for the full characterization of a failure mechanism. This thesis focuses on studying the crack initiation as the first step of several failure mechanisms. This study is based on the application of the coupled stress and energy criteria, proposed by Leguillon (2002) in the framework of the finite fracture mechanics introduced by Hashin (1996), see Taylor (2007) or Chapter 3 of this thesis for a review. This criterion often enables the prediction of crack initiation by semianalytical expressions, which is very useful to study the influence of the different parameters governing the problem or to be employed as part of a future global failure criterion. 1.3 Objectives This thesis is part of a number of works carried out within the “Grupo de Elasticidad y Resistencia de Materiales” (GERM) with the objective of understanding the failure mechanisms in composites from the physical point of view. The long term objective is the development of a failure criteria established on a well understood physical basis. Previous works focused on studying the first steps of the main failure mechanisms by means of micro and meso scale models. However, as discussed previously, in spite of that crack initiation is considered the first step of some of these failure mechanisms, it is not still fully understood. In view of these previous studies the objectives of this thesis can be divided into two groups: on the one hand, a set of general objectives, and on the other hand, a set of objectives wich are common to each failure mechanism studied with their application being particularized for each one. The general objectives are: •Contribute to the development of the coupled stress and energy criterion of the finite fracture mechanics since it is the main tool employed for the analysis of the crack initiation in this thesis. The particular characteristics of the problems for which this criterion is applied here require a further development of this criterion. In this sense, the contribution of this thesis is twofold. First, the original theoretical formulation is generalized in order to take into account certain phenomena, which are relevant for the 81. Introduction problems studied in this thesis but were neglected in the original formulation. Second, a set of methodologies are developed with the purpose of enhancing the application of this criterion. •Evidence that the coupled stress and energy criterion is suitable for the prediction of crack initiation at several scales in composites in the context of the definition of a new failure criterion for composites in the terms discussed previously. Thus, it is essential to show that the coupled criterion is able to obtain analytical or semianalaytical expressions for the prediction of crack initiation, these expressions being simple enough to be implemented in a global analysis. The simplicity of these expressions is also required for the physical interpretation of the results. Even more important, it is necessary to verify that the predictions given by the coupled criterion agrees with experiments or other accepted techniques. •Generate new knowledge about the nature of the first step in the failure mechanisms studied. The results have to enable the ability of predicting the conditions under which crack initiation occurs, i.e. the critical loads leading to it along with the characteristics of the crack immediately after the initiation. In addition it is necessary to connect it with the previous works in order to integrate the new knowledge. The particular objectives, enumerated in what follows, are actually a detailed description of the last objective of generating new knowledge about the failure mechanisms. They are discussed here because although they are particularized for each failure mechanism, their formulation is common in several of the problems studied in this thesis, •Obtain theoretical models with analytical or semianalytical expressions to characterize the crack initiation as a function of the elastic, strength, fracture and geometric properties. It is particularly relevant that these properties correspond to well-known physical properties which can be obtained by well established experiments. As described in Chapter 3, the characterization of the crack initiation from the perspective of the coupled criterion consists in obtaining the critical load for the initiation on the one hand and the crack geometry immediately after its initiation. •Study the variation of the previous results with the main parameters of the problem. In addition, given the large number of parameters, it is necessary to evaluate the relevance of different parameters for this variation with the aim of identifying the main parameters governing the problem •Propose indirect experimental procedures to obtain the fracture and strength properties based on the theoretical models developed. Some fracture and strength properties governing the problems studied are difficult to be obtained experimentally although they are physically founded. The difficulty is originated by the small scales of the necessary tests. 1.4. Outline of the thesis 9 1.4 Outline of the thesis This thesis is structured in 8 chapters where the previous objectives are developed. Excluding this first chapter of introduction and the last one describing the conclusions and future developments, the thesis outline can be divided into two parts: the first part studies the phenomenon of crack initiation and the criterion employed here for its prediction, the application of this criterion to the different failure mechanisms and scales studied is described in the second part. The general objectives described previously are not equivalently developed in the two parts. The first objective is mainly developed in the first part, whereas the third objective is worked in the second part. The second objective is present in both parts. The first part is composed by Chapters 2 and 3. In Chapter 2, the controversial concept of crack initiation is discussed and an interpretation in the context of this thesis is provided. In addition, this chapter describes briefly the main tools employed to predict crack initiation. In Chapter 3, the coupled criterion of the finite fracture mechanics is discussed in the context developed in the previous chapter. Both the original formulation and the new contributions are described from a practical perspective leading to a global generalized formulation. The different techniques associated to the application of the coupled criterion are also detailed including some originally proposed in this thesis. The second part is composed by Chapters 4 to 7 and deals with the application of the coupled criterion to the different problems studied. The chapters are sorted by the scale of the problem studied, going from the smallest to the largest one. In Chapter 4, the problem of crack initiation at the fiber-matrix interface when the composite material is subjected to transverse load is analyzed. Firstly, the preference of the different forms of the crack initiation predicted by other methods in the literature is analyzed from the perspective of the coupled criterion. Second, the results predicted for the case of uniaxial transverse loading are compared with the results given by a well-accepted method as the cohesive zone model. Finally the influence of a secondary transverse load is analyzed focusing on how this influence varies with the value of the main parameters. In Chapter 5, a similar problem is studied: the crack initiation at the interface between a spherical inclusion and the surrounding matrix when the material is subjected to tension. The results found are compared with preliminary experiments found in the literature. Chapter 6 deals with the problem of crack initiation in the off-axis ply in cross-ply laminates. This is a classical problem and a large number of particular criteria have been proposed to interpret some phenomena typically found in experiments. The objective here is the interpretation from the perspective of a general criterion as the coupled criterion. First, a simple but reasonable 2D model of the problem is proposed, which is later generalized to take into account 3D phenomena, on the one hand and thermal effects on the other hand. Finally, some experiments with cross-ply laminates carried out in the context of the thesis are described and the results obtained are compared with the prediction of the theoretical model. In Chapter 7, the crack initiation at a reentrant corner in the presence of an interface is analyzed. The theoretical model enables an 10 1. Introduction analysis of the variation of the failure typology as a function of different problem parameters. CHAPTER 2 From damage to fracture: crack initiation Crack initiation under static loads has been a controversial topic for a long time in spite of being, a priori, the first step of some failure mechanisms. The reason can be found in the combination of historical causes and the difficulties inherent to the physical characterization of the process. This chapter is devoted to introduce how the concept of crack initiation is understood and predicted in the context of this thesis. First, some historical notes on the fracture mechanics will be reviewed in Section 2.1 to present the main problems that the study of crack initiation have from the point of view of the classical fracture mechanics. Subsequently, the two main problems of the study of crack initiation will be discussed in Sections 2.2 and 2.3, describing how they have been solved. 2.1 Crack initiation from the classical fracture mechanics perspective The conditions under which a crack in a body propagates have been studied for near a century leading to a new field: the fracture mechanics. Traditionally, the criterion proposed by Griffith (1921), which is widely used yet, is considered the first1milestone of this field. This criterion is motivated by the works carried out by Griffith and his colleagues about the lifetime of aircraft elements at the Royal Aircraft Factory in the context of the World War I. They observed that lifetime of rods at the aircraft engines was strongly increased when their external surfaces were polished. An attempt to explain this observation was made by the stress 1Currently it is known, thanks to Rossmanith (1995), that, some years before, Wieghardt (1907) proposed a theory for crack propagation in terms of the stress intensity factor equivalent to that proposed by Griffith (1921). The reference to Griffith’s works is kept here by their historical relevance and the importance of the Griffith’s energy perspective in the context of this thesis 11 12 2. From damage to fracture: crack initiation analysis of scratches in the rod surfaces, modeling them as elliptical flaws. Using the maximum stress criterion of fracture, widely used by his contemporaries, this analysis led to the conclusion of that the maximum stress expected was almost independent of the depth of scratches beyond a certain depth. However, this result was contrary to the facts observed by the engineers when rods were polished. The result of this first analysis and his subsequent experiments with glass fibers containing artificial flaws motivated Griffith to reject the maximum stress criterion and to propose his new criterion. The idea behind the Griffith criterion is simple and revolutionary: a certain amount of energy is necessary to break the bonds joining the atoms or molecules, and this energy comes from the potential elastic energy released in the elastic body and from the work done by the external forces due to the propagation of the crack. Thus, a crack is expected to grow when the energy released by an infinitesimal increase of the crack length reaches the energy necessary to break the inter-atomic or inter-molecular bonds at an infinitesimal distance of the crack tip2. In addition, Griffith assumed that the energy necessary to open a new surface is proportional to the amount of bonds to break, hence proportional to the area of the new surface. The energy necessary to generate a new surface per unit area is known as “fracture energy” γf, whereas the energy necessary to open a new crack per unit area is the “fracture toughness” Gc= 2γf. The elastic potential energy released per unit of new crack area is the “energy release rate” (ERR) G. Thus, the Griffith criterion for propagation is written as, G≥Gc,(2.1) which summarizes the Griffith criterion. The application of this new criterion to a straight crack of length 2a, within a semi-infinite plate subjected to a constant and remote tension σ∞, gives that the crack is propagated when σ∞reaches a critical value σ∞ c, σ∞ c∝p1/a. (2.2) This dependence allowed Griffith to explain the influence of the scratches on the lifetime of engines rods such as the experimental results obtained in glass fibers. After the seminal papers by Griffith (1921, 1924), a large period followed without a significant development of the Griffith criterion because his works were ignored by the engineering community for almost 30 years coincident in part with the decrease of attention to the problem during the first part of the interbellum, see Figure 2.1. In the meantime Westergaard (1939) obtained the power of singularity of the stresses around the crack tip, σ∝1 √r(2.3) where ris the distance to the crack tip. 2Actually, Griffith (1921) proposed his criterion as a generalization of the “theorem of minimum energy” by taking into account the surface potential energy due to the inter-molecular or inter-atomic bonds. To keep these historical notes as simple and self-contained as possible, the equivalent explanation given was chosen. 2.1. Crack initiation from the classical fracture mechanics perspective 13 (a) (b) Figure 2.1: Use over time of the terms (a) “fracture” and (b) “fracture mechanics” normalized with the total number of mentions. Data are extracted from the Google Books Ngram Viewer developed by Michel et al. (2011) and setting smoothing to 2. According to Erdogan (2000), the Griffith criterion was not immediately accepted in the engineering community mainly due to two reasons: a) In metals, which is, along with concrete, the main material family used in large structures, the fracture energy due to inter-atomic bonds is several orders of magnitude lower that the actual fracture energy due to energy dissipation in other irreversible processes as yielding and b) the stresses obtained by a linear analysis around the crack tip are very unrealistic for metals because yielding is expected at these points. Both problems were solved at the end of 1940’s and the 1950’s. On the one hand, Irwin (1948); Orowan (1949) proposed to correct the value of fracture toughness by adding the terms of the dissipated energy due to other 14 2. From damage to fracture: crack initiation irreversible processes. On the other hand, Irwin (1957) demonstrated that the energy release rate is well approximated by a linear elastic analysis if the plastic zone around the tip is small enough even tough the stresses near the crack tip are unrealistic. The resolution of these problems allowed to apply the Griffith criterion to engineering materials. The combination of this fact with a succession of several catastrophic failures of large metallic structures motivated an extensive development of the linear elastic fracture mechanics from then on. A large and intense amount of contributions followed dealing with the characterization of the stress tensor around the crack tip, the definition of the “stress intensity factor” as the coefficient of the singular term of the asymptotic solution near the crack tip, the relation between this factor and the energy release rate. In addition, the first methods to obtain experimentally the fracture toughness were developed in order to measure the main magnitude of the new field. Next decades3, the efforts were devoted to develop the non-linear fracture mechanics which provides with tools designed to study problems with nonlinear effects which cannot be neglected. A huge effort was also spent to obtain elastic solutions for geometries with cracks of practical importance. Subsequently, the development of the computational methods enabled to solve more complex problems such as to simulate nonlinear phenomena. As shown in the previous historical notes, the answer to the question of which are the conditions necessary for crack propagation has been studied more or less intensely during a whole century. However the very related question of how and under which conditions a crack initiates has drawn much less attention. In fact, at least in the case of static crack initiation, few pages dedicated to this topic can be found in the majority of books of fracture mechanics. Crack initiation is often studied as a part of damage mechanics. This traditional undervaluing of this, a priori, step on the fracture process can be explained by the addition of two reasons: •Many mechanicians claim that crack initiation has not any sense since flaws, scratches, voids and/or other types of defects exist within any material system right after its manufacturing. Thus, according to this perspective, in brittle and quasi-brittle materials, the failure initiation would be given by the statistical distribution and the lengths of these defects, which are very linked to the manufacturing processes and material nature. Hence, the study of crack initiation under static loads would be spurious and of no relevance. •Griffith’s criterion is not able to predict crack initiation at least according to its classical formulation. For the example of a crack in a semi-infinite plate, this result can be directly derived from (2.2). For the crack length 3These lines are not claimed to be an exhaustive historical review of fracture mechanics but a brief introduction to provide a context to the subject of crack initiation. The reader interested in the subjects can find excellent works covering it in the literature, see e.g. Erdogan (2000); Cotterell (2010) 2.2. Do cracks initiate or exist before loading? 15 corresponding to crack initiation (a→0+), the critical remote tension σ∞ c→ ∞. Hence, no crack will appear in the lack of an initial one according to the classical Griffith criterion. This result is general for any problem. Note that both reasons are of a very different nature. Whereas the first one discusses the physical and practical sense of studying the crack initiation, the second one shows a limitation of the Griffith criterion. In what follows, this chapter is consecrated to delimit in which sense crack initiation is going to be understood in the context of this thesis and to briefly present how both problems have been solved previously. 2.2 Do cracks initiate or exist before loading? Griffith verified its theory by testing glass with artificial flaws larger than those existing in the material according to his assumptions. Next decades, an effort followed dealing with the observation of the “natural” flaws and its characterization. Unfortunately, observing directly these flaws was impossible in the glass used in the experiments by Griffith (1921). The reason was not that they do not exist but the limited means of the optical science and the typical small size of flaws in glass. As pointed out by e.g. Lawn (1993) the typical flaw sizes depends strongly on the material, ranging from nanometric in pristine optical fibers and whiskers, submicroscopic in glass and milimetric in rocks. With the development of the modern electron microscopy, flaws have been observed in the majority of materials of interest, see e.g. micrometric flaws in composites by Miller and Wingert (1979), submicroscopic flaws in Kevlar by Wagner (1986) and nanometric voids in alumina films by Huang et al. (2004). The origin of these flaws is very variated, some of the most important are: •Chemical origin: The exposition of the external surfaces to chemical spices generates chemical micro-reactions. This can lead to generate surface flaws. In the case of materials composed by several phases as composites or concrete, the micro-reaction between the aggregates and the stochastic characteristic of this reaction can also generate micro-flaws at the interface between phases or in the vicinity. In the majority of cases, these reactions are highly promoted by temperature, hence a large part of these flaws appear during the manufacturing process. •Thermal origin: Temperature change and gradients generates high stresses, which promotes the incompatibility of deformations promoting the appearance of flaws. In addition, the solidification of granular materials as metals generates the appearance of voids and bubbles. •Radiation origin: Low-energy radiation favors chemical micro-reactions at the external surfaces, generating new phases or even voids. This is particularly important in polymers since it is well known that radiation can generate chain scission or crosslinking. In the case of chain scission, a weaker zone is generated, which is equivalent to a flaw in some manner. 22 2. From damage to fracture: crack initiation (a) Example of a body with a cohesive law prescribed along a line. (b) Example of cohesive law Figure 2.3: Cohesive zone model Actually, CZM can be considered, with some conceptual differences, to be an application of the damage mechanics described previously with the damage confined into a finite number of surfaces. The dimensionless separation ˆ δcan be seen as a damage variable which governs the constitutive law. In CZM, the damaged zone is denoted process zone since for cracked bodies this is the zone where the irreversible processes occur, see Figure 2.4. This figure shows the typical zones of a cracked body modeled with CZM and the physical sense for two materials for which CZM is extensively employed. Observe the particular microstructural processes which justify the assumption of a continuous separation law and the existence of cohesive tractions. In concrete, the interaction between the constituents promotes basically the generation of unconnected microcracks among other phenomena, which reduces partially the ability of supporting tractions, see e.g. Otsuka and Date (2000) for a full description of the process zone in concrete. In composites, the reason for a process zone is also due to the different behavior of the constituents. In addition to the reason described for concrete, in composites, long fibers can act as traction bridges maintaining an important part of the supporting ability. Previous examples to justify the existence of cohesive tractions show that the cohesive law depends strongly on the material through the microstructure and its failure behavior. Thus, apart from the failure properties as strength and fracture energy, the form of the law depends also on the material. Hence, an extensive amount of forms have been proposed to be used for certain materials or problems. For the sake of simplicity, in what follows, the discussion about the forms is reduced to pure mode I. Some of the most typical examples are: •Constant: This is surely the simplest form for a cohesive law. Dugdale (1960) proposed to model the plastic zone in the vicinity of the crack tip 2.3. Predicting crack initiation 23 process zone undamaged zone stress-free crack concrete fiber reinforced composite Figure 2.4: Different zones which can be found along a surface with a cohesive law prescribed compared to examples of the failure at the microstructure scale for concrete and long fiber reinforced composites. by limiting tractions to the yield strength σy. If this is prescribed within a range of relative displacements [0, δc] between faces, this is de facto a cohesive law with the form represented in Figure 2.5(a). This idea enabled to avoid the unrealistic singularity of stresses at the crack tip in metals. The law is fully defined by σyand the fracture toughness G1c because δc=G1c/σy. •Linear softening: It was proposed by Hillerborg et al. (1976) for its use to model fracture in concrete inspired in experimental results. According to this law, see Figure 2.5(b), no separation occurs for tractions below the tensile strength. Once it is reached, a linear softening follows up to δc. Similarly to the previous cohesive law, this is fully defined by σcand G1c since δc= 2G1c/σc. •Triangular: This is essentially similar to the previous cohesive law with the 24 2. From damage to fracture: crack initiation (a) Dugdale (b) Linear softening (c) Triangular (d) Bilinear (e) Cubic (f) Exponential Figure 2.5: Examples of cohesive laws addition of a linear hardening part with a finite slope instead of the vertical first part of the previous one, see Figure 2.5(c). In general, the finite initial slope introduces a new independent parameter, which has to be defined. In many practical problems, this initial slope is introduced only because of computational reasons and it is set high enough to neglect the influence of the value on the results. However, the initial slope has been reported to be important on the tendency of some results, see Section 4.2 for a discussion in the context of one of the problems studied in this thesis. •Bilinear softening: This cohesive law, represented in Figure 2.5(d), is a generalization of the linear softening by defining two different slopes for the softening part. These two zones are inspired in experimental results in concrete, see e.g. Petersson (1981). An additional parameter is necessary to fully define this cohesive law in comparison with the linear softening law. Hence, the traditional fracture energy is divided into initial fracture energy and total fracture energy, see Jenq and Shah (1985). •Cubic polynomial: The form of this cohesive law is given by a cubic polynomial which vanishes for δ= 0 and δ=δc, the last one being a double root, see Figure 2.5(e). This law was proposed by Needleman (1987) to study crack initiation at interfaces. Only two parameters, σcand G1c, have to be defined to fully characterize this law. A wide variety of generalizations of 2.3. Predicting crack initiation 25 this law to take into account mixed mode has been proposed, due to the relevance of fracture mode-mixity in interface cracks. •Exponential: This law is represented in Figure 2.5(f) and is characterized by an exponential form. This was proposed by Needleman (1990) to model crack initiation at interfaces. The exponential curve is derived from the comept of universal binding energy correlation reported by Rose et al. (1981). The suitability of each cohesive law has been extensively studied, see e.g. Alfano (2006) for a critical review of cohesive laws used in composites and Park and Paulino (2011) for a general recent review. 2.3.4 Finite fracture mechanics Finite fracture mechanics is based on releasing other hypothesis of the Griffith criterion: the assumption of that crack growth is infinitesimal. If this assumption is released and the energy balance is formulated in an incremental manner, the resulting balance is able to predict initiation of cracks with finite length. Based on this idea, several methods, either based on this energetic balance or on stress conditions, have been proposed and used in a intermittent manner along the last half-century, see e.g. Neuber (1958); Tszeng (1993); Hashin (1996); Leguillon (2002); Taylor et al. (2005); Cornetti et al. (2006). The common framework of these methods is also named “Theory of critical distances” by other authors. Under the term finite fracture mechanics, a large set of criteria for crack initiation are framed. They all are based on assuming that a critical length, either material- or structural-dependent, governs the crack initiation. A large part of them have been used in an intermittent manner for a long time. Actually, the term finite fracture mechanics was initially proposed by Hashin (1996) to refer to an energy-based criteria which assumes a crack onset of a finite length. However, the term has been also employed since then by the engineering community to refer to either other criteria based on a critical length or to the complete framework. In the context of this thesis, the original criterion used by Hashin (1996) is denoted as “incremental energy criterion”, being described in Section 2.3.4.1, whereas the term “finite fracture mechanics” is reserved for the general framework. Although every criterion within this framework uses a critical length one way or another to predict crack initiation, this is not necessarily associated to a length with a physical sense. Only in some cases, this critical length is associated to the crack length predicted immediately after the onset. The main general advantage of the criteria for crack initiation proposed within this framework, in comparison with the methods described previously, is that they are based generally on linear analyses. Thus, the criteria within the finite fracture mechanics are more adequated for applications which require the combination of a high level of simplicity, e.g., industrial applications or multiscale analyses, along with a good agreement with experiments, as reported in the majority of problems. However, as pointed out by some authors, see e.g. Taylor (2007), the other side of the coin is that, at least at the moment of writing these 26 2. From damage to fracture: crack initiation lines, the majority of these criteria are not fully physically fonded. They are well based on typical physical laws and its generalization, but a rigorous interpretation of why they works has not been provided, due to the inherent difficulty on deducting general conclusions of the complex nonlinear and stochastic nature of the physics governing the crack initiation. In spite of this fact, these criteria are claimed as suitable tools to provide of useful and simple expressions to predict crack initiation. s t r e s s c r i t e r i o n e n e r g y c r i t e r i o n C T O D c r i t e r i o n coupled criterion hybrid criterion mixed criterion finite fracture mechanics inherent flaw mechanics incremental energy criterion quantized fracture mechanics finite fracture mechanics pointwise stress criterion point method averaged stress criterion line method JS criterion Figure 2.6: Schematic of the main criteria in the framework on the finite fracture mechanics Figure 2.6 shows a general outline of the different criteria within finite fracture mechanics. They are classified by which type of condition or conditions implement. Due to the recent and intense development of these criteria, each criterion have been referred to with several names in the literature. Alternative names found in the literature are written in italics. In the following, the main criteria are described following the structure proposed by Maim´ı et al. (2013) by splitting the criteria into two groups: those whose critical length is either fixed or variable. 2.3.4.1 Fixed-critical-length criteria This category includes a diverse series of criteria with a common point: they are based on a critical length lc, which depends on the material properties. The form in which lcis obtained differs for each criterion. In spite of the inherent diversity on the nature of each criteria, in the majority of cases the estimation of lcis based on setting it so that the criterion matches with the classical criteria for the problems for which they are assumed to work. Basically, these problems 2.3. Predicting crack initiation 27 correspond to a body with a sufficiently large crack on the one hand and a tensile specimen on the other. The classical criteria to match with, are, respectively, the Griffith criterion and the tensile stress criterion. Thus, in these cases the critical length depends mainly on the typical material properties governing both problems: the fracture toughness Gcand the tensile strength σc. In the following, the main criteria based on a fixed lcare divided into two groups depending on if they are founded on either a stress or an energy analysis. Stress-based criteria The stress-based criteria were the first criteria to be used to predict failure and crack initiation. However, they were rejected by Griffith (1921) because they contradict the experimental evidences of a strong size effect on failure of brittle materials with initial defects. The reason is that initial stress criteria were based on the maximum value of stresses found in the whole body. Thus, the scaling of the geometry does not modify the stress state at the point where the maximum is found, and as a direct consequence any size effect cannot be predicted. However, if the stresses are regarded along a fixed length and this length is not scaled with the geometry, because, e.g., it is considered a material property, the stresses along the critical length lcdo change with the geometry scaling, hence a size effect can be predicted. Once it is accepted that it is necessary to examine stresses along a critical length, the next question arises: which stress value of the whole length has to be compared with the strength material properties. Different answers to this question produce different criteria. The two most relevant criteria are described in the following. crack initiation Figure 2.7: Schematic of the process of crack initiation according to the criteria in the framework on the finite fracture mechanics Pointwise stress criterion According to the review by Taylor (2007), the pointwise stress criterion, also known as point method, was initially proposed by Peterson (1938, 1959) based on his works about fatigue in the 1930’s. Unlike the 28 2. From damage to fracture: crack initiation usual strategies to obtain lcdescribed previously, these authors initially related the critical length with microstructural material distances as grain size or interatomic distances. This criterion is based on the condition for the crack onset of that the stresses at all the points situated at a segment of length lchave to exceed or equal the tensile strength σc, σ(x)≥σc∀x∈(0, lc),(2.8) where xis the coordinate which varies along the segment, see Figure 2.7. When this condition is fulfilled a crack onset occurs. This criterion is usually applied along segments where stress field is monotonic, e.g., stress concentrators as notches or cracks. In this case it is only necessary to verify this condition at the point situated at a distance lcof the stress concentration point. Thus, the condition (2.8) can be rewritten more simply as, σ(lc) = σc.(2.9) This form of expressing this criterion causes that this criterion is also known as point method (PM). Note that only the solution at one point of a linearelastic problem is required. This value can be easily obtained by means of either analytical methods or, if required, well established computational methods as finite elements. The critical length corresponding to this criterion lPSC ccan be derived by imposing the agreement with the Griffith criterion in the case of large cracks, lPSC c=EGc 2πσ2 c ,(2.10) which can be obtained by evaluating the stresses at r=lPSC cgiven by the Westergaard (1939) solution for a cracked infinite plate, applying the Griffith criterion based on the Irwin relation between Gcand the critical stress intensity factor Kc, and equaling the resulting stress to the tensile strength σc. Averaged stress criterion The averaged stress criterion was introduced by Neuber (1958) based on his studies in Germany contemporaries to Peterson’s. Similarly to the previous criterion, this was proposed initially for fatigue and subsequently reinvented for problems with static loading by Novozhilov (1969). Instead of imposing that every value of stresses along the segment has to exceed or equal to σc, this criterion compares the average of stresses along the segment with σc. Thus, the condition for the crack onset is Zlc 0 σ(x)dx=σc.(2.11) This expression has been extended to three-dimensional problems by Nazarov (2004). 2.3. Predicting crack initiation 29 The critical length corresponding to this criterion lASC ccan be obtained taking the averaged stress of the Westergaard (1939) solution giving lASC c=2EGc πσ2 c .(2.12) Energy-based criteria This category includes those criteria based on the combination of an energy condition and a fixed length. Incremental energy criterion This criterion proposes a condition based on the same energy balance used by Griffith (1921) but formulated in a incremental manner. The critical length lcis the incremental jump of crack length and is normally associated to a certain material characteristic length at the scale employed for the study. The condition can be written as, Zlc 0 G(a)da=Gclc,(2.13) where ais an intermediate crack length used to obtain the integral term. This criterion has been used in an intermittent manner for a long time without an apparent connection between the authors employing it. This criterion was implemented by Parvizi et al. (1978) to explain the size effect found in the transverse crack onset in cross-ply laminates. They identified lcwith the inner-ply thickness, see Chapter 6 for an analysis of the problem. Subsequently, Tszeng (1993) applied the same criterion to the crack onset at the interface between a stiff spherical inclusion and the surrounding matrix. In this case, lcwas associated to an arbitrary fixed debond angle. Three years later, Hashin (1996) examined the evolution of the crack density as function of the external load in the problem of transverse cracking studied by Parvizi et al. (1978). This analysis provided an analytic expression to predict the crack density based on this criterion. Similarly to Parvizi et al. (1978), Hashin (1996) identified lcwith the inner-ply thickness. In addition, he proposed the name of finite fracture mechanics for this criterion, now used to denote the complete category of methods as discussed previously. More recently, Pugno (2004) proposed a similar method, called quantized fracture mechanics, and applied it to the atomistic scale. This application leads to the interesting result of the existence of prohibited strength bands if lcis associated to multiples of the inter-atomic distance. Inherent flaw model This criterion is based on assuming the existence of flaws which grow in a stable manner as “non-Griffith cracks”, according to the exact words used by Dvorak and Laws (1986). This stable growth ends when flaws reach a critical length lc, which is a material property. Above this length, flaws become governed by the Griffith criterion and if its conditions are fulfilled, the crack grows unstably. Note that, from the point of view of the result obtained, this is often equivalent to assume the existence of a crack in the material with length lc. This criterion was initially proposed by El Haddad et al. (1979) for 30 2. From damage to fracture: crack initiation fatigue problems based on the length for which a crack arising from a notch has a minimum value of G. Subsequently to this work, Dvorak and Laws (1986) presented a theory to predict transverse crack initiation in cross-ply laminates, which is based on an equivalent idea. Next decades, many works based on the vision of Dvorak and Laws (1986) of transverse cracking have been presented. In summary, this criterion for crack initiation can be written as, G(σ∞ c, lc) = Gc.(2.14) 2.3.4.2 Variable-critical-length criteria Many authors have claimed that the critical length is not a material property, see Maim´ı et al. (2013) for a review, but it also depends on the other problem parameters, being, in general, a structural parameter. In this sense, the criteria included in this category are based on a critical length which is a result of the analysis and not a material property. JS criterion This criterion was proposed by Jenq and Shah (1985) to study crack initiation in concrete. It is essentially similar to the inherent flaw model described previously but with a variable critical length. The basic assumption is similar: a defect or damaged zone grow stably (even tough the evaluation of the Griffith criterion predicts an unstable growth) up to a critical length for which the defect propagation becomes governed by the Griffith criterion. The critical length is given by the combination of the Griffith criterion and the crack tip opening displacement (CTOD) criterion proposed by Wells (1961) and Cottrell (1961). The CTOD criterion was proposed to account for the nonlinear processes near stress singular points when their effect cannot be neglected. The combination of both criteria leads to a system of two equations with two unknowns: the critical length lcand load σ∞ c. This criterion can be written as, G(σ∞ c, lc) = Gc,(2.15a) CTOD (σ∞ c, lc) = CTODc.(2.15b) Coupled stress and energy criterion Multiples evidences exist in the literature of that in many problems, a simultaneous fulfillment of stress and energy criteria is necessary to initiate a crack, see Chapter 3 for a revision. Based on these evidences and within the context of the finite fracture mechanics, Leguillon (2002) proposed a coupled stress and energy criterion to predict crack initiation. The two criteria impose two conditions for two unknowns: the critical length and load, hence the problem is determinated. If the pointwise criterion is taken as the stress criterion following the original proposal by Leguillon (2002), the system given by this criterion can be written as, σ(σ∞ c, x)≥σc∀x∈(0, lc),(2.16a) 2.3. Predicting crack initiation 31 Zlc 0 G(σ∞ c, a)da=Gclc.(2.16b) Note that the two equations correspond to the previous equations defining the pointwise (2.8) and energy criteria (2.13). However, in this case lcis an unknown. Many modifications of this criterion have been proposed to study problems of a very different nature. Since this is the criterion employed in this thesis, Chapter 3 is consecrated to present it in more detail. 38 3. The coupled criterion of the finite fracture mechanics (2008b). The related phenomenon of crack deflection in an homogeneous material has been also studied by applying the coupled criterion by Leguillon and Murer (2008a) obtaining a good agreement with experiments. Crack initiation in composites has been also studied using the coupled criterion at different scales, from nano to macro scale. In the majority of cases with available experiments, a good agreement have been found with the predictions obtained. In what follows the most relevant works are briefly reviewed sorted by the scale of the analysis. At the nanoscale, Salviato et al. (2013) presented, to the author’s knowledge, the first work implementing the coupled criterion to this scale. They utilized it combined with surface elasticity to study the debonding strength of a spherical nanoparticle subjected to a hydrostatic tension. The implementation enables a relatively simple evaluation of the influence of the interface properties on the process of debonding. At the microscale, the coupled criterion has been implemented to study the initiation of the those failure mechanisms which have the origin at this scale. This is very common in long-fiber reinforced composites due to the typical fiber radius in common composites. Regarding to the well-known matrix or interfiber failure, Mantiˇc (2009) developed a model to predict the onset of a debond at the fibermatrix interface when the composite is subjected to tension transverse to the fiber direction. This enables to predict a strong effect of the fiber radius on the critical tension leading to the interface debond. This model has been generalized to take into account a more general loading state. First, in the context of this thesis it was generalized to take into account a biaxial load inside the plane perpendicular to the fiber axis, see Section 4.3. Second, Carraro and Quaresimin (2014) evaluated the influence of a shear stress out of the plane transverse to the fiber axis. The effect of a pure transverse compression was also investigated by Quesada et al. (2009). This work actually focuses on stiff inclusions in soils but the model can be directly applied to the failure of fibers. At the mesoscale, transverse cracking has been briefly revisited by Leguillon (2002) and a general theoretical model has been proposed in this thesis, see Chapter 6, showing a good agreement with experiments. An interesting characteristic of the application of the coupled criterion for this problem is that the stress and energy criteria act decoupled. The related phenomenon of free-edge delamination has also analyzed by several authors, see e.g. Martin et al. (2010), by means of the coupled criterion. At the macroscale, the key problem of failure at an open hole has been widely investigated, see e.g. Zhang and Li (2008); Leite et al. (2010); Camanho et al. (2012); Martin et al. (2012). These models predict a size effect of the hole radius on the global strength, which agrees with experiments in some materials. A similar model was proposed by Catalanotti and Camanho (2013) to predict the net-tension failure of mechanically fastened joints. The coupled criterion enables to obtain a semianalytical expression which is very useful from an engineering point of view. Andersons et al. (2010) proposed a more general model to predict the failure due to the presence of internal U-notches. Other key problem at the macroscale is the failure of adhesive joints, several models have been proposed 3.2. Leguillon’s hypothesis 39 enabling the prediction of their failure, see e.g. Weissgraeber and Becker (2011, 2013). Although this revision focuses mainly on the applications for composites, the range of applications for which the coupled criterion has been shown to be a useful tool is very wide. This can be applied to diverse problems as snow mechanics, see Cardu et al. (2008), or the generation of crack pattern due to thermal effects, see Leguillon (2013). 3.2 Leguillon’s hypothesis Leguillon’s hypothesis is the key concept behind the coupled criterion. It postulates that the simultaneous fulfillment of both a stress and an energy condition is necessary for the crack initiation. This hypothesis combines two types of criteria commonly utilized separately in brittle and quasibrittle materials. Stress criteria are commonly used to predict failure in the absence of stress singularities, whereas the energy criterion is employed in the presence of cracks. Thus, this hypothesis enables to unify the two different visions of the failure process claiming that these cases classically associated to a single type of criterion, either on stresses or on energy, are extreme cases of the general condition of fulfilling both criteria. Experimental and conceptual evidences supporting Leguillon’s hypothesis have been presented by Leguillon (2002); Taylor et al. (2005); Cornetti et al. (2006). These evidences are far from being a closed and general proof justifying the coupled criterion. The rigorous physical justification of the criterion is therefore pending. In what follows these evidences are briefly discussed. The experiments carried out by Parvizi et al. (1978) in cross-ply specimens shows that the first transverse crack occurs for a certain critical strain which depends on the inner-ply thickness. In fact, a threshold thickness is observed, the behavior being very different above or below this threshold. Above this threshold thickness the critical strain corresponding to the first transverse crack initiation does not vary with the thickness. On the contrary, below this threshold thickness the critical strain increases strongly when the inner-ply thickness decreases. The presence of two differentiated scenarios motivates the idea that two different criteria are being superimposed. Leguillon (2002) showed that the two behaviors could correspond to the two stress and energy criteria. Subsequent experiments showed a similar behavior, see Chapter 6 for a full analysis of this problem. The requirement of fulfilling a stress criterion for crack initiation is evidenced by Leguillon (2002) employing the example of a bar subjected to a uniform strain. If only an energy condition is utilized to predict the failure, the predicted critical strain decreases with the bar length. The reason is that the elastic potential energy stored in the bar is directly proportional to the bar length and the square of the strain, whereas the dissipated energy during an abrupt fracture of the bar is given only by the fracture energy and the bar-section area. After the abrupt failure, all the potential elastic energy is released, thus, if the bar length increases, the strain level necessary to fulfill the energy balance decreases. This implies that 40 3. The coupled criterion of the finite fracture mechanics for sufficiently large lengths, the critical strain vanishes, which is not reasonable. The addition of the stress condition proposed by Leguillon’s hypothesis solves this problem because above a certain length, the stress criterion prevails, fixing the critical load to the value of the tensile strength. From a rigorous point of view, not all the elastic energy stored can be actually employed on breaking the bar if dynamics is taken into account given the finite velocity of the elastic waves. The reason is that it is unrealistic to assume that the energy density at points far from the breaking surface can be taken into account for the energetic balance given the high speed of the crack onset process. Thus, the stress criterion can be partially interpreted as a condition to take into account this limitation. At the other extreme, for short bars below a certain length, the energy criterion prevails and an increase of the critical strain is expected for shorter bars. The transition between the criteria governing the failure is evidenced by the problem of the crack initiation at a reentrant corner. By varying the corner angle, the problem goes from a sharp crack, whose failure is governed by the energy criterion, to a straight edge without stress concentration, whose failure is ruled by the stress criterion. Thus, it is reasonable to think that the cases with intermediate angles have to be governed by a combination of both criteria, the crack and the straight edge being the extreme cases of a unifying theory. The combination of Leguillon’s hypothesis and the assumption of finite length of the crack at onset enables to explain this transition and agrees well with experiments as demonstrated in Leguillon (2002) and many other subsequent works, see a review of the analysis of the failure at reentrant corners in Section 3.1. It is interesting to remark the behavior predicted for short cracks. It is well known that the linear elastic fracture mechanics is not valid for short cracks below a certain length since the nonlinear region in the vicinity of the crack becomes of the same order of the crack. Experimental methods, see e.g. Taylor et al. (2005) for a review of several experiments, shows that the critical remote stress predicted by the coupled criterion goes from a LEFM-like variation for large cracks to a constant value for short cracks. This well-known phenomenon can be interpreted as an evidence of the transition between two scenarios governed by the stress and the energy criteria as was proposed by Taylor et al. (2005); Cornetti et al. (2006). They found that the transition observed in experiments agrees with the prediction of the coupled criterion. In what follows, the two types of criteria are analyzed and the different formulations proposed for them are discussed. In addition, the practical evaluation of each one is described and the tools used are described. 3.3 Stress criterion In brittle and quasibrittle materials, a stress criterion is commonly employed to predict failure in the absence of stress singularities as cracks or damaged zones at the macro scale. The stress criteria are normally based on predicting phenomena which occur at the lowest scales through a macroscopic magnitude as the stress tensor. Since the processes behind the stress criteria depend strongly 3.3. Stress criterion 41 on the material microstructure, the different stress criteria are associated to the materials or type of materials. Many of the stress criteria are based on phenomenological laws, often founded on processes which have not been understood yet. From a practical point of view, these criteria are formulated by defining a critical value for a certain combination of the components of the stress tensor, see e.g. the Mohr-Coulomb criterion for soil mechanics or the recent failure criteria for composites, see also Par´ıs (2001); Hinton et al. (2002) for reviews. In the context of the finite fracture mechanics, the stress criterion is evaluated before the crack onset at the entire surface ∆Scwhere the crack will initiate, see Figure 3.2. In general, the stress criteria can be written by defining f(σij (P)) of the stress tensor σij for i, j = 1,2,3 at the points Pcontained in the future crack surface ∆Scand a critical value for this function fc, f(σij (P),∆Sc)≥fc, P ∈∆Sc.(3.1) Note that neither f(σij (P),∆Sc) nor fchave to be necessarily scalar functions since the stress criterion can be expressed as a combination of several scalar conditions. Figure 3.2: Arbitrary solid employed for the formulation of the stress criterion. Several stress criteria have been used for the coupled criterion since the publication of the first works employing it. In the majority of cases, they are adaptations of the classical stress criteria commonly used out of the context of the finite fracture mechanics. In the context of the finite fracture mechanics, the key condition for defining a stress criterion from a classic one is that the prediction given for an extreme problem governed exclusively by the stress criterion has to agree with the classical prediction. A classical problem governed by the stress criterion within a certain range of length is a cylindrical bar under tension. This requirement for the stress criteria enables the resulting coupled criterion to be claimed as a generalization of the stress and energy criteria. The first stress criteria were based on a critical value of the normal component of the traction vector associated to the normal to the surface ∆Sc. The 42 3. The coupled criterion of the finite fracture mechanics critical value can be identified with the tensile strength of the material given by a tensile test. Thus, in what follows these stress criteria will be denoted tensile stress criteria. The adaptation of these criteria to the context of the finite fracture mechanics for static problems was carried out by Novozhilov (1969) and previously by others for fatigue problems. They were used as stress criteria with a fixed critical length, see Section 2.3.4.1 for a review. In the context of the coupled criterion, Leguillon (2002) proposed to use the “pointwise stress criterion”, see Section 2.3.4.1, in combination with the energy criterion. This criterion postulates that the normal tractions σat all the points Pin ∆Schave to exceed or equal the tensile strength, σ(P) = ni(P)σij(P)nj(P)≥σc,∀P∈∆Sc(3.2) where ni(P), i= 1,2,3 are the components of the unit vector normal to ∆Scin P. In the terms expressed in (3.1), fccan be identified with σcand the function f(σij) can be expressed as, f(σij (P)) = min P∈∆Sc (ni(P)σij(P)nj(P)) (3.3) The condition for σof equaling or exceeding σcat all the points of the future crack surface can be physically understood as a trigger condition for each point. Thus, a point fulfilling the stress criterion would be ready to be broken if the energy criterion is also fulfilled. At the microstructure, the fulfillment of this condition can be associated to several states depending on the material. In general, it can be interpreted to be associated to a damage level which is high enough at a lower scale to enable the onset of a macroscopic crack. An alternative form of the tensile criterion is the “averaged stress criterion”, see Section 2.3.4.1. Cornetti et al. (2006) proposed to compare the averaged value of σat ∆Scwith the tensile strength. Note that in the case of the tensile test of a bar, this criterion also agrees with the previous one and predicts the expected value for this extreme problem governed by the stress criterion. According to this idea, (3.1) can be written as ¯σ(P)≥σc, P ∈∆Sc(3.4) where ¯ ·represents the average value. In this case, the stress criterion can be interpreted as the competition between two forces: The driven force for the crack onset and the strength force against the onset. The driven force would be the sum of the normal tractions which would act as a single force. The strength force would be the sum of the tensile strength of each point at the surface. The condition would impose that the driven force equals or exceed the strength one to allow the crack onset. An alternative interpretation, not necessarily incompatible with the previous one, is the following: at micro scale, it can be considered that the stresses are redistributed and their value become uniform. This criterion avoids the question arising at the pointwise criterion about how the normal tractions at some points can exceed considerably the tensile strength without the presence of macroscopic irreversible effects. In this case, the uniforming 3.3. Stress criterion 43 process of the averaged stress criterion avoids the existence of points with a stress level which exceeds the tensile strength significantly if a redistribution is assumed. However, the fact that the stress exceed considerably the tensile strength at a point does not have to be rejected necessarily since a post tensile strength behavior is possible although it cannot be observed in standard tensile tests. On the other hand, the average criterion presents a conceptual difficulty when it is applied to the onset of cracks which are not plane: the concept of stress redistribution is not as obvious as in the plane case because the direction of the traction vector depends on the point. Thus, the combination in a single total force would require additional assumptions. Both stress criteria have been employed within the context of the coupled criterion to predict the crack initiation in a wide variety of problems. Several works used both criteria in order to compare their predictions, see e.g. Cornetti et al. (2012). The difference between the critical loads predicted by the two criteria is often low, at least for the problems studied. Moreover, when the critical load is compared with experimental results, the difference between the two predictions is lower that the typical dispersion of the experiments. The stress criteria described previously are based exclusively on the normal tractions to ∆Sc, i.e. neglecting the influence of the shear component of the traction vector. This is acceptable for the majority of problems dealing with homogeneous bodies subjected mainly to tension and formed with materials with a tensile strength higher than the shear strength. In these cases, the angle of crack initiation observed usually agrees very approximately with the plane without shear component of the traction vector, see e.g. Yosibash et al. (2006). However, the shear stresses can become relevant in the presence of an interface weaker than the bulk material. A weak interface is a preferential surface for crack initiation, avoiding an optimum alignment of the angle of initiation as observed in the absence of interfaces by Yosibash et al. (2006). As a consequence, the influence of the shear stresses can become relevant if the loads promote their presence at the interface. On the other hand, even for homogeneous bodies, the role of the shear stresses should be taken into account for materials with a lower shear than tensile strength or subjected to compression. This is the typical scenario in soil mechanics. In the context of the application of the coupled criterion to predict interface crack initiation, Mantiˇc (2009) proposed to use a Mohr-Coulomb criterion, typically employed in soil mechanics, as a future development of his model, which was initially based on a tensile stress criterion. A Mohr-Coulomb criterion can be expressed in similar terms to (3.2) by defining a equivalent stress σMC eq (P) as, σMC eq (P) = σ(P) + |τ(P)| µ(3.5) where τis the shear stress at the interface and µ=τc/σcis the ratio of shear τc to tensile σcstrength. Note that, according to this definition, the value of σeq tends to σwhen τvanishes or if τcis much higher than σc. In terms of this σeq, 44 3. The coupled criterion of the finite fracture mechanics the pointwise stress criterion (3.2) can be rewritten as, σMC eq (P)≥σc,∀P∈∆Sc.(3.6) whose treatment in the context of the coupled criterion is totally equivalent to the employment of the original tensile criterion. An averaged Mohr-Coulomb criterion can also be defined by modifying (3.4) similarly, ¯σMC eq (P)≥σc, P ∈∆Sc.(3.7) For the problem studied by Mantiˇc (2009) subjected to a mixed-mode loading, Carraro and Quaresimin (2011, 2014) proposed a quadratic stress criterion to take into account the influence of the shear stresses. In the terms employed here, the quadratic stress criterion can be expressed defining a equivalent stress σq eq as σq eq(P) = sσ2(P) + τ(P) µ2 (3.8) Similarly to the Mohr-Coulomb criterion, this expression can be used either as a pointwise stress criterion, see e.g. Chapter 7 or an averaged one, see e.g. Carraro and Quaresimin (2014). At a conceptual level, the quadratic criterion solves one of the main problems of the Mohr-Coulomb criterion when it is used with dominant tension: the nonsoft transition between the critical curve σ, τ from τ > 0 to τ < 0, see the knee at σ=σcin Figure 3.3. In fact, through a simple analysis with Mohr circles, which is in the basis of the original Mohr-Coulomb theory, it can be demonstrated that this criterion, in the form defined in (3.5), is not able to predict a fracture plane perpendicular to the load direction under uniaxial loads in homogeneous brittle materials. In order to predict the expected fracture plane, the peak has to be rounded using the Mohr circle corresponding to a uniaxial state. This adds a complexity in the form of the stress criterion and the derived analyses. Thus, in addition to using the quadratic criterion for the materials or interfaces presenting this fracture behavior, it can be employed as a good approximation for materials which show a Mohr-Coulomb-like fracture behavior. In the presence of weak interfaces, the previous analysis with Mohr circles is not longer valid since the weak interface avoids the prediction of a unexpected angle for the fracture plane. In addition, the analysis with Mohr-circles of the stresses at the interface has not sense since not all the components of the stress tensor have to be continuous through the interface. Tran et al. (2012); Garc´ıa and Leguillon (2012) proposed to employ a polynomial expression as a generalization of the previous criteria, which can be expressed by defining an equivalent stress σpol eq as, σpol eq (P) = m sσm(P) + τ(P) µm (3.9) 3.3. Stress criterion 45 where the exponent mis considered a material or interface property. Note that the values m= 1 and m= 2 correspond to the Mohr-Coulomb and the quadratic criteria, respectively. The previous polynomial criterion for even values of the exponent mpresents an unrealistic behavior for σ < 0. If mis even, this expression prescribes that a compression promotes the failure decreasing the critical value of τ. On the contrary, if mis odd, a compression increases the required value of τfor failure. In general, it is more realistic that a compression makes more difficult the failure, therefore, the required value for τshould increase for σ < 0. Carraro and Quaresimin (2011, 2014) avoided this problem assuming that failure was impossible in the presence of compressions. Actually, it is, taken to extremes, an application of the idea of that compressions difficult the failure. A softer correction of this behavior can be obtained if the expression (3.9) is slightly modified as, σpol eq (P) = m ssgn (σ(P)) |σ(P)|m+|τ(P)| µm+ (3.10) where h·i+denotes the positive part of a real number. Figure 3.3: Critical curves predicted by the different stress criteria proposed to include the influence of the shear stresses at ∆Scbefore the crack onset. The stress criterion can be expressed in a general form including the different visions described previously. A monotonic and proportional increase of the external loads is assumed independently of the nature of the load: either prescribed displacement, prescribed traction or a combination of them. In the context of 46 3. The coupled criterion of the finite fracture mechanics this thesis, a linear relation between the external loads and the stresses generated is assumed. For the sake of simplicity, a certain tension σ0is considered as a reference of the external loads. Thus, the solution of stresses, strains and displacements is linear with σ0. The stress criterion can be written in a general and dimensionless form as, σ0 σc≥s(σeq,∆ˆ Sc) (3.11) where s(σeq,∆Sc) = 1 f(σeq/σ0,ˆ Sc),(3.12) ∆ˆ Scbeing the surface of the crack after the onset ∆Scexpressed in dimensionless coordinates. Since the function sis necessarily dimensionless in (3.11), expressing ∆Scin dimensionless terms is the most natural form. Observe that, in the definition of the function fin (3.1), the dependency on the stress tensor σij has been substituted by the equivalent stress σeq because it summarizes the different forms of combining the components of the stress tensor σij described previously. On the other hand, the form of the function fdiffers among the pointwise criterion, fPW(σeq,∆ˆ Sc) = min ˆ P∈∆ˆ Scσeq ˆ P (3.13) and the averaged criterion, fAV(σeq,∆ˆ Sc) = R∆ˆ Scσeq ˆ Pdˆ S R∆ˆ Scdˆ Sc .(3.14) where ˆ Pis the point Pexpressed in the dimensionless coordinates. In view of the previous general expression, the application of the stress criterion requires: on the one hand the knowledge of the typical strength properties of the material or interface and on the other hand the elastic solution of the problem before the crack onset. The first can be obtained for the majority of cases by well established experimental procedures because the required properties are well known. The form used to obtain the elastic solution depends strongly on the complexity of the problem under study. In some cases, the problem can be simplified to enable the employment of analytical solutions, see e.g. Sections 4.3.1 and 5.1.1. In other cases, with the crack onset sufficiently enclosed in the vicinity of a certain point, an approximate solution can be obtained by employing a matched asymptotic analysis, see e.g. Section 7.2. If neither analytical nor approximated solutions are available, the solution can be extracted from a computational analysis by means of, e.g., the finite element method, see Section 6.2.2. 3.4 Energy criterion The energy criterion employed in the coupled criterion is an adaptation to the context of the finite fracture mechanics of the differential energetic balance pro- 3.4. Energy criterion 47 posed by Griffith. Since a crack onset of a finite extension is assumed in FFM, the energy criterion is based on an incremental energetic balance between the states before and after the crack onset. In the context of the FFM, this has been used previously to the formal proposal of the coupled criterion by some authors using a fixed crack length at the onset, see Section 2.3.4.1 for a review. The incremental energetic balance can be expressed as ∆Π + ∆Ek+ ∆Γ = 0 (3.15) where ∆Π and ∆Ekare, respectively, the change in potential elastic and kinetic energy between the states before and after the crack onset. ∆Γ is the energy dissipated at the irreversible processes associated to the crack onset. Figure 3.4 represents the two states between which the previous changes are evaluated. (a) (b) Figure 3.4: Arbitrary solid employed for the formulation of the energy criterion at the situations immediately (a) before and (b) after the crack onset. In the context of this thesis, the initial state is considered quasistatic, implying, ∆Ek≥0.(3.16) To the author’s knowledge, this is a common assumption to all the works employing the coupled criterion presented in the literature. Under this assumption, the balance (3.15) can be written as, −∆Π ≥∆Γ.(3.17) This condition can be physically interpreted as the requirement of releasing enough elastic energy to fulfill the required energy which has to be dissipated during the crack onset. In what follows, the two terms are analyzed describing the different manners proposed to calculate or approximate their values. 54 3. The coupled criterion of the finite fracture mechanics mechanisms to the total energy dissipated is much more relevant per unit area than the contribution of the cleavage of chemical bonds. In fact, it is well known that the value experimentally obtained for Gcin these materials is actually the sum of the contribution of all these phenomena to the dissipation energy per unit area of crack growth. Depending on the material, some of the irreversible processes contributing to Gccould be irrelevant when a crack onset of a finite extension is assumed. Similarly, other processes can be relevant in FFM which are neglected by a value of Gcobtained for a quasistatic crack growth. Due to these reasons, the usage in the context of FFM of a Gcobtained for LEFM has to be considered as an approximation of the actual energy dissipated per unit area of crack onset. In the context of LEFM, it is normally assumed that Gcvaries with the fracture mode mixity ψ. This variation is due to the dependence on ψof the irreversible processes contributing to Gc. For instance, in metals, the yielding around the crack tip varies with the fracture mode mixity, therefore the contribution of this process to Gcdepends on. Since these irreversible processes and the relevance of their contribution to Gcare very linked to the microstructure, the variation of Gcdepends strongly on the material nature. As a consequence, the phenomenological laws for Gcare very particular of the material for which they are obtained. In homogeneous and isotropic materials, the cracks tends to propagate near pure mode I according to the local symmetry criterion by Barenblatt and Cherepanov (1961); Erdogan and Sih (1963), or following other criteria leading to very similar results, see e.g. experiments for PMMA by Maccagno and Knott (1989). In the case of ductile materials, the behavior differs slightly, observing a more relevant contribution of the mode II, see e.g. experiments for aluminum by Hallb¨ack and Nilsson (1994). For interface cracks, the presence of an interface with weaker properties promotes the propagation along the interface even in highly mixed fracture modes, avoiding the crack deviation following the local symmetry criterion. Due to this, the variation of Gcwith ψis particularly relevant for the propagation of interface cracks. This fact has motivated efforts to design mixed-mode fracture specimens with interface cracks to measure the variation of Gc. Experiments show a strong variation of Gcwith ψ, see e.g. Hutchinson and Suo (1992). In the context of the FFM, the influence of the fracture mixity on ∆Γ is controversial, in particular for interface cracks. On the one hand, the extrapolation of the microstructural causes of the variation in LEFM to FFM is not direct. On the other hand, the definition of fracture mode mixity is associated to a continuous crack growth, therefore additional assumptions are required to generalize this concept to FFM. In what follows, these controversial problems are discussed and the main contributions in the literature are presented. A part of the mechanicians claim that, in the context of the FFM, Gcis a surface energy which does not depend on the fracture-mode mixity. The main arguments supporting this idea were detailed by Carraro and Quaresimin (2014). According to their discussion, the crack onset in FFM is supposed to occur suddenly, therefore, the concept of fracture-mode mixity at the crack tip is not valid since the first crack tip appears after the crack onset. According to Carraro and 3.4. Energy criterion 55 Quaresimin (2014) and in the context of the problem of fiber-matrix debonding, in the absence of a crack tip, none of the phenomena contributing to the variation of the dissipated energy with the fracture-mode mixity are expected to occur. An example of them is the plastic deformation of the matrix which requires the presence of the singularity at the crack tip, therefore, in the absence of a crack tip, the dependence on the mixity due to its influence on the plastic deformation cannot be extrapolated to the context of FFM. Although some of physical reasons for the influence of the fracture-mode mixity do not have a clear sense in FFM, other physical phenomena which do contribute may remain valid. For instance, the microvoid mechanics, a common phenomenon in the basis of crack initiation, varies with ψas reported by Shi et al. (1994). Additionally, fractographies show that the morphology of the surface roughness varies with ψ, see e.g. Lane (2003). As a consequence, the effective surface changes and therefore the dissipated energy per unit of macroscopic area Gcvaries. Moreover, even in the absence of a crack tip, the effect of friction is expected to occur. This effect is strongly affected by ψ, thus, its contribution to the dissipated energy is an additional source of variation of Gcwith ψ. The influence of ψdue to these phenomena and others, see Evans et al. (1990), motivates Mantiˇc (2009) to propose an expression to approximate the influence of ψon ∆Γ. This expression is inspired by the formula employed to obtain ∆Π trough the integration of ERR, see Section 3.4.1.1. Thus, according to Mantiˇc (2009), the dissipated energy can be approximated by, ∆Γ = Z∆Sc Gc(ψ)dS(3.31) where ψis the fracture-mode mixity of a crack advancing from a void surface to ∆Scfollowing a certain growth sequence. Unlike the integral for the ERR, in this case the value obtained of the dissipated energy does depend on the assumption about the growth sequence. In the context of the plane strain study of fibermatrix debonding when subjected to uniaxial transverse tension, Mantiˇc (2009) assumed a “natural” growth sequence: the crack propagates symmetrically with respect to the tension direction and begins at the stress concentration. However, for other problems, the hypothesis of growth can be difficult to be established. This is particularly problematic for 3D problems. In the framework of his problem, Mantiˇc (2009) proposed to use the phenomenological Hutchinson-Suo law to approximate Gc, Gc=G1c 1 + tan2[(1 −λ)ψ](3.32) where G1c is the fracture toughness in pure mode 1 and λis a mode-mixity sensitivity parameter. In order to avoid the incongruence between the assumption of sudden crack onset in the FFM and the need of a crack tip to measure the fracture mode mixity and its influence on the dissipated energy, Garc´ıa and Leguillon (2012) proposed to base the approximation of the dissipated energy on the stress mixity before the crack onset. Thus, ∆Γ is approximated by the expression (3.31), taking ψ 56 3. The coupled criterion of the finite fracture mechanics as, ψ(P) = arctanτ(P) σ(P)(3.33) where σand τare the normal and shear tractions at points Pin ∆Scbefore the crack onset. This approximation avoids to assume a slow crack growth. In addition, the value of ∆Γ obtained following this idea does not depend on an arbitrary crack sequence. As a consequence, this can be employed for any problem including 3D problems without the difficulties described for the previous approximation. Nevertheless, even assuming that the dissipated energy depends on the stress mixity before the crack onset, the fact of employing phenomenological laws for Gcobtained experimentally as a function of the fracture-mode mixity for quasistatic crack growth is a controversial hypothesis which is difficult to physically justify. In general, the expression in (3.30) can be generalized to take into account any possible variation of the value of Gcin ∆Scand not only due to the discussed dependence on ψ. Thus, the next expression can be used to account for the variation of Gcon any property depending on the point P∈∆Sc, ∆Γ = Z∆Sc Gc(P)dS. (3.34) This expression can be normalized, defining a dimensionless dissipated energy ∆ˆ Γ as, ∆ˆ Γ = ∆Γ G1cL2=Z∆ˆ Sc ˆ Gc(ˆ P)d ˆ S. (3.35) where ˆ Gc=Gc/G1c and G1c is a value of the fracture toughness taken as reference as, e.g., the fracture toughness for a quasistatic growth in pure opening fracture mode. Lis the length of the solid geometry taken as reference in the dimensional analysis for ∆Π, see (3.19). ∆ ˆ Sc,ˆ Sand ˆ Pdenote the geometric entities ∆Sc,Sand Pexpressed in the normalized coordinates with L. Previous approximations of the dissipated energy in FFM are based on extrapolating the values of the fracture toughness obtained for a crack growth in LEFM. It is a very practical approximation because it enables to use properties of LEFM which can be measured by employing standard tests. However, the nature of the irreversible processes contributing to the energy dissipation can be very different in FFM and LEFM. An attempt to take into account this difference was carried out by Zhang and Li (2008) through the definition of a FFM fracture toughness GFFM c. They assume that the dissipated energy per unit area in the presence of a crack tip is different from its value under uniform tensile stress. The first correspond to the LEFM fracture toughness Gc, whereas the second is an additional material parameter, being denoted Gu c. The basis of this difference is that energy dissipation occurs at a band of a certain width hin the vicinity of the fracture surface. Thus, it is reasonable to think that hdiffers in the presence of a crack tip from its value for uniform tensile stress. In an intermediate case, as a non-singular stress concentration, Zhang and Li (2008) proposed a FFM 3.4. Energy criterion 57 fracture toughness GFFM cas an interpolation between the two values, GFFM c=αGc+ (1 −α)Gu c(3.36) where αis a parameter depending on the level of the stress concentration. The authors proposed to obtain it as, α=hcrack h(3.37) where hcrack is the width of the dissipating band for a crack and his the width of the equivalent zone for the stress concentration. Once the value of GFFM cis obtained, it can be employed directly in (3.30). From a conceptual perspective, this idea is, a priori, a more congruent approximation to the FFM assumptions. However, its general application presents practical problems. The main cause is that, to the author’s knowledge, no experimental method has been proposed to measure the fracture toughness under tensile stress Gu c. In addition, the concept behind the value of his complex enough and its approximation is not straightforward. 3.4.3 General form of the energy criterion The expression of the energy criterion (3.17) can be rewritten by substituting the dimensionless forms of the released energy (3.19) and the dissipated energy (3.35), giving −σ2 0L3 E∆ˆ Π∆ˆ Sc≥G1cL2∆ˆ Γ∆ˆ Sc.(3.38) Although it is not detailed in previous expression for the sake of brevity, ∆ˆ Π and ∆ˆ Γ depends also on the dimensionless elastic and geometric parameters of the solid before the crack onset. The energy criterion in (3.38) can be rewritten as a condition over the reference of external loads σ0as, σ0≥rG1cE Lg∆ˆ Sc(3.39) where g(·) is a dimensionless function defined as the ratio of the dimensionless dissipated energy to the released energy, g∆ˆ Sc=−∆ˆ Π∆ˆ Sc ∆ˆ Γ∆ˆ Sc.(3.40) The function g(·) can be interpreted as a measurement of the efficiency of a certain crack geometry at onset ∆ ˆ Scfrom the perspective of the energy criterion. Note that g(·) is scale-independent since the geometric parameters on which this function depends are dimensionless. On the contrary, the energy criterion does 58 3. The coupled criterion of the finite fracture mechanics depends on the scale since Lappears explicitly in (3.39). Thus, for a certain solid and ∆ ˆ Sc, the energy criterion prescribes a critical value for σ0, which is inversely proportional to the square root of the length of reference L. This is the typical size effect predicted by other energy criteria as Griffith’s criterion. The reason for this size effect is the same in both cases: the dissimilarity between the scales of releasing energy (L3) and dissipating energy (L2). 3.5 Coupled criterion In previous sections, the two stress and energy criteria have been discussed separately and the main methodologies to be formulated and approximated have been described. This section describes the form in which both criteria are combined according to Leguillon’s hypothesis, described in Section 3.2. In the first part, the combination of both criteria is studied from the practical perspective of their application. In the second part, the typical failure behavior predicted by the coupled criterion is discussed. 3.5.1 General form of the coupled criterion According to Leguillon’s hypothesis, described in Section 3.2, the critical load σc 0leading to the onset of a crack of finite extension is given by the minimum value of σ0fulfilling both the stress and energy criteria. It means that it is necessary to minimize the combination of the conditions obtained for the stress (3.11) and energy (3.39) criteria. In order to express both criteria in the same terms, the expression corresponding to the energy criterion (3.39) is normalized with the tensile strength σc, which was taken in the stress criterion as a reference of strength. Thus, the energy criterion is expressed as, σ0 σc≥γrg∆ˆ Sc,(3.41) where γis a dimensionless brittleness number defined in the context of the FFM by Mantiˇc (2009) as, γ=1 σcrG1cE L.(3.42) This brittleness number is analogous to others being proposed in the literature in other contexts, see Mantiˇc (2009) for a comparative review. Once both criteria have been expressed in the same terms, the coupled criterion can be formulated as an optimization problem given by the next expression, σc 0 σc = min ∆ˆ Schmax s∆ˆ Sc, γg ∆ˆ Sci.(3.43) Thus, the coupled criterion is expressed as a minimization problem, the objective function being the maximum of the dimensionless expressions of the two stress and energy criteria. The apparition of the function max(·,·) is the mathematical 3.5. Coupled criterion 59 expression of the requirement of fulfilling both conditions since the critical load is given by the most limiting condition, i.e. the condition requiring the maximum value for σc 0/σc. The minimization is carried out over the dimensionless crack surface ∆ ˆ Sc. Thus, the optimum value of ∆ ˆ Scin the minimization process corresponds to the crack geometry immediately after the onset. The morphology given by ∆ ˆ Scdoes not correspond necessarily to the crack observed after the onset since an unstable growth could occur. In general, the minimization of (3.43) over the dimensionless morphology of the crack ∆ ˆ Scis a problem of calculus of variations. The rigorous solution of problems formulated following the expression (3.43) presents a very high complexity. This fact is incongruent with the claimed advantage of the coupled criterion of being able to obtain analytical or semianalytical expressions by simple procedures to predict the crack initiation in problems with practical interest. Motivated by this reasoning, all the works applying the coupled criterion found in the literature assume additional hypothesis on the crack geometry over which the minimization is carried out. The most common assumptions are: •The crack onset is contained in a certain surface. This is typical in symmetric problems where the crack is assumed to be contained by the plane of symmetry, see e.g. the corner studied by Leguillon (2002). In the presence of anisotropy on the fracture and strength properties, the weaker plane is also supposed to contain the crack onset, see e.g. the problem in Section 6.2. In the presence of interfaces with weaker properties, the crack can be supposed to appear at the interface. In general, this type of assumption reduces drastically the complexity of the problem because the dimensions of the domain for ∆ ˆ Scare reduced. •The problem can be studied as a plane problem under the typical conditions given by the elasticity theory. The assumption of plane problem in the context of the FFM is also an assumption over the morphology of the crack since actually it implies that the crack morphology after the onset is invariable through the direction normal to the plane of study. This hypothesis also simplifies greatly the formulation of the coupled criterion. Moreover, the combination of this hypothesis with the previous one, reduces the minimization to find the optimum segment contained in a line. In fact, the majority of the problems found in the literature employing the coupled criterion are plane problems. •A certain point of the initial geometry is necessarily contained in ∆ ˆ Sc. This is a very common hypothesis in the presence of stress concentrations. The stress concentration point is normally supposed to be contained in ∆ ˆ Sc given that the stress level is maximum there. •The crack onset is symmetric with respect to the initial planes of symmetry. Note that this is a more restrictive assumption than the hypothesis of crack contained in the plane of symmetry. For instance, in the problem of 60 3. The coupled criterion of the finite fracture mechanics a crack onset at a holed plate under tensile stress, see e.g. Li and Zhang (2006); Zhang and Li (2008); Leite et al. (2010); Camanho et al. (2012), the assumption of crack contained by the plane of symmetry implies that the crack appears at either only one or the two symmetric stress concentration points at the hole perimeter, whereas the assumption of symmetric crack onset prescribes that the crack onset occurs simultaneously at the two stress concentration points. Although this hypothesis has been intensely employed, it is controversial since it has been demonstrated to be incorrect for some problems, see e.g. Section 4.1. The assumption of some of the previous hypotheses enables to reduce the optimization problem presented in (3.43) to an optimization over a finite set of scalar variables. In fact, the majority of works found in the literature employing the coupled criterion are based on the optimization over one scalar variable. Moreover, the complexity is often reduced since the variation of stress and energy expressions in (3.43) with the scalar variable is monotonic, with opposed tendency for the stress and energy criterion for many of the problems studied in the literature. Thus, the optimization becomes a problem of solving a nonlinear equation. In some problems, the assumptions enumerated previously cannot be easily applied. The cause is that different hypotheses can be taken under reasonable evidences. In these cases, the problem can be studied using separately the different assumptions and comparing the critical load obtained for each assumption. Thus, applying (3.43), the most accurate assumption will be the one for which the critical load predicted is minimum. In some cases, this result depends on the elastic, geometric or fracture parameters of the problem. For instance, Martin et al. (2008) studied the competition between deflection and penetration of a crack approaching an interface between dissimilar materials. They studied several hypotheses about the situation of the crack onset, e.g., the crack being contained either in the interface or in the material initially without crack. The comparison of the critical loads obtained for each assumption enables to predict the preferential type of crack as a function of the problem parameters. 3.5.2 Failure behavior predicted by the coupled criterion Previous sections have focused on characterizing the crack onset by predicting the critical load leading to the onset of a crack with finite length. In addition, the morphology of the new crack immediately after the onset is obtained as a result of the optimization process. Once the process of crack onset is characterized, it is necessary to evaluate the whole loading process predicted by the coupled criterion. Thus, this section describes the stress-strain behavior predicted for an arbitrary solid subjected to monotonically increasing loads. Figure 3.8 present examples of the typical force-displacement curve predicted by the coupled criterion for solids subjected to a monotonic increase of the external load level. The perfectly linear elastic equilibrium path is plotted with dot-dashed lines. In addition, the equilibrium paths predicted by the LEFM, 3.5. Coupled criterion 61 (a) (b) (c) (d) Figure 3.8: Examples of force-displacement curves predicted by the coupled criterion for a case (a,b) with and (c,d) without global failure immediately after the onset and (a,c) force-controlled and (b,d) displacement-controlled tests. The main steps of the behavior predicted have been denoted as: (0-1) linear-elastic loading, (1-2) crack onset, (2-3) unstable crack growth after the onset, (3-) subsequent stable growth. given by the condition G=Gc, are plotted with dashed lines. Note that, in the cases plotted, the LEFM and the linear elastic equilibrium paths do not intersect. The reason is that these cases correspond to solids without preexisting cracks. As discussed previously, in the absence of cracks, the LEFM predicts an infinite critical load for crack initiation. As a consequence, the linear elastic and the LEFM equilibrium paths are asymptotically tangent for u→+∞or F→+∞. In general, the coupled criterion predicts initially a linear behavior following the dot-dashed line from states 0 to 1. For a certain value of either Fc(force controlled) or uc(displacement controlled) corresponding to the critical value of the load of reference σc 0, both criteria are simultaneous fulfilled. Then, a crack onset occurs leading to an abrupt jump from states 1 to 2 on either uor F depending if the test is force or displacement controlled, see Figures 3.8(a) and 3.8(c) or Figures 3.8(b) and 3.8(d) respectively. Immediately after the crack onset the state is: •For a force-controlled test, the force is F2=Fcand the displacement corresponds to u2. 62 3. The coupled criterion of the finite fracture mechanics •For a displacement-controlled test, the force is F2and the displacement corresponds to u2=uc. Note that the dissipated energy during the crack onset corresponds to the area of the triangle (0-1-2) filled with undulated lines and formed by the elastic line of loading up to the critical load (0-1), the line corresponding to the jump predicted by the coupled criterion (1-2) and a fictitious line of unloading from the situation (2) immediately after the crack onset to the origin (0). If the dissipated energy was computed by using the same Gcemployed to plot the LEFM equilibrium path, the energy dissipated by the abrupt onset predicted by the coupled criterion is equivalent to the energy dissipated in the fictitious process predicted for the crack initiation by the LEFM. This process corresponds to a path which follows the linear elastic equilibrium up to an infinite load, for which the LEFM predicts crack initiation, and returns following the LEFM equilibrium path up to its intersection with the fictitious elastic unloading path from the state 2. Thus, the area enclosed by these paths, which is filled with bars in Figure 3.8, is the same that the area of the triangle (0-1-2). The subsequent growth after the crack onset can be predicted by the LEFM. Thus, the stability of the growth depends strongly on the evolution of the LEFM equilibrium path. In some cases, see e.g. Figures 3.8(a) and 3.8(b), no mechanisms exists avoiding the unstable growth of the crack up to the global failure. This is the typical behavior of plates under tensile stress with stress concentrators as corners or holes. In these cases, the crack grows unstably after the crack onset up to reach the free edge, leading to the global failure of the specimen without any additional increase of the load. The equilibrium paths predicted by the LEFM for these problems normally end at the origin. The cause is that, when the crack approaches the free edge, the force or displacement necessary for the crack propagation vanishes. On the contrary, in other problems, the presence of certain phenomena avoids the sequence of unstable growth and global failure. For instance, when a crack approaches an interface between two dissimilar materials, the material containing initially the crack being softer, the value of G→0 when the crack approaches the interface. As a consequence, the growth after the crack onset becomes stable after (or not) a first unstable growth. In fact, according to the LEFM, the equilibrium path tends to a new straight line corresponding to the stiffness of a specimen with a crack totally arrested, see Section 4.1 for more details of this behavior for an actual problem. It is not reasonable to think that this stable growth will go on ad infinitum due to the same idea employed to reject an unlimited evolution along the linear elastic path. For a certain load level, a new failure mechanism will appear leading to a new crack onset, e.g., in the case of the crack approaching an interface, the new mechanisms are the onset of a crack at the interface or the penetration through the interface, see Martin et al. (2008). CHAPTER 4 Failure initiation in long-fiber reinfonced composites under transverse loads Composites reinforced by long fibers are commonly used as a structural material in lightweight structures at present. In aerospace applications, where lightweight is a key aspect of the design, their level of structural responsibility has significantly increased as they are massively used in primary structures. However, our understanding of the damage mechanisms occurring in these composites on different scales is still insufficient. Thus, it is necessary to generate more knowledge about these mechanisms in order to avoid the present high level of uncertainty in the failure loads predicted in the design. In particular, the failure criteria still do not predict satisfactorily failures of unidirectional laminae under combined transverse loads (with loads perpendicular to the fiber-axis), as was highlighted in a series of coordinated studies (known as the world-wide failure exercises), e.g. Hinton et al. (2004); Hinton and Kaddour (2013). A reason for this is that the current failure criteria are still not sufficiently physically based from the microscopic point of view. One of the most complex failure mechanisms on micro scale in these composites is associated to the matrix failure, also called inter-fiber failure. In particular, the tension dominated mechanism follows a well described sequence of stages, see Hull and Clyne (1996) and Par´ıs et al. (2007): i) failure is initiated at the fibermatrix interface as small debonds, ii) the interface cracks grow along the interface until a certain arrest angle and then iii) kink out the interface towards the matrix, iv) coalescence of growing matrix cracks generates a macrocrack which may cause the failure of the unidirectional lamina. 63 70 4. Failure initiation in FRC under transverse loads reduce the number of numerically solved cases required to obtain G, the Vaschy- Buckingham Π theorem, Vaschy (1892); Buckingham (1914), is applied. Thus, as is demonstrated in 4.A, a dimensionless ERR ˆ Gcan be defined as, ˆ G(θd, n, α, β) = E∗ a(σ∞)2G(θd, n, σ∞, a, E∗, α, β),(4.8) where E∗is the harmonic mean of the effective elasticity moduli 1 E∗=1 21−ν2 1 E1 +1−ν2 2 E2.(4.9) As the first part of the present study is carried out for glass/epoxy, the dependence on αand βwill be omitted for the sake of brevity. The elastic problem for each value of θdand nis solved by the Boundary Element Method (BEM) code developed by Graciani et al. (2005). This code uses continuous linear boundary elements and is optimized for the solution of frictionless contact problems of interface cracks. The BEM model is defined by a square cell of matrix significantly larger than the circle at its center representing the fiber. Boundary conditions at interface nodes are either “perfectly bonded” or “in contact”, depending on the values of θdand n. A mesh suitably refined at the crack tips is used, with element lengths at the interface varying from 2◦, far from the crack tips, to 0.00004◦, at the crack tips. Normal traction σ∞>0 is imposed at two parallel external edges of the square cell, whereas the other two edges are free. A detailed description of the geometry, mesh and boundary conditions can be found in Garc´ıa et al. (2012). The values of interface tractions and relative displacements computed by BEM in the vicinity of the crack tips are used to estimate ˆ Gby the Virtual Crack Closure Technique (VCCT) with the aid of a Chebyshev-Gauss quadrature, see Graciani et al. (2010); Garc´ıa et al. (2012) for details. Figure 4.5 shows the computed values of ˆ Gas a function of θdand n, and includes the analytical solution by Toya (1974) for checking purposes. The values of ˆ Gcomputed for θd.65◦and n= 1 are very similar to the analytical values. However, for larger values of θd, a significant contact zone appears at the debond tip, that is not captured by the analytic solution assuming the open model of interface cracks, leading to an overestimation of ˆ Gby Toya’s solution. A comparison of ˆ Gvalues for n= 1 and 2 shows that they are larger for n= 1 and any value of θd>0. The difference is quite relevant for larger values of θddue to the shielding effect between the two debonds for n= 2. Moreover, ˆ Gfor n= 2 tends to zero for θd= 90◦, since it corresponds to a symmetric situation of two shear cracks approaching each other along the same line. The previous qualitative arguments are independent of the bimaterial properties, so the differences between the cases n= 1 and 2 observed in Figure 4.5 can be extrapolated to other bimaterials. The values of ˆ GIand ˆ GII, corresponding to a virtual crack increment angle of δθ = 0.5◦, are also plotted in Figure 4.5. The difference between both postfailure configuration is much larger in mode II than in mode I. The reason is 4.1. Symmetry on the crack initiation 71 Figure 4.5: ˆ Gas a function of θdfor glass/epoxy, obtained by the VCCT applied with a virtual crack increment angle of δθ = 0.5◦to the BEM results for each value of θd(points) and n= 1 and 2, compared with the Toya solution for n= 1. that the shielding effect in the case n= 2 is not sufficiently strong for the small debonds where the mode I is dominant. 4.1.2.2 Dissipated energy The energy dissipated during the debond onset due to the associated irreversible processes can be estimated by integrating the interface fracture toughness corresponding to an intermediate debond semiangle θdas proposed by Mantiˇc (2009), ∆Γ = 2nZ∆θ 0 Gc(ψ(θd, n))dθd,(4.10) where the angle ψis a stress-based measure of fracture mode mixity at the debond tip for θd. This expression is analogous to that for the released energy (4.7), but in (4.10), in opposite to (4.7), the value of the integral does depend on the sequence of growth chosen for the intermediate debond. In the present case, the debond growth sequence described above for the computation of ∆Π is assumed in accordance with Mantiˇc (2009), see Section 3.4.2 for a discussion on this issue. The dependence of Gcon ψcan be approximated e.g. by the phenomenological law due to Hutchinson and Suo (1992), which provides a good agreement 72 4. Failure initiation in FRC under transverse loads Figure 4.6: Fracture-mode-mixity angle ψas a function of the debond semiangle θdand both post-failure configurations for λ= 0.3, θl= 0.1◦and glass/epoxy. with available experiments for interface cracks, Gc=G1c ˆ Gc=G1c(1 + tan2[(1 −λ)ψ]),(4.11) where G1c is the interface fracture toughness in pure mode I, λis a fracture mode-sensitivity parameter, with a typical range 0.2≤λ≤0.35, and ˆ Gcis a dimensionless interface fracture toughness. The fracture-mode-mixity angle ψcan be calculated by using the interface stresses obtained from the computational model tan ψ(θd, θl) = τ(θd+θl) σ(θd+θl),(4.12) where θlis the distance from the debond tip where the fracture mode mixity is measured. Following Mantiˇc (2009), θl= 0.1◦is taken. Figure 4.6 shows the value of ψas a function of θdfor the two post-failure configurations. In general the angle ψis very similar for both configurations, though slightly larger for n= 2 in general. The difference become larger for the extreme where θd→90◦since a closed crack with ψ&90◦is expected for n= 1, whereas ψ→180◦for ν2>0 and n= 2 due to the symmetry of the limit configuration and the Poisson’s effect of the matrix. ˆ Gcis plotted in Figure 4.7(a) as a function of θd, for the above values of ψand λ= 0.3. As could be expected, in view of Figure 4.6 and (4.11), ˆ Gcis slightly greater for n= 2 than for n= 1 and this difference increases with θd. Figure 4.7(b) also shows the percentage difference in ˆ Gcbetween n= 1 and n= 2, confirming that for typical values of λ, the above observations hold. Anyway, the relative difference in ˆ Gcappears to be in general smaller than in ˆ G, the latter providing the main contribution to the difference in the energetic balance between both configurations. 4.1. Symmetry on the crack initiation 73 (a) (b) Figure 4.7: (a) dimensionless interface fracture toughness ˆ Gcand (b) percentage difference ∆ˆ Gc= 100 ·ˆ G(2 debonds) c−ˆ G(1 debonds) c/ˆ G(1 debond) cbetween the dimensionless interface fracture toughness for the symmetric ˆ G(2 debonds) cand asymmetric ˆ G(1 debond) cconfigurations as a function of the debond semiangle θdfor λ= 0.3, θl= 0.1◦and glass/epoxy. 4.1.2.3 Final expression of the energy criterion Both terms in the energetic balance (4.6) have been analyzed and computed above. This subsection aims to collect the main results in order to achieve an expression analogous to that obtained for the stress criterion in (4.4). Introducing the expression of the released energy (4.7) and the dissipated energy (4.10) in the energetic balance (4.6) and taking into account the definitions of ˆ Gin (4.8) and ˆ Gcin (4.11), this balance leads to 2n(σ∞)2a E∗Z∆θ 0 ˆ G(θd, n)dθd≥2nG1c Z∆θ 0 ˆ Gc(ψ(θd, n))dθd.(4.13) This inequality can be rewritten to the form of a condition for the normalized remote tension σ∞/σcnecessary to originate a debond of semiangle ∆θ, giving a minimum remote tension semiangle σ∞ σc≥γpg(∆θ, n),(4.14) where γ=1 σcrG1cE∗ a(4.15) is a brittleness number, cf. Mantiˇc (2009), and g(∆θ, n) = R∆θ 0ˆ Gc(ψ(θd, n))dθd R∆θ 0ˆ G(θd, n)dθd (4.16) is a dimensionless function which measures the ratio of the dimensionless dissipated to released energies of the debond and depends on the parameters that 74 4. Failure initiation in FRC under transverse loads Figure 4.8: Ratio gof the dimensionless dissipated to released energy as a function of the debond semiangle ∆θ, for λ= 0.3, θl= 0.1◦and glass/epoxy. determines the post-failure situation, i.e. ∆θand n, and also on the dimensionless material parameters α,βand λ. According to Figure 4.8, which shows g computed by numerical integrations, g(∆,1) < g(∆,2) with both the absolute and relative difference of both values increasing with ∆θat least for ∆θ≥1◦. It is interesting to observe that ghas a minimum for ∆θ= ∆θE min <90◦. In the limit ∆θ→0◦,g→+∞in accordance with the classical Griffith criterion for an infinitesimal advance of a crack. 4.1.3 Coupled criterion Both stress and energy criteria have been analyzed previously in an independent manner leading to two conditions on the remote tension σ∞required for a debond onset. Both criteria are combined in this section assuming the hypothesis by Leguillon (2002): the critical remote tension σ∞ coriginating a debond onset is given by the minimum remote tension σ∞for which both criteria are fulfilled simultaneously for a critical value of ∆θ. An analysis of the monotonicity of the functions on the right-hand sides of (4.4) and (4.14), corresponding to the stress and energy criteria, respectively, is required to implement Leguillon’s hypothesis in a semianalytical procedure. For both post-failure configurations, the function s(∆θ) in (4.4) is increasing with ∆θ, whereas g(∆θ, n) in (4.16) is decreasing with ∆θup to a minimum ∆θE min. At the plane (∆θ, σ∞/σc) for ∆θ≤∆θE min, the two curves given by these criteria have either one or none at all intersection point, s(∆θ) = γpg(∆θ, n), depending on the value of γ, which leads to the following two scenarios: •Scenario A: If both curves have one intersection point for ∆θ≤∆θE min, the minimum remote tension fulfilling both criteria for the debond onset σ∞ cis 4.1. Symmetry on the crack initiation 75 given by this intersection point (∆θc, γpg(∆θc, n)), ∆θcbeing the critical semiangle of debond and σ∞ c/σ =γpg(∆θc, n) the critical remote tension for the debond onset. •Scenario B: If both curves have none intersection point for ∆θ≤∆θE min, the remote tension fulfilling both criteria for the debond onset σ∞ cis given by the minimum value of g, the critical semiangle of debond being given by ∆θE min, and the critical remote tension by σ∞ c/σc=γpg(∆θE min, n). The scenario is determinated by the value of γsince the energy criterion curve is proportional to this value. A threshold value of γcan be defined as γth =s(∆θE min) g(∆θE min(n), n),(4.17) which separates the scenario A (γ≤γth) from B (γ > γth). Thus, for low values of γ, the debond onset (θcand σ∞ c/σc) is governed by the combination of both criteria, the critical remote tension being essentially given by the stress criterion, see Mantiˇc and Garc´ıa (2012), whereas the debond semiangle by the energy criterion. On the contrary, for large values of γ, the debond onset is governed by the energy criterion only. An unstable growth of the debond is possible after its onset. This is studied here by means of the classical LEFM. Thus, the condition for a further growth of the debond is given by the Griffith criterion as G(σ∞ c, θd, n)≥Gd(ψ(θd, n)),for ∆θc≤θd≤θa,(4.18) where this unstable growth will be arrested for an arrest semiangle θaverifying still G(θa, n) = Gc(ψ(θa, n)),but dG(θd, n) dθdθd=θa <dGc(ψ(θd, n)) dθdθd=θa . (4.19) The critical values characterizing the debond as σ∞ c/σc, ∆θcand θaare computed by algorithms developed in Mantiˇc (2009); Mantiˇc and Garc´ıa (2012). The critical and arrest semiangles are plotted in Figure 4.9 as functions of γ and the post-failure configuration. Note that ∆θcis very similar for both postfailure configurations in scenario A whereas for scenario B, both corresponds to the value of ∆θE min, which is quite different for both post-failure configurations. On the contrary, the arrest angle is very different in both cases due to the very different values of ˆ Gfor large θd, see Figure 4.5. Figure 4.10 shows σ∞ c/σcas a function of γfor both post-failure configurations. σ∞ cis quite constant for small γ, but it increases strongly for γ&1. In particular, it is a linear function of γin scenario B, as shown above. Qualitatively the behavior of σ∞ cis similar for both post-failure configuration in both scenarios, although a larger absolute difference is found in scenario B, as will be pointed out in Section 4.1.4. 76 4. Failure initiation in FRC under transverse loads Figure 4.9: Critical and arrest semiangles, ∆θcand θa, respectively, as functions of the brittleness number γ,λ= 0.3, θl= 0.1◦and glass/epoxy. Figure 4.10: Critical remote tension σ∞ cas a function of the brittleness number γfor both post-failure configurations, λ= 0.3, θl= 0.1◦and glass/epoxy. 4.1. Symmetry on the crack initiation 77 Figure 4.11: Critical remote tension σ∞ cas a function of the fiber radius afor both post-failure configurations, λ= 0.3, θl= 0.1◦and glass/epoxy. For given material properties, the dependence of the results on γcan be interpreted as a size effect by defining a reference radius a0=G1cE∗ σ2 c ,(4.20) and relating a dimensionless radius a/a0and γby a a0 =1 γ2.(4.21) A strong size effect can be observed in Figure 4.11, showing σ∞ c/σcas a function of a/a0, where σ∞ c/σcincreases strongly for small values of a/a0, see Mantiˇc (2009) or Section 4.2 for a detailed analysis of this size effect. 4.1.4 Analysis and discussion of results This subsection focuses on the main question which motivated the present section: the loss or conservation of symmetry in the post-failure configuration in a single-fiber specimen under transverse tension. First, both configurations are compared by computing the difference of their critical tensions predicted by the coupled criterion of the FFM for glass-epoxy and also for other bimaterials. Then, the stress-strain curves predicted by this criterion for both post-failure configurations are plotted and interpreted. Finally, some experimental results found in the literature are briefly revisited. 4.1.4.1 Difference in critical loads for symmetric and asymmetric post-failure configurations It is assumed that the preferential post-failure configuration, which can be expected to appear in experiments, corresponds to the configuration requiring a 78 4. Failure initiation in FRC under transverse loads Figure 4.12: Percentage difference ∆σ∞ c= 100·σ∞(2 debonds) c−σ∞(1 debond) c/σ∞(1 debond) c between the critical remote tensions for the symmetric and asymmetric post-failure configurations, taking λ= 0.3, θl= 0.1◦and glass/epoxy. lower critical remote tension for the debond onset. Thus, in view of Figures 4.10 and 4.11, the asymmetric configuration is preferential for glass-epoxy. The percentage difference between the critical tensions predicted for both configurations, ∆σ∞ c(%) = 100 ·σ∞ c(n= 2) −σ∞ c(n= 1) σ∞ c(n= 1) ,(4.22) is shown in Figure 4.12, where it is clearly seen that ∆σ∞ c>0 for γ > 0. Therefore, the asymmetric configuration is preferential independently of γ. However, ∆σ∞ cvaries between two limit values. In the limit γ→0+, which corresponds to scenario A, ∆σ∞ c= 0. The reason is that in this limit case the critical remote tension is governed by the stress criterion, which is equivalent for both postfailure configurations, see Section 4.1.1. At the other extreme, for large values of γ, corresponding to scenario B, ∆σ∞ cis given by its maximum value max ∆σ∞ c(%) = ∆σ∞ cγ≥max n=1,2γth(n)= 100 · sg(∆θE min(n= 2)) g(∆θE min(n= 1)) −1!, (4.23) obtained in view of the expression σ∞ c/σc=γpg(∆θE min, n) in scenario B. max ∆σ∞ cobviously corresponds to scenario B governed by the energy criterion, as this is the source of difference between both configurations. The main reason for this is the shielding effect between both debonds in the symmetric configuration, as discussed in Section 4.1.2. Since ∆σ∞ c, the key result of this work, is bounded by the above described two limit values, the behavior of a bimaterial is essentially characterized by the 4.1. Symmetry on the crack initiation 79 λ= 0 0.1 0.2 0.3 0.4 0.5 1 Bimaterial α β glass/epoxy 0.919 0.229 2.53 % 3.16 % 4.02 % 5.13 % 6.63 % 8.78 % 15.1 % carbon/epoxy 0.624 0.136 4.26 % 5.03 % 5.97 % 7.08 % 8.39 % 9.93 % 15.8 % A 0 0 8.29 % 9.06 % 9.91 % 10.8 % 11.8 % 12.8 % 16.5 % B 1 0 4.56 % 5.26 % 6.11 % 7.18 % 8.48 % 10.0 % 16.1 % C 1 0.5 0.85 % 1.65 % 2.63 % 4.31 % 7.16 % 9.84 % 14.2 % Table 4.2: Maximum percentage difference in critical tensions max ∆σ∞ c(4.23) between the symmetric and asymmetric configurations for several bimaterials and values of λ. Virtual bimaterials corresponds to: A: Similar elastic properties for fiber and matrix. B: Rigid fiber (E1E2) and incompressible matrix (ν2= 0,5). C: Rigid fiber (E1E2) and matrix with ν2= 0. upper limit (4.23), as the lower limit is null independently of the bimaterial. To generalize the previous study to other materials, this upper limit is computed for several bimaterials and values of the fracture-mode-mixity sensitivity parameter λ. Table 4.2 shows the maximum percentage difference, max ∆σ∞ c, predicted for two usual composites, glass-epoxy and carbon-epoxy, and three virtual bimaterials corresponding to some extreme values of the Dundurs parameters αand β. The bimaterial A corresponds to the extreme case of identical elastic materials, whereas the bimaterials B and C correspond to cases with the fiber much stiffer than the matrix. The difference between them is given by the Poisson ratio of the matrix, ν2= 0 for B and 0.5 for C. This table also shows the variations of max ∆σ∞ cwith λto evaluate the influence of the sensitivity of the fracture toughness ˆ Gcon the fracture-mode-mixity angle ψ. According to Table 4.2, λhas a strong influence on the values of max ∆σ∞ c. The reason for this is that a variation of λmodifies the position of the steeply increasing part of ˆ Gc-curve. For large values of λ, which correspond to a lower sensitivity of ˆ Gcto variations of ψ, the steeply increasing part of ˆ Gc-curve moves toward large values of θd. At the extreme λ= 1, ˆ Gcis a constant; this case was considered by Carraro and Quaresimin (2014). Therefore, in view of the definition of gin (4.16), if the increasing part of ˆ Gcis translated towards large values of θd, the value ∆θE min of the minimum of gis expected to increase. Thus, since the shielding effect between two symmetric debonds increases for large debonds, max ∆σ∞ cincreases with increasing λ. This argument will be repeatedly used in the following. The influence of the elastic bimaterial properties on the value of max ∆σ∞ cis elucidated in Table 4.2 by analyzing the extreme cases A, B and C of αand β, cf. Mantiˇc and Garc´ıa (2012). This influence decreases with increasing λ. In fact, the value of ∆σ∞ cfor λ= 1 is very similar for all the bimaterials studied. It means that the main source of difference between bimaterials is the different evolution of the fracture mode mixity for each bimaterial. This is particularly remarkable by comparing the results for the virtual bimaterials with a rigid fiber (α= 1): B (β= 0) and C (β= 0.5). It can be shown, following Dundurs (1967), that for a bimaterial with α= 1, stress solution only depends on the Poisson’s ratio of the matrix, which is ν2= 0.5 and ν2= 0 for B and C, respectively. In view of 86 4. Failure initiation in FRC under transverse loads 4.2.1 CZM: formulation and predictions The problem of fiber-matrix debonding caused by transverse loads is studied under the assumption of plane strain. Then, the problem geometry is reduced, in the plane perpendicular to the fiber axis, to a circular inclusion with radius asurrounded by a matrix perfectly bonded along its interface, see Figure 4.1. Both matrix and fiber are assumed to be linear-elastic materials. Mechanical parameters are chosen to represent an aluminium alloy 2124-SiC whisker-reinforced material, i.e., Em= 60 GPa, Ef= 340 GPa, νm= 0.3, νf= 0.18, where subscripts (f) and (m) refer, respectively, to the fiber and matrix. Initially, it is considered the case of a regular rectangular packing with a low volumetric fiber content equal to v= 18%, leading to b/a = 4.18, see the the geometry modeled in the FEM simulation, Figure 4.16(a), and the FEM mesh in Figure 4.16(b). Note that the geometry shown in this figure is not necessarily a representative volume element (RVE) of the material microstructure, but rather the smallest element of the composite in the case of perfect periodicity of the material microstructure (a) (b) Figure 4.16: Scheme of the FEM simulation: (a) geometry and boundary conditions and (b) mesh used for v= 18% The fiber-matrix interface failure is modeled by a CZM describing a nonlinear behavior of the interface. The CZM is defined by a constitutive relation for a zero-thickness interface written in terms of cohesive tractions and relative displacements. Two pure fracture modes, Mode I (opening) and Mode II (sliding) are suitably coupled in the case of mode mixity. In the sequel, σand τdenote the normal and tangential cohesive tractions, where the adjectives normal and tangential refer to the local axes of the curvilinear reference system along the interface. Correspondingly, a relative normal displacement (also called normal gap), gN, and a relative tangential displacement (tangential gap), gT, are admitted. The normal gap gNmust be non negative, since overlapping of fiber and matrix is not physically allowed. Different CZM constitutive laws can be adopted depending on the materials 4.2. Comparison with a cohesive zone model 87 under consideration, see Elices et al. (2002); Paggi and Wriggers (2011) for a critical examination. In any case, regardless of the shape of the traction-gap relation, the CZM introduces a nonlinearity into the problem. Moreover, the satisfaction of the unilateral contact constraint is another source of nonlinearity. In the present study, the FEM is employed to solve the problem numerically. The corresponding nonlinear problem is discretized using standard 2D plane strain elements with quadratic shape functions. The curvilinear interface is discretized applying the virtual node technique, firstly proposed in Boso et al. (2005) for contact problems and then extended to nonlinear fracture mechanics applications in Carpinteri et al. (2008). As a main advantage with respect to the classic interface element formulation Schellekens and de Borst (1993), no fixed coupling between FEM nodes along the interface has to be enforced a priori, which can be a limitation in the case of large displacements. The kinematic variables gN and gTare determined using a node-to-segment contact strategy Wriggers (2002) and the CZM relation is adopted for the computation of the cohesive tractions. Depending on the sign of gN, an automatic switching procedure is adopted to choose between cohesive and contact models for the normal direction. In the case of a negative gN, a penalty formulation is used to enforce the contact constraint Carpinteri et al. (2005, 2008); Paggi et al. (2006). For the CZM relation, the model by Tvergaard Tvergaard (1990) is selected, often used for particle-matrix debonding and similar interface problems. The normal and tangential cohesive tractions vs. normal and tangential relative displacements are computed as follows (see also Figure 4.17 for their visualization): σ=σc gN lNc P(λ),(4.27a) τ=µσc gT lTc P(λ),(4.27b) with gN≥0 and P(λ) = (27 4(1 −λ)2,for 0 ≤λ≤1 0,otherwise (4.28a) λ=sgN lNc 2 +gT lTc 2 .(4.28b) The parameters lNc and lTc are, respectively, the critical relative opening and sliding displacements, σcand τcare the maximum interface normal and tangential tractions, µ=τc/σc, and λis the dimensionless effective relative displacement. The energy dissipated after complete separation by cohesive elements under Mode I (the interface fracture toughness in Mode I), G1c, is calculated by integrating the cohesive law, G1c =27 4σc·lNc ·Z1 0 λ(1 −λ)2dλ=27 48σclNc,(4.29) 88 4. Failure initiation in FRC under transverse loads (a) Normal cohesive tractions (b) Tangential cohesive tractions Figure 4.17: Tvergaard CZM law used to describe the nonlinear behavior of the fiber-matrix interface. considering that in a cohesive element monotonically loaded in pure Mode I the dimensionless effective opening displacement is λ=gN/lNc. The energy dissipated after complete separation in pure Mode II, can be obtained analogously as G1c in (4.29): G2c = (27/48)τclTc. The energy dissipated by the cohesive elements in mixed mode, Gc, can be evaluated as a function of the stress-based fracture mode mixity ψσ= tan−1τ σ by the following integral, Gc(ψσ) = ZlNcr1−gT lTc 2 0 σgN lNc ,gT lTc dgN+ZlTcr1−gN lNc 2 0 τgN lNc ,gT lTc dgT. (4.30) This expression can be calculated using the next change of variables gN lNc =λcos ψ(4.31a) gT lTc =λsin ψ(4.31b) where ψis a effective displacement based fracture mode mixity which can be related to ψσfrom (4.27) as tan ψ=gT/lTc gN/lNc =1 µtan ψσ.(4.32) Introducing (4.32) in (4.27), (4.28) and (4.30), the expression for Gc(ψ) is obtained Gc=27 48 σclNc cos2ψ+τclTc sin2ψ(4.33) and using trigonometric identities and the relation between ψand ψσ(4.32), the expression of the equivalent interface fracture toughness in the Tvergaard cohesive law as a function of the stress based fracture mode mixity ψσis obtained 4.2. Comparison with a cohesive zone model 89 Assuming a fixed stress-based fracture mode mixity ψσ= tan−1(τ/σ) during the loading, the energy dissipated by a cohesive element can be expressed as a function of ψσ, Gc(ψσ) = G1c µ2+G2c G1c tan2ψσ µ2+ tan2ψσ!(4.34) where G2c= (27/48)τclTc, the interface fracture toughness in Mode II, can be obtained analogously as G1c in (4.29). It is interesting to compare this expression to the Hutchinson and Suo law Hutchinson and Suo (1992) commonly used in applications (e.g. Mantiˇc (2009); T´avara et al. (2011); Correa et al. (2008,?); Mantiˇc, T´avara, Bl´azquez, Graciani, and Par´ıs (Mantiˇc et al.)). Figure 4.18 shows the above expression of Gc(ψσ)/G1c for several values of µand G2c/G1c compared to the Hutchinson and Suo law. Note that whereas Gc(ψσ) from the Tvergaard’s CZM depends on µ, the Hutchinson and Suo law does not. As a consequence the difference between both depends on the value of µ. In general, two main differences can be observed: Hutchinson and Suo law is convex in the whole domain whereas Tvergaard CZM law has an inflection point between convex and concave parts because G0 c(ψσ= 0◦) = G0 c(ψσ= 90◦) = 0. Figure 4.18: Interface fracture toughness Gcas a function of the stress based fracture mode mixity ψσextracted from the Tvergaard CZM for several values of G2c/G1c and µ= 0.5 (dashed lines), µ= 1 (dotted-dashed lines) and µ= 2 (dotted lines) in comparison to the Hutchinson and Suo phenomenological law Hutchinson and Suo (1992) (solid line). In the present work the same values as in Carpinteri et al. (2005) are used: µ= 1, lTc =lNc = 0.03 µm and σc=τc= 300 MPa. In this case, since lNc =lTc and σc=τcit is possible to demonstrate that the energy dissipated after complete separation Gcis independent of the stress path with value, Gc=G1c =G2c= 5.0625 J/m2.(4.35) 90 4. Failure initiation in FRC under transverse loads (a) afrom 5 µm to 1.5 µm (b) afrom 1.1 µm to 0.5 µm Figure 4.19: Homogenized stress-strain curves by changing the fiber size for ν= 18%. Results courtesy of Professor M. Paggi published in Garc´ıa et al. (2014). Imposing horizontal (direction 1) displacements on the vertical boundaries of the unit square cell as shown in Figure 4.16(a), the homogenized stress-strain response can numerically be computed. As in Paggi and Wriggers (2011), the average stress σ1is determined as the sum of the horizontal reactions divided by the length of the vertical cell side b. Similarly, the average strain ε1is evaluated as the imposed relative displacement 2δdivided by the original length of the horizontal side of the cell b. For the same volumetric fiber content, different fiber radii aare considered, obtaining the family of curves shown in Figure 4.19. A strong nonlinear behavior can be observed. This is associated to the progressive decohesion of the interface leading to two separated debonds symmetric with respect to the x2-axis as shown in Figure 4.20. Reducing the fiber radius from 5µm to 2.55 µm, a transition from snap-back to snap-through instability is observed in the post-peak response, see Figure 4.19(a). A further reduction of the fiber radius leads to hardening behavior, see Figure 4.19(b). At the other extreme when fibers are large, see Figure 4.19(a), all curves approach to the same asymptotic behavior before the peak. To capture the post-peak branch in the case of large fibers, a simple displacement control is not sufficient and the position of the real interface crack tip has to be used as the control parameter monotonically increasing with time Carpinteri et al. (2005). These curves show a pronounced fiber-size effect that will be discussed in Section 4.2.3. The effect of the fiber volumetric content vcan be analyzed by changing it and repeating the above virtual uniaxial tensile tests. The homogenized stressstrain curves are shown in Figure 4.21 for two different fiber radii a= 1.5µm and a= 3.6µm. The progress of interface decohesion leads to a nonlinear stress-strain response. For small fiber radii, the post-peak response substantially depends on the volumetric content and may present a hardening (v= 17.8% and 35% in Figure 4.21(a)) or a snap-through branch (v= 50% in Figure 4.21(a)) by increasing v. On the other hand, for larger fiber sizes, the post-peak branch does not change type (always snap-through as in Figure 4.21(b) or snap-back for larger radius). From the numerical results shown in Figure 4.19 it is possible to define the 4.2. Comparison with a cohesive zone model 91 Figure 4.20: Fiber-matrix interface decohesion predicted by the CZM-FEM simulation. (a) a= 1.5µm (b) a= 3.6µm Figure 4.21: Homogenized stress-strain curves by changing the fiber volumetric content v. Results courtesy of Professor M. Paggi published in Garc´ıa et al. (2014). critical stress for the onset of fiber-matrix debonding, σ1c, as the value of the homogenized stress corresponding to a local maximum in the stress-strain curve. This maximum is, however, only present at curves for a≥2.55 µm. For smaller fiber radii (curves for a≤1.5µm), this local maximum is not present and the critical stress is estimated in correspondence of the point where the curvature of the longitudinal stress-strain curve changes sign, which corresponds to a similar mechanism. Examining the critical stress for the onset of debonding, the results are summarized in Figure 4.22 for the two fiber radii and the three volumetric contents considered in Figure 4.21. For a small radius (a= 1.5µm), the ratio σ1c/σcis a decreasing function of v. For larger fibers (a= 3.6µm and so forth), the critical stress ratio σ1c/σcbecomes almost independent of the volumetric content. This result is important in practical applications, since the fiber size is often in the micrometer scale in existing composite materials Chawla (1998). On the other 92 4. Failure initiation in FRC under transverse loads hand, these results suggest that the volumetric content may be relevant for the critical stress in nano-scale fiber-reinforced composites. Figure 4.22: Effect of the fiber volumetric content on the dimensionless critical stress. Results courtesy of Professor M. Paggi published by Garc´ıa et al. (2014). 4.2.2 FFM coupled criterion: formulation and predictions This subsection summarizes the FFM analysis of this problem carried out by Mantiˇc (2009) and highlight the modifications made to be coherent with the cohesive law studied previously. The FFM model is based on the next hypotheses assumed in Mantiˇc (2009): •Dilute fiber packing with a low fiber volumetric content. This allows using Goodier’s Goodier (1933) and Toya’s Toya (1974) analytical solutions for an unbounded matrix as reasonable approximations. Small variations of the critical stress for small volumetric contents in the FEM results for CZM shown in Figure 4.22 and large enough fibers supports this hypothesis. •Only one debond is supposed to appear at the interface symmetrically situated with respect to the loading direction (x1-axis) (see Figure 4.23) as discussed in Section 4.1. Thus, the problem configuration symmetry with respect to the x2-axis before the crack onset is broken by the one crack onset. Nevertheless, this is the situation usually observed in experiments Zhang et al. (1997). Finally, due to the symmetry with respect to the x1-axis, only the upper half domain will be studied, all the angles being measured from the positive x1-axis, see Figure 4.23. Stress criterion is invoked assuming no pre-existing cracks. It states that an interface crack onset of semiangle ∆θcannot occur if an equivalent stress σeq 4.2. Comparison with a cohesive zone model 93 Figure 4.23: The elastic problem after the interface crack onset. (function of the interface traction vector) at some point with semiangle θat the expected crack path is below a critical value σc. Hence, the crack onset requires σeq(θ)≥σc,∀θ∈[0,∆θ],(4.36) where σeq(θ) depends on the particular stress criterion used. In the case of the tensile stress criterion used in Mantiˇc (2009) and Section 4.1, σeq =σ, where σis the normal traction at the interface. In the present work, in view of the experimental evidences in Ogihara and Koyanagi (2010) and in order to take into account the influence of the tangential tractions τprescribed by the cohesive law, the Mohr-Coulomb criterion is employed, σeq =σ+|τ| µwith µ > 0.(4.37) Note that, the limit cases µ→ ∞ and µ→0+, respectively, correspond to the tensile and shear criteria. The value µ= 1 is adopted here in accordance with setting of the CZM law used, which corresponds to τc=σc. Introducing the analytical Goodier’s solution, rewritten following Mantiˇc (2009) in terms of the Dundurs bimaterial parameters αand β, into (4.36) leads to σ1 σc≥s(θ, µ, α, β) = 1 k−msin2θ+msin θcos θ µ ,∀θ∈[0,∆θ],(4.38) where kand mare elastic bimaterial parameters and s(θ, µ, α, β) is a function giving the minimum value of σ1/σcrequired to fulfill the Mohr-Coulomb criterion at a certain interface point θ, k(α, β) = 1 2 1 + α 1 + β 2 + α−β 1 + α−2βand m(α, β) = 1 + α 1 + β.(4.39) It is easy to check that ∂s(θ, µ, α, β)/∂θ < 0 for θ= 0◦and µ < ∞, whereas ∂s(θ, µ, α, β)/∂θ > 0 for θ > θσ min, where θσ min =1 2arctan 1 µ.(4.40) 94 4. Failure initiation in FRC under transverse loads Thus, s(θ, µ, α, β) is not a monotonic function for θ∈[0◦,90◦], and consequently a debond (in fact, two debonds symmetrically situated with respect to the x1- axis) could be predicted without including the interface point θ= 0◦. The reason is that according to the Mohr-Coulomb criterion the first point fulfilling this stress criterion, for µ < ∞, when increasing σ1is not θ= 0◦(see Figure 4.24 and a discussion at the end of this section). In order to use the Toya analytical solution for the energy criterion, the hypothesis of “symmetric debond onset with respect to x1-axis” has to be assumed. Taking into account this assumption, an onset of semiangle ∆θcannot occur if the maximum of the required values for σ1/σcfor points θ∈[0◦,∆θ] is not exceed. Hence, the crack onset requires σ1 σc≥¯s(∆θ, µ, α, β) = max 0≤θ≤∆θ[s(θ, µ, α, β)] ,(4.41) where ¯s(∆θ, µ, α, β) is the minimum value of σ1/σcwhich fulfills the stress criterion for an interface crack onset with semiangle ∆θ. Figure 4.24 shows functions discussed above. The difference between the tensile stress criterion employed in Mantiˇc (2009); Mantiˇc and Garc´ıa (2012) where the tangential traction influence was neglected, and functions s(θ) and ¯s(∆θ) is remarkable for bimaterial properties used here. Note that the difference between the last two functions is reduced to a narrow range for small semiangles. Figure 4.24: Graphical representation of the stress criteria for µ= 1. The (incremental) energy-criterion is based on an energy balance between the state prior to the debond and after the appearance of a finite debond of an angle 2∆θ, ∆Π(∆θ)+∆Ek(∆θ) + ∆Γ(∆θ)=0,(4.42) where ∆Π and ∆Ek, respectively, denote the change in the potential and kinetic energy, whereas ∆Γ is the energy dissipated during the crack onset. According 4.2. Comparison with a cohesive zone model 95 Figure 4.25: Plots of the coupled stress (stress criterion for µ= 1) and (incremental) energy criterion. to the (classical) fracture mechanics, ∆Π = −2R∆θ 0G(θd)adθd, where G(θd) is the Energy Release Rate (ERR) for a debond of semiangle θd. ∆Ek≥0 since the initial state is assumed to be quasi-static. Regarding to the dissipated energy ∆Γ, this can be estimated assuming an interface fracture toughness Gc(ψσ) which, in general, depends on the fracture-mode-mixity angle ψσat the interface crack tip. However, according to (4.35) and related discussion in Section 4.2.1, it is assumed that the interface fracture toughness is a constant1independent of ψσand write ∆Γ = G1c2a∆θ. Taking into account all these assumptions, the energy balance (4.42) can be expressed as a condition for the crack onset by employing Toya’s explicit expression Toya (1974) of the ERR, rewritten as G(θd, σ1, a, α, β, E∗) = σ2 1aˆ G(θd, α, β)/E∗in Mantiˇc (2009): σ1 σc≥γpg(∆θ;α, β),(4.43) where g= ∆θ/ R∆θ 0ˆ G(θd;α, β)dθdand γis a dimensionless brittleness number introduced in Mantiˇc (2009) as a generalization of Carpinteri’s brittleness number sto interface cracks Carpinteri (1981, 1982), γ=1 σcrG1cE∗ a=ra0 a,(4.44) with a0defining an interface characteristic length. The dimensionless function g, introduced in Mantiˇc (2009), characterizes the variations of the ratio of the 1In general the interface fracture toughness may depend strongly on the fracture mode mixity at the crack tip. This dependence can easily be introduced in the FFM model Mantiˇc (2009) and also in the CZM. 102 4. Failure initiation in FRC under transverse loads Figure 4.30: Dimensionless effective opening displacement λat the interface from the CZMFEM results at the load leading to the onset of debonding (a= 5 µm). Results courtesy of Professor M. Paggi published by Garc´ıa et al. (2014). with θ= 0◦. Thus, it seems interesting to discuss the influence of this hypothesis on the predictions of the coupled criterion. In particular this is going to be analyzed for the extreme aa0since there the stress criterion governs the value of σ1c. By minimizing the stress criterion function s(θ;µ;α, β) in (4.38) the most favorable point to failure at the fiber-matrix interface is given by θσ min defined in (4.40). Figure 4.29(a) shows the angle θσ min as a function of µ. Note that this angle is far from θ= 0◦for moderate values of µ. In particular, for µ= 1 used in this comparison, θσ min = 22.5◦. This reasonably agrees with the CZM predictions when looking for the effective opening displacement distribution λ corresponding to the load leading to the onset of debonding, Figure 4.30 plots an example for a= 5 µm. As expected, the most critical point is not at θ∼ =0◦, but at θ∼ =27◦. Therefore, in FFM a debond without including the interface point θ= 0◦would be possible for sufficiently small values of the debond angle. It can be shown Mantiˇc (2009) that, in general, the debond angle at the onset tends to zero for pa0/a →0, corresponding to large values of the fiber radius. Furthermore, σ1c depends only on the stress criterion (4.38) as discussed previously and, thus, is given directly by its minimum value: σ1c σc =2 2k+mq1 + 1 µ2−1.(4.51) Figure 4.29(b) shows this value of the critical stress and the value σ1c/σc= 1/k predicted by the Mohr-Coulomb criterion (4.41) assuming a symmetric debond with respect to x1-axis, for pa0/a →0. Note that the difference is large for small and moderate values of µ. However, no difference in the predicted critical stress is expected for small fiber radii since this case corresponds to larger debond angles which necessarily include the interface point θ= 0◦. 4.2. Comparison with a cohesive zone model 103 The use of the Mohr-Coulomb criterion releasing the assumption of symmetric debond with respect to x-axis in FFM improves the agreement with CZM predictions for (pa0/a .1), see Figures 4.26 and 4.28. 4.2.4 Limits of validity of CZM and FFM methods The scaling laws obtained according to both approaches described in Sections 4.2.1 and 4.2.2 and compared in Section 4.2.3 present some limits of validity. This section focuses only on the limits of validity derived from some conceptual incongruences. A complete study of validity would require experiments to compare, which are not available at the moment. Regarding the numerical predictions based on CZM, a discretization issue arises for capturing the mechanical response in the case of large fibers. In this case, the process zone size becomes relatively very small with respect to the fiber radius and a suitably refined mesh is necessary to approximate adequately the distribution of cohesive tractions obtaining mesh independent results. Similar convergence problems have been pioneeringly put into evidence by Carpinteri Carpinteri (1989a,b) in the case of cohesive crack propagation in three-point bending concrete beams. Provided that the size of the finite elements is small as compared to the process zone size, the horizontal asymptote in Figure 4.26 can be correctly captured. In general, the FFM model has no problem for large fibers. In fact, as demonstrated in Section 4.1 for large fibers which correspond to brittle configurations, FFM predicts that the critical load leading to the failure is essentially governed by the stress criterion. This implies that critical load can directly be predicted from the stress state prior to the crack onset. As a consequence the analysis becomes simpler since the elastic state prior to the crack onset can be obtained quite easily. In some cases, as in the present work an analytical solution is available, thus an explicit expression of the critical load can be extracted. Additionally, small stresses in large part of specimen make the linear elastic approximation more accurate. Regarding the opposite asymptote for very small fiber sizes, a deviation of the actual results from the CZM-based scaling laws can be due to the fact that the large stresses required are not compatible with the linear elastic behavior assumed for the matrix. In fact, releasing the assumption of linear elasticity for the matrix and considering an elasto-plastic behavior, the interplay between cohesive fracture and matrix plasticity could take place. As compared to the same simulations with purely linear elastic behavior (Figure 4.19), it is possible to note that for large fibers the response is not so different, since only the snapback branches are smooth out at least for a < 5µm. This implies that cohesive crack propagation prevails over plasticity. For small fibers, on the other hand, the hardening response is no longer visible due to a large amount of plasticity in the matrix. This has an important consequence on the scaling law, which can be deviated from that predicted by the CZM. In this context, the ratio between the yield strength and the interface tensile strength, σy/σcis an important quantity for the definition of the upper threshold to the CZM scaling law. Ratios larger 104 4. Failure initiation in FRC under transverse loads than unity, admissible in the case of poor quality of fiber-matrix cohesion, may increase the validity of the CZM predictions that would be admissible in the range σ1c/σc< σy/σc. This behavior agrees with the experimental evidences which show that finer particles limit the onset of cavities and microcracks Zahid et al. (1998) enhancing superplasticity. The explanation, following these results, would be that below a given fiber size, plasticity of the matrix predominates over fiber-matrix debond, so superplasticity properties are improved. Regarding to the FFM model for small fibers, some problems related to those found in CZM appear as well. FFM is used here by supposing the cracks follows a determinated path. For instance, in the present work, the crack path supposed is the fiber-matrix interface. This implies that other possible preferential paths or failure points are not taken into account. In particular, for small fibers the FFM model predicts critical loads which exceed the interface strength by two or three times as can be seen in Figure 4.26. In this case as demonstrated by the FEM analysis, a plastic zone is expected to appear in the matrix. Therefore, in the case of FFM, limits of validity are very dependent on the ratio σy/σc Additional problems can be found in CZM for very small fibers since the present model considers constant material lengths lNc and lT c. As a result of this, for very small fibers, the radius acan become similar to lNc. In this case, the numerical analysis would not provide representative results. 4.3 Crack initiation under biaxial transverse loads In the present work, an extension of the theoretical procedure developed by Mantiˇc (2009) to the case of tension dominated remote biaxial transverse loads is developed. This study is mainly motivated by the effect of the secondary compression in addition to a dominating tension shown in the preliminary experiments by Par´ıs et al. (2003) and Correa (2008). The FFM analysis is carried out for dilute fiber packing where the influence of adjacent inclusions is almost negligible, whereas for dense fiber packing, a numerical method should be employed based on other solutions, e.g. Kushch and Mishnaevsky Jr. (2010). In addition, several models developed under the assumption of dilute fiber packing have demonstrated being useful to explain experimental results even for densely packed composites, see Correa (2008). After a revision of the solution of the elastic inclusion perfectly bonded to the matrix in Section 4.3.1, the elastic problem of an inclusion with a debond is analyzed in Section 4.3.2, where a general analytical solution is particularized for this problem. The coupled criterion is developed and applied in Section 4.3.3 obtaining results for the critical crack length and remote load at the onset. Section 4.3.4 describes the influence of the remote secondary load and the inclusion size on the results obtained. Finally a new experimental procedure for an indirect measurement of the strength and fracture interface properties is presented in Section 4.3.5. 4.3. Crack initiation under biaxial transverse loads 105 4.3.1 Stresses in a single inclusion under a remote biaxial transverse load Stress based failure criteria usually consider the stress state prior to the damage appearance. Hence the aim of this section is to study the stresses at the fibermatrix interface under remote biaxial load transverse to the fiber axis. Consider a circular cylindrical inclusion of radius aembedded in an infinite matrix and perfectly bonded along its lateral interface. Let (x, y, z) and (r, θ, z) be suitably defined cartesian and cylindrical coordinate systems, the z-axis being coincident with the inclusion (longitudinal) axis. Remote uniform biaxial transverse load (σ∞ x, σ∞ y) is applied parallel to the the two axes, xand y, transverse to the cylindrical inclusion axis, see Figure 4.31. Figure 4.31: The inclusion problem configuration before the crack onset. An analytic solution for stresses in this problem was deduced by Goodier (1933). As was shown by Hardiman (1954) the stresses inside the inclusion are constant. Following Mantiˇc (2009), these inclusion stresses can be rewritten in terms of the Dundurs bimaterial constants αand β(Dundurs (1967, 1969)) as follows: σ(1) xσ(1) xy σ(1) xy σ(1) y!=σ∞ xk0 0k−m+σ∞ yk−m0 0k,(4.52) where k(α, β) = 1 2 1 + α 1 + β 2 + α−β 1 + α−2βand m(α, β) = 1 + α 1 + β,(4.53) with the Dundurs bimaterial constants being defined as α=µ1(κ2+ 1) −µ2(κ1+ 1) µ1(κ2+ 1) + µ2(κ1+ 1) and β=µ1(κ2−1) −µ2(κ1−1) µ1(κ2+ 1) + µ2(κ1+ 1),(4.54) with µk=Ek/(2(1 + νk)) and κk= 3 −4νk,Ekand νkdenoting the Young’s modulus and Poisson’s ratio, respectively. It can be shown that 0 ≤k≤5/3 and 0≤m≤2. 106 4. Failure initiation in FRC under transverse loads Let a dimensionless load-biaxiality parameter ηbe defined as the ratio of remote stresses η=σ∞ y σ∞ x .(4.55) Then, normal and shear stresses, σand τ, acting along the interface (r=a) can be expressed as a function of the polar angle θ(see Figure 4.31) and the parameter ηas σ(θ) σ∞ x =σr(θ) σ∞ x =k+ (k−m)η−(1 −η)msin2θ, (4.56a) τ(θ) σ∞ x =−σrθ(θ) σ∞ x = (1 −η)msin θcos θ. (4.56b) In view of the problem symmetry only angles 0◦≤θ≤90◦will be considered for the sake of simplicity. In the following and without loss of generality it will be assumed that σ∞ x>0 and σ∞ x≥σ∞ y, i.e. η≤1. The derivative of σ(θ) evaluated from (4.56a), ∂σ(θ) σ∞ x ∂θ = (η−1)msin (2θ).(4.57) This shows that normal stress is a decreasing function of θfor θ∈[0◦,90◦] and any η≤1. According to this and the expression in (4.56a), σ(θ) achieves its maximum value at θ= 0◦: σmax =σ(θ= 0◦) = σ∞ x·k+σ∞ y·(k−m).(4.58) Note that kand k−mrepresent, respectively, the relative contribution of remote stresses σ∞ xand σ∞ yto this maximum value of σ(θ). Following (4.56a) and (4.58), the influence of σ∞ yon the normal stresses σ(θ) is given by the ratio k/m. In particular, tension σ(θ= 0◦)>0 is generated by a remote compression σ∞ y<0 (tension σ∞ y>0) for k/m < 1 (k/m > 1), assuming small σ∞ x&0. Recalling that σ∞ x>0, the semiangle θ0for which the interface normal stresses vanish is given from (4.56a) by θ0(η;α, β) = arcsin sk m+k m−1η 1−η.(4.59) According to this expression and the analysis in 4.B, the angle θ0∈[0◦,90◦] does not exist for all the considered values of η∈(−∞,1] and all admissible values of k m∈[3/4,+∞), see Mantiˇc (2009). The condition of vanishing derivative of (4.56a) with respect to ηgives the angle θηfor which the interface normal stress is independent of η, θη(α, β) = arccosrk m= 90◦−θ0(η= 0).(4.60) 4.3. Crack initiation under biaxial transverse loads 107 Table 4.3: Examples of isotropic bimaterials constants (1, inclusion; 2, matrix) Bimaterial E1(GPa) ν1E2(GPa) ν2 Glass/epoxy 70.8 0.22 2.79 0.33 Carbon/epoxy 13.0 0.20 2.79 0.33 α β ε E∗(GPa) Glass/epoxy 0.919 0.229 -0.074 6.01 Carbon/epoxy 0.624 0.136 -0.044 5.09 k m k/m θη(◦)η0 Glass/epoxy 1.44 1.56 0.9205 16.3 0.086 Carbon/epoxy 1.32 1.43 0.9200 16.4 0.087 This expression makes sense only if k≤m. Then, θηdivides the interface sector 0≤θ≤90◦into two regions. In the first region (θ < θη) where ∂σ ∂η <0, σ decreases with increasing the remote secondary load σ∞ y. On the contrary, in the second region (θ > θη) where ∂σ ∂η >0, σrises with increasing the remote secondary load σ∞ y. Hence, θηis a key angle to evaluate the influence of load biaxiality. The values of the above defined and used constants characterizing the interface stress distribution for two typical fiber reinforced composites are presented in Table 4.3. The glass/epoxy bimaterial will be used as an example in the present work. Plots of normal stress distributions along the interface obtained from (4.56a) for glass/epoxy are shown in Figure 4.32. Figure 4.32: Distribution of the normal stresses along the interface for several values of ηand glass/epoxy. 108 4. Failure initiation in FRC under transverse loads 4.3.2 The solution for a crack at the interface of a single inclusion under a remote biaxial transverse load Energy based fracture criteria consider a cracked configuration. Hence the aim of this section is to analyze the problem of a partial debond at the fiber-matrix interface. Under certain assumptions, a classic elastic solution particularized for the present problem provides closed form expressions for a fracture mode mixity and the energy release rate, their dependence on the key problem parameters being pointed out. Consider the problem configuration from the previous section altered by the presence of a crack at the interface. In view of the fact that the maximum of σ(θ) is achieved at θ= 0◦, see (4.57), it will be assumed that this crack is symmetrically situated with respect to the x-axis, with a semidebond angle θd≥0 and an infinite length in the z-axis direction, see Figure 4.33. Figure 4.33: The interface crack problem configuration. A more general problem has been analytically studied by Toya (1974) and several other authors using the open model of interface cracks. As will be seen later on, the validity of the analytic solutions based on the open model is limited and, thus, computational methods should sometimes be used employing the contact model of interface cracks, see Section 4.1 for an example on this problem and Par´ıs et al. (2007) for a review. The interface stresses at a point placed ahead of the crack tip at the polar angle θ=θd+θl,θl>0◦, see Figure 4.33, can be expressed by particularizing Toya’s solution for stresses2and rewriting it in terms of the Dundurs parameters, σ(θ, θd, η, β)−iτ(θ, θd, η, β) = −σ∞ x 2 1−α 1−βχ(θ, θd, β)p(θ, θd, η, β),(4.61) where χ(θ, θd, β) and p(θ, θd, η, β) are defined in 4.C). It should be noticed that the stresses along the interface are independent of the inclusion radius a. Ratio of 2The following values of the parameters used by Toya: φ= 0 and ε∞= 0 are taken. 4.3. Crack initiation under biaxial transverse loads 109 the interface shear and normal stresses ahead the crack tip at a small reference length (either geometry or material based) gives a measure of fracture mode mixity of an interface crack. Thus, the angle ψat a reference angle θl, measured from the crack tip (see Mantiˇc (2009) for a discussion about this reference angle for a similar problem), is defined as: tan ψ(θd;θl, η, ε) = τ(θd+θl) σ(θd+θl).(4.62) This angle will be used as a suitable measure of the fracture mode mixity. Figure 4.34 shows the evolution of ψ(θd) for different values of the load-biaxiality parameter η. Figure 4.34: Examples of the evolution of the fracture mode mixity angle ψ(obtained from Toya’s solution of the open model of interfacial cracks) taking θl= 0.1◦, for different values of ηand glass/epoxy. The ERR of the interface crack propagating at its upper crack tip at an angle θdcan be expressed, rewriting Toya’s expression as previously, by G(θd;σ∞ x, σ∞ y;a;E∗, α, β) = (σ∞ x)2a E∗ˆ G(θd;η;α, β),(4.63) where E∗is the harmonic mean of the effective elasticity moduli 1 E∗=1 21−ν2 1 E1 +1−ν2 2 E2(4.64) and ˆ Gis a dimensionless normalized ERR whose expression is presented in 4.D. According to expression (4.63), the ERR varies linearly with the ratio a/E∗and quadratically with the remote load σ∞ x. Figure 4.35 shows the evolution of the normalized ERR ˆ G(θd), and also of its asymptotes for θd≈0◦given by (4.124), for different values of the load-biaxiality 110 4. Failure initiation in FRC under transverse loads Figure 4.35: Examples of the normalized ERR (obtained from Toya’s solution of the open model of interfacial cracks) and its asymptotes for different values of ηand glass/epoxy. parameter η, see 4.E. Validity of these plots is limited by the validity of the open model of interface cracks. Notice, in relation to Figure 4.34, that compressions ahead of the crack tip correspond to |ψ|>90◦and can become relevant for η < 0. These compressions may have associated a relevant overlapping of crack faces close to the crack tip. This is not physically admissible, so it may invalidate Toya’s solution for some values of θdand η. Due to the above mentioned overlapping, an additional unrealistic term, corresponding in some sense to Mode I, appears in the computation of the ERR, causing some overestimation of ˆ G. This overestimation can be studied by using the relation between the ERR based fracture mode mixity and the stress based fracture mode mixity according to Mantiˇc and Par´ıs (2004). This relation allows partitioning ˆ Ginto two components ˆ G(θd) = ˆ GI(θd, δθ) + ˆ GII(θd, δθ), for a given virtual-crack-step angle δθ, as shown in 4.F, leading to, ˆ GI,II (θd, δθ) = 1 2ˆ G(θd)(1 ±F(ε) cos(2(ψ(θd, θl) + ψ0(δθ/θl, ε))),(4.65) where θlis the reference angle for ψ(4.62) and the oscillation index εis given in terms of βin (4.122). Figure 4.36 shows the individual components of the ERR corresponding to a small virtual-crack-step angle δθ = 0.5◦. These plots allow clarifying the range of validity of Toya’s expression of ERR for larger values of θdand different values of η. Decreasing values of ηdecreases the range of the values of θd, where 4.3. Crack initiation under biaxial transverse loads 111 Toya’s expression of ERR is valid. This figure also confirms that the cause of the strong increase of ˆ Gfor large values of θdand η < 0 is associated to a fictitious contribution of ˆ GIdue to a large overlapping. (a) η=−2 (b) η=−1 (c) η= 0 (d) η= 1 Figure 4.36: Plots of the individual components of the ERR associated to δθ = 0.5◦for different values of the load-biaxiality parameter ηand glass/epoxy. (a) η=−2. (b) η=−1. (c) η= 0. (d) η= 1. In order to clarify the influence of the remote secondary load σ∞ yon the values of ˆ G, it is useful to study the variation of the derivative of ˆ G(4.124) with respect to the load-biaxiality parameter ηat θd∼ =0◦ d2ˆ G dηdθdθd=0 =2π(k−m) (k+ (k−m)η)1+4ε2 cosh (πε).(4.66) This expression shows again the importance of the parameter k/m. In fact, the sign of the variation of the asymptotic slope of ˆ Gat θd∼ =0◦with ηis directly characterized by sign d2ˆ G dηdθdθd=0 = sign k m−1k m+k m−1η.(4.67) 118 4. Failure initiation in FRC under transverse loads σ∞,E cx(4.83). Therefore, two scenarios are possible as described in the following. Without loss of generality, only configurations where ˜ θ0(4.118) is defined will be analyzed. In scenario A, both criteria, (4.136) and (4.78) are fulfilled as equalities. This implies that the curves of both criteria have an intersection for a semiangle of the crack after the onset ∆θ= ∆θc, called critical semidebond angle, giving the minimum value of σ∞ cx, ∆θc= ∆θE c= ∆θσ c≤∆θE min.(4.85) As the function s(θ, η) on the right-hand side of the stress criterion is increasing with θ(strictly increasing up to ˜ θ0), it occurs if γqg(∆θE min, η)≤s(∆θE min, η).(4.86) where γis a dimensionless parameter defined as (see Mantiˇc (2009)) γ=1 σcrG1cE∗ a.(4.87) The value of ∆θccan be computed by solving the following nonlinear equation: γpg(∆θc, η) = s(∆θc, η).(4.88) Thus, the value of the critical load originating a crack is easily calculated, using the previously obtained value of ∆θc, from 1 η σ∞ cy σc =σ∞ cx σc =γpg(∆θc, η).(4.89) In scenario B, where condition (4.86) is not fulfilled, the minimum load originating a crack is always associated to ∆θc= ∆θE min,(4.90) because of the increasing character of gfor ∆θ > ∆θE min. Hence, 1 η σ∞ cy σc =σ∞ cx σc =γqg(∆θE min, η),(4.91) the interface crack onset being essentially governed by the energy criterion. In fact, σ∞ cx=σ∞,E cx. Let a threshold value of γbe defined from the equality in (4.86) as γth(η;α, β;λ, θl) = s(∆θE min, η) pg(∆θE min, η).(4.92) It is easy to see, that γth separates scenario A (0 < γ ≤γth) from scenario B (γ > γth). If ∆θE min ≥θ0then γth = +∞in view of (4.135), which means that only scenario A is possible. 4.3. Crack initiation under biaxial transverse loads 119 If (k/m > 1and η > 1/η0)or k/m ≤1then Find min ∆θg(∆θ)def =g(θE min) If γqg(θE min)< s(θE min) Solve the next equation for ∆θ < θE min: γpg(∆θ) = s(∆θ) θc=the solution ∆θof this equation Else θc=θE min Endif Compute the critical load σ∞ cxby σ∞ cx σc=γpg(θc) and σ∞ cyby σ∞ cy σc=η·σ∞ cx σc Else No debond is possible under the present hypotheses Endif End Figure 4.40: Computational procedure for the evaluation of θc,σ∞ cxand σ∞ cy, assuming η≤1. According to the definition of the two scenarios, the critical values of θc,σ∞ cx and σ∞ cycan be computed by the procedure shown in Figure 4.40. The above defined dimensionless structural parameter γ(4.87) can be referred to as stress oriented brittleness number (see Mantiˇc (2009) and further references therein). The adjective ”stress oriented” corresponds to the fact that the critical load (remote stress) is in some sense proportional to this number, see (4.89) and (4.91). In fact, in scenario B (with γ > γth) the critical load is linearly proportional to γ. The brittleness number γis governing brittle-to-tough transition in the fiber-matrix debond onset, small values of γcorresponding to brittle and large values to tough configurations (cf. Kushch et al. (2011)). Whereas the values of the critical angle and load in tough configurations, usually associated to scenario B, are simply described by (4.90) and (4.91), the asymptotic behavior of these values in brittle configurations with vanishing values of γ(γ→0+), associated to scenario A, requires a further analysis. Looking at equation (4.88), defining ∆θc, for γ→0+, it holds g(∆θc, η)→ ∞ because s(∆θc, η) on the right hand side of (4.88) is bounded from below by its positive minimum value (4.71) for a fixed η. Taking into account condition (4.85) and definition of gin (4.79), see also Figure 4.39 and approximation (4.126), it is obtained that lim γ→0+θc= 0◦.(4.93) In the same manner, an asymptotic expression can be extracted as demonstrated in 4.E. 120 4. Failure initiation in FRC under transverse loads When evaluating the critical load for γ→0+, which implies ∆θc→0+, the following approximation is obtained, see (4.71): σ∞ cx σc &s(0◦, η)>0 for γ→0+.(4.94) Combining this equation with the definition of ηin (4.55), the relation between σ∞ cxand σ∞ cycan be approximated as (k−m)·σ∞ cy σc +k·σ∞ cx σc &1 for γ→0+.(4.95) This linear relation between critical stresses is given, in fact, by the stress criterion (4.136) when considered for a small angle ∆θ. Figure 4.41, computed using the computational procedure in Figure 4.40, presents the effect of the brittleness number γon the critical semiangle ∆θc, the arrest semiangle θa(defined later in Section 4.3.3.4) and on the critical remote load σ∞ cx. This figure is a nice illustration of the above mentioned brittle-to-tough transition in the fiber-matrix debond onset. It can be seen that the behavior of ∆θcand σ∞ cxagrees with the above analytic predictions, namely (4.127) with (4.94) for small and (4.90) with (4.91) for large values of γ. Figure 4.42 shows the safe regions and failure envelopes in (σ∞ cx, σ∞ cy) plane for glass/epoxy bimaterial computed for different values of γby applying the procedure described in Figure 4.40. According to this figure, an increase of remote secondary load σ∞ yincreases the critical load σ∞ cxfor sufficiently small values of γ(recall that k/m < 1 and η0>0 for glass/epoxy). However, a nonmonotonic boundary curve of the safe region is observed for greater values of γ in Figure 4.42. In fact, it is observed that the curve which joins the points with the maximum critical remote load σ∞ cxshows that the maximum is situated at η= 1 just for γ→0+. For moderate values of γ, the maximum critical remote load σ∞ cxcorresponds to η < 0. The reason for this behavior is clarified in Figure 4.43 where two situations are explained. In the first case, the value of γis relatively small, Figure 4.43(a), which is associated to small values of ∆θc, as previously demonstrated. For small values of ∆θc, the effect of the secondary load σ∞ yis the same for both the stress criterion and energy criterion curves, both curves descending when ηreduces. In the second case, the value of γis larger, Figure 4.43(b), and the values of ∆θcare greater than θη, see (4.60) and the discussion below. Then, for values of γoriginating θc> θη, an increase of the remote secondary load σ∞ ymakes less restrictive the stress criterion and more restrictive the energy criterion. Thus, the monotony of the function σ∞ cx(σ∞ cy) can be broken down as observed in Figure 4.42 for moderate values of γ. The straight line defined by (4.95) represents, according to Figure 4.42, a limit of failure envelope curves for γ→0+. Note that the failure envelope curves for γ→0+in Figure 4.42 show the most relevant influence of the secondary load σ∞ yon the value of the critical load σ∞ cx. Figure 4.42 also shows the ”threshold curve” which separates scenarios A and B. It is interesting to remark that greater values of γcorrespond to a larger 4.3. Crack initiation under biaxial transverse loads 121 (a) (b) Figure 4.41: (a) Semiangles ∆θc, ∆θE min and θa, and (b) Critical remote tension σ∞ cxas functions of the brittleness number γ, taking λ= 0.3, θl= 0.1◦and glass/epoxy. range of failure behavior governed by scenario B. On the contrary, the presence of a remote secondary compression σ∞ y<0 leads to scenario A, for small and moderate values of γ. Figure 4.44 studies the influence of αand βvalues on the biaxial safe region for γ→0+, for selected theoretical (but possible) bimaterials and also for usual composites. From Dundurs’ α−βparallelogram it is seen that the most common bimaterials have very similar properties in the debond onset problem. A more extensive list of α−βvalues for real bimaterials can be found in Suga et al. (1988) and Schmauder and Meyer (1992). The safe region in the limit case γ→0+is defined by the intersection of 122 4. Failure initiation in FRC under transverse loads Figure 4.42: Critical biaxial loads originating a crack for different values of γ, taking λ= 0.3, θl= 0.1◦and glass/epoxy. the semiplanes including the origin of coordinates and limited by the straight line defined by (4.95) and the symmetric one with respect to the bisector of the coordinate axes. From (4.95), the position of the corner point of the safe region (η= 1) is given by σ∞ cx σc =σ∞ cy σc =1 2k−m.(4.96) The slope of the linear relation in (4.95) characterizes the influence of the secondary load σ∞ yon the critical load σ∞ cx. This slope can be expressed as: lim γ→0+∂σ∞ cx ∂σ∞ cy=η0(4.97) where η0is defined in (4.115) and its range in (4.116), see also Figure 4.49. 4.3. Crack initiation under biaxial transverse loads 123 (a) γ= 0.4 (b) γ= 1 Figure 4.43: Stress and energy criteria curves, taking λ= 0.3, θl= 0.1◦and glass/epoxy for two different values of γ. (a) γ= 0.4, (b) γ= 1 Hence, this slope is only dependent on the elastic bimaterial properties and does not depend on the interface properties. In view of the range of possible values for the slope (4.116), the safe region is always convex, cf. Figure 4.44(b). For k/m < 1 the slope (4.97) is positive and an increase in the secondary load σ∞ yincreases the critical load σ∞ cxnecessary to originate a debond, see Figure 4.44. This is the case of the two bimaterials defined in Table 4.3, i.e. glass/epoxy and carbon/epoxy. However, an opposite effect is predicted for k/m > 1, see Figure 4.44. In fact, this dependence matches up with the effect of the secondary load 124 4. Failure initiation in FRC under transverse loads 1 3 -1 0.6 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 2 ¥ 0.7 AA BB DD EE FF 0.5 Carbon/epoxy Boro/epoxy Aramid/epoxy Glass/polyester Glass/epoxy -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 C -1.0 -0.5 0.0 0.5 1.0 -0.4 -0.2 0.0 0.2 0.4 Α Β (a) A B C SAFE FAILURE -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 Σcx ¥Σc Σcy ¥ Σc SAFE FAILURE D E F -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 Σcx ¥Σc Σcy ¥ Σc FAILURE SAFE Glass/epoxy Boro/epoxy Aramid/epoxy Glass/polyester Carbon/epoxy -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 Σcx ¥Σc Σcy ¥ Σc (b) Figure 4.44: (a) α−βdiagram for bimaterials in plane strain with isovalue curves for γ→0+ corresponding to: solid lines with values of σ∞ cx/σc=σ∞ cy/σc, and dashed lines with values of ∂σ∞ cx/∂σ∞ cy. (b) Some biaxial failure envelopes for γ→0+for selected points in the α−β diagram and usual composites. A: k/m = 0.75, B: k/m = 1, C: k/m →+∞, D: k/m = 1.25, E: k/m = 1.25, F: k/m = 1.25, Carbon/epoxy: k/m = 0.9200, Glass/polyester: k/m = 0.9201, Glass/epoxy: k/m = 0.9205, Aramid/epoxy: k/m = 0.9217, Boro/epoxy: k/m = 0.9204 σ∞ ypredicted by Goodier’s solution for the interface point θ= 0◦, see (4.52), which shows that a compression or tension is expected at θ= 0◦when a secondary load σ∞ y>0 is applied for k/m < 1 or k/m > 1, respectively. The present results show that the influence of the secondary load σ∞ yon the critical load σ∞ cxis at most moderate in usual composites. Taking into account that in the case γ→0+, analyzed in Figure 4.44, the values of σ∞ cxare the most sensitive to the values of σ∞ y, in general for k/m < 1 the influence of σ∞ yon the value of σ∞ cxexists but it is small or at most moderate. For bimaterials as glass/epoxy and carbon/epoxy with k m<1, these results agree with Par´ıs et al. (2003). According to these authors a secondary compression σ∞ ymakes easier 4.3. Crack initiation under biaxial transverse loads 125 the debond onset for these bimaterials and reduces the critical load σ∞ cxas shown in Figure 4.44(b). However, for bimaterials with k m>1 the effect is opposite, a secondary compression σ∞ y<0 increases the critical load σ∞ cx. 4.3.3.4 Post-crack-onset evolution After the onset of a new crack, an unstable growth of the crack is possible depending on the relation between G(θd) and Gc(θd) for θd≥θcand according to the criterion of the classical (infinitesimal) interface fracture mechanics (see, e.g. Par´ıs et al. (2007); Mantiˇc et al. (2006)). Thus, the condition for the further crack growth will be G(θd, η)≥Gc(ψ(θd, η)), θd≥θc.(4.98) The crack will stop growing at an arrest angle θa≥∆θcverifying G(θa, η) = Gc(ψ((θa, η)) if for angles θd&θacriterion (4.98) is not fulfilled. The stability of the post-onset growth of the crack is different for two scenarios A and B separated by γth two post-onset scenarios being possible: •For γ < γth,G(θc, η)> Gc(ψ(∆θc, η)) and the crack is expected to grow in an unstable manner up to an arrest angle θa>∆θE min, which can be shown similarly as in Mantiˇc (2009). •For γ≥γth, ∆θc= ∆θE min,G(∆θc, η) = Gc(ψ(∆θc, η)) and the derivative dG/dθd|θd=∆θE min ≤dGc/dθd|θd=∆θE min , see (4.82) and the related discussion in Section 4.3.3.2. Therefore, assuming strict inequality in (4.82) (which, in fact, has been verified in all present calculations), no unstable crack growth is usually expected after the crack onset and θa= ∆θE min. Figures 4.41(a) and 4.45 were computed by implementing the above ideas. The values of Gand Gcand their derivatives are compared for θd≥∆θcin order to find the arrest semiangle θa. According to these figures, a long unstable crack growth after the crack onset is predicted for small values of γ(brittle configurations), whereas short (or zero) unstable crack growth is predicted for large vales of γ(tough configurations). 4.3.3.5 Applicability of the open model of interface cracks The applicability of the theoretical model developed is limited by its assumptions, perhaps the most restrictive being the usage of the open model of interface cracks. Toya’s (1974) solution assumes negligible overlapping of traction-free crack faces. The angle of the overlapping zone at the crack tip can be estimated by the formula deduced by Hills and Barber (1993) and generalized by Graciani et al. (2007), which rewritten for the present case is defined as the largest value of θI(θd, η) = θl·exp [((2n−1/2) π−ψ(θd, η, θl)signε+ arctan(2 |ε|)) /|ε|],(4.99) lower than the semidebond angle θd, with nbeing an integer. 126 4. Failure initiation in FRC under transverse loads Figure 4.45: Semiangles ∆θc, ∆θE min and θaas functions of the biaxiality parameter ηfor different values of γ, taking θl= 0.1◦,λ= 0.3 and glass/epoxy. Figure 4.46 shows the evolutions of ˆ Gfor glass/epoxy and different values of the load biaxiality parameter η. Additionally, for γ= 1.5, corresponding to relatively tough configurations, the values of ∆θc, ∆θE min and θacomputed by the present model are indicated. Finally, angles θI,1% for which the overlapping zone represent 1% of the crack length, i.e. θI/2θd= 0.01, providing a reasonable limit of validity of the open model, are also presented in Figure 4.46. This figure shows that all the values of ∆θcare lower than the reference limit θI,1%, therefore the open model is acceptable for the evaluation of ∆θcand the critical load σ∞ cx. However, it might not be fully acceptable when computing θafor large negative values of η. Note that, the above discussed limit on the semiangle θdis mainly due to somewhat inaccurate evaluation of ˆ Gbecause of a large overlapping zone at the crack tip, see also Figure 4.36 and the related discussion. A correct procedure for the evaluation of ˆ Gis such cases would require employing the contact model of interface cracks as in Par´ıs et al. (2007) and Correa (2008). 4.3.4 Size effect of the inclusion radius aon the crack onset and its variations with the load biaxiality Analogously to the uniaxial case, see e.g. Section 4.2, a size effect in the present debond onset problem can be understood as a dependence of the critical remote load (σ∞ cx,σ∞ cy) and critical semiangle ∆θcon the only geometric parameter in the present problem, the inclusion radius a. The objective of this section is to study the variation of this size effect predicted by the uniaxial model in the biaxial case. 4.3. Crack initiation under biaxial transverse loads 127 Figure 4.46: Plots of evolution of values of dimensionless ERR limited by the semiangles θd with θI 2θd= 0.01 which estimate the validity of the model, and representation of values of semiangles ∆θc, ∆θE min,θ0and θa, taking λ= 0.3, θl= 0.1◦,γ= 1.5 for glass/epoxy. Let a bimaterial characteristic length a0be defined in terms of the interface properties σcand G1c and the harmonic mean of effective Young moduli E∗, a0=G1cE∗ σ2 c .(4.100) Then, the ratio a/a0and γare related by γ=ra0 a.(4.101) A threshold value of acan be defined in terms of the threshold value γth (4.92), ath =a0 γ2 th .(4.102) scenario A taking place for a≥ath and B for a<ath. In fact, all the above analyses of the results obtained by the coupled stress and energy criterion taking γ as a governing parameter could be rewritten in terms of the ratio a/a0, see Martin et al. (2008) for a similar approach. For sufficiently large values of a, which correspond to small values of γ, the critical semiangle θcand the critical remote load σ∞ cxcan be approximated by the following expressions, see Section 4.3.3, ∆θc∼ =2cosh2(πε) π(1 + 4ε2) a0 a(4.103a) 134 4. Failure initiation in FRC under transverse loads G=|K|2/(E∗·cosh2(πε)), this alternative brittleness number is expressed as γK=γ·cosh (πε) = K1c σc 1 √a.(4.109) Recall that the expression of γKin terms of K1c reminds the classical definition of the brittleness number sin homogeneous materials by Carpinteri (1981). Note that, ε= 0 for a crack in a homogeneous material, thus γK=γin this case. The way how the remote secondary load σ∞ yinfluence the critical value of remote tension σ∞ cxdepends on the value of the ratio k/m =1 2 2+α−β 1+α−2β, defined in terms of the Dundurs elastic bimaterial parameters αand β. In particular, for γ1, a remote secondary compression σ∞ ydecreases or increases σ∞ cxif k/m < 1 or k/m > 1, respectively. This result, for k/m < 1, is coherent with the hypothesis proposed and experimentally verified by Par´ıs et al. (2003) for a particular carbon/epoxy composite. For moderate or larger values of γ,γ&1, this model predicts an almost negligible influence of the secondary compression σ∞ yon the critical tension σ∞ cx for the glass/epoxy bimaterial studied, having a slightly flat maximum for η. 0 (see Figure 4.42). This observation is related to the fact that the critical semidebond angles θcpredicted for these values of γare sufficiently large to make the influence of the secondary compression σ∞ ymore complex. However, the latter conclusions should be accepted with a caution in view of the range of model applicability, which appears to be very suitable for brittle configurations but to a lesser extent for tough ones. In addition to the inclusion-matrix debond onset mechanism studied in the present work, other failure mechanisms can occur in the inclusion-matrix system under remote transverse loads. This is, for example, the case of the dominating compressive load studied by the coupled stress and energy criterion in Quesada et al. (2009), where parallel cracks in the inclusion and the matrix are predicted. Another example of inclusion-matrix debond configurations not allowed by the present assumption of the debond symmetrically situated with respect to the principal directions of the remote load were studied in Correa et al. (2008). Experimental tests of specimens subjected to remote transverse compressions show debonds originating at interface positions with large shear stresses. Thus, in order to complete the picture of failure envelopes shown in Figure 4.42, such configurations should be studied in a similar way as done in the present work. Other model was proposed by Carraro and Quaresimin (2014) evaluating the influence of a out-of-plane shear stress on the main secondary transverse load. A full theoretical model could be generated if all of these partial models are combined. Appendices to Chapter 4 4.A Dimensional analysis of the energy release rate of a debond at the fiber-matrix interface The value of the ERR of a debond at a loaded specimen is a functional depending on the stress tensor σij and the displacement vector uiin the whole specimen, G=f(σij(x, y), ui(x, y)).(4.110) The solution for stresses and displacements depends on the problem geometry, which includes the debond angle, the elastic properties of the material and boundary conditions. Assuming that the length of the contact zone is independent of the remote tension σ∞>0, since this is a frictionless receding contact problem, σij can be rewritten as the product of σ∞and a function ˆσij depending on the point (x, y) and the bimaterial elastic properties (E1, ν1, E2, ν2) σij =σ∞ˆσij(x, y, E1, ν1, E2, ν2, a),(4.111) and analogously ui=σ∞ˆui(x, y, E1, ν1, E2, ν2, a).(4.112) It is known from the LEFM theory that the ERR value is a homogeneous and linear function of a product of displacements and stresses, G= (σ∞)2h(ˆσij,ˆui)=(σ∞)2h(θd, a, n, E1, ν1, E2, ν2),(4.113) where the terms x,y,iand jhave been omitted since the ERR value cannot depend on them. Then, in order to obtain the dimensionless parameters which govern the value of G, two independent dimensional parameters to normalize this expression should be selected since this is a static mechanical problem, see Barenblatt (1996) for details. Taking the fiber radius aand E∗defined in (4.9) and taking into account that the elastic parameters can be reduced to two under the conditions discussed by Dundurs (1967), the value of Gcan be expressed as G=(σ∞)2a E∗ˆ G(θd, n, α, β).(4.114) 135 136 4. Failure initiation in FRC under transverse loads 4.B Domain of existence of the angle θ0where the normal traction vanishes A threshold parameter η0can be defined as η0(α, β) = k m−1 −1.(4.115) It will be useful to know the range of this parameter given in terms of the values of k/m, η0∈(−1,0] | {z } k m≥1 ∪(0,1/3] | {z } 3/4≤k m<1 .(4.116) Then, the expression on the right-hand side of (4.59) makes sense for η < 1 only if η≤η0if 3 4≤k m≤1, 1 η0≤η≤η0if k m>1.(4.117) As can be seen in Figure 4.49, only tensions (compressions) take place along the whole interface for η > η0(for η < 1/η0and k m>1). It will be convenient to extend the definition of θ0defining ˜ θ0as follows ˜ θ0(η;α, β) =    θ0,∃θ0,see (4.117), 90◦, η0≤η≤1, @, η < 1/η0and k/m > 1. (4.118) Figure 4.49: Domain of definition of an angle θ0∈[0◦,90◦] along the interface where the normal tractions vanish as a function of the elastic bimaterial parameter k/m and the biaxiality parameter ηfor σ∞ x>0. 4.C. Functions χand pof the Toya’s solution 137 4.C Functions χand pof the expression of Toya for stresses along the interface with a partial debond χ(θ, θd, β)=(eiθ −eiθd)−(1/2)−iε(eiθ −e−iθd)−1/2+iε,(4.119) p(θ, θd, η, β) =q(θd, η, β)(eiθ −(cos θd−2εsin θd)) −1 + α 1−α(1 −η)e2ε(π−θd)−iθ× ×(cos θd+ 2εsin θd−e−iθ), (4.120) with q(θd, η, β) = h(1 + η)(1 −(cos θd−2εsin θd)e2ε(θd−π)) + +1 2(1 + α)(1 + 4ε2)(1 −η) sin2θd/ /h3 + α−(1 −α)(cos θd−2εsin θd)e2ε(θd−π)i−1 + η 1−α. (4.121) where i=√−1 is the imaginary unit, and ε= (1/2π) ln(1 −β)/(1 + β) (4.122) is the so-called oscillation index (see Table 4.3). 4.D Expression of the dimensionless ERR ˆ G ˆ G(θd, η, α, β) =π 4e−2ε(π−θd)(α−1)21+4ε2× ×||c(θd, η, α, β) + d(θd, η, α, β)||2·sinθd, (4.123a) with c(θd, η, α, β) = −0.5(1 + α)1+4ε2(η−1)sin(θd)2+ + (1 + η)1 + e−2ε(π−θd)(2εsin(θd)−cos(θd)))i/ /h3 + α+e−2ε(π−θd)(α−1)(cos(θd)−2εsin(θd))i, (4.123b) and d(θd, η, α, β) = 1−e2ε(π−θd)−iθd(1 + α)(η−1) + η α−1.(4.123c) 4.E Asymptotic analysis of ˆ Gand Gfor a vanishing critical semiangle ∆θc For a vanishing crack semiangle, the derivatives of ˆ Gand Gwith respect to the crack semiangle and semilength respectively are, dˆ G dθdθd=0 = (k+ (k−m)η)2π1+4ε2 cosh2(πε),(4.124) 138 4. Failure initiation in FRC under transverse loads dG d(aθd)θd=0 = [(k+ (k−m)η)σ∞ x]2π1+4ε2 E∗cosh2(πε).(4.125) This expressions agree with the analogous derivative of ERR seen in Rice (1988) for a crack at a straight interface. It is instructive to notice that the factor (k+ (k−m)η) agrees with the concentration factor of normal tractions σ(θ) obtained from Goodier’s solution at θ= 0◦, see (4.56a) and (4.58). On the other hand, the function gin (4.79) can be approximated for small values of ∆θ. As ψ(θd) is small for small θd, thus tan2(1 −λ)ψis negligible with respect to the unity and Gc(ψ)'G1c. Then, using (4.124), for small ∆θ, g(∆θ, η;α, β;λ, θl)&˜g(∆θ;η;α, β) = 2 ˆ G0(0◦;η;α, β)·∆θ= =cosh2(πε) π(1 + 4ε2) (k+ (k−m)η)2 2 ∆θ, (4.126) where the asymptotic approximation function ˜gis smaller than the exact function g, see Figure 4.39. Then, combining the approximation of gin (4.126) and (4.88)–(4.70), θccan be approximated by θc∼ =2(k+ (k−m)η)2 ˆ G0(0) γ2=2cosh2(πε) π(1 + 4ε2)γ2for γ→0+.(4.127) Note that, the asymptotic solution is independent of the load biaxiality parameter η. Therefore, for small values of γ, the effect of the remote secondary load σ∞ y on the semiangle of the crack originated is negligible. 4.F Partitioning of ˆ Ginto two components ˆ GIand ˆ GII ERR based measure of fracture mode mixity, angle ψG, is defined in terms of GI and GII as follows tan2ψG=GII GI =ˆ GII ˆ GI .(4.128) According to Mantiˇc and Par´ıs (2004), the ERR and the SIF based measures of mode mixity, phase angles ψand ψG, respectively, can be related by the following expression: cos(2ψG) = F(ε) cos [2(ψ+ψ0(δθ/θl, ε))] ,(4.129) where F(ε) = ssinh(2πε) 2πε(1 + 4ε2),(4.130) and 2ψ0(δθ/θl, ε)=2εln(δθ/2θl) + ϕ(ε)−arctan(2ε) (4.131) 4.G. Expressions for computational implementation 139 with ϕ(ε) = arg Γ(1/2 + iε) Γ(1 + iε),(4.132) Γ(·) being the gamma function. Combining (4.128), (4.129) and some trigonometric identities, the value of ˆ GI(θd, δθ) and ˆ GII (θd, δθ) can be expressed in terms of ψ(θl) as, ˆ GI,II(θd, δθ) = 1 2ˆ G(θd)(1 ±F(ε) cos(2(ψ(θd, θl) + ψ0(δθ/θl, ε))).(4.133) 4.G Expressions for computational implementation Combining the upper limit given by θ0in (4.73), those for the existence of θ0 in (4.117) with the stress condition in (4.70), and the condition of minimal load (4.71), a maximum semiangle of a debond for a given load according to the stress criterion is defined as θσ cσ∞ x σc , η;α, β=             @,(η≤1/η0and k/m > 1) or σ∞ x σc<1 k+(k−m)η, 180◦, η > η0and σ∞ x σc≥1 k(1+η)−m, arcsin rk+(k−m)η−σc σ∞ x (1−η)m,in other cases. (4.134) In the same sense, the dimensionless function s(θ) can be generalized for θ∈[0◦,180◦] as s(θ, η;α, β) =        1 k+(k−m)η−(1−η)msin2θ,∃˜ θ0and θ < ˜ θ0, +∞,∃θ0and θ≥θ0, 1 k(1+η)−m, η > η0and θ≥90◦, +∞, η ≤1/η0and k/m > 1. (4.135) Then, the present stress criterion can be written in a general form as follows: an interface debond onset of an angle 2∆θ(symmetrical with respect to the x-axis) is possible if σ∞ x σc≥s(∆θ, η),(4.136) θσ crepresenting the upper limit for the semiangles ∆θverifying (4.136) for a given remote load. 4.H Proof of inequality dG dθd≤dGc dθdfor θd= ∆θE c= ∆θE min For the sake of simplicity, the following notation will be used G(θd) = G(θd;σ∞ x, η;a;E∗, α, β),(4.137a) 140 4. Failure initiation in FRC under transverse loads Gc(θd) = Gc(ψ(θd, η), G1c, λ).(4.137b) Recall that ∆θE cis defined as the minimum positive angle for which the equality in (4.77), or equivalently in (4.78), hold. Then, if ∆θE c= ∆θE min, Z∆θ 0 G(θd)dθd<Z∆θ 0 Gc(θd)dθdfor 0 <∆θ < ∆θE min,(4.138) and Z∆θE min 0 G(θd)dθd=Z∆θE min 0 Gc(θd)dθd.(4.139) By subtracting (4.138) from (4.139), Z∆θE min ∆θ G(θd)dθd>Z∆θE min ∆θ Gc(θd)dθdfor 0 <∆θ < ∆θE min.(4.140) Let G(θd) and Gc(θd) be approximated by Taylor polynomials centered at θd= ∆θE min, G(θd)≈G(∆θE min) + dG dθdθd=∆θE min (θd−∆θE min) + O(θd−∆θE min)2, (4.141a) Gc(θd)≈Gc(∆θE min) + dGc dθdθd=∆θE min (θd−∆θE min) + O(θd−∆θE min)2. (4.141b) Then, introducing (4.141) in (4.140) and integrating the Taylor polynomials, the following inequality is obtained, denoting h= ∆θE min −∆θ: G(∆θE min)h−dG dθdθd=∆θE min h2 2+Oh3> Gc(∆θE min)h−dGc dθdθd=∆θE min h2 2+Oh3. (4.142) for 0 <h<∆θE min. Hence, taking into account that G(∆θE min) = Gc(∆θE min) from (4.81) and considering (4.142) for vanishing h > 0, dG dθdθd=∆θE min ≤dGc dθdθd=∆θE min ,(4.143) otherwise (4.142) could not hold for a sufficiently small h > 0. CHAPTER 5 Failure initiation at spherical-particle reinforced composites Particle-reinforced composites are extensively used in a variety of industrial applications. Their main advantages over the fiber-reinforced composites are costeffectiveness and manufacturing flexibility since they can be machined with the majority of traditional manufacturing processes designed for metals. In particular, spherical particles are usually employed as a reinforcement to enhance tensile strength, stiffness, tear and abrasion resistances of polymeric matrices, see e.g. Fu et al. (2008); Shen et al. (2002), or to increase fracture toughness of ceramic matrices, see Dlouhy and Boccaccini (1996). The manner in which the reinforcement modifies the failure properties of the unreinforced matrix is closely related to the reinforcement geometry, the contrast in elastic properties between reinforcement and matrix, and the mechanical behavior of the interface between them. The dependence between the macroscopic properties of the composite and these parameters have been observed in several experiments, see Fu et al. (2008) for a review. In particular, experiments show that the particle size affects significantly the tensile strength. In the specific case of reinforcements at a micro scale, composites with smaller particles present a higher tensile strength, see Gent and Park (1984); Yoshinobu et al. (1992); Cho et al. (2006). This is due to the influence of the microstructure on the first stages of the failure mechanisms. In fact, microscope observations by Cho et al. (2006) show that failure initiates as debonds at the particle-matrix interface and subsequently grow along the interface up to an angle for which the debonds kink out of the interface to coalesce with others. The importance of these composites and the observed influence of the reinforcement and interface properties on the strength have encouraged a wide variety 141 142 5. Failure initiation at composites with spherical particles of analysis at the reinforcement scale which is usually micro or nano. The problem of a spherical particle embedded in a matrix has been intensively studied since the seminal works by Goodier (1933) and Eshelby (1957) who introduced solutions and formalisms for this geometry assuming isotropic elastic materials. The solution of the basic problem of a spherical particle embedded in an infinite matrix, assuming a perfect particle-matrix interface, subjected to a uniaxial tension is fundamental in understanding micromechanics of particle-reinforced composites. When the spherical particle is stiffer than the matrix, this solution shows a stress concentration at the poles of the sphere axis parallel to the tension direction. As a consequence, these are a priori preferred points for a failure initiation in the form of a debond at (or a void near) the interface. A broad variety of works have been presented dealing with the interface debond onset and growth by different approaches, e.g. stress analysis assuming an elastoplastic matrix, see e.g. Gent (1980); Tszeng (2000), cohesive zone models for the interface, see e.g. Needleman (1987); Tan et al. (2007); Lee and Pyo (2007, 2008); Othmani et al. (2011) or using an energy criterion assuming a material-dependent critical debond angle by Tszeng (1993). This chapter aims to develop a model to predict the debond onset at the particle-matrix interface as a function of the particle, matrix and interface properties employing the coupled criterion of the finite fracture mechanics (FFM) described in Chapter 3. Some works applying the coupled criterion dealing with similar problems to the present one have been presented for long fiber inclusions, see Chapter 4 and Carraro and Quaresimin (2014), for long fiber inclusions, and in Zappalorto et al. (2011), for nano-spherical inclusions subjected to hydrostatictension loading. The chapter is organized as follows: first, the theoretical model is developed in Section 5.1 in accordance with the formulation proposed in Chapter 3, i.e. initially the conditions imposed by the stress and energy criteria are separately obtained in Sections 5.1.1 and 5.1.2 respectively, being combined in Section 5.1.3. The model is particularized for glass/vinylester employed by Cho et al. (2006) and a preliminary comparison with their experiments is carried out in Section 5.2. 5.1 A theoretical model based on the finite fracture mechanics Consider a piece of a composite reinforced by spherical particles subjected to a remote uniaxial tension σ∞>0 as shown in Figure 5.1(a). Under the assumption of dilute packing (with a low reinforcement volumetric fraction), the present work focuses on the debonding process at the interface of a single particle by neglecting the influence of the neighboring ones. Thus, the geometry under study is simplified to that shown in Figure 5.1(b): a spherical particle of radius aperfectly bonded to an infinite surrounding matrix. In the context of the present model, the spherical particles are assumed to be stiffer than the surrounding matrix. Both particles and matrix are assumed to be isotropic and linear elastic. For the sake of illustration of the model, a glass/vinylester com- 5.1. A theoretical model based on the finite fracture mechanics 143 posite used in experiments in Cho et al. (2006) is taken as an example here, see material properties in Table 5.1. As will be justified in Section 5.1.1, preferred points for debonding are situated at the poles of the sphere axis parallel to the load direction. In view of this, for a certain critical value of the remote load σ∞ c, two debonding modes will be studied in the following: the onset of either a single debond at one of the poles or two symmetric debonds at both poles, see Figure 5.1(c). As both geometries and loads are axisymmetric, the problem is solved as an axisymmetric one, see Graciani et al. (2005), measuring all the angles from the symmetry axis. Figure 5.1: Schematic of the models employed to study the process of debond initiation at microparticles-reinforced composites under the assumptions of the finite fracture mechanics. (a) Particle-reinforced composite specimen subjected to tension, Spherical particle (b) perfectly bonded to the surrounding matrix or (c) partially debonded at the interface. Following the FFM approach Leguillon (2002); Hashin (1996); Cornetti et al. (2006), next subsections will be devoted to obtain the independent conditions imposed by the stress and energy criteria for the debond onset. Finally, and 246 6. Transverse cracking in cross-ply laminates The second derivative of gcan be calculated from the first one, obtaining, g00 =Iˆ G−∆ˆaI0 ˆ G0 I2 ˆ G−2Iˆ GI0 ˆ GIˆ G−∆ˆaI0 ˆ G I4 ˆ G ,(6.132) Iˆ G−∆ˆaI0 ˆ G= 0 for ∆ˆa= ∆ˆamin,g because g0(∆ˆamin,g) = 0. Then g00 (∆ˆamin,g) = Iˆ G−∆ˆaI0 ˆ G0 I2 ˆ G ,(6.133) and knowing that I2 ˆ G>0, the sign of the second derivative is reduced to sign (g00 (∆ˆamin,g)) = −sign (G0(∆ˆamin,g)) ,(6.134) where the fundamental theorem of calculus has again been applied. This relation implies that the (first) minimum point of g(where g00(∆ˆamin,g)> 0) is situated in the region where ˆ Gis a decreasing function ( ˆ G0<0), i.e. after the (first) maximum of ˆ G, ∆ˆamin,g >ˆamax,ˆ G,(6.135) where ˆamax,ˆ Gcorresponds to the (first) maximum of ˆ G. 6.F Proof of that ∆ˆamin,g is a lower bound for the arrest length Proposition: ˆaa≥∆ˆamin,g Proof: In scenario B, the proof has been given in Section 6.1.4. In scenario A, let ∆ˆaα cdenote the length of the crack at the onset and εc,α yy the critical strain of a determinated configuration α. Let εβ yy denote the minimum strain required by the energy criterion for a given value of ∆ˆaβ∈(∆ˆaα c,∆ˆamin,g), it means the value of the right-hand side of (6.17) for ∆ˆa= ∆ˆaβ. Taking into account that g(∆ˆa) is a decreasing function for ∆ˆa < ∆ˆamin,g, εc,α yy (∆ˆaα c)> εβ yy(∆ˆaβ),(6.136) and introducing from (6.121) the dependence of Gon εyy, Gˆa, εc,α yy (∆ˆaα c)> G ˆa, εβ yy(∆ˆaβ),∀ˆa∈(0,1) (6.137) and from (6.30) that G∆ˆaβ, εβ yy(∆ˆaβ)> Gcfor ∆ˆaβ<∆ˆamin,g implies that G∆ˆaβ, εc,α yy (∆ˆaα c)> Gc,for ∆ˆaα c<∆ˆaβ<∆ˆamin,g.(6.138) Then, the condition to arrest the crack (6.33) is not fulfilled for values of the length lower than ∆ˆamin,g. Therefore, ∆ˆamin,g, which is independent of the fracture and strength properties, is a lower bound of the arrest length. CHAPTER 7 Crack initiation at a weak surface in a V-notch Crack initiation at stress singular points, e.g. corner singularities, has been studied for decades and is still an open subject. They are preferential failure points and their presence has a strong effect on the global strength of structures. Despite this importance, the problem of predicting the critical load originating the failure and its direction is still an open problem, the scientific debate is not closed on this point, especially for complex loadings. One of the reasons is that the linear elastic fracture mechanics (LEFM) itself is not able to predict the onset of a crack at such a location. Several criteria have been developed based on very different approaches. They mainly consider that the generalized stress intensity factor (GSIF) is the critical parameter to predict failure at corner singularities. This idea is based on many experimental evidences, see e.g. Reedy Jr and Guess (1993); Seweryn (1994); Dunn et al. (1997). In the context of the finite fracture mechanics theory (FFM), a new model based on the coupled criterion was proposed by Leguillon (2002) to predict the onset of a crack at corner singularities. It enables to obtain the critical value of the GSIF under symmetric loading as shown in previous reference or by other authors, see e.g. Carpinteri et al. (2008). The predictions agree quite well with the experimental results. Moreover, one of the main advantages of this new criterion is that the critical GSIF can be expressed as a function of two commonly used material properties: the tensile strength σcand the fracture toughness Gc of the material. Yosibash et al. (2006) and Priel et al. (2008) generalized this criterion to corner singularities under non-symmetric loads. In this case the two first singular terms of Williams’ expansion must be accounted for. The leading one corresponds to a symmetric mode and the second one corresponds to an antisymmetric mode which is activated only if the loading is not symmetric. Then, the generalization developed here allows obtaining the critical load, the critical length and the 247 248 7. Crack initiation at a weak surface in a V-notch angle of the new created crack, which is not situated necessarily symmetric (i.e. along the bisector). Despite the non-symmetry of the problem the influence of the shear stresses in the stress criterion and the effect of the stress mixity at the crack tip in the energy criterion are neglected in these approaches. This is because the new crack is assumed to grow in pure mode 1 (or very close to, depending on the choice of the criterion, see Erdogan and Sih (1963)). This angle is very close to that where the normal stresses and the energy release rate reach their maximum value. Moreover, shear stresses at this angle vanish as can be derived from the equilibrium equations. This hypothesis is consistent with the criterion of local symmetry, see Barenblatt and Cherepanov (1961); Erdogan and Sih (1963), usually employed to predict the crack path in LEFM. This is formulated for homogeneous solids having homogeneous properties of fracture and strength and it gives a very good agreement with experimental results for corner points as have been shown by Yosibash et al. (2006) and Priel et al. (2008). However, the hypothesis of pure mode 1 after the crack onset is not longer valid in the presence of a preferential crack path imposed by an interface or a weak surface with weaker fracture and strength properties. The most simple example of such a preferential direction is the presence of a weak adhesive joint between similar or dissimilar materials. In this case, the joint is generally a preferential path for the crack despite the facts that in general the crack does not grow in pure mode 1 and that the shear stresses do not vanish. Thus, both shear stresses and the variation of the fracture toughness with the stress mixity can play a crucial role. Recent experiments (Toda et al. (2001); Ogihara and Koyanagi (2010)) have shown the importance of the shear stresses in failure of adhesive joints. As a consequence, in the cases where a high ratio of shear to normal stresses is expected, shear stresses should not be neglected. Similarly, the so called mode-mixity cannot be neglected in the energy criterion because the fracture toughness can depend strongly on the mixity as shown by several experiments, see e.g. Hutchinson and Suo (1992) or Banks-Sills et al. (2000). Therefore, it is necessary to generalize the criterion presented by Yosibash et al. (2006) including the shear stresses and mode-dependent fracture toughness in order to study a crack initiating and growing along the weak surface. The crack initiation at a corner in the presence of an interface is a common problem in composites and can be associated with the first step of some failure mechanisms. A well studied example is the failure initiation at the multimaterial corner which can be found in double lap joints, see Barroso (2007); Vicentini (2012). The model developed in this chapter can be easily extended to account for the study of this particular case. Regarding the computational methods, the influence of shear stresses and fracture mode-mixity has been implemented in some models. A general law taking into account these dependencies for cohesive zone models (CZM) for interfaces is presented in Tvergaard and Hutchinson (1993). In this model, the normal and shear stress laws depend on the normal and tangential separations. In the limit where the tangential (or normal) separation vanishes, the normal (or shear) stress peak is the tensile (or shear) strength and the work of separation is the fracture toughness in pure mode 1 (or 2). In the general case the proposed 7.1. The singular stress field at a v-notch 249 model couples both effects. Based on this pioneering work, several cohesive laws have been proposed and implemented during the last years, among them it is worth mentioning the work by Freed and Banks-Sills (2008) which proposed a new CZM which coincides with the well known law of Hutchinson and Suo (1992) if the mixity is defined as the ratio of tangential to normal separation. In the similar way, T´avara et al. (2010) proposes a linear elastic brittle interface model (LEBIM). In the context of the coupled criterion criterion of the FFM, Mantiˇc (2009) implements a mode-dependent toughness in order to apply the criterion to particular problems of interfaces where a high mixity exists and proposes also as a future development the introduction of Mohr-Coulomb type law in order to take into consideration the influence of the shear stresses on the failure. This chapter proposes the generalization of Yosibash et al. (2006) and Priel et al. (2008) for the case where a weak surface exists on a V-notched specimen under non-symmetric loading. The singular stress field at a v-notch is shortly described in Section 7.1. Stress and energy criteria considering the shear stresses and mode-dependent fracture toughness are implemented to the present problem in Sections 7.2 and 7.3 respectively. Both conditions are combined in Section 7.4 where the first general expressions are obtained. In Section 7.5, these expressions are applied to a particular problem, the simulation of the Arcan test on V-notched specimens embedding a weak surface starting from the notch root. The solution for the crack onset is a competition between two possible paths: the crack is initiated along the weak surface if it is sufficiently weak or in the bulk at an optimum angle calculated by Yosibash et al. (2006). Finally, both possibilities are studied in Sections 7.6.1 and 7.6.2 and compared for a particular case of material and weak-surface properties in Section 7.6.3. 7.1 The singular stress field at a v-notch Stress singularities around a corner point are nowadays a well-studied problem. The v-notch corner induces a stress singularity in its vicinity, whose level depends strongly on the v-notch opening angle ω. With stress-free boundary condition along the two notch faces, it is a weaker singularity than that of a crack (corresponding to ω= 0◦). In plane elasticity and under general mixed modes loading conditions, the displacement field around this point (Wieghardt (1907); Williams (1952)) can be expressed as U(r, θ) = U(0) + k1rλ1u1(θ) + k2rλ2u2(θ) + ··· (7.1) where U(r, θ) is the displacement vector at the polar coordinates (r,θ) with the origin at the corner root Oand θ= 0◦corresponding to the bisector (see Figure 7.1). The term U(0) holds for the displacement of the corner root and is constant. This term is irrelevant and does not have any importance in the subsequent developments but is present for consistency. The second term k1rλ1u1(θ) is the contribution to the displacement field due to the symmetric loads. The factor k1is the corresponding generalized stress intensity factor (GSIF). It is proportional to the remote load, units are MPa.m1−λ1. The singularity expo- 250 7. Crack initiation at a weak surface in a V-notch nent λ1lies in the domain 1/2≤λ1≤1 and depends strongly on the v-notch opening ω. It is 1/2 and 1 respectively for a crack (ω= 0◦) and a straight edge (ω= 180◦). The vector u1(θ) is a symmetric angular shape function with dimensions MPa−1. This may seem surprising, this is due to the fact that the elastic material properties are implicitly embedded in the denominator of u1(θ). The third term k2rλ2u2(θ) corresponds to the antisymmetric mode. Analogously, k2,λ2and u2(θ) are the GSIF, singularity exponent and angular shape function of the antisymmetric mode. Properties of these parameters are similar to that described for the symmetric case except that λ2can be larger than 1 roughly for ω≤105◦. Figure 7.1: Focus on the vicinity of the corner point, the shaded side corresponds to the solid part. The dashed line is the bisector of this region. Applying the constitutive elastic law provides an analogous expansion of the stress field in the vicinity of the origin σ(r, θ) = k1rλ1−1s1(θ) + k2rλ2−1s2(θ) + ··· (7.2) where σis the stress tensor at point (r, θ). The functions s1(θ) and s2(θ) are the dimensionless associated angular functions derived from (7.1). As a consequence the GSIF units are MPa.m1−λi, they vary with the exponent value. This also explains the previous remark on the unit of vectors u1(θ) and u2(θ). Note that both terms are unbounded for r→0 if the exponent λi<1 (i= 1,2). As mentioned above, this is true for the first exponent λ1for any opening angle smaller than 180◦, whereas the second exponent λ2can be higher than 1 for widely opened corners (roughly ω > 105◦). As a consequence, in this special case, stresses in the vicinity of the corner can be not singular under a pure antisymmetric loading (since k1= 0 in this case). The exponents λ1,λ2and the angular functions u1(θ), u2(θ) are solutions to an eigenvalue problem (Leguillon and Sanchez-Palencia (1987)), see Table 7.1 for some values of exponents. 7.1. The singular stress field at a v-notch 251 The values of the GSIF k1and k2are proportional to the remote load and can be extracted, in general, from a numerical analysis of the global elastic solution in the vicinity of the corner, see e.g. Leguillon and Sanchez-Palencia (1987); Labossiere and Dunn (1999). ω(◦) 0 30 60 90 120 150 180 λ10.5 0.502 0.512 0.545 0.616 0.752 1 λ20.5 0.598 0.730 0.906 1.149 1.484 2 Table 7.1: Exponents λ1and λ2for different values of the notch opening ω As in LEFM, a combined equivalent GSIF is proposed here as k2 eq(r) = k2 1r2λ1−1 | {z } (k∗ 1)2 +k2 2r2λ2−1 | {z } (k∗ 2)2 (7.3) This equivalent parameter keq has the same units as the usual SIF of a crack, e.g. MPa.mm1/2. It enables to define new dimensionless parameters ˆaand ˆ bas ˆa=k1 keq rλ1−1/2(7.4a) ˆ b=k2 keq rλ2−1/2(7.4b) Note that ˆaand ˆ bcan be related by the equation (7.3). Combining (7.3) and (7.4), next relation is obtained: ˆa2+ˆ b2= 1 (7.5) As a consequence the pure modes can be easily defined: •In pure mode 1: ˆa= 1 ⇒ˆ b= 0 and k2= 0. •In pure mode 2: ˆ b= 1 ⇒ˆa= 0 and k1= 0. One can rewrite the expansions (7.1) and (7.2) functions of these new parameters U(r, θ) = U(0) + keqr1/2ˆau1(θ) + ˆ bu2(θ)+··· (7.6) and σ(r, θ) = keqr−1/2ˆas1(θ) + ˆ bs2(θ)+··· (7.7) These two relations are apparently similar to the usual expansions for a crack simply modified by the parameters ˆaand ˆ b. However, the dependence on λ1and λ2has not disappeared, it is implicitly embedded in ˆaand ˆ bwhich depend also on the distance rto the corner point. These expressions should allow defining a mixed-mode criterion based on a critical value for the equivalent SIF keq with an expression which is totally symmetric with respect to the two modes. It is necessary to remark the symmetric 252 7. Crack initiation at a weak surface in a V-notch aspect of this formulation which differs from what is done usually. Pure symmetric mode is traditionally characterized by a vanishing mixity parameter whereas the pure antisymmetric mode is obtained when this parameter tends to infinity (a convenient way of presenting this property is to consider an angle varying from 0◦to 90◦and its tangent, see below in Section 7.3.2). Another new expression was presented recently by Hills and Dini (2011) as a function of a length with a very interesting physical meaning: a point closer or farther to the corner than this reference length will have a stress dominated either by the symmetric or the antisymmetric modes respectively. However this length is defined by normalizing with k1and this may cause some inaccuracy when studying quasi antisymmetric modes, what it is sought to avoid. 7.2 Stress criterion The stress criterion is based on the assumption of a critical value of the tensile or shear stress all along the line where the crack is expected to appear. Originally Leguillon (2002) proposed a stress criterion based on the assumption of a critical value σcso-called tensile strength for the normal tractions. An abrupt onset is possible over a finite extension lif for all points xin this extension σ(r)≥σc,for 0 < r ≤l(7.8) where σ(r) corresponds to the normal traction acting to the plane of the new crack prior to its onset. Later, other authors have proposed different definitions of the stress criterion as Cornetti et al. (2006) who proposed the use of the averaged value of the normal traction over the finite crack extension and comparing it to the critical value σc. All these definitions are based on the hypothesis of pure or nearly mode 1. In the mixed-mode case, it is necessary to take into account the shear stresses τ(r) along this line; whereas, none of the above definitions included this possible influence. To this aim, Mantiˇc (2009) proposed to use the Mohr-Coulomb criterion, which is intensively employed in soils mechanics, to predict interface debonding in composite materials. Experimental evidences of the influence of τhave been presented in Toda et al. (2001) and Ogihara and Koyanagi (2010) for glass/epoxy interfaces. In the mixed-mode case, it is necessary to take into account both σand τ. To generalize the Mohr-Coulomb criterion, the following relationship between σand τis classically used ξ=σ(r) σcp +|τ(r)| τcp ≥1 (7.9) where σ=σθθ and τ=τrθ can be extracted from (7.7). σcand τcdenote respectively the tensile and the shear strengths respectively. The shear component τ is involved regardless of its sign. The parameter pis related to the shape of the failure curve as shown in Figure 7.2. Thanks to this definition, the main failure curves used in the literature can be obtained for different values of p(Figure 7.2. Stress criterion 253 Figure 7.2: The stress criterion evaluated for several values of the parameter p. 7.3). This failure curves are concave for p≥1. In particular, failure curve for p= 1 and p= 2 corresponds respectively to the Mohr-Coulomb criterion and an elliptical failure curve. Finally p→ ∞ gives a rectangular safe region which corresponds to impose an uncoupled fulfillment of the pure mode stress criteria to failure: σ≥σcor τ≥τc. The choice of the power law (7.9) is not essential. It is convenient for analytical calculations, it may be noted that simply considering to different exponents for the tension and the shear terms complicated greatly the calculations. Of course, the exponent pneeds to be identified, 2 is often selected, see Seweryn et al. (1997). Under the form (7.9), the criterion corresponds to a tension dominated situation. However, with a slight change in refeqnFig-SCdef, the model could be obviously be extended by considering that a compression may act on the failure zone, see Section 3.3. Transcribing (7.7) into the components of interest of the stress tensor leads to σ(r, θ) = keqr−1/2ˆas1(θ) + ˆ bs2(θ)+··· (7.10a) 254 7. Crack initiation at a weak surface in a V-notch τ(r, θ) = keqr−1/2ˆat1(θ) + ˆ bt2(θ)+··· (7.10b) where s1and s2(respectively t1and t2) hold for the dimensionless normal (respectively tangent) angular functions extracted directly from s1and s2. Figure 7.3 shows the values of these parameters as functions of the polar angle θfor several values of the opening angle ωin the half domain θ > 0◦. Clearly s1 and t2are decreasing functions of θbecause the maximum value of the normal stresses under a symmetric loading and shear stresses under an antisymmetric loading are expected along the bisector. Moreover, it is foreseeable that: •s2(θ= 0) = 0 and t1(θ= 0) = 0 because of the symmetry and antisymmetry, respectively. •s1(θ= 180◦−ω/2) = s2(θ= 180◦−ω/2) = t1(θ= 180◦−ω/2) = t2(θ= 180◦−ω/2) = 0, due to the stress-free boundary conditions (Figure 7.3). Moreover, the eigenvectors u1(θ) and uw(θ) are normalized in order to have s1(θ= 0◦) = 1 and t2(θ= 0◦) = 1. Note that, for a crack, these last equalities slightly differ from the classical notation by a multiplicative factor √2π. The combination of relations in (7.10) together with (7.9) amounts to a new way to express the stress condition keqr−1/2hˆas1(θ) + ˆ bs2(θ)pτp c+|ˆat1(θ) + ˆ bt2(θ)|pσp ci1/p ≥τcσc(7.11) Crack onset over a length lis allowed if the above inequality holds true for all the values of r≤l. If both σ(r, θ) and |τ(r, θ)|are decreasing functions of r, it suffices to fulfill this condition at r=l. As already mentioned, this assumption is not valid for τif ω > 105◦, because λ2>1 and then τnow increases with r and the condition τ≥τccan never be satisfied all along the crack path: failure cannot occur under a pure shearing mode if ω > 105◦. Thus the final formulation splits into two parts, below and above the transition value ω= 105◦(see Table 7.1), below both tension and shear are used whereas only tension is involved above. In conclusion, the stress condition imposes, in general, a maximum value of the crack onset length las a function of the load (implicit through the ki’s and then keq). When the remote load increases, keq increases, then lincreases expanding the range of allowed lengths of the crack onset according to this criterion. 7.3. Energy criterion 255 (a) (b) (c) (d) Figure 7.3: Dimensionless angular functions (a) s1, (b) s2, (c) t1, and (d) t2for several values of the opening angle ω= 0◦,30◦,60◦,90◦(except for s2and t2). The functions s1and t2(resp. s2and t1) are symmetric (resp. antisymmetric) with respect to θ= 0◦. Results courtesy of Professor D. Leguillon published by Garc´ıa and Leguillon (2012). 7.3 Energy criterion The Griffith criterion is originally based on an energy balance between an initial equilibrium state and another one with an infinitesimal crack extension, ∆Π + ∆Ek+ ∆Γ = 0 (7.12) where ∆Π and ∆Ek, are the changes in potential and kinetic energy and where ∆Γ denotes the energy dissipated in the process of fracture. Assuming a quasistatic initial state, ∆Ek≥0, the above balance is rewritten as, −∆Π ≥∆Γ (7.13) where the terms ∆Π and ∆Γ are developed in the following. 262 7. Crack initiation at a weak surface in a V-notch and s1(0) = t2(0) = 1. Thus, the equation (7.24) for lcan be simplified leading to l=¯ Gad c (σad cτad c)2(ˆaτad c)p+|ˆ bσad c|p2/p ˆa2A1(0) + ˆ b2A2(0) ,(7.32) and the expression of the critical GSIF keq is reduced to kf eq =s¯ Gad c ˆa2A1(0) + ˆ b2A2(0),(7.33) Finally kf 1and kf 2write kf 1= ˆa ¯ Gad c ˆa2A1(0) + ˆ b2A2(0)!1−λ1   σad cτad c (ˆaτad c)p+|ˆ bσad c|p1/p    2λ1−1 ,(7.34a) kf 2=ˆ b ¯ Gad c ˆa2A1(0) + ˆ b2A2(0)!1−λ2   σad cτad c (ˆaτad c)p+|ˆ bσad c|p1/p    2λ2−1 .(7.34b) The corresponding load intensity can be calculated by means of (7.31). The computation of these values for a particular problem are described in Section 7.6. This will point out the importance of taking into account the mixture of modes. 7.5.2 Crack initiation in the bulk at the optimum angle In this case the crack initiation angle αis a priori unknown. The critical GSIF depends on αby (7.25) through the Aj’s and ˆaand ˆ b kf eq =s¯ Gh c ˆa2A1(α) + ˆaˆ bA12(α) + ˆ b2A2(α),(7.35) where ˆaand ˆ bdepends on lwhich varies itself on αthrough the following equation: l=¯ Gh c (σh cτh c)2(ˆa2s1(α) + ˆ bs2(α))p(τh c)p+|ˆat1(α) + ˆ bt2(α)|p(σh c)p2/p ˆa2A1(α) + ˆaˆ bA12(α) + ˆ b2A2(α).(7.36) The actual direction of fracture is obtained by minimizing the value of the critical force (7.31) with respect to α. According to the expression which uses kf 1and kf 2, the optimization of the critical load Rfis equivalent to minimize kf 1or kf 2 because k0 1and k0 2does not depend on l. Unfortunately, both methodologies meet a difficulty, the optimization is ill-conditioned in case of near pure modes. 7.6. Predictions of the critical load at failure in an Arcan test 263 One or the other expression is unusable and it is not possible to build a general formula valid in all cases. In order to keep the formulation general, the third equality providing Rfin (7.31) will be employed. In this case it is necessary to note that the optimization of Rfis not equivalent to minimize kf eq. The reason is that k0 eq does depend on the critical length lcwhich can differ for different values of α. Therefore, it is necessary to minimize the ratio kf eq/k0 eq kf eq k0 eq =v u u t ¯ Gh c(α, l) ((k0 1)2l2λ1−1+ (k0 2)2l2λ2−1)ˆa2A1(α) + ˆaˆ bA12(α) + ˆ b2A2(α).(7.37) Then, the following numerical procedure should be implemented to obtain the optimum angle αand the minimum value of the critical force Rf: •Compute by a numerical code, e.g. linear FEM, the problem of interest assuming an arbitrary load intensity R0without crack, •Extract the GSIF k0 1and k0 2from the solution •Obtain the angular functions si,ti(i= 1,2) solving the eigenvalue problem and the Aj’s (j= 1,2,12) from a numerical computation (see e.g. Yosibash et al. (2006)), •Minimize the expression (7.37), lbeing obtained solving the nonlinear equation (7.36). An iterative algorithm should be implemented to obtain αand the ratio (7.37), •Compute the critical resultant force using (7.31). 7.6 Predictions of the critical load at failure in an Arcan test By way of example of application, the procedure developed in the above Sections 7.5.1 and 7.5.2 is applied to the prediction of the failure load in compact shear tension (CTS) V-notched specimens. Figure 7.5 shows the mixed-mode loading device proposed by Richard and Benitz (1983). The CTS specimen is assumed to be made of PMMA and the jaws are from steel. Properties of both materials are listed in Table 7.6, they are extracted from Leguillon et al. (2009) and Tran et al. (2010). The mode mixity is varied by modifying the loading angle βof the applied remote force Rfrom a pure symmetric mode for β= 0◦to a pure antisymmetric one for β= 90◦. Note that the angles βand the previously mentioned ψ(7.20) measured at the distance lare likely closed to each other but there is no reason that they coincide except for 0◦and 90◦. In order to obtain the parameters k1and k2corresponding to given R0and βa numerical analysis is carried out using FEM. It is quite tricky because the notched specimens are not symmetric with respect to the load direction and any applied force creates a bending moment. For this reason, during the tests 264 7. Crack initiation at a weak surface in a V-notch Figure 7.5: Arcan experimental setup and specimen Property E(GPa) ν σc(MPa) τc(MPa) G1c (MPa.mm) G2c/G1c PMMA 3.250 0.3 75 225 0.350 5 Adhesive - - 36.5 16.5 0.200 2 Steel 200 0.3 - - - - Table 7.2: Material properties the jaws are allowed to rotate around a horizontal axis. Therefore, meshing only the CTS specimen with prescribed boundary conditions along the top and bottom faces is not sufficient (Figure 7.6 middle). In view of this conclusion, the entire setup (CTS and jaws) is modeled using FEM (Figure 7.6 left). A linear elastic simulation assuming perfect bonding (i.e. continuity of displacement and stress vectors) is sufficient to extract k1and k2using path independent integrals (Leguillon and Sanchez-Palencia (1987); Labossiere and Dunn (1999)). The main geometric characteristics of the simulated specimen are shown in Figure 7.5. The mesh employed is represented in Figure 7.6. Triangular linear elements are used, their maximum and minimum diameters are respectively 11 mm and 40 µm (near the corner, Figure 7.6 right). A 1 mm displacement is prescribed at the loading points situated along the outer part of the jaws. Finally, k1and k2are extracted from the FE solution, see Table 7.3 for several values of the opening of the corner ωant the load angles β. The contrast between k1and k2depends strongly on βproving the efficiency of this type of test to experiment specimens under mixed-mode loading. The data needed to apply the model developed in Section 7.5 are now avail- 7.6. Predictions of the critical load at failure in an Arcan test 265 Figure 7.6: The finite element mesh of the entire structure β(◦)ω= 0◦ω= 30◦ω= 60◦ω= 90◦ k1k2k1k2k1k2k1k2 0 0.0971 0 0.0974 0 0.0978 0 0.0959 0 30 0.0827 0.0223 0.0821 0.0210 0.0809 0.0172 0.0774 0.0119 60 0.0455 0.0369 0.0443 0.0339 0.0423 0.0269 0.0389 0.0179 75 0.0228 0.0399 0.0220 0.0363 0.0208 0.0287 0.0190 0.0189 83 0.0115 0.0402 0.0111 0.0366 0.0105 0.0288 0.0095 0.0189 86 0.0058 0.0401 0.0055 0.0364 0.0052 0.0287 0.0047 0.0188 90 0 0.0397 0 0.0361 0 0.0285 0 0.0187 Table 7.3: GSIF values extracted from the numerical results in Modulef for several values of β and ω. The values of kiare expressed in MPa.mm(1−λi) able. The parameter pwhich defines the stress criterion (7.9), will be set to p= 2. As discussed in the previous section, the critical load for either a failure at the weak surface or in the bulk material are separately determined. A subsequent comparison between these two critical loads will give the critical load at failure. 7.6.1 Crack initiation at the weak surface Results presented in the following are computed using the procedure described in Section 7.5.1. Figure 7.7 shows the safe and failure regions as a function of the GSIF values k1and k2, safe is below and unsafe above the curves. The different subfigures correspond to 4 different values of the opening angle ω. Different values of ωlead to different exponents and thus different units for kf 1and kf 2. As a consequence the different failure curves cannot be plotted together. The coupling between kf 1and kf 2decreases when the value of the opening angle ω increases. In the limit, when λ2>1 for an opening angle ω≈105◦, the shear stresses at the weak surface (where t1= 0) is not longer singular at the corner point and the influence of mode 2 vanishes. Hence kf 1become totally independent of kf 2. Figure 7.8 shows the values of critical loads as a function of the load angle β. Different subfigures corresponds to the different values of ω. The minimum value corresponds to the symmetric load β= 0◦when crack initiates at the weak surface in pure mode 1. For β > 0◦the mixture of modes causes an increase of 266 7. Crack initiation at a weak surface in a V-notch (a) ω= 0◦(b) ω= 30◦ (c) ω= 60◦(d) ω= 90◦ Figure 7.7: Failure (above) and safe (below) regions as a function of the GSIF values for p= 2 and (a) ω= 0◦, (b) ω= 30◦, (c) ω= 60◦, (d) ω= 90◦. the critical load. This figure shows also the comparison between the critical load obtained by the model presented and other simplified models. It is worth noting the moderate consequences of ignoring mode 2 and the shear stresses, except, as expected, for high load angles where mode 2 and the shear stresses become predominant. The simplified models seem to have an admissible accuracy, say up to β≃50◦, their predictions are no longer conservative above this value and even meaningless in pure mode 2. In Figure 7.8, the reference load R∗ fis the remote load Rfnormalized to 1 for ω= 0◦and β= 0◦(a crack in pure mode 1). 7.6.2 Crack initiation in the bulk at the optimum angle Under the assumption of crack initiation in the bulk material, the angle of the crack onset αis a priori not known and is calculated by means of the optimization scheme described in Section 7.5.2. Following this scheme, the angle α, the critical load Rfand the critical GSIF kf 1and kf 2are computed. 7.6. Predictions of the critical load at failure in an Arcan test 267 (a) ω= 0◦(b) ω= 30◦ (c) ω= 60◦(d) ω= 90◦ Figure 7.8: Critical loading as a function of the loading angle βfor (a) ω= 0◦, (b) ω= 30◦, (c) ω= 60◦and (d) ω= 90◦. Simplified model (k2= 0) corresponds to the initial criterion taking into account the single mode 1 and tensile stress (Leguillon (2002)); simplified model (t1=t2= 0) to the improved model proposed by Yosibash et al. (2006) taking into account two modes but neglecting the influence of the shear stresses. Figure 7.9 shows the optimum angle αas a function of the load angle β for several values of the opening angle ω. This is equivalent to the principle of local symmetry (Barenblatt and Cherepanov (1961) and Erdogan and Sih (1963)). The value calculated here is compared to the predictions by Yosibash et al. (2006) (dashed line) which corresponds to neglecting the shear stresses, i.e. taking t1=t2= 0. The results show that this generalization of the model gives similar values of the optimum angle but it is strongly dependent on the material properties used (namely in the present case τPMMA cσPMMA cand GPMMA c2 GPMMA c1 ). Then, in the case of the full model, according to the results shown in Figure 7.9, the optimum angle is similar to the one predicted by the model when neglecting the shear stresses. This means that, in this case, the full model predicts 268 7. Crack initiation at a weak surface in a V-notch Figure 7.9: Predictions of the crack initiation angle αas a function of the load angle βfor several values of the opening angle ω. Full model: solid line; simplified model (t1=t2= 0): dashed line. a nearly pure mode 1 crack onset. Thus, as described above, this angle corresponds to a maximum in normal tractions and vanishing shear stresses. Hence, the value estimated of the critical resultant by the full model will be very similar to the results presented in Yosibash et al. (2006). For brevity these results are not presented in this subsection but in the next one where the comparison between the two hypotheses is carried out. 7.6.3 Global prediction of failure load The global prediction of the failure load can be obtained by comparing the critical loads under the two hypotheses. Figure 7.10 shows this critical load as a function of the opening ωand the load βangles. It shows that the failure occurs in general at the weak surface except for the combination of low values of ωand high values of βwhere the antisymmetric mode is predominant. In this case, the competition is between a failure in pure mode 1 at the optimum angle and a nearly pure mode 2 at the weak surface. As already mentioned ω= 0◦corresponds to a situation where the failure is governed by the energy criterion, in this case, due to the similarity between the values of Gh c1 and Gad c2 (see Table 7.6), the crack onset occurs in pure mode 1 at the optimum angle. On the contrary, ω= 90◦ corresponds to a situation much more governed by the stress condition and the failure is initiated at the weak surface. This is why the crack onset occurs at the weak surface, the tensile strength of the PMMA being higher than the shear strength of the weak surface. In other cases with low or moderate values of βthe 7.7. Concluding remarks 269 (a) (b) Figure 7.10: (a) Comparison of the critical load obtained by the two hypotheses: Crack initiation either at the weak surface for α= 0◦or in the bulk. (b) Values of βand ωfor which the failure is predicted in accordance with each hypothesis. mode 1 contribution at the weak surface is sufficient to ensure the onset at the weak surface due to its weaker strength and fracture properties as can be seen in Figure 7.10. These results confirm previous ones, particularly the principle of local symmetry, but they are obviously closely related to the fracture properties of the material in question, i.e. higher constants in shear than in tension. It seems that they could be questioned in the opposite case of a hypothetical material with failure properties significantly lower in shear than in tension. 7.7 Concluding remarks A semi-analytical coupled stress and energy criterion has been developed in order to predict the failure of V-notched specimens containing weak surfaces between similar materials under complex loading. The procedure introduced requires first to solve a nonlinear equation for the crack length at onset. Then, the knowledge of this length enables to calculate the critical load to failure using a straightforward expression.In fact, this criterion goes much further and can be used in all situations where there is a competition between tension and shear. Two dimensionless parameters have been defined which measure the contribution of each mode as a function of the length of the crack onset. The obtained expression is totally symmetric with respect to the two modes and reduces as expected to the known expressions in the pure mode cases. An equivalent combined GSIF which depends on the distance to the corner root has been defined. This has allowed defining the failure as a critical value of this parameter in a similar way to the Irwin criterion in LEFM. A new expression of the stress condition has been introduced in order to consider the effect of the shear stresses. This is generated by a general law which 270 7. Crack initiation at a weak surface in a V-notch couples the critical normal and shear stresses. This law has a free parameter p. It gives the relationship between tensile and shear stress at failure defining a curve in the plane σ−τ. Some typical curves can be generated for different values of pas the Mohr-Coulomb or the elliptical failure curves. In the studied examples, a value of p= 2 has been used but the expression of the criterion remains of course valid for any other value of p. Regarding to the energy condition, a new expression based on the phenomenological law of Hutchinson and Suo has been introduced in order to consider the influence of the mixity on the fracture toughness. An average toughness is defined as the mean dissipated energy per unit of surface along the line defined by the crack at onset. This mean value is necessary because the mode mixity depends on the distance to the corner root and therefore it is the same for the fracture toughness. The criterion has been applied to a particular example: The simulation of an Arcan test on a V-notched specimen in PMMA with a weak surface under several loading angle. In the case of the weak surface, the coupling between the critical GSIF of mode 1 and that of mode 2 decreases when the opening angle of the notch increases. The reason is that the difference between the singularity orders of the two modes increases when the opening angle increases and mode 2 is becoming less important. The competition between the crack onset in the homogeneous material and at the weak surface has been analyzed for the particular values of strength and fracture properties considered here. The results shows that the onset is generally promoted at the weak surface except if two combined conditions hold: the solution is governed by the energy condition (low values of the opening angle) and the onset at the weak surface is in almost pure mode 2 whereas it is almost in pure mode 1 in the homogeneous material. The reason is that in cases governed by the energy condition the value of G2c of the weak surface can be close to the value of the G1c in the homogeneous material. In conclusion, the failure is predicted in general at the weak surface except for the cases that combine high mixed-mode loading and low values of the opening angle. A comparison has been carried out between the results of this criterion and simplified versions which have been presented in the literature. This demonstrates that even in the case of moderate mixed-mode loading, mode 2 must not be neglected because the predictions overestimates strongly the critical load to failure. If shear stresses are neglected but still two modes are taken into account, the difference between the predictions is less pronounced but it cannot be assumed in general. However, no additional comparison is proposed herein with other approaches such as the averaged stress criterion or CZM, it is beyond the scope of this work. A preliminary analysis was given by Leguillon and Yosibash (2003) in the symmetric case. The mixed criterion (Leguillon (2002)) was compared with point stress criterion of McClintock (1958) and the average stress criterion of Novozhilov (1969). The main difference concerns the characteristic length that is used. Nevertheless the criteria are not highly sensitive of this length which anyway are of the same order of magnitude, thus differences in the failure load prediction remain small: about 15 % for ω= 120◦and less for smaller open- 7.7. Concluding remarks 271 ing angles. Indeed the difference lies in the fact that these authors assume the characteristic lengths to be intrinsic and related to the microstructure, reasoning which is refuted by the coupled criterion at the present work. 278 8. Conclusions and future developments in certain aspects of the coupled criterion, the main ones are outlined in the following. The coupled criterion is based mainly on two hypotheses: the assumption of finite length for the crack after the onset and the so-called Leguillon’s hypothesis: the condition of fulfilling simultaneously both the stress and energy criteria. Several examples have been presented in the literature showing that the coupled criterion gives good approximations of the results obtained in experiments. However, currently the physical basis and justification of both hypotheses are still not fully clear. Some partial evidences of the requirement postulated in Leguillon’s hypothesis have been presented, see a discussion in Section 3.2, but they do not represent a full justification of why this criterion works even in very different material systems. Thus, it is necessary to deepen in the understanding of the physical basis of the coupled criterion. A sound physical justification of the coupled criterion would be very useful from a practical point of view as well. For example, it would enable to define range of materials, scales and loads for which the coupled criterion can be applied. In this thesis, the influence of the fracture-mode mixity on the formulation of the coupled criterion has been widely discussed since the presence of interfaces in composites makes the mixity to be, a priori, very relevant in the characterization of the crack initiation in some typical failure mechanisms. In spite of the work carried out, the influence of this mode mixity is still controversial. Thus, it is necessary to generate more knowledge about the influence of the mixity in the onset of a crack of a finite length assumed in the context of the finite fracture mechanics. Given that several forms of generalizing the original coupled criterion to take into account the mixity have been proposed, it would be interesting to design an experiment for measuring this influence in order to evaluate the adequacy of the different methods proposed. Fracture mechanics and fatigue are traditionally two closely related disciplines. Thus, some of the methods within the context of the finite fracture mechanics described in Section 2.3.4 have been (even previously) applied by other authors for the study of fatigue initiation of cracks in the absence of an initial crack. Some first attempts to generalize the coupled criterion to were proposed by Pugno et al. (2007); Leguillon and Murer (2012). Given the relevance of fatigue in the failure initiation process, it would be interesting to work on this generalization, in particular in the experimental comparison of the results predicted by these approaches. In addition, some of the concepts which are employed in the coupled criterion in its static formulation should be revisited for fatigue. The coupled criterion requires for its application to obtain an analytical or approximate solution for a set of elastic problems as described in Chapter 3. If neither analytical nor approximate solutions in closed form are available, the elastic solutions required are approximated by computational procedures. The numerical solution is extracted from a suitable numerical model and is used in the application of the coupled criterion. However, there is not exist an interaction between the application of the coupled criterion and the numerical model. This strategy limits the complexity of the problems which can be studied, given that for a sufficiently complex problem the number of elastic solutions to be approx- 8.2. Future developments 279 imated strongly increases, particularly due to the uncertainty about the crack geometry and position. Thus, it is necessary to generate a numerical method able to integrate the numerical solution along with the formulation of the coupled criterion. It would enable the extension of the coupled criterion to more complex problems, which cannot be studied by the coupled criterion currently. It could be argued that the need of a numerical method for the coupled criterion is against one of the main advantages of the coupled criterion: its ability to generate simple expressions for predicting crack initiation. However, the hypothetical numerical method proposed here could be employed as a first step in the analysis of a global problem providing the preferential solutions. Once the number of solutions to be studied is reduced, the problem can be studied by the usual strategy leading to semianalyticial expressions for the prediction of crack initiation. With regards to the second objective of verifying that the coupled criterion is a suitable tool for the study of the initiation of failure mechanisms in composites, a set of additional developments are proposed in the following. The experimental validation of the results predicted by the coupled criterion can be found in the literature for a wide range of problems. However, it is necessary to validate some models for certain problems with no experimental data to compare. This lack of experimental validation is found particularly for problems which are studied at the smallest scales, e.g., the problem studied in Chapter 4. The experimental validation of these models is considered very necessary to show the applicability of the coupled criterion also beyond certain small scales and certain materials. This is a consequence of the lack of a full physical justification of the coupled criterion discussed previously. Even in those models which have been compared with experiments, the comparison is normally carried out with the variation of one parameter. In these cases, additional experiments are necessary to validate the variation of the results predicted when other parameters are varied in order to verify that the coupled criterion account for them adequately. A particular experiment is motivated by the divergence found between the predictions by the coupled criterion and a cohesive zone model described in Section 4.2 for certain values of the governing parameters. For the problem of crack initiation at the fiber-matrix interface when the composite is subjected to transverse loading, the two criteria diverges in the asymptotic limit of the critical load for small fibers. The coupled criterion predicts that the critical load is proportional to the square root of the inverse of the fiber radius, whereas the cohesive zone model gives a variation which is linear with the inverse of the fiber radius. This divergence is an evidence of a deep difference in the physical basis which governs the asymptotic behavior in this extreme, as was discussed in Section 4.2. Thus, new experiments are required to verify which is the actual asymptotic behavior. Given that the divergence between the two models is caused by a divergence in the physical basis as has been demonstrated, this experiments would give also an information about the basis behind the failure initiation in these situations. Regarding to the third objective which focuses on the characterization of 280 8. Conclusions and future developments the first steps of some failure mechanisms in composites, a set of particularized future developments are proposed in what follows for each one of the particular problems of crack initiation studied in this thesis. In the case of the problem of crack initiation at the fiber-matrix interface when the composite is subjected to transverse loading, several future developments are proposed in view of the conclusions described in Section 4.4. First, the study of interface crack under biaxial loads has to be completed for the case of dominant transverse compression. Then, a full critical curve for biaxial transverse load could be generated. In addition, this curve could be combined with an approach proposed by Carraro and Quaresimin (2014) to generate a model able to evaluate the influence of three of the six independent components of the stress tensor on the initiation of this failure mechanism. The effect of the residual thermal stresses originated due to the fabrication process could be introduced in this model using the methodology employed in the problem of transverse cracking in cross-ply laminates. Once the initiation of the crack is characterized, it is necessary to study the subsequent steps of the failure mechanism, in particular the crack growing along the interface and the kinking out towards the matrix. The previous developments are adequate under the hypothesis of low volumetric fiber-content since this enables to neglect the influence of the neighboring fibers. Rigorously, this hypothesis is not accurate in most usual composites, which does not necessarily imply that the results obtained cannot be a good approximation for real composites. To evaluate how the neighboring fibers affect to the results obtained by these simplified models, it is necessary to develop multi-fiber models. Given the complexity of the geometry of the crack initiation in these models, this problem requires to be solved by a numerical method which has to be developed as discussed previously in this section. Besides the new models, it is necessary to verify experimentally the main results obtained by the models already developed, in particular those with a certain practical relevance, e.g. the size effect of the fiber radius or the influence of a secondary transverse load. With regards to the problem of crack initiation at the interface between a spherical inclusion and the surrounding matrix, some analogous future developments can be proposed. First, the subsequent steps after the crack initiation have to be modeled as an interface crack growth and kink out towards the matrix. The problem of neighboring inclusions is not as relevant as in the case of fibrous composite because the volumetric content is usually lower. In a similar way of the future development proposed for the previous problem, the influence of the reasidual thermal stresses can be introduced here similarly. In addition, new experiments are required for this problem since the available experiments were not detailed enough to obtain a satisfactory conclusion. In the problem of transverse crack initiation in cross-ply laminates, the future developments should mainly focus on the extension of the model to take into account other failure mechanisms along with the next steps of the type of failure whose initiation is modeled. A new model can be developed by combining the model developed by Martin et al. (2010) for the delamination due to the singularity at the free edge and the model developed here. These would enable to predict under which conditions each failure mechanism is preferential. Going beyond the 8.2. Future developments 281 initiation of the first transverse crack, the evolution of the crack density with the external strain is also a very relevant problem due to its influence on the permeability of the laminate and even on the global stiffness in some composites. Thus, it would be interesting to extend this model to predict the evolution of the crack density and compare the results with the models generated by employing other criteria. After the transverse cracking, the subsequent step of the failure mechanism is well known: the delamination at the interface between the transverse and the longitudinal ply due to the stress concentration near the crack tip. This step can also be studied by combining the coupled criterion and matched asymptotics. Finally, the combination of the models of transverse cracking, free-edge delamination and interface delamination could provide of a full characterization of this failure mechanism. For the model developed to predict crack initiation at reentrant corners in the presence of a weak interface between two subdomains, the future developments are mainly based on its experimental validation and its application to problems of practical interest. The experimental validation can be carried out by an Arcan test as that employed to show the application of the model. In addition, it would be necessary to design a set of experiments to estimate the material and interface properties required by the model, as fracture toughness and strength in pure opening and shearing modes. In addition, the model can be extended to account for problems with multi-material corners with any number of materials and interfaces. 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