A fully physically based explanation of the scale effect in composites

María Luisa Velasco López A fully physically based explanation of the scale effect in composites iii A fully physically based explanation of the scale effect in composites María Luisa Velasco López Universidad de Sevilla Grupo de Elasticidad y Resistencia de Materiales Ph.D. Thesis Advisors: Elena Correa Montoto Federico París Carballo Sevilla, 2021 iv A fully physically based explanation of the scale effect in composites María Luisa Velasco López Thesis for the Degree of Doctor of Philosophy with International Mention for the Universidad de Sevilla. Industrial Engineer. Group of Elasticity and Strength of Material, School of Engineering, Universidad de Sevilla. Advisors: Elena Correa Montoto Federico París Carballo Sevilla, March 2021 v A mis padres, Luisa y Domingo, ya que, sin sus esfuerzos, sacrificios y valores, no hubiera sido posible vi vii Acknowledgments First, I would like to express my gratitude to my advisors, Elena Correa and Federico París, for their continuous support, their guidance and all the opportunities that they have given me throughout these years. I would like to thank all my colleagues of the Group of Elasticity and Strength of Materials (GERM) for their help in different things, some of which are not visible in this work but that have been essential for the development of this Thesis: José Cañas, Alejandro Estefani, Luis Távara, José Reinoso, Israel García, Alberto Barroso, Juan Carlos Marín, Enrique Graciani, Vlado Mantič, Antonio Blázquez and Rocío Parejo. I wish to extend my thanks to all colleagues of the Laboratory of Elasticity and Strength of Materials (LERM) for their assistance in carrying out the experimental part of this work: Antonio Cañas, José Ramón Rodriguez, Patricia Zumaquero, Miguel Muñoz and Isabel Del Príncipe. I would also like to acknowledge the collaboration of María Inmaculada Valverde, María José Peña, Elena Arroyo, Paula Caballos, Victoria Gabriela Navarrete and Javier Navarro, Next, I would like to thank my colleagues in the L3 office for their support every day: Jesús Justo and Serafín Sánchez, and the new incorporations, Ninfa Vidal and María Herrera. I would also like to thank Teresa Aranda for her constant companionship. Last year, I had the opportunity of meeting Dr. Pedro Camanho from the Faculdade de Engenharia of the Universidade do Porto during my stay in Porto. I would like to express my gratitude to him for the interesting discussions and suggestions. I would like to thank the Spanish Ministry of Education, Science and Sports (Project MAT2016- 80879-P and MAT2013-45069-P) for its financial support. I want to thank all my friends, particularly Marina Torres and Manuela Castro, for sharing so many good moments that encouraged me to keep going. I would also like to thank my parents Luisa and Domingo, and my sister Teresa for their support and care. My sincere thanks to my grandmother, María, for teaching me to never give up. Finally, my special thanks to Vicente for taking care of me more than I myself. viii ix Contents List of figures .............................................................................................................. xiv List of tables ............................................................................................................. xxiii 1.-Introduction ............................................................................................................. 1 1.1.- Research Background .................................................................................................. 1 1.2.- Objectives .................................................................................................................... 6 1.3.- Contents ...................................................................................................................... 7 2.- Tools and Materials .............................................................................................. 12 2.1.- Numerical Models...................................................................................................... 12 2.2.- Materials ................................................................................................................... 14 2.3- Experimental Tests. Equipment and Procedures for Testing, Damage Detection and Observations. .................................................................................................................... 15 3.- Global summary of the numerical results ............................................................. 16 3.1.- Progressive Mechanism of Damage ........................................................................... 17 3.1.1.- Micromechanical view of the onset ................................................................................... 17 3.1.2.- Micromechanical view of the propagation of damage ........................................................ 18 3.2.- Explosive Mechanism of Damage .............................................................................. 22 3.3.- Discussion on the different mechanism of damage involved in the failure of the 90 degrees layer in [0,90n]s laminates ..................................................................................... 25 4.- Experimental Evidence .......................................................................................... 31 5.- Discussion of the results ........................................................................................ 41 6.- Conclusions ........................................................................................................... 45 7.- Future Developments ............................................................................................ 48 References .................................................................................................................. 49 APPENDED PUBLICATIONS ......................................................................................... 55 PAPER A: Microscopical observations of inter-fibre failure under tension ................. 57 A.1.- Introduction ....................................................................................................... 60 A.2.- Coupons manufacturing and testing.................................................................. 61 A.3.- Preparation of samples ...................................................................................... 62 A.4.- Microscopic observation .................................................................................... 63 A.4.1.- Inspection at low magnification ............................................................................. 63 A.4.2.- Inspection at high magnification ............................................................................ 66 xvi PAPER B Figure B. 1.-Sequential stages of expected damage at micromechanical level in a 90° ply. ...... 82 Figure B. 2.- Multiscale 2D model of a [0,90n]S laminate under tension with a cell in the central part of the 90° lamina. ........................................................................................................... 83 Figure B. 3.- Damaged cell in the 90° lamina with a debonding between fibre and matrix. ...... 83 Figure B. 4.- Detail of the model involving a debonding between fibre and matrix and an associated kinked crack. ......................................................................................................... 85 Figure B. 5.- General view of the discretization carried out in the model. ............................... 86 Figure B. 6.- Discretization of the local cell corresponding to a debonding case. ..................... 87 Figure B. 7.-Discretization performed in the neighbourhood of the crack tip to analyse the first phase of damage (isolated debondings between fibre and matrix). ........................................ 89 Figure B. 8.- Discretization of the local cell corresponding to the kinked matrix crack case. .... 90 Figure B. 9.- Predictions on G values obtained with the multiscale model (surrounded by matrix) and with the single fibre model. ................................................................................. 92 Figure B. 10.- Radial stress and relative displacements in the neighbourhood of the crack tip for a debonding of 30º using a hexagonal cell (HC) and a circular cell (CC). .................................. 93 Figure B. 11.- Radial stress and relative displacements in the neighbourhood of the crack tip for a debonding of 50º using a hexagonal cell (HC) and a circular cell (CC). .................................. 93 Figure B. 12.- Values of G, GI and GII for the whole range of debondings using hexagonal cells (HC) and circular cells (CC). ..................................................................................................... 94 Figure B. 13.- Evolution of G, and their counterparts GI and GII, for different lengths of the kinked crack. .......................................................................................................................... 95 Figure B. 14.- Undeformed and deformed shapes of the model (only the fibre and surrounding matrix are represented). ........................................................................................................ 96 Figure B. 15.- Values of G, GI and GII for different values of the t0º thickness. .......................... 97 PAPER C Figure C. 1.-a) Multi-scale BEM model of a [0,90n]S laminate with an inner cell involving fibre and matrix, b) detail of the cell. ............................................................................................ 109 Figure C. 2.- Validation of the multi-scale BEM model........................................................... 110 Figure C. 3.-Evolution of s rr and s r q along the fibre-matrix interface for different values of n. ............................................................................................................................................ 111 Figure C. 4.- Role of the 0° lamina in G predictions. .............................................................. 112 Figure C. 5.- Scale effect in the first stage of damage for a [0,90n]S laminate. ........................ 113 Figure C. 6.- Scheme of the observations by Saito et al [9] for: a) ns=1, b) ns=2. .................... 114 Figure C. 7.- Results corresponding to the three relative configurations of the secondary fibre and the two potential cases of relative location of the damage. ........................................... 116 Figure C. 8.- Multi-scale model for the 45° (OS) case. ........................................................... 117 Figure C. 9.- Evolution of G for the 45° (OS) case (2t90=340µm for n=1). ............................... 117 Figure C. 10.- Multi-scale model for the 90° (SS) case. .......................................................... 118 Figure C. 11.- Evolution of G for the 90° (SS) case (2t90=340µm for n=1). .............................. 118 Figure C. 12.-Evolution of G for the cases 45° (OS) and 90° (SS) with n=1 using the multi-scale model (2t90=340µm for n=1). ................................................................................................ 119 xvii Figure C. 13.- Observations (left side) and corresponding schemes (right side) of damage associated with: a) the 45° (OS) case, b) a chain of 45° (OS) cases, c) combinations of 45° (OS) cases with 90º (SS) cases. ..................................................................................................... 120 PAPER D Figure D. 1.- Multiscale BEM model to study the isolated debondings between fibres and matrix as a function of the thickness of the 90 degrees ply: a) general view, b) detail of the micromechanical cell. ........................................................................................................... 128 Figure D. 2.- Evolution of G of the debonding crack as a function of the thickness of the 90 degrees ply........................................................................................................................... 129 Figure D. 3.-Micrography showing isolated debondings between fibres and matrix. ............. 129 Figure D. 4.- a) Scheme of the first phase of damage: isolated debondings, b) scheme of the second phase of damage: connection between debondings leading to a meso-transverse crack, c) scheme of the onset of the second phase of damage studied in this paper. ...................... 130 Figure D. 5.- BEM multiscale modelling of the composite involving kinking of the debonding. ............................................................................................................................................ 131 Figure D. 6.- Configurations studied...................................................................................... 131 Figure D. 7.- Influence of the model used in the values of G of the kinked crack. .................. 131 Figure D. 8.- Scale effect, influence of the thickness of the 90 degrees lamina in the energy release rate for different lengths of the transverse crack. ..................................................... 132 PAPER E Figure E. 1.- (a) Single fibre (b) two fibres (c) three fibres in and infinite matrix model with interfacial damage. .............................................................................................................. 143 Figure E. 2.- Integrated multiscale model of a [0,90n]S laminate with interfacial damage in the micro-cell. ............................................................................................................................ 143 Figure E. 3.- Evolution of G, GI and GII with the semidebonding angle for the single fibreinfinite matrix model and the integrated multiscale model. ............................................................. 145 Figure E. 4.- Plane Stress (PSe) and Plane Strain (PSa) predictions for G, GI and GII (integrated multiscale model). ................................................................................................................ 146 Figure E. 5.- Values of (a) Dur and (b) srr at both sides of the debonding crack tip for PSe and PSa problems (integrated multiscale model). ....................................................................... 147 Figure E. 6.- Two damaged fibres in an infinite matrix model. ............................................... 148 Figure E. 7.- Models corresponding to the q =0° case. ........................................................... 149 Figure E. 8.- GI/G0, GII/G0 and G/G0 distributions for Ref, USF, SS and OS cases corresponding to q =0° configuration. .............................................................................................................. 150 Figure E. 9.- Models corresponding to the q =45° case. ......................................................... 150 Figure E. 10.- GI/G0, GII/G0 and G/G0 distributions for Ref, USF, SS and OS cases corresponding to q =45° configuration. ........................................................................................................ 151 Figure E. 11.- Models corresponding to the q =90° case. ....................................................... 151 Figure E. 12.- GI/G0, GII/G0 and G/G0 distributions for Ref, USF, SS and OS cases corresponding to q =90° configuration. ........................................................................................................ 152 Figure E. 13.- GI/G0 distribution versus q d for r2=2.42r1, 3r1, 5r1, 10r1 corresponding to q =90° OS configuration and single fibre model (Ref). ...................................................................... 155 xviii Figure E. 14.- GII/G0 distribution versus q d for r2=2.42r1, 3r1, 5r1, 10r1 corresponding to q =90° OS configuration and single fibre model (Ref). ........................................................................... 155 Figure E. 15.- G/G0 distribution versus q d for r2=2.42r1, 3r1, 5r1, 10r1 corresponding to q =90° OS configuration and single fibre model (Ref). ........................................................................... 156 Figure E. 16.- GI/G0 distribution versus q d for r2=2.42r1, 3r1, 5r1, 10r1 corresponding to 45° OS configuration and single fibre model (Ref). ........................................................................... 156 Figure E. 17.- GII/G0 distribution versus q d for r2=2.42r1, 3r1, 5r1, 10r1 corresponding to 45° OS configuration and single fibre model (Ref). ........................................................................... 156 Figure E. 18.- G/G0 distribution versus q d for r2=2.42r1, 3r1, 5r1, 10r1 corresponding to 45° OS configuration and single fibre model (Ref). ........................................................................... 157 Figure E. 19.- Cases studied for Ld=12°, 32°, 64°, 100° and 140°. Single fibre and USF cases are taken as references. ............................................................................................................. 158 Figure E. 20.- GI/G0 distribution versus q d for Ld=12°, 32°, 64°, 100° and 140° corresponding to q =90° SS configuration and single fibre (Ref) and USF models. ............................................. 158 Figure E. 21.- GII/G0 distribution versus q d for Ld=12°, 32°, 64°, 100° and 140° corresponding to q =90° SS configuration and single fibre (Ref) and USF models. ............................................. 159 Figure E. 22.- G/G0 distribution versus q d for Ld=12º, 32º, 64º, 100º and 140º corresponding to q =90º SS configuration and single fibre (Ref) and USF models. ............................................. 159 Figure E. 23.- Observations of damage associated with the combination of 45° OS case and 90° OS case: a) original micrograph, b) corresponding schemes of the micrograph. .................... 160 Figure E. 24.- Observations of damage associated with the 90° SS case: a) original micrograph, b) corresponding schemes of the micrograph. ...................................................................... 161 Figure E. 25.- Observations of generalized damage. .............................................................. 161 Figure E. 26.- Observation of a matrix rich area surrounded by high fiber density. ................ 162 Figure E. 27.- Observations of damage in a matrix rich area: a) original micrograph, b) corresponding schemes of the micrograph. .......................................................................... 162 PAPER F Figure F. 1.- Integrated multiscale model of a [0,90n]S laminate. ........................................... 176 Figure F. 2.- Detail of the cell with the fibre surrounded by matrix with the presence of damage consisting on a debonding between fibre and matrix, having the debonding a potential separation zone between the lips of the crack and a contact zone. ...................................... 176 Figure F. 3.- Phases of the assumed progressive mechanism of failure represented schematically on a micromechanical cell: a) appearance of isolated debondings, b1) kinking of the debonding crack penetrating into the matrix, b2) progression of the damage through the 90-degree lamina. ................................................................................................................ 179 Figure F. 4.- Evolution of the energy release rates G, GI and GII of a debonding crack as a function of the semidebonding angle for different [0,90n]S laminates ................................... 180 Figure F. 5.- Schematic generation of isolated debondings (under a vertical load) in thick and thin laminates in terms of the 90-degree ply thickness. ........................................................ 180 Figure F. 6.- Isolated debondings (under a horizontal load) in a) thin, [0,90]S, and b) thick, [0,904]S, laminates, in terms of the 90-degree ply thickness. ................................................. 181 Figure F. 7.- Stress state in the neighbourhood of the debonding crack tip. .......................... 182 Figure F. 8.- Evolution of  in the neighbourhood of the debonding crack tip for a semidebonding angle of 70 degrees and at two distances of the crack tip. ........................... 182 xix Figure F. 9.- Evolution of sq for three different thicknesses of the 90-degree ply, qd being 70 degrees and R being 0.001 rf. ............................................................................................... 183 Figure F. 10.- Damage in the 90-degree lamina consistent in a through-the-matrix crack from a debonding crack. .................................................................................................................. 184 Figure F. 11.- Integrated multiscale model with a through-the-matrix crack kinked from a debonding crack. .................................................................................................................. 184 Figure F. 12.- Values of G corresponding to different lengths a of the transverse crack and different thicknesses of the 90-degree ply, represented by the distance d from the crack tip to the 0-degree lamina. ............................................................................................................ 185 Figure F. 13.- Two configurations where the absence, a), and presence, b), of the Cook-Gordon effect is appreciated. ............................................................................................................ 185 Figure F. 14.- Damage in the 90-degree lamina consisting of isolated debondings aligned in such a manner that they may connect and produce a complete transverse crack. ................ 186 Figure F. 15.- Multi-isolated debonding configurations as precursor of a transverse crack in the 90-degree ply: a) stresses involved and b) displacements involved. General configuration corresponding to the presence of n debondings and detailed case for n=4. .......................... 187 Figure F. 16.- Evolution of the values of G for different multidebondings configurations and different thicknesses of the 90-degree lamina, represented by the distance d of the debonding closer to the 0-degree lamina to this lamina. ........................................................................ 188 Figure F. 17.- Sudden appearance of a transverse damage in the 90-degree lamina.............. 189 Figure F. 18.- Evolution of the capacity of release energy as a function of t90 by the appearance of a complete transverse crack. ............................................................................................ 190 Figure F. 19.- Capacity of energy release in the 90-degree lamina for the appearance of a transverse crack per unit of crack generated. ....................................................................... 190 Figure F. 20.- Deformation required to produce a complete transverse crack as a function of the 90-degree ply thickness. ................................................................................................. 191 Figure F. 21.- Evolution of G values for [0,90]S laminate, indicating the total energy release associated with WIpro and WIIpro for the case of a 60-degree semidebonding. ........................ 193 Figure F. 22.- Values of sr and srq in the pristine state along the length of the 60-degree semidebonding. ................................................................................................................... 194 Figure F. 23.- Values of Dur and Duq corresponding to a 60-degree semidebonding. ............. 195 Figure F. 24.- Values of wIexp and wIIexp for each element (per unit of length) of the boundary element model for the case of a 60-degree semidebonding. ................................................ 195 Figure F. 25.- Evolution of W, WI and WII for the progressive and explosive mechanisms of failure associated with half of the total debonding. .............................................................. 196 Figure F. 26.- a) Scheme of the extension of the damage coming from a damaged fibre to a second undamaged fibre located in its neighbourhood, b) Scheme of the deformed micromechanical cell corresponding to the presence of a semidebonding angle in the originally undamaged fibre qdu=80 degrees. ........................................................................................ 197 Figure F. 27.- Evolution of G, GI and GII in the progressive view of damage involving a second fibre as a function of the semidebonding angle qdu in the originally undamaged fibre. ......... 198 Figure F. 28.- Experimental results corresponding to [04, 90n,04] laminates made of AS4/8552. ............................................................................................................................................ 200 Figure F. 29.- Configuration of the damage in a [04,9016,04] laminate. Picture obtained with SEM. .................................................................................................................................... 200 Figure F. 30.- Damage observed in three specimens of a [04, 902,04] laminate consisting of transverse cracks with sometimes slight indications of the appearance of delamination associated. ........................................................................................................................... 201 xx Figure F. 31.- Damage observed in two specimens of a [04,90,04] laminate consisting of generalized transverse damage without delamination associated. ....................................... 202 Figure F. 32.- Transverse section of two [04,90,04] laminates whose 90-degree lamina is made with UTPs of (a) 20 gsm and (b) 150 gsm. ............................................................................. 203 Figure F. 33.- Experimental results corresponding to [04,90,04] laminates made of UTPs of USNxxx. ................................................................................................................................ 203 Figure F. 34.- Damage observed in [04,90,04] with a thickness of the 90-degree lamina corresponding to 450 gsm. ................................................................................................... 204 Figure F. 35.- Damage observed in a 50-gsm lamina for different levels of deformation: a) 0.8%, b) 1.34%. .............................................................................................................................. 205 Figure F. 36.- Schematic manifestation of the scale effect in [0,90]S laminates. ..................... 206 BOOK CHAPTER Figure G. 1.- A [04,908]S laminate tested under tension showing a transverse crack in the 90 degrees ply: a) under loading, b) unloaded. .......................................................................... 222 Figure G. 2.- Representation of a [0,90n]S laminate with a micromechanical cell in the 90 degrees lamina. .................................................................................................................... 223 Figure G. 3.- Cell at micromechanical level in the 90 degrees ply involving due to symmetry a total debonding 2 q d between fibre and matrix. .................................................................... 223 Figure G. 4.- Morphology of an interface crack between fibre and matrix under shear loading. ............................................................................................................................................ 224 Figure G. 5.- Evolution of G and counter parts GI and GII as a function of the debonding length for different thicknesses of the 90 degrees ply. .................................................................... 228 Figure G. 6.- Isolated debondings in a 90 degrees lamina of a [04,90,04] laminate made of USN150 (0 degrees laminas) and USN50 (90 degrees lamina) material. ................................ 229 Figure G. 7.- Configuration of two fibres oriented 45 degrees with respect to the loading direction. ............................................................................................................................. 230 Figure G. 8.- Evolution of G and counter parts GI and GII for the case of two fibres oriented 45 degrees with respect to the loading direction. ...................................................................... 230 Figure G. 9.- Experimental observation where the presence of a morphology of the damaged fibres oriented 45 degrees appears. It corresponds to a [03, 903]S laminate made of AS4/8552 material. .............................................................................................................................. 231 Figure G. 10.- Configurations of two fibres oriented 90 degrees with respect to the loading direction. ............................................................................................................................. 231 Figure G. 11.- Evolution of G and counter parts GI and GII for the case of two fibres oriented 90 degrees with respect to the loading direction. ...................................................................... 232 Figure G. 12.- Experimental observation on a [03, 903]S laminate made of AS4/8552 material where the presence of a morphology of the damaged fibres oriented 90 degrees appears. .. 232 Figure G. 13.- Comparison between the values of G corresponding to the 45 and 90 degrees morphologies of two fibres damaged configurations. ........................................................... 233 Figure G. 14.- a) model used for the study of the cracking of the matrix, b) detail of the kinked crack abandoning the fibre-matrix interface. ........................................................................ 234 Figure G. 15.- Evolutions of G for different sizes of the cell to validate the BEM model using 90 degrees ply properties for solids S2, S3 and S4. .................................................................... 235 Figure G. 16.- Influence of assigning different properties to solids S2, S3 and S4 of the model. ............................................................................................................................................ 236 xxi Figure G. 17.- Influence of the size of the cell in the model. .................................................. 237 Figure G. 18.- Normal displacements of the faces of the debonding crack corresponding to the case of a length of the kinked crack equal to the radius of the fibre...................................... 238 Figure G. 19.- Normal stresses ahead the debonding crack tip corresponding to the case of a length of the kinked crack equal to the radius of the fibre. ................................................... 238 Figure G. 20.- Illustration of Cook-Gordon Mechanism: a) stop of the crack before reaching the interface with the stiff material, b) damage along the interface before the transverse crack reaches it. ............................................................................................................................ 239 Figure G. 21.- Stopping of the transverse crack in the 90 degrees ply near the interface with the 0 degrees ply. The picture corresponds to a [02,904]S laminate. ............................................ 239 Figure G. 22.- Influence of the properties of the fibres. ........................................................ 240 Figure G. 23.- Configuration to study the effect of the thickness of the 90 degrees ply on the energy released by the kinked crack. .................................................................................... 241 Figure G. 24.- Example of a transverse crack mainly running across the matrix in the 90 degrees lamina of a laminate made of AS4/8552 material. ................................................................ 242 Figure G. 25.- Evolutions of G for different lengths of the kinked crack as a function of the thickness of the 90 degrees lamina....................................................................................... 242 Figure G. 26.- Longitudinal stresses s 90 ° (before damage) and longitudinal displacements u90° in the transverse crack (after cracking) of the 90 degrees lamina of a [0,90n]S laminate. ........... 244 Figure G. 27.- Normal displacements of the faces of the transverse crack in the 90 degrees lamina. ................................................................................................................................. 245 Figure G. 28.- Evolution of G versus the length of a transverse crack in the 90 degrees ply. .. 246 Figure G. 29.- Damage consistent on an isolated debonding which is going to appear in a explosive or progressive way. ............................................................................................... 247 Figure G. 30.- Normal s and tangential t components of the stresses in the pristine state along the interface crack (60 degrees semidebonding), stresses participating in the release of energy during an explosive mechanism of damage. ......................................................................... 247 Figure G. 31.- Relative normal and tangential displacements along the faces of the 60 degrees semidebonding. ................................................................................................................... 248 Figure G. 32.- Evolution of G and counterparts GI and GII associated with a progressive scheme of damage. ........................................................................................................................... 248 xxii xxiii List of tables Table 1.- Carbon-epoxy system, properties obtained from Soden et al [56]. ........................... 14 Table 2.- Glass-fibre system, properties obtained from Sandino et al [22]. ............................. 14 Table 3.- Wpro, WIpro, WIIpro for semidebonding angles of 50, 60 and 70 degrees associated with a progressive damage mechanism. ........................................................................................... 28 Table 4.- Wexp, WIexp, WIIexp for semidebonding angles of 50, 60 and 70 degrees associated with an explosive damage mechanism. .......................................................................................... 30 PAPER A Table A. 1.- Denomination of the coupons and corresponding testing load. ............................ 62 Table A. 2.- Number of macro-cracks as a function of the loading level................................... 63 Table A. 3.- First-ply and ultimate failure predictions for a selection of failure criteria. ........... 64 Table A. 4.- q d: Dependence on Stage III typology. .................................................................. 71 Table A. 5.- q d: Dependence on the loading level .................................................................... 73 PAPER B Table B. 1.- Carbon-epoxy system used. ................................................................................. 83 Table B. 2.- Glass-epoxy system used...................................................................................... 84 Table B. 3.- Subtended central angle and length of the elements in the neighbourhood of the crack tip. ................................................................................................................................ 89 PAPER Table C. 1.- Carbon-epoxy system used: AS4/3501-6, Soden et al [22]. ................................. 108 Table C. 2.- Glass-epoxy system used, data obtained directly from tests. .............................. 108 PAPER D Table D. 1.- Properties of the materials involved in the model. ............................................. 130 PAPER E Table E. 1.- Properties of the materials involved in the models represented in Figs. E.1 and E.2. ............................................................................................................................................ 143 Table E. 2.- Properties of the glass-epoxy system used. ........................................................ 149 Table E. 3.- Predictions of the final value of the semidebonding angle after the period of unstable growth at the interface. ......................................................................................... 154 PAPER F xxiv Table F. 1.- Properties of the carbon-epoxy material AS4/8552. ........................................... 177 Table F. 2.- Wpro, WIpro and WIIpro for the case of a progressive mechanism of failure. ............ 193 Table F. 3.- Values of Wexp, WIexp and WIIexp for 50, 60 and 70 degrees of semidebondings corresponding to an explosive mechanism of damage. ......................................................... 196 BOOK CHAPTER Table G. 1.- Details of BEM discretizations carried out. ......................................................... 226 Table G. 2.- Properties of the carbon-epoxy material AS4/8552. ........................................... 226 Table G. 3.- Results for W, WI and WII associated with both, explosive and progressive, mechanisms of damage for a semidebonding of 60 degrees. ................................................ 249 xxv 7 - To characterize the different damage mechanisms that can appear in the weakest lamina of a laminate (typically that oriented 90 degrees with respect the loading direction) in cross-ply laminates subjected to uniaxial tension, both at the micro-mechanical level (analysing the fibrematrix debonding, its kinking and penetration into the matrix and how all this can influence nearby undamaged fibres) and at the meso-mechanical level (studying the direct appearance of a transverse crack). The appearance of these two mechanisms must be connected with the thickness of the 90 degrees layer, thus involving the scale effect. - To develop numerical models able to explain the damage observed in testing laminates involving ultra-thin plies. - To perform experimental tests to support the predictions obtained with the numerical models carried out. It is important to note that this study does not intend to provide quantitative predictions of when and where damage will appear and propagate, but rather to understand the mechanism of damage at the micro- and meso-mechanical level associated with the scale effect. In this context, it is important to mention several hypotheses assumed in the development of this Thesis which are connected to the previous comment. First of all, only nominal stresses derived from the external loading are considered, curing stresses not having been incorporated to the reasoning. It is based on the previous studies carried out by París et al [39] and Correa et al [16, 17] in which it was shown that the presence of curing stresses does not qualitatively affect the micromechanical results used in this Thesis, but only quantitatively. Thus, if a quantitative evaluation of properties based on the mechanisms of damage identified in this Thesis were going to be performed in the future, the presence of these curing effects should be taken into consideration. The excellent agreement between predictions obtained from the applications of nominal stress and the experimental observations are an additional support of this hypothesis. In second place, it is obvious that all problems are three-dimensional although a twodimensional view of the problem may afford a sufficient knowledge on the question under consideration. This is exactly the situation covered here, where the observations obtained to feed the numerical predictions are obtained from the lateral sides of the specimen, paying then the maximum attention in the representation of what happens in these lateral sides by means of a two-dimensional model. 1.3.- Contents The contents of this Thesis are directly associated with the following chapters. First, chapter 2, entitled Tools and Materials, describes the numerical tool used (BEM), the materials employed, and the equipment used in testing and observing damage in the specimens. Chapter 3, Global summary of the numerical results, covers all the numerical predictions obtained from the different models developed. Obviously, these models were developed in conjunction with the experimental observations, but it has been preferred in this presentation to group all of them in a chapter, following the sequence of reasoning that justifies the development of all models. Chapter 4, Experimental evidence, covers all the experimental results obtained in this Thesis, following the same idea previously mentioned with numerical predictions. 8 The results obtained are discussed in Chapter 5, Discussion of the results, followed in Chapter 6 by the Conclusions. A list of the publications derived from the development of this PhD Thesis are included now. First the papers invoked in the compendium are listed in a chronological way, considering the date of publication of the contribution, Journal name or chapter in a book. In this case, the name of the contribution, authors and data of the Journal or book appears. The abstract of the contribution is also included. PAPER A [40] - Name of the contribution: Microscopical observations of inter-fibre failure under tension. - Authors: Correa Elena, Valverde María Inmaculada, Velasco María Luisa, París Federico. - Data of the Journal: Composites Science and Technology, volume 155, pages 213-220, 2018. - Abstract: The numerical study of the inter-fibre failure at micromechanical level predicts the appearance of different stages in the development of this damage mechanism; these studies have also allowed the main features of each stage (such as the interfacial debond length and the kinking angle) to be identified. The development of experimental studies aiming to check the relevance of the aforementioned numerical results is crucial. Based on this, this research focused on the tensile test under different loading levels of specimens manufactured from carbon-epoxy cross-ply symmetrical laminates. The microscopic observation of the 90° layers leads to the analysis of the appearance of the transverse cracks as a function of the load, the identification of the previously numerically predicted stages of the mechanism of damage, the measurement of key parameters and the evaluation of the influence of nearby fibres. A clear connection between the numerical and experimental results has been found. PAPER B [41] - Name of the contribution: BEM multiscale modelling involving micromechanical damage in fibrous composites. - Authors: Velasco María Luisa, Graciani Enrique, Távara Luis, Correa Elena, París Federico. - Data of the Journal: Engineering Analysis with Boundary Elements, volume 93, pages 1- 9, 2018. - Abstract: Composite laminates are materials where different scales are required for the understanding of the damage and for the calculation of structures made of these materials. The appearance of ultra thin plies of composites has opened the possibility of delaying the onset of damage, which at a first stage appears at a micromechanical level, debondings in between fibre and matrix in the weakest lamina of the laminate. A multiscale BEM model is developed in the present paper with the final purpose of being able to study the effect that the relative size of the laminas of the laminate plays in the appearance of this initial damage. The model presents many difficulties derived from the different scales involved in it and from the non-linear nature of the problem under study. The approach followed involves the solution of the whole problem, with the different scales involved in it, at once. The solution obtained is checked with another already obtained with a much simpler model. The multiscale model developed has been 9 proved to be very efficient, accurate and robust, having been applied to simulate the first stages of damage in the light of the scale effect that is trying to be studied. PAPER C [42] - Name of the contribution: Micromechanical study on the influence of scale effect in the first stage of damage in composites. - Authors: París Federico, Velasco María Luisa, Correa Elena. - Data of the Journal: Composites Science and Technology, volume 160, pages 1-8, 2018. - Abstract: The variation in the apparent strength of a lamina in a laminate depending on the stacking sequence and thicknesses of the laminas of the laminate has been a matter of interest since the initiation of the extension of the applicability of composites. This fact led to the concept of in-situ strength, the problem itself being covered as a scale effect. In this paper this question is revisited moving towards the level where the damage appears, that is the micromechanical level. As the origin of the effect under consideration, the variation of thickness, takes place at a different level (mesomechanical), a multi-scale model is developed. It is possible in this model to vary the thickness of the lamina under consideration and to observe its effect on the damage. In this paper, only the first stage of damage, which appears in form of debondings between fibres and matrix, is taken into consideration. The analysis carried out shows that there is no scale effect at this first stage of the damage, which is corroborated by experimental evidences. PAPER D [43] - Name of the contribution: Numerical study of the progression of the micromechanical debonding damage in composites. - Authors: Velasco María Luisa, París Federico, Correa Elena. - Data of the Journal: Key Engineering Materials, volume 774, pages 644-649, 2018. - Abstract: This paper deals with the study of the actual progression of the damage in the 90 degrees lamina of a composite. It has been proved and observed that isolated debondings between fibres and matrix are the first manifestation of damage in the weakest lamina, the 90 degrees lamina in a [0,90]S laminate. It was also numerically supported that this first phase was independent of the thickness of the 90 degrees lamina, not being then affected by the “scale effect”. The continuation of this first phase of damage is the objective of the present paper. To this end, a multiscale model is created involving the debonding between fibre and matrix and studying the kink of this crack, abandoning the fibre-matrix interface and entering into the matrix to produce a meso-transverse crack in the 90 degrees ply. The study is based on the application of Fracture Mechanics to an incipient kinked crack that starts from a debonding between fibre and matrix. It is concluded that this second phase of damage, playing with the thickness of the 90 degrees lamina, is not affected by the scale effect, as the variation of the energy release rate of the kinked crack is not significantly influenced by the variation of the thickness of the lamina. PAPER E [44] - Name of the contribution: Interaction between fibres in the transverse damage in composites. - Authors: Velasco María Luisa, Correa Elena, París Federico. - Data of the Journal: Engineering Fracture Mechanics, volume 239, 107273, 2020. 10 - Abstract: A micromechanical study is carried out taking the initial expected damage that is manifested in a composite laminate by the debonding between the fibres and matrix in the 90° lamina as a reference. Questions such as the influence in the appearance of damage of the inter-fibre distance and the orientation of the fibres with respect to the loading direction are studied. Additionally, the effect of the presence of damage at a secondary fibre in the subsequent appearance of further debonding is also analysed. Finally, the conclusions reached are supported by experimental evidence obtained by testing [0,90n]S laminates. PAPER F [45] - Name of the contribution: The scale effect in composites: An explanation physically based on the different mechanisms of damage involved in failure - Authors: París Federico, Velasco María Luisa, Correa Elena. - Data of the Journal: Composites Structures, volume 257, 113089, 2021. - Abstract: The strength of a lamina of a laminate was considered from almost the very beginning of the application of composites as a value depending on the lamina thickness. This was generically called scale effect, the value of the stress at the appearance of first damage in a lamina being identified as the in situ strength. The ultrathin plies have allowed this first damage, typically transverse cracks in the 90-degree laminas, to be delayed. This study presents a completely physically based explanation of the scale effect in composites based on the key identification of two different failure mechanisms in the 90-degree lamina of a [0,90n]S laminate (here called progressive and explosive) that are triggered as a function of the morphology of the laminate. Their characterization is based on energy concepts and does not require either properties of difficult interpretations or fitting parameters. The study extends from plies of conventional thickness to ultra-thin plies. BEM models have been performed, correlating the predictions with experimental microscopic observations. Developing a physically based explanation of the scale effect is the first step towards identifying the parameters involved in the failure initiation. Once identified, these parameters can be measured. BOOK CHAPTER [46] - Name of the contribution: Modelling fibre/matrix interface debonding and matrix cracking in composites laminates. - Authors: París Federico, Velasco María Luisa, Correa Elena. - Data of the book: Multi-Scale Continuum Mechanics Modelling of Fibre-Reinforced Polymer Composites, W.V. Paepegem (Ed.), Woodhead Publishing Series in Composites Science and Engineering, pages 243-274, 2021. - Abstract: The objective of this chapter is to show the modelling of composite materials at the micro-meso level, involving the nonlinearities associated with the damage that may appear in a composite laminate. The potential damage includes debondings between the fibres and matrix where the initial damage occurs in the weakest lamina of a laminate, and the cracking of the matrix that appears when a debonding abandons the fibre–matrix interface, kinking and penetrating into the matrix. Integrated multiscale models based on the boundary element method are developed, allowing us to analyze the whole laminate in a model while also having the possibility of modelling the damage object of study at the micromechanical level in this chapter. The incidence of the features of the model in the representativity of the results is first analyzed, correlating 11 the predictions with the experimental observations from testing laminates. The models carried out will follow a line of association with particular problems regarding the use of composites, with a special emphasis on the study of the scale effect in composite laminates, emphasizing the correspondence between the model performed to simulate the damage and the actual mechanism of damage that a laminate experiences. Additionally, a list of the publications derived from this Thesis, not invoked in the compendium, is also included, indicating just the name of the contribution, authors and details of the contribution. List of contributions not covered in the compendium 1.- XI Congreso Nacional de Materiales Compuestos (MATCOMP’15). “Predicción de fallo en laminados no convencionales: Estudio experimental”. María Luisa Velasco, Jesús Justo, Juan Carlos Marín, Federico París (Universidad de Sevilla). 2.- 17th European Conference on Composite Materials (ECCM17). “Numerical and Experimental study on the failure of non-conventional laminates”. María Luisa Velasco, Federico París, Juan Carlos Marín, Jesús Justo, Antonio Barroso, Enrique Graciani. School of Engineering (Universidad de Sevilla). 3.- XII Congreso Nacional de Materiales Compuestos (MATCOMP’17). “Estudio Micromecánico del Efecto Escala en Laminados de Material Compuesto”. María Luisa Velasco, Federico París, Elena Correa (Universidad de Sevilla). 4.- 35 Encuentro del Grupo Español de Fractura. “Interacción de daño entre fibras en materiales compuestos aplicando Mecánica de la Fractura de Grietas de Interfase”. María Luisa Velasco, Elena Correa, Federico París (Universidad de Sevilla). 5.- 36 Encuentro del Grupo Español de Fractura. “Progresión de daño intralaminar en materiales compuestos: un análisis basado en la mecánica de la Fractura”. Federico París, María Luisa Velasco, Elena Correa (Universidad de Sevilla). 6.- 36 Encuentro del Grupo Español de Fractura. “Influencia de la microestructura en el daño transversal en materiales compuestos”. María Luisa Velasco, Federico París, Elena Correa (Universidad de Sevilla). 7.- XIII Congreso Nacional de Materiales Compuestos (MATCOMP’19) “Efecto escala en materiales compuestos: estudio del daño intralaminar a través de un modelo micromecánico”. Federico París, María Luisa Velasco, Elena Correa (Universidad de Sevilla). 8.- XIII Congreso Nacional de Materiales Compuestos (MATCOMP’19). “Estudio experimental de la aparición del daño en laminados de material compuesto”. María Luisa Velasco, Federico París, Elena Correa (Universidad de Sevilla). 12 2.- Tools and Materials 2.1.- Numerical Models All problems are three-dimensional, although the majority of them accept two-dimensional representations, useful for increasing the capacity of resolution of the problem but that have to be taken with the corresponding caution, since some effects or mechanisms of damage cannot be recorded by a two dimensional model. In this work, all problems studied have been represented by a two-dimensional model, e.g. as shown in Figure 2, studying what occurs in the x-y plane of a [0,90n]S laminate without taking account of the z direction. As expressed by Figure 2, a symmetry plane at y=0, has been considered. This implies to assume that the damage between fibre and matrix appears only at one side of the fibre and presents a symmetrical pattern with respect to y=0, as can be observed in Figure 3. This corresponds to what has been overwhelmingly observed in the experiments carried out. Figure 2.- Two-dimensional model used for study the damage mechanisms in a [0,90n]S laminate. Figure 3.-Detail of the cell with the fibre surrounded by matrix with the presence of damage consisting on a debonding. The models carried out are based on the Boundary Element Method (BEM) where Linear Elasticity is only required. In most cases, the nature of the problems under consideration has required a multiscale model involving the presence of isotropic and orthotropic behaviour, multibody formulation and capacity of dealing with contact problems, as damages involving debondings and cracks connecting debondings are simulated. All these features are contemplated in Graciani et al [47], whose BEM code is used in this Thesis. The used selfdeveloped BEM code is written in Fortran 90 and compiled in a Linux based system. This code is compiled to run in a vectorised manner i.e. able to use multiple processors within a High Performance Computing (HPC) System. 13 Not too many papers have been published on the use of multiscale models using BEM. Sfantos and Aliabadi [48] used a multiscale approach for modelling material degradation and fracture of polycrystalline ceramics, although in a different context than that used here. Notice that most problems requiring a double scale of attention were modelled coupling BEM-FEM, using FEM on the zone of detailed analysis and BEM to model zones with no special interest, e.g. Fernandes el al [49]. In this work, the zone of interest requires a very accurate solution and therefore BEM has to be used for it. A recent paper, Rodríguez et al [50], presented a multiscale dynamic failure analysis of 3D composite laminate using BEM and MCZM (Multi-scale Cohesive Zone Model). Due to the fact that in some models a debonding crack between fibre and matrix, Foundations of Interfacial Fracture Mechanics are required. Two models are employed at the crack tip because of the crack morphology. One is the so-called open crack model, proposed by Rice [51] where the lips of the crack are open and the contact zone is very small, interpenetrations associated with a small length will take place and a zone will appear where the stresses oscillate if the crack is considered completely open. The other is the so-called contact crack model proposed by Comninou [52] where there is a contact zone without interpenetrations and oscillations of the stresses, then being a correct solution from a mechanical point of view. Additionally, the refined discretization required for the interface cracks in the open crack model to overcome the presence of interpenetrations does not have implications, as it has in the use of FEM, on the discretization of the domain, as only a refined discretization is required along the boundary in the neighbourhood of the crack tip of the interface crack. More details about numerical aspects of the procedure employed to carry out numerical models can be found in Paper B [41]. Once a debonding reaches a certain length, it might kink and propagate throughout the matrix. In this case it is necessary to apply the Foundations of the Classical Fracture Mechanics. For both situations, the concept of Energy Release Rate, G, at the crack tip is involved in the different damage mechanisms that can appear in the failure of the 90 degrees laminas, employing G as the parameter that characterizes the failure process in this work. G is evaluated following Irwinˈs Virtual Closure Technique [53] for the frictionless case, considering both the Mode I fracture contribution, GI (eq. (1)), and the Mode II fracture contribution, GII (eq. (2)), where (r, θ) represent a local system centred at the fibre, θd is a semidebonding angle, 𝑟" is the fibre radius, and σrr and σrθ are the normal and tangential tractions along the interface, respectively, while Δur and Δuθ are the opening and sliding relative displacements between the crack faces, respectively. 𝐺$=1 ∆𝜃) * [𝜎--(𝑟",𝜃)+𝜃′)∆𝑢-(𝑟",𝜃)+∆𝜃)−𝜃′)] ∆67 89𝑑𝜃′ (1) 𝐺$$ =1 ∆𝜃) * ; 𝜎-6 < 𝑟",𝜃)+𝜃= > ∆𝑢6 < 𝑟",𝜃)+∆𝜃)−𝜃= >? 𝑑𝜃′ ∆67 8 (2) On account of the fact that BEM considers the displacements and the components of the stress vector along the boundary as independent variables, the values of the stress involved in the 14 calculation of G are then directly obtained and not by deriving the displacements field as in the case of Finite Element Method. The results of some numerical models included in this study, e.g. in Paper E [44], have been presented in dimensionless form by dividing the value of G by G0= (1+Km)σ02aπ/8µm, following Toya [54] and Murakami [55], where Km=3-4νm, µm is the shear modulus of the matrix and σ0 denotes the value of the applied tension. To make all the results obtained along this Thesis comparable, all the numerical problems are solved with boundary conditions that reproduce a nominal strain of 2%, unless specified otherwise. Since there are no predictive purposes in this Thesis, as already outlined, the value of the strain applied has no particular importance. 2.2.- Materials Different materials have been used during the development of this Thesis. In the case of the carried out numerical models, two materials with different properties have been employed: a carbon-epoxy system and a glass-epoxy system. The first one is a material with similar properties than AS4/8552 (widely used in the aeronautical industry), whose properties appear in Table 1 and are obtained from Soden et al [56]. These properties have been employed when analysing the damage mechanisms with the multiscale model, due to the fact that it is necessary to define the 0 degrees layer as orthotropic in the x-y plane, whereas the 90 degrees layer is isotropic at the same plane and, finally, in the micromechanical cell, the matrix is modelled as isotropic and the fibre as transversely isotropic. Table 1.- Carbon-epoxy system, properties obtained from Soden et al [56]. Material Properties 0˚ ply (orthotropic) E11=135GPa, E22=8.75GPa, E33=8.75GPa, ν12=0.3,ν13=0.3, ν23=0.4, G12=4.75GPa 90˚ ply (isotropic) E(E22=E33)=8.75GPa, ν=0.4 Matrix (epoxy, isotropic) E=4.2GPa, ν=0.32 Fibre (carbon, transversely isotropic) E (E22)=15GPa, ν=0.2 The second material, a glass-epoxy system, has been used for micromechanical models where one or two fibres are embedded in an infinite matrix with the purpose of comparing with previous numerical results [11-22]. The properties of this material obtained from [22] are included in Table 2. Table 2.- Glass-fibre system, properties obtained from Sandino et al [22]. Material Properties Matrix (epoxy, isotropic) E=2,79GPa, ν=0.33 15 Fibre (glass, transversely isotropic) E=70,9GPa, ν=0.22 In the case of experimental tests, two different composite materials have been employed too. The first one is a carbon-epoxy conventional pre-preg AS4/8552 provided by Hexcel® with an actual thickness of the lamina measured on the fabricated laminates after autoclaving of 170 µm (190 µm being the thickness of the uncured pre-preg provided by the manufacturer). The second one is a high-strength carbon epoxy composite with different thicknesses of the prepreg, from conventional to ultra thin thicknesses, and is composed of an epoxy resin system, SKYFLEX K51, and carbon fibre, Porofil® TR50S, supplied by Triple-H-Composites. This material is denominated as, for instance, USN150, where the number refers to the lamina grammage measured in units of grams per square metre (which corresponds approximately with lamina thickness equal to 150 µm). USN150, USN100, USN50, USN30 and USN20 (corresponding to 150, 100, 50, 30 and 20 µm of thickness) are used in the experimental results presented in this Thesis. When ultra-thin plies have been analysed with optical techniques, some important features were detected. The first one was the irregular transverse section of the fibres in USN20 and USN30 material, as can be observed in Figure 4. Figure 4.- Micrographs of two specimens belonging to different [04,90,04] laminates with 90 degrees layer thickness of a) 20 gsm and b) 30 gsm. Additionally, in the micrographs of Figure 4, a low fibre content throughout the thickness can also be clearly appreciated. For this reason, the fibre volumes of these materials (USN20 and USN30) were calculated through physical tests, obtaining a lower value in each case (25% and 30%, respectively) than that provided by the supplier over 55%. These features must be taken into account when experimentally analysing the mechanisms of damage within these materials. 2.3- Experimental Tests. Equipment and Procedures for Testing, Damage Detection and Observations. The results obtained from the numerical studies in this Thesis interact with experimental observations, giving mutual support between them. Cross-ply specimens have been manufactured from different carbon fibre reinforced composites that have been tested under a) b) 16 uniaxial tension to detect the appearance of damage in the 90 degrees layer as well as its possible extension in the form of damage like delamination between 0 and 90 degrees plies. The laminates have been manufactured in a clean room where the temperature, humidity and content of air particles are controlled. The stacked pre-pregs have been autoclaved (SCHOLZBA-E44313-1) applying different curing cycles depending on the material used. Then, the composite laminates have been cut into 250mm×15mm specimens by a water cutting machine using a diamond disk. To detect the appearance of the damage in the specimens, in most cases, the acoustic emission is a sufficiently accurate tool when the damage consists of a transverse crack in the 90 degrees layer. However, when the first damage that appears in the material consists of debondings connected (without being a full transverse crack) or unconnected, the noise produced by this damage is so tiny that may be imperceptible. In this case, small load increments have been applied to detect by direct observation the appearance of the first damage. A universal testing machine, Instron 4483, has been used to test the specimens and to measure the strain suffered by the specimens during each test, an extensometer INSTRON 2630-112 with an opening of 50 mm has been added to the specimen to this end. To study the damage found in the specimens a surface treatment in the observation area is needed. In the case of specimens where the damage generated is observed with an optical microscope, specifically with a binocular inverted microscope EPIPHOT 200, they are sanded and polished before testing. The result of the polishing and sanding process must allow the fibres corresponding to the 90 degrees layer to be clearly observed in the microscope at 1000× magnification. All details about these processes can be found in Paper A [40]. On the other side, in the case of the specimens inspected with a high-resolution scanning electronic microscope, FEI TENEO SEM, they have been covered with electrically conductive particles after testing. In all cases at the instant of detection of the damage, the test is stopped and the specimen is removed from the testing machine and inspected along its total length at optical or electronic microscope, obtaining micrographs of the damage found. 3.- Global summary of the numerical results Thanks to the accumulated knowledge two mechanisms of damages have been identified to potentially act to produce intralaminar damage in the 90 degrees layer; in this Thesis, one is called ‘progressive’ mechanism of damage, and the other ‘explosive’ mechanism of damage. The first ideas that arose in the development of this Thesis were connected to thought of a progressive failure, in which, a first debonding would be followed by a crack kinking and its subsequent penetration into the matrix. The progression of this damage throughout the lamina or the coalescence of individual isolated debondings would lead to a transverse crack in the lamina. This mechanism of damage will be studied in Section 3.1. As some of the findings from the numerical simulations did not supported the appearance of a scale effect, a look to the first explanation of the scale effect based on a global balance of energy was done. This approach would imply the sudden appearance of the transverse crack at once, 23 almost a zero value of the deformation producing the first damage. This approach is going to be detailed in the following analysis. If we want to calculate the total energy released by the sudden appearance of a crack, the procedure would be to make the stresses at the pristine state to work on the displacements of the final cracked layer. Applying this reasoning to the laminate object of study of this Thesis, [0,90n]s, as represented in Figure 14, the value of the stresses along the loading direction in the 90 degrees layer is a constant value that can be calculated employing the Classical Laminate Theory (CLT). BEM is used to obtain the displacements of the lips of the transverse crack lips. Figure 14.- Stresses before damage, σ90°, and displacements, u90°, after the sudden appearance of a transverse crack in the 90 degrees layer. Since the purpose of this study is to analyse the mechanism of damage as a function of the 90 degrees layer thickness, the integral of u90° along the thickness, which controls the energy release, is represented in Figure 15. Figure 15.- Evolution of the integral of u90° along the thickness of the 90 degrees layer. 24 If the magnitude represented in Figure 15 is now divided by the 90 degrees layer thickness, the result is the capacity of energy release per unit of crack created, which is shown in Figure 16. Figure 16.- Capacity of energy release per unit of crack generated in the 90 degrees layer for the appearance of a transverse crack. It can be clearly observed that there is a linear evolution between the capacity of releasing energy per unit of crack created, with respect to the 90 degrees layer thickness. This clearly indicates that, when the thickness tends to zero, the amount of energy available to generate a transverse crack tends also to zero. Thus, the balance of energy to generate a complete transverse crack, corresponding, due to symmetry, to one half of the crack, is produced by means of eq. 3, where 𝛾 is the energy required to generate a free surface in the 90 degrees layer, considered as an equivalent homogeneous material. 2 B 1 2𝜎C8° * 𝑢C8°9𝑑𝑠 FGH° 8 I =2𝛾𝑡C8° (3) If, for instance, a value of 𝛾 equal to 20 J/m2, which was employed by Saito et al [32] in their numerical studies with FEM, is taken, the value of σ90° can be calculated for a thickness given applying the CLT and, consequently, the strain, ε, required to satisfy the balance of energy can be obtained. The evolution of this strain is presented in Figure 17 as a function of the 90 degrees layer thickness. 25 Figure 17.- Evolution of the strain required to produce a complete transverse crack as a function of the 90 degrees layer. For the case of ultra thin plies (values of the 90 degrees layer thickness closer to the vertical axis in Figure 17), the necessary strain to create a transverse crack is very high. Focusing only on this aspect of the curve, this is a clear manifestation of the scale effect. To sum up, the presence of the scale effect has not been detected in the micro-mechanical studies of the different phases of what has been denominated in this Thesis progressive mechanism of failure of the 90 degrees layer (debonding, kinking and progression of the kinked crack to create a complete transverse crack). However, the manifestation of the scale effect does occur at the meso-mechanical study by the direct appearance of a transverse crack in the 90 degrees layer created from the pristine material, what has been in this Thesis an explosive mechanism of failure. 3.3.- Discussion on the different mechanism of damage involved in the failure of the 90 degrees layer in [0,90n]s laminates The two approaches followed to describe the damage in the 90 degrees layer lead apparently to contradictory conclusions, the one called explosive mechanism of failure (associated with a meso-mechanical view) supporting the scale effect phenomenon, and the other one named progressive mechanism of failure (associated with a micro-mechanical view) not supporting it. The first question that must be elucidated is if, in both approaches, associated with both mechanisms of failure, there is the same amount of energy involved, and above everything, if the fracture modes mixity is the same in both cases. To analyse this question, two different problems, each of them with two different approaches corresponding to the two views of the mechanisms of failure considered, are going to be studied. The first problem refers to the case of the generation of a complete transverse crack in the 90 degrees layer of the laminate. The case studied corresponds to that represented in Figure 14 26 with n=2 (i.e. a thickness of t90 ° equal to 240 µm). The energy released is going to be calculated with two different procedures. The first procedure corresponds to that followed in Section 3.2, in other words, it corresponds to a view of the final situation (what has been called catastrophic failure) making to work the stresses in the pristine state, obtained from CLT, to work over the displacements obtained from a BEM model. These displacements were calculated with an applied external u90 ° corresponding in this particular case to a nominal strain of 4% and are represented in Figure 18. Figure 18.- Normal displacements of the faces of the transverse crack in the 90 degrees layer. The work, w, associated with the displacements represented in Figure 18 and their corresponding stresses in the pristine state is calculated by means of eq. 4: 𝑤=1 2𝜎C8° * 𝑢C8° FGH° 8𝑑𝑠 (4) Due to the double symmetry of the problem, eq. (4) represents only one-fourth of the total energy released for the transverse crack. The value of w corresponding to the complete crack results to be 0.41 J/m. An alternative way of calculating w would be associated with what has been called a progressive mechanism of failure. Thus, the transverse crack would grow gradually, starting from a small transverse crack centred in the middle of 90 degrees lamina layer, until it reaches the interface with the 0 degrees layer. The evolution of G at the crack tip of transverse cracks of different lengths is represented in Figure 19. 27 Figure 19.- Evolution of G versus the length of a transverse crack in the 90 degrees layer. The results of the integration of this evolution of G for the whole crack (where one half is represented in Figure 19) produces the value 0.40 J/m, which is very close to that formerly encountered by de progressive mechanism of damage, the differences being associated with the numerical calculations. The agreement between the two procedures is based on the fact that in both cases, mode I of fracture is uniquely involved. However, this similarity between the two calculations associated with the two mechanism of failure does not always occur as is shown in the following study. The second problem consists now in analysing the multi-scale model with a single fibre in a hexagonal cell of matrix located in the 90 degrees ply and with corresponding 0 degrees lamina (see Figure 1), selecting the configuration [0,90n]s with a value of n equal to 1. This problem has been analysed, as the previous one, in two different ways, applying the two mechanisms of failure previously considered, progressive and explosive. In the case of the progressive mechanism of damage, the evolution of G along the semidebonding between fibre and matrix and its corresponding parts associated with mode I and mode II have already been obtained. For this reason, it is immediate to calculate the total energy released, and corresponding components during the generation of this damage. The values of the semidebonding angle taken into consideration are 50, 60 and 70 degrees. 50 degrees correspond to the maximum value of G evolution, and 70 correspond to the value of θd where nominally a contact zone appears, generating a pure mode II of fracture. It is obvious that an estimation of GC could be performed, but to avoid having to estimate values of fitting parameters, the aforementioned interval of θd is considered representative (see Correa et al [18] for predictions and Paper A [40] for an experimental evidence). The evolutions of G, GI and GII for the case [0,90n]s with n=1 are shown in Figure 20, to compare the values of the total energy released associated with semidebondings of 50, 60 and 70 degrees. The striped areas, corresponding to the case of 60 degrees, associated with modes I and II, have been indicated in the figure as WIpro and WIIpro, respectively. 28 Figure 20.- Evolution of G and its components GI and GII associated with WIpro and WIIpro for the case of a 60 degrees semidebonding angle in a progressive damage scheme. The values of Wpro, WIpro, WIIpro for the three semidebondings considered are summarized in Table 3. Table 3.- Wpro, WIpro, WIIpro for semidebonding angles of 50, 60 and 70 degrees associated with a progressive damage mechanism. PROGRESSIVE MECHANISM θd, semidebonding angle 50° 60° 70° WI pro (10-3 J/m) 0,025237 0,026644 0,026677 WII pro (10-3 J/m) 0,025208 0,038086 0,051323 Wpro (10-3 J/m) 0,050445 0,064730 0,078000 Since mode I dominates in the case of small semidebonding angles and disappears once a contact zone between the faces of the semidebonding crack arises, the values of WIpro are small and practically the same in the three cases studied, whereas the values of WIIpro increase significantly with the semidebonding in the range of θd considered. These values associated with mode II are considerably higher than those associated with mode I. Now, in the case of the explosive mechanism of damage, the same variables for the same configurations are evaluated with an explosive approach, i.e. the entire semidebonding angle appears at once. The equations to calculate the values of WIexp and WIIexp are shown in eq. (4) and eq. (5), respectively, where σr (radial stress) and σrθ (tangential stress) are obtained along the semidebonding angle in a pristine state of the material, and whose values have been represented in Figure 21 for a value of the semidebonding angle of 60 degrees. These stresses will work on the displacements, Δur (radial displacements) and Δuθ (circumferential 29 displacements), represented in Figure 22 for a value of semidebonding angle also of 60 degrees, and obtained in the debonded state. 𝑊$$ MNO =1 2 * 𝜎-6 67 8∙𝛥𝑢6 9 𝑑𝜃 (4) 𝑊$MNO =1 2 * 𝜎- 67 8∙𝛥𝑢- 9 𝑑𝜃 (5) Figure 21.- Values of s r and s r q in the pristine state along the length of the 60 degrees semidebonding angle. 30 Figure 22.- Values of D ur and D u q corresponding to a 60 degrees semidebonding angle. It is clear from the evolutions of stresses and displacements in Figure 21 and Figure 22, respectively, that what has been called explosive mechanism is dominated by mode I, the small contribution of mode II appears due to the curved morphology of the debonding. Table 4 summarizes the values of Wexp, WIexp, WIIexp for 50, 60 and 70 degrees of semidebondings in the case of considering an explosive mechanism of damage. Table 4.- Wexp, WIexp, WIIexp for semidebonding angles of 50, 60 and 70 degrees associated with an explosive damage mechanism. EXPLOSIVE MECHANISM θd, semidebonding angle 50° 60° 70° WI exp (10-3 J/m) 0,050597 0,062540 0,073014 WII exp (10-3 J/m) 0,001767 0,002922 0,004329 Wexp (10-3 J/m) 0,052364 0,065462 0,077343 Figure 23 shows the evolutions of W, WI and WII for the two mechanisms of damage under consideration, i.e. the values represented have been taken from Table 3 and Table 4, in order to compare them. 31 Figure 23.- Evolution of W, WI and WII for the progressive and explosive mechanisms of damage for the semidebonding angle values of 50, 60 and 70 degrees. Although in both mechanisms of damage total amount of energy, W, involved is the same, as corresponds to a view based on the Linear Elastic Fracture Mechanic approach, the explosive mechanism is clearly dominated by mode I, whereas in the progressive mechanism mode I and mode II are clearly involved. In view of the results shown in Figure 23, it can be said that, the reach of a final common damaged situation following two different mechanism of damage does not lead, in the case under consideration, to an equivalent situation from an energetic point of view. It is not a matter of whether a micro- or a meso-mechanical model is used. It is a matter associated with the way in which the damage is generated. It opens the possibility of having different, maybe opposite, conclusions from two studies conducted using the two different failure approaches previously mentioned, progressive and explosive. As will be shown later on in this Thesis, this will be one of the key factors for the search of a physically based explanation of the scale effect. 4.- Experimental Evidence Some of the results of experimental programmes carried out during this Thesis are going to be presented in order to contrast the numerical predictions obtained in the previous section. Prior to showing the experimental observations of the specimens belonging to the different laminates manufactured, it is useful to make a classification of the different damages that can be found in the 90 degrees layer in cross-ply laminates. This is shown in Figure 24, where the typology of the damage can extend from an almost negligible damage to a more advanced damage that may have catastrophic consequences. Figure 24 is used also to introduce the terminology employed in the rest of the Thesis for the different patterns of damage encountered. 32 Figure 24.- Typologies of damage that can be found in the 90 degrees layer in cross-ply laminates subjected to tensile load: a) isolated debondings, b) transverse generalized damage, c) transverse crack with premonitory delamination damage, d) transverse crack with total marked delamination. Thus, Figure 24.a), represents a pattern of isolated debondings whose appearance would have no effects on the integrity of the laminate. Figure 24.b) shows how these isolated debondings connect giving rise to what is denoted in this Thesis as transverse generalize damage. Figure 24.c) shows a complete transverse crack with slight indications of damage (associated with a future delamination), which consist of some debondings in the 90 degrees layer, close to the interface between layers and parallel to the direction of the load. Figure 24.d), represents a transverse crack now with a neat delamination between 0 and 90 degrees layers. A further explanation of configurations represented in Figure 24.c) and d) can be found in París et al [58, 59]. It is very important to note that the types of damage shown in Figure 24 do not appear necessarily in a sequential way, that is, we can find one or another without having detected the previous one, all this, as will be seen later on, depending on the configuration of the laminate. The testing programme has been conducted with cross-ply laminates and with two types of material described in Section 2, the emphasis in the tests having been put on the detection of the first damage in each case. The first material employed is AS4/8552, a material with conventional ply thickness given by the manufacturer of 190 µm. This material has been taken for both 0 and 90 degrees layers. The results obtained with the use of conventional thicknesses are shown in Figure 25, where the thickness of the central 90 degrees layer (2t90 in coherence with Figure 1) is represented in the horizontal axis, whereas the strain at which the first damage appears, or we have been able to detect it, is plotted in the vertical axis). a) b) c) d) 39 Figure 33.- Damage observed in a [04,90,04] laminates with a thickness of 90 degrees layers corresponding to 150 gsm. In the case of a 90 degrees layer thickness of 100 gsm, the type of damage found consists in isolated debondings following a crack path as can be seen in Figure 34. Figure 34.- Damage observed in a [04,90,04] laminate with a thickness of 90 degrees layer corresponding to 100 gsm. 40 The first detected damage for a thickness of the 90 degrees layer of 50 gsm, consisted in some isolated debondings as shown in Figure 35.a) If the load continued to increase, new isolated debondings would appear following a crack path, Figure 35.b), but nevertheless, these debondings would never coalesce to form a complete transverse crack. Figure 35.- Damage observed in a [04,90,04] laminate with a thickness of the 90 degrees layer corresponding to 50 gsm for different levels of strain: a) 0.8%, b) 1.34%. Finally, the smallest thickness of the 90 degrees layer used with ultra thin plies is 20 gsm and the first damage detected is shown in Figure 36 where unconnected ‘debondings’ appear. In this case, it should be taken into account the irregular section of the fibres (commented in Section 2), which might have an influence on the damage mechanisms. Figure 36.- Damage observed in a [04,90,04] laminate with a thickness of the 90 degrees layer corresponding to 20 gsm. (a) (b) 41 Notice that a complete transverse crack is never observed when using thicknesses with a unique ultra thin ply. Only isolated debondings appear, whose number may increase with the level of strain applied. If the strain continued increasing the laminate may suffer a total failure associated with the rupture of the fibres of the 0 degrees layer, without a complete transverse crack in the 90 degrees layer, only isolated debondings. It is very important to insist in the fact that although Figure 25 and Figure 31 are equivalent in the sense that both have strains associated with appearance of damage in the 90 degrees layer in the vertical axis, both have different meaning. Thus, whereas complete transverse damage (with the mentioned nuances about involving delamination) with no previous observations of isolated debondings appear for great thickness of the 90 degrees layer (those corresponding to composites of conventional thickness), only isolated debondings appear in the whole range of ultra thin thicknesses, these debondings never reaching a damage that could be associated with a complete crack in the 90 degrees layer. In conclusion, from the results of the experimental programs it can be deduced that there are different mechanisms of damage depending on the 90 degrees layer thickness, this being a clear manifestation of the phenomenon under study in this Thesis. 5.- Discussion of the results In the numerical studies carried out in a previous section, the existence of two mechanisms of damage that have been named progressive and explosive, has been taken into consideration, avoiding the denominations micro and meso because these denominations might not be representative of what is happening in the laminate, since it is not a question of whether the mechanism of damage is actually micro or meso, but rather of how the mechanism of damage really is (what has been observed experimentally in the laboratory) and how our models of damage represent it. In the progressive mechanism of damage, in which it was assumed that the damage begins with a debonding between the fibres and the matrix, that kinks and penetrates into the matrix to create a transverse crack, all damage phases analysed are not influenced, in accordance with the numerical analysis carried out, by the 90 degrees layer thickness. However, in the case of the explosive mechanism of damage, which consists on the direct appearance of the transverse crack from the pristine state of a material, the influence of the 90 degrees layer thickness is clearly identified by the predictions. With respect to the experimental studies, tests oriented to detect the first appearance of damage in the 90 degrees layer in crossply laminates have been designed and carried out. The matter is now to integrate all this information (both numerical and experimental) to give a physically based explanation of the scale effect in composites without involving fitting parameters in the explanation. The presence of two mechanisms of damage has experimentally been found in the different 90 degrees layer of the laminates and material employed. One presents only isolated debondings that may never coalesce to get a complete transverse crack in the 90 degrees layer before the total breakage of the laminate takes place. The other consists in the sudden appearance of a transverse crack in the 90 degrees layer (without previous observations of isolated debondings), which may be or not be accompanied by different manifestations of delamination of the interface between the 0 and 90 degrees layers. The explanation of the appearance of one or the 42 other mechanism of damage will be illustrated with the help of a generic schematic curve for a general material (EFGJH, similar to that represented in Figure 17) showing the evolution of the strain that satisfies the explosive mechanism of damage involving a global energy balance in the 90 degrees layer, see Figure 37. With reference to what has been previously called progressive mechanism of damage, the first manifestation of damage associated with this mechanism is the appearance of isolated debondings. When it has been possible to detect these isolated debondings (e.g. in the case of laminate with thicknesses of the 90 degrees layer compound of a single ultra-thin ply), they have appeared at a level of the strain of an order of that associated with YT, the nominal value of strength of a 90 degrees lamina obtained from a test. This is a value clearly identified for a composite material when we look at it as a homogeneous orthotropic material. This test presents a high scatter, see Figure 25 and Figure 31 for laminates of conventional and ultra-thin thicknesses respectively. Thus, instead of a particular value, in this case it has been preferred to draw an area, defined by points AFCDGB in Figure 37, associated with YT. Notice that this interval has been drawn horizontally along the whole interval of values of the thickness, because as explained in Paper C [42], in accordance with the numerical predictions, there is the same probability of generating an isolated debonding in a [0,90]S laminate, independently of the thickness of the 90 degrees layer. At this point, it should be clear that for any thickness of the 90 degrees layer, when the applied deformation reaches the value of ε(YT), isolated debondings should start to appear. It is now the moment to remember that it was said that the same probability of getting a debonding between fibre and matrix does not imply the same possibility of observing it, as will be explained in what follows. Figure 37.- Schematic representation of a possible physically based explanation of the scale effect in [0,90]s laminates. On the other hand, with reference to what has been called explosive mechanism of damage, the EFGJH curve represents for a certain material applying the Fracture Mechanics Theory, the values that must reach the applied strain to release the energy required to break the internal links of the 90 degrees layer, considered as a homogeneous material (isotropic in the plane A B C D E F G H I J K L M N P !"(%) 2'90°(mm) ≈ !"(,-) !"(.-) 43 under consideration). It has to be emphasized that in the context of this mechanism of damage the material is characterized by a mesomechanical property, the apparent fracture toughness of the 90 degrees layer, g or GIC. Thus, once we have reflected the presence of the two mechanisms of damage detected (both characterized for identifiable parameters of a composite, YT for the progressive mechanism of damage and GIC of the 90 degrees layer for the explosive mechanism of damage) the two potential evolutions that a specimen can follow in a test, associated with two representative thicknesses of the 90 degrees layer (referred here as high and low values of the thickness), will be considered in what follows. 1.- High values of the thickness of the 90 degrees layer. Evolution along line IJK. If a laminate with a thick 90 degrees layer is taken into consideration (represented by point I in Figure 37), it is obvious that small values of e allow the curve EFGJH to be reached, then implying that the energy necessary to break the material is available, but , why the failure does not appear in this case? EFGJH is based on Fracture Mechanics, which aims to predict the behaviour of the material (in this case the behaviour of the laminate and in the particular case of the 90 degrees layer) in presence of a damage that can be assimilated to a crack. However, at point J there is no damage at all and, consequently, Fracture Mechanics Theory is not applicable or, in simple words, there is no point where the damage can arise, although if there were the possibility of having a small damage, then there would be energy enough to propagate it immediately. This last situation is what does happen when, increasing the strain applied, the material enters into the band AFCDGB, represented by point K in Figure 37. It is important to recall here that YT represents (at mesomechanical level) the value of the external load that is able to produce the release of the amount of energy enough to generate a debonding between fibre and matrix, what is considered to be the weakest point of the material in the plane under consideration. For the corresponding value of the strain associated with YT there was, in the test of a 90 degrees specimen, energy enough to unchain the propagation of an incipient damage (as a debonding) that led the 90 degrees specimen to the total failure. Going back to the laminate under consideration, once a debonding in the most risky position (a question widely studied experimentally in Paper A [40] and numerically in Paper E [44]) appears, and considering that from the value of the strain associated with point J there was energy enough to propagate a damage which could have a small size in the 90 degrees layer, now, only the presence of one debonding, just one, will unchain the explosive mechanism of damage of the 90 degrees layer, see Figure 27 where the damage pattern corresponding to the sudden appearance of a transverse crack is observed. This is due to the energy accumulated in the material by continuing to increase the strain from point J. Obviously, in accordance with the knowledge previously accumulated, there would be no possibility in the type of laminates with high values of the thickness of the 90 degrees layer studied in this section, of observing isolated debondings, and in fact we have never observed them in experimental testing. As soon as the material enters in the AFCDGB zone, situation represented by a generic point K in Figure 37, the appearance of the first debonding implies the generation of a transverse crack damage with the possibility of delamination if the thickness of the 90 degrees layer is extremely high. Thus, if the specimen is removed from the test machine at a point K, a complete crack, with the nuances already mentioned, is observed, whereas if the 44 specimen is removed from the test machine at a strain lower than that associated with point K no damage at all is observed. 2.- Low values of the thickness of the 90 degrees layer. Evolution along line LMNP. If a laminate with a thin 90 degrees layer is taken now into consideration (represented by point L in Figure 37), it is obvious now that it is impossible to reach the EFGJH curve without crossing the zone of isolated debondings represented by area AFCDGB in Figure 37. When the strain reaches the value that define the area AFCDGB (point M in Figure 37) isolated debondings will appear, as the material is in the zone where this happens, see Figure 37 where the damage pattern corresponding to Figure 35.a) has been included. Why the appearance of these isolated debondings does not lead to the appearance of a whole transverse failure of the layer? Obviously because now point M is under the EFGJH curve and there is no energy available, at this level of strain applied, to produce a transverse crack. If starting now at point M and the strain is increased until reaching a point like N in Figure 37, no transverse crack appears because the EFGJH curve has not yet been reached and, consequently in this case, in presence of a debonding, there is no energy available to generate an explosive failure consisting in the direct appearance of a transverse crack in the 90 degrees layer. It is nevertheless possible that new isolated debondings appeared when the strain increases, and it would be even possible to observe a configuration with many isolated debondings that do not connect between them to generate a complete transverse crack, see Figure 37 where the damage pattern corresponding to Figure 35.b) has been included. It is important to highlight that a situation like the last described, a 90 degrees layer full of isolated debondings, does not alter the stiffness of the whole laminate and could be considered as an acceptable pattern of damage for the performance of the laminate. The most natural evolution when increasing the strain is that before reaching curve EFGJH, a value of the strain, defined in Figure 37 as e (XT) is reached. This value corresponds to the strain that appears in the testing of a 0 degrees layer at the instant of failure, and of course this happens, as observed in the experiments, without the appearance of a transverse crack damage in the 90 degrees layer. The situation defined in this case for a laminate whose behaviour is visualized starting at point L in Figure 37, clearly allows isolated debondings to be observed, as illustrated in Figure 37 for values of the deformation corresponding to points M or N. Although what has been described along the line LMNP is what has been typically observed in the tests carried out, it is obvious that point N can reach curve EFGJH before the strain reaches the value e (XT). In this particular case, a whole transverse crack in the 90 degrees layer may appear before the total breakage of the laminate. The occurrence of one or the other situation will obviously depend on the material and on the thickness of the 90 degrees layer, the thinnest layer being the most unlikely to reach the EFGJH curve before reaching e (XT). It is important to note that only two extreme situations of damage, from the total set of potential configurations, have been considered in the aforementioned explanations: isolated debondings and transverse crack in the 90 degrees layer. In this second case, if we have a thickness of the 90 degrees layer near to those defining points F or G in Figure 37, the transverse crack does not present delamination or even debondings associated with delamination, París et al [58, 59]. However, if the thickness of the 90 degrees layer is high, there is an excess of energy to produce 45 the transverse crack, which originates the appearance of delamination or debondings associated with it between the 0 and 90 degrees layers. There is an important message in the explanation given that goes beyond a physically based explanation of the scale effect in composites. Why is it necessary to have a double criterion, see for instance Mantič and García [60], one based on energy and another on stress? Most of the phenomena appearing in materials (damage or changes in behaviour) are associated with energy. The most direct example is plasticity appearing in metals, which appears when distortion energy (associated with the change in shape) reaches a critical value (the one producing the irreversible relative displacements between planes associated with the presence of dislocations). This fact does not prevent, in spite of being a phenomenon controlled by energy, from having values associated with the stress, as the deviator modulus or the octaedric shear stress, that also define the damage under consideration, in this case the start of plastic behaviour, for the same value of the loading. What does happen in composites for the question under consideration? Everything is controlled by energy. What is called strength of the 90 degrees layer, nominally YT, is a very convenient way to describe the appearance of a phenomenon: the breakage of a specimen of a composite with the fibres oriented 90 degrees with respect to the loading direction. This breakage starts by a debonding between fibres and matrix controlled by the energy, or equivalently by YT, and it is a characteristic value of the material independently of where it is placed, as has been shown in this paper when selecting different thickness in the 90 degrees layer. YT is not an in situ value. The problem starts when the same concept is tried to be used to define the failure of a 90 degrees layer in a laminate, due to the fact that, in presence of certain thicknesses, to reach the level of energy required to activate the debonding (which is not an in-situ value), does not produce the total transverse failure of the 90 degrees layer, a phenomenon associated with a different mechanism of damage, not controlled by YT. The question was solved creating the convenient and simple concept of in-situ strength, assigning different properties to the same material depending on where it was placed. The present explanation assigns to the energy the role of being associated with the appearance of damage under consideration in a composite. Since composites are heterogeneous materials, there are different mechanisms of damage involved in the behaviour of the laminate (two in this particular case) and consequently different energy values need to be taken into account. One of the mechanisms was traditionally thought to be controlled in a convenient way by the value of a stress. The corresponding allowable was then taken as an in-situ property to consider other failures not associated with the same mechanism of damage. Figure 26 6.- Conclusions The most important goal accomplished in this Thesis has been the obtaining of a physically based explanation of the scale effect in composite materials without employing fitting parameters, only concepts associated with Fracture Mechanics or Interfacial Fracture Mechanics have been used together with the corroboration of experimental observations. The key factor was the identification of the two mechanisms of failure that can appear in the 90 degrees layer in [0,90n]s laminates subjected to uniaxial tension. 46 With regard to the two mechanisms of damage experimentally identified, isolated debondings can be observed as a first appearance of damage in the 90 degrees layer in one of them, what has been called progressive mechanism; whereas in the other, the direct appearance of a transverse crack with or without delamination damage, called explosive mechanism, is the first manifestation of damage. In the numerical studies carried out, on the one hand, a mechanism of progressive damage has been investigated using an integrated multi-scale model in different phases of damage (debonding, kinking and propagation of the damage through the matrix) where no indication of the scale effect was detected, i.e. there was no influence of the 90 degrees layer thickness in the different phases of damage. On the other hand, an explosive mechanism of damage in which the direct appearance of the transverse crack using a meso-mechanical model has been studied, where clearly the presence of the scale effect is detected, although this does not satisfy what happens when the 90 degrees layer is very thick because the values of the predictions of the strain that produce the failure are very small and they do not correspond to the actual behaviour of the laminate. So then, why is one damage mechanism capable of capturing the scale effect and the other not? The question was to elucidate if both mechanisms of damage studied were, from a fracture mechanic point of view, equivalent or not: i.e. if in both mechanisms there is or not the same amount of energy released involved and if the nature of the energy released (mode I and mode II), was the same or not. What has been clearly clarified with the studies carried out in this Thesis is that both mechanisms are not equivalent, as both involve the same amount of energy released but they are not equivalent in the mode of releasing it. Thus, it has been shown that the progressive mode of failure has a greater implication of mode II, which cancel any equivalence between both mechanisms. In other words, to perform a micromechanical model putting cohesive elements between fibres and matrix to take into account the damage not only does not guarantee that an explosive mechanism of failure is being properly modeled, on the contrary, after the studies carried out it is clear that it does not reproduce such mechanism. This step was compulsory to proceed with the study prosecuted and gives a neat answer to the question of why micromechanical models that involve progressive mechanisms of failure cannot explain the scale effect. Another important point of the explanation given, which in fact goes further of the field of composites, is the question of the necessity of employing a double criterion. Here it has been shown that the two mechanisms of damage involved in the problem are controlled by energy, although it is very convenient, as in many cases of continuum mechanics or structural design, to express the instant in what the mechanism is activated by means of a component of the stress tensor (or combination of stress components). This fact, additionally to highlighting the role of the energy in the damage processes of materials, opens the door to reconsider problems in which the double criterion has been successfully used to predict the appearance and propagation of damage in all materials and configurations of geometry and loading. These ideas also open a reflection on the “in-situ strength” concept. And what it is more, open even the question of acceptance of the strength concept for composites, at least in what refers to YT. The concept of “in-situ strength” was originally viewed, at least by some authors, as a convenient way to deal with something not fully understood, what in some way was associated with the idea that the strength was a physical property of a material. It has been shown in this Thesis that 47 what are physical properties of the material are variables associated with the energy involved in the mechanisms of damage and what has been regularly called ‘strength’ is a value of the stress associated with the critical values of the energy that controls, in the corresponding form, the failure of the material, in a determined configuration, typically that associated to a simple tension test (or similar). Thus, if strength is associated with the appearance of the first damage in the 90 degrees layer, it is not adequate at all to refer the strength as an-situ value, as debondings appear always for an identified value of the deformation applied, e (YT). On the contrary, if strength is associated with the appearance of a damage that can be recognized as a transverse crack, then the concept of in-situ strength can be applied as a convenient way of reflecting the level of damage involved in the concept. Other conclusions that are relevant for the study of damage mechanisms in composite materials have also been obtained throughout the development of this Thesis, they can be found in the papers appended to this work and are summarised in what follows. - From the multiscale models it is deduced that the appearance of an isolated debonding, as a first damage that arises in the 90 degrees layer, is not affected by the thickness of the layer. For this reason, there is the same probability of nucleation of isolated debondings in a thick or thin 90 degrees layer but this does not imply the same probability of observing them. This conclusion has been essential to achieve the physically based explanation about the scale effect. - The BEM multiscale numerical models developed have proven to be extremely accurate, based on the comparison with previous results and supported by experimental evidence. In addition, the ability of the model to deal with the first stages of the damage as a function of the variables that define the scale effect in composites has also been proved to be very robust and agile. - The experimental micromechanical studies carried out made it possible to observe, by means of optical microscopy, the micromechanical damage promoted by tensile loads at the 90 degrees layer of cross-ply laminates. The different stages of the mechanism of damage predicted by previous numerical studies were identified. The final extension of the debond at the interface was measured, also analysing its dependence on the loading level, the presence of neighbouring fibres as well as the kinking type. In addition, specific configurations of two damaged fibres (in terms of their relative orientation and distance) with a critical effect on the damage evolution were identified. - The numerical micromechanical studies, employing two-fibres models in order to search critical configurations, have shown that the presence of a damaged fibre (a fibre housing a debonding at its matrix interface) may have a shielding or an accelerative effect on the appearance of damage in the undamaged fibres placed in the neighbourhood of the damaged fibre depending on the relative orientation between both fibres. The configurations with an accelerative effect have been identified and present a clear physical meaning. Thus, in presence of a first debonding oriented normal to the load applied, the second appears in a way that represents a damage path associated with the generation of a transverse crack, normal to the load applied. Experimental evidence has been found supporting this conclusion. - Another interesting conclusion obtained from the micromechanical numerical studies is the effect of the distance between the fibres, concluding that there is no interaction between the damaged and the initially undamaged fibres when they are separated approximately by a distance of four fibre diameters. For this reason, only interactions in the local neighbourhood of the fibre where the first debonding appeared are expected. 48 7.- Future Developments In this Thesis, a physically based explanation of the scale effect has been presented. Now, new challenges appear and corresponding research for each of them could be developed. Due to the fact that the physically based explanation given for the scale effect does not require fitting parameters or the assumption of the existence of previous damage, and the observed mechanisms of damage at the micro and meso level have been identified and characterized, we are in the adequate scenario to identify the values of the parameters involved in the failure. Thus, in what has been called the progressive mechanism of failure, the fracture toughnesses GIC and GIIC of the interface fibre-matrix are involved and can be evaluated from a test. Additionally, what is involved in the explosive mechanism of failure is the apparent fracture mode I toughness of the 90 degrees layer, considering the lamina in the plane under study as an equivalent isotropic and homogeneous material. This value can also be measured from tests now that the damage mechanisms involved in the failure have been identified. The determination of these parameters would be very useful for the design of composites structures and for their predictions of failure. It is obvious that to decrease the thickness of a 90 degrees layer is positive to delay the appearance of damage but at the same time may increase the manufacturing costs. Thus, it would be very useful, from a structural design optimization point of view with composite laminates, to determine the thickness value of the 90 degrees layer for a particular material in which the breakage of the fibres of the 0 degrees layer and the appearance of the transverse crack in the 90 degrees layer occur at the same time (i.e. the appearance of the first transverse crack in this 90 degrees layer would be produced for a strain value of ε(XT) involving consequently, the complete failure of the laminate). In this way, the structural capacity of the layer could be extended to the limit, since, as has been mentioned several times in this Thesis, the appearance of transverse cracks with incipient damage associated with delamination in the 90 degrees layer can lead to premature and catastrophic failure of the laminate. Further, but not less important, the identification of the mechanisms of damage in connection with the necessity of the employing a double criterion based on stress and energy, opens the possibility of revisiting other problems to investigate if there is an energy based explanation about the necessity of involving a stress on the failure. Finally, the extension of this study with ultra thin plies to fatigue loading is clearly required. 55 APPENDED PUBLICATIONS 56 57 PAPER A: Microscopical observations of inter-fibre failure under tension The original version of this paper can be found in www.elsevier.com • DOI: 10.1016/j.compscitech.2017.12.009 • Journal Name: Composites Science and Technology • ISSN: 0266-3538 • Journal Citation Reports (2017). Impact Factor:5,16 -Materials, Science, Composites: Q1 58 59 Microscopical observations of inter-fibre failure under tension Elena Correa, María Inmaculada Valverde, María Luisa Velasco, Federico París Departamento de Mecánica de Medios Continuos y Teoría de Estructuras, Grupo de Elasticidad y Resistencia de Materiales, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092, Sevilla, España Abstract The numerical study of the inter-fibre failure at micromechanical level predicts the appearance of different stages in the development of this damage mechanism; these studies have also allowed the main features of each stage (such as the interfacial debond length and the kinking angle) to be identified. The development of experimental studies aiming to check the relevance of the aforementioned numerical results is crucial. Based on this, this research focused on the tensile test under different loading levels of specimens manufactured from carbon-epoxy cross-ply symmetrical laminates. The microscopic observation of the 90º layers leads to the analysis of the appearance of the transverse cracks as a function of the load, the identification of the previously numerically predicted stages of the mechanism of damage, the measurement of key parameters and the evaluation of the influence of nearby fibres. A clear connection between the numerical and experimental results has been found. Keywords: debonding; transverse cracking; optical microscopy; cross-ply. 60 A.1.- Introduction Matrix/Inter-fibre failure arises in plies subjected to transverse loads, being characteristic, among others, of 90° plies belonging to cross-ply laminates subjected to loads parallel to their 0° plies. Previous numerical studies allowed the identification of three main stages in the mechanism of damage at micromechanical level, Figure A. 1. In particular, in the case of tensile loads (París et al [1]), damage initiates with the appearance of small debonds at the fibre-matrix interfaces aligned with the applied load (Stage I); these debonds grow unstably in Mixed Mode along the interfaces until reaching an extension (semi-debonding angle, q d) of 60°-70° (Stage II); at this position, a change to stable growth in pure Mode II takes place (caused by the appearance of a physically relevant contact zone at the crack tip), favouring the kinking of the crack towards the matrix (Stage III). The crack finally propagates along the matrix (unstably and in Mode I) following the direction perpendicular to the load applied. The coalescence of this type of cracks coming from different fibres gives rise to transverse macro-cracks in the ply, eventually leading to its catastrophic failure. Figure A. 1.- Stages of the mechanism of damage. París et al [1]. The effect that the presence of a secondary load (París et al [2], Correa et al [3]) has on the evolution of these stages, the consideration of thermal residual curing stresses (Correa et al [4]), and the case of cyclic loading (Correa et al [5]), have also been studied numerically. The important role that micromechanical studies can play in helping first in the understanding of the mechanism of failure and second in verifying and proposing failure criteria was point out in these works. In particular, this outstanding question was deeply analysed in París [6]. A common pattern of these studies is the usage of a single fibre (assumed isolated) Boundary Element model. More recent works, focusing on Stage II of the mechanism of damage, also include a large number of fibres, see for instance Sandino et al [7] (two-fibres models), Távara et al [8] (ten-fibres models) and Távara et al [9] (one hundred-fibres models). FEA has also been used to study this problem; see for instance Hojo et al [10], Kushch et al [11] y Saito et al [12]. At this point, it turns out crucial to check, by means of experimental studies, the validity of the results derived from the numerical models, specifically, by means of microscopic observation of tested samples. Previous works from different authors focusing on the experimental micromechanical observation of the matrix failure can be found in the literature; among them it can be stood out the already classical works by Zhang et al [13] (single fibre tests under tension) and Gamstedt STAGE I a ≈10º q d=60-70º STAGE II STAGE III 61 and Sjögren [14] (based on Zhang but studying the effect of cyclic load), and also those from Saito et al [15] (analysing the effect of the 90º ply thickness in cross ply laminates), Baral et al [16] (studying the effect of the fibre type), Hobbiebrunken et al [17] (comparing the behaviour between high and low curing temperature resins in three points bending tests) or Genz et al [18] (evaluating the presence of thermal residual stresses). In the present work, an optical microscope is used to observe the 90° plies of cross-ply specimens tested under longitudinal tensile load (which will be thus transverse to the 90° plies), in order to check if the numerically predicted micromechanical stages of the matrix/inter-fibre failure are in fact present and can be distinguished. The identification of the differences that these stages show in real laminates with a high fibre volume content versus the numerical single (and isolated) fibre models, is also pursued. In addition, the measurement of the final extension of the crack at the interface that gives rise to the kinking towards the matrix ( q d= 60°-70°, from the numerical models), as well as the study of the different damage typologies, the analysis of the influence of the applied loading level and the effect of the presence of neighbouring fibres, are also objectives of the study. A.2.- Coupons manufacturing and testing [03,903]s tensile coupons were manufactured from carbon-epoxy (AS4-8552) pre-preg, reinforcing their ends with glass-epoxy tabs. Their dimensions are detailed in Figure A. 2. Figure A. 2.- Coupons dimensions. A first set of 5 coupons was selected for testing under tension. The results provided by the tensile strength (X) show a mean value of 942.47 MPa, a standard deviation of 41.25 MPa and a variation coefficient of 4.38%, therefore implying an excellent result in terms of dispersion. Subsequent testing campaigns were performed under maximum loads lower than the previously measured X, in order to generate damage in the 90° plies without reaching the catastrophic failure of the whole laminate. In particular, the loads applied can be grouped in 5 levels: 80%X, 60%X, 40%X, 30%X and 20%X. The number and denomination of the coupons tested as well as the exact loading level applied are included in Table A. 1. 62 Table A. 1.- Denomination of the coupons and corresponding testing load. Coupons denomination Testing load (%X) 80-1 84.2 80-2 82 60-1 62.4 60-2 59.5 60-3 60.5 40-1 42.6 40-2 40.3 40-3 39.2 30-1 31.5 30-2 29 30-3 27.6 20-1 21 A.3.- Preparation of samples The central part of the specimens was cut in two halves of approximately 28 mm length; each pair was encapsulated in resin (using the VersoCit-2 kit). After that, the samples were polished using Saphir 20 polisher (automatic head and programmable) from NEURTEK Instruments, LaboDoser liquide dispenser and LaboPol-5 as peripheral polisher from Struers. The result of the polishing process must allow the fibres corresponding to the 90° plies to be clearly observed in an optical microscope at 1000x magnification. A suitable polishing process should also avoid the appearance of marks that could interfere in the later microscopical inspection. To this particular end, a specific polishing process was designed and tested, consisting of four steps. 40N pressure and 50rpm speed are selected for all steps, the disc type, refrigerator liquid and time used are detailed next: Step 1: Disc: SiC #320. Liquide: Water. Time: 5’. Step 2: Disc: MD-Largo. Liquide: DiaPro Allegro/Largo 9 μm (2). Time: 6’. Step 3: Disc: MD-Dac. Liquide: DiaPro Dac 3 μm (1). Time: 5’. 63 Step 4: Disc: MD-Dac. Liquide: DiaPro Nap B1 1 μm (3). Time: 4’. A.4.- Microscopic observation Once the samples have been polished the microscopic inspection is undertaken. There is a double objective here: first, using low magnification (25x), the analysis of the appearance of transverse cracks in the 90° plies as a function of the load applied. Second, the stages of the mechanism of damage previously identified by the numerical models want to be detected, along with the experimental values of their main parameters; to this end, observation at high magnification (500x and 1000x) is required. A.4.1.- Inspection at low magnification A first inspection is made at 25x in order to count the number of transverse macro-cracks that appears in each specimen, and thus its loading level dependence. In Table A. 2 the number of cracks observed per cm in each coupon is detailed. The most relevant conclusion is that for loads lower than 30%X no transverse macro-cracks are observed in the samples. A later inspection at greater magnification will also show, Section A.4.2, that no damage of any type is detected below this loading level. It is also observed that the number of cracks encountered increases as the loading level does. Notice in any case the reduced number of samples chosen for the lowest loading levels. Table A. 2.- Number of macro-cracks as a function of the loading level. %X Number of macro-cracks/cm 84.2 9 82 9.2 62.4 6.4 59.5 4 60.5 3.6 42.6 3.5 40.3 1.2 39.2 0.8 31.5 2.3 29 0 64 27.6 0 21 0 Although not being directly the objective of this paper, making use of the Classical Laminate Theory (CLT) and selecting a suitable failure criterion it is possible to predict the stress associated with first-ply failure (XFPF). Then, it is interesting at this point to check if the lowest stress value for which damage has been detected in the material (31.5%X in this case, see Table A. 2) is accurately predicted by the Laminate Theory in combination with failure criteria. To this end, a selection of failure criteria (Maximum Stress, Tsai-Hill (Azzi and Tsai [19]), Tsai-Wu [20], Maximum Strain, Hashin [21], Hashin and Rotem [22] and Puck [23]) has been performed, and their XFPF predictions calculated, see Table A. 3. Table A. 3.- First-ply and ultimate failure predictions for a selection of failure criteria. Failure Criterion XFPF (MPa) XUF (MPa) XFPF/XUF (%) Maximum Stress 374.4 1112.9 33.6 Tsai-Hill 374.1 1109.5 33.7 Tsai-Wu 368.1 1149.1 32.0 Maximum Strain 367.2 1124.6 32.7 Hashin 374.4 1112.9 33.6 Hashin-Rotem 374.4 1112.9 33.6 Puck 374.4 1113.8 33.6 First of all, it can be observed that all failure criteria show similar predictions with a mean value of 372 MPa. This value can be compared with the lowest experimental load for which transverse cracks are observed, 31.5%X in this case, which corresponds to a stress value of 297 MPa. Taking into account that the number of coupons included in the 30%-40%X group is small (see Table A. 2) and also that all of them show a very reduced number of transverse cracks, a good agreement between the analytical prediction and the experimental result is found. The appearance of the first crack at a value lower than that predicted by the criteria for the failure of the ply is, on the other hand, a very coherent fact, as the appearance of one transverse crack does not mean the total failure of the ply. The appearance of transverse cracks in a 90° ply is a continuous (not sudden) phenomenon, and it can be checked that (as mentioned above), for the first ply failure predicted by the criteria, only a very reduced number of cracks appear in the specimen, this fact not implying a significant reduction in the contribution of the 90° ply to the properties and performance of the laminate. Another interesting feature arises when using the CLT to also predict the stress associated with the ultimate failure (XUF) and then to calculate the ratio between XFPF and XUF (Table A. 3), i.e. 71 A.5.1.- Effect of Stage III typology It seems also interesting to study the distribution of the debonding angle as a function of Stage III typology (Figure A. 6). The values obtained for q d (mean value, µ , standard deviation, s, and variation coefficient, VC) as a function of Stage III typology, along with the number of measures n carried out, are included in Table A. 4. Table A. 4.- q d: Dependence on Stage III typology. q d Stage III typology µ (º) s (º) VC (%) n (A). Theoretical 48.1 10.6 22.0 611 (B1+B2). Deviated 53.4 12.8 23.9 187 (C). Bilaterally tangential 59.0 9.1 15.4 15 (D). Gradual transition 40.8 12.5 30.6 30 (E). Splitting 60.8 27.2 44.8 6 From the number of measures n it is deduced that typology A is, by far, the most frequent and its corresponding q d results are very similar to the global ones presented in Figure A. 10. In the rest of the cases, excepting typology D, q d values increase with reference to the global results. Notice in any case that the number of measures n is very small for typologies C, D and E. Finally, with reference to the sub-ranges distribution (Figure A. 11) and taking the global results as a base, it can be checked that for typologies B and C there is a translation to upper values of q d, so that the most preferential sub-range is in these cases [50°-70°] (closer to the numerical prediction [1]). Figure A. 10.- q d: values distribution. 2% 18% 38% 25% 10% 6% 1% q d sub-ranges <30º [30º-40º] [40º-50º] [50º-60º] [60º-70º] [70º-80º] >80º 72 Figure A. 11.- Effect of Stage III typlogy on q d sub-ranges. A.5.2.- Effect of neighbouring fibres Another interesting factor to analyse is the dependence of the obtained results with the fibres density in the local neighbourhood of the Stage III crack to be analysed and, in particular, with the possibility of considering the fibre that houses the debond as isolated or not. The objective in this case is the search of a scenario that can be more easily compared with that used by numerical models. It has been possible to perform 126 measures on what has been considered as isolated fibres. The requirement imposed in this work to classify them as isolated fibres has only been the inexistence of nearby fibres at a very short distance (in particular, one fibre radius distance in the most unfavourable case), at least, in the damaged side of the interface. The high fibre volume fraction of the material employed and the necessity of having a significant number of measures in order to derive any conclusion conditioned the selection of such short distance. The results show that q d values are distributed within the interval [22.4°-76.8°], with a mean value of 49.3°, a standard deviation of 20.8° and a variation coefficient of 21.2%; these values do not present a significant difference with reference to the global results. In contrast, notice that the distribution range is shorter by its upper side than that corresponding to the global results. Moreover, most measures are included again in the [40°-60°] sub-range but in a greater percentage than in the global case, 68%, which means a slight translation to greater values and thus closer to the numerical predictions. Finally, it should be noticed that, by means of two-fibre models, Sandino et al [7] showed that the disappearance of the effect of the presence of a neighbouring fibre on the growth of a debond housed on the fibre under study, occurred when the distance between both fibres was equivalent to a number of fibres that depended on their angular position. This number of fibres 2% 18% 43% 24% 8% 4% 1% A. q d sub-ranges <30º [30º-40º] [40º-50º] [50º-60º] [60º-70º] [70º-80º] 2% 14% 25% 30% 16% 10% 3% B. q d sub-ranges <30º [30º-40º] [40º-50º] [50º-60º] [60º-70º] [70º-80º] 20% 40% 20% 20% C. q d sub-ranges [40º-50º] [50º-60º] [60º-70º] [70º-80º] 20% 34% 23% 13% 10% D. q d sub-ranges <30º [30º-40º] [40º-50º] [50º-60º] [60º-70º] 73 was in the range 3 to 8, which could be considered as an alternative and upper limit to that chosen in this work to label a fibre as ‘isolated’. Presumably, the usage of this limit could have lead to provide experimental results closer to the numerical prediction [1], but it would be of application only in materials with fibre volume fractions lower than that presented in this work. A.5.3.- Effect of the loading level The global effect of the loading level is summarized in Table A. 5, where q d is presented together with n corresponding to each loading level. It is observed that the greatest n value is found for 60%X, a fact mainly associated with the greater number of samples tested at this loading level versus the 80%X one. In addition, the mean value increases versus the global results (although not significantly) as the loading level decreases. Table A. 5.- q d: Dependence on the loading level q d %X µ (º) s (º) CV(%) n 80 48.7 12.5 25.8 128 60 48.9 11.7 23.9 509 40 50.0 11.3 22.6 173 30 54.4 10.4 19.4 39 When the results corresponding to the case of isolated fibre and non-isolated fibre are compared as a function of the loading level, it can be observed that this tendency towards greater values of the debonding angle (commented in the previous paragraph) is more pronounced in the isolated fibre case. A.6.- Conclusions The study carried out has made it possible to observe, by means of optical microscopy, the micromechanical damage promoted by tensile loads at the 90° plies of a [03,903]s laminate. The different stages of the mechanism of damage (predicted by previous numerical studies, Stages I, II and III) have been identified, as well as damages of less relevance. Five new Stage III typologies have been detected, although their occurrence is less frequent than the theoretical one (that predicted by numerical models). The final extension of the debond at the interface that gives rise to Stage III has been measured, also analysing its dependence on the loading level, the presence of neighbouring fibres as well as the Stage III typology. The mean value of the debonding angle experimentally measured is lower than that predicted by singlefibre numerical models but is in accordance with the twofibre models conclusions and with the expected effect of the presence of friction in the interface crack contact. 74 A-References [1] F. París, E. Correa, V. Mantič, Kinking of Transversal Interface Cracks Between Fiber and Matrix, J. App. Mech. 74 (2007) 703-716. [2] F. París, E. Correa, J. Cañas, Micromechanical view of failure of the matrix in fibrous composite materials, Compos. Sci. Technol. 63 (2003) 1041-1052. [3] E. Correa, F. París, V. Mantič, Effect of the presence of a secondary transverse load on the inter-fibre failure under tension, Eng. Fract. Mech. 103 (2013) 174–189. [4] E. Correa, V. Mantič, F. París, Effect of thermal residual stresses on matrix failure under transverse tension at micromechanical level: A numerical and experimental analysis, Compos. Sci. Technol. 71 (2011) 622-629. [5] E. Correa, E.K. Gamstedt, F. París, V. Mantič, Effects of the presence of compression in transverse cyclic loading on fibre–matrix debonding in unidirectional composite plies, Composites Part A 38 (2007) 2260–2269. [6] F. París. A study of failure criteria of fibrous composite materials. NASA/CR-2001-210661 (2001). [7] C. Sandino, E. Correa, F. 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Fiedler, Evaluation of interfacial strength in CF/epoxies using FEM and in-situ experiments, Composites Part A 37 (2006) 2248- 2256. [18] M. Genz, D. Armentrout, P. Rupnowski, L. Kumosa, E. Shin E, J. Sutter, M. Kumosa, Inplane shear testing of medium and high modulus woven graphite fiber reinforced/polyimide composites, Compos. Sci. Technol. 64 (2004) 203-220. [19] V.D. Azzi, S.W. Tsai, Experimental Mechanics 5 (1965) 283-288. [20] S.W. Tsai, E.M. Wu, J. Comp. Mat. 5 (1971) 58-80. [21] Z. Hashin, Failure criteria for unidirectional fiber composites, J Appl. Mech. 47 (1980) 329- 334. [22] Z. Hashin, A. Rotem, A fatigue failure criterion for fiber reinforced materials, J. Comp. Mat. 7 (1973) 448-464. [23] A. Puck, Festigkeitanalyse von Faser-Matrix-Laminaten, Modelle für die Praxis (Strength analysis of fibre-matrix/laminates, models for design practice), Carl Hanser Verlag, Munich, 1996. [24] F. París, M.L. Velasco, E. Correa, Micromechanical study of the influence of scale effect in the first stage of damage in composites, Compos. Sci. Technol. submitted for publication. [25] V. Mantič, Interface crack onset at a circular cylindrical inclusion under a remote transverse tension. Application of a coupled stress and energy criterion, Int. J. Solids Struct. 46(6) (2009) 1287–1304. [26] V. Mantič, I.G. García, Crack onset and growth at the fibre–matrix interface under a remote biaxial transverse load. Application of a coupled stress and energy criterion, Int. J. Solids Struct. 49(17) (2012) 2273–2290. [27] E. Correa, F. París F, V. Mantič, Effect of friction on inter-fibre failure under tension at micromechanical level, 2nd ECCOMAS Thematic Conference on the Mechanical Response of Composites, Imperial College, London, 2009. 76 77 PAPER B: BEM multiscale modelling involving micromechanical damage in fibrous composites The original version of this paper can be found in www.elsevier.com • DOI: 10.1016/j.enganabound.2018.03.012 • Journal Name: Engineering Analysis with Boundary Elements • ISSN: 0955-7997 • Journal Citation Reports (2018). Impact Factor:2,193 -Engineering Multidisciplinary: Q2 (30/88) - Mathematics, Interdisciplinary Applications: Q1 (26/105) 78 79 BEM multiscale modelling involving micromechanical damage in fibrous composites María Luisa Velasco, Enrique Graciani, Luis Távara, Elena Correa, Federico París Grupo de Elasticidad y Resistencia de Materiales Escuela Superior de Ingeniería, Universidad de Sevilla Camino de los Descubrimientos s/n, 41092 Sevilla, Spain Abstract Composite laminates are materials where different scales are required for the understanding of the damage and for the calculation of structures made of these materials. The appearance of ultra thin plies of composites has opened the possibility of delaying the onset of damage, which at a first stage appears at a micromechanical level, debondings in between fibre and matrix in the weakest lamina of the laminate. A multiscale BEM model is developed in the present paper with the final purpose of being able to study the effect that the relative size of the laminas of the laminate plays in the appearance of this initial damage. The model presents many difficulties derived from the different scales involved in it and from the non-linear nature of the problem under study. The approach followed involves the solution of the whole problem, with the different scales involved in it, at once. The solution obtained is checked with another already obtained with a much simpler model. The multiscale model developed has been proved to be very efficient, accurate and robust, having been applied to simulate the first stages of damage in the light of the scale effect that is trying to be studied. Keywords: BEM, Multiscale model, Composites, Damage, Scale effect 80 B.1.- Introduction Composites have evolved in last decades from being an interesting material used in structures of low responsibility in aeronautical components to be the preferred material for primary structures of modern aircrafts. Although composites have, as one of its main features, an excellent behaviour in terms of damage tolerance, it is obvious that the best possible design is that involving no damage in the material in the expected range of load at which the component has to work. With reference to Carbon Fibre Reinforced Plastics (CFRP), the material object of the present study, the weakest laminas, those oriented 90 ° with relation to the load direction, present a premature presence of damage, typically viewed as transverse through the thickness of the lamina cracks, not involving the breakage of any fibre. The appearance of this damage was experimentally shown to be dependent on the relative thickness of the 90º lamina with regards to the 0º laminas, see for instance Parvizi et al [1] and Flaggs and Kural [2]. The availability of having ultra thin plies (UTPs) to design and manufacture composite laminates opens the possibility to delay the appearance of this damage, amplifying the range of work of the material. The understanding of the mechanism of damage of these 90º laminas and its dependence on the relative thicknesses of the 0 and 90º laminas becomes of crucial importance to deal with comfort in using these UTPs. The necessity of limiting the difficulty associated with structural calculations has traditionally led to simplify the representation and modelling of materials. Thus, metals are typically represented as homogeneous (omitting the representation of grains) and isotropic (representing the case of non-preferential orientation of the grains), whereas a lamina of fibre reinforced composites is represented as homogeneous (omitting in this case the representation of fibres and matrix) though orthotropic (to represent the macromechanical behaviour associated with the presence of the fibres aligned along one direction). However, there is a need for many structural engineering problems (in order to fully understand really them), of descending to the scales where the damage/failure/degradation is taking place, it being necessarily compatible with the modelling, at least of a part of the structure, at the scale at which it is studied. This fact, less common in metals, is very typical in composites, in which the calculations to design the composite structure are made at the level of assumed homogeneity, although the damage appears at the level of fibre and matrix and what is more common at the interfaces between them, a level of detail (micromechanical) not covered in the aforementioned homogeneous representation of a ply of a composite. Associated with this question is the fact that a micromechanical study typically requires a very accurate determination of variables (displacements and stresses) associated with the damage under study (debondings between fibres and matrix, cracks connecting debondings,…), in order to evaluate the magnitudes controlling the damage (the energy release rate of the crack). This calculation has to be additionally carried out on a very reduced area (in terms of several orders of difference in scale) of the whole problem under study. 87 discretization (PD) appears in between brackets. In this particular case of a progressive discretization the length ratio in consecutive elements has been maintained between 1 and 1.2. Details of the mesh in the local scale will be shown in the next subsections in which the two different local cells will be described separately. The element length (in microns) is also indicated at each corner and in the zones in which a uniform discretization is used. Element length is identical in all elements sharing a corner node and conforming discretizations are employed in all contact zones or bonded interfaces. Notice that the ratio between the size of the model ( ) and the size of the smallest element ( ) is extremely high. Since the accurate evaluation of the boundary integrals when the collocation point is extremely far away from the smallest element is impossible, the use of a local and global scale is mandatory even if the material of solids S2 and S3 was identical. The use of the local model reduces the length ratio between the largest and the smallest elements of each solid allowing an accurate evaluation of the boundary integrals to be carried out. Finally, perfect bonding conditions are employed in the interface between S3 and S4 and between S3 and the local cells described in the following. B.3.1.- Interface crack case Figure B. 6 represents a scheme of the discretization carried out in the local cell for a case having only a debond crack. Notation employed follows the same rules described in the previous section. Figure B. 6.- Discretization of the local cell corresponding to a debonding case. The most important information associated with the discretization carried out in the local cell is that regarding to the mesh employed in the neighbourhood of the crack tip. m10·53µ m10~7µ - 88 The debonding crack, being an interface crack, can be modelled in accordance with either an open model, Williams [28], which would involve interpenetrations near the crack tip of the faces of the crack, or in accordance with a contact model, Comninou [29], which would involve the appearance of a contact zone near the crack tip. In our case the BEM code employed allows the presence of a contact zone to be developed and consequently BEM solution directly satisfies the conditions of Comninou's model, given that if interpenetrations tend to appear (using initially open conditions), the iterative procedure employed will adjust the incompatibilities found. Notice that the contact zone to avoid interpenetrations may be extremely small or relatively large depending on the fracture mode mixity, which varies with the length of the crack. However, this fact does not affect the discretization to be carried out, since the evaluation of the ERR has to be carried out at a scale along a distance of growth of physical relevance. Consequently if this distance scale is larger than the contact zone, the presence of extremely small contact zone does not affect the ERR associated to crack propagation at the scale of interest. Since friction between crack faces is neglected, energy release rate can be evaluated according to Irwin's Virtual Crack Closure Technique, and decomposed into fracture mode I contribution, , and fracture mode II contribution, : (1) where is a local co-ordinate system centred with the fibre, and are, respectively, the normal and tangential tractions along the interface, while and are, respectively, the opening and sliding between crack faces. The previous experience indicates, París et al [8], that depending on the debonding angle, the interface crack fracture mode evolves from having an extremely small contact zone (for small debondings ) to showing a significant contact zone (for large debondings ). When the length of the contact zone is larger than (with being the fibre radius), the scale in which the ERR is being evaluated, pure fracture mode II takes place. Figure B. 7 shows a scheme of the discretization carried out (the same for all the cases studied) at both sides of the crack tip and Table B. 3 gives the values of the angle a ij subtended by the element joining nodes i and j, g ij denoting the length of the element connecting nodes i and j. Since the accuracy of the solution in the closest vicinity of the crack tip is spoiled by the presence of the singularity, the solution of the closest nodes to the crack tip is not employed, and integration is carried out from node 5 up to node 16 (see Table B. 3), which means that the integrals in (1) are effectively carried out in the range . G I G II G ò ò D D ¢¢ -D+D ¢ + D + + ¢¢ -D+D ¢ + D =+= d d dRuR dRuRGGG dddr d ddrdrr d III q qq q qqqqqqs q qqqqqqs q 0 0 )],(),([ 1 )],(),([ 1 ),( q r rr s q s r r uD q uD °60< d q °60> d q d Ra q D=D R °5.0°0001.0 =D< ¢ <d qq 89 Figure B. 7.-Discretization performed in the neighbourhood of the crack tip to analyse the first phase of damage (isolated debondings between fibre and matrix). Table B. 3.- Subtended central angle and length of the elements in the neighbourhood of the crack tip. Note that the problem under consideration is nonlinear, as it implies the contact/separation of surfaces of fibre and matrix. This fact does not interfere with the intrinsic difficulty of solving of the problem, but it is significant during the process of definition of a suitable discretization. Thus, the crack was artificially closed along this process, adjusting the discretization, in order to get a smooth evolution of stresses in the neighbourhood of the crack tip. Once the discretization had been adjusted with the crack closed, it worked properly with faces of the crack open or partially open in all cases (different debondings). For the different debond angles studied, the mesh is identical to that shown in Figure B. 6, corresponding to the q d=60° case, except for the fact that the mesh in the neighbourhood of the crack tip is shifted to its corresponding position. When the HPC system already described is used, the computation times are quite reasonable, being of the same order independent from the configuration studied (different length of the debonding). In the cases in which a contact zone is detected, only a few iterations are needed 90 to obtain convergence and only a reduced number of contact equations has to be rebuilt in each iteration. To give an idea of the complexity of the problem, with reference to the case of a debonding of 40°, the total number of elements is 2070, leading to a total system of equations of 4728, with a minimum size of an element of 4,09·10-7 microns and a maximum size of element of 50 microns. When this model is run in the HPC system, which includes 8 cpus, an iteration takes around 40 seconds. Usually, to get a converged solution of the whole problem, only a reduced number of iterations (9 in the case described here) is needed, the total solution time being around 4-5 minutes. It is well known how sensitive is BEM, particularly in the presence of singularities, to the discretization carried out, and although the final proof of having a correct discretization will be obtained in the next section comparing with previous results, it has to be mentioned here that this study is not carried out with prediction purposes but with the aim to have a model to compare results associated to different configurations of a laminate. Thus, once the discretization employed is proved to work properly, it will be employed for all cases, then supporting the validity of the comparison for the problem under consideration. B.3.2.- Kinked matrix crack case The local cell described in the previous subsection corresponds to the modelling of the first stage of damage, that is, the debonding between fibre and matrix. This damage is assumed to continue, once the debonding crack stops, kinking and entering into the matrix following a direction normal to the applied load, what implies for the kinked crack to work under mode I, París et al [8]. The model described in the previous sections has been adapted, as shown in Figure B. 8, for the study of stage III of damage represented in Figure B. 1. Figure B. 8.- Discretization of the local cell corresponding to the kinked matrix crack case. Although at the global scale, the models representing the two stages of damage have different sizes, the details of the mesh at the global scale of the model corresponding to the third stage 91 identified in Figure B. 1 are omitted for the sake of brevity, since the mesh is made following a similar scheme to that described in Figure B. 5 for the model of the first and second stages. In the local scale, in order to easily assess stresses ahead of the crack tip, a fictitious interface along the predicted path of the kinked crack has been created, starting at the position of the original crack tip of the debonding and aligned normally to the applied load. Then, for a particular length of the assumed kinked crack, the closest neighbourhood of its crack tip is treated as described in the previous section. Contact conditions are employed between crack faces (from the symmetry plane to the crack tip) and perfect bonding conditions are applied along the fictitious interface where the crack has not extended. Other feasible option to study how the interface crack gets into the matrix (homogeneous material) without including a fictitious interface could be the use of the so-called Dual-BEM [30] or the use of a Symmetric Galerkin Scheme [31,32]. However, due to the fact that in the problem under study the direction of progression of the crack is a priori known, the method selected, in spite of the necessity of using a fictitious interface, was estimated to be the optimum solution. Moreover, the use of those schemes (dual BEM or SGBEM) would imply the use of hypersingular integral equations as well as their corresponding fundamental solutions. B.4.- Results B.4.1.- Validation of the model The key result, as will be used in further studies to discriminate on the role of the scale effect in the damage of composite laminates, is the energy release rate G. As the first question to be addressed is the ability of the multiscale model developed to get accurate results, those obtained with the present model will be compared with the ones obtained with a simpler model, a single fibre surrounded by matrix, París et al [8]. To make the comparison effective, all the bodies, except for the fibre, appearing in the multiscale model will adopt the properties of the matrix, thus making the comparison advisable. Figure B. 9 represents the values of G obtained with the multiscale model and those obtained with the model of a single fibre surrounded by matrix, both adimensionalized using a reference number G0, París et al [8]. 92 Figure B. 9.- Predictions on G values obtained with the multiscale model (surrounded by matrix) and with the single fibre model. Only the values of the total energy release rate are shown for the sake of clarity. It can be clearly observed that they are almost undistinguishable, proving then the validity of the multiscale model developed. B.4.2.- Multiscale model results Once the validity of the model has been proved, one of the questions that attracts immediate attention is the shape of the hexagonal cell. Hexagonal cells are typically used when representing composites micromechanically as they are reproducible leading to a complete representation of the material at a plane normal to the fibres direction, allowing also in a simple way to reproduce the volume fibre ratio at the level of the cell. However, the fibres are in fact distributed in the material in a complete random way, and consequently any shape of the cell would be acceptable as will represent the actual situation of a certain fibre in the material. As the hexagonal cell has corners, this fact may introduce some alterations in the solution. Notice that in the case of having the corner, as in Section B.4.1, located in a unique media, it would not produce any effect when using a FEM approach, but would do when using a BEM approach. However, having the corner of the cell in between dissimilar media (with matrix at one side and the homogeneous model of the 90° ply at the other side), it produces a nominal singular stress state associated to a multimaterial corner, Barroso et al [33], BEM being much more sensitive than FEM to this situation. Then, an alternative circular cell is now going to be used in order to check if the presence of corners in the hexagonal cell produces significant alterations in the results. Figure B. 10 (for a 30° debonding) and Figure B. 11 (for a 50° debonding), represent the evolutions of the radial stress s r ahead of the crack tip and the relative displacements D ur of the faces of the crack near the crack tip, values used to calculate G. It can be clearly observed the similarity of the results found for the two cells, hexagonal (HC) and circular (CC). 93 Figure B. 10.- Radial stress and relative displacements in the neighbourhood of the crack tip for a debonding of 30º using a hexagonal cell (HC) and a circular cell (CC). Figure B. 11.- Radial stress and relative displacements in the neighbourhood of the crack tip for a debonding of 50º using a hexagonal cell (HC) and a circular cell (CC). As a general view of the situation, Figure B. 12 represents the values of G predicted with the use of both shapes of the cell for the values of G and their counter parts GI and GII. 94 Figure B. 12.- Values of G, GI and GII for the whole range of debondings using hexagonal cells (HC) and circular cells (CC). When comparing the predictions of the results obtained with the two shapes of the cell, it has to be mentioned that they correspond obviously to different discretizations of the boundary of the cell, as uniform discretizations are performed for the circular cell, whereas a progressive discretization (involving more elements) is used in the case of hexagonal cells near the corners of the boundary of the cell. In any case the results are quite similar, having taken in Figure B. 10 and Figure B. 11 two representative situations involving one of them the case of greatest difference in the predictions. All of this supports the use of the BEM model initially represented in Figure B. 2 and Figure B. 3 to study the scale effect in composites by means of the energy release rate of the debonding crack, as the results are not affected by the shape of the cell. Once the presence of corners in the cell has been proved to have no influence in the results of interest, it justifies the use of a rectangular cell for the case of including the kinked crack in the model. Once the ability of the multiscale model to accurately predict the values of the parameter used to study the scale effect has been proved, the model generated to study the third phase of damage, Figure B. 4 and Figure B. 8, will now be considered. The case corresponding to a debonding of 65 ° has been selected, as this debonding is in the interval where, as justified in París et al [8], the abandonment by the debonding crack of the interface fibre-matrix, kinking into the matrix, is more plausible. Different lengths of the kinked crack have been considered, Figure B. 13 representing the evolution of the values of G of the kinked crack, and their counterparts GI and GII, as a function of its length for a carbon epoxy-system. 95 Figure B. 13.- Evolution of G, and their counterparts GI and GII, for different lengths of the kinked crack. First of all, the evolution obtained with the multiscale model is comparable to that obtained by París et al [8] for a glass-epoxy system, using a single fibre model surrounded by matrix. Several things are noticeable in this evolution. First, coherently with the decision taken to build the model, negligible values of GII are obtained, leading to almost a pure mode I associated to the kinked crack. This is coherent with the reasoning that led in París et al [8] to select the 60-70° range for the debonding as that more plausible for the kinking of the interface crack to occur. Additionally, it is also noticeable the increasing character of G (and obviously of GI) with the increase of the length of the kinked crack. This leads to conclude that once the kinking of the debonding crack arises, the growth of the kinked crack will be unstable, based on the increasing character of G and on the unaltered character of the fracture mode of the kinked crack (mode I) along its path. Notice that a maximum kinked crack length of 2R has been considered in Figure B. 13, although further extensions of the crack have been explored confirming the same behaviour. This study has required the use of a bigger than in previous cases. The same cell has been used for all cases considered involving a kinked crack of different length, which makes the results comparable among them. The aforementioned lack of stability of the growth of the kinked crack in the matrix detected by the present numerical model is in agreement with the experimental evidence presented in Correa et al [34]. It is also important to have a look of the deformed shape of the model obtained involving the kinked crack, Figure B. 14, where a magnification factor of 1.2 has been employed. 96 Figure B. 14.- Undeformed and deformed shapes of the model (only the fibre and surrounding matrix are represented). In addition to the coherence of the deformed shape obtained (only the fibre and surrounding matrix are represented in order to concentrate in the area of interest), what can be observed is that independently of the length of the kinked crack chosen (the deformed shape represented corresponds to a length of the kinked crack of 2R (7.5 microns)), no interaction between the lips of the two cracks is detected. As soon as a kinked crack is modelled, no matter how short it is, the two faces of the parent debonding crack separate along the whole length, and thus the contact zone that existed prior to kinking for this particular interface crack size (65°) disappears. This result is also in agreement with París et al [8]. B.4.3.- Incidence of the 0° ply thickness in the scale effect Once the ability of the multiscale model developed to work satisfactorily has been proved, a first application of the model to the problem to which the model has been developed for, that is the scale effect in composites, is now going to be developed. The scale effect is sometimes referred to by means of the thickness of the 90º ply, Parvizzi et al [1], or by a number n representing the proportion of 90º plies with regards to 0º plies in a [0,90n]S laminate, Flaggs and Kural [2]. Three different thicknesses, t0° = 85µm, 170 µm and 340 µm, which correspond to n=0.5, 1 and 2, will be considered in this study. The values of G, GI and GII for the three thicknesses mentioned and for the whole interval of debondings considered are represented in Figure B. 15, with reference to the multiscale model considering only the first stage of damage. 103 PAPER C: Micromechanical study on the influence of scale effect in the first stage of damage in composites The original version of this paper can be found in www.elsevier.com • DOI: 10.1016/j.compscitech.2018.03.004 • Journal Name: Composites Science and Technology • ISSN: 0266-3538 • Journal Citation Reports (2018). Impact Factor:6,309 - Materials, Science, Composites: Q1(2/25) 104 105 Micromechanical study on the influence of scale effect in the first stage of damage in composites Federico París, María Luisa Velasco, Elena Correa Departamento de Mecánica de Medios Continuos y Teoría de Estructuras Grupo de Elasticidad y Resistencia de Materiales Escuela Técnica Superior de Ingeniería Universidad de Sevilla Camino de los Descubrimientos s/n 41092, Sevilla, Spain Abstract The variation in the apparent strength of a lamina in a laminate depending on the stacking sequence and thicknesses of the laminas of the laminate has been a matter of interest since the initiation of the extension of the applicability of composites. This fact led to the concept of insitu strength, the problem itself being covered as a scale effect. In this paper this question is revisited moving towards the level where the damage appears, that is the micromechanical level. As the origin of the effect under consideration, the variation of thickness, takes place at a different level, that is mesomechanical, a multi-scale model is developed. It is possible in this model to vary the thickness of the lamina under consideration and to observe the effect on the damage. In this paper, only the first stage of damage, which appears in form of debondings between fibres and matrix, is taken into consideration. The analysis carried out shows that there is no scale effect at this first stage of the damage, which is corroborated by experimental evidences. Keywords: Composites, In-situ strength, Scale effect, Damage, Interfacial Fracture Mechanics 106 C.1.- Introduction From almost the beginning of the explosion of application of composites, a phenomenon associated with the strength of laminas placed in a laminate was experimentally identified. Thus, Parvizzi et al [1] and Flaggs and Kural [2], among others, found that the appearance of damage in the 90° lamina of a [0,90n]S laminate was a function of the value of n, the delay in the appearance of the first transversal damage in the 90° lamina being associated with the lowest values of n used. This fact, not easy to be understood, was associated with the different role that the restriction of the 0° lamina may play in the behaviour of the 90° lamina, not covered by the Classical Laminate Theory. This phenomenon gave rise to identify the term of “in situ strength”, which indicates that the actual strength was not only a property of the material but a function of the configuration of the laminate where the lamina was, the total phenomenon being known as a “scale effect”. When analysing the scale effect, the emphasis may be put either in the value of n in a [0, 90n]S laminate (see for instance [2]) or in the value of the thickness of the 90° lamina (see for instance [1]). There exists a link, but not a complete and biunique connection, between the two approaches to deal with the problem. Notice that with the definition of the laminate given, the thickness of the 0° plies is taken as a reference and the value of n defines the thickness of the 90° lamina in relation to the 0° lamina. All of this lead to the fact that many different laminates have been historically studied and tested (with different procedures to detect the appearance of first damage in the 90° lamina). The results might neither always be comparable nor leading to a unique view of the matter. There have been many trials to give an explanation to this phenomenon, those of Dvorak and Laws [3] and Li and Wisnom [4], being widely recognized, among others. Both approaches, while different conceptually speaking, have in common that are based on the fact that the laminas are considered as equivalent homogeneous materials. Again in this context and more recently, García et al [5] have given a new insight into the problem by using a double stress and energy criterion, based on Finite Fracture Mechanics, which adds a convincing explanation on the scale effect identified by Parvizzi et al [1]. It is also worth to mention that the challenges of dealing with in-situ strength of embedded laminae through different approaches have been highlighted in Kaddour et al [6]. In contrast with the previous approaches that consider an homogenized view of the laminas, it is reasonable to accept that the most plausible explanation for the initiation of the damage indicates that it appears in a 90° lamina as debondings between fibre and matrix. By coalescence, these debondings, may give rise to a mesomechanical crack that propagates across the thickness of the lamina stopping near the interface with the 0° lamina. When approaching this interface, the cracks provoke damage at micromechanical level that could be viewed, at mesomechanical level, as an incipient delamination, able to grow giving rise to an apparent and noticeable delamination between 0° and 90° laminas París et al [7, 8]. The appearance of ultra thin plies has alerted on the possibility of having now an extra increase of the apparent strength of these 90° laminas, significantly delaying the appearance of the first identified damage in the laminate. This would allow a longer period of behaviour without damage, what for a designer of actual composite structures would compensate the extra work in manufacturing which derives from the use of ultra thin plies. Several attempts have been recently made to deal with a view of the scale effect at micromechanical level. Saito et al [9] 107 performed an experimental study complemented later with a numerical simulation at micromechanical level, Saito et al [10], allowing debondings between fibres and matrix as the mechanism of damage. More recently, Arteiro et al [11] and Herráez et al [12] have used finite element models, both at micromechanical level involving cohesive approaches, to try to give an explanation to the scale effect. In the first study the authors claim to have performed a numerical study that support the scale effect, whereas in the second one they claim that the numerical model developed does not support the presence of the scale effect. All these models require the use of several material and interfaces properties that are not identified in the literature, and thus must be fitted using the models. Additionally the use of many fibres in the model may partially hide the actual nature of the damage and the causes that control their onset and propagation. In this paper, it is assumed that the mechanism of damage starts with micromechanical fibrematrix debondings that may coalesce by giving rise to a mesomechanical transverse crack. The first stage of this damage mechanism (that associated to the debonding between fibre and matrix) is going to be analysed in this paper in light of the scale effect. To this end, a multi-scale model is developed using the Boundary Element Method (BEM) and the concepts derived from Interfacial Fracture Mechanics (IFM), as the damage consists of an interface crack running between fibres and matrix in a lamina of a laminate. The model developed does not require any property other than those of the stiffness of the materials, fibre, matrix and lamina, as it has no predictive objectives in terms of damage and load associated. This model has only the objective of understanding the relation, at micromechanical level, between the first stage of damage and the scale effect, which only requires predictions on the stress state. The effect of the presence of additional fibres surrounding the damaged fibre under consideration will be studied following the previous studies by Sandino et al [13, 14]. Damaged fibre refers in what follows to a fibre with an interface damaged. Finally, experimental evidence supporting the predictions obtained with the numerical methods is presented. C.2.-Tools and materials As debonding between fibre and matrix is going to be the focus of attention in this study, Interfacial Fracture Mechanics (IFM) is the adequate tool to characterize the behaviour of these interface cracks. The background of IFM and its application to the problem under consideration can be found in París et al [15, 16]. The variable to be used as representative of the behaviour of this damage is the Energy Release Rate G of the debonding crack. Both the case of the open crack model, Rice [17], and the case of contact crack model, Comninou [18], require very accurate values of displacements of the faces of the crack as well as the stress state in its neighbourhood. It is significant to remark that both models need to be applied when modelling the different lengths of the debonding to simulate the growth of a debonding crack. Thus, for small values of the debonding it will be oriented almost normal to the load applied, mode I being dominant and the open model being of application. On the contrary, when the debonding reaches a certain value, a contact zone between the faces of the debonding crack appears, mode II being the unique existent, and consequently, the contact model being of application. In the transition between these two extreme situations, a mixed mode appears evolving from mode I to mode II, the open model being of application along this period of growth, París et al [16]. 108 All matters associated with IFM, for both models, lead to the necessity of having a very fine discretization in the neighbourhood of the crack tip. The Boundary Element Method (BEM), París and Cañas [19], has been selected as the adequate tool to deal with the multi-scale problem under consideration. The level of accuracy at which the solution in the neighbourhood of the debonding crack tip must be obtained to get accurate values of the Energy Release Rate (ERR), without having the associated refined domain discretization, can be only reached by BEM. A program for multi-body modelling considering contact between the boundaries of the bodies involved in the model will be used, Graciani et al [20]. The main difficulties derived from this approach come from the necessity of having extremely fine discretizations near the crack debonding tip. This leads to have very similar integral equations involved in the final system of equations to be solved, see Velasco et al [21]. As a summary, the analysis carried out presents the following features: is based on linear elastic behaviour and no thermal stresses are considered, the fibres are assumed to have perfect bonding out of the damaged zone, no interphase layer has been used between fibres and surrounding matrix, and no failure is aimed to be predicted, as previously stated. Table C. 1 shows the properties of the carbon-epoxy material taken as reference for this study, which corresponds to that employed in Soden et al [22], to facilitate comparisons with previous studies. Table C. 2 gives the properties of a glass-epoxy system also used in this study. Table C. 1.- Carbon-epoxy system used: AS4/3501-6, Soden et al [22]. Material Properties 0º ply (orthotropic) E11=135GPa, E22=8.75GPa, E33=8.75GPa, ν12=0.3,ν13=0.3, ν23=0.4, G12=4.75GPa 90º ply (isotropic) E (E22=E33)=8.75GPa, ν (ν23)=0.4 Matrix (epoxy, isotropic) E=4.2GPa, ν=0.32 Fibre (carbon, transversely isotropic) E (E22=E33)=15GPa, ν (ν23)=0.2 Table C. 2.- Glass-epoxy system used, data obtained directly from tests. Material Properties Matrix (epoxy, isotropic) E=2.79GPa, ν=0.33 Fibre (glass, isotropic) E =70.9GPa, ν =0.22 C.3.- Multi-scale model Figure C. 1 describes the multi-scale model used in this study of a [0,90n]S laminate with a cell involving fibre and matrix. The thickness of each 0° ply is t0 and the total thickness of the 90° ply is 2t90. The outer 0° lamina, Figure C. 1.a), is modelled as homogeneous and orthotropic in the 109 x-y plane, whereas the central 90° lamina is modelled as homogeneous and isotropic. At the centre of the 90° lamina, a heterogeneous cell involving a fibre surrounded by matrix is incorporated, the interface between fibre and matrix being able to model the presence of a debonding crack, FFigure C. 1.b). The model applies explicit symmetry with respect to axis x, which means that the damage appearing in y>0 is also simulated for y<0, a situation corresponding to what is experimentally observed. No symmetry is applied with respect to axis y, as only damage at one side (in our case at side x>0 in Figure C. 1) is going to be modelled, a situation also generally observed in experiments, Correa et al [23]. Thus, consequently, the model developed incorporating symmetry involves, Figure C. 1, a 0° ply of thickness t0 and a central 90° ply of thickness t90=n t0. In all cases considered in this paper t0 has been taken equal to 170 µm. Figure C. 1.-a) Multi-scale BEM model of a [0,90n]S laminate with an inner cell involving fibre and matrix, b) detail of the cell. In the model studied in this paper a hexagonal shape, typically used in micromechanical representations, see for instance Hull and Clyne [24], has been given to the boundary of the cell. The influence of this shape in the results, almost negligible, is studied in Velasco et al [21]. The areas of matrix and fibre in the cell have been designed in this model to maintain the nominal fibre volume of the material, 55% in this case. The radius of the fibre has been taken as rf = 3.75 µm, the thickness of the 0° ply is 170 µm, and the laminate has been taken long enough to have regularized stress states. The test is simulated to be run by means of displacements control, applying e x = 0.5%, see [21] for details of the discretization. a) b) 110 The BEM model presents so many numerical complications that the first step must be to compare the values obtained from the model represented in Figure C. 1 with those obtained from a simpler model. To this end, all bodies involved in the model, except for the fibre, have received the properties of the matrix in order to compare with the results obtained by París et al [15, 16] using a single fibre model surrounded by infinite matrix. This case is quite well represented by the model of Figure C. 1, with the properties indicated. The comparison is done for the glass-epoxy system (the system used in [15, 16]) and the predictions on the values of G, adimensionalized (see [16]), for different debondings appear in Figure C. 2. In this particular case and in all cases studied in this paper, G is calculated by the crack closure technique, Irwin [25], using an angle of growth of 0.5°. Figure C. 2.- Validation of the multi-scale BEM model. The evolutions of G, GI and GII, see [14, 15] for a deeper explanation, indicate that for small debondings (the crack being nominally oriented normal to the direction of the load applied), the growth occurs under dominant mode I. It changes to mixed mode as the debonding increases, reaching a pure mode II for an interval of the semidebonding angle between 60° and 70°. Thus, this is the expected interval of semidebonding angle to be observed in damaged composites as the end of the unstable growth of the debonding, in accordance with the evolutions of G, GI and GII, observed in the figure. Results shown in Figure C. 2 validate, in what refers to discretization and numerical problems, the multi-scale BEM model, as no differences are appreciated with previous simpler models. The first stage of damage under investigation in this paper, involving debondings between fibre and matrix, will be presumably controlled by stresses along the interface between fibre and matrix and by the capacity of releasing energy by a potential crack running along the interface. Thus, to clarify the role of these stresses in the onset of damage, the evolution of s rr and s r q along the interface for the pristine case of the material is studied for different values of n. Figure C. 3 represents such evolutions for n=0.1, 0.5, 1, 3. 111 Figure C. 3.-Evolution of s rr and s r q along the fibre-matrix interface for different values of n. It can be seen that there is no influence at all, even for very small values of n, in the values of s rr and s r q along the interface. This indicates that the role of the stresses in the onset of damage between fibre and matrix is not affected by the scale effect. The other variable involved in the damage under consideration is the capacity of releasing energy by the debonding crack, which will be investigated by means of the energy release rate G. The first task to be done now, as there are many results obtained from a single fibre embedded in matrix, is to clarify the role of the 0º lamina in the predictions of G. Figure C. 4 represents, in this case for the carbon-epoxy system, the comparison obtained in two different cases. On the one hand, that from the multi-scale model with the laminate, and corresponding properties of the laminas, for a value of n=1. On the other hand, the one with all bodies modelled as matrix. The results clearly show the role of the stiffness of the 0º lamina, with a generalized increase of G values for the whole interval of debonding angles studied. The reasons for these results lay on the control on displacements carried out, which for the fixed value of deformation imposed, produces a significant increment in stresses, which in turns produces the coherent increases in G observed in Figure C. 4. The effect of the thickness of the 0° ply is studied in Velasco et al [21]. 112 Figure C. 4.- Role of the 0° lamina in G predictions. Once the model has been validated and the coherence of the results understood, the scale effect on the first stage of a [0,90n]S laminate is studied by means of Figure C. 5. In this, the values of G corresponding to the whole interval of values of the semidebonding angles between fibre and matrix are shown for different values of the parameter n. Values of n equal to 1 and 3 are studied, as Flaggs and Kural [2] found scale effect for values, using the same nomenclature, of n=1 and 2. Notice that an apparently extreme case, as that corresponding to n=0.1, has been included for the sake of completeness. This case would correspond to a 90° ply of 17 µm of thickness in the model studied (double due to the symmetry), that corresponds to a typical ultra thin ply, which in fact would only admit 4/5 fibres across its thickness. It can be concluded that, based on the present model, the scale effect does not significantly affect the appearance of the first stage of damage in the 90° ply: the appearance of unconnected debondings between fibre and matrix. 119 Figure C. 12.-Evolution of G for the cases 45° (OS) and 90° (SS) with n=1 using the multi-scale model (2t90=340µm for n=1). To support the representativity of the configurations studied, the laminates formerly mentioned, which were manufactured to observe the scale effect on the first phase of damage, are now inspected again. Although it was mentioned that no isolated debondings were observed prior the formation of a meso transverse crack, once these transverse cracks appear, a region of local damage is observed near these cracks. In these damaged areas is easy to identify the patterns studied in this section. Thus, Figure C. 13 represents several of the damaged configurations found, that in turn correspond with those found by Saito et al [9], already schematized in Figure C. 6. 120 Figure C. 13.- Observations (left side) and corresponding schemes (right side) of damage associated with: a) the 45° (OS) case, b) a chain of 45° (OS) cases, c) combinations of 45° (OS) cases with 90º (SS) cases. The pattern most commonly found is that corresponding to what we have called 45° (OS) that can appear in an isolated way, Figure C. 13.a), or combined forming a chain of debondings reproducing a meso transverse crack normal to the applied load, Figure C. 13.b). The case we have called 90° (SS), also prone to arise, appears commonly combined with the 45° (OS) case, as represented in Figure C. 13.c). Now it is interesting to make a reflection on the predictions found with the multi-scale model and the experimental observations. Although the objective of the present section was to clarify if the absence of scale effect predicted at the level of the isolated debondings was confirmed in presence of a previously damaged fibre, which has been concluded after Figure C. 9 and Figure C. 11, there are other noticeable questions in the results obtained. Thus, it is easy to check that the evolution of G for the case of a debonding crack in presence of an already damaged fibre (both in the 45° and 90° configurations) is significantly greater (both in terms of values associated with the initial debonding taken as reference, 20°, as in the maximum values reached) than the evolution of G for the case of the first debonding, see Figure C. 5, Figure C. 9 and Figure C. 11. This, in practical terms, means that it will not be easy to find completely isolated debondings. The appearance of one of them (which may be due to poor interface properties of the affected fibre in connection with a more risky position of the fibre in the lamina) will immediately generate in the most relative risky positions (45° and 90°) a significant increase in the capacity of releasing energy. Thus, consecutive damage in the appropriate fibre surrounding the first damaged fibre would take place. This is clearly the case of the damage presented by Saito et al [9], Figure C. 6 in this paper, and also the case of the new observations presented in this paper, as can be seen in Figure C. 13 where different sequences of debondings connected in configurations oriented 45° and/or 90° appear. 121 Notice that the phenomenon under consideration may be also affected by the relative distance between neighbouring fibres, as many of the damage areas observed involve fibres that are almost in contact. This question has been already addressed in Sandino et al [13]. C.5.- Conclusions A micromechanical study of the scale effect on the first phase of damage in a weak ply (oriented 90° with respect to the load in the present paper) has been carried out. First, a multi-scale model containing a cell in which fibre and matrix are modelled, has been developed and used to understand the role of the 0° lamina in the context of results obtained from single fibres surrounded by matrix. The model has then been used to prove that there is no scale effect on this phase of the damage (isolated debondings between fibres and matrix) of a 90° ply. The reason lies on the fact that the evolution of the energy release rate, G, of a debonding crack is not altered by the distance between the 0° lamina and the fibre susceptible of suffering the debonding. Additionally, the effect of the presence of already damaged fibres has also been considered, not finding any indication of alteration of the former conclusions. All of this indicates that the appearance of the first isolated debondings between fibre and matrix (not being at that moment correct to talk of a mesomechanical damage as the stiffness of the laminate is not yet altered) is controlled by the own internal structure of the material. The debondings appear at places where either the toughness of the interface or the presence of defects initiates the damage. The numerical predictions, supported by experimental observations, indicate that the appearance of a debonding favours the subsequent appearing of new debondings in fibres located in the neighbourhood of the first damaged fibre, particularly in those oriented 45° and 90°. 122 C-References [1] A. Parvizi, K.W. Garrett, J.E. Bailey, Constrained cracking in glass fibre-reinforced epoxy cross-ply laminates, J. Mat. Sci. 13 (1978) 195–201. [2] D.L. Flaggs, M.H. Kural, Experimental Determination of the In Situ Transverse Lamina Strength in Graphite/Epoxy Laminates, J. Comp. Mat. 16 (1982) 103–116. [3] G.J. Dvorak, N. Laws, Analysis of Progressive Matrix Cracking In Composite Laminates II. First Ply Failure, J. Comp. Mat. 21 (1987) 309–329. [4] D.S. Li, M.R. Wisnom, Evaluating Weibull Parameters for Transverse Cracking in Cross- Ply Laminates, J. Comp. Mat. 31 (1997) 935–951. [5] I.G. García, V. Mantič, A. Blázquez, F. París, Transverse crack onset and growth in crossply [0/90n]s laminates under tension. Application of a coupled stress and energy criterion, Int. J. Sol. Struct. 51 (2014) 3844-3856. [6] A S Kaddour, M J Hinton, P A Smith, S Li, A comparison between the predictive capability of current matrix cracking, continuum damage and fracture criteria for fibre reinforced composite laminates: Part A of WWFE-III, J Compos Mater, V 47 (20-21) (2013) 2749-2779. [7] F. París, A. Blázquez, L.N. Mccartney, V. Mantič, Characterization and evolution of matrix and interface related damage in [0/90]s laminates under tension. Part I: Numerical predictions, Comp. Sci. Tech. 70 (2010) 1168–1175. [8] F. París, A. Blázquez, L.N. Mccartney, A. Barroso, Characterization and evolution of matrix and interface related damage in [0/90]s laminates under tension. Part II: Experimental evidence, Comp. Sci. Tech. 70 (2010) 1176–1183. [9] H. Saito, H. Takeuchi, I. Kimpara, Experimental Evaluation of the Damage Growth Restraining in 90° Layer of Thin-ply CFRP Cross-ply Laminates, Adv. Comp. Mat. 21 (2012) 57–66. [10] H. Saito, H. Takeuchi, I. Kimpara, A study of crack suppression mechanism of thin-ply carbon-fiber-reinforced polymer laminate with mesoscopic numerical simulation, J. Comp. Mat. 48 (2014) 2085–2096. [11] A. Arteiro, G. Catalanotti, A.R. Melro, P. Linde, P.P. Camanho, Micro-mechanical analysis of the in situ effect in polymer composite laminates, Comp. Struct. 116 (2014) 827–840. [12] M. Herráez, D. Mora, F. Naya, C. S. Lopes, C. González, J. LLorca, Transverse cracking of cross ply laminates: a computational micro mechanics perspective. Comp. Sci. Tech. 110 (2015) 196-204. [13] C. Sandino, E. Correa, F. París, Numerical analysis of the influence of a nearby fibre on the interface crack growth in composites under transverse tensile load, Eng. Fract. Mech. 168 (2016) 58–75. [14] C. Sandino, E. Correa, F. París, Composite materials under transverse biaxial loads: Study of the influence of a nearby fibre on the interface crack growth under tension, in: 17th Eur. Conf. Comp. Mat. (ECCM17); Munich, 2016. 123 [15] F. París, E. Correa, J. Cañas, Micromechanical view of failure of the matrix in fibrous composite materials, Comp. Sci. Tech. 63 (2003) 1041–1052. [16] F. París, E. Correa, V. Mantič, Kinking of Transversal Interface Cracks Between Fiber and Matrix, J. Appl. Mech. 74 (2007) 703–716. [17] J.R. Rice, Elastic Fracture Mechanics Concepts for Interfacial Cracks, J. Appl. Mech. 55 (1988) 98–103. [18] M. Comninou, The interface crack, J. App. Mech. 44 (1977) 631–636. [19] F. París, J. Cañas, Boundary element method: fundamentals and applications, Oxford University Press, 1997. [20] E. Graciani, V. Mantič, F. París, A. Blázquez, Weak formulation of axi-symmetric frictionless contact problems with boundary elements. Application to interface cracks, Comp. Struct. 83 (2005) 836-855. [21] M.L. Velasco, E. Graciani, L. Tavara, E. Correa, F. París, BEM multiscale modelling involving micromechanical damage in fibrous composites. Submitted for publication to Eng. Anal. with Boundary Elements. [22] P.D. Soden, M.J. Hinton, A.S. Kaddour, Lamina properties, lay-up configurations and loading conditions for a range of fibre-reinforced composite laminates, Comp. Sci. Tech. 58 (1998) 1011–1022. [23] E. Correa, M.I. Valverde, M.L. Velasco, F. París, Microscopical observations of inter-fibre failure under tension. Submitted for publication in Comp. Sci. Tech. [24] D. Hull, T.W. Clyne, An Introduction to Composite Materials, Second edition, Cambridge Press, 1996 [25] G.R. Irwin, Analysis of stresses and strains near the end of a crack transversing a plate, J. Appl. Mech. 24 (1957) 361–364. 124 125 PAPER D: Numerical study of the progression of the micromechanical debonding damage in composites The original version of this paper can be found in www.scientific.net/KEM.774.644 • DOI: 10.4028/www.scientific.net/KEM.774.644 • Journal Name: Key Engineering Materials • ISSN: 10139826 • SCImago Journal Rank (2018). Impact Factor: 0.18 - Materials Science (miscellaneas): Q3 - Mechanical Engineering: Q3 - Mechanics of Materials: Q4 126 127 Numerical Study of the progression of the micromechanical debonding damage in composites María Luisa Velasco, Federico París, Elena Correa Departamento de Mecánica de Medios Continuos y Teoría de Estructuras Grupo de Elasticidad y Resistencia de Materiales Escuela Técnica Superior de Ingeniería Universidad de Sevilla Camino de los Descubrimientos s/n 41092, Sevilla, Spain Abstract This paper deals with the study of the actual progression of the damage in the 90 degrees lamina of a composite. It has been proved and observed that isolated debondings between fibres and matrix are the first manifestation of damage in the weakest lamina, the 90 degrees lamina in a [0,90]S laminate. It was also numerically supported that this first phase was independent of the thickness of the 90 degrees lamina, not being then affected by the “scale effect”. The continuation of this first phase of damage is the objective of the present paper. To this end, a multiscale model is created involving the debonding between fibre and matrix and studying the kink of this crack, abandoning the fibre-matrix interface and entering into the matrix to produce a meso-transverse crack in the 90 degrees ply. The study is based on the application of Fracture Mechanics to an incipient kinked crack that starts from a debonding between fibre and matrix. It is concluded that this second phase of damage, playing with the thickness of the 90 degrees lamina, is not affected by the scale effect, as the variation of the energy release rate of the kinked crack is not significantly influenced by the variation of the thickness of the lamina. Keywords: Composites, Damage, BEM modelling, Scale Effect 128 D.1.- Introduction Composite laminates must have plies with fibres in several direction, at least 0, 90 and 45 degrees. 90 degrees plies are clearly those plies prone to have the first manifestation of damage, which consists in debondings between fibres and matrix. This damage would not be of particular concern if it does not grow to generate a complete transverse crack, and what is worst, to originate, when this transverse crack approaches a 0 degrees ply, a damage like delamination, París et al [1,2], that it is a very serious problem in terms of the performance of the laminate. The delay in the appearance of the first manifestation of damage has been traditionally a clear objective when designing a composite laminate. The effects on this damage of the decrease of the thickness of the 90 degrees lamina (a phenomenon known as the “scale effect”) was long time ago observed, Parvizzi et al [3] and Flaggs and Kural [4], among others. Several explanations with homogenized view of the material have been tried, see Dvorak and Laws [5] and Li and Wisnom et al [6], or García et al [7] more recently, among others. Some micromechanical views have also been tried with contradictory conclusions, claiming to having supported scale effect, Arteiro et al [8], or on the contrary claiming that the scale effect does not exist, Herráez et al [9]. París et al [10], have shown by means of a numerical analysis, supported by observations, that the first manifestation of the damage in the 90 degrees lamina of a [0,90n]S laminate, i.e.: the debondings between fibre and matrix, is independent of the scale effect. Thus, using the multiscale model represented in Figure D. 1.a), with a detail of the cell represented in Figure D. 1, the variation of the evolution of the energy release rate as a function of the thickness of the 90 degrees ply can be carried out, as represented in Figure D. 2. Figure D. 1.- Multiscale BEM model to study the isolated debondings between fibres and matrix as a function of the thickness of the 90 degrees ply: a) general view, b) detail of the micromechanical cell. a) b) 231 Figure G. 9.- Experimental observation where the presence of a morphology of the damaged fibres oriented 45 degrees appears. It corresponds to a [03, 903]S laminate made of AS4/8552 material. Figure G. 10 represents the configuration studied next where the secondary fibre is oriented 90 degrees with respect to the loading direction. Notice that, as has been already said, the presence of these two debondings represents also, as in the case of Figure G. 7 but in a different manner, the prelude of the generation of a transverse crack in the 90 degrees lamina. Figure G. 10.- Configurations of two fibres oriented 90 degrees with respect to the loading direction. The evolutions of G, GI and GII are represented in Figure G. 11 as a function of the semidebonding angle and for different values of n. primary fibre matrix secondary fibre (SF) b=2mm 0 degrees lamina 90 degrees lamina 140⁰≈ θd t90⁰= n· t0⁰ t0⁰ x ux yux 232 Figure G. 11.- Evolution of G and counter parts GI and GII for the case of two fibres oriented 90 degrees with respect to the loading direction. Again, as in the case of the configuration of Figure G. 7, there is no change from a qualitative point of view in the evolutions of G, GI and GII with respect to the case studied in Figure G. 5 involving just one fibre in the cell. However, significant differences are observed from a quantitative point of view now with respect to the single fibre case but also with respect to the 45 degrees case. Again, a noticeable increase in the values of G’s is observed in relation to the case of the unique fibre formerly studied, having then the same meaning than in the 45 degrees case studied. This configuration is also widely observed in tested specimens as can be seen, as an example, in Figure G. 12. Figure G. 12.- Experimental observation on a [03, 903]S laminate made of AS4/8552 material where the presence of a morphology of the damaged fibres oriented 90 degrees appears. The comparison between the 45 and 90 degrees cases produces some interesting conclusions, see Figure G. 13. The area under the curve corresponding to the 45 degrees case is greater than the area under that corresponding to the 90 degrees case. Additionally, whereas the curve corresponding to the 45 case has a peak in the small interval between 40 and 50 degrees of semidebonding, that corresponding to 90 degrees case present a plateau value from 50 to 80 233 degrees of semidebonding. All this on the one side indicates that 45 degrees morphology is more prone to appear, and on the other side indicates that the morphology associated with the 90 degrees case would be prone to present longer debondings. Figure G. 13.- Comparison between the values of G corresponding to the 45 and 90 degrees morphologies of two fibres damaged configurations. G.4.- Integrated multiscale modelling of a laminate: study of the matrix cracking Once the debondings between fibre and matrix have been characterized, the matrix cracking coming from the debondings is now being studied. Obviously, the first question is to predict the direction of kinking, what has been already done in París et al [6]. The direction found under tension based on circumferential stresses in the neighbourhood of the crack tip of the debonding crack is that producing a transverse matrix crack normal to the loading direction working then under mode I of fracture. The competition between the progression of the debonding in a stable manner working in mode II, in accordance with results of Figure G. 5, and the kinking of the debonding giving rise to a crack penetrating into the matrix in the direction mentioned were already studied in París et al [6]. The emphasis in this Section is put on the progression of a kinked crack. First the aspects that affect the representativity of the model are going to be considered and then the characterization of the progression of the crack as a function of the thickness of the 90 degrees lamina will be presented. G.4.1.- Cell with a single fibre with a kinked crack: influence on the predictions of the properties of the materials involved in the model The model studied in this section is presented in Figure G. 14.a). The cell used presents a debonding that will be taken in this section in the interval between 120 and 140 degrees (in this case q d = 65 degrees), and a kinked crack is modelled taking the normal direction to the load 234 applied, Figure G. 14.b). 5 solids are involved in the model. Solid S1 represents the fibre. S2 and S3 represent the matrix surrounding the fibre. The division between S2 and S3 is artificial, just to allow the kinked crack be a part of the boundaries of the two solids (S2 and S3 in this case) involved in the BEM model, as the kinked crack appears along the line separating solids S2 and S3, represented in Figure G. 14.b) with a size equal to the radius of the fibre. Solid S4 represents the 90 degrees lamina and finally S5 the 0 degrees lamina. Figure G. 14.- a) model used for the study of the cracking of the matrix, b) detail of the kinked crack abandoning the fibre-matrix interface. Different studies affecting the morphology of the model presented in Figure G. 14 are going to be carried out to check the representativity of the predictions obtained from the model. a) Checking the consistency of the model BEM is a very precise numerical method, particularly in presence of contact and singularities in the stress state, as happens in the case studied, but a particular care must be taken in the discretizations carried out, Velasco et al [23]. Thus, the first thing to be checked is the consistency of the model varying parameters that without altering the problem, just may affect to the accuracy of the integrations involved. In this case what is going to be done is to identify the properties of solids S2, S3 and S4, using for them the properties of the 90 degrees ply, and to change the size of the cell, calculating the values of G for different lengths of the kinked crack. 235 Figure G. 15.- Evolutions of G for different sizes of the cell to validate the BEM model using 90 degrees ply properties for solids S2, S3 and S4. It is obvious that with the identification of the properties of solids S2, S3 and S4 the difference between the two problems studied, just changing the size of the cell, is artificial and the same solution ought to be found for both cases. The results shown in Figure G. 15 support then the adequacy of the discretizations carried out. b) Checking the influence of the properties assigned to solids representing the 90 degrees lamina and the matrix in the cell Once the representativity of the model has been checked, it is possible to play with the properties of the model trying to learn from the predictions obtained when varying some of the parameters involved in it. In this section the results associated with three cases varying the properties assigned to the 90 degrees lamina and to the matrix surrounding the cell are going to be compared. The properties of solids S1 and S5 are maintained unchanged and first the properties of matrix are assigned to solids S2 and S3 and those of the 90 degrees lamina to solid S4 (Matrix+90ply in Figure G. 16), second the three solids receive the properties of the matrix (Matrix in Figure G. 16) and third the three solids receive the properties of the 90 degrees lamina (90 ply in Figure G. 16). The results, indicating the three cases studied, are shown in Figure G. 16. In all cases the size of the cell is maintained 40x20 µm. 0" 20" 40" 60" 80" 100" 120" 140" 0" 0.5" 1" 1.5" 2" 2.5" G"(J/m2)" Length"of"the"kinked"crack/"Fibre"radius" 90"ply"(cell=20)" 90"ply"(cell=40)" 90"ply"(cell=40x20"µm)" 90"ply"(cell=40x40"µm)" S1 S4 S5 S3S2 236 Figure G. 16.- Influence of assigning different properties to solids S2, S3 and S4 of the model. The main aspects of the results shown in Figure G. 16 to be pointed out are that first the values of G represented correspond to a pure mode I, and second that in all cases the values of G increase with increasing values of the length of the kinked crack, both aspects being fully coherent with the relative orientation of crack and external loading. The most noticeable difference between the predictions of the three cases are the significantly lower values obtained for the case in which the threes solids are modelled with the values of the matrix. This fact is fully coherent because having the matrix a stiffness of about one half of the stiffness of the 90 degrees lamina, a significant part of the external displacement is absorbed by the own body, then producing smaller values of the displacement of the faces of the crack, which in turns produces smaller values of the energy released. Notice that the values of G corresponding to the case where S2, S3 and S4 are modelled as matrix are more or less one half of those corresponding to the other two cases. In a limiting case in which S2, S3 and S4 had an infinite value of the stiffness, all the external displacements must be absorbed by the displacements of the faces of the crack which would imply higher values of the energy released. It is clear that the growth of a kinked crack through the 90 degrees ply would be governed in the right hand side of the damage control equation by GC, associated with a nominal toughness of the 90 degrees lamina in a meso scheme, or with GC values of the interface and of the matrix in a micro scheme. However, on the left hand side of the control equation, it has been shown how properties affect the values of G. Thus, when G increases (with increasing values of the stiffness of the 90 degrees ply), the appearance of transverse damage in the 90 degrees lamina is more plausible. c) Influence of the size of the cell in the crack direction In the previous calculations the role of the stiffness of the material surrounding the fibre has been considered. Now, it is interesting to check and quantify the role of the size of the cell in the direction of the crack. Thus, the size of the cell along the symmetry axis is maintained fixed, whereas the size along crack direction (vertical in Figure G. 14) is altered, taking in this case two values, 20 and 40 µm. Figure G. 17 first represents the evolutions of G for these two sizes of the 237 cell (40x20 µm and 40x40 µm), modelling the matrix surrounding the fibre and the 90 degrees ply surrounding the cell. Figure G. 17.- Influence of the size of the cell in the model. Let us concentrate in the results corresponding to continuous lines for the cases of 40x20 µm and 40x40 µm sizes of the cell and regular values of the properties used in this section. Notice first of all the strong influence that from a quantitative point of view the size of the cell in the crack direction has, the values of G being clearly greater when the size of the cell in the direction of the crack increases. This might be considered to enter in conflict with previous results. Thus, in Figure G. 16, to put more matrix in the model implied a decrease in the values of G. However, now, when the cell changes from 40x20 µm to 40x40 µm, then putting more matrix, the values of G increase. The explanation for this apparently contradictory result is that now the amount of less rigid material in the direction of the applied load (the reason invoked to explain Figure G. 16 results) is maintained constant. The difference in the obtained values of G for sizes 40x20 and 40x40 µm of the cell is easily understood showing the evolutions of the normal displacements of the faces of the crack, Figure G. 18, as well as the normal stresses involved in the energy released, Figure G. 19, corresponding again to the two cases studied, 40x20 and 40x40 µm. Clearly both displacements and stresses are higher for the 40 µm case in comparison to the 20 µm case. 0" 50" 100" 150" 200" 250" 0" 0.5" 1" 1.5" 2" 2.5" 3" 3.5" 4" G"(J/m2)" Length"of"the"kinked"crack/"Fibre"radius" Matrix+90"ply"(cell=20)" Matrix+90"ply"(cell=40)" Matrix+90"ply"(Eply90=42"Gpa,"cell=20)" Matrix+90"ply"(E90°ply=42"GPa,"cell=40x20"µm)" Matrix+90"ply"(cell=40x40"µm)" Matrix+90"ply"(cell="40x20"µm)" 238 Figure G. 18.- Normal displacements of the faces of the debonding crack corresponding to the case of a length of the kinked crack equal to the radius of the fibre. Figure G. 19.- Normal stresses ahead the debonding crack tip corresponding to the case of a length of the kinked crack equal to the radius of the fibre. In any case, it is somehow shocking the increasing character of the evolutions of G with the length of the kinked crack in the interval studied, for the two sizes of the cell. To try to understand the different phenomena observed, particularly in Figure G. 17, it is important to notice that there are two effects involved in the problem under consideration. First there is the effect of the stiffness of the material surrounding the crack under consideration (in this case the debonding between fibre and matrix) in the direction of the load applied, and second, there is the Cook-Gordon effect that is explained in what follows, Figure G. 20. Node positions near to the crack tip (µm) ∆" (µm) A B Matrix + 90 ply (cell=40×20 µm) Matrix + 90 ply (cell=40×40 µm) Kinked crack tip A B C 0.0544 µm 0.0544 µm Node positions near to the crack tip (µm) u(µm) A B 239 When a crack grows inside a body of low stiffness (compliant material in Figure G. 20) approaching a more rigid material (stiff material in Figure G. 20), a mechanism known as Cook- Gordon [27] appears, consisting in the decrease of the energy released by the crack when approaching the interface with the stiff material, which implies the stopping of the crack before arriving to the interface at a distance b of the interface, as represented in Figure G. 20.a). The crack can grow more, but requiring the application of an increase in the external load, in this case being able of generating a certain damage along the interface before the crack reaches it. This is represented in Figure G. 20.b) by an interfacial crack of length 2d. This situation is of application to the models studied as can be appreciated in Figure G. 14.a), where the kinked crack approaches solid S4 that may be stiffer than solids S2 and S3, where the kinked crack is. Figure G. 20.- Illustration of Cook-Gordon Mechanism: a) stop of the crack before reaching the interface with the stiff material, b) damage along the interface before the transverse crack reaches it. There are many evidences of the Cook-Gordon effect in the case of the damage under consideration in this chapter. Figure G. 21 shows the stop of the transverse crack before reaching the interface with the 0 degrees ply. Figure G. 21.- Stopping of the transverse crack in the 90 degrees ply near the interface with the 0 degrees ply. The picture corresponds to a [02,904]S laminate. More evidences can be found in París et al [28], where the damage in the interface schematized in Figure G. 20.b) can be seen materialized either by transverse debondings in the 90 degrees Compliant material Stiff material ab Compliant material Stiff material d d a) b) a+∆a 240 ply (debondings between fibres of the 90 degrees ply and matrix) or/and by longitudinal debondings in the 0 degrees ply (debondings between fibres of the 0 degrees ply and matrix). Now we are in a disposition to understand the apparently unconditional growing character of G in Figure G. 17 (continuous lines). Thus, observing now the evolution of G with the increase of the kinked crack represented by the dashed lines, it can be clearly observed for the case of the cell of size 40x20 µm, how the energy released starts to decrease as soon as the crack tip is approaching the interface. In the case of the 40x40 µm cell, the effect is not observed because the cell has double length and the crack tip is not yet, for the lengths considered, close enough to the 0 degrees ply interface to suffer Cook-Gordon effect. An artificial case has also been included in Figure G. 17 to go further in the understanding of the effect. The value of the Young modulus of the 90 degrees ply has been artificially increased till 42 GPa (for a size of 40x20 µm of the cell), which immediately produces an increase, already explained, of the values of G. Additionally, the Cook-Gordon effect is clearly manifested, not only by the proximity to the 0 degrees ply, but now the decrease is more noticeable because the crack is growing towards a significant much more stiff material, solid S4 in the model. As in the previous case the important consequence of this analysis is the enormous influence that small variations in the model may have in the quantitative results obtained, what is a key factor when trying to get the properties of a material based on predictions obtained from a certain model. d) Influence of the properties of the fibre The influence of the properties of the fibre is now going to be studied. Properties corresponding to glass and carbon fibres are going to be considered. The properties taken for glass fibre are E=7.08 104 MPa and n =0.22, whereas those of the carbon fibre appeared in Table G. 2. These properties have been considered to affect only to solid S1, to isolate exactly the role of the stiffness of the fibre in the plane under consideration. The results are shown in Figure G. 22. Figure G. 22.- Influence of the properties of the fibres. 247 Figure G. 29.- Damage consistent on an isolated debonding which is going to appear in a explosive or progressive way. With reference to the first way of appearance of damage, explosive way, Figure G. 30 represents the evolution of s and t in the pristine (undamaged) state along the interface that will debond having taken the semidebonding angle, for a comparison with the progressive damage, of a value of 60 degrees in this case. All the results associated with this case correspond to a nominal deformation of the laminate of 1%. Figure G. 30.- Normal s and tangential t components of the stresses in the pristine state along the interface crack (60 degrees semidebonding), stresses participating in the release of energy during an explosive mechanism of damage. Figure G. 31 represents the evolution of the opening of the crack D ur and relative tangential displacements D u q along the faces of the 60 degrees semidebonding considered. x y Deformed(Shape( !"=60° 0" 20" 40" 60" 80" 100" 120" 0" 10" 20" 30" 40" 50" 60" 70" σ"(MPa)" θd" Sigma" tau" Angular position along the 60 degrees semidebonding ! " !, " (MPa) 0" 20" 40" 60" 80" 100" 120" 0" 10" 20" 30" 40" 50" 60" 70" σ"(MPa)" θd" Sigma" tau" 248 Figure G. 31.- Relative normal and tangential displacements along the faces of the 60 degrees semidebonding. With reference to the second way of appearance of damage, Figure G. 32 represents the evolution of G and counterparts GI and GII corresponding to the same morphology but with a progressive scheme. The areas of the total energy released under modes I and II corresponding to a 60 degrees semidebonding are indicated in the figure by means of WI and WII. Figure G. 32.- Evolution of G and counterparts GI and GII associated with a progressive scheme of damage. Table G. 3 summarizes the results obtained with both procedures in terms of the total energy released W and the corresponding counterparts WI and WII. In the case of the explosive 0" 20" 40" 60" 80" 100" 120" 140" 160" 0" 10" 20" 30" 40" 50" 60" 70" u(nm)" θd" ur" uθ" ∆ur ∆u" ∆ur, ∆u"#(nm) Angular position along the 60 degrees semidebonding# 249 mechanism the values are directly obtained from stresses and displacements corresponding to undamaged and damaged configurations, whereas in the progressive mechanism the values are obtained from the integration of the evolutions of G, GI and GII from Figure G. 32. Table G. 3.- Results for W, WI and WII associated with both, explosive and progressive, mechanisms of damage for a semidebonding of 60 degrees. It is clear now that the two procedures followed, similar to those previously carried out for the transverse crack in the 90 degrees ply at mesomechanical level, do not produce in this occasion the same results. Thus, whereas the explosive mechanism is fully dominated by mode I and consequently controlled by GIC, the progressive mechanism is a mixed mode where knowing that GIIC >> GIC, makes it dominated by mode II. Thus, the implications of both calculations, and obviously the two procedures associated with them, are not equivalent at all. The implication of this result is that to perform calculations at micromechanical level trying to justify observed behavior is not only a matter of putting fibres and matrix and use of a tool to simulate the generation of damage employing Fracture Mechanics or Interfacial Fracture Mechanics, or tools derived from these concepts like a cohesive approach, linear elastic brittle interfaces or phase field, or any other. The hub of the issue is to be sure that the modelling procedure employed reproduces the mechanism of failure that the material presents. G.6.- Conclusions The interfibre intralaminar damage in a lamina of a composite laminate is studied in this chapter. This damage appears in a premature way in the weakest laminae of a laminate, typically at those oriented 90 degrees with respect to the orientation of the load. As a laminate may suffer different loads, each lamina is potentially susceptible of suffering this type of damage. The damage has been assumed to be composed of debondings between fibres and matrix and by the breakage of the matrix typically appearing connecting debondings between fibres. The treatment of each type of damage has been studied in separate sections. With reference to the debondings, this damage has been treated by means of Interfacial Fracture Mechanics. The study of different [0,90n]S laminates has manifested the unstable character of a debonding between fibre and matrix till reaching a total length corresponding to about 120 degrees and the numerical predictions indicate that this fact is independent of the thickness of the 90 degrees ply, the appearance of this type of damage having been observed when using ultra-thin plies. The representativity of the studies carried out in connection with the debondings has been correlated with experimental results for the cases of more than one fibre, finding very close similarities between observations and predictions. !"=60° PROGRESSIVE MECHANISM EXPLOSIVE MECHANISM WI(10-3 J/m) 0,006667 0,031100 WII (10-3 J/m) 0,009520 0,001470 W(10-3 J/m) 0,016187 0,032570 250 With reference to the breakage of the matrix, its onset has been connected to the evolution of a debonding that abandons the interface, kinking and penetrating into the matrix in a direction normal to the loading. The progression of this kinked crack as a function of the different parameters involved in the model has been studied as well as the connection between the predictions and the thickness of the 90 degrees ply. No effect of this type has been found, just a slight dependence on the 90 degrees ply thickness associated with the Cook-Gordon effect when the tip of the transverse crack is approaching the interface with the 0 degrees lamina. Thus, no connection with the scale effect has been detected. A final consideration on the correlation between the predictions obtained with a model and the actual mechanism of failure of a determined material or structure is discussed, alerting on the possibility of performing a numerical analysis which is associated with a particular mechanism of failure that does not represent the actual mechanism of failure of the material or structure. 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