Interaction between fibres in the transverse damage in composites

Depósito de Investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published by Elsevier in Engineering Fracture Mechanics, Vol. 239, on November 2020, available at: https://doi.org/10.1016/j.engfracmech.2020.107273 © 2020 Elsevier. En idUS Licencia Creative Commons CC BY-NC-ND Interaction between fibres in the transverse damage in composites María Luisa Velasco, Elena Correa, 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, Spain 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. Keywords: Composites; debonding; inter-fibre failure; mixed mode fracture; transverse failure E.1.- Introduction The use of composites has experienced a significant increase in many industries where high properties in terms of stiffness and strength as well as a low weight play a significant role in the performance of a structure. Metallic materials may present either ductile or brittle behaviour. Steels, considered ductile materials, may have brittle behaviour when the octaedric part of the stress state is dominant. Other metallic materials as cast iron do not have the possibility of suffering large deformations before failure, then presenting always brittle behaviour. These brittle materials, which are studied satisfactorily as homogeneous as most of metals, present catastrophic failure with no previous warning, although the failure is quite easy to predict since it is generally governed by the maximum values of the stress or strain. Instead, composites present at failure a very complicated behaviour. First, because composites are not homogeneous materials and, as a consequence, many mechanisms of damage take part in the failure process with interactions between them. Additionally, the loads at which each mechanism (breakage of fibres, debonding between fibres and the matrix, delamination between the plies, deformation of the matrix, etc.) appears may be quite different, combining the sustainability of an external load with the presence of a certain level of damage that can be considered acceptable for the structure. It is then necessary to determine the precise behaviour of a material placed in a certain structure from the pristine state until final failure. París [1] performed a study on existing failure criteria and concluded that a particular criterion should be physically based if a convenient level of confidence wanted to be reached. In this sense, it is necessary to focus on the level where the damage appears, which in most cases requires investigation at a micromechanical level. At this scale, debonding along the fibre-matrix interfaces, as well as their connections, can be considered the weakest links in a pristine material and is where the study of the damage should be performed. To develop the necessary knowledge at this micromechanical level, following the pioneering work by París et al. [2], París, Correa and their co-workers [3-12] presented a series of papers on the micromechanical modelling of damage in composites based on single fibre BEM models. Thus, a clear correlation between the predictions and experimental observations under transverse tension was established by París et al. [3]. París et al. [4] studied the case of bidimensional loading connecting the micromechanical predictions with those derived from failure criteria. Correa et al. [5-6] extended the study to failure under compression, again connecting the predictions of the study with the experimental observation of failure. Correa et al. [7] provided a micromechanical explanation of the observed progression of damage under cyclic loading. Correa et al. studied the effect of thermal residual stresses on matrix failure under transverse tension in [8] and compression in [9] by developing experimental tests to correlate the numerical predictions. Correa et al. also studied the effect of the presence of a secondary transverse load on interfibre failure under tension [10] and compression [11]. A complete view of the mechanism of damage under tension can be found in [12]. Previous works from different authors focusing on the numerical study of matrix/interfibre failure at different levels and employing various tools and approaches can be found in the literature. In particular, and in the framework of BEM numerical analysis, Távara et al. [13] employed one hundred-fibre models. Mantič [14] focused on a finite fracture mechanics approach by analysing the initiation of interface cracks under transverse tension, whereas Carraro and Quaresimin [15] also employed this approach for the case of transverse and antiplane stresses. FEA has also been employed for similar studies; specifically and among others, Hojo et al. [16] analysed the microscopic interfacial normal stress states for transversely loaded unidirectional composites in multifibre models; Kushch et al. [17] focused on the progressive debonding in fibre reinforced composites based on the cohesive-zone model at the interface for single fibre and many fibre models; Saito et al. [18] studied ultrathin plies in cross-ply laminates; and, more recently, Zhuang et al. [19, 20] implemented a multifibre model to study the kinking of a fibre-matrix interface crack. Some authors have also studied matrix/interfibre failure by means of experimental techniques, with the objective of observing the micromechanical generation of matrix failure. In particular, the pioneering work from Zhang et al. [21] focused on single fibre tests under tension, whereas Gamstedt and Sjrogren [22] observed damage generation in the context of cyclic loading. Additional studies include the works of Saito et al. [23], who analysed the effect of a 90° ply thickness on the appearance of damage in crossply laminates; Baral et al. [24], who studied the effect of the fibre type; Hobbiebrunken et al. [25], who analysed the behaviour at both high and low curing temperatures; Genz et al. [26], who evaluated the presence of thermal residual stresses; and Martyniuk et al. [27], who employed X-ray microtomography in single fibre specimens to evaluate debonding initiation and propagation. Focusing on the previous investigation performed by the authors, as the beginning of the damage is assigned to the assumed weakest part of the heterogeneous internal structure of the materials, i.e., the interface between the fibres and the matrix, Interfacial Fracture Mechanics Theory has been the essential tool for analysing the results obtained using both the open model by Rice [28] and the contact model by Comninou [29]. The numerical method selected was the Boundary Element Method (BEM) París and Cañas [30]. BEM seems the most adequate method for the direct treatment of the complicated stress state that appears in the neighbourhood of a crack between dissimilar materials, avoiding cumbersome domain discretizations. Details of the application of this method for the particular problem under study are included in the work of París et al. [31]. The objective of this paper is to analyse some particular features that have not been previously studied to gain knowledge on the appearance of damage in the weakest layer of a laminate, i.e., the layer that is oriented at 90° with respect to the nominal external load, where the first manifestations of damage appear. In particular, in this paper, the development of fibre-matrix debonding is studied in the presence of a nearby fibre that is already damaged. Two-fibre configurations are modelled by means of the BEM and the effect of the interfibre distance and the fibre relative orientation on the progression of damage is analysed. This study is motivated by the revision of previous experimental work, whose main conclusions were included in Correa et al. [32]; in that investigation, and for a different purpose, a considerable amount of microscopic evidence of interface cracks associated with interfibre failure in cross-ply laminates was obtained. The later analysis of that collection of images encouraged us to think that there could be specific configurations of two damaged fibres (in terms of their relative orientation and distance) with a critical effect on the damage evolution. A deeper investigation into this topic showed renewed importance due to the present relevancy of ultrathin plies, whose main purpose is to produce a delay in the appearance of the aforementioned manifestations of initial damage in the weakest layer of a laminate, as was pointed in the work of París et al [33]. The structure of the paper is as follows. First, the appropriateness of the model used in the study is discussed in Section E.2. Then, in Section E.3, a study is carried out to evaluate the effect of the presence of a neighbouring fibre that has been previously damaged on a fibre with a growing interface crack. The study considers different orientations of the previously damaged fibre with reference to the fibre under study and the external load. A similar study is carried out in Section E.4 that focuses on the distance between the two fibres. A fixed length of the debonding corresponding to a total angle of 140° has been chosen in Sections E.3 and E.4 for the previously damaged fibre, based on former numerical and experimental studies. Section E.5 is devoted to studying the influence of the presence of smaller debonding lengths on the conclusions obtained from the two previous sections. Finally, Section E.6 shows experimental observations from a testing campaign, specifically performed by the authors, supporting the numerical predictions. A Conclusions section closes the paper. E.2.- Modelling damage in the 90° layer The selection of a model for studying a certain problem is always a compromise between, on one hand, the accuracy and representativeness of the results, and, on the other hand, the own complexity of the model required for the highest efficiency (in terms of properties involved, computing requirements, etc.). A discussion on the selection of an adequate model for obtaining the best balance for the problem under consideration in this paper follows. The first manifestations of damage associated with transverse failure (fibre-matrix debonding) studied in this paper are assumed to occur in the 90° layer of a [0, 90n]s cross-ply laminate. Thus, a model for checking the role of the micromechanical internal structure of a composite in the evolution of debonding needs to be developed. As explained in the Introduction, the local interaction between a reduced number of fibres (two damaged fibres in this particular case) requires further study; thus, to isolate this problem and highlight it, the discussion is reduced to two possible models. First, a simple model of an infinite matrix surrounding just the minimum number of fibres required for the analysis of the target local effect is analysed, see Fig. E.1, where several general examples of its application for different problems are included. Second, an integrated multiscale model is analysed where both layers of the laminate (0° and 90°) are represented (and, thus, their elastic properties considered) and a matrix inner micro-cell with the minimum number of fibres required is included, see Fig. E.2, where for the sake of simplicity only an example of a single fibre micro-cell has been plotted. In both cases, symmetry in the xz plane is considered and the actual length of the debonding angle is 2  d. (a) (b) (c) Figure E.1.- (a) Single fibre (b) two fibres (c) three fibres in and infinite matrix model with interfacial damage. Figure E.2.- Integrated multiscale model of a [0,90n]S laminate with interfacial damage in the micro-cell. It is worth noting that an integrated multiscale model is much more complicated, from a numerical point of view, than an infinite matrix model, independently of the number of individual fibres modelled in each configuration. This circumstance is particularly pronounced in BEM models, where an integrated multiscale model often requires sizes of the elements that differ significantly, involving integrations along the elements from points extremely close to them. Some of these features have been studied by Velasco et al. [34]. Therefore, the use of an integrated multiscale model is only recommended when a parameter that is involved in the model but not considered in an infinite matrix model is a key factor in the problem under consideration or when the comparison between the results of both models leads to different conclusions. This could be, for instance, the case of the effect of the thickness of the 90° layer for the study of the scale effect in composites (París et al [35]). As the direct use of a simplified infinite matrix model might create a lack of confidence in the representativeness of the results, a comparison between the two aforementioned models was first carried out. To this end and for the sake of simplicity, the models represented in Figs. E.1a and E.2 (where a single damaged fibre is considered) were solved, and their results were compared. Table E.1 compiles the  d fibre matrix x y s 0 s 0  d: semidebonding angle x y t90 ⁰ =170 µm 0 degrees lamina 90 degrees lamina fibre matrix semidebonding angle b= 5mm t0 ⁰ =170 µm properties of the materials involved corresponding to a carbon epoxy system in both configurations with a fibre volume fraction of 57%. It is clear that because different properties are involved in both models, the models are not expected to produce exactly the same results from a quantitative point of view, similarities from a qualitative point of view, if obtained, are sufficient for the acceptance of the simplified infinite matrix model for the study under consideration. Table E.1.- Properties of the materials involved in the models represented in Figs. E.1 and E.2. 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 (transversely 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 A comparison between both models is performed in terms of the evolution of the Energy Release Rate ( G ) of a semidebonding crack between the fibre and matrix. G is evaluated following Irwin's Virtual Crack Closure Technique [36] for the frictionless case, considering both the Mode I fracture contribution, I G , and the Mode II fracture contribution, II G :                  d d dRuR dRuRGGG dddr d ddrdrr d III    s  s  0 0 )],(),([ 1 )],(),([ 1 (1) where ),(  r is a local system centred at the fibre, R is the fibre radius, and rr s and  s r are the normal and tangential tractions along the interface, respectively, while r u and  u are the opening and sliding relative displacements between the crack faces, respectively. In accordance with the previous comment about the sufficiency of the similarity of the qualitative evolution of the damage under consideration, i.e., the evolution of G with the semidebonding angle  d, the comparison between the predictions of both models is carried out as indicated in Fig. E.3, where the relative values of G (and corresponding GI and GII values) with respect to the maximum values of G, Gmax, are presented for each model. Obviously, the predictions of both models coincide, with a value of 1 at the point of the maximum value of G, which is located between 50° and 60° of the semidebonding angle axis. Figure E.3.- Evolution of G, GI and GII with the semidebonding angle for the single fibreinfinite matrix model and the integrated multiscale model. First, since the evolutions of G for the two models under consideration have already been studied and their interpretation has been given, see for instance [3, 4, 33], the objective here is only to provide a comparison between the predictions derived from the two models. The similarity of the evolutions obtained in both cases for the relative values of G, GI and GII are immediately clear. The unique difference appears in the semidebonding angle at which the maximum value of G is obtained or, in other words, at the instant at which the change from the mixed mode to pure Mode II takes place. A reduction in this angle from approximately 60° for the single fibre-infinite matrix model to approximately 50° in the integrated multiscale model can be observed, and the rest of the parameters of the evolution are coherent with this observation. This change in the angle is consistent further than with the different properties involved in both models, with the role that the extremely high stiffness of the 0° lamina (involved in the integrated multiscale model) plays with respect to the low stiffness of the matrix (the unique material involved in the infinite matrix model). Even more significant is the stiffness of the 90° layer in the modelled plane, with a higher value than that of the matrix, which plays a significant role in the change observed due to the proximity of the matrix to the damage modelled, the debonding crack between the fibre and matrix, and its orientation with respect to the applied load. Therefore, based on the qualitative similarity of the results provided by both models, an infinite matrix model is employed in Sections E.3, E.4, and E.5. Another matter of interest with reference to the model under study is the type of plane problem used. The most employed option, as can be checked in all previous references, is to consider plane strain (PSa). Generalized Plane Strain in the mid xy plane of the laminate, with reference to Figure E.2, could in fact be accepted as the most correct way to represent the actual situation by means of a 2D study. However, at the lateral edges of the laminate it is obvious that, due to the boundary conditions, szi(i=x,y,z)=0. This fact clearly discards at the edges the plane strain condition, that generally requires a non-null szz, and in contrast suggests the presence of a plane stress state (PSe); this is a commonly accepted conclusion when considering a crack in a homogeneous isotropic plate: plane strain in the central zone and plane stress at the lateral edges, although, in fact, the stress state at the points belonging to the lateral edges is really 3D. Strictly speaking, in the rest of the paper PSe refers to the use of generalized plane stress using Filon hypothesis and associated average values, 0 0,2 0,4 0,6 0,8 1 1,2 010 20 30 40 50 60 70 80 90 100 110 120 130 140 150 G /Gmax Semidebonding angle, θd G (Laminate) GI (Laminate) GII (Laminate) G carbono GI carbono GII carbono G(Multiscale) GI(Multiscale) GII (Multiscale) G(Infinite matrix) GI(Infinite matrix) GII (Infinite matrix) since the pure plane stress problem satisfying szi(i=x,y,z)=0 is an extremely complicated problem. Nevertheless, since the goal is to compare the numerical predictions obtained from a plane model with experimental observations of the damage at the lateral edges of a specimen (Section E.5), it seems adequate to first compare the results derived from plane strain and generalized plane stress hypotheses that are applied to the same problem. To this end, Fig. E.4 shows the G, GI and GII distributions of the debonding that appears in the integrated multiscale model for both the PSe and PSa cases. From the comparison of the distributions presented in Fig E.4, it is clear that the predictions for both cases are similar from a qualitative point of view. The differences come again from the different fields of stresses and displacements obtained for the PSe and PSa conditions. As an example, Fig. E.5 shows the srr and ur distributions at both sides of the crack tip and at the points used in the calculation of G for both cases. The values correspond to a semidebonding angle of 50°. Figure E.4.- Plane Stress (PSe) and Plane Strain (PSa) predictions for G, GI and GII (integrated multiscale model). (a) 0 1 2 3 4 5 6 7 020 40 60 80 100 120 140 160 G(J/m2) Semidebonding angle, θd GI (PSe) GI (PSa) GII (PSe) GII (PSa) G (PSe) G (PSa) GI(PSe) GI(PSa) GII (PSe) GII (PSa) G (PSe) G(PSa) 0 0,5 1 1,5 2 2,5 3 3,5 4 49,4 49,5 49,6 49,7 49,8 49,9 50 50,1 Δur(nm) Semidebonding angle, θd PSa PSe When all the G distributions for the three  cases from Sections E.3.2.1, E.3.2.2 and E.3.2.3, are compared, it can be estimated that the highest accelerative effect is found for the  =45° OS crack location, followed by  =90° and the OS/SS crack location. This reasoning is based both on the position of Gmax and on the area under the G curve. This would imply that, experimentally, these three configurations should be observed most frequently. The method of comparing this accelerative effect is an open discussion since there are alternative parameters, such as the G value of the smallest debonding angle considered (i.e., at crack onset, 10° in this case), involved in the potential approaches used for the prediction of the damage under consideration. For instance, based on this last option, it could be predicted that in the  =90° case, the SS crack configuration would present a more accelerative effect than the OS configuration. Finally, it is also relevant to observe the translations in the position of Gmax in the different cases and configurations with reference to the single fibre case. Looking in detail at the GII and GI curves, it can be seen that the position of Gmax approximately coincides in most cases with the position of GII, max. The prediction of the unstable growth of an interface crack and thus its final length at the interface can be made by comparing the values of the G distribution with the fracture mode-mixity dependent fracture toughness, Gc(  K), as proposed by Hutchinson and Suo [40] for instance. This exercise has been performed here using the same parameters employed in [37] for the Gc trend, and their results have been included in Table E.3. From these results, it is clear that the 0° OS and 45° USF cases produce the longest debonding lengths at the interface, whereas the 0° USF case leads to the shortest debonding length. Focusing on the previous predictions of which should be the most frequently observed configurations, i.e.,  =45° OS and  =90° OS/SS, the results presented in Table E.3 show that both 90° cases would have associated greater extensions of the interface crack at the primary fibre (especially in the OS case) than the 45° case. Table E.3. Predictions of the final value of the semidebonding angle after the period of unstable growth at the interface. Case (  2=0º) Final  d REF 70º USF 52º SS 58º OS 110º Case (  2=45º) Final  d REF 70º USF 112º SS 79º OS 65º Case (  2=90º) Final  d REF 70º USF 84º SS 85º OS 95º E.4.- Damage interaction for two neighbouring fibres: effect of the interfibre distance In this section, the distance r2 (Fig. E.6) between the primary and the secondary fibres is varied to predict the end of the influence of the damaged secondary fibre in the G distribution of the interface crack at the primary fibre. E.4.1. Model description Based on the results of the previous section, two of the situations most prone to damage, i.e., the 45° OS and the 90° OS cases, were selected for the subsequent analysis. The models are similar to those previously employed in Section 3 (Fig. E.6), with the unique difference being that now the distance between the fibres, defined by r2, varies to analyse its role. E.4.2. Results A selection of r2 values are chosen. Specifically, r2 =2.42r1 (the original distance employed in Section 3), 3r1, 5r1 and 10r1 are considered. The GI, GII and G distributions for every r2 value have been calculated, and their dimensionless results are presented together with the single fibre (Ref) case in Figs. E.13, E.14 and E.15 for the  90° OS case and in Figs. E.16, E.17 and E.18 for the 45° OS case. Figure E.13.- GI/G0 distribution versus  d for r2=2.42r1, 3r1, 5r1, 10r1 corresponding to  =90 OS configuration and single fibre model (Ref). Figure E.14.- GII/G0 distribution versus  d for r2=2.42r1, 3r1, 5r1, 10r1 corresponding to  =90 OS configuration and single fibre model (Ref). 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 020 40 60 80 100 120 140 160 GII/G0 Semidebonding angle, θd GII/G0 (REF) GII_r2_2.42 GII_r2_3 GII_r2_5 GII_r2_10 Ref 2.42r1 3r1 5r1 10r1 Figure E.15.- G/G0 distribution versus  d for r2=2.42r1, 3r1, 5r1, 10r1 corresponding to  =90º OS configuration and single fibre model (Ref). Figure E.16.- GI/G0 distribution versus  d for r2=2.42r1, 3r1, 5r1, 10r1 corresponding to 45 OS configuration and single fibre model (Ref). Figure E.17.- GII/G0 distribution versus  d for r2=2.42r1, 3r1, 5r1, 10r1 corresponding to 45 OS configuration and single fibre model (Ref). 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 20 40 60 80 100 120 140 160 G/G0 Semidebonding angle, θd G/G0 (Ref) G_r2_2.42 G_r2_3 G_r2_5 G_r2_10 Ref 2.42r1 3r1 5r1 10r1 0 0,1 0,2 0,3 0,4 0,5 0,6 020 40 60 80 100 120 140 160 GI/G0 Semidebonding angle, θd GI/G0 (Ref) GI_r2_2.42 GI_r2_3 GI_r2_5 GI_r2_10 Ref 2.42r1 3r1 5r1 10r1 0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 020 40 60 80 100 120 140 160 GII/G0 Semidebonding angle, θd GII/G0 (Ref) GII_r2_2.4 2 GII_r2_3 GII_r2_5 GII_r2_10 Ref 2.42r1 3r1 5r1 10r1 Figure E.18.- G/G0 distribution versus  d for r2=2.42r1, 3r1, 5r1, 10r1 corresponding to 45 OS configuration and single fibre model (Ref). Comparing the G distributions presented in Figs. E.15 and E.18, it can be concluded that for both  cases, the effect of the damaged secondary fibre on G of the primary fibre interface crack vanishes for r2=10r1, which is equivalent to saying that a separation of 4 fibres between the primary and the secondary fibre is sufficient to cancel their mutual influence. In other words, the G distribution of an interface crack at a primary fibre in the presence of a secondary damaged fibre at a distance r2>10r1 coincides with the G distribution of an interface crack at a primary fibre surrounded by the matrix. Sandino et al. [37] found that the effect that an undamaged secondary fibre has on the G distribution of an interface crack at the primary fibre vanishes for r2=7r1. Therefore, if the secondary fibre is debonded, its effect reaches a longer distance than if it remains undamaged. This result is in agreement with the evidence presented in Figs. E.10 and E.12 for both cases of  , where for the reference distance r2=2.42r1, the G curve of the USF configuration is closer to the single fibre one than the G curves of the OS configurations, indicating that for the OS cases, there exist greater differences that need to be suppressed by the distance than in the USF case. 5.- Exploring the effect of the extension of the damage in a previously damaged nearby fibre In Sections E.3 and E.4, the extension of the damage at the secondary fibre corresponded to a debonding angle of 140°. This arc-length, Ld =140°, represents the numerically predicted final situation prior to kinking of an interface crack subjected to transverse tension, according to París et al [3], which was the reason for its choice in all the models that included a damaged secondary fibre. Nevertheless, the situation where Ld =140° may not be the most relevant, again in terms of alterations in the G distribution of the primary fibre. This observation is clarified in this section. To this end, one of the cases analysed in Section E.3, in particular the  =90° SS case, is studied for a selection of Ld values, specifically 12°, 32°, 64°, 100° and 140°. Note that values larger than 140° have not been chosen. This decision is based on the work of Correa et al. [32], where more than 800 Ld measures revealed that only 7% of them were greater than 140° and 1% were greater than 160°. The model developed for this section is similar to those previously used in Sections E.3 0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 020 40 60 80 100 120 140 160 G/G0 Semidebonding angle, θd G/G0 (Ref) G_r2_2.42 G_r2_3 G_r2_5 G_r2_10 Ref 2.42r1 3r1 5r1 10r1 and E.4, with only the parameter Ld varying to reproduce the different cases, as shown in Fig. E.19. Note that the USF case corresponds to the SS case for Ld equal to zero. Figure E.19.- Cases studied for Ld=12, 32, 64, 100 and 140. Single fibre and USF cases are taken as references. As commented on in Section E.3, the USF curves are below the corresponding single fibre distributions, meaning that the presence of a secondary undamaged fibre inhibits damage generation in the primary fibre. This result is in agreement with the conclusions presented by Sandino et al. [37]. Observing Figs. E.20, E.21 and E.22, it can be deduced that for small debonding angles at the secondary fibre, the effect is similar to the undamaged (USF) case. As Ld increases, the level of the curve approximates the reference case (this happens, for instance, for Ld =64°), meaning that for a certain Ld value, the interface crack at the primary fibre behaves as if it was surrounded by an infinite matrix. Moreover, as the extension of the damage increases, the energetic distributions increases above the reference case, leading to an accelerative effect of the damaged secondary fibre on the interface crack propagation. Figure E.20.- GI/G0 distribution versus  d for Ld=12, 32, 64, 100 and 140 corresponding to  =90 SS configuration and single fibre (Ref) and USF models. Secondary fiber whithout damage (USF)  d  Semidebonding angle Matrix Fibre Secondary fiber with damage on the same side (SS)  d  Semidebonding angle Matrix Fibre 140  d  Semidebonding angle Fibre Reference Case (Ref) Matrix σ0σ0σ0σ0σ0σ0 0,00 0,03 0,06 0,09 0,12 0,15 0,18 0,21 0,24 0,27 010 20 30 40 50 60 70 80 90 100 110 120 130 140 150 GI/G0 Semidebonding angle, θd GI/G0 (Ref) GI/G0 (SS0) GI/G0 (SS12) GI/G0 (SS32) GI/G0 (SS64) GI/G0 (SS100) GI/G0 (SS140) Ref USF Ld=12º Ld=32º Ld=64º Ld=100º Ld=140º Figure E.21.- GII/G0 distribution versus  d for Ld=12, 32, 64, 100 and 140 corresponding to  =90 SS configuration and single fibre (Ref) and USF models. Figure E.22.- G/G0 distribution versus  d for Ld=12º, 32º, 64º, 100º and 140º corresponding to  =90º SS configuration and single fibre (Ref) and USF models. Although the aforementioned tendency is presented by the three energetic distributions, GI, GII and G, it is again evident that the qualitative shape of G is due to Mode II, whose contribution is more relevant than that of Mode I. The most outstanding result is, in any case, the fact that the largest damage extension, Ld=140°, produces the maximum alteration in the G values and results in the most accelerative effect on the interface crack propagation at the primary fibre. This conclusion supports the decision made in Sections E.3 and E.4, where Ld was established at 140° for the studies presented in those sections. E.6.- Experimental evidence To support the representativeness of the results obtained from the models developed, cross-ply symmetrical laminates have been manufactured and specimens obtained from them tested, in order to observe the appearance of damage in the weakest lamina. The stacking sequence selected was [04, 90n]S with n values of 1 and 4. The material used to manufacture the specimens was a unidirectional carbon-epoxy prepreg with a ply thickness of 190 m, specifically AS4/8552 Hexcel whose principal stiffness properties are E11= 141 GPa, E22= 10 GPa and 12= 0,3. This material is cured in 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 GII/G0 Semidebonding angle, θd GII/G0 (Ref) GII/G0 (SS0) GII/G0 (SS12) GII/G0 (SS32) GII/G0 (SS64) GII/G0 (SS100) GII/G0 (SS140) Ref USF Ld=12º Ld=32º Ld=64º Ld=100º Ld=140º 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 010 20 30 40 50 60 70 80 90 100 110 120 130 140 150 G/G0 Semidebonding angle, θd G/G0 (Ref) G/G0 (SS0) G/G0 (SS12) G/G0 (SS32) G/G0 (SS64) G/G0 (SS100) G/G0 (SS140) Ref USF Ld=12º Ld=32º Ld=64º Ld=100º Ld=140º an autoclave at 180ºC for 120 minutes. The specimens, whose dimensions are detailed in the Spanish standard UNE-EN 2561[41], were subjected to uniaxial tension using a universal testing machine INSTRON 4483, applying acoustic emission to identify the appearance of damage. Once the specimens were tested until first damage detection (which arises for a stress value of 800 MPa and 500 MPa, for n=1 and n=4, respectively), they were cut, encapsulated, sanded and polished with the ultimate aim to observe the damage generated by using an optical microscope, more specifically a binocular inverted microscope EPIPHOT 200. The numerical results shown in Figs. E.7-E.12 indicate that there are always configurations in which, in the presence of an already damaged neighbouring fibre, the amount of energy released by an interface crack at the fibre under study is greater than that corresponding to the presence of a neighbouring undamaged fibre. This implies that, if debonding takes place for any reason (for example, poor local bonded conditions), there may be fibres in the neighbourhood of this first damaged fibre in a clear disposition to experience debonding at the same loading level; in particular, debonding is most likely to occur in the path that reproduces a meso-crack oriented perpendicularly to the applied load (working nominally in the mesoscale under Mode I). All this implies that isolated debonds would rarely appear. As a consequence, damaged configurations with more than one debond would be expected. Specifically, based on the numerical predictions of the previous sections, those referred to as 45° OS, 90° OS, or even 90° SS are more likely to arise. As an example, Fig. E.23 shows a set of three debonds involving one 45° OS configuration (the most frequently observed) and one 90° OS configuration (frequently observed). These particular observations shown in Figs. E.23 and E.24 can be complemented by a general view, such as that shown in Fig. E.25. In this figure a sequence of debonds that are not yet connected (generalized damage) and that would eventually lead to a transverse crack is observed. The 45° and 90° configurations can be easily identified in that sequence. 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. . a) b) Figure E.24.- Observations of damage associated with the 90 SS case: a) original micrograph, b) corresponding schemes of the micrograph. Figure E.25.- Observations of generalized damage. In general, it is observed that the damage always appears in areas where there is a high concentration of fibres. However, it is easy to find rich matrix areas within the transverse layer, that is, areas without fibres; a clear example of this is represented in Fig. E.26, where pores that are not connected to the damage under investigation produced by external actions, can be observed. The appearance of local damage in the form of debonding is less frequent in these areas. Even when damage in the form of debonding is observed in a rich matrix area, as shown in Fig. E.27, its effect on a second fibre is detected only in the close neighbourhood of the damaged fibre. Those fibres located far from the area where the damage first appears seem to be protected by the distance effect studied in Section E.4. a) b) Figure E.26.- Observation of a matrix rich area surrounded by high fiber density. Figure E.27.- Observations of damage in a matrix rich area: a) original micrograph, b) corresponding schemes of the micrograph. E.7.- Conclusions A study on the damage that first appears in a laminate has been carried out using the Boundary Element Method to model the damage configurations. As the weakest part of the laminate is typically the interface between a fibre and matrix, the first damage is assumed to take place in laminas with the fibres oriented at 90° with respect to the directions of the load. The damage considered is the debonding between the fibre and matrix, and Interfacial Fracture Mechanics are used as the appropriate tool to deal with the generation and propagation of the damage. First, a comparison has been made between different models concluding that either a plane stress or a plain strain model of a single fibre surrounded by a matrix is adequate to obtain representative results for the problems under consideration with a reasonable level of complexity in the model. The study of the problem of the presence of a damaged fibre (a fibre housing debonding at its matrix interface) has demonstrated that it may have a shielding or an accelerative effect on the undamaged fibres placed in the neighbourhood of the damaged fibre depending on the relative orientation between both fibres. In particular, if the total energy released by the onset and propagation of debonding in a) b) the undamaged fibre is greater than that associated with the initial appearance of a first debonding in a pristine material, an accelerative effect is foreseen. The configurations prone to produce this effect have been identified and present a clear physical meaning, as the appearance of the second debond is connected to the incipient generation of damage in the 90° layer associated with a transverse crack, normal to the direction of the load in a [0,90n]s laminate. The same problem was studied but with a varying distance between the damaged and undamaged fibres showing that no interaction appears for fibres separated approximately by a 4-fibre distance. Then, the effect was clearly reduced by the interfibre distance, and only interactions in the local neighbourhood of the fibre where the first debond appeared were expected. The problem of exploring the effect of the extension of the debonding length at the first damaged surrounding fibre in the previously obtained conclusions showed that a total debonding angle of 140° oriented symmetrically with respect to the loading direction produced the greatest Energy Release Rate values of the interface crack. This result gives representativeness to the conclusions obtained in the previous sections. The experimental observations support the predictions previously mentioned. Thus, no isolated debonding was observed. The presence of debonding was typically accompanied by additional debonding, which was always within the local neighbourhood and preferentially oriented in a position that tended to initially generate a full crack in the 90° lamina. This crack was perpendicular to the orientation of the load and gave rise to the generation of a macrocrack under Mode I in the mesoscopic view of the damage. As previously established, the objective of this paper is to improve the existing knowledge of the appearance of damage in the weakest layer of a laminate that oriented 90° with respect to the nominal external load, where the first manifestations of damage will appear. Isolated debonding may not be of concern from the point of view of a full composite structure. However, this initial damage may lead to the generation of a transverse crack that may cause associated delaminations between the plies, which is a much more serious damage form from a structural point view. As a starting point, the knowledge generated in this study will help to understand the delay in the appearance of translaminar damage in ultrathin plies, which will be the next objective of following publications by the authors. ACKNOWLEDGEMENTS.- This work has been supported by the Spanish Ministry of Education, Science and Sports (Project MAT2016-80879-P). The authors would also like to acknowledge the collaboration of Elena Arroyo in the experimental work.