Models of Representative Volume Elements (RVEs) of cross-ply laminates with di erent geometric con gurations and damage states are studied. Debond growth is characterized by the estimation of the Mode I and Mode II Energy Release Rate (ERR) using the Virtual Crack Closure Technique (VCCT). It is found that the presence of the 0=90interface and the thickness of the 0layer have no e ect, apart from laminates with ultra-thin 90plies where it is however modest. The present analysis support the claim that debond growth is not a ected by the ply-thickness e ect. Keywords: Polymer-matrix Composites (PMCs), Fibre/matrix bond, Debonding, Finite Element Analysis (FEA) 1. Introduction Since the development of the spread tow technology or \FUKUI method" [1], signi cant e orts have been directed toward the characterization of thin-ply laminates [2, 3, 4, 5, 6, 7, 8, 9, 10, 11] and their application to mission-critical structures in the aerospace sector [12]. 5 At the lamina level, the use of thin-plies leads to more regular and homoge- neous microstructures [6, 9]. Measurements of ply level properties (tensile and compressive modulus, Poisson's ratio, ultimate tensile strength,tensile onset of damage, interlaminar shear strength) on Uni-Directional (UD) specimens ([0 m] Preprint submitted to Composites Part A July 2, 2019 DRAFTand [90 m]) revealed no remarkable di erence with average properties available in 10 the literature for the same type of ber, nor showed any particular dependence on the ply thickness [9]. Only an increase of the ultimate compressive strength in the ber direction was observed with very thin plies ( 4 ber diameters), although with very scattered values. The authors claim the increase to be due to the ber arrangement's increased regularity which prevents the onset of ber 15 microbuckling [9]. A number of researchers [2, 3, 4] has reported improvements in fatigue life with the use of thin-plies , which are explained as a consequence of delayed propagation of free edge delaminations and intralaminar cracks. Several researchers have analyzed the e ect of thin-plies on damage development under static [3, 4, 5, 6, 7, 8, 9], fatigue [2, 3, 4, 5, 9] and impact loadings [3, 4, 5, 9]. It 20 seems apparent that thin-ply laminates possess an increased ability to delay, and in some cases even suppress, the onset and propagation of intralaminar cracks (called often transverse or matrix or micro-cracks). The rst stage in the appearance of transverse cracks is known to be the occur- rence of ber/matrix interface cracks (also referred to as debonds), which grow 25 along the ber arc direction, then kink out of the interface and coalesce forming a transverse crack [13]. Di erent approaches have been applied to model the initiation and growth of debonds. The Cohesive Zone Model (CZM) has been used to mimic the propagation of debonds along ber interfaces; coupled with a failure criterion for the matrix, it has provided simulations of the growth of 30 transverse cracks starting from a virgin material [14, 15, 16, 17]. The main advantages of this approach are the possibility to observe the development of a simulated crack path and to record a load-displacement curve to be com- pared with experimental measurements. However, various observations cast a doubt about the applicability of the CZM: the triaxiality of the matrix stress 35 state in the inter- ber region that is linked with a cavitation-like failure of the polymer [18]; the locality and mode dependency of the interface failure [19]; the problematic use at the microscopic level of properties measured in UD specimens at the laminate level [15]. A second approach that obviates these drawbacks is the application of Linear Elastic Fracture Mechanics (LEFM) arguments to the 40 2 DRAFTstudy of debond growth. The analysis focuses on the evaluation of Mode I and Mode II Energy Release Rate (ERR) at the crack tip by means of the Virtual Crack Closure Technique (VCCT) [20] or the J-Integral method [21]. The stress and strain elds, required for the ERR computation, can be solved by applica- tion of di erent methodologies such as analytical solutions [22], the Boundary 45 Element Method (BEM) [23] or the Finite Element Method (FEM) [24]. This approach presents nonetheless some limitations: it describes propagation of the debond and not its initiation; the role of friction in the contact zone is still an open issue; consensus is still lacking on a proper criterion for crack propagation in mixed mode. Finite fracture mechanics [25] is one way to address the initia- 50 tion problem. Di erent studies have followed the LEFM approach and analyzed models of one or two bers in an e ectively in nite matrix [26, 27, 28, 29, 30] and of an hexagonal cluster of bers in an e ectively in nite homogenized UD composite [31, 24]. The problem of debond growth along the ber-matrix in- terface in a cross-ply laminate has been only treated very recently in [32, 33], 55 where authors embed a single partially debonded ber in an e ectively in nite homogenized 90ply bounded by homogenized 0layers. Thus, the e ect of debond-debond interaction and of the relative proximity of a 0=90interface on debond ERR in cross-ply laminates is yet to be addressed. The present work is devoted to this problem. Models of Repeating Unit Cells (RUCs) are de- 60 veloped to represent laminates with di erent degrees of damage in the 90ply (here only in the form of debonds). The number of fully bonded bers across the thickness of the 90ply is varied in order to investigate the e ect of the proxim- ity of the 0=90interface. The thickness of the bounding 0layers is also used as a parameter of the study. The stress and strain elds are solved with the 65 Finite Element Method in Abaqus [34] and the debond (crack) is characterized by its Mode I and Mode II ERR calculated with the VCCT. 3 DRAFT2. RVE models & FE discretization 2.1. Introduction & nomenclature In the present work, we investigate debond development under in-plane lon- 70 gitudinal tension in [0 mk2L;90 k2L;0 mk2L] laminates. The interaction between debonds in the presence of an interface with a sti layer is studied with the use of di erent Repeating Unit Cells (RUCs) (see Figures 1 and 2 in Sec. 2.2), in which only the central ber is partially debonded. Repetition of the composite RUC occurs along the in-plane laminate 0-direction (corresponding to spec- 75 imen axial direction and RUC horizontal direction in Figures 1 and 2), thus representing a cross-ply laminate with a thin or even ultra-thin 90ply in the middle. All the RUCs present regular microstructures with bers placed according to a square-packing con guration characterized by the repetition of the same one- 80 ber unit cell of size 2 L2L, whereLis a function of the ber volume fraction Vfand the ber radius according to L=Rf 2r Vf: (1) Each ber in the model has the same radius Rf, equal to 1 m. This speci c value has no physical meaning per se and it has been selected for simplicity. It is useful to observe that, in a linear elastic solution as the one described in the 85 present article, the ERR is proportional to the geometrical dimensions of the model and thus re-evaluation of the ERR for bers of any size requires just a multiplication. Furthermore, it is worth to point out that Vfis the same in the one- ber unit and in the overall RUC, i.e. no clustering of bers is considered. The thickness of the 90ply depends on the number kof ber rows present 90 across the thickness (the vertical or zdirection in Figures 1 and 2) according to t90=k2L: (2) On the other hand, the thickness of 0layers can be assigned freely as a multiple of the 90ply thickness as 4 DRAFTt0=mt90 (3) wheremis an arbitrary integer. Thus, the thickness ratio mrepresents one additional parameter for the investigation. 95 In the following, let us consider in-plane coordinates xandy, and assume that the laminate 0-direction is aligned with the x-axis. In the presence of a load in thex-direction, the strain in the y-direction is small, due to the very small Poisson's ratio of the laminate. Debonds are present only in the 90layer and are considered to be signi cantly longer in the ber direction than in the arc di- 100 rection [35]. Therefore we use 2D models under the assumption of plane strain, de ned in the xzsection of the composite. The study presented in this paper thus applies to long debonds and its focus is on understanding the mechanisms of growth along their arc direction. The laminates are assumed to be subject to tensile strain, which is applied in the form of a constant displacement in the 105 x-direction along both vertical boundaries of the RUC as shown in Figure 3. We assume damage to be present only in the central \row" of bers of the 90 layer in the form of multiple debonds appearing at di erent regular intervals along the loading (horizontal) direction. The number of bers npresent in the horizontal direction of the RUC (Figures 1 and 2) controls the distance, 110 in terms of fully bonded bers, between consecutive debonds: if the RUC has n bers in the horizontal direction, two consecutive debonds are separated by n1 undamaged bers. The RUCs considered are thus Representative Volume Elements (RVEs) of cross-ply laminates with a certain distribution of debonds in the middle 90layer. 115 In summary, the models are di erentiated by: rst, the spacing between debonds along the horizontal direction in the 90layer, which corresponds to the number nof bers in the RUC's horizontal direction; second, the thickness of the middle 90ply measured in terms of the number kof ber rows in the vertical direction; third, the factor mwhich provides the thickness of 0layers as a multiple of 120 the 90ply thickness. It thus seems natural to introduce a common notation 5 DRAFTfor the RUCs as nkmt90. An additional family of RUCs is considered, in which: only one partially debonded ber is present; the 0layer is absent; di erent combinations of displacement boundary conditions are applied to the upper surface. The application of cou- 125 pling of horizontal displacements uxalong the right and left sides allows for repetition along the horizontal direction. When the upper boundary of the RUC is left free, we de ne the 1 1free model. If coupling of the vertical displacements uzis applied to the upper boundary (coupling condition), we de- ne instead the 1 1coupling model. In the case a linear distribution of the 130 horizontal displacement uxis applied to the upper boundary (H-condition), the model is refered to as 1 1H. Finally, when the linear distribution of the hor- izontal displacement uxis superimposed to the condition of coupling of the ver- tical displacements uzon the upper boundary, we have the 1 1coupling +H. Further details about this family of RUCs and the corresponding laminate RVE 135 can be found in [36]. 2.2. Description of modelled Representative Volume Elements (RVEs) The rst family of Representative Volume Elements (RVEs) is represented in Figure 1. It represents a set of [0 m12L;90 12L;0 m12L] laminates with an ultra-thin 90layer, constituted by a single row of bers across the thickness. 140 Debonds appear at regular intervals measured in terms of number n1 of fully bonded bers present between them, which in turn correspond to the number of bers along the horizontal direction of the RVE as highlighted in Fig. 1. They are thus the n1mt90models, where m= 1;10 andnis an integer 1 (n= 1 corresponds to the case of a debond appearing on all the bers in 145 the central 90layer). These models are geometrically extreme, but allow to focus on the interaction between debonds and the inter-ply 0=90interface. Furthermore, the spread tow technology is today capable of producing cross-ply laminates with the central 90layer thickness only 4 5 times the ber diameter, as shown for example in [6], which may in future give practical relevance even 150 to such extreme case. 6 DRAFTThe second set of models considers instead cross-ply laminates with a central 90ply of variable thickness, measured in terms of number kof ber rows \stacked" in the vertical direction in Figure 2. Once again, debonds appear in the central row only at regular intervals measured in terms of number n1 155 of fully bonded bers present between them, as highlighted in Fig. 2. These models are thus the nkmt90models, where m= 1;10,k > 1 andn is an integer 1 (n= 1 corresponds to the case of a debond appearing on all bers of the central ber row in the 90layer). By increasing the number n of bers in the horizontal direction in the RUC, decreasing levels of damage 160 (debonds spaced further apart and the interaction between debonds becomes less important) are considered to be present in the laminate. By increasing the numberkof ber rows, the thickness of the 90layer is increased and the e ect of the relative proximity of the inter-ply 0=90interface can thus be studied. Finally, by increasing the factor m, the thickness of the 0layers is increased for 165 a given thickness of the 90, which allows the investigation of the 0ply-block e ect [37]. 2.3. Finite Element (FE) discretization The RUCs are discretized and solved with the Finite Element Method (FEM) using the commercial FEM package Abaqus [34]. The total length and height of 170 a RUC are determined by the number of bers nin the horizontal direction, the number of ber rows kacross the thickness and the thickness ratio m(see Sec. 2.1 and Sec. 2.2). The debond appears symmetrically with respect to the xaxis (see Fig. 3) and we characterize it with the angular size  (the full debond size is thus 2). In the case of large debond sizes ( 6080), a region of size  175 to be determined by the solution itself appears at the crack tip. In this region, called the contact zone , the crack faces are in contact and slide on each other. Due to existence of the contact zone, frictionless contact is considered between the two crack faces to avoid interpenetration and allow free sliding. Symmetry with respect to the xaxis is applied on the lower boundary. Kinematic coupling 180 on thex-displacement is applied along the left and right boundaries of the model 7 DRAFTin the form of a constant x-displacement "xnL, corresponding to laminate x- strain "xequal to 1%. The FEM model is discretized using second order, 2D, plane strain triangular (CPE6) and rectangular (CPE8) elements. In the crack tip neighborhood, a 185 re ned regular mesh of quadrilateral elements with almost unitary aspect ratio is needed to ensure a correct evaluation of the ERR. The angular size of an element in this re ned region close to the crack tip is by design equal to 0 :05. The crack faces are modeled as element-based surfaces with a frictionless small- sliding contact pair interaction. The Mode I, Mode II and total Energy Release 190 Rates (ERRs) (respectively GI,GIIandGTOT) represent the main result of the numerical analysis. They are computed using the VCCT [20] implemented in a custom Python routine. Glass ber and epoxy are considered throughout this article, and it is assumed that their response always lies in the linear elastic domain. The e ective UD properties are computed using Hashin's Concentric 195 Cylinder Assembly model [38] with the self-consistency scheme for the out-of- plane shear modulus of Christensen [39]. The properties used are listed in Table 1. The model was validated with respect to BEM results of [40, 29]; considerations about the order of accuracy can be found in [36]. 3. Results & Discussion 200 3.1. E ect of the proximity of the 0=90interface and of the thickness of the 0layer on debond ERR for highly interactive debonds We rst focus our attention on the model 1 1mt90, which represents a particular case of the family n1mt90. It corresponds to a cross- ply laminate in which the central 90ply is constituted by only one ber row, 205 in which each ber possesses a debond appearing on alternating sides. The model thus represents an extreme idealization, in the sense that: rst, the central 90layer is the thinniest that can be conceived, which allows us to investigate the direct e ect of the proximity of the 0=90interface on debond ERR; second, a very particular damage state is present for which every ber 210 8 DRAFTis partially debonded from the sorrounding matrix, corresponding to the most severe damage state that can occur in the 90ply when considering debonds as the only mechanism of damage. We are thus focusing on the presence of the 0=90interface and on the thickness of the 0layer, by considering the ratio m=t0 t90of ply thicknessess as a free parameter. 215 In Figures 4 and 5 it is possible to observe respectively the Mode I and Mode II ERR for models 1 1mt90withm= 1;10;100. Mode I ERR is practically una ected by the 0layer thickness, only a marginal increase 1% can be seen whenmis increased from 1 to 10. No further observable change is present when mis increased to 100. Moreover, the contact zone onset, which corresponds to 220 the rst value of  such thatGI= 0, is always equal to 70irrespective of the value of m. A more remarkable, albeit small, e ect of the 0layer thickness can be observed for Mode II when mis increased from 1 to values 10. For open cracks, i.e. when no contact zone is present and thus  is smaller than 70, increasing the 0layer thickness causes a reduction of Mode II ERR; while 225 for closed cracks, when a contact zone is present and  >70, the increase in thickness leads to an increase in ERR. In order to understand the interaction mechanism between the 0=90interface and the debond, Mode I and Mode II ERR are reported respectively in Figures 4 and 5 for models 1 1free, 11H, 11coupling and 11coupling +H. 230 These RUCs all present equivalent boundary conditions and it is here useful to recall their characteristics: in model 1 1free the upper bounday is left free; coupling conditions on the vertical displacements uzare applied to the upper boundary in model 1 1coupling (coupling condition); in model 11Ha linearly distributed horizontal displacement uxis applied to the upper 235 boundary (H-condition); in model 1 1coupling +Hcoupling conditions on the vertical displacements uzand a linearly distributed horizontal displacement uxare imposed together on the upper boundary. Given that the presence of a 0layer provides two constraints: rst, it tends to keep the 90layer boundary straight; second, it forces a more homogeneous horizontal displacement at the 240 90layer boundary; the equivalent boundary conditions of 1 1coupling , 9 DRAFT11Hand 11coupling +Hrepresent an extreme case respectively of the rst constraint (1 1coupling ), the second constraint (1 1H) and the two together (1 1coupling +H). The case 1 1free constitutes instead the base case (absence of 0layer), on which comparisons are built. 245 Observing Figure 4, it is possible to notice that the values of GIfor the 1  1free and the 1 1coupling models represent respectively a lower and an upper bound for the 1 1mt90RVEs: this is true with respect to the value of GIas well as of contact zone onset (60for 11free, 70for 11mt90, 80for 11coupling ). When the H-condition is added to 250 the 11free and the 1 1coupling models, thus obtaining the 1 1H and 11coupling +Hmodels,GIdecreases while the value of  at contact zone onset remains unchanged (60for 11free and 1 1H, 80for 11coupling and 11coupling +H). Moreover, it is possible to observe that the values of GIof 11coupling +Hare much closer to but always greater 255 than those of 1 1mt90RVEs, thus constituting a more representative upper bound for the latter. Analogous considerations are drawn with regard to Mode II (see Fig. 5). For small debonds,  30, no signi cant di erence in GIIcan be seen between 11free and 11Hand between 1 1coupling and 11coupling +H 260 in this region. With respect to 1 1mt90RVEs, the rst pair (1 1free and 11H) represents the lower bound while the second pair (1 1coupling and 11coupling +H) the upper bound. For 30<60, 11Hand 11coupling +Hprovide signi cantly lower values of GIIthan respectively 11free and 1 1coupling .GIIvalues of 1 1Hare very close to 265 111t90, even coincident for  = 60. On the other hand, GIIvalues of 11coupling are very close to 1 1mt90withm10 and even coincident for = 50. For 60<110, the situation changes. 1 1free and 11coupling provides values of GIIclose to each other, even coincident for = 70. Values of GIIof 11Hand 11coupling +Hare signi cantly 270 larger than both 1 1free and 11coupling . Furthermore, GIIvalues of 11Hcoincide with those of 1 1mt90withm10. Mode II ERR of 10 DRAFT111t90is instead close, but not coincident, to that of 1 1coupling . For  >110,GIIis the same for all models and reaches 0 at a debond size of around 130. 275 These results help to understand the e ect of the 0=90interface on debond ERR. Two constraining mechanisms are present in the case of 0=90interface that are absent in the free surface case: rst, the boundary of the 90layer remains straighter (e ect modelled by the coupling condition in 1 1coupling ); second, the x-strain on the 90layer boundary is more uniform (e ect modelled 280 by the H-condition in 1 1H). For small debonds ( <6070), the presence of the 0=90interface causes an increase of GIand a decrease of GIIwith respect to the free surface case. For Mode I, the fact that the 90layer boundary remains straight (coupling condition) forces the debond to open more than in the free case, thus increasing 285 GI. However, the uniformity of the x-strain on the 90layer boundary reduces the local (in the debond neighborhood) x-strain magni cation and contains the increase in GI. This corresponds in Figure 4 to the fact that Mode I ERR for 11mt90is always higher than 1 1free but lower than 1 1coupling , and it is best approximated by 1 1coupling +H. For Mode II in the case of 290 small debonds, the presence of the 0layer keeps the 0=90interface straighter and reduces the vertical contraction of the matrix, which contributes for the most part to Mode II in this range, thus leading to a decrease of GII. The small e ect of 0layer thickness on Mode II (Fig. 5) can be explained in terms of local bending sti ness: a thinner 0layer (t0 t90= 1) does not keep the 90layer 295 boundary as straight as thicker 0layers (t0 t9010). In the caset0 t90= 1, the 90layer boundary deforms in a way that is similar to the free surface case, but smaller in magnitude. This corresponds to the fact that for  <6070, in Figure 5: 1 11t90is best approximated by 1 1H(curved 90layer boundary but uniform x-strain at the 90layer boundary that disfavors GII), 300 11mt90;m10 is best approximated by 1 1coupling (straight 90 layer boundary). For debonds larger than 70, the presence of the 0=90interface causes an 11 DRAFTincrease of GIIwith respect to the free surface case. The uniform x-strain distribution on the 90layer boundary determined by the presence of the 0305 layer causes, with respect to the free case, the matrix x-strain to be higher in thex0 neighborhood and lower around x L, in order to keep the average "xat 1%. Given that for large debonds Mode II ERR is determined mostly by the magnitude of the x-strain gap (between the matrix x-strain and the ber x-strain), an increase of GIIis thus observed in the presence of the 0=90310 interface. Again, the observed e ect of the 0layer thickness on Mode II for  > 6070(Fig. 5) can be discussed in terms of local 0layer bending sti ness. In the free case, it is the curvature of the material around the ber that causes the x-strain reduction and thus a lower GII. Thicker 0layers (t0 t9010) prevent this 90boundary deformation to a greater extent than 315 the thinner t0=t90case: thex-strain (and thus GII) increase is greater for t0 t9010 thant0 t90= 1. 3.2. E ect of the proximity of the 0=90interface and of the thickness of the 0layer on non-interactive debonds in a one- ber row 90ply We turn now our attention to models n1mt90, which correspond 320 to a cross-ply laminate in which the central 90ply is constituted by only one ber row where multiple partially debonded bers are present with n1 fully bonded bers between them and debonds appear on alternating sides of consec- utive damaged bers (see Figure 1). As observed in a previous work [36], the presence of fully bonded bers between partially debonded ones in the loading 325 direction has a strong e ect on debond ERR and controls the interaction be- tween debonds. When nis increased, both Mode I and Mode II increase: the addition of sti er elements, in the form of fully bonded bers, increase the strain applied to the damaged unit and thus causes higher values of ERR. Looked from this perspective, i.e. moving from the most to the least severe state of damage, 330 this e ect is referred to as \strain magni cation" [36]. There seems to exist a characteristic distance, measured in terms of fully bonded bers, above which a change in the number of undamaged bers a ects only marginally, or even not at 12 DRAFTall, debond ERR. This distance, generally n20, marks the transition between a non-interactive solution ( n>20) and an interactive one ( n<20). The \strain 335 magni cation" e ect thus represents the transition from the interactive to the non-interactive solution. If in Sec. 3.1 we studied the e ect of the proximity of the 0=90interface and of the thickness of the 0layer on interactive debonds (11:::), we analyze in the present section the e ect of the 0=90interface and of the 0layer thickness on non-interactive ones ( n1:::withn>20). 340 Comparing Fig. 6 with Fig. 4 and Fig. 7 with Fig. 5, it is possible to observe how, as previously described, increasing the number of fully bonded bers be- tween consecutive debonds in the loading direction leads to an increase in Mode I and Mode II ERR. The peak GIincreases from 1 :93J m2 in 111t90to 3:42J m2 in 2111t90, while the peak GIIfrom 0:86J m2 to 3:04J m2 . 345 The value of  at contact zone onset remains however the same (70). The e ect of the 0layer thickness is instead non-existent: values of both GI andGIIare coincident for 21 11t90and 21 110t90. In agreement with the introductory considerations of this section and the re- sults in [36], it is possible to observe in Figures 6 and 7 that 21 1free and 350 211coupling (in which the horizontal displacement uxis left uncostrained on the upper boundary) show both the highest values of Mode I and Mode II ERR as well as the maximum increase with respect to the interactive case (1 1free and 1 1coupling ). When the H-condition is applied to the upper bound- ary, thus constraining the magnitude of the strain magni cation e ect, both the 355 magnitude of Mode I and Mode II ERR as well as their relative increase with respect to the interactive case are signi cantly reduced. 21 1coupling +H represents, when considering both Mode I and Mode II ERR, the best approxi- mation to the results of 21 1mt90. The mechanisms at play are the same as in Sec. 3.1: by keeping the 0=90interface straight (coupling condition), the 360 0layer favors an increase in GIand decrease in GIIfor small debonds and an increase in GIIfor large debonds; by applying a uniform x-strain on the 90 layer boundary (H-condition), the 0layer promotes a more uniform x-strain in the 90layer and acts against the strain magni cation e ect, reducing debond 13 DRAFTERR. Results in Fig. 6 and Fig. 7 show that the latter e ect (H-condition) is 365 dominant. It seems reasonable to conclude that debond growth is favored (i.e. debond ERR is higher) in the presence of strain or stress concentrations (as for example in the presence of a free surface or only coupling conditions on the ver- tical displacement), while more uniform strain and stress elds as those created by the proximity of the 0=90interface reduce both Mode I and Mode II ERR 370 and thus tend to prevent debond growth. 3.3. E ect of the presence of ber rows with no damage on the debond- 0=90 interface interaction After having investigated the e ect of the proximity of the 0=90interface and of the thickness of the 0layer on debond ERR for di erent cases of debond- 375 debond interaction in the same ber row, we address in this section the e ect of the presence of ber rows with only fully bonded bers between debonds and the 0=90interface. In other words, we are separating the debond from the 0=90interface by inserting rows of fully bonded bers in between. We consider only the case m= 1, i.e.t0=t90, given that increasing the 0380 layer thickness does not result in any remarkable e ect on ERR as shown in Sec. 3.1 and Sec. 3.2. Following the same philosophy of Sec. 3.1 and Sec. 3.2, we analyze the e ect of the presence of ber rows with no damage on debond ERR: rst, when the central ber row possesses only partially debonded bers, which represents the most severe damage state for these RUCs and the solution 385 for interactive debonds (models 1 k1t90in Figures 8 and 9); second, the case of debonds separated by n1 fully bonded bers in the central ber row, which corresponds to the least severe state of damage and to the solution for non-interactive debonds (models 21 k1t90in Figures 10 and 11). Observation of Fig. 8, Fig. 9, Fig. 10 and Fig. 11 reveals that no di erence can 390 be seen in Mode I and Mode II ERR by increasing the number kof rows with undamaged bers when k3, which means that debond ERR does not change once at least 1 row of undamaged bers is present between the debond and the 0=90interface. A signi cant change is visible only when k= 1, which 14 DRAFTmeans that no row of undamaged bers is present between the debond and the 395 0=90interface. This change, from k3 tok= 1, corresponds in particular to a reduction of both GIandGII. The results of Figures 8, 9, 10 and 11 imply that the mechanisms of debond-0=90interface interaction described in Sec. 3.1 and Sec. 3.2 are actually very localized and that debond ERR is a ected by the presence of the 0=90interface only when no fully bonded ber is placed 400 in between. Given that the number kof bers in the RUC vertical direction corresponds to the thickness of the 90ply measured in terms of number of ber rows present through its thickness, the results presented here point to another conclusion: the ply-thickness e ect does not seem to apply to debond growth, unless an ultra-thin ply constituted by only one ber row ( k= 1) 405 is considered. Analogous results can be found in [32, 33], where the authors investigate the ply-thickness e ect on debond growth in cross-ply laminates using: rst, a single centrally-placed partially debonded ber with surrounding matrix corresponding to Vf= 55%, embedded from all sides in a homogenized 90ply bounded by homogenized 0layers; second, one partially debonded ber 410 placed in the center and a second partially debonded ber placed at an angle 2 with respect to the horizontal direction with surrounding matrix corresponding toVf= 55%, embedded from all sides in a homogenized 90ply bounded by homogenized 0layers. The thickness of the 0layer is chosen as reference and a  0 p;90 rp Slaminate is considered. Carbon-epoxy and glass-epoxy systems are 415 both studied. The thickness of the 90ply,t90=rt0, varies from r= 3 (thick 90ply,>100 ber diameters) to r= 0:1 (thin 90ply,45 ber diameters). No measurable ply-thickness e ect was observed. Experimental support to the claim that the ply-thickness e ect has no in uence on debond growth can be also found in the literature, in [6]. The authors conducted in-situ observations of 420 edge micro-cracks with an optical microscope on [0 2;90 n;0 2] carbon ber-epoxy laminates with n= 1;2;4, corresponding to a 90ply thickness of respectively 40 [m] (68 ber diameters), 80 [ m] (1216 ber diameters) and 160 [m] (2432 ber diameters). For n= 1, i.e. the case of a very thin 90 ply, isolated debonds appear at a lower value of the applied strain than in thicker 425 15 DRAFTplies (at 0:4% vs 0:7%) while growth and coalescence of debonds is suppressed and no transverse crack can be observed even at a strain of 1 :5%. The ply- thickness e ect was thus observed in [6] for transverse cracks, i.e. coalescence of debonds was delayed to higher strains and even suppressed, but not for debond growth. The analysis presented in this article brings new arguments to the claim 430 that the ply-thickness e ect does not in uence the growth of debonds. 4. Conclusions Di erent models of Repeating Unit Cell, representing di erent cross-ply lam- inates, have been studied in order to investigate the e ect of the presence of the 0layer and of its thickness on debond Energy Release Rate for interactive and 435 non-interactive debonds. A particular damage state is studied, in which only the central row of bers of the 90ply possesses debonds. The thickness of the 90ply is measured in terms of the number of ber rows in the layer; the 0layer is on the other hand modelled as a homogenized material, the thick- ness of which is a multiple of the 90ply thickness. In order to investigate 440 the mechanisms of the debond-0=90interface interaction, Mode I and Mode II ERR of cross-ply RUCs are compared with those of RUCs with equivalent boundary conditions on the upper boundary: free surface; coupling conditions on the vertical displacements; an applied linear distribution of the horizontal displacement; coupling conditions on the vertical displacements superimposed 445 to an applied linear distribution of the horizontal displacement (this last com- bination represents the most extreme e ect of the 0layer on debond growth). It has been found that: { by forcing the 0=90interface to remain approximately straight and con- trolling the uniformity of the horizontal displacements in the composite 450 (and thus in the 90ply), the presence of the 0layer causes more ho- mogeneous local (i.e. in the debond neighborhood) strains, reducing the ERR at the debond crack tip; 16 DRAFT{ when increasing the thickness of the 0layer, the e ect of the presence of the 0layer on debond ERR remains the same as in the case t0=t90; 455 { no e ect of the 90layer thickness, measured in terms of number of ber rows, is observed; a reduction in ERR takes place only when the thickness is reduced to only one ber row. The results reported in this article strengthen the claim that the ply-thickness e ect does not in uence the growth of individual debonds, as previously sug- 460 gested in the literature [6, 17, 32, 33]. Acknowledgements Luca Di Stasio gratefully acknowledges the support of the European School of Materials (EUSMAT) through the DocMASE Doctoral Programme and the European Commission through the Erasmus Mundus Programme. 465