The e ects of crack shielding, nite thickness of the composite and ber content on ber/matrix debond growth in thin unidirectional composites are investi- gated analyzing Representative Volume Elements (RVEs) of di erent ordered microstructures. Debond growth is characterized by estimation of the Energy Release Rates (ERRs) in Mode I and Mode II using the Virtual Crack Closure Technique (VCCT) and the J-integral. It is found that increasing ber content, a larger distance between debonds in the loading direction and the presence of a free surface close to the debond have all a strong enhancing e ect on the ERR. The presence of fully bonded bers in the composite thickness direction has in- stead a constraining e ect, and it is shown to be very localized. An explanation of these observations is proposed based on mechanical considerations. Keywords: Polymer-matrix Composites (PMCs), Thin-ply, Energy Release Rate, Debonding, Finite Element Analysis (FEA) 1. Introduction Stimulated by the ever more stringent requirements in terms of weight and mechanical performances of the aerospace industry, in recent years the com- posite community has returned its attention to the mechanisms of intralam- inar crack initiation with a focus on thin-ply laminates. Alternative design 5 1Corresponding author: luca.di.stasio@ltu.se . Preprint submitted to Theoretical and Applied Fracture Mechanics July 12, 2019 DRAFTapproaches are now considered based on this non-conventional laminate in ap- plications ranging from cryogenic pressure vessels [1], to airplanes' wings [2], and even reusable space launchers [3]. Thin-ply laminates are the result of a technological innovation, the spread tow technology , which consists in opening or spreading the tows, in which bers 10 (carbon, glass, aramid, basalt among others) are usually shipped in, into very thin tapes used for laminate production. Ply thicknesses of less than 50 m can nowadays be mass-produced, and record thicknesses of around 20 25m, or45 times the average ber's diameter, have been achieved. In its cur- rent form the technique, sometimes referred to as \FUKUI method", was rstly 15 proposed towards the end of the 1990s [4] and perfected in the subsequent decade [5, 6]. Several experimental investigations on thin ply laminates have highlighted their main properties [7, 8, 9, 10, 11, 12, 13]: increased ber content; more uniform packing of bers; delay and even suppression of intralaminar cracking (called 20 also transverse-, matrix- or micro-cracking) and delamination. A very insight- ful work documenting how these phenomena are a ected by the morphology of thin-ply laminates is the microscopic study of Saito & al. [14], which focuses on the e ect of ply thickness on the onset and propagation of intralaminar crack- ing. In their investigation, tensile tests were performed on carbon ber/epoxy 25 [02;90n;02]thin-ply laminates for n= 1;2;4 and the crack density was mea- sured with a digital microscope at several levels of applied tensile strain in the range between 0% and 1 :5%. Furthermore, they performed microscopic obser- vations on the specimen's edge at each level of strain. They observed the onset of ber/matrix interface cracks (referred to as debonds in the following) at lower 30 levels of strain in thinner plies, while at the same time coalescence of debonds and through-the-thickness propagation of transverse cracks in thin plies were delayed and even suppressed as ply thickness decreased. In particular, they reported the rst onset of debonds at 0 :4% forn= 1;2 and 0:7% forn= 4. For n= 1, however, at "= 1:5% coalescence of debonds had started to take place 35 but the crack had not completely propagated through the thickness, while for 2 DRAFTn= 2 andn= 4 the latter alreay happened at a value of strain respectively of 1:3% and 1%. Our inability to explain these observations with the currently accumulated knowledge demonstrates the necessity of further investigation of interactions between debonds and studies of the constraining (or accelerating) 40 e ect of presence of bonded bers, free and constrained boundaries in the vicin- ity of a partially debonded ber. Early studies on the e ect of ply thickness on the onset and propagation of transverse cracks were conducted on glass ber/epoxy cross-ply laminates by Bailey, Parvizi and collaborators [15, 16, 17], who rstly observed the bene cial 45 e ect of thickness reduction on the delay of transverse cracking. They further- more pointed the attention to the appearance of debonds at the ber/matrix interface and their subsequent coalescence as the mechanism at the origin of transverse cracks [18]. Moreover, they identi ed the main mechanical driver of the damage process in the mismatch of elastic properties, and particularly of 50 Poisson's ratios, between bers and matrix [19]. A full understanding of dam- age onset and propagation in thin-ply laminates thus requires comprehension of the mechanisms governing its very rst stage, i.e. the ber/matrix interface crack. First results were obtained through analytical models in the case of a single ber with an arc crack (debond) in an in nite matrix under transverse 55 tension by England [20] and Perlman & Sih [21], who obtained the stresses at the interface and calculated the stress intensity factors at the crack tip, and by Toya [22], who evaluated the Energy Release Rate (ERR). Drawing upon the results for the straight bi-material interface crack by Comninou [23], the ef- fect of crack face contact in ber-matrix debonding was investigated in [24, 25]. 60 In [26], it was showed in terms of ERR why the case of a single asymmetric debond is more likely to be observed under remote transverse tension than two symmetric debonds on the same ber. The e ect of di erent types and com- binations of loads on debonding have been studied for the single ber model: compression [27], residual thermal stresses [28], and biaxial con gurations with 65 di erent combinations of tension and compression [29, 30]. The e ect of the presence of nearby bonded bers on the debonding of a ber embedded in an 3 DRAFTin nite matrix has been studied under uniaxial transverse tension [31], biaxial tension [32] and uniaxial transverse compression [33]. The e ect of inter- ber distance on debond growth has been studied for a partially debonded ber at 70 the center of a hexagonal cluster inside a homogenized UD composite in the case of fully bonded neighbouring bers [34] and of two partially debonded bers out of the surrounding six [35]. An understanding of crack shielding and nite ply thickness e ects on debond growth in non-homogenized microstructural models of UDs seems thus to be lacking: this is the problem that we want to address 75 in the present work. Mode I and Mode II energy release rates will be analyzed using stress elds calculated with the FEM for a variety of Repeating Unit Cell (RUC) of the composite with square packing of bers under transverse tensile loading. These RUCs represent composites with di erent distances between partially debonded bers and a varying number of bonded bers between them, 80 which allows to study the e ect of crack shielding on the ERR. In the ply thick- ness direction, the varying number of perfectly bonded ber rows exposes the e ect of the proximity of the free boundary of the composite on debond growth. Finally, using coupling of thickness direction displacements on horizontal bound- aries of the RUC, the accelerating e ect of the interaction between debonds of 85 bers located on the same vertical line is studied. 2. RVE models & FE discretization 2.1. Introduction & Nomenclature In this paper, we analyze debond development in unidirectional (UD) com- posites subjected to in-plane transverse tensile loading. The interaction between 90 debonds in UD composites is studied developing models of di erent Repeating Unit Cells (RUC) of laminates (see Fig. 1 to Fig. 3) where only the central ber in the cell has a damage in the form of a ber/matrix interface crack (debond). The composite RUC may be repeating in the in-plane transverse direction only (representing an ultra-thin composite) or repeating also in the 95 composite thickness direction, representing an in nite composite in a limiting 4 DRAFTcase. Thus, the conditions at the UD composite's upper and lower bound- aries are one of the parameters for the investigation. The used RUCs allow for considering the composite with debonds as a sequence of stacked damaged and undamaged ber rows, each row with only one ber in the thickness direc- 100 tion. Since all of these RUCs feature regular microstructures with bers placed according to a square-packing con guration, they are Representative Volume Elements (RVE) of composites with a certain distribution of debonds. Intro- ducing in-plane coordinates xandy, wherexis in the transverse direction of the UD composite under consideration, the strain in the y-direction due to a 105 load in the x-direction is small, caused in turn by the very small minor Pois- son's ratio of the UD composite. Additionally, debonds are considered to be signi cantly longer in the ber direction than in the arc direction. Therefore, we use 2D models under the assumption of plane strain, de ned in the xz section of the composite. Thus, the analysis presented applies to long debonds, 110 with a focus on understanding the mechanisms of growth along their arc di- rection. The composites are subjected to transverse tensile strain, applied as a constant displacement in the x-direction along the vertical boundary of the RUC as shown in Figures 1 to 4. As the models are di erentiated by the num- ber of rows of bers and by the spacing between debonds along the vertical and 115 horizontal directions, the corresponding RUCs can be distinguished from each other based on the number nof bers in the horizontal direction and kin the vertical direction. Furthermore, the horizontal surfaces can be either free or vertical displacement coupling can be applied. We thus introduce a common notationnkfree andnkcoupling to denote a RUC with nk bers 120 and, respectively, a free upper surface or with kinematic coupling applied to it. The speci c combinations of particular choices of n,k, and boundary conditions are detailed in Section 2.2, together with the description of the corresponding models of damaged composite they are representing. 5 DRAFT2.2. Models of Representative Volume Element (RVE) 125 The rst two models feature, as shown in Fig. 1, an ultra-thin UD lami- nate with only one row of bers across its thickness, k= 1. This is quite an extreme model from the microstructural point of view; however, it allows to focus the analysis on the interaction between debonded bers placed along the x-direction. Furthermore, as the horizontal surfaces are considered free, the in- 130 teraction is stronger in this case than in any other, making the trends very clear and the predictions of this model rather conservative. In retrospective, if only 20 years ago such a model would have been considered too abstracted from the physical reality, the recent advancements in the spread tow technology make this approach appealing also as a limiting case for practical considerations. 135 xz n=3k=1 (a) Single row of bers with a debond appearing every n bers: model n1free (n= 3 in the gure). xz n=1k=1 (b) Single row of bers with debonds appearing on each ber: model 1 1free . Figure 1: Models of ultra-thin UD composites with a single \row" of bers and debonds repeating at di erent distances. In the sub-model of Fig. 1a, every nth ber in the composite is partially debonded on alternating sides of the ber. The symmetries of the model allow the use of the upper part of the RUC, as highlighted in Fig. 1 to 3. Following the notation introduced in Section 2.1, we will refer to this model as n1free. In the sub-model n= 1, Fig. 1b, a debond appears on each ber on alternating 140 6 DRAFTsides and the corresponding RUC contains only one ber. We will refer to this model as 1 1free. xz n=1k=3 (a) Multiple rows of bers with debonds appearing on each ber beloging to the central row: model 1kfree (k= 3 in the gure). xz n=3k=3 (b) Mutiple rows of bers with a debond appearing every n bers within the central row: model nkfree (n= 3 and k= 3 in the gure). Figure 2: Models of UD composites with di erent \rows" of bers and debonds repeating at di erent distances. The second set of models in Fig. 2 and Fig. 3 considers laminates with multiple rows of bers across the thickness: a nite number of rows in the rst two sub-models in Fig. 2; an in nite number in the model of Fig. 3. In Fig. 2a, 145 7 DRAFTthe RUC contains n= 1 ber in the x-direction, k bers across the thickness and the central ber is debonded. This model will be referred to in the following as 1kfree. Thinking in terms of rows, in this model we have a central row where each ber is debonded. This row is surrounded from each side by (k1)=2 rows with perfectly bonded bers. In the sub-model in Fig. 2b, each nth ber 150 in the central row is debonded and this row is surrounded by (k1)=2rows of undamaged bers from each side. We will refer to this model as nkfree (because the horizontal boundary of the RUC is free of any constraint). xz n=1k=1 Figure 3: Model of UD composites with an in nite number of \rows" of bers and debonds appearing on each ber: model 1 1coupling . Finally, the model in Fig. 3 represents an UD composite with an in nite number of rows; all of them with partially debonded bers. As all bers have 155 debonds, the corresponding RUC is made of a single partially debonded ber with kinematic coupling conditions applied to the upper boundary to assure periodicity. This model is referred to as 1 1coupling . 8 DRAFT2.3. Finite Element (FE) discretization Each RUC is discretized using the Finite Element Method (FEM) within 160 the Abaqus environment, a commercial FEM package [36]. The length land heighthof the model are determined by the number of bers nin the horizontal direction and kacross the thickness (see 2.2) according to Eq. 1: l= 2nL h =kL; (1) where 2Lis the length of a one- ber unit, see Fig. 4, de ned as a function of the ber volume fraction Vfand the ber radius according to 165 L=Rf 2r Vf: (2) The ber radius Rfis assumed to be the same for each ber in the model and equal to 1 m. The latter value is not physical and it has been chosen for simplicity. It is worth to note at this point that, in a linear elastic solution as the one presented here, the ERR is proportional to the geometrical dimensions and recalculation of the ERR for bers of any size, thus, requires a simple 170 multiplication. Furthermore, notice that the relationships in Eqs. 1 and 2 ensure that the local and global Vfare everywhere equal. The debond is placed symmetrically with respect to the xaxis (see Fig. 4) and we characterize it with an angular size of  (the full debond size is thus 2). For large debond sizes ( 6080), a region of variable size  appears 175 at the crack tip in which the crack faces are in contact and slide on each other. Due to its appearance, frictionless contact is considered between the two crack faces to allow free sliding and avoid interpenetration. Symmetry with respect to thexaxis is applied on the lower boundary. The upper boundary is in general free, except for the model 1 1coupling (Fig. 3) which requires kinematic 180 coupling of vertical displacements also on the upper side. Kinematic coupling on thex-displacement is applied along the left and right sides of the model in the form of a constant x-displacement "xl, corresponding to transverse strain "xequal to 1%. 9 DRAFT xz ∆θL L L¯ux=¯εxl ¯ux=−¯εxl ΩfΩm Rf O∆ΦFigure 4: Schematic of the model with its main parameters. Table 1: Summary of the mechanical properties of ber and matrix. Estands for Young's modulus,for shear modulus and for Poisson's ratio. Material E[GPa ][GPa ][] Glass ber 70.0 29.2 0.2 Epoxy 3.5 1.25 0.4 The model is meshed using second order, 2D, plane strain triangular (CPE6) 185 and rectangular (CPE8) elements. A regular mesh of quadrilateral elements with an almost unitary aspect ratio is required at the crack tip. The angular size of an element in the crack tip region is always equal to 0 :05. The crack faces are modeled as element-based surfaces and a small-sliding contact pair interaction with no friction is established between them. The Mode I, Mode II and total 190 Energy Release Rates (ERRs) (respectively referred to as GI,GIIandGTOT) represent the main output of the FEM analysis; they are evaluated using the VCCT [37] implemented in a custom Python routine and, for the total ERR, the J-integral [38] is obtained by application of the Abaqus built-in functionality. A glass ber-epoxy system is considered throughout this paper, and it is assumed 195 that their response lies always in the linear elastic domain. The properties used 10 DRAFTare listed in Table 1. 2.4. Validation of the model The model is validated in Fig. 5 against the results reported in [39, 31], obtained with the Boundary Element Method (BEM) for a single ber with a 200 symmetric debond placed in an in nite matrix. This situation is modeled using the 11free RVE withVf= 0:0079%, which corresponds to a RUC's length and height of respectively 200Rfand100Rf. 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 15005·10−20.10.150.20.250.30.350.40.450.50.550.60.650.70.75 ∆θ/bracketleftBigg◦/bracketrightBiggG/G0[−]GI, BEM GII, BEM GTOT, BEM GI, FEM GII, FEM GTOT, FEM Figure 5: Validation of the single ber model for the in nite matrix case with respect to the BEM solution in [31]. To allow for a comparison, the results are normalized following [31] with respect to a reference Energy Release Rate G0de ned as 205 G0=1 +km 8m2 0Rf (3) 11 DRAFTwhereis the shear modulus, kis the Kolosov's constant de ned as 3 4 for plane strain conditions, Rfis the ber radius and the index mrefers to the properties of the matrix. 0is the stress at the boundary, computed as the average of the stress extracted at each boundary node along the right side (arithmetic average as nodes are equispaced by design along both the left and 210 right sides). The agreement is good: the di erence between the BEM solution, which is considered more accurate, and the FEM solution does not exceed 5%. The ERRs' maxima are in the same positions and the size of the contact zone is the same. Nevertheless, an analysis of phenomena leading to less than 5% di erences in ERR would not be reliable and, therefore, it is not recommended. 215 3. Results & Discussion 3.1. E ect of Fiber Volume Fraction As shown in Figs. 6 and 7, respectively for Mode I and Mode II, the ber con- tent has a drastic e ect on the Energy Release Rate at the tip of the ber/matrix interface crack. The e ect of four levels of ber volume fraction are compared, 220 30%, 50%, 60% and 65%, on two microstructural models: a 11 11free (every 11th ber in the central ber row is partially debonded and, on the top of this row, we have 5 undamaged ber rows), Figs. 6a and 7a, and a 21 21free (every 21th ber in the central ber row is partially debonded and, on the top of this row, we have 10 undamaged ber rows), Figs. 6b and 7b. 225 Comparing Fig. 6a with 6b, and Fig. 7a with 7b, we can observe that the ERRs' values are very similar for RUCs with 11 11 and 21 21 bers, though they are slightly higher for the larger RUC where the next debonded ber and the free surface are further away from the debonded ber. From these results we conclude that both RUCs are large enough to represent a single debonded ber 230 in an in nite array of bonded bers. Obviously, there exists a speci c e ect of the ber content. For Mode I, Fig. 6, the maximum value of the ERR increases by5:2 times when Vfchanges from 30% to 65%. The debond's angular size for which the peak value occurs remains unchanged at 20, but forVf= 60% 12 DRAFT 010203040506070809000.511.522.533.544.55 ∆θ/bracketleftBigg◦/bracketrightBiggGI J/m2 Vf= 30% Vf= 50% Vf= 60% Vf= 65%(a) Model 11 11free. 010203040506070809000.511.522.533.544.55 ∆θ/bracketleftBigg◦/bracketrightBiggGI J/m2 Vf= 30% Vf= 50% Vf= 60% Vf= 65% (b) Model 21 21free. Figure 6: A view of the e ect of ber volume fraction on Mode I ERR in two exempli cative models, subject to an applied transverse strain "xof 1%. and 65% the values of Mode I ERR are rather similar when measured at 10235 and at 20, approximately creating a plateau. Furthermore, increasing the ber volume fraction delays the onset of the contact zone, which corresponds in Fig. 6 to the rst value of  for whichGIis equal to zero. For Vf= 30%, the contact zone rst appears for a debond of 60, similarly to what happens in the single ber in in nite matrix model (Fig. 5). For higher ber contents, the contact 240 zone's onset is delayed to a debond's size approximately equal to 70. 010 30 50 70 90 110 130 15000.511.522.533.544.55 ∆θ/bracketleftBigg◦/bracketrightBiggGII J/m2 Vf= 30% Vf= 50% Vf= 60% Vf= 65% (a) Model 11 11free. 010 30 50 70 90 110 130 15000.511.522.533.544.55 ∆θ/bracketleftBigg◦/bracketrightBiggGII J/m2 Vf= 30% Vf= 50% Vf= 60% Vf= 65% (b) Model 21 21free. Figure 7: A view of the e ect of ber volume fraction on Mode II ERR in two exempli cative models, subject to an applied transverse strain "xof 1%. 13 DRAFTFor Mode II, Fig. 7, there is a distinct maximum in the curve and its shape does not depend on the ber content. The maximum value of the ERR increases by2:1 times when Vfchanges from 30% to 65%. The e ect is thus similar to Mode I, but with a signi cantly lower magnitude. Similar to Mode I, the 245 debond's size for which the peak value of Mode II occurs remains unchanged, at 60. It is worthwhile to notice that the ratio of Mode II to Mode I peak values ismax(GII) max(GI)2:2 0:92:4 forVf= 30%, while it is 4:7 4:71 forVf= 65%. The general increasing trends observed in Figs. 6 and 7 are related to the fact that, given that the global and local Vfare everywhere identical in the models 250 presented, an increase in ber content corresponds to a decrease in the average distance between bers. Thus, the distances for the decay of the local stress and strain elds in the matrix domain become shorter, leading to higher stresses in general and causing higher values at the crack tip. The di erence in relative magnitude between Mode I and Mode II and the delay in the contact zone's 255 onset are instead due to the interplay between two di erent mechanisms, both caused by the ordered microstructural arrangement of the model. In the models considered, a fully bonded ber is always placed along the horizontal direction, aligned with the partially debonded ber and exactly in front of the debond. By increasing Vf, the former moves closer to the latter and for small debonds 260 this causes a magni cation of the x-strain at the crack tip. For small debonds (2030) in fact, the crack tip is approximately normal to the x-direction and thus an increase in "xcauses an increase in GI. On the other hand, for large debonds ( 7080) the crack growth direction is almost aligned with the x-axis, thus a magni cation in the x-strain translates into an increase of Mode 265 II ERR. However, this increasing e ect on GIIis partially counteracted by the presence of a fully bonded ber on top of the debonded ber and aligned with it. As bers are more rigid than the surrounding matrix, the presence of the former will restrain horizontal displacements, thus hampering strong increases in GII for large debonds. Furthermore, due to the mismatch in the Poisson's ratios, 270 the fully bonded ber placed above generates an upward-directed component of the vertical displacement eld in the matrix, which tends to open the debond 14 DRAFTand causes the delay in the contact zone's onset. The interplay between these mechanisms is governed by the average inter- ber distance and, in turn, by the ber volume fraction. 275 These observations are in agreement with the results reported in [31], where the e ect on the ERR of a partially debonded ber of two fully bonded nearby bers, placed symmetrically with respect to the loading direction, is studied for di erent angular positions (denoted as 2) and radial distances in a model with an e ectively in nite matrix ( Vf0:09%). The e ect of the former is studied 280 for a constant value of the radial distance between the debonded and bonded bers, which corresponds to a local Vlocal f of62% assuming hexagonal pack- ing. They report an increase in both Mode I and Mode II ERR with respect to the single ber case when the two bers are placed at an angle of respectively 25;30;140;150;155, i.e. closest to the loading direction. Notice 285 that for 25and155the two fully bonded bers are almost in contact, with an inter- ber distance of 0:04 times their radius. This result con rms the considerations made in the previous paragraph about the x-strain magni ca- tion caused by the presence of fully bonded bers along the loading direction. The e ect is further analyzed and discussed in Sec. 3.2 and Sec. 3.4. In the 290 range40130instead, the presence of the other bers causes a reduction of the ERR and, particularly in the range 80120, results are very close and almost insensitive to variations in 2, which supports the previous conclusion about the e ect of a fully bonded ber on top the partially debonded one. This e ect is treated in more detail in Sec. 3.3. 295 Comparing the results from [31] with those presented in this paper, an hypoth- esis can be furthermore formulated about the robustness of the results of the present article with respect to deviations in ber position: it seems reasonable to assume a tolerance to deviations of max. 30with respect to the loading direction and of max. 20with respect to the through-the-thickness direction. 300 The e ect of the local ber content is also investigated in [31], by changing the radial distance between the partially debonded ber and the fully bonded ones. They observe that the further the fully bonded bers are placed from the 15 DRAFTcentral one, i.e. the lower the local Vf, the lower is their e ect on the ERR. The magnitude of the e ect is however small: the maximum increase of the 305 total ERR is of 1:15 times for 2= 30and 150when increasing Vlocal f from 28% to 62%; the total ERR decreases by a factor of 0:62 for2= 60, 0:74 for2= 90and0:5 for2= 120when increasing Vlocal f from 28% to 62%. Analogous results can be found in [34], where the authors consider a centrally-placed partially debonded ber sorrounded by an hexagonal cluster 310 inside an homogenized UD composite. They observe a reduction in the ERR when the local ber volume fraction is increased, i.e. when the spacing between bers is reduced. The strongest change is reported for Mode II, which decreases by a factor of 0:73 when the local ber volume fraction is decreased from 66% to 78%. Thus, the trends presented in [31, 34] are in agreement with our 315 results on the e ect of Vfand support the considerations made so far. The stark di erence in magnitude however highlights the contrast between the ef- fect of the local ber volume fraction of a cluster of bers inside an in nite medium and of the global Vfof long-range microstructural arrangements, such the ones considered in this article. The similarity in trends with the concurrent 320 di erence in magnitudes can be explained in relation to the characteristics of the elastic solution computed. In the rst case the local ber volume fraction controls the distance, with respect to the central partially debonded ber, at which a localized perturbation zone appears in the far- eld elastic solution; in the second case the global Vfdetermines the characteristic lengths of a global 325 periodic solution. 3.2. Interaction between debonds in UD composites with a single row of bers The interaction of debonds appearing at regular intervals in an ultra-thin UD composite with a single row of bers is studied for Mode I (Fig. 8) and Mode II (Fig. 9) and ber content equal to 30% (Figs. 8a and 9a) and 60% (Figs. 8b 330 and 9b). The models treated are 3 1free, 51free, 71free, 111free, 211free, 1011free and 201 1free, corresponding respectively to a debond every 3rd, 5th, 7th, 11th, 21st, 101stand 201st ber 16 DRAFT(Fig. 1a). Given that the upper surface of the UD row is left free, the interaction with the debonded ber in the next RUC is stronger than in any other case and 335 the results of this section are thus the most conservative in terms of debond's growth: the ERRs should be the largest. The e ect is enhanced in composites with highVfand especially for GII: atVf= 60% the highest GIIvalue for the 2011free composite in Fig. 9b is more than 3 times higher than the GII value value for the 21 21free composite in Fig. 7b. Even the maximum is 340 shifted to larger angles. The GIvalue is for some cases only 30% higher. From both Fig. 8 and Fig. 9, it can be seen that the presence of a debond close to the analyzed debond decreases the strain magni cation e ect discussed in Sec. 3.1 and thus reduces the value of the ERR. This phenomenon is called \crack shielding" [26]. 345 0 10 20 30 40 50 60 70 80 9000.10.20.30.40.50.60.70.8 ∆θ/bracketleftBigg◦/bracketrightBiggGI J/m2 1×1−free 3×1−free 5×1−free 7×1−free 11×1−free 21×1−free 101×1−free 201×1−free (a)Vf= 30%. 0 10 20 30 40 50 60 70 80 9000.511.522.533.544.5 ∆θ/bracketleftBigg◦/bracketrightBiggGI J/m2 1×1−free 3×1−free 5×1−free 7×1−free 11×1−free 21×1−free 101×1−free 201×1−free (b)Vf= 60%. Figure 8: E ect of the interaction between debonds appearing at regular intervals on Mode I ERR in an UD with a single row of bers at di erent levels of ber volume fraction Vf, subject to an applied transverse strain "xof 1%. For Mode I, the presence of a free surface, and inversely the absence of a fully bonded ber along the vertical direction, implies the absence of the counteracting upward-oriented vertical component of the displacement eld due to the mismatch in Poisson's ratios. This in turn translates into the constancy of the value of  corresponding to contact zone's onset, always equal to 60. 350 ForVf= 30%, Mode I is reduced when the spacing between debonds (in terms 17 DRAFTof number of fully bonded bers between them in our models) decreases, but the magnitude of change is signi cant only in the range when the spacing is reduced from a debond every 5th ber to one every 3rd. For comparison, the di erence of peak GIvalues forVf= 30% between 5 1free and 31free 355 is0:2J m2(around 30% of the lower value), while between 201 1free and 51free is0:05J m2(around 7% of the lower value). A similar observation can be made for Vf= 60%, but for larger spacings: no di erence can be seen between the case of a debond placed every 101thand every 201th ber. These observations suggest the existence of characteristic distance dependent on the 360 ber volume fraction which governs the interaction between debonds: in low Vf composites ( Vf= 30%) the convergence to a non-interactive solution is faster (less interaction between debonded bers in neighboring RUCs). 010 30 50 70 90 110 130 15000.511.522.533.5 ∆θ/bracketleftBigg◦/bracketrightBiggGII J/m2 1×1−free 3×1−free 5×1−free 7×1−free 11×1−free 21×1−free 101×1−free 201×1−free (a)Vf= 30%. 010 30 50 70 90 110 130 150024681012141618202224262830 ∆θ/bracketleftBigg◦/bracketrightBiggGII J/m2 1×1−free 3×1−free 5×1−free 7×1−free 11×1−free 21×1−free 101×1−free 201×1−free (b)Vf= 60%. Figure 9: E ect of the interaction between debonds appearing at regular intervals on Mode II ERR in an UD with a single row of bers at di erent levels of ber volume fraction Vf, subject to an applied transverse strain "xof 1%. Without costraint on the upper surface, the strain magni cation e ect cre- ates a larger displacement gap in the x-direction, which increases Mode II for 365 larger debonds. When debonds are far apart, the series of rigid elements in the ultra-thin composite row (constituted by fully bonded bers and their sur- rounding matrix) creates higher x-strains than in average in the element with the debonded ber, which in turn generates higher tangential displacements at the 18 DRAFTcrack tip for larger debonds. Conversely, when debonds are closer (smaller num- 370 ber of rigid elements between them), the strain concentration in the debonded element is more similar to the applied strain (the magni cation is reduced) and the tangential displacement component at the crack tip decreases for large  . This is the mechanism behind the change in the value of  for which the peak ofGIIoccurs: from 70to 50at 30%, and from 80to 40at 60% going 375 from the higher to the smaller spacing of debonds. Di erently from Mode I, the presence of a characteristic distance is harder to establish. For Vf= 30% (Fig. 9a), it seems reasonable to establish it at around 100 fully bonded bers between each debond. For Vf= 60% (Fig. 9b), the di erence between models 1011free and 201 1free is still sizable, thus preventing the estab- 380 lishment of such characteristic distance. It is possible to observe, however, that the change between 101 1free and 201 1free is signi cantly smaller than between 21 1free and 101 1free (2J m2 vs 11J m2 ), thus sug- gesting the existence of the characteristic distance outside the range studied. Nevertheless, one should question whether the single row composite with free 385 surface is an appropriate RUC for de ning the upper bound for GII:GIImay be more a ected by the free surface than by the e ect of the interaction between debonds in the row. 3.3. In uence of rows of fully bonded bers on debond's ERR in the middle row The e ect of the presence of rows of fully bonded bers on debond's growth in 390 the central row with all bers partially debonded is studied for Mode I (Fig. 10) and Mode II (Fig. 11) and ber content equal to 30% (Figs. 10a and 11a) and 60% (Figs. 10b and 11b). The models treated are 1 3free, 15free, 17free, 111free, 121free, 1101free and 1201free, corresponding to a UD composite with respectively 3, 5, 7, 11, 21, 101 and 201 395 rows of bers (Fig. 2a). The results shown strengthen the arguments made in Sec. 3.1 and Sec. 3.2. It can, in fact, be seen in Fig. 10 that an increasing number of bonded ber rows across the thickness delays the onset of the contact zone to a debond of 70in 19 DRAFT 0 10 20 30 40 50 60 70 80 9000.10.20.30.40.50.60.7 ∆θ/bracketleftBigg◦/bracketrightBiggGI J/m2 1×3−free 1×5−free 1×7−free 1×11−free 1×21−free 1×101−free 1×201−free(a)Vf= 30%. 0 10 20 30 40 50 60 70 80 9000.20.40.60.811.21.41.61.822.2 ∆θ/bracketleftBigg◦/bracketrightBiggGI J/m2 1×2−free 1×3−free 1×4−free 1×6−free 1×11−free 1×51−free 1×101−free (b)Vf= 60%. Figure 10: In uence of rows of fully bonded bers on debond's growth in Mode I ERR in a centrally located row of debonded bers at di erent levels of ber volume fraction Vf, subject to an applied transverse strain "xof 1%. size, due to the introduction of an additional positive component of the vertical 400 displacement which translates into an opening displacement at the debond's tip. Comparing Fig. 9b with Fig. 11b, we observe that the presence of bonded ber rows signi cantly reduce the GIIand its maximum is shifted back to 60, thus con rming the hypothesis in Section 3.2 that the absence of GIIconvergence with the increasing distance in a single-row composite is caused more by the 405 free surface than by the interaction between debonds. 010 30 50 70 90 110 130 15000.10.20.30.40.50.60.70.80.911.1 ∆θ/bracketleftBigg◦/bracketrightBiggGII J/m2 1×3−free 1×5−free 1×7−free 1×11−free 1×21−free 1×101−free 1×201−free (a)Vf= 30%. 010 30 50 70 90 110 130 15000.10.20.30.40.50.60.70.80.911.1 ∆θ/bracketleftBigg◦/bracketrightBiggGII J/m2 1×2−free 1×3−free 1×4−free 1×6−free 1×11−free 1×51−free 1×101−free (b)Vf= 60%. Figure 11: In uence of rows of fully bonded bers on debond's growth in Mode II ERR in a centrally located row of debonded bers at di erent levels of ber volume fraction Vf, subject to an applied transverse strain "xof 1%. 20 DRAFTThe results of both Mode I and Mode II show that the introduction of an increasing number of fully bonded ber rows doesn't change the ERR calculated at the crack tip after adding more than one row (the convergence is very fast). A small e ect, mostly on Mode I, of the number of bonded ber rows can be 410 observed at low ber content (Figs. 10a and 11a), while for high ber content the smaller model with only one ber row above the partially debonded one is already representative. 3.4. E ect of multiple rows of bonded bers on debonding in the central row of a UD composite with di erent distances between debonded bers 415 The ERR of debonds appearing at regular intervals in the central row of bers in UD composites is a ected by the number of rows with bonded bers. The e ect is investigated using di erent combinations of horizontal debond spac- ing, controlled by the number of bonded bers in the central row of the RUC, and the number of rows of bonded bers on top of it. The following models have 420 been studied: 3 3free, 53free, 55free, 73free, 75free, 77free, 113free, 115free, 117free, 1111free, 213free, 215free, 217free, 2111free, 2121free, 1013free, 1015free, 1017free, 10111free, 2013free, 2015free, 2017free, 20111free (Fig. 2b). 425 The results shown in Fig. 12 con rm the observations discussed in Sec. 3.2 and Sec. 3.3: the presence of fully bonded ber rows on top of the central row with debonded bers reduces the interaction with the free surface and thus has a restraining e ect on the ERR, that counteracts the magni cation due to an increasing number of fully bonded bers in the horizontal direction. The 430 interplay is further modulated by the ber content. Observing Fig. 12, it is possibe to note how the free surface interaction decays fast: the presence of 5 ber rows across the thickness is already sucient to prevent any signi cant e ect of additional ber rows on the ERR of a debond in the central row. The results in Fig. 13 show instead the e ect of increasing the distance 435 between two consecutive debonds in the central row of a UD composite of given 21 DRAFT 010203040506070809000.511.522.533.54 ∆θ/bracketleftBigg◦/bracketrightBiggGI J/m2 21×3−free 21×5−free 21×7−free 21×11−free 21×21−free(a)GI. 010 30 50 70 90 110 130 15000.511.522.533.544.55 ∆θ/bracketleftBigg◦/bracketrightBiggGII J/m2 21×3−free 21×5−free 21×7−free 21×11−free 21×21−free (b)GII. Figure 12: E ect on Mode I and Mode II ERR of the presence of an increasing number of rows of fully bonded bers in UD composites with debonds appearing every 10th ber (model 21kfree).Vf= 60% and "x= 1%. 010203040506070809000.511.522.533.5 ∆θ/bracketleftBigg◦/bracketrightBiggGI J/m2 5×5−free 11×5−free 21×5−free 101×5−free 201×5−free (a)GI. 010203040506070809010011012013014015000.511.522.533.544.55 ∆θ/bracketleftBigg◦/bracketrightBiggGII J/m2 5×5−free 11×5−free 21×5−free 101×5−free 201×5−free (b)GII. Figure 13: E ect on Mode I and Mode II ERR of increasing the spacing between debonds appearing in the central row of bers in a UD composite with a xed number of rows across the thickness. Vf= 60%,k= 5 and"x= 1%. thickness. In agreement with the observations of Section 3.2, increasing the distance between debonds (measured in terms of fully bonded bers between them) causes an increase in the ERR in both Mode I and Mode II. For both Mode I and Mode II, it is possible to observe the existence of a characteristic 440 distance which de nes the limit between the interactive and the non-interactive solution. Furthermore, comparing Figure 13a and 13b, it is possible to notice 22 DRAFTthat Mode I is less sensitive than Mode II to the horizontal spacing of debonds. 3.5. Comparison with the single ber model with equivalent boundary conditions The single ber RUC (1 1free or 11coupling ) corresponds to the most 445 damaged state of the composite, i.e. the state in which all bers have debonds. The 11free model represents an ultra-thin UD composite with a single row of partially debonded bers. The 1 1coupling model, where the displacement coupling is used to enforce periodic boundary conditions, represents an in nite composite. 450 The comparison of the 1 1free model with one row multi- ber models n1free in Figure 8 and Figure 9 show that the former provides in general the lowest value of the ERR (the highest crack shielding case). The 11coupling model is compared with 1 3free and 1201free models in Fig. 14. In all three models the distance between debonds in the 455 x-direction is the same and the di erence is in the vertical direction. The 11coupling model describes the interaction between debonds in di erent rows of debonded bers whereas the 1 kfree models describe the e ect of the proximity of the composite's free surface. The Mode I ERR in the 1 3free and 1201free model is very similar to the 1 1coupling model, which 460 leads to a rather surprising conclusion. In both models we have, on the top of the central one, a large amount of bers (bonded in two cases and debonded in the third case). It appears that the e ect of bonded and debonded bers on the central debond is the same. This implies that, for Mode I ERR, the interaction between debonds in elements placed on top of each other is small. 465 The same comparison for Mode II shows a sizeable di erence in the range 5090, while the results almost coincide for smaller values of  . The lower values ofGIIof the 1 1coupling model in the range 5090are due to the shielding e ect of a debond of the same size in the ber just above the central one (modeled by the coupling boundary condition), which leaves the 470 strip of matrix between the two bers free to deform away from both of them due to the Poisson's e ect and thus favors Mode I and reduces Mode II. This 23 DRAFT 010203040506070809000.20.40.60.811.21.41.61.822.2 ∆θ/bracketleftBigg◦/bracketrightBiggGI J/m2 1×1−coupling 1×3−free 1×201−free(a)GI. 010 30 50 70 90 110 130 15000.10.20.30.40.50.60.70.80.91 ∆θ/bracketleftBigg◦/bracketrightBiggGII J/m2 1×1−coupling 1×3−free 1×201−free (b)GII. Figure 14: Comparison of the ERR between the single ber model with coupling conditions along the upper boundary and the 1 kfree model.Vf= 60% and "x= 1%. translates into the delay in the appearance of the contact zone, particularly evident in Fig. 14a. 4. Conclusions & Outlook 475 Several models of Repeating Unit Cell, representative of di erent microstruc- tural arrangements of a unidirectional (UD) composite, have been studied in order to investigate the e ect on ber/matrix interface crack growth of the pres- ence of partially debonded and/or fully bonded bers. Regular microstructures based on square-packing arrangements of bers have been loaded in transverse 480 tension, with debonds appearing in the central row of bers at regular intervals measured in terms of number of fully bonded bers between them. This central row is embedded in-between a varying number of rows with perfectly bonded bers. The surface of the composite is either traction-free or with imposed vertical displacement constraint imitating a periodic structure in the composite 485 thickness direction. The ber volume fraction is the same everywhere and the ber distribution is uniform by design, which establishes a direct relationship between ber content and inter- ber distance. The main conclusions of this work are summarized here. 490 24 DRAFT1. With a decreasing number of fully bonded bers between two partially debonded bers in the central row, the ERR decreases. It seems to exist a characteristic distance between debonds which de nes the transition to a non-interactive solution. However, this distance depends on the number of perfectly bonded ber rows surrounding the central row. 495 2. The presence of a free surface close to the debond leads to higher Mode I and Mode II ERRs and a shift of the peak Gvalues to larger debonds. 3. The presence of bers (fully or partially bonded) in the composite thick- ness direction, along the same vertical line as the analyzed central ber, appears to have a restraining e ect on both GIandGII. The free compos- 500 ite surface e ect on the ERR decays very fast: adding more than 2 fully bonded bers below and above the central row leads to stable constant values of ERR. 4. The presence of a debond in the ber above the central partially debonded one only delays the appearance of the contact zone, while no signi cant 505 e ect on the ERR has been observed. 5. Increasing the ber content (decreasing the inter- ber distance), magni es in general the e ects described in the previous points. 6. The results and conclusions presented agree well with previous observa- tions reported in the literature [31, 34]. A mechanical explanation of 510 the observed trends has been presented based on the mismatch in elastic properties, particularly Poisson's ratios, and the positions of bers and debonds with respect to the loading direction. Acknowledgements Luca Di Stasio gratefully acknowledges the support of the European School 515 of Materials (EUSMAT) through the DocMASE Doctoral Programme and the European Commission through the Erasmus Mundus Programme. 25 DRAFT