AN ANALYSIS OF COMPETING TOUGHENING MECHANISMS IN LAYERED AND PARTICULATE SOLIDS

Transcription

1 AN ANALYSIS OF COMPETING TOUGHENING MECHANISMS IN LAYERED AND PARTICULATE SOLIDS Giovanni Noselli *, Vikram S. Deshpande and Norman A. Fleck Cambridge University Engineering Department, Trumpington Street, Cambridge, CB2 1PZ, UK 26 February 2013 Abstract The potency o various toughening mechanisms is explored or layered solids and particulate solids, containing one or more semi-ininite cracks. First, the enhancement in toughness due to a parallel array o cracks in an elastic solid is explored and the stability o co-operative cracking is quantiied. Second, toughening by crack kinking is analysed or a layered solid containing a pre-existing crack. The asymptotic problem o continued, sel-similar crack advance versus 90 symmetric kinking is considered. The two competing crack paths involve rupture o cohesive zones, and each crack is endowed with a uniorm strength and a critical displacement (at which the traction drops to zero). Third, the evolution o the crack path (and associated crack growth resistance curve) is determined or a regular array o elastic, hexagonal grains bonded together by cohesive zones o inite strength and inite work o separation. Regardless o the initial coniguration o a semi-ininite parent crack, with varying degrees o initial crack tip branching, we ind that a single dominant crack ensues. The study concludes with the analysis o a mode I crack in a multi-layer stack o elastic and elastic-plastic solids. Plastic dissipation in the ductile layers conveys macroscopic toughening but the degree o toughening is sensitive to the relative height o the layers. The study demonstrates that multiple cracking is a diicult mode to activate under quasi-static conditions and the kinking mechanism, when activated, is potent in enhancing the toughness o a solid. Keywords: toughening mechanisms, multi-layered composites, particulate solids * Telephone: ; Fax: ; giovanni.noselli@eng.cam.ac.uk 1

2 1. Introduction Layered and particulate solids are commonly employed as engineering materials, with applications ranging rom lightweight structural parts to high strength coatings. Commonly, high strength composites are brittle, and there is a need to improve their macroscopic toughness by suitable tailoring o topology and interacial properties. A number o toughening strategies are available, and a major thrust o the current paper is to compare the potency o toughening or some o these strategies. We consider 3 toughening mechanisms in turn: (i) toughening by crack multiplication, (ii) crack kinking, and (iii) toughening by plastic dissipation Toughening by crack multiplication Monolithic ceramics can contain a zone o crack-tip shielding by micro-cracking, see Fig. 1a. This mechanism gives rise to a small elevation in macroscopic toughness, as discussed by Kreher and Pompe (1981), Evans and Faber (1984), Rose (1986), Hutchinson (1987) and Shum and Hutchinson (1990). The eect is modest as the main crack tends to ollow the most brittle path. For this mechanism to operate it is essential that the micro-cracks arrest at grain boundaries or particle interaces and be highly stable in the arrested state. The micro-cracked zone reduces the crack tip stress intensity actor K tip by two mechanisms: (i) the micro-cracked zone is more compliant then that o the remote material, and (ii) the micro-cracks release local residual stresses when they orm and this leads to a dilatational strain in the micro-cracked zone. However, the degree o crack tip shielding by the micro-cracked zone is only modest. Hutchinson (1987) has shown that the reduction in K tip is between 30% and 40% or heavily micro-cracked zones in ceramic materials, as observed by Rühle et al. (1990). We begin by exploring the macroscopic toughening that arises rom a parallel array o cracks. Consider a stack o n parallel semi-ininite cracks under remote mode I K-ield, see Fig. 1b. The magnitude o the crack tip stress intensity actor, and the mode-mix, will vary rom crack to crack. Suppose that the cracks were able to adopt a coniguration (in terms o relative spacing and relative crack tip position) such that each crack tip is in a state o mode I and has the same tip value o K. Then, the cracks could, in-principle, advance in a steady-state coniguration and the remote applied energy release rate G would be n times the value G tip or each crack tip. But this degree o toughening assumes that it is possible or such an array o cracks to be conigurationally stable and to adopt a coniguration with an identical mode I K-ield or each crack tip. We shall explore whether this is possible or the simple cases o n = 2 and n = 3. 2

3 1.2. Crack kinking Crack tip kinking gives the possibility o toughness elevation or both layered and particulate composites. The case o a layered solid with a pre-existing crack transverse to several layers is shown in Fig. 1c,d. The presence o weak or brittle interaces between the layers leads to the possibility o crack tip kinking and thereby to crack delection along the interace. For elastic-brittle interaces as shown in Fig. 1c, the macroscopic toughness remains constant or decreases when the crack kinks along the interace, as discussed by He and Hutchinson (1989a). Alternatively, the interace may be elastic-plastic in nature, see Fig. 1d, and crack kinking by the progressive ailure o this interace can give rise to macroscopic toughening (Cao and Evans 1991; Chan et al. 1993; He et al. 1993; He et al. 1994; Shaw et al. 1996). Now consider intergranular racture o a particulate elastic solid, with racture occurring along the interaces between grains, see Fig. 1e. Potential sources o toughening are crack branching and the simultaneous advance o multiple, connected cracks. Shabir et al. (2011) recently conducted a numerical study on the quasi-static racture o polycrystalline aggregates. They considered an assembly o elastic, randomly-shaped grains, and the grains were connected by cohesive zones. The progressive ailure o a specimen containing a single edge crack was addressed, and their quasi-static simulations revealed the progression o a single dominant crack, with minor deviation in path rom the initial plane o cracking. A major issue is whether dynamic crack growth is needed in order to achieve multiple branching: Xu and Needleman (1994) observed such kinking behaviour in their simulations o dynamic intergranular crack growth. In the current paper, an attempt is made to gain more insight into the racture behaviour o particulate solids by exploring the possibility o multiple crack growth in the quasi-static regime rom a multiply-branched pre-crack Toughening by plastic dissipation It is generally accepted that plastic dissipation in the vicinity o a crack tip in a ductile solid causes a multiplicative enhancement in macroscopic toughness. This has been demonstrated by Tvergaard and Hutchinson (1992): they calculated the crack growth resistance curve (the so-called R-curve) or a mode I crack in an elastic-plastic solid, with crack advance modelled by a cohesive zone o inite strength and o inite racture energy. They ound that the macroscopic toughness increases with crack advance, until a steady-state is achieved. The steady-state toughness Gss scales linearly with the cohesive zone energy (as demanded on dimensional grounds), but the degree o toughening increases dramatically with an increase in the ratio o cohesive zone strength to yield strength o the solid. 3

4 A closely related problem has been addressed by Suo et al. (1993) and by Tvergaard (1997). They replaced the cohesive zone by an elastic strip o mode I toughness G 0 and o inite height h, see Fig. 1. Again, an R-curve is predicted, with the level o macroscopic toughening sensitive to the value o h. We shall extend this approach to the case o a parallel stack o alternating elastic and elastic-plastic layers, with a mode I crack lying on the mid-plane o one elastic strip. Thereby, the degree o toughening within constrained elastic-plastic layers is assessed or a layered composite Toughening in Nature: the role o structural hierarchy Polymer-ceramic composites are ubiquitous in Nature, such as in shell, bone and horn (Currey 1977; Lin and Meyers 2005; Bertoldi et al. 2008). Despite the relatively poor mechanical properties o their constituents 1, these structural biomaterials exhibit a remarkable level o macroscopic toughness and possible reasons or this have been suggested. Several toughening mechanisms have been identiied, such as crack bridging, crack blunting, crack tip shielding by the nucleation o micro-voids, and crack delection along visco-plastic interaces (Jackson et al. 1988; Jackson et al. 1990; Wang et al. 2001; Gao et al. 2003; Espinosa et al. 2011; Shao et al. 2012). Remarkably, these natural composites are organized in hierarchical microstructures; or example, 3 to 7 levels o structural hierarchy exist in shell and bone, respectively. The precise evolutionary reasons or this remain unclear. One attractive view, as advanced by Gao et al. (2006) and Zhang et al. (2011), is that structural hierarchy conveys damage tolerance Overall scope o the study The scope o the present study is as ollows. In section 2, the enhancement in toughness due to a parallel array o cracks in an elastic solid is explored and the stability o co-operative cracking is quantiied. In section 3, toughening by crack kinking is analysed or a layered solid containing a pre-existing transverse crack. The competition between continued crack advance and 90 symmetric kinking is also considered. The two competing crack paths contain cohesive zones, and each cohesive zone is endowed with a strength and toughness. The sensitivity o macroscopic toughness to the relative properties o the two cohesive zones is determined. Further analysis o the signiicance o crack path upon macroscopic toughness is carried out or intergranular crack advance or a regular array o elastic, hexagonal grains bonded together by cohesive zones o inite strength and racture 1 Nacre, mother o pearl, is a typical example o a natural structural composite, composed by brittle platelets o aragonite (95 vol%) and thin layers o biopolymers (5 vol%), organized in a brick-and-mortar microstructure. 4

5 energy. In section 4, the role o plastic dissipation in controlling toughness o multi-layers is determined: the crack growth resistance curve is predicted or alternating layers o elastic and elastic-plastic solids. 2. Toughening o an elastic-brittle solid by multiple cracking We begin by exploring the possibility o macroscopic toughening or an array o 2 or 3 parallel cracks under remote mode I loading. I these cracks grow simultaneously then macroscopic toughening will exist, but are they conigurationally stable? To address this two geometries are analysed: an ininite, elastic-isotropic solid contains either 2 or 3 parallel, semi-ininite cracks, and is subjected to a remote mode I K-ield, see Fig. 2. To address the issue o conigurational stability, the crack tips are given a relative o-set, as parameterised by an angle α. The challenge is to determine the K-ield at each crack tip as a unction o α Finite element analysis Plane strain inite element analyses have been perormed in order to extract the crack tip K as a unction o the remote mode I K, with the commercial sotware ABAQUS Standard Consider an isotropic elastic solid o Young s modulus E and Poisson s ratio ν. A remote mode I K-ield is applied by imposing appropriate displacements u i on the outer boundary o the mesh. With x i taken as Cartesian co-ordinates centred on the average position o the crack tips, and with the associated polar co-ordinates ( r, θ ), the displacement ield imposed on the outer periphery o the mesh is K r θ u1 = cos ( κ cos θ ) 2µ 2π 2 K r θ u2 = sin ( κ cos θ ) 2µ 2π 2 where µ = E / [2(1 + ν )] is the shear modulus, and κ equals 3 4ν or a state o plane strain. Throughout this study we assume ν = 0.3. The inite element meshes are highly reined in the region near the crack tips 2 and comprise CPE4 (plane strain, our-nodes bilinear) elements; converged results were obtained by taking the outer dimension o the mesh to be 250 times the crack spacing. For both geometries the crack tip mode I and mode II stress intensity actors ( K I and K II, respectively) are computed by means o the domain-integral method as (1) 2 The dimension o the square inite elements near the crack tips has been ixed to 1/20 the vertical distance between the semiininite cracks. 5

6 implemented in ABAQUS. Note that the stress intensity actors or tips A and B are dierent, and their relative magnitude gives the requisite inormation on their conigurational stability Results We assume that the direction o crack advance or tips A and B o Fig. 2 is dictated by the local angle arctan( K / K ) φ = II I or each crack. I φ = 0, the crack advances in a sel-similar manner, i φ > 0 the crack tip kinks downwards and i φ < 0 the crack tip kinks upwards. Whether crack growth occurs or not at each crack tip is also dictated by the energy release rate G (1 ν )( KI KII) / E = + in relation to the material toughness G IC : kinking will occur when G attains the value G = G IC ( φ), where the unction ( φ ) has been given by Lo (1978) and He and Hutchinson (1989b). For present purposes it suices to consider the relative value o G and φ at each crack tip A and B, and to normalize the crack tip values or G by the remote energy release rate 2 2 G = (1 ν )( K ) / E. Energy release rates are reported in dimensionless orm G / G in Fig. 3a and in Fig. 4a as a unction o α, or the two- and threecrack geometry, respectively. Consider irst the results or a pair o cracks, o geometry as given in Fig. 2a. When α is between zero and π / 2 (such that crack tip A is behind tip B), we ind that the value o G or tip A is less than that or tip B, see Fig. 3a. Consequently, tip B will advance in preerence to tip A, destabilizing the coniguration such that α reduces urther. The phase angle φ or tip B is small and negative implying that the tip B will veer upwards away rom the neighbouring crack A 3. In contrast, the geometries π / 2 < α < π have the eature that tip A will kink beore tip B and will kink downwards away rom tip B. Again, the crack tips diverge in path, and in an unstable manner such that the longer crack grows in preerence to the shorter crack. The exception is or α = π / 2. Then, the two cracks possess the same value o G, and φ equals 0.3 or tip A and φ equals -0.3 or tip B, see Fig. 3b: the two cracks kink away rom each other. To complete the picture, consider a slight perturbation in crack length such that α slightly exceeds π / 2. Then the crack tip A is slightly ahead o tip B. It will advance in preerence to tip B due to its higher value o G, and it will diverge (kink) away rom tip B. We conclude that there is a strong tendency or 3 Recall that φ < 0 implies upwards kinking, whereas φ > 0 will give downwards kinking, as explained by Lo (1987) or example. 6

7 crack growth to shut down or the lagging crack whereas the leading crack will kink away rom its passive neighbour. A similar picture emerges rom the case o 3 cracks (Fig. 2b); see computational results in Fig. 4. When α < π / 2, crack tip A lags that o B and has a much reduced value o G (Fig. 4a). Consequently, the tips B will advance and will kink outwards and away rom the mid-plane o crack A. When α is large (say in the range 3 π / 4 to π ), the crack tip A is ar ahead o the tips B and cracks o type B have a negligible inluence upon it: the crack advances at G G and φ 0. A cross-over point does exist such that G or tip A equals that or tips B at α 1.79, corresponding to the case where tip A is slightly ahead o tip B. But this coniguration is unstable since a slight advance in tip A will lead to an increased value o α and thereby to an increased value o G or tip A and to a reduced value o G or tips B. Again, we conclude that the crack coniguration is unstable, with the propensity or crack advance to occur by a single crack, and the shut down o growth by the lagging cracks. In passing, we note that the values or G or the 2 cracks o Fig. 2a and or the 3 cracks o Fig. 2b always sum to G, as demanded by path independence o the J-integral (with J reducing to G in elastic analysis). Consequently, or geometries where G values are equal or the crack tips we have G / G = 1/ 2 or the pair o cracks (Fig. 2a) and G / G = 1/ 3 or the triplet o cracks (Fig. 2b). We emphasise that φ 0 or each crack tip in the conigurations where the G values are identical: the cracks will kink away rom each other. 3. Toughening due to interacial crack kinking Crack delection can play a signiicant role in the ailure o particulate, layered and ibre-reinorced composites, recall Fig. 1c-e. Crack kinking can arrest the orward growth o a crack, and act as a potent crack blunting mechanism, particularly when the parent crack kinks into a ductile interace, see Fig. 1d. We shall consider two prototypical problems: (i) crack kinking in a layered solid, as shown in Fig. 5, and (ii) multiple crack branching in a particulate composite, as shown in Fig. 1e. In both cases, the challenge is to obtain the degree o toughening associated with the deviation o crack path rom the initial crack plane Crack kinking versus sel-similar growth in a layered solid Crack growth in layered composites or in ibre composites can involve sel-similar crack advance ahead o the parent mode I crack or kinking at an angle o φ = ± π / 2, as shown in Fig. 5. This competition in direction o crack growth has been analysed in the elastic-brittle case by He and Hutchinson (1989a). 7

8 Here, we consider ductile interaces such that the parent crack has a tensile band o strength σ y and critical opening u p ; likewise, the crack can kink by the yielding o a shear band o shear strength critical shear displacement τ y and ailure at a u k, see Fig. 5. For simplicity, an ideally plastic response in mode I opening is assumed or the tensile band, and a constant shear resistance exists in the shear band. In the inite element simulations reported below the cohesive zone laws include an elastic loading branch such that the normal traction σ equals y σ at an opening displacement u = u. Similarly, or the shear bands, or computational reasons, the initial y p p response is elastic until the shear traction τ attains the yield value τ y at a shear displacement y k uk = u. The values o y y up u k are taken to be suiciently small in relation to ( up, u k ) that the cohesive zones can be (, ) treated as rigid, ideally plastic in behaviour. Then, the racture energy or the normal band equals σ y up, whereas the racture energy or the shear band equals y uk τ. We are interested in the plane strain, small scale yielding problem such that the parent crack is loaded by a remote mode I stress intensity actor K. This is achieved by loading the outer boundary o a square mesh by the asymptotic ield (1), as explained in the previous section. There is now the potential or the cohesive zones to shield each other such that the energy release rate at the onset o crack advance G exceeds both σ u and y p y uk τ. We explore this behaviour by detailed inite element calculations o the problem as shown in Fig. 5. Dimensional analysis dictates that and equivalently G σ y p 1, u = yu τ p y u k σ (2) G σ y p 2, u = yu τ k y u k τ where the unctions 1 and 2 are to be determined, and the additional independent groups o y (3) σ / E and Poisson s ratio ν are held ixed at and 0.3, respectively. We assume that the parent crack advances by kinking when τ u < σ u in accordance with relations (2) and (3). Otherwise, sel-similar crack advance y k 2 y p 1 occurs by ailure o the tensile band. 8

9 The plane strain inite element calculations were again perormed using ABAQUS Standard The mesh comprises our-nodes bilinear elastic elements (o type CPE4 in ABAQUS notation). Both the tensile cohesive zone (directly ahead o the parent crack) and the two shear cohesive zones (orthogonal to the parent crack) are modelled by COH2D4 elements. The elastic, ideally plastic constitutive laws or the cohesive zones were implemented by means o a user subroutine UMAT. The shear bands are constrained not to open, while the penetration band has no shear traction by symmetry. Numerical experimentation revealed that adequate accuracy is obtained by using at least 20 cohesive elements (o type COH2D4) to model each racture process zone. The plane strain elements (CPE4) were graded in dimension, and were o side length equal to that o the cohesive elements near the crack tip. Symmetry dictates that only the upper hal o the geometry o Fig. 5 was analysed. Small scale yielding conditions were ensured by taking the side length o the mesh to exceed 100 times the larger o the two plastic zones at the crack tip Degree o toughening or crack penetration versus crack kinking A series o inite element simulations were perormed in order to identiy the regime o ( σ / τ, u / u ) y y p k parameters space or which the crack advances by penetration in the tensile band or by kinking along the shear band. The number o required simulations is drastically reduced upon noting that penetration occurs when u p attains u p in the tensile band beore u k attains u k in the shear band, and vice versa. Thus, it suices to increment K max or a given value o σ y / τ y, and to monitor the maximum opening u and the maximum shear p displacement max k u as a unction o K. With increased loading by K max max, a steady-state ratio o u / u is p k attained and this ratio identiies the value o u / u at which penetration and kinking occur simultaneously. p k This procedure is used to identiy the boundary between penetration and kinking in Fig. 6. Contours o G / ( σ y up ) 1 and o G / ( τ u ) are included in the igure, and are also plotted in Fig. 7a as a unction y k 2 o σ y / τ y. In order to interpret Fig. 7a, we irst make use o Fig. 6 to identiy whether penetration or kinking occurs and then select the appropriate result or G in Fig. 7a. Likewise, the lengths o the plastic zone or the tensile band l p and or the shear band 9 l k are plotted in Fig. 7b, and are shown as solid curves or the case o penetration and as dashed curves or kinking. Note that the normalisation o ( l, l ) is dierent or the case where penetration or kinking occurs irst. The procedure is again p k

10 to make use o Fig. 6 in order to determine whether kinking or penetration occurs or any given values o σ y / τ y and u / u. We note rom Fig. 6 that the boundary between penetration and kinking is more sensitive to the value p k o σ y / τ y than to the ratio o u / u. Consequently, the ordinate is plotted as ( u u ) / ( u + u ) such that up / k u can range rom zero to ininity. p k p k p k To gain additional physical insight, the relative size o the tensile and shear plastic zones is sketched in Fig. 6 or the trajectory A B C or the choice up = uk. There is a marked increase in the size o the shear plastic zone as σ y / τ y increases rom zero magnitude at point A to a large value at point C. It is clear rom both Fig. 6 and rom the plot o l σ / ( u E) in Fig. 7b that no shear plastic zone exists or the regime σ y / τ y < 2.49, p y p regardless o the value o u / u. Only penetration is possible within this regime. p k The above analysis extends the previous work o Chan et al. (1993), and o He et al. (1994): these previous studies assumed a shear band o ininite ductility. He et al. (1994) employed a linear spring model or the tensile band whereas Chan et al. (1993) neglected the presence o a mode I cohesive zone directly ahead o the parent crack. In broad terms, the same trends are observed as in the present study. For example, He et al. (1994) showed that the penetration toughness G increases with diminishing τ y, while Chan et al. (1993) revealed that the level o tensile stress ahead o the main crack reduces with diminishing τ y Crack path and racture toughness in particulate solids The above two sections have separately explored the role o multiple cracking and o crack branching with plasticity in giving rise to macroscopic toughening. We proceed to explore the combined eect o multiple crack branching and crack tip plasticity or a particulate composite. The challenge is to determine whether a parent mode I crack can give rise to multiple crack branching at its tip, assuming various degrees o initial crack branching. The geometry is sketched in Fig. 8; the solid is made up o elastic grains, connected by thin (zero thickness) elastic-plastic interaces, and contains a semi-ininite parent crack under remote mode I loading. The grains are assumed to be hexagonal in shape with edge length d, and elasticisotropic o Young s modulus E and Poisson s ratio ν. As beore, small scale yielding and plane strain conditions are assumed, with a remote mode I K-ield imposed, see Fig. 8 and recall (1). 10

11 The commercial sotware ABAQUS has been employed to perorm the computations under quasi-static conditions. Each hexagonal grain is meshed with 96 CPE4R (reduced integration, plane strain, our-nodes bilinear) inite elements, resulting in 4 elements along the side o each hexagon, whereas each interace between the grains has been deined by means o 4 cohesive elements COH2D4. A rectangular domain is employed, o dimension at least 50 times that o the crack tip plastic zone, in order to ensure that small scale yielding conditions prevail. Some initial calculations were perormed using the implicit version o ABAQUS but given the large size o the computations, the computational time o those runs was prohibitive. The calculations presented in this study are perormed using the explicit solver o ABAQUS in a multi-processor mode (the explicit solver does not require matrix inversion and is thus better suited to parallel execution compared to the implicit solver). Care was taken to ensure that the applied rate o loading was suiciently slow in the explicit calculations to ensure that material inertial eects are negligible and the results correspond to a quasi-static situation Speciication o the traction-separation law The interace between neighbouring grains is idealised by a sotening cohesive zone law, with cross-coupling between the normal and the tangential components o traction and o displacement. Consider a representative interace with unit normal e n and unit tangential direction e s. Then, the displacement jump u across the interace can be expressed in component orm ( un, u s) as u = un en + us e s and similarly the traction t can be written in component orm ( σ, τ ) as t = σ en + τ e s. We assume an elastic-plastic response o the cohesive zone such that the displacement rate uɺ can be additively decomposed into elastic and plastic components e uɺ and p uɺ, respectively, giving ɺ ɺ e ɺ p. (4) u = u + u The traction rate tɺ is linear and taken to be co-directional in such that e uɺ, and is related by a scalar spring stiness k Plastic collapse occurs when an eective traction e t = k u ɺ ɺ. (5) p attains a yield value t y. Upon writing eective displacement rate uɺ n and uɺ e is introduced as 2 2 e + (6) t σ τ p uɺ s as the plastic work conjugates to σ and τ, respectively, an 11

12 p 2 p 2 uɺ e ( uɺ n ) + ( uɺ s ). (7) We proceed in the usual way or a plastic solid: assume normality o plastic low such that p te tɺ uɺ e n = σ h and (8) p te tɺ uɺ e s = τ h in terms o a hardening modulus h. We shall subsequently consider the sotening case with h negative. Upon making use o (6) and (7) we can write h as The incremental constitutive law ollows as h dt = e. (9) due or plastic loading, and 2 k ( t u) tɺ ɺ = kuɺ t 2 (10) te ( k + h) tɺ = kuɺ (11) or elastic unloading. The rate o dissipated energy per unit area o interace is ɺ ɺ ɺ. (12) G0 = t up = te ue It remains to speciy the relation between t y and u e. A sotening law is adopted here, such that ty = t0 + hue (13) in terms o an initial strength t 0 and a linear sotening parameter h, o negative magnitude. The plastic work o separation G 0 is obtained by integration o (12) to obtain via (13) 2 t G 0 0 =. (14) 2h The above constitutive law or the interace has been implemented in ABAQUS by means o a user subroutine VUMAT and employing the elastic predictor-plastic corrector algorithm, see Lubliner (1990) Dimensional analysis and results Two reerence quantities are now introduced and will be used or normalisation o the numerical results: a reerence stress intensity actor 12

13 and a reerence material length E G K , = (15) ν which sets the length scale or the plastic zone size. R 0 2 K0 = π t 0 2, (16) The predicted R-curve response, K versus the arc length o crack extension a, has the ollowing dimensionless orm K a R0 t0 k d = 3,,,, K0 R0 d E E in terms o an unknown unction 3. Ater suicient crack extension, a steady-state value o stress intensity actor (17) Kss is attained. Our primary aim is to explore the dependence o the R-curve upon selected initial conigurations o the crack tip, as shown in the inserts o Figs. 9 and 11, or the choice t0 E = 0.003, k d / E = 100, ν = 0.3 and or the two choices R 0 / d = 1 and R 0 / d = 7. The case R 0 / d = 1 corresponds to the case where the plastic zone is comparable to the particle size, whereas the plastic zone size much exceeds the particle size at R 0 / d = 7. The initial crack tip coniguration comprises the ollowing 4 choices: a single crack, a branched crack, a doubly branched crack and a micro-crack o length d directly ahead o a singly-branched parent crack tip. The predicted crack paths and associated R-curves are given in Figs. 9 and 10, respectively, or the choice R 0 / d = 1. Likewise, the predicted crack paths and associated R-curves are reported in Figs. 11 and 12 or the choice R0 / d = 7. Note that the displacement ield in Figs. 9 and 11 is magniied by a actor 10, while the red colour reers to traction-ree resh crack surace. The predicted crack trajectories and R-curves are remarkably independent o the magnitude o crack tip toughness and o the initial coniguration o the crack tip. For the case (a) o an unbranched initial crack the R-curve begins at K K0 or R 0 / d = 1, and K 1.25K0 or R 0 / d = 7. Initial branching o the crack leads to a somewhat higher value o K at the initiation o crack growth. We note that in all cases a single crack advances, despite the act that several crack tips may exist in the initial coniguration 4. Further, there is only a minor elevation o crack growth resistance with crack advance. This is 4 This contrasts with dynamic racture where crack branching is ubiquitous, Lawn (1993). 13

14 consistent with the act that the cohesive zone is sotening in nature and thereby promotes plastic localisation o deormation as also observed by Shabir et al. (2011). 4. The mode I crack growth resistance o multi-layered, elastic-plastic composites Multi-layered solids, comprising an elastic brittle phase and a much tougher elastic-plastic phase are increasingly used in engineering application, rom the joining o metallic parts by an adhesive layer to electronic devices. Failure within the elastic layer is o general concern, and the question arises: to what degree does the presence o the elastic-plastic layers enhance the macroscopic toughness? A prototypical plane strain geometry is shown in Fig. 13a: an elastic strip o height h contains a semi-ininite crack, and is sandwiched between two elastic-perectly plastic hal-planes. This problem has been previously investigated by Suo et al. (1993) and Tvergaard (1997). The racture energy or the central strip is denoted by G 0, and σ y is the yield strength o the elastic-plastic phase. Both the elastic strip and the elastic-plastic substrates have a Young s modulus and a Poisson s ratio E and ν, respectively. Displacements are imposed on the outer periphery according to the ield (1), so that the magnitude o the external loading is given by the remote stress intensity actor K. The onset o crack growth occurs when the remote stress intensity actor K attains the value = 2. Upon increasing K, the crack advances through the elastic strip until a steady-state K 0 EG 0 / (1 ν ) is attained, such that the crack advances at K equal to Kss. The numerical analyses, carried out by Suo et al. (1993) under plane strain and small scale yielding conditions, revealed that the macroscopic steady-state toughness Kss / K0 is a maximum as h tends to zero, see or instance Fig. 5 in Suo et al. (1993). These results are now generalized in order to compute the mode I crack growth resistance o an elastic-plastic, multi-layered composites, consisting o alternating elastic and elastic-perectly plastic strips o height h and H, respectively, see Fig. 13b. Again, the semi-ininite crack is placed along the centre-line o the central elastic layer. The R-curve o the layered solid, K versus a, now depends upon a set o our dimensionless parameters, K a h H 4,,, y, K0 R0 R0 R ε = 0 (18) where the reerence length R 0 is now taken to be the plane strain plastic zone size or K equal to K 0, 14

15 2 1 K R 0 0 =. 3π σ y (19) 4.1. Speciication o the inite element model and numerical results The inite element computations, perormed with ABAQUS Standard , required highly reined meshes in the process zone near the crack tip; CPE4 (plane strain, our-nodes bilinear) inite elements have been used and converged results have been obtained taking their dimension equal to R 0 / 30. Moreover, at least 5 elements have been employed through the thickness o the elastic and elastic-plastic strips and, to enorce the small scale yielding conditions assumed in the computations, the outer dimension o the meshes exceeds 50 times the maximum extension on the plastic zone at ailure. By symmetry considerations about the central horizontal axis, only the upper hal o the geometry has been meshed, with zero traction imposed on the crack plane and zero normal displacement on the remaining portion o the symmetry plane ahead o the initial crack tip. We remark here that the results by Suo et al. (1993) have been obtained employing an iterative inite element procedure, as developed by Dean and Hutchinson (1980), in which the steady-state condition o quasi-static crack growth is directly satisied, whereas, in our analyses, R-curves have been determined explicitly. For this purpose, the strategy o Tvergaard (1997) has been used, based on the evaluation o the J-integral at the crack tip, J tip, with crack growth obtained by successive node-release when the condition Jtip = G0 is satisied. Resistance curves or the multi-layered composite have been computed taking ε y = 0.003, ν = 0.3, h / R 0 = {0.58, 0.33, 0.18} and H / R 0 ranging rom 0 up to 100. Selected results are shown in Fig. 14 as dasheddotted and dashed curves, or H / R 0 = {0.85,1.70} respectively, and are compared with the solid curve or H / R 0 = corresponding to the problem as sketched in Fig. 13a. We conclude that H / R 0 = provides an upper limit on the R-curve or any height o elastic strip h / R 0. A simple interpretation is as ollows. Most o the elevation in toughness is due to non-proportional loading o the plastically deormed material immediately adjacent to the central elastic strip. Thus a reduction in the magnitude o h / R 0 elevates the toughness. A smaller eect is observed or H / R 0 : there is a small elevation in toughness as H / R 0 is increased rom 1.7 to ininity. The relative sensitivity o toughening to h / R 0 and to H / R 0 is urther explored in Fig. 15; the igure gives the sensitivity o the steady state toughness Kss / K0 to H / R 0 or selected values o h / R 0. A sketch o the initial 15

16 stationary plastic zone is shown in the insert to the igure or h / R 0 = 0.18, and or both small and large H / R 0. Since most o the plastic dissipation takes place in material elements close to the crack tip, Kss / K0 increases sharply when h / R 0 is reduced rom 0.33 to 0.18 and when H / R 0 is increased rom zero to about unity. For any given value o h / R 0, a threshold or H / R 0 exists above which the racture toughness o the layered solid is constant: this is set by the condition that the active plastic zone just reaches the outer boundary o the irst elasticplastic layer adjacent to the crack. At the other extreme, as H / R 0 tends to zero, no toughening by plastic dissipation is possible and Kss / K0 tends to unity. 5. Discussion and concluding remarks In this study, inite element computations have been perormed in order to explore the relative potency o toughening mechanism by crack branching, crack multiplication and by plasticity in layered solids. In most cases, the predicted degree o toughening is only moderate, as it is remarkably diicult to attain a state o crack multiplication when a cohesive zone o inite tip toughness exists at the crack tip: unloading occurs in the wake o a single dominant crack. The study thus suggests that multiple crack branching requires additional physics such as dynamic eects at the crack tip. The competition between crack penetration and crack kinking at an interace is also explored, and it is shown that the direction o crack advance is sensitive to both the relative strength and relative toughness o the cohesive zones associated with penetration or kinking. The kinking mechanism is shown to be able to signiicantly increase the macroscopic toughness o the solid: it is the most potent o the mechanisms considered here in terms o enhancing toughness. Acknowledgements The authors are grateul to DARPA or their inancial support through grant W91CRB titled A microcellular solids approach to thermo-structural materials with controlled architecture. Reerences [1] ABAQUS Theory Manual, version 6.10 (2010) Hibbitt, Karlsson and Sorensen, Inc., Pawtucket, R.I., USA. [2] Bertoldi K, Bigoni D and Drugan WJ (2008) Nacre: an orthotropic and bimodular elastic material. Compos Sci Technol 68: [3] Cao HC and Evans AG (1991) On crack extension in ductile/brittle laminates. Acta Metall Mater 39: [4] Chan KS, He M-Y and Hutchinson JW (1993) Cracking and stress redistribution in ceramic layered composites. Mater Sci Eng A 167:

17 [5] Currey JD (1977) Mechanical properties o mother o pearl in tension. Proc R Soc Lond B 196: [6] Dean RH and Hutchinson JW (1980) Quasi-static crack growth in small-scale yielding. In: Fracture Mechanichs. Proceedings o the Twelth National Symposium o Fracture Mechanics, ASTM STP 700, American Society or Testing and Materials, pp [7] Evans AG and Faber KT (1984) Crack-growth resistance o microcracking brittle materials. J Am Ceram Soc 67: [8] Espinosa HD, Juster AL, Latourte FJ, Loh OY, Gregoire D and Zavattieri PD (2011) Tablet-level origin o toughening in abalone shells and translation to synthetic composite materials. Nature Commun 2:1-9. [9] Gao H, Ji B, Jäger IL, Arzt E and Fratzl P (2003) Materials become insensitive to laws at the nanoscale: lesson rom nature. Proc Natl Acad Sci USA 100: [10] Gao H (2006) Application o racture mechanics concepts to hierarchical biomechanics o bone and bone-like materials. Int J Fract 138: [11] He M-Y and Hutchinson JW (1989a) Crack delection at an interace between dissimilar elastic materials. Int J Solid Structures 25: [12] He M-Y and Hutchinson JW (1989b) Kinking o a crack out o an interace. J Appl Mech 56: [13] He M-Y, Heredia FE, Wissuchek DJ, Shaw MC and Evans AG (1993) The mechanics o crack growth in layered materials. Acta Metall Mater 41: [14] He M-Y, Wu B and Suo Z (1994) Notch-sensitivity and shear bands in brittle matrix composites. Acta Metall Mater 42: [15] Hutchinson JW (1987) Crack tip shielding by micro-cracking in brittle solids. Acta Metall Mater 35: [16] Jackson AP, Vincent JFV and Turner RM (1988) The mechanical design o nacre. Proc R Soc Lond B 234: [17] Jackson AP, Vincent JFV and Turner RM (1990) Comparison o nacre with other ceramic composites. J Mater Sci 25: [18] Kreher W and Pompe W (1981) Increased racture toughness o ceramics by energy-dissipative mechanisms. J Mater Sci 16: [19] Lawn B (1993) Fracture o brittle solids. Cambridge University Press. [20] Lin A and Meyers MA (2005) Growth and structure in abalone shell. Mater Sci Eng A 390: [21] Lo KK (1978) Analysis o branched cracks. J Appl Mech 45: [22] Lubliner J (1990) Plasticity theory. Pearson Education. [23] Rose LRF (1986) Microcrack interaction with a main crack. Int J Fract 31: [24] Rühle M, Evans AG, McMeeking RM, Charalambides PG and Hutchinson JW (1987) Microcrack toughening in aluminia/zirconia. Acta Metall Mater 35: [25] Shabir Z, Van der Giessen E, Duarte CA and Simone A (2011) The role o cohesive properties on intergranular crack propagation in brittle polycrystals. Modelling Simul Mater Sci Eng 19:1-21. [26] Shao Y, Zhao H-P, Feng X-Q and Gao H (2012) Discontinuous crack-bridging model or racture toughness analysis o nacre. J Mech Phys Solids 60: [27] Shaw MC, Clyne TW, Cocks ACF, Fleck NA and Pateras SK (1996) Cracking patterns in metal-ceramic laminates: eects o plasticity. J Mech Phys Solids 44: [28] Shum DKM and Hutchinson JW (1990) On toughening by microcracks. Mech Materials 9: [29] Suo Z, Shih C and Varias AG (1993) A theory or cleavage cracking in the presence o plastic low. Acta Metall Mater 41: [30] Tvergaard V and Hutchinson JW (1992) The relation between crack growth resistance and racture process parameters in elastic-plastic solids. J Mech Phys Solids 40: [31] Tvergaard V (1997) Cleavage crack growth resistance due to plastic low around a near-tip dislocation-ree region. J Mech Phys Solids 45:

18 [32] Wang RZ, Suo Z, Evans AG, Yao N and Aksay IA (2001) Deormation mechanisms in nacre. J Mater Res 16: [33] Xu X-P and Needleman A (1994) Numerical simulations o ast crack growth in brittle solids. J Mech Phys Solids 42: [34] Zhang Z, Zhang Y-W and Gao H (2011) On optimal hierarchy o load-bearing biological materials. Proc R Soc Lond B 278:

19 List o igures captions Fig. 1: A schematic representation o potential toughening mechanisms: (a) stress-induced micro-cracking in monolithic ceramics; (b) crack multiplication; (c) crack kinking in layered solids along a brittle interace; (d) crack kinking in layered solids along a ductile interace; (e) crack delection and crack branching in particulate solids; and () plastic dissipation in a ductile phase. Fig. 2: An elastic, isotropic solid containing (a) 2 and (b) 3 semi-ininite, parallel cracks. The geometry depicted in (b) is symmetric about the central horizontal axis, whereas that in (a) is not. In both cases, the coniguration o the cracks is uniquely deined by the angle α, and displacements are imposed on the outer periphery o the solid according to the asymptotic mode I K-ield centred on the average position o the crack tips. Fig. 3: (a) The dimensionless energy release rates G / G (b) and phase angles φ = arctan( KII KI) as a unction o α or the 2 cracks geometry shown in Fig. 2a. Results are reported or both the cracks labeled A and B. Fig. 4: (a) The dimensionless energy release rates G / G and (b) phase angles φ = arctan( KII KI) as a unction o α or the 3 cracks geometry shown in Fig. 2b. Results are reported or both the central crack labeled A and or the two cracks labeled B. Fig. 5: A layered solid containing a semi-ininite initial crack remotely loaded by means o the mode I asymptotic K-ield. The competition between racture by orward crack growth and kinking is modeled via the tensile and shear cohesive zones depicted in the diagram. Fig. 6: Map illustrating the regimes o dominance o orward crack growth (penetration) and crack kinking. Contours o the predicted macroscopic toughness are included or selected values o G / ( σ yup ) and G / ( τ yuk ) along with sketches illustrating the relative sizes o the orward and kinking plastic zones or the 3 cases labelled A through C on the map. Fig. 7: The predicted (a) toughness and (b) lengths o the plastic tensile and shear bands as a unction o σ y / τ y. The dashed and solid curves pertain to situations that lie within the kinking and penetration regimes, respectively o the map in Fig. 6. Fig. 8: A cracked particulate solid made up o hexagonal elastic grains o dimension d and glued together by zero thickness elastic-plastic interaces. The solid is loaded by applying on its boundary the asymptotic mode I K-ield. The eective tractionseparation law or the interaces is depicted in the inset. 19

20 Fig. 9: Finite element predictions o the crack paths (shaded in red) or the particulate solid with t 0 / E = 0.003, kd / E = 100 and R0 / d = 1. Four initial crack conigurations are considered in (a) through (d) as depicted in the insets. The displacement ield is magniied by a actor 10 in order to better illustrate the crack paths. Fig. 10: Predictions o the R-curves or the particulate solid with t 0 / E = 0.003, kd / E = 100 and R0 / d = 1. Results or our cases corresponding to the initial crack conigurations (a) through (d) shown in Fig. 9 are included. Fig. 11: Finite element predictions o the crack paths (shaded in red) or the particulate solid with t 0 / E = 0.003, kd / E = 100 and R0 / d = 7. Four initial crack conigurations are considered in (a) through (d) as depicted in the insets. The displacement ield is magniied by a actor 10 in order to better illustrate the crack paths. Fig. 12: Predictions o the R-curves or the particulate solid with t 0 / E = 0.003, kd / E = 100 and R0 / d = 7. Results or our cases corresponding to the initial crack conigurations (a) through (d) shown in Fig. 11 are included. Fig. 13: (a) Sketch o the crack in an elastic layer sandwiched between elastic-plastic solids as analyzed by Suo et al. (1993) and Tvergaard (1997). (b) Sketch o the cracked multi-layered elastic-plastic composite comprising alternating elastic and elastic-plastic layers o height h and H, respectively. The crack lies within an elastic layer as depicted in (b). Both geometries are subjected to a remote mode I K-ield. Fig. 14: Predictions o the R-curves or the multi-layer composite o Fig.13b or the selected values o h / R 0 and H / R 0. For each value o h / R 0 we have also included the predicted R-curve or the geometry o Fig. 13a: these results are identical to those presented by Suo et al. (1993) and Tvergaard (1997). Fig. 15: Predictions o the steady-state macroscopic toughness Kss / K0 in the multi-layered composite o Fig.13b as a unction o H / R 0 or selected values o h / R 0. Sketches depicting the plastic zones or large and small values o H are included in the inset. 20

21 List o igures Fig. 1: A schematic representation o potential toughening mechanisms: (a) stress-induced micro-cracking in monolithic ceramics; (b) crack multiplication; (c) crack kinking in layered solids along a brittle interace; (d) crack kinking in layered solids along a ductile interace; (e) crack delection and crack branching in particulate solids; and () plastic dissipation in a ductile phase. 21

22 Fig. 2: An elastic, isotropic solid containing (a) 2 and (b) 3 semi-ininite, parallel cracks. The geometry depicted in (b) is symmetric about the central horizontal axis, whereas that in (a) is not. In both cases, the coniguration o the cracks is uniquely deined by the angle α, and displacements are imposed on the outer periphery o the solid according to the asymptotic mode I K-ield centred on the average position o the crack tips. Fig. 3: (a) The dimensionless energy release rates G / G (b) and phase angles φ = arctan( KII KI) as a unction o α or the 2 cracks geometry shown in Fig. 2a. Results are reported or both the cracks labeled A and B. 22

23 Fig. 4: (a) The dimensionless energy release rates G / G and (b) phase angles φ = arctan( KII KI) as a unction o α or the 3 cracks geometry shown in Fig. 2b. Results are reported or both the central crack labeled A and or the two cracks labeled B. Fig. 5: A layered solid containing a semi-ininite initial crack remotely loaded by means o the mode I asymptotic K-ield. The competition between racture by orward crack growth and kinking is modeled via the tensile and shear cohesive zones depicted in the diagram. 23

24 Fig. 6: Map illustrating the regimes o dominance o orward crack growth (penetration) and crack kinking. Contours o the predicted macroscopic toughness are included or selected values o G / ( σ yup ) and G / ( τ yuk ) along with sketches illustrating the relative sizes o the orward and kinking plastic zones or the 3 cases labelled A through C on the map. Fig. 7: The predicted (a) toughness and (b) lengths o the plastic tensile and shear bands as a unction o σ y / τ y. The dashed and solid curves pertain to situations that lie within the kinking and penetration regimes, respectively o the map in Fig

25 Fig. 8: A cracked particulate solid made up o hexagonal elastic grains o dimension d and glued together by zero thickness elastic-plastic interaces. The solid is loaded by applying on its boundary the asymptotic mode I K-ield. The eective tractionseparation law or the interaces is depicted in the inset. Fig. 9: Finite element predictions o the crack paths (shaded in red) or the particulate solid with t0 / E = 0.003, kd / E = 100 and R0 / d = 1. Four initial crack conigurations are considered in (a) through (d) as depicted in the insets. The displacement ield is magniied by a actor 10 in order to better illustrate the crack paths. 25

26 Fig. 10: Predictions o the R-curves or the particulate solid with t0 / E = 0.003, kd / E = 100 and R0 / d = 1. Results or our cases corresponding to the initial crack conigurations (a) through (d) shown in Fig. 9 are included. Fig. 11: Finite element predictions o the crack paths (shaded in red) or the particulate solid with t0 / E = 0.003, kd / E = 100 and R0 / d = 7. Four initial crack conigurations are considered in (a) through (d) as depicted in the insets. The displacement ield is magniied by a actor 10 in order to better illustrate the crack paths. 26