Interface failure modes explain non-monotonic size-dependent mechanical properties in bioinspired nanolaminates

Bioinspired discontinuous nanolaminate design becomes an efficient way to mitigate the strength-ductility tradeoff in brittle materials via arresting the crack at the interface followed by controllable interface failure. The analytical solution and numerical simulation based on the nonlinear shear-lag model indicates that propagation of the interface failure can be unstable or stable when the interfacial shear stress between laminae is uniform or highly localized, respectively. A dimensionless key parameter defined by the ratio of two characteristic lengths governs the transition between the two interface-failure modes, which can explain the non-monotonic size-dependent mechanical properties observed in various laminate composites.

Laminated composite architectures have been widely used to make tough and damage tolerant materials which are intrinsically brittle [1][2][3][4] . It is well accepted that interface failure by delamination or debonding and/or friction between adjacent laminae plays an important role in enhancing toughness and damage tolerance [5][6][7][8][9][10][11] . Usually laminated composites are continuous, i.e. ceramic-metal laminates 12,13 , metal-intermetallic multilayers 14,15 , and multilayered ceramics with weak interlayer 1,16-18 etc. When the strength of the weak interlayer bond is below a critical value compared to the strength of the lamina, the nucleated cracks in each lamina are deflected onto the interface followed by interface failure [5][6][7] , then it becomes a discontinuously laminated structure. Within the structure, the penetrations of these cracks into adjacent discontinuous laminae that usually lead to a straight propagation path and a catastrophic failure are significantly mitigated. Instead, a non-catastrophic fracture called as a "graceful failure" with the highly meandering crack path is observed 1,[15][16][17][18][19] . Biological materials, i.e. nacre and bone are such good examples to achieve excellent mechanical performance by controlling interface failure via delicate hybrid discontinuous laminate design, a so called brick-and-mortar microstructure 20,21 which is the incorporation of inorganic platelets with optimal aspect ratio and nanometers thick organic weak interlayer. Therefore, fabrication of biomimetic discontinuously laminated composites opens a new way towards superb mechanical properties [21][22][23][24] .
It is known that weak interface can arrest the cracks in the laminae, deflect them onto the interface and facilitate interface failure. The weak interlayer is unnecessary to be soft organic component 25 . Recent studies further show that there are significant size-dependent mechanical properties in discontinuously laminated composites under uniaxial stretch. They are definitely related to the interface failure since the unique stress transfer mechanism in these composites is the lamina/platelet in tension and the interface in shear. Once the maximum shear stress at the edge of the laminae is beyond the interface strength the interface starts to fail and the failure zone can propagate through the overlapping domain which varies with the platelet size and the platelet arrangement [26][27][28][29][30][31][32][33][34][35][36] . Theoretic analysis indicates that the aspect ratio of the platelet in nacre has a characteristic value, at which the platelet and the interface simultaneously undergo uniform failure and that is why nacre exhibits optimal mechanical properties 26,[37][38][39][40] . In contrast, another experimental results report that the polymer/nanoclay nacre-mimetics exhibit high toughness and interface failure with low aspect ratio nanoplatelets, while they exhibit superior stiffness and strength with large aspect ratio nanoplatelets, in accompany with the failure mode changing from interface failure to platelet fracture 33 . The failure mode transition depends on where the maximum stress first reaches the failure point. The effects of the interfacial elasto-plasticity and the overlapping length on the stress field developed in the laminated structure are identified 34 . More interestingly, a non-monotonic size-dependency of the mechanical response is observed in the case of even no platelet fracture. The stiffness, strength, failure strain and toughness can be synergistically increased with the increase of the overlapping length in discontinuously overlapped ply carbon/epoxy composites, and they show a ductile behavior wherein the interface fails progressively although their components are brittle 35 . However in 3D-printed laminated composites under uniaxial stretch the measured failure strain is found to decrease with the increase of the overlapping length, and they exhibit a brittle behavior wherein the propagation of the interface failure is catastrophic 36 . These experimental data seems contradictory. A unified explanation about the size effect of the interface failure in such structures is needed. In most previous theoretic efforts based on interfacial fracture mechanics, the interface failure is considered as a crack. Although when the crack deflects onto the interface has been studied extensively, it remains unclear how the interface failure proceeds and how its propagation correlates the size-dependent mechanical behavior, as well as the so called "graceful failure" with enhanced work of failure.
In this paper, we aim to theoretically elucidate the size effect of the interface failure in such structures mentioned in the literatures. We use a shear lag model to calculate the stress field in the discontinuously laminated composite 34,41-44 under uniaxial stretch. The interface failure zone is considered as a debonding zone wherein the shear stress drops to zero when the failure criterion is met. We then adopt a stress-based criterion to characterize the condition of the interface failure 5 instead of the energy-based criterion usually used in the interfacial fracture mechanics 6 . The consistency of the both criterions is identified 45 . In fact the shear lag model with the stress-based failure criterion shows its capability to study the interface failure in layered nanoceramic composites compared with molecular modeling 46 . Our analytical results for the regularly staggered laminated structure show that the non-monotonic size effect of the interface failure is attributed to the propagation mode transition of the interface failure zone, which is a function of the overlapping length. At a small overlapping length, the interface exhibits catastrophic failure corresponding to the brittle feature, while it shows a steady and progressive failure above a critical overlapping length. The size-dependent mechanical response in the randomly staggered laminates is also discussed. The staggering randomness is found to suppress the single-sided interface failure on the interface with short overlapping length and to promote the double-sided interface failure which significantly increases the failure strain and the work of failure. The calculated step-like stress-strain response that exhibits the feature of "graceful failure" in these randomly staggered laminates is obtained. Its formation can be roughly explained as the result of superposition of propagation of interface failure involving different overlapping lengths.

Results
Mechanical response in regularly staggered laminates. We consider the interface failure process of a typical discontinuous laminate under uniaxial stretch ε c as shown in Fig. 1. The discontinuous laminate can be a continuous laminate after the crack propagation in each lamina is arrested at the interface between lamina (see Fig. 1(a)) or inorganic platelets bonded by weak interlayer with nacre-like brick-and-mortar structure. The representative volume element (RVE) of such structure depends on the size of the platelets l p and their arrangements. When the arrangement is regularly staggered with the offset s = 0.5 ( Fig. 1(b)), the two-dimensional RVE is simplified as shown in Fig. 1(c) and a shear-lag model can be analytically solved to obtain the stress field in the structure. While if the arrangement is randomly staggered, the RVE becomes complex, and the shear-lag model is numerically solved via an overdamping relaxation method to obtain the stress field 34 . In the nonlinear shear-lag model, the interface is assumed to be elastic with τ = G int γ linking the shear stress τ and strain γ by the shear modulus G int and it undergoes brittle failure after the shear stress is above its shear strength τ f with the failure strain γ e f for simplicity. The mechanical equilibrium is governed by for the platelet #1 and for the platelet #2 with E p , h l , u 1 , u 2 the Young's modulus, the thickness of the platelet and the is a characteristic strain with h int the thicknesses of the interlayer, l d is the overlapping length defined in Fig. 1(c is a critical length that characterizes the distribution of the shear stress on the interface, i.e. the shear stress is almost uniform as l d ≤ 0.5l 0 . = a a l / p is the dimensionless length of the interface failure zone. At a given uniaxial stretch ε c the value of a is determined by the following equation When the overlapping length is smaller than a critical value, the failure of the structure is brittle, however it exhibits a ductile behavior after the overlapping length is larger than a critical value. The yield strain ε y is defined at which interface failure starts to appear, namely, the value of a changes from zero to nonzero.  Figure 3 plots the applied stretch as a function of the size of the interface failure zone at different values of the dimensionless key parameter l d /l 0 . It shows that when l d /l 0 is small, the maximum applied stretch reaches at = a 0 and this implies that once the interface failure occurs its propagation is catastrophic, while if l d /l 0 is sufficiently large, the maximum applied stretch reaches at → . a 0 25, and this signifies that there is a steady and progressive interface failure process. Figure 4 shows that the yield strain monotonically decreases till the value of ε 0 with the increase of l d /l 0 , however, the failure strain firstly decreases, then increases and finally converges toward 2ε 0 at sufficiently large value of l d /l 0 . The result in Fig. 4 indicates that on the one hand, the failure strain could be much larger than the yield strain where a progressive non-catastrophic interface failure contributes a ductile behavior in the laminate although the platelets and the interlayer are intrinsically brittle, on the other hand, in the structure with l d /l 0 < 1.95 the failure strain is very close to the yield strain, and it exhibits a brittle behavior. The reason is that the shear stress on the interface is almost uniform and the interface tends to undergo uniform rupture. In the case of large l d /l 0 , the shear stress is highly localized. Figure 5 plots the shear stress profile along the interface with the increase of uniaxial stretch at l d /l 0 = 15. The result clearly shows that the interface is separated into three distinct zones: failure zone, localized shear zone, and zero shear stress zone. With the increase of the uniaxial stretch, the failure zone can grow steadily without the drop of the tensile stress in the platelets at the expense of decreasing the zero shear stress zone until it disappears. Therefore it is the steady profile of the localized shear stress that guarantees a progressive interface failure. Our simple model thus elucidates two distinct interface failure modes:    can explain why the failure strain in the 3D-printed laminate 36 decreases while the failure strain in the discontinuously overlapped ply carbon/epoxy laminate 35 increases with the increase of the overlapping length. Based on estimating the value of l d /l 0 in the two laminates 35,36 , we find that l d /l 0 < 2 (l 0 = 5/λ with the interface plasticity taken into account 34,36 , l d = L, l d /l 0 ≈ λL/5) in the former while l d /l 0 ≈ 21 (l 0 = 37 mm, l 0 = 7.64 mm) in the latter 35 . From Fig. 4, the two laminates are exactly located the two sides of the deflection point l d /l 0 ≈ 2, respectively. Therefore their failure strains show different size-dependency.
Mechanical response in randomly staggered laminates. The overlapping length depends on not only the size of the platelets but also their arrangements. In the real case as shown in Fig. 1(a), the discontinuous laminates are more or less randomly staggered. Hence the overlapping length is not uniform any more but widely distributed. We need to discuss the effect of the staggering randomness on the size-dependent mechanical response. We introduce a distributed offset s to characterize the staggered arrangement. For example, the random staggering can be generated by a normal distribution with the mean = .
s 0 5 and a standard deviation Δs. In this setting, the average overlapping length l d is equal to 0.5l p , the large value of Δs implies the overlapping length is very unevenly distributed. The regularly staggered laminate as we discussed in the above section is the limit of Δs = 0. If the value of the offset s has a distribution, the interface failure processes between adjacent platelets could be not simultaneous any more. The mechanical equilibrium configuration of the structure under uniaxial stretch is hard to be analytically solved but can be obtained by minimizing the total energy of the structure E tot including the elastic energy in the all layers and the interface energies between all adjacent layers. The strong elastic interactions between adjacent layers can be taken into account by modeling the interlayer as a cohesive interface at which the interface energy is a functional of the displacement jump across the interface. The total energy can thus be written as a functional of the displacement in each layer u (i) (See the details in the Supplementary Material Appendix B). Based on a gradient flow directed relaxation model, we assume that the energy minimization process is governed by the Ginzburg-Landau kinetic equation 34 . After the randomly staggered structure is generated (see the Figure S2 in the Supplementary Material), the evolution of the kinetic equation could track the stress field and the failure process in the laminate.
As we have checked (see the Figure S3 in the Supplementary Material), in the case of regularly staggered laminates with the offset s ≠ 0.5, i.e. s = 0.3, the overlapping length can be 0.3l p or 0.7l p , the maximum interfacial stress appears at the interface zone with smaller overlapping length. Therefore the interface failure in regularly staggered laminates with the offset s ≠ 0.5, i.e. s = 0.49, prefers to occur only on the interface zone with smaller overlapping length instead of double-sided failure, and the enhancement of the resultant failure strain due to the single-sided interface failure drops to one-half, compared to the case of double-sided failure in all the interfaces (see the Figure S4 in the Supplementary Material). Figure 6 plot the calculated stress-strain curves of the ten-layers discontinuous laminated structure with a distributed offset in these layers s = All the curves demonstrate a step-like or saw-tooth load-displacement response replicating the so called "graceful failure" widely observed in many tough laminates 1,15-19 . Figure 6(a) shows that the laminate with smaller average overlapping length has higher failure strain, in consistent with the previous study for the nacreous laminates using finite element method 48 wherein the aspect ratio of the platelets is not big such that the shear stress is uniform with high stress-transfer efficiency 26 . Figure 6(b) indicates that the laminate with larger average overlapping length has higher failure strain, in good agreement with the previous experiment for the discontinuous laminate with ultrahigh aspect ratio of the platelet 35 . The results in Fig. 6 also demonstrate that the interface failure in the staggered laminate with a distributed offset tends to be a progressive layer-by-layer failure process. However, the overall mechanical response in the laminate cannot be simply viewed as the result of the superposition of propagation of interface failure involving different overlapping lengths (in which the average stress is σ σ , and the average strain is the applied stretch ε c ) since there are strong elastic interactions between adjacent layers. Figure 7 further shows the snapshots of the laminated structure during the interface failure process at the points D-F and O-P marked in Fig. 6(b). Figure 7D-F demonstrate that the strain hardening feature of the stress-strain curve in Fig. 6(b) is due to the fact that the adjacent interface starts to carry much more load with the increase of the applied stretch until it also fails completely. Figure 7O-Q elucidate that a sharp stress drop and the appearance of the platform in the stress-strain curve in Fig. 6(b) is due to the unstable and progressively stable propagation of the interface failure, respectively. In addition, we note that the single-sided interface failure dominating in the regularly staggered laminates with s ≠ 0.5 is mitigated in the randomly staggered laminates especially at the large value of l l / d 0 . The results in Fig. 7 clearly show that with the increase of l l / d 0 formation of double-sided interface failure is more favorable and it is not sensitive to the staggering arrangement any more. To check whether large value of l l / d 0 favors the double-sided interface failure or not, we estimate the points (l l , d 0 ) in many typical discontinuous laminates. The input data is from the Table 1 in the Supplementary Material Appendix D. The result in Fig. 8 indicates that with the increase of l l / d 0 the single-sided interface failure mode exactly tends to change into the double-sided interface failure mode.
Obviously the double-sided interface failure could provide more tensile strain capacity and work of failure in comparison with the case of single-sided interface failure. Figure 9 plots the calculated failure strain and the work of failure as a function of l l / d 0 in randomly staggered laminates compared with those of regularly staggered ones. The result in Fig. 9 demonstrates that the failure strain and the dimensionless work of failure in the randomly staggered laminate can be larger than those in regularly staggered laminates with the offset s ≠ 0.5 wherein single-sided interface failure is dominant. The correlation between the large value of

Discussion
In the above modeling and simulation, the interface is assumed to be brittle. For the interface involving significant plastic yielding 38 , we could model the interface to be elastic perfectly plastic 34 . Our calculated results indicate that interface plasticity promotes to homogenize the interfacial shear stress and the localization of the interfacial shear stress only appears in the nacreous composites with larger overlapping length. Interestingly the two size-dependent interface failure modes are still observed. It is understandable since the transition of the interface failure modes mainly depends on when the interfacial shear stress between laminae is uniform or highly localized. As we expect, the transition point l d /l 0 = 1.95 shown in Fig. 4 shifts to the position with larger value of l d /l 0 in the presence of notable interface plasticity as we have checked. In addition, we should note that the above conclusions are limited to the case without platelet fracture. They are valid only if the maximum tensile stress in the platelets is smaller than the fracture strength. The maximum tensile stress in the platelets for the case of regularly staggered laminates with s = 0.5 is σ ε = ( )  26 . Therefore, only if the parameters in the discontinuous laminate guarantee the condition σ σ ≤ p m g the platelet fracture will not occur. In addition, the real interface between adjacent platelet usually involves plasticity, friction and interlock, these factors lead to significant stress hardening and thus enhanced probability of the platelet fracture, as is observed in the nacre-mimetics composed of platelets with ultrahigh aspect ratios 33 .
In summary, using nonlinear shear-lag analysis on the discontinuous laminate we have shown that there are two kinds of propagation modes of the interface failure: unstable or progressively stable interface failure, dependent on whether the shear stress at the interface is uniform or localized. The transition from the uniform shear stress to the localized shear stress is determined by the critical ratio of the overlapping length to the characteristic length of the localized shear zone. If the value of the ratio is larger than two, the shear stress tends to be localized, and the interface failure tends to be progressively stable. These results have explained the non-monotonic size-dependent mechanical properties reported in various laminate composites. We expect that the size effect of the interface failure may provide guideline for rational design of strong and tough bioinspired laminates.