Introduction

Laminated composite architectures have been widely used to make tough and damage tolerant materials which are intrinsically brittle1,2,3,4. 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 tolerance5,6,7,8,9,10,11. Usually laminated composites are continuous, i.e. ceramic-metal laminates12,13, metal-intermetallic multilayers14,15, and multilayered ceramics with weak interlayer1,16,17,18 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 failure5,6,7, 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 observed1,15,16,17,18,19. 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 microstructure20,21 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 properties21,22,23,24.

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 component25. 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 arrangement26,27,28,29,30,31,32,33,34,35,36. 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 properties26,37,38,39,40. 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 fracture33. 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 identified34. 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 brittle35. 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 catastrophic36. 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 composite34,41,42,43,44 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 failure5 instead of the energy-based criterion usually used in the interfacial fracture mechanics6. The consistency of the both criterions is identified45. 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 modeling46. 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 εc 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 lp 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 field34. In the nonlinear shear-lag model, the interface is assumed to be elastic with τ = Gintγ linking the shear stress τ and strain γ by the shear modulus Gint and it undergoes brittle failure after the shear stress is above its shear strength τf with the failure strain for simplicity. The mechanical equilibrium is governed by for the platelet #1 and for the platelet #2 with Ep, hl, u1, u2 the Young’s modulus, the thickness of the platelet and the displacement in the platelets #1 and #2, respectively. After the stress fields are solved (see the details in the Supplementary Material Appendix A), the average stress in the RVE has the form

Figure 1
figure 1

Schematic illustrating (a) the image adapted from ref. 47 for discontinuous Al2O3/LaPO4 laminates after cracking in each layer, (b) sketch of the discontinuously laminated structure for modeling, (c) the shear-lag model for the interface failure process.

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where is a characteristic strain with hint the thicknesses of the interlayer, ld is the overlapping length defined in Fig. 1(c), and is a critical length that characterizes the distribution of the shear stress on the interface, i.e. the shear stress is almost uniform as ld ≤ 0.5l0. is the dimensionless length of the interface failure zone. At a given uniaxial stretch εc the value of is determined by the following equation

Figure 2 plots the calculated effective stress-strain curve of the RVE in Fig. 1(c) under different values of ld by substituting the value of in eq. (2) into eq. (1). The result demonstrates a size-dependent mechanical response. 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 εy is defined at which interface failure starts to appear, namely, the value of changes from zero to nonzero.

Figure 2
figure 2

The calculated stress-strain curve under different overlapping length in regularly staggered discontinuous laminates.

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The failure strain of the RVE with is defined as the maximum attainable applied stretch before the complete failure which poses an energetic instability. We noted that this criterion is consistent with the elastic energetic instability under the strain-controlled boundary condition since in the case of σc > 0 is the same as .

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 ld/l0. It shows that when ld/l0 is small, the maximum applied stretch reaches at and this implies that once the interface failure occurs its propagation is catastrophic, while if ld/l0 is sufficiently large, the maximum applied stretch reaches at , 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 ε0 with the increase of ld/l0, however, the failure strain firstly decreases, then increases and finally converges toward 2ε0 at sufficiently large value of ld/l0. 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 ld/l0 < 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 ld/l0, the shear stress is highly localized. Figure 5 plots the shear stress profile along the interface with the increase of uniaxial stretch at ld/l0 = 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: the catastrophic and non-catastrophic interface failures. The former occurs when the shear stress on the interface is uniform and uniform interface failure is favorable. The latter takes place when the shear stress on the interface is localized and progressive interface failure dominates. Control of the interface mode depends on the value of ld/l0 which determines the distribution of the interfacial shear stress. The size-dependent interface failure mode can explain why the failure strain in the 3D-printed laminate36 decreases while the failure strain in the discontinuously overlapped ply carbon/epoxy laminate35 increases with the increase of the overlapping length. Based on estimating the value of ld/l0 in the two laminates35,36, we find that ld/l0 < 2 (l0 = 5/λ with the interface plasticity taken into account34,36, ld = L, ld/l0 ≈ λL/5) in the former while ld/l0 ≈ 21 (l0 = 37 mm, l0 = 7.64 mm) in the latter35. From Fig. 4, the two laminates are exactly located the two sides of the deflection point ld/l0 ≈ 2, respectively. Therefore their failure strains show different size-dependency.

Figure 3
figure 3

Plot of the applied stretch as a function of the size of the interface failure zone at different values of ld/l0.

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Figure 4
figure 4

Plots of the yield and failure strains as a function of ld/l0 in regularly staggered discontinuous laminates.

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Figure 5
figure 5

The sequential profile of the localized shear stress at the interface with the increase of the applied strain at ld/l0 = 15.

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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 and a standard deviation Δs. In this setting, the average overlapping length is equal to 0.5lp, 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 Etot 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 , where Γis a constant34. 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 0.3lp or 0.7lp, 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 = [0.1, 0.06, 0.4, 0.2, 0.45, 0.4, 0.25, 0.2, 0.4] under different values of . All the curves demonstrate a step-like or saw-tooth load-displacement response replicating the so called “graceful failure” widely observed in many tough laminates1,15,16,17,18,19. 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 method48 wherein the aspect ratio of the platelets is not big such that the shear stress is uniform with high stress-transfer efficiency26. 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 platelet35. 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 , and the average strain is the applied stretch εc) 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 . The results in Fig. 7 clearly show that with the increase of 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 favors the double-sided interface failure or not, we estimate the points () 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 the single-sided interface failure mode exactly tends to change into the double-sided interface failure mode.

Figure 6
figure 6

The calculated stress-strain curves demonstrating a so called “graceful failure” process in the ten-layers randomly staggered discontinuous laminate with a distributed offset s = [0.1, 0.06, 0.4, 0.2, 0.45, 0.4, 0.25, 0.2, 0.4] under different values of for (a), and for (b).

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Figure 7
figure 7

The snapshots of the laminated structure with propagation of the interface failure at the points D–F and O–P marked in Fig. 6(b).

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Figure 8
figure 8

Two characteristic lengths coordinate the transition map from single-sided to double-sided interface failure wherein the input data is from the Table 1 in the Supplementary Material.

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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 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 with 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 and synergistically increased failure strain, strength, and work of failure can now be understood based on the mechanism of the size-dependent interface failure.

Figure 9
figure 9

The calculated (a) failure strain and (b) effective toughness as a function of the average overlapping length in randomly staggered laminates compared with those of regularly staggered ones.

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Discussion

In the above modeling and simulation, the interface is assumed to be brittle. For the interface involving significant plastic yielding38, we could model the interface to be elastic perfectly plastic34. 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 ld/l0 = 1.95 shown in Fig. 4 shifts to the position with larger value of ld/l0 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 . Based on the Griffith theory, the fracture strength is saturated to the theoretic strength σth if the platelet’s thickness is below than a critical value with γ the surface energy of the platelet26. Therefore, only if the parameters in the discontinuous laminate guarantee the condition 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 ratios33.

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.

Additional Information

How to cite this article: Song, Z.Q. et al. Interface failure modes explain non-monotonic size-dependent mechanical properties in bioinspired nanolaminates. Sci. Rep. 6, 23724; doi: 10.1038/srep23724 (2016).