Patterns of load, elastic energy and damage in network models of architected composite materials

Abstract

We investigate the role of architected thin films in the interfacial failure properties of bi-layer composites. Our results show that, while graded structures can be used to prescribe failure at the interface, they do not offer significant advantages in terms of fracture toughness. Hierarchically patterned layers can localize failure at the interface and simultaneously enhance interface toughness, by enforcing a buffer region where elastic energy is dissipated in the form of diffuse damage, so that no stress concentration can drive crack growth. To analyze these mechanisms, the associated patterns of local load redistribution and the soft deformation modes, we develop a network formalism that brings together concepts of discrete differential geometry and spectral graph theory.

Figures (8)

• Figure 1: Network model of a bi-layer composite, and details of the construction of a hierarchical H top layer. Figure partially adapted from Greff2024_SciRep. The case of $s=4$ is shown. (a) Nodes and edges are distributed in a 3D cubic lattice. The boundaries at $x=L$ (shown in gray) and $y=L$ (not shown) are periodic. The boundaries at $z=\pm(L_z+1/2)$ (square nodes) are not periodic. Edges at the interface ($z=0$) are plotted in yellow; (b) In order to generate a hierarchical H top layer, in the region with $z>0$, $x/y$-edges are recursively removed to form cuts. The case of a deterministic hierarchical cut pattern is shown. Cuts of levels $h=1$, $h=2$, $h=3$ are shown in red, blue, green respectively. Periodic boundaries are introduced as additional highest-level cuts (pale green). (c) 3D view of the deterministic cut structure of (b). (d) 3D view of the final H cut pattern, obtained by shuffling cut positions.
• Figure 2: Schematic representation of the T/S bi-layer composite model. The top layer T can be of type H (hierarchical), G (graded) and R (random). The substrate layer S is of type R. Microstructures of each layer type are plotted by displaying the portions of the layer that are removed. H layers exhibit patterns of extended cuts of different heights, originating from the bottom boundary and ensuring that the layer is sparser near that boundary. G layers exhibit the same density profile as H, but matter is removed in the form of randomly distributed voids. R layers too exhibit randomly distributed voids, but the density is homogeneous. The globally averaged sparsity/density is the same for all systems.
• Figure 3: (a) Average constitutive behavior of the edges in the substrate layer, for different values of the parameter $c$. The average failure threshold is chosen such that curves of different stiffness have, on average, the same area (work of failure). (b) Typical stress-strain curve under displacement control (thick dot-dashed black line), constructed by enveloping the stress-strain curve obtained from a quasi-static simulation (thin solid blue line).
• Figure 4: Fracture strength and fracture profiles. Data are for systems with $c=1$ (a), $c=2.0$ (b), and $c=0.5$ (c). Peak stress $\sigma_\mathrm{p}$ and specific work of failure $w_\mathrm{f}$ are plotted as functions of notch sizes $a$. The probability $p(z)$ of observing cracks at height $z$ is computed for the un-notched systems ($a=0$).
• Figure 5: Energy profiles. The average fraction $E_\mathrm{h}/E$ of elastic energy stored in $x/y$-edges is plotted against the $z$ coordinate, in case of $c=1$, for different notch sizes $a$, in pristine systems. For $a=0$$x/y$-edges carry no load, and thus store no energy. For any $a>0$, instead, $x/y$-edges store energy as they participate in the stress redistribution. This role is however suppressed in hierarchical top layers (H/R, for $z>0$).