The fracture resistance of elastic networks increases with the density of defects like a random walk

Abstract

Disordered spring networks are a well-established model system to study fracture in a wide range of materials, from ceramics to polymer networks and mechanical metamaterials, across length scales from the atomistic to the macroscopic. A central quantity characterizing fracture is the apparent fracture energy $G^c$, which measures the resistance to the propagation of a preexisting dominant crack. While it is well established that disorder can increase $G^c$ through crack arrest by local inhomogeneities, its dependence on the degree of disorder remains poorly understood. Here, we study the effect of varying concentrations of missing bonds on crack propagation of an otherwise perfect two-dimensional triangular network of springs. For a given network with a fixed concentration of missing bonds, the apparent fracture energy $G^c(a)$ increases with crack advance $a$. This behavior can be explained by mapping the effect of the missing bonds onto an equivalent local fracture energy landscape $\Gamma^{loc}(a)$ and applying established theories linking planar crack arrest with fluctuations in $\Gamma^{loc}(a)$. For increasing fraction of missing bonds $\nu$, the standard deviation of the fluctuations of $\Gamma^{loc}$ increases with $\sqrt{\nu}$, which we explain by considering a random-walk-like superposition of perturbations caused by individual missing bonds. We demonstrate that as a consequence of crack arrest by fluctuations in $\Gamma^{loc}$, the average $G^c(a)$ follows the same $\sqrt{\nu}$ scaling. Furthermore, we observe that the probability density of $\Gamma^{loc}$ has an exponential tail leading to a logarithmic increase of $G^c(a)$ with crack advance $a$. Our results quantitatively link microstructural disorder to macroscopic fracture energy and paves the way for quantitative predictions of the fracture energy in a wide variety of materials.