Fracture in Disordered Heterogeneous Materials as a Stochastic Process

Fracture processes in heterogeneous materials comprise a large number of disordered spatial degrees of freedom, representing the dynamical state of a sample over the entire domain of interest. This complexity is usually modeled directly, obscuring the underlying physics, which can often be characterized by a small number of physical parameters. In this paper, we derive a closed-form expression for a low dimensional model that reproduces the stochastic dynamical evolution of time-dependent failure in heterogeneous materials, and efficiently captures the spatial fluctuations and critical behavior near failure. Our construction is based on a novel time domain formulation of Fiber Bundle Models, which represent spatial variations in material strength via lattices of brittle, viscoelastic fiber elements. We apply the inverse transform method of random number sampling in order to construct an exact stochastic jump process for the failure sequence in a material with arbitrary strength distributions. We also complement this with a mean field approximation that captures the coupled constitutive dynamics, and validate both with numerical simulations. Our method provides a compact representation of random fiber lattices with arbitrary failure distributions, even in the presence of rapid loading and nontrivial fiber dynamics.