  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.
