  In variational phase-field modeling of brittle fracture, the functional to be
minimized is not convex, so that the necessary stationarity conditions of the
functional may admit multiple solutions. The solution obtained in an actual
computation is typically one out of several local minimizers. Evidence of
multiple solutions induced by small perturbations of numerical or physical
parameters was occasionally recorded but not explicitly investigated in the
literature. In this work, we focus on this issue and advocate a paradigm shift,
away from the search for one particular solution towards the simultaneous
description of all possible solutions (local minimizers), along with the
probabilities of their occurrence. Inspired by recent approaches advocating
measure-valued solutions (Young measures as well as their generalization to
statistical solutions) and their numerical approximations in fluid mechanics,
we propose the stochastic relaxation of the variational brittle fracture
problem through random perturbations of the functional. We introduce the
concept of stochastic solution, with the main advantage that point-to-point
correlations of the crack phase fields in the underlying domain can be
captured. These stochastic solutions are represented by random fields or random
variables with values in the classical deterministic solution spaces. In the
numerical experiments, we use a simple Monte Carlo approach to compute
approximations to such stochastic solutions. The final result of the
computation is not a single crack pattern, but rather several possible crack
patterns and their probabilities. The stochastic solution framework using
evolving random fields allows additionally the interesting possibility of
conditioning the probabilities of further crack paths on intermediate crack
patterns.
