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 is occasionally recorded but not explicitly investigated in the literature.
In this project, we focus on the search for not just one particular solution, but the simultaneous description of all possible solutions (local minimizers), along with the probabilities of their occurrence.
To this end, the stochastic relaxation of the variational brittle fracture problem through random perturbations of the functional is proposed, giving rise to the concept of stochastic solution represented by random fields or random variables with values in the classical deterministic solution spaces.
In the numerical simulations, a simple Monte Carlo approach to compute approximations to such stochastic solutions is used. The final result of the computation is not a single crack pattern, but rather several possible crack patterns and their probabilities.
