Stochastic phase-field modeling of brittle fracture: computing multiple\n crack patterns and their probabilities

Abstract

In variational phase-field modeling of brittle fracture, the functional to be\nminimized is not convex, so that the necessary stationarity conditions of the\nfunctional may admit multiple solutions. The solution obtained in an actual\ncomputation is typically one out of several local minimizers. Evidence of\nmultiple solutions induced by small perturbations of numerical or physical\nparameters was occasionally recorded but not explicitly investigated in the\nliterature. In this work, we focus on this issue and advocate a paradigm shift,\naway from the search for one particular solution towards the simultaneous\ndescription of all possible solutions (local minimizers), along with the\nprobabilities of their occurrence. Inspired by recent approaches advocating\nmeasure-valued solutions (Young measures as well as their generalization to\nstatistical solutions) and their numerical approximations in fluid mechanics,\nwe propose the stochastic relaxation of the variational brittle fracture\nproblem through random perturbations of the functional. We introduce the\nconcept of stochastic solution, with the main advantage that point-to-point\ncorrelations of the crack phase fields in the underlying domain can be\ncaptured. These stochastic solutions are represented by random fields or random\nvariables with values in the classical deterministic solution spaces. In the\nnumerical experiments, we use a simple Monte Carlo approach to compute\napproximations to such stochastic solutions. The final result of the\ncomputation is not a single crack pattern, but rather several possible crack\npatterns and their probabilities. The stochastic solution framework using\nevolving random fields allows additionally the interesting possibility of\nconditioning the probabilities of further crack paths on intermediate crack\npatterns.\n