  A probabilistic approach to phase-field brittle and ductile fracture with
random material and geometric properties is proposed within this work. In the
macroscopic failure mechanics, materials properties and exactness of spatial
quantities (of different phases in the geometrical domain) are assumed to be
homogeneous and deterministic. This is unlike the lower-scale with strong
fluctuation in the material and geometrical properties. Such a response is
approximated through some uncertainty in the model problem. The presented
contribution is devoted to providing a mathematical framework for modeling
uncertainty through stochastic analysis of a microstructure undergoing
brittle/ductile failure. Hereby, the proposed model employs various
representative volume elements with random distribution of stiff-inclusions and
voids within the composite structure. We develop an allocating strategy to
allocate the heterogeneities and generate the corresponding meshes in two- and
three-dimensional cases. Then the Monte Carlo finite element technique is
employed for solving the stochastic PDE-based model and approximate the
expectation and the variance of the solution field of brittle/ductile failure
by evaluating a large number of samples. For the prediction of failure
mechanisms, we rely on the phase-field approach which is a widely adopted
framework for modeling and computing the fracture phenomena in solids.
Incremental perturbed minimization principles for a class of gradient-type
dissipative materials are used to derive the perturbed governing equations.
This analysis enables us to study the highly heterogeneous microstructure and
monitor the uncertainty in failure mechanics. Several numerical examples are
given to examine the efficiency of the proposed method.
