A Proper Generalized Decomposition (PGD) approach to crack propagation in brittle materials: with application to random field material properties

Abstract

Understanding the failure of brittle heterogeneous materials is essential in many applications. Heterogeneities in materialproperties are frequently modeled through random fields, which typically induces the need to solve finite element problemsfor a large number of realizations. In this context, we make use of reduced order modeling to solve these problems at anaffordable computational cost. This paper proposes a reduced order modeling framework to predict crack propagation in brittlematerials with random heterogeneities. The framework is based on a combination of the Proper Generalized Decomposition(PGD) method with Griffith’s global energy criterion. The PGD framework provides an explicit parametric solution forthe physical response of the system. We illustrate that a non-intrusive sampling-based technique can be applied as a post-processing operation on the explicit solution provided by PGD. We first validate the framework using a global energy approachon a deterministic two-dimensional linear elastic fracture mechanics benchmark. Subsequently, we apply the reduced ordermodeling approach to a stochastic fracture propagation problem.