Analysis and simulations for a phase-field fracture model at finite strains based on modified invariants

Abstract

Phase-field models have already been proven to predict complex fracture patterns for brittle fracture at small strains. In this paper we discuss a model for phase-field fracture at finite deformations in more detail. Among the identification of crack location and projection of crack growth the numerical stability is one of the main challenges in solid mechanics. Here we present a phase-field model at finite strains, which takes into account the anisotropy of damage by applying an anisotropic split of the modified invariants of the right Cauchy-Green strain tensor. We introduce a suitable weak notion of solution that also allows for a spatial and temporal discretization of the model. In this framework we study the existence of solutions and we show that the time-discrete solutions converge in a weak sense to a solution of the time-continuous formulation of the model. Numerical examples in two and three space dimensions illustrate the range of validity of the analytical results.