Phase field modeling of ductile fracture at finite strains: A variational gradient-extended plasticity-damage theory

Abstract

This work outlines a rigorous variational-based framework for the phase field modeling of ductile fracture in elastic–plastic solids undergoing large strains. The phase field approach regularizes sharp crack surfaces within a pure continuum setting by a specific gradient damage modeling with geometric features rooted in fracture mechanics. It has proven immensely successful with regard to the analysis of complex crack topologies without the need for fracture-specific computational structure such as finite element design of crack discontinuities or intricate crack-tracking algorithms. Following the recent work Miehe et al. (2015), the phase field model of fracture is linked to a formulation of gradient plasticity at finite strains. The formulation includes two independent length scales which regularize both the plastic response as well as the crack discontinuities. This ensures that the damage zones of ductile fracture are inside of plastic zones, and guarantees on the computational side a mesh objectivity in post-critical ranges. The novel aspect of this work is a precise representation of this framework in a canonical format governed by variational principles. The coupling of gradient plasticity to gradient damage is realized by a constitutive work density function that includes the stored elastic energy and the dissipated work due to plasticity and fracture. The latter represents a coupled resistance to plasticity and damage, depending on the gradient-extended internal variables which enter plastic yield functions and fracture threshold functions. With this viewpoint on the generalized internal variables at hand, the thermodynamic formulation is outlined for gradient-extended dissipative solids with generalized internal variables which are passive in nature. It is specified for a conceptual model of von Mises-type elasto-plasticity at finite strains coupled with fracture. The canonical theory proposed is shown to be governed by a rate-type minimization principle, which fully determines the coupled multi-field evolution problem. This is exploited on the numerical side by a fully symmetric monolithic finite element implementation. The performance of the formulation is demonstrated by means of some representative examples.