Phase field modeling of fracture in multi-physics problems. Part I. Balance of crack surface and failure criteria for brittle crack propagation in thermo-elastic solids

This work presents a generalization of recently developed continuum phase field models for brittle fracture towards fully coupled thermo-mechanical and multi-physics problems at large strains. It outlines a rigorous geometric approach to the diffusive crack modeling based on the introduction of a balance of regularized crack surface, governed by a crack phase field. The regularized crack surface functional is based on a crack surface density function, that describes the macroscopic crack surface in the bulk material per unit of the reference volume. The approach overcomes difficulties associated with the computational realization of sharp crack discontinuities, in particular when it comes to complex crack topologies. The formulation proposed is essentially a gradient damage theory, however, equipped with critical ingredients rooted in fracture mechanics. A modular concept is outlined for the linking of the diffusive crack modeling with complex multi-field response of the bulk material, where focus is put on the model problem of finite thermo-elasticity. This concerns a generalization of crack driving forces from the energetic definitions towards stress-based criteria, the constitutive modeling of heat conduction across cracks and convective heat exchanges at crack faces based on additional constitutive functions. This is achieved by approximating surface load integrals of the sharp crack approach by distinct volume integrals. We demonstrate the performance of the phase field formulation of fracture at large strains by means of representative numerical examples.