Multi‐field modeling of thermomechanical coupled fracture problems

In this contribution we focus on a novel variationally consistent formulation for thermomechanical coupled fracture problems. To be specific, we aim on a phase‐field approach to fracture such that the resulting system consists of three fields, the deformation map, the thermal field as well as the crack phase‐field. In the sense of Griffith energy criterion for brittle fracture, a crack initiates or continues upon the attainment of a critical crack energy density. This critical energy exhibits strong temperature dependence, which is approximated by a linear interpolation within our framework. The material behaviour is governed by a modified Helmholtz free energy function in terms of the elastic part of the deformation gradient and the temperature. Postulating that fracture requires a local state of tension, we apply an anisotropic decomposition of the principle stretches of the deformation gradient and reduce the tensile contributions by a suitable degradation function, see [1] for more details. Accordingly, we are able to derive local constitutive relations for the first Piola‐Kirchhoff stress tensor, the entropy density and the phase‐field driving force in variationally consistent manner, see [2]. A further constitutive relation is the Duhamel's law of heat conduction. In case of fracture, the conduction degenerates locally such that we achieve a pure convection problem and the heat transfer depends on the crack opening width. Here, we formulate the conductivity tensor in terms of the phase‐field parameter. Eventually, various numerical examples will show the accuracy and capability of the proposed multi‐field approach to fracture.