A non-isothermal thermodynamically consistent phase field framework for structural damage and fatigue

We present a general thermodynamically consistent non-isothermal non-local framework for the evolution of damage, fatigue and fracture in materials under the hypothesis of small deformation. The approach is based on the principle of virtual power (PVP), the balance of energy and the second law of thermodynamics in the form of the generalized Clausius-Duhem inequality for the entropy. In addition to the usual physical fields, the model uses the phase field approach to describe the evolution of both damage and fatigue. The kinematic descriptor (phase field) for damage is considered a continuous dynamical variable whose evolution equation is obtained by the PVP. The kinematic descriptor (another phase field) for fatigue is a continuous internal variable whose evolution equation is considered as a constitutive relation to be determined in a thermodynamically consistent way. The behavior of particular material classes can be specified by their corresponding free-energy potentials (which gives the reversible parts of the involved thermodynamic forces) and their associated pseudo-potentials of dissipation (which gives the irreversible parts of the involved thermodynamic forces). To exemplify our general framework, we present the case of an isotropic linear elastic material with viscous dissipation and constant specific heat. The corresponding case of irreversible damage is also presented by using penalization. The considered damage and fatigue phase field approach is a framework from which other methods in the literature may be recovered. The model is approximated by the nodal high-order finite element method with explicit fourth-order Runge-Kutta time integration. Results for one-dimensional examples are presented and conclusions are addressed.