A variational formulation of thermomechanical constitutive update for hyperbolic conservation laws

In this paper, a variational framework is proposed for the constitutive update of thermomechanical constitutive models in the special case where their input results from quantities directly updated by hyperbolic conservation laws. Both a continuum and a consistent first order accurate discrete settings are derived. The originality of this work lies in that the constitutive update is driven by the rates of some strain measure and the internal energy density in the continumm setting, leading to a rate-type description of the local constitutive problem, and by the updated values at some discrete time of these strain measure and internal energy density in the discrete setting. These quantities are updated by the solution of a system of discrete conservation laws including the first principle of thermodynamics, ensuring that the right shock speeds will be computed. This point is of crucial importance when simulating impact on structures for instance. The proposed variational approach is illustrated for thermo-hyperelastic-viscoplastic solid media, especially using the parameterization of the flow rule direction based on pseudo-stresses proposed by Mosler \& co-workers. The proposed discrete variational solver is then coupled with the second order accurate flux difference splitting finite volume method, which permits to solve the set of conservation laws. Comparisons are performed on a set of test cases with numerical solutions obtained with finite elements coupled to an explicit time-stepping and to a temperature-driven variational constitutive update. They allow to show the good behavior of the proposed approach