On the variational formulation of nonlinear coupled thermomechanical constitutive equations and updates

A variational formulation of the coupled thermo-mechanical boundary-value problem for general dissipative solids is presented. The coupled thermo-mechanical boundary value problem under consideration consists of the equilibrium problem for a deformable, inelastic and dissipative solid with the heat conduction problem appended in addition. The variational formulation allows for general dissipative solids, including finite elastic and plastic deformations, non-Newtonian viscosity, rate sensitivity, arbitrary flow and hardening rules, as well as heat conduction.We showed that a joint potential function exists such that both the conservation of energy and the balance of linear momentum equations follow as Euler-Lagrange equations. This variational formulation of the thermo-mechanical boundary-value problem is then extended to the incremental framework.This allows to rewrite the update algorithm for the unknown fields over a given time (load) step as an optimization problem. The resulting variational update leads to an eÿcient numerical fully-coupled (i.e. monolithic) finite element formulation of thermo-mechanical problems