Mixed Lagrangian formalism for temperature-dependent dynamic thermoplasticity

A variational temperature-dependent formulation is proposed and developed for dynamical problems of thermoplasticity. The variational method is based on a Hamiltonian approach using a weak form of the mixed thermoplastic governing equations. Interestingly, within this framework, when an exponential dependence on temperature is assumed for the material properties (e.g., viscous dashpot and yielding force), the correct viscous and plastic dissipated energy terms are obtained for the entropy-energy equation. With this in place, a discrete variational calculus approach is adopted to represent the nonlinear discrete equations of motion. Next, several case studies are performed with a lumped-parameter thermoplastic model to investigate the exponential dependence of the material properties on temperature. The thermoplastic model is subjected to cycling external forces and external heat sources, for which continuous softening of the material properties is observed per cycle. This suggests that the proposed multiphysics mixed variational formulation may be used to capture a range of complex rate-dependent thermoplastic material behavior. Finally, a discussion on the convexity of the proposed thermoplastic formulation is included.