On the continuum thermodynamic rate variational formulation of models for extended crystal plasticity at large deformation

The purpose of this work is the unified formulation and generalization of selected models for extended, gradient, or “higher-order” crystal plasticity via the application of a recently developed rate variational approach to the formulation of continuum thermodynamic models for history-dependent, inelastic systems. The investigation here includes models which were not originally formulated in a thermodynamic or “work-conjugate” fashion. The approach is based on the formulation of rate potentials for each model whose form is determined by (i) energetic processes via the free energy, (ii) kinetic processes via the dissipation potential, and (iii) the form of the evolution relations for the internal-variable-like quantities upon which the free energy and dissipation potential depend. For the case of extended crystal plasticity, these latter quantities include for example the inelastic local deformation, or dislocation densities. The stationarity conditions of the corresponding rate functional then yield volumetric and surficial balance-like field relations determining in the current context for example the form of momentum balance or that of the generalized glide-system flow rule. With the help of this approach, we derive thermodynamically consistent forms of specific models for extended crystal plasticity. Since most of these were formulated for small deformation, we also investigate their generalization to large deformation with the help of, e.g., form invariance. Among other things, the current rate variational approach implies that, beyond the form of the free energy itself, it is form of the evolution relations for the dislocation densities which is important in determining whether or not higher-order model quantities like the glide-system back stress can be formulated in a thermodynamic fashion.