On thermodynamic- and variational-based formulations of models for inelastic continua with internal lengthscales Original Research Article

The purpose of this work is the thermodynamic and variational formulation of models for multiscale inelastic materials whose behaviour is influenced by the evolution of inelastic microstructure (e.g., dislocation substructure) and the corresponding material or internal lengthscales. To this end, the direct thermodynamic formulation is based on the total energy balance extended to account for the contribution of microscopic processes (e.g., hardening) to the storage, flux and dissipation of energy in the material. Extension of standard invariance arguments as based on the Euclidean frame-indifference of the energy balance to this case yield in an analogous fashion the form of the evolution/field relations. In addition, exploitation of the dissipation principle for a class of general rate-dependent inelastic materials with microstructure yields in this extended setting reduced thermodynamic form for the constitutive relations and so for the balance/evolution field relations. The variational formulation of the model is based on the constitutive modeling of the residual dissipation-rate density via dissipation potential. Together with the free energy density and boundary conditions, this potential determines a rate functional from which and corresponding variational formulation of the model is obtained. In the last part of the work, the rate-based variational formulation of the model is extended to incremental form over a finite time-interval, resulting in a pseudo-hyperelastic form of the model. As has been shown in a number of works, such a form is useful for the investigation of the stability of (homogeneous) deformation processes in inelastic materials. The paper ends with an application of the incremental variational formulation to the case of a materially homogeneous one-dimensional hardening and softening inelastic bar.