A Norton-Hoff model with an elastic and inelastic constitutive relationships dependent on temperature

TL;DR

This paper proves the existence of weak solutions for a thermo-visco-elastic model with temperature-dependent elastic and inelastic relationships, incorporating thermal effects into Norton-Hoff plasticity law, a novel approach in this field.

Contribution

It introduces a generalized thermo-visco-elastic model with temperature-dependent properties and demonstrates the existence of weak solutions using advanced mathematical tools, filling a gap in previous literature.

Findings

Established existence of weak solutions for the model.

Extended the Norton-Hoff law to include thermal effects.

Applied advanced mathematical techniques for nonlinear analysis.

Abstract

The aim of this paper is to prove the existence of weak solution for a quasi-static evolution of thermo-visco-elastic model with Norton-Hoff law of plasticity. The dependence on temperature occurs both in the elastic constitutive equations (generalised Hooke's law) and in describing the evolution of visco-elastic strain. These thermal effects have not been previously considered. The approximations of the considered models did not allow in literature such a general models. The main idea of the article is the revocation to R. Temam articles on the plasticity from eighties of the previous century and to write down the equations related to the plastic deformations in the same way. For the obtained equations we propose approximations in a flow rule. Thanks to this manner of writing the equations, we show the existence of a weak solution. To characterize the weak limits in nonlinearities occurring in the system at different levels of approximation, the following tools are used: Young measures, Boccardo's and Gallou{\"e}t's approach, truncation methods for heat equation used by D. Blanchard and Minty-Browder trick.