On renormalized solutions for thermomechanical problems in perfect plasticity with damping forces

Abstract

We consider the quasi-static evolution of the thermo-plasticity model in which the evolution equation law for the inelastic strain is given by the Prandtl-Reuss flow rule. The thermal part of the Cauchy stress tensor is not linearised in the neighbourhood of a references temperature. This nonlinear thermal part imposed to add a damping term to the balance of the momentum, which can be interpreted as external forces acting on the material.In general the dissipation term occurring in the heat equation is integrable function only and the standard methods can not be applied. Combining truncation techniques and Boccardo-Gallouët approach with monotone methods we prove an existence of renormalised solutions.

On a thermo‐visco‐elastic model with nonlinear damping forces andL1temperature data

“…It is worth noting that the case where F is linear has been studied before in previous articles. 7,9,20,29 It is assumed that considered body is subjected to i.a. thermal expansion; therefore, the total stress tensor consists of two parts…”

“…The majority of articles that consider inelastic deformations do not analyze the non-negativity of the temperature at all, as can be observed in previous works. 7,8,[18][19][20][21][22][23][24][25] For the sake of clarity, in the following sections of this chapter, we present the laws of physics that enable to formulate the equations, which describe the body evolution in our thermo-visco-elastic system. In addition, we show that our model is consistent with the laws of thermodynamics.…”

“…What is more important, none of the problems describes the thermo‐visco‐elastic model in three dimensional space with the additional nonlinear term in the equation of motion (see previous studies 10,16,17 ). The majority of articles that consider inelastic deformations do not analyze the non‐negativity of the temperature at all, as can be observed in previous works 7,8,18–25 …”

“…This additional term on the right‐hand side of Equation () is often called the damping force. It is worth noting that the case where $$$ \mathrm{F} $$$ is linear has been studied before in previous articles 7,9,20,29 …”