Thermo-visco-elasticity for Norton–Hoff-type models with homogeneous thermal expansion

Abstract

In this work we study a quasi-static evolution of thermo-visco-elastic model with homogeneous thermal expansion. We assume that material is subject to two kinds of mechanical deformations: elastic and inelastic. Inelastic deformation is related to a hardening rule of Norton-Hoff type. Appearance of inelastic deformation causes transformation of mechanical energy into thermal one, hence we also take into the consideration changes of material's temperature.The novelty of this paper is to take into account the thermal expansion of material. We are proposing linearisation of the model for homogeneous thermal expansion, which preserves symmetry of system and therefore total energy is conserved. Linearisation of material's thermal expansion is performed in definition of Cauchy stress tensor and in heat equation. In previous studies, it was done in different way. Considering of such linearisation leads to system where the coupling between temperature and displacement occurs in two places, i.e. in the constitutive function for the evolution of visco-elastic strain and in the additional term in the heat equation, in comparison to models without thermal expansion. The second coupling was not considered previously. For such system of equations we prove the existence of solutions. Moreover, we obtain existence of displacement's time derivative, which has not been done previously.2000 Mathematics Subject Classification. 74C10, 35Q74, 74F05.

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

“…Additionally, solutions were obtained only in the renormalised sense. In works [28,29,34] and [27], Gwiazda et al proposed a very complicated two-level Galerkin approximation for the visco-elastic strain tensor. With the help of this approximation, they were able to consider a flow rules which depends on the temperature.…”

“…The standard unknowns in this theory are the displacement field u : Ω × [0, T] → R 3 and the stress tensor σ : Ω × [0, T] → S 3 , where S 3 denotes the set of symmetric 3 × 3-matrices. A very popular description is described in Alber's book [1], which was very often used in recent years, see for example [2,17,23,27,29,43]. In this approach the inelastic part of the infinitesimal strain tensor ε(u) = sym(∇ x u) is described by an additional internal variable ε p i.e.…”

“…where G is a given constitutive function, which in our model depends on T and θ. We are going to deal with a generalization of the Norton-Hoff type constitutive function which is known and very popular in the literature, see [9,23,27,29,47]. More precisely, we replace equation (1.8) by…”

“…The nonlinear term is usually approximated in the linear theory by a linear term c 0 div u t with the physical argument that the temperature in the deformation process remains close to the reference temperature, see e.g. [6,27,31,41]. The first mathematical analysis of thermoelasticity including the nonlinear heat equation was done in [12] and later in [13].…”

“…Moreover, in these articles the coupling with thermal effects occurs only in an elastic constitutive relationship (the constitutive function G does not depend on θ). Additionally, it is worth to emphasize the works [28,29,34] and [27,41], where the authors deal with thermo-visco-elasticity systems. However, in these works the thermal expansion does not appear, which means that the Cauchy stress tensor does not depend on temperature function.…”