  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.
