  The thermodynamical model of viscoelastic deformable solids at finite strains
with Kelvin-Voigt rheology with a higher-order viscosity (using the concept of
multipolar materials) is formulated in a fully Eulerian way in rates.
Assumptions used in this paper allow for a physically justified free energy
leading to non-negative entropy that satisfies the 3rd law of thermodynamics,
i.e. entropy vanishes at zero temperature, and energy-controlled stress. This
last attribute is used advantageously to prove the existence and a certain
regularity of weak solutions by a simplified Faedo-Galerkin
semi-discretization, based on estimates obtained from the total-energy and the
mechanical-energy balances. Some examples that model neo-Hookean-type materials
are presented, too.
