Thermodynamics of viscoelastic solids, its Eulerian formulation, and existence of weak solutions

The thermodynamic model of viscoelastic deformable solids at finite strains is formulated in a fully Eulerian way in rates. Also, effects of thermal expansion or buoyancy due to evolving mass density in a gravity field are covered. The Kelvin–Voigt rheology with a higher-order viscosity (exploiting the concept of multipolar materials) is used, allowing for physically relevant frame-indifferent stored energies and for local invertibility of deformation. The model complies with energy conservation and Clausius–Duhem entropy inequality. Existence and a certain regularity of weak solutions are proved by a Faedo–Galerkin semi-discretization and a suitable regularization. Subtle physical limitations of the model are illustrated on thermally expanding neo-Hookean materials or materials with phase transitions.