  Isothermal visco-elastodynamics in the Kelvin-Voigt rheology with is
formulated in the spatial Eulerian coordinates in terms of velocity and
deformation gradient. A generally nonconvex (possibly also frame-indifferent)
stored energy is admitted. The model involves a nonlinear 2nd-grade nonsimple
(multipolar) viscosity so that the velocity field is well regular. To simplify
analytical arguments, volume variations of the solid material material are
assumed only rather small so that the mass density is constant, exploiting the
concept of semi-compressible materials. Existence of weak solutions is proved
by using the Galerkin method combined with a suitable regularization, using
nontrivial results about transport by smooth velocity fields.
