  We consider nonlinear viscoelastic materials of Kelvin-Voigt type with stored
energies satisfying an Andrews-Ball condition, allowing for non convexity in a
compact set. Existence of weak solutions with deformation gradients in $H^1$ is
established for energies of any superquadratic growth. In two space dimensions,
weak solutions notably turn out to be unique in this class. Conservation of
energy for weak solutions in two and three dimensions, as well as global
regularity for smooth initial data in two dimensions are established under
additional mild restrictions on the growth of the stored energy.
