On a doubly nonlinear model for the evolution of damaging in viscoelastic materials

We consider a model describing the evolution of damage in visco-elastic materials, where both the stiffness and the viscosity properties are assumed to degenerate as the damaging is complete. The equation of motion ruling the evolution of macroscopic displacement is hyperbolic. The evolution of the damage parameter is described by a doubly nonlinear parabolic variational inclusion, due to the presence of two maximal monotone graphs involving the phase parameter and its time derivative. Existence of a solution is proved in some subinterval of time in which the damage process is not complete. Uniqueness is established in the case when one of the two monotone graphs is assumed to be Lipschitz continuous