35D40 Viscosity solutions
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This paper focuses on
rate-independent damage in elastic bodies. Since the driving energy is nonconvex,
solutions may have jumps as a function of time, and in this situation it is known that the classical concept
of energetic solutions for rate-independent systems
may fail to accurately describe the
behavior of the system at jumps.
Therefore, we resort to the (by now well-established) vanishing viscosity approach to rate-independent modeling
and approximate the model by its viscous regularization.
In fact, the analysis of the latter PDE system presents
remarkable difficulties, due to its highly nonlinear character.
We tackle it by combining a variational approach to a class of abstract doubly nonlinear evolution equations, with
careful regularity estimates tailored to this specific system relying on a q-Laplacian type gradient regularization of the damage variable.
Hence, for the viscous problem we conclude the existence of weak solutions satisfying a
suitable energy-dissipation inequality that is the starting point for the vanishing viscosity analysis.
The latter leads to the notion of (weak) parameterized
solution to our rate-independent system,
which encompasses the influence of viscosity in the description of the jump regime.