Calculus of Variations and Geometric Measure Theory

D. Knees - R. Rossi - C. Zanini

A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains

created by rossi on 14 Nov 2013
modified by knees on 19 Dec 2017


Published Paper

Inserted: 14 nov 2013
Last Updated: 19 dec 2017

Journal: Nonlinear Analysis Series B: Real World Applications
Volume: 24
Pages: 126-162
Year: 2015


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.