Vanishing‐viscosity solutions to a rate‐independent two‐field gradient damage model

Abstract

A rate-independent damage model which features two damage variables coupled through a penalty term in the stored energy is considered. Since the energy functional is nonconvex, solutions may be discontinuous in time. This calls for suitable notions of (weak) solutions which allow for jumps. We resort to a vanishing-viscosity approach based on an 𝐿 2 (Ω)−arclength parametrization, where parametrized solutions arise as a limit of the (reparametrized) graphs of the viscous solutions in the extended state space. This enables us to prove that vanishing-viscosity solutions exist and belong to the class of parametrized solutions. We show that the latter can be characterized in various different ways. These alternative formulations highlight the influence of the viscous effects at the jump points, while at the continuity points, the evolution displays a rateindependent behavior. As it turns out, the behavior of the system at each jump point is described by an ordinary differential equation in Banach space during which the physical time is constant.