Approximating dynamic phase-field fracture in viscoelastic materials with a first-order formulation for velocity and stress

Abstract

We investigate a model for dynamic fracture in viscoelastic materials at small strains. While the sharp crack interface is approximated with a phase-field method, we consider a viscous evolution with a quadratic dissipation potential for the phase-field variable. A non-smooth constraint enforces a unidirectional evolution of the phase-field, i.e. material cannot heal. The viscoelastic equation of motion is transformed into a first order formulation and coupled in a nonlinear way to the non-smooth evolution law of the phase field. The system is fully discretized in space and time with a discontinuous Galerkin approach for the first-order formulation. Based on this, existence of discrete solutions is shown and, as the step size in space and time tends to zero, their convergence to a suitable notion of weak solution of the system is discussed.