Energy-Dissipation Balance for a Dynamic Viscoelastic Problem with Memory in Domains with Growing Cracks

The principle of conservation of energy implies that a crack growing in a viscoelastic body must satisfy the dynamic energy-dissipation balance, an equality involving the energy dissipated by viscosity and by crack growth. Unfortunately, some models of evolution of a viscoelastic body, like the frequently used Kelvin-Voigt model, imply that the dynamic energy-dissipation balance prevents crack growth. This unrealistic result forces us to use different models to describe the evolution of a viscoelastic body in the context of dynamic fracture mechanics. In this paper we consider an example of dynamic viscoelastic problem with memory in a two-dimensional domain with a crack growing with constant velocity along a straight line. Through a careful analysis of the singularity of the solutions around the crack tip we show that for suitable values of the material constants there exist solutions that satisfy the energy-dissipation balance. This may suggest that we should use suitable models with memory in the context of dynamic fracture mechanics of viscoelastic bodies.