Nonlinear dynamic fracture problems with polynomial and strain-limiting constitutive relations

Abstract

We extend the framework of dynamic fracture problems with a phase-field approximation to the case of a nonlinear constitutive relation between the Cauchy stress tensor T, linearised strain ε(u) and strain rate ε(u t ). The relationship takes the form ε(u t ) + αε(u) = F (T) where F satisfies certain p-growth conditions. We take particular care to study the case p = 1 of a 'strain-limiting' solid, that is, one in which the strain is bounded a priori. We prove the existence of long-time, large-data weak solutions of a balance law coupled with a minimisation problem for the phase-field function and an energy-dissipation inequality, in any number d of spatial dimensions. In the case of Dirichlet boundary conditions, we also prove the satisfaction of an energy-dissipation equality.