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

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 $\mathbb{T}$, linearised strain $\boldsymbol{\epsilon}(\mathbf{u})$ and strain rate $\boldsymbol{\epsilon}(\mathbf{u}_t )$. The relationship takes the form $\boldsymbol{\epsilon}(\mathbf{u}_t) + \alpha\boldsymbol{\epsilon}(\mathbf{u}) = F(\mathbb{T})$ 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 {\it 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.