  In this work, we study the finite difference approximation for a class of
nonlocal fracture models. The nonlocal model is initially elastic but beyond a
critical strain the material softens with increasing strain. This model is
formulated as a state-based perydynamic model using two potentials: one
associated with hydrostatic strain and the other associated with tensile
strain. We show that the dynamic evolution is well-posed in the space of
H\"older continuous functions $C^{0,\gamma}$ with H\"older exponent $\gamma \in
(0,1]$. Here the length scale of nonlocality is $\epsilon$, the size of time
step is $\Delta t$ and the mesh size is $h$. The finite difference
approximations are seen to converge to the H\"older solution at the rate $C_t
\Delta t + C_s h^\gamma/\epsilon^2$ where the constants $C_t$ and $C_s$ are
independent of the discretization. The semi-discrete approximations are found
to be stable with time. We present numerical simulations for crack propagation
that computationally verify the theoretically predicted convergence rate. We
also present numerical simulations for crack propagation in precracked samples
subject to a bending load.
