  We formulate a nonlocal cohesive model for calculating the deformation state
inside a cracking body. In this model a more complete set of physical
properties including elastic and softening behavior are assigned to each point
in the medium. We work within the small deformation setting and use the
peridynamic formulation. Here strains are calculated as difference quotients.
The constitutive relation is given by a nonlocal cohesive law relating force to
strain. At each instant of the evolution we identify a process zone where
strains lie above a threshold value. Perturbation analysis shows that jump
discontinuities within the process zone can become unstable and grow. We derive
an explicit inequality that shows that the size of the process zone is
controlled by the ratio given by the length scale of nonlocal interaction
divided by the characteristic dimension of the sample. The process zone is
shown to concentrate on a set of zero volume in the limit where the length
scale of nonlocal interaction vanishes with respect to the size of the domain.
In this limit the dynamic evolution is seen to have bounded linear elastic
energy and Griffith surface energy. The limit dynamics corresponds to the
simultaneous evolution of linear elastic displacement and the fracture set
across which the displacement is discontinuous. We conclude illustrating how
the approach developed here can be applied to limits of dynamics associated
with other energies that $\Gamma$- converge to the Griffith fracture energy.
