  Fracture involves interaction across large and small length scales. With the
application of enough stress or strain to a brittle material, atomistic scale
bonds will break, leading to fracture of the macroscopic specimen. From the
perspective of mechanics fracture should appear as an emergent phenomena
generated by a continuum field theory eliminating the need for a supplemental
kinetic relation describing crack growth. We develop a new fast method for
modeling quasi-static fracture using peridynamics. We apply fixed point theory
and model stable crack evolution for hard and soft loading. For soft loading we
recover unstable fracture. For hard loading we recover stable crack growth. We
show existence of quasistatic fracture solutions in the neighborhood of stable
critical points for appropriately defined energies. The numerical method uses
an analytic stiffness matrix for fast numerical implementation. A rigorous
mathematical analysis shows that the method converges for load paths associated
with soft and hard loading. For soft loading the crack becomes unstable shortly
after the stress at the tip of the pre-crack reaches the material strength.
