  This paper concerns the analysis and implementation of a novel iterative
staggered scheme for quasi-static brittle fracture propagation models, where
the fracture evolution is tracked by a phase field variable. The model we
consider is a two-field variational inequality system, with the phase field
function and the elastic displacements of the solid material as independent
variables. Using a penalization strategy, this variational inequality system is
transformed into a variational equality system, which is the formulation we
take as the starting point for our algorithmic developments. The proposed
scheme involves a partitioning of this model into two subproblems; phase field
and mechanics, with added stabilization terms to both subproblems for improved
efficiency and robustness. We analyze the convergence of the proposed scheme
using a fixed point argument, and find that under a natural condition, the
elastic mechanical energy remains bounded, and, if the diffusive zone around
crack surfaces is sufficiently thick, monotonic convergence is achieved.
Finally, the proposed scheme is validated numerically with several bench-mark
problems.
