Eigenfracture approximation of quasi-static crack growth in brittle materials

We study an approximation scheme for a variational theory of quasi-static crack growth based on an eigendeformation approach. We consider a family of energy functionals depending on a small parameter $\epsilon$ and on two fields, the displacement field and an eigendeformation field that approximates the crack in the material. By imposing a suitable irreversibility condition and adopting an incremental minimization scheme, we define a notion of quasi-static evolution for this model. We then show that, as $\epsilon \to 0$, these evolutions converge to a quasi-static crack evolution for the Griffith energy of brittle fracture, characterized by irreversibility, global stability, and an energy balance.