Griffith energies as small strain limit of nonlinear models for nonsimple brittle materials

We consider a nonlinear, frame indifferent Griffith model for nonsimple brittle materials where the elastic energy also depends on the second gradient of the deformations. In the framework of free discontinuity and gradient discontinuity problems, we prove existence of minimizers for boundary value problems. We then pass to a small strain limit in terms of suitably rescaled displacement fields and show that the nonlinear energies can be identified with a linear Griffith model in the sense of $\Gamma$-convergence. This complements the study in [Arch. Ration. Mech. Anal. 225 (2017), 425-467] by providing a linearization result in arbitrary space dimensions.