Functionals Defined on Piecewise Rigid Functions: Integral Representation and Γ-Convergence

We analyse integral representation and Gamma-convergence properties of functionals defined on piecewise rigid functions, that is, functions which are piecewise affine on a Caccioppoli partition where the derivative in each component is constant and lies in a set without rank-one connections. Such functionals account for interfacial energies in the variational modeling of materials which locally show a rigid behavior. Our results are based on localization techniques for Gamma-convergence and a careful adaption of the global method for relaxation (Bouchitte et al. in Arch Ration Mech Anal 165:187-242, 2002; Bouchitte et al. in Arch Ration Mech Anal 145:51-98, 1998), to this new setting, under rather general assumptions. They constitute a first step towards the investigation of lower semicontinuity, relaxation, and homogenization for free-discontinuity problems in spaces of (generalized) functions of bounded deformation.