  We analyze integral representation and $\Gamma$-convergence properties of
functionals defined on \emph{piecewise rigid functions}, i.e., 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 (Bouchitt\'e et al. 1998, 2001) 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.
