  We analyze the $\Gamma$-convergence of sequences of free-discontinuity
functionals arising in the modeling of linear elastic solids with surface
discontinuities, including phenomena as fracture, damage, or material voids. We
prove compactness with respect to $\Gamma$-convergence and represent the
$\Gamma$-limit in an integral form defined on the space of generalized special
functions of bounded deformation ($GSBD^p$). We identify the integrands in
terms of asymptotic cell formulas and prove a non-interaction property between
bulk and surface contributions. Eventually, we investigate sequences of
corresponding boundary value problems and show convergence of minimum values
and minimizers. In particular, our techniques allow to characterize relaxations
of functionals on $GSBD^p$, and cover the classical case of periodic
homogenization.
