  We consider a class of nonconvex energy functionals that lies in the
framework of the peridynamics model of continuum mechanics. The energy
densities are functions of a nonlocal strain that describes deformation based
on pairwise interaction of material points, and as such are nonconvex with
respect to nonlocal deformation. We apply variational analysis to investigate
the consistency of the effective behavior of these nonlocal nonconvex
functionals with established classical and peridynamic models in two different
regimes. In the regime of small displacement, we show the model can be
effectively described by its linearization. To be precise, we rigorously derive
what is commonly called the linearized bond-based peridynamic functional as a
$\Gamma$-limit of nonlinear functionals. In the regime of vanishing
nonlocality, the effective behavior the nonlocal nonconvex functionals is
characterized by an integral representation, which is obtained via
$\Gamma$-convergence with respect to the strong $L^p$ topology. We also prove
various properties of the density of the localized quasiconvex functional such
as frame-indifference and coercivity. We demonstrate that the density vanishes
on matrices whose singular values are less than or equal to one. These results
confirm that the localization, in the context of $\Gamma$-convergence, of
peridynamic-type energy functionals exhibit behavior quite different from
classical hyperelastic energy functionals.
