Linearization and localization of nonconvex functionals motivated by nonlinear peridynamic models

Abstract

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 $Γ$-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 $Γ$-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 $Γ$-convergence, of peridynamic-type energy functionals exhibit behavior quite different from classical hyperelastic energy functionals.