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 of 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 exhibits behavior quite different from classical hyperelastic energy functionals.

我们考虑位于连续介质力学近场动力学模型框架中的一类非凸能量泛函。能量密度是非局部应变的函数，其描述基于材料点的成对相互作用的变形，因此相对于非局部变形是非凸的。我们应用变分分析来研究这些非局部非凸泛函的有效行为与两种不同状态下建立的经典和近场动力学模型的一致性。在小位移的情况下，我们表明模型可以通过其线性化有效地描述。准确地说，我们严格推导出通常所说的线性化键基近场动力学泛函作为非线性泛函的\(\Gamma \)极限。在非局域性消失的情况下，非局域非凸泛函的有效行为由积分表示来表征，该积分表示是通过相对于强\(L^p\)拓扑的\(\Gamma \)收敛获得的。我们还证明了局域拟凸泛函密度的各种性质，例如框架无差异和矫顽力。我们证明密度在奇异值小于或等于 1 的矩阵上消失。这些结果证实，在\(\Gamma \)收敛的背景下，近场动力学型能量泛函的局域化表现出与经典超弹性能量泛函截然不同的行为。