Hyperelasticity as a Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-limit of peridynamics when the horizon goes to zero

Peridynamics is a nonlocal model in Continuum Mechanics, and in particular Elasticity, introduced by Silling (2000). The nonlocality is reflected in the fact that points at a finite distance exert a force upon each other. If, however, those points are more distant than a characteristic length called horizon, it is customary to assume that they do not interact. We work in the variational approach of time-independent deformations, according to which, their energy is expressed as a double integral that does not involve gradients. We prove that the Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-limit of this model, as the horizon tends to zero, is the classical model of hyperelasticity. We pay special attention to how the passage from the density of the non-local model to its local counterpart takes place.