Comparison between solutions to the linear peridynamics model and solutions to the classical wave equation

Abstract

In this paper, we consider an equation inspired by linear peridynamics and we establish its connection with the classical wave equation. In particular, given a horizon $\hat{I}´>0$ accounting for the region of influence around a material point, we prove existence and uniqueness of a solution $u_{\hat{I}}´$ and demonstrate the convergence of $u_{\hat{I}}´$ to solutions to the classical wave equation as $\hat{I}´\to 0$. Moreover, we prove that the solutions to the peridynamics model with small frequency initial data are close to solutions to the classical wave equation.