Linearization in nonlocal elasticity via Gamma-convergence

Abstract

Linearization is a classical tool in continuum mechanics that connects nonlinear models of solids to their linear counterparts through the small-strain limit, as in the passage from finite to linear elasticity. Here, we study a nonlocal analogue of this limit process in the context of state-based peridynamics, starting from a class of models in nonlocal hyperelasticity subject to nonlocal Dirichlet boundary conditions. Recent formal calculations in the literature indicate that the linearized versions of these models are closely related to the widely used Eringen models. The main purpose of this note is to make this connection rigorous by establishing $Γ$-convergence of the associated energy functionals and a corresponding compactness result that together imply strong convergence of minimizers. Our proof relies on a local-to-nonlocal transfer argument that reduces the problem to a well-known local linearization result.