Quasistatic elastoplasticity via Peridynamics: existence and localization

TL;DR

This paper extends peridynamics to quasistatic elastoplasticity, establishing existence of solutions and proving convergence to classical local models as the interaction range diminishes.

Contribution

It introduces a variational formulation for quasistatic elastoplasticity within peridynamics and proves convergence to local models via $ ext{Gamma}$-convergence.

Findings

Existence of solutions for the nonlocal elastoplastic model

Convergence of the nonlocal model to classical elastoplasticity as interaction range shrinks

A variational time discretization scheme for the quasistatic evolution

Abstract

Peridynamics is a nonlocal continuum-mechanical theory based on minimal regularity on the deformations. Its key trait is that of replacing local constitutive relations featuring spacial differential operators with integrals over differences of displacement fields over a suitable positive interaction range. The advantage of such perspective is that of directly including nonregular situations, in which discontinuities in the displacement field may occur. In the linearized elastic setting, the mechanical foundation of the theory and its mathematical amenability have been thoroughly analyzed in the last years. We present here the extension of Peridynamics to linearized elastoplasticity. This calls for considering the time evolution of elastic and plastic variables, as the effect of a combination of elastic energy storage and plastic energy dissipation mechanisms. The quasistatic evolution problem is variationally reformulated and solved by time discretization. In addition, by a rigorous evolutive $\Gamma$-convergence argument we prove that the nonlocal peridynamic model converges to classic local elastoplasticity as the interaction range goes to zero.