Minimizers of Nonlocal Polyconvex Energies in Nonlocal Hyperelasticity

TL;DR

This paper establishes the existence of minimizers for nonlocal polyconvex energy functionals in hyperelasticity, extending classical elasticity results to a nonlocal setting that accommodates cavitation and fracture.

Contribution

It introduces a nonlocal gradient model inspired by Riesz' fractional gradient, proving existence of minimizers under polyconvexity in a bounded domain context.

Findings

Existence of minimizers for nonlocal polyconvex energies

Compatibility with cavitation and fracture phenomena

Extension of classical elasticity results to nonlocal models

Abstract

We develop a theory of existence of minimizers of energy functionals in vectorial problems based on a nonlocal gradient under Dirichlet boundary conditions. The model shares many features with the peridynamics model and is also applicable to nonlocal solid mechanics, especially nonlinear elasticity. This nonlocal gradient was introduced in an earlier work, inspired by Riesz' fractional gradient, but suitable for bounded domains. The main assumption on the integrand of the energy is polyconvexity. Thus, we adapt the corresponding results of the classical case to this nonlocal context, notably, Piola's identity, the integration by parts of the determinant and the weak continuity of the determinant. The proof exploits the fact that every nonlocal gradient is a classical gradient. Contrary to classical elasticity, this existence result is compatible with cavitation and fracture.