$\Gamma$-convergence involving nonlocal gradients with varying horizon: Recovery of local and fractional models

TL;DR

This paper provides a rigorous mathematical analysis of nonlocal hyperelasticity models, showing how they converge to classical local and fractional models as the interaction range varies, using $ ext{Gamma}$-convergence techniques.

Contribution

It establishes the $ ext{Gamma}$-convergence of nonlocal hyperelastic models to local and fractional models under varying horizon limits, filling a gap in the asymptotic analysis of such models.

Findings

Nonlocal gradients converge to classical gradients as horizon vanishes.

Nonlocal gradients converge to fractional Riesz gradients as horizon diverges.

The paper proves $ ext{Gamma}$-convergence of associated functionals in both regimes.

Abstract

This work revolves around the rigorous asymptotic analysis of models in nonlocal hyperelasticity. The corresponding variational problems involve integral functionals depending on nonlocal gradients with a finite interaction range $\delta$, called the horizon. After an isotropic scaling of the associated kernel functions, we prove convergence results in the two critical limit regimes of vanishing and diverging horizon. While the nonlocal gradients localize to the classical gradient as $\delta\to 0$, we recover the Riesz fractional gradient as $\delta\to \infty$, irrespective of the nonlocal gradient we started with. Besides rigorous convergence statements for the nonlocal gradients, our analysis in both cases requires compact embeddings uniformly in $\delta$ as a crucial ingredient. These tools enable us to derive the $\Gamma$-convergence of quasiconvex integral functionals with varying horizon to their local and fractional counterparts, respectively.