Derivation of variational membrane models in the context of anisotropic nonlocal hyperelasticity

TL;DR

This paper develops a theory for anisotropic nonlocal gradients in hyperelasticity, deriving effective membrane models through $ ext{Gamma}$-convergence that interpolate between nonlocal and local behaviors in thin structures.

Contribution

It introduces a unified framework for anisotropic nonlocal gradients with ellipsoidal interaction regions, extending existing models to include direction-dependent interaction ranges.

Findings

Derived a $ ext{Gamma}$-convergence result for nonlocal thin-film energies.

Showed the limit functional matches classical membrane energy.

Recovered local models when all interaction radii vanish.

Abstract

Motivated by the analysis of thin structures, we study the variational dimension reduction of hyperelastic energies involving nonlocal gradients to an effective membrane model. When rescaling the thin domain, isotropic interaction ranges naturally become anisotropic, leading to the development of a theory for anisotropic nonlocal gradients with direction-dependent interaction ranges. Unlike existing nonlocal derivatives with finite horizon, which are defined via interaction kernels supported on balls of positive radius, our formulation is based on ellipsoidal interaction regions whose principal radii may vanish independently. This yields a unified framework that interpolates between fully nonlocal, partially nonlocal, and purely local models. Employing these tools, we present a $\Gamma$-convergence analysis for the nonlocal thin-film energies. The limit functional retains the structural form of the classical membrane energy, and the classical local model is recovered precisely when all interaction radii vanish.