A homogenization result for weak membrane energies

TL;DR

This paper establishes a $ ext{Gamma}$-convergence result for space-dependent weak membrane energies with truncated quadratic potentials, showing the limit surface energy matches spin systems and the bulk energy remains purely elastic.

Contribution

It provides a novel $ ext{Gamma}$-convergence analysis for weak membrane energies with space-dependent truncations, linking surface energies to spin systems.

Findings

Surface energy density matches that of spin systems.

Bulk energy density remains purely elastic despite truncation.

Limit energies are unaffected by the truncation levels.

Abstract

We prove a $\Gamma$-convergence result for space dependent weak membrane energies, that is for 'truncated quadratic potentials', that are quadratic below some threshold (depending on the pair of points that we are considering) and constant above. We prove that the limit surface energy density coincides with the one for spin systems, while the bulk energy density is not affected by the different levels of truncation and coincides with an purely elastic energy density.