Wrinkling in the Lam\'e problem: a $\Gamma$-convergence approach

TL;DR

This paper analyzes wrinkling patterns in a thin elastic annulus under radial stretching using a $ ext{Gamma}$-convergence approach, revealing the asymptotic behavior of energy and wrinkle distribution.

Contribution

It establishes a $ ext{Gamma}$-convergence result for the energy functional, characterizing the limiting measure-valued energy and wrinkle frequency distribution.

Findings

Derived the asymptotic energy functional as $h o 0$

Proved existence of minimizers for the limiting functional

Described the distribution of wrinkle frequencies via a measure constraint

Abstract

We study wrinkling patterns in a thin elastic annulus subjected to radial stretching within the framework of the F\"oppl--von K\'arm\'an theory. Building on the analysis of the Lam\'e problem in Bella and Kohn, we investigate the asymptotic regime $h\to0$ and establish a $\Gamma$-convergence result for suitably rescaled energies after subtraction of the relaxed membrane energy. The limiting functional is a scalar convex measure-valued energy coupled with a constraint on the marginal of the limiting measure, describing the distribution of wrinkle frequencies. We also prove existence and qualitative properties of minimizers of the limiting functional.