Minimal Energy for Geometrically Nonlinear Elastic Inclusions in Two Dimensions

TL;DR

This paper studies the minimal energy scaling of elastic inclusions in two dimensions with nonlinear elasticity, combining rigidity arguments and explicit constructions to understand optimal shapes and energies.

Contribution

It introduces a new analysis of energy scaling for nonlinear elastic inclusions, extending previous methods with a two-well rigidity approach and explicit shape constructions.

Findings

Derived lower energy bounds using rigidity and covering arguments

Constructed explicit lens-shaped inclusion configurations for upper bounds

Identified optimal scaling laws for elastic inclusions in 2D

Abstract

We are concerned with a variant of the isoperimetric problem, which in our setting arises in a geometrically nonlinear two-well problem in elasticity. More precisely, we investigate the optimal scaling of the energy of an elastic inclusion of a fixed volume for which the energy is determined by a surface and an (anisotropic) elastic contribution. Following ideas from \cite{CS} and \cite{KnuepferKohn-2011}, we derive the lower scaling bound by invoking a two-well rigidity argument and a covering result. The upper bound follows from a well-known construction for a lens-shaped elastic inclusion.