Minimal Energy for Geometrically Nonlinear Elastic Inclusions in Two Dimensions

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.

22 pages, 4 figures, comments welcome

References in corpus (4)

• On the Energy Scaling Behaviour of a Singularly Perturbed Tartar Square
• On Scaling Laws for Multi-Well Nucleation Problems without Gauge Invariances
• On a probabilistic model for martensitic avalanches incorporating mechanical compatibility
• Asymptotic shape of isolated magnetic domains

Cited by in corpus (2)

• On Scaling Laws for Multi-Well Nucleation Problems without Gauge Invariances
• On the Scaling of the Cubic-to-Tetragonal Phase Transformation with Displacement Boundary Conditions