Geometric rigidity in variable domains and derivation of linearized models for elastic materials with free surfaces

TL;DR

This paper extends geometric rigidity estimates to variable domains, enabling the derivation of linearized elastic models for materials with free surfaces, including applications to crystalline films and void formation.

Contribution

It generalizes the geometric rigidity estimate to variable domains and uses it to rigorously derive linearized models for elastic materials with free surfaces.

Findings

Established a domain-independent geometric rigidity estimate.

Proved compactness results in $GSBD^2$ for elastic deformations.

Derived linearized models via $ ext{Gamma}$-convergence for specific materials.

Abstract

We present a quantitative geometric rigidity estimate in dimensions $d=2,3$ generalizing the celebrated result by Friesecke, James, and M\"uller to the setting of variable domains. Loosely speaking, we show that for each $y \in H^1(U;\mathbb{R}^d)$ and for each connected component of a smooth open, bounded set $U \subset \mathbb{R}^d$, the $L^2$-distance of $\nabla y$ from a single rotation can be controlled up to a constant by its $L^2$-distance from the group $SO(d)$, with the constant not depending on the precise shape of $U$, but only on an integral curvature functional related to $\partial U$. We further show that for linear strains the estimate can be refined, leading to a uniform control independent of the set $U$. The estimate can be used to establish compactness in the space of generalized special functions of bounded deformation ($GSBD^2$) for sequences of displacements related to deformations with uniformly bounded elastic energy. As an application, we rigorously derive linearized models for nonlinearly elastic materials with free surfaces by means of $\Gamma$-convergence. In particular, we study energies related to epitaxially strained crystalline films and to the formation of material voids inside elastically stressed solids.