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

Abstract

We present a quantitative geometric rigidity estimate in dimensions $d=2,3$ generalizing the celebrated result by Friesecke, James, and Müller [Comm. Pure Appl. Math. 55 (2002), 1461–1506] 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 an 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 $\mathrm{SO}(d)$, 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 ($\mathrm{GSBD}$) 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.