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We present a quantitative geometric rigidity estimate in dimensions $d=2,3$generalizing the celebrated result by Friesecke, James, and M\"uller to thesetting of variable domains. Loosely speaking, we show that for each $y \inH^1(U;\mathbb{R}^d)$ and for each connected component of a smooth open, boundedset $U \subset \mathbb{R}^d$, the $L^2$-distance of $\nabla y$ from a singlerotation can be controlled up to a constant by its $L^2$-distance from thegroup $SO(d)$, with the constant not depending on the precise shape of $U$, butonly on an integral curvature functional related to $\partial U$. We furthershow that for linear strains the estimate can be refined, leading to a uniformcontrol independent of the set $U$. The estimate can be used to establishcompactness in the space of generalized special functions of boundeddeformation ($GSBD^2$) for sequences of displacements related to deformationswith uniformly bounded elastic energy. As an application, we rigorously derivelinearized models for nonlinearly elastic materials with free surfaces by meansof $\Gamma$-convergence. In particular, we study energies related toepitaxially strained crystalline films and to the formation of material voidsinside elastically stressed solids.

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