Calculus of Variations and Geometric Measure Theory

M. Friedrich - L. Kreutz - K. Zemas

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

created by kreutz on 16 Aug 2021


Submitted Paper

Inserted: 16 aug 2021
Last Updated: 16 aug 2021

Pages: 51
Year: 2021

ArXiv: 2107.10808 PDF


We present a quantitative geometric rigidity estimate in dimensions $d=2,3$ generalizing the celebrated result by Friesecke, James, and Müller 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 $SO(d)$, 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 ($GSBD$) 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.