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
modified by zemas on 13 Jul 2024

[BibTeX]

Accepted Paper

Inserted: 16 aug 2021
Last Updated: 13 jul 2024

Journal: Ann. Inst. H. Poincaré Anal. Non Linéaire (C)
Pages: 51
Year: 2024

ArXiv: 2107.10808 PDF

Abstract:

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.


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