We present a quantitative geometric rigidity estimate in dimensions d = 2, 3 generalizing the celebrated result by Friesecke, James, and Müller [48] to the setting of variable domains. Loosely speaking, we show that for each y ∈ H 1 (U ; R d ) and for each connected component of an open, bounded set U ⊂ R d , the L 2 -distance of ∇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 ∂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 Γ-convergence. In particular, we study energies related to epitaxially strained crystalline films and to the formation of material voids inside elastically stressed solids.
In this note, we present a short alternative proof of an estimate obtained by Mantel, Muratov and Simon in (Arch Rational Mech. Anal. 239 (2021), 219-299) regarding the rigidity of degree ±1 conformal maps of 𝕊 2 , that is, its Möbius transformations.
The purpose of this paper is to exhibit a quantitative stability result for the class of Möbius transformations of S n−1 when n ≥ 3. The main estimate is of local nature and asserts that for a Lipschitz map that is apriori close to a Möbius transformation, an average conformal-isoperimetric type of deficit controls the deviation (in an average sense) of the map in question from a particular Möbius map. The optimality of the result together with its link with the geometric rigidity of the special orthogonal group are also discussed.
We study an atomistic model that describes the microscopic formation of material voids inside elastically stressed solids under an additional curvature regularization at the discrete level. Using a discrete-to-continuum analysis, by means of a recent geometric rigidity result in variable domains (Friedrich et al 2021 arXiv:2107.10808) and Γ-convergence tools, we rigorously derive effective linearized continuum models for elastically stressed solids with material voids in three-dimensional elasticity.
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