2021/07/22 by Manuel Friedrich, Friedrich, Manuel, Leonard Kreutz +3
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Analysis Techniques #Analysis of PDEs (math.AP) #Elasticity and Material Modeling #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2107.10808
openalex publication_date 2021/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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 ∈ H1(U;ℝd) and for each connected component of a smooth open, bounded set U ⊂ ℝd, the L2-distance of ∇ y from a single rotation can be controlled up to a constant by its L2-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 (GSBD2) 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.