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

Friedrich M, Kreutz L, Zemas K (2025)

Publication Type: Journal article

Publication year: 2025

Journal

Annales de l'Institut Henri Poincaré - Analyse Non Linéaire Elsevier Masson / Institute Henri Poincaré

Book Volume: 42

Pages Range: 1093-1163

Journal Issue: 5

DOI: 10.4171/AIHPC/136

Abstract

We present a quantitative geometric rigidity estimate in dimensions d D 2; 3 generalizing the celebrated result by Friesecke, James, and Müller [Comm. Pure Appl. Math. 55 (2002), 1461–1506] to the setting of variable domains. Loosely speaking, we show that for each y ∈ H 1(U; Rd ) and for each connected component of an 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 (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.

Authors with CRIS profile

Manuel Friedrich Professur für Angewandte Mathematik (Angewandte Analysis)

Involved external institutions

Technische Universität München (TUM) DE Germany (DE) Rheinische Friedrich-Wilhelms-Universität Bonn DE Germany (DE)

How to cite

APA:

Friedrich, M., Kreutz, L., & Zemas, K. (2025). Geometric rigidity in variable domains and derivation of linearized models for elastic materials with free surfaces. Annales de l'Institut Henri Poincaré - Analyse Non Linéaire, 42(5), 1093-1163. https://doi.org/10.4171/AIHPC/136

MLA:

Friedrich, Manuel, Leonard Kreutz, and Konstantinos Zemas. "Geometric rigidity in variable domains and derivation of linearized models for elastic materials with free surfaces." Annales de l'Institut Henri Poincaré - Analyse Non Linéaire 42.5 (2025): 1093-1163.

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