Elasticity in Curved Topographies: Exact Theories and Linear Approximations

Abstract

Almost all available results in elasticity on curved topographies are obtained within either a small curvature expansion or an empirical covariant generalization that accounts for screening between Gaussian curvature and disclinations. In this paper, we present a formulation of elasticity theory in curved geometries that unifies its underlying geometric and topological content with the theory of defects. The two different linear approximations widely used in the literature are shown to arise as systematic expansions in reference and actual space. Taking the concrete example of a 2D crystal, with and without a central disclination, constrained on a spherical cap, we compare the exact results with different approximations and evaluate their range of validity. We conclude with some general discussion about the universality of non-linear elasticity.

arXiv comment16 pages, 6 figures

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Cited by in corpus (7)

• Ground States of Crystalline Caps: Generalized Jellium on Curved Space
• On duality between Cosserat elasticity and fractons
• Fracton-elasticity duality on curved manifolds
• Exact Solution for Elastic Networks on Curved Surfaces
• A simple, general criterion for onset of disclination disorder on curved surfaces
• Predicting the Characteristics of Defect Transitions on Curved Surfaces
• General Solution for Elastic Networks on Arbitrary Curved Surfaces in the Absence of Rotational Symmetry