Elasticity in Curved Topographies: Exact Theories and Linear Approximations

TL;DR

This paper develops a unified elasticity theory for curved geometries, compares exact and approximate solutions in 2D crystals on spherical caps, and discusses the scope of linear approximations and non-linear elasticity.

Contribution

It introduces a comprehensive formulation of elasticity in curved spaces that integrates geometric and topological aspects with defect theory, clarifying the origins of common linear approximations.

Findings

Exact solutions for 2D crystals on spherical caps are derived.

Linear approximations are shown to be systematic expansions with specific validity ranges.

Discussion on the universality and limitations of non-linear elasticity is provided.

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.