Linearization in incompatible elasticity for general ambient spaces

TL;DR

This paper develops a limit elastic model for bodies in non-Euclidean spaces, linking elastic energy to curvature discrepancies and extending linearization techniques to manifold-valued configurations.

Contribution

It introduces a $ ext{Gamma}$-convergence framework for incompatible elasticity in general ambient spaces and relates the limit energy to curvature differences, confirming a conjecture in the field.

Findings

Limit model obtained via $ ext{Gamma}$-convergence for non-Euclidean elastic bodies.

Relation established between elastic energy and linearized curvature discrepancy.

Rigidity property holds for round spheres, enabling compactness results.

Abstract

Motivated by recent interest in elastic problems in which the target space is non-Euclidean, we study a limit where local rest distances within an elastic body are incompatible, yet close to, distances within the ambient space. Specifically, we obtain, via $\Gamma$-convergence, a limit elastic model for a sequence of elastic bodies $(M,g_\varepsilon)$ in an ambient space $(S,s)$, for Riemannian metrics $g_\varepsilon$ and $s$ such that $g_\varepsilon \to s$. Furthermore, we relate the minimum of the limit problem to a linearized curvature discrepancy between $g_\varepsilon$ and $s$, using recent results of Kupferman and Leder. This relation confirms a linearized version of a long-standing conjecture in elasticity regarding the relation between the elastic energy and the curvature of the underlying space. The main technical challenge, compared to other linearization results in elasticity, is obtaining the correct notion of displacement for manifold-valued configurations, using Sobolev truncations and parallel transport. We show that the associated compactness result is obtained if $(S,s)$ satisfies a quantitative rigidity property, analogous to the Friesecke--James--M\"uller rigidity estimate in Euclidean space, and show that this property holds when $(S,s)$ is a round sphere.