Linearization in Incompatible Elasticity for General Ambient Spaces
Raz Kupferman, Cy Maor
Source abstract
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 [Formula: see text]-convergence, a limit elastic model for a sequence of elastic bodies [Formula: see text] in an ambient space [Formula: see text], for Riemannian metrics [Formula: see text] and [Formula: see text] such that [Formula: see text]. Furthermore, we relate the minimum of the limit problem to a linearized curvature discrepancy between [Formula: see text] and [Formula: see text], using recent results of Kupferman and Leder [ Forum Math. Sigma, 13 (2025), e51]. 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 [Formula: see text] satisfies a quantitative rigidity property, analogous to the Friesecke–James–Müller rigidity estimate in Euclidean space, and show that this property holds when [Formula: see text] is a round sphere.
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