Linearization in incompatible elasticity for general ambient spaces
Raz Kupferman, Cy Maor
TL;DR
We address elastic bodies in a non-Euclidean ambient space with metrics $\mathfrak{g}_\varepsilon$ converging to $\mathfrak{s}$ and leading discrepancy $\mathfrak{h}$, deriving a Γ-limit $E_0(u) = \frac{1}{4}\int_{\mathcal{M}}|\mathcal{L}_u\mathfrak{s}-\mathfrak{h}|^2 \, d\operatorname{Vol}_{\mathfrak{s}}$ that encodes a linearized curvature discrepancy. The analysis introduces a manifold-valued displacement framework built on Lipschitz truncations and parallel transport, enabling a compactness theorem under a geometric rigidity property that holds for spheres (and, in newer work, for closed spaces with $\mathcal{M}=\mathcal{S}$). The minimization of $E_0$ is governed by projecting $\mathfrak{h}$ onto the image of the deformation operator, giving $\min E_0 = \frac{1}{4}\|P_\perp\mathfrak{h}\|^2_{L^2}$ and linking to curvature variation $\dot{\Re}_{\mathfrak{s}}$; this yields a linearized version of a curvature-elastic energy conjecture. The results extend linearization in elasticity to non-Euclidean targets and quantify how curvature discrepancies drive the limit energy, with implications for manifold-valued elasticity and potential generalizations of FJM-type rigidity.Overall, the paper provides a rigorous bridge between incompatibility energy, Γ-convergence, and linearized curvature in a non-Euclidean setting.
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 $Γ$-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--Müller rigidity estimate in Euclidean space, and show that this property holds when $(S,s)$ is a round sphere.