Non-Euclidean elasticity for rods and almost isometric embeddings of geodesic tubes

TL;DR

This paper extends the theory of nonlinear elasticity to non-Euclidean settings, analyzing the asymptotic behavior of elastic energies of thin tubes around geodesics in Riemannian manifolds, and characterizes the limiting energy functional.

Contribution

It generalizes Euclidean rod models to Riemannian manifolds, establishing a $ ext{Gamma}$-convergence result for elastic energies of thin tubes and deriving explicit formulas for the limiting energy.

Findings

Proved compactness for sequences with bounded scaled energy.

Derived the $ ext{Gamma}$-limit of the scaled elastic energy.

Expressed the minimum of the limiting energy in terms of curvature tensors.

Abstract

We consider a geodesic $\gamma$ of length $2L$ in an oriented Riemannian manifold $(\mathcal M, g)$ and a thin tube $\Omega^*_h$ around $\gamma$ of radius $h$. We study an 'elastic' energy per unit volume $E_h(u)$ of maps $u$ from $\Omega^*_h$ into another oriented Riemannian manifold $(\tilde {\mathcal M},\tilde g)$. The energy $E_h$ is based on the squared distance of the differentials $du$ from the set of orientation preserving linear maps between the corresponding tangent spaces. We prove a compactness result for sequences of maps $u_h$ for which $h^{-4} E_h(u_h)$ remains bounded and we study the $\Gamma$-Limit of $h^{-4} E_h(u_h)$ as $h \to 0$ with respect to a suitable notion of convergence for $u_h$ that involves certain blow-ups in the radial direction. This $\Gamma$-convergence result ge\-ne\-ra\-lizes work by Mora and M\"uller on the limiting energy of thin rods in the Euclidean setting. We also obtain an expression for the minimum of the limiting energy as a specific quadratic functional in the difference of the pullbacks of the curvature tensors of $\mathcal M$ and $\tilde{\mathcal M}$ along the curves $\gamma$ and $u \circ \gamma$, respectively, thus answering a question by Maor and Shachar, J. Elasticity 134 (2019), pp. 149--173.