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

Abstract

We consider a geodesic $\gamma$ of length $2L$ in an oriented Riemannian manifold $(\mathcal{M},g)$ and a thin tube ${\Omega }_h^{\ast }$ around $\gamma$ of radius $h$. We study an 'elastic' energy per unit volume $E_h(u)$ of maps $u$ from ${\Omega }_h^{\ast }$ 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 Müller 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.