Geometric Theory of Elastic Curves Bounded on Non-Euclidean Surfaces

Slender elastic objects with inherent anisotropy confined to surfaces are ubiquitous in biology, with examples ranging from DNA wrapping around histones to actin forming contractile rings on the cell membrane during cytokinesis. In the continuum limit, the conformation of these one-dimensional (1D) objects is governed by both their elasticity and the geometry and mechanics of the embedding surface. Here we present a geometric theory of 1D elastic curves constrained on rigid surfaces, derived from the first principles. Our coarse-grained model emerges naturally in terms of the Young's modulus, Poisson's ratio and area moments of inertia. In addition to bending and twisting, typical in existing phenomenological theories, our constitutive model leads to an Almansi strain-measure term bounded in tension, which is the intuitive choice for elastic filaments. Our simulations determine the equilibrium conformations of inextensible curves with an array of intrinsic curvatures and torsions on different positive and negative Gaussian curvature surfaces. We are also extending our simulations to extensible curves for comparisons with the inextensible case. For any given smooth curve and surface, our theory is also capable of finding the global energy minimum of the bound elastic curve.