Equilibrium analysis of surface-constrained elastic rods: Unveiling contact and internal forces through local geometry

Abstract

Confinement scenarios of thin elastic rods are prevalent in both natural and engineered systems. The accurate quantification of the mechanical interplay between confined rods and their confining surfaces remains a formidable challenge, primarily due to the intricate nonlinear nature of thin rods and their contact with surfaces. Here, we present a theoretical framework designed to characterize equilibrium states of thin elastic rods constrained to general surfaces, accounting for the influence of both conservative force and rod inhomogeneity. The framework proposes a general scheme to directly unveil contact and internal forces of confined rods through the consideration of local geometry, based on balance equations for local forces and moments. Characteristics of the contact force are contingent upon whether the deformed rod exhibits vanishing geodesic curvature. Case studies involving closed rods constrained to spherical, cylindrical, and torus surfaces are conducted to elucidate geometric effects on various rod behavior, encompassing equilibrium configurations, internal forces and moments, and contact forces. The outcomes of our study offer fundamental insights into how geometric and material properties lead to the intricate mechanical interactions between one- and two-dimensional materials.