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.

中文翻译：

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表面约束弹性杆的平衡分析：通过局部几何揭示接触力和内力

细弹性杆的限制场景在自然系统和工程系统中都很普遍。精确量化受限杆与其受限表面之间的机械相互作用仍然是一项艰巨的挑战，这主要是由于细杆及其与表面接触的复杂非线性特性。在这里，我们提出了一个理论框架，旨在描述约束于一般表面的细弹性杆的平衡状态，同时考虑保守力和杆不均匀性的影响。该框架提出了一种通用方案，基于局部力和力矩的平衡方程，通过考虑局部几何形状，直接揭示受限杆的接触力和内力。接触力的特性取决于变形杆是否表现出测地曲率消失。涉及受限于球形、圆柱形和圆环面的封闭杆的案例研究旨在阐明对各种杆行为的几何影响，包括平衡配置、内力和力矩以及接触力。我们的研究结果为几何和材料特性如何导致一维和二维材料之间复杂的机械相互作用提供了基本见解。