Mechanical insight into the packing of slender objects within confinement is essential for understanding how polymers, filaments, or wires organize and rearrange in limited space. Here we combine theoretical modeling, numerical optimization, and experimental studies to reveal spherical packing behavior of thin elastic rod loops of homogeneous or inhomogeneous stiffness. Across varying loop lengths, a rich array of configurations including circle, saddle, figure-eight, and more intricate patterns are identified. A theoretical framework rooted in the local equilibrium of force and moment is proposed for the rod loop deformation, facilitating the determination of internal and contact forces experienced by the rods during deformation. For the confined homogeneous rod loops, their stable and metastable configurations are well described using proposed Euler rotation curves, which offer a concise and effective approach for configuration prediction. Moreover, formulated analysis on the stability and critical force for homogeneous rod loops on great circles of the spherical confinement are performed. For inhomogeneous rod loops with two segments of differing stiffness, the stiffer segment exhibits less deviation from the great circle, while the softer segment undergoes more pronounced deformation. These findings not only enhance our understanding of buckling and post-buckling phenomena but also offer insights into filament patterning within confining environments.

中文翻译：

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球形约束内弹性杆环的变形、形状变换和稳定性


对限制内细长物体堆积的机械洞察对于理解聚合物、细丝或电线如何在有限空间内组织和重新排列至关重要。在这里，我们结合理论建模、数值优化和实验研究来揭示均匀或不均匀刚度的薄弹性杆环的球形堆积行为。在不同的环长度上，可以识别出丰富的配置，包括圆形、马鞍形、8 字形和更复杂的图案。为杆环变形提出了一个植根于力和力矩局部平衡的理论框架，有助于确定杆在变形过程中所经历的内力和接触力。对于受限齐质棒环，使用提出的欧拉旋转曲线很好地描述了它们的稳定和亚稳态构型，这为构型预测提供了简洁有效的方法。此外，还对球形约束大圆上均匀杆环的稳定性和临界力进行了公式化分析。对于具有不同刚度的两个段的不均匀杆环，较硬的段表现出与大圆的较小偏差，而较软的段经历更明显的变形。这些发现不仅增强了我们对屈曲和屈曲后现象的理解，而且还提供了对受限环境中的细丝图案的见解。