Multistable structures have widespread applications in the design of deployable aerospace systems, mechanical metamaterials, flexible electronics, and multimodal soft robotics due to their capability of shape reconfiguration between multiple stable states. Recently, the snap-folding of rings, often in the form of circles or polygons, has shown the capability of inducing diverse stable configurations. The natural curvature of the rod segment (curvature in its stress-free state) plays an important role in the elastic stability of these rings, determining the number and form of their stable configurations during folding. Here, we develop a general theoretical framework for the elastic stability analysis of segmented rings (e.g., polygons) based on an energy variational approach. Combining this framework with finite element simulations, we map out all planar stable configurations of various segmented rings and determine the natural curvature ranges of their multistable states. The theoretical and numerical results are validated through experiments, which demonstrate that a segmented ring with a rectangular cross-section can show up to six distinct planar stable states. The results also reveal that, by rationally designing the segment number and natural curvature of the segmented ring, its one- or multiloop configuration can store more strain energy than a circular ring of the same total length. We envision that the proposed strategy for achieving multistability in the current work will aid in the design of multifunctional, reconfigurable, and deployable structures.

中文翻译：

---

通过自然曲率编程实现分段环的多稳定性


多稳态结构由于其在多个稳定状态之间进行形状重构的能力，在可部署航空航天系统、机械超材料、柔性电子和多模态软机器人的设计中具有广泛的应用。最近，通常呈圆形或多边形形式的环的卡扣折叠已显示出诱导多种稳定构型的能力。杆段的自然曲率（无应力状态下的曲率）对这些环的弹性稳定性起着重要作用，决定了折叠过程中其稳定构型的数量和形式。在这里，我们开发了基于能量变分方法的分段环（例如多边形）弹性稳定性分析的通用理论框架。将该框架与有限元模拟相结合，我们绘制出各种分段环的所有平面稳定构型，并确定其多稳态的自然曲率范围。理论和数值结果通过实验得到验证，表明具有矩形横截面的分段环可以显示多达六个不同的平面稳定状态。研究结果还表明，通过合理设计分段环的分段数和自然曲率，其单环或多环结构可以比相同总长度的圆环存储更多的应变能。我们预计，在当前工作中实现多稳定性的拟议策略将有助于多功能、可重构和可部署结构的设计。