AI 中文总结
研究折叠弹性带中 snapping-through 对称性，通过改变褶皱位置调整对称破缺叉形分叉和鞍结分叉顺序，结合实验等方法绘制分叉顺序交换相图，确立分叉重排序为编程 snapping-through 路径的几何机制并提供设计原则。
AI 中文摘要
细长弹性结构中的 snapping-through 通常被视为稳定构型之间的突然转变，但其转变过程可能有根本差异。结构可能在保持对称性的情况下 snapping-through，也可能先失去对称性并经过非对称状态再达到最终构型。本文表明，通过对折叠弹性带中相互竞争的分叉进行重排序，可以对 snapping-through 对称性进行编程。研究了具有两个关于中点对称放置的局部褶皱的弹性带，发现改变褶皱位置会改变两个不稳定性的相对顺序：一个是对称破缺的叉形分叉和一个在对称保持分支上的鞍结分叉。当叉形分叉先发生时，带在 snapping-through 之前失去对称性并遵循非对称路径；反之，当鞍结分叉先发生时，带在保持对称分支上失去稳定性，导致对称保持转变。通过实验、离散微分几何模拟和数值延续，绘制了分叉顺序的这种交换并构建了预测非对称和对称 snapping-through 状态之间切换的相图。一个降阶双质量冯·米塞斯桁架模型捕捉了与对称破缺和对称保持不稳定性之间的一般竞争相同的机制。这些结果将分叉重排序确立为一种用于对细长弹性结构中的 snapping-through 路径进行编程的几何机制，为多稳态系统、变形结构和基于不稳定性的机械设备提供了一种设计原则。
英文摘要
Snap-through in slender elastic structures is often viewed as a sudden transition between stable configurations, yet the pathway taken during this transition can differ fundamentally. A structure may snap while preserving symmetry, or first lose symmetry and pass through an asymmetric state before reaching its final configuration. What selects between these pathways remains less well understood, especially when the geometry and loading are themselves symmetric. Here, we show that snap-through symmetry can be programmed by reordering competing bifurcations in folded elastic ribbons. We study an elastic ribbon with two localized folds placed symmetrically about the midpoint and show that varying the fold position changes the relative order of two instabilities: a symmetry-breaking pitchfork bifurcation and a saddle-node bifurcation on the symmetry-preserving branch. When the pitchfork bifurcation occurs first, the ribbon loses symmetry before snapping and follows an asymmetric pathway. Conversely, when the saddle-node bifurcation occurs first, the ribbon loses stability while remaining on the symmetric branch, resulting in a symmetry-preserving transition. Combining experiments, discrete differential geometry simulations and numerical continuation, we map this exchange in bifurcation ordering and construct a phase diagram that predicts the switch between asymmetric and symmetric snap-through regimes. A reduced-order double-mass von Mises truss model captures the same mechanism as a generic competition between symmetry-breaking and symmetry-preserving instabilities. These results establish bifurcation reordering as a geometric mechanism for programming snap-through pathways in slender elastic structures, offering a design principle for multistable systems, morphing structures and instability-based mechanical devices.
Comments19 pages, 7 figures
