Euclidean Frustrated Ribbons
Euclidean Frustrated Ribbons
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DOI：
10.1103/physrevx.11.011062
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发表时间：
2021-03-29
影响因子：
12.5
通讯作者：
Sharon, Eran
中科院分区：
文献类型：
--
作者：
Siefert, Emmanuel;Levin, Ido;Sharon, Eran
薄板中的几何挫折在生物学中普遍存在，并在技术上变得越来越重要。以往的研究将挫折的根源归结为违反了高斯的埃格罗里姆定理。这种“高斯挫折感”展示了丰富的现象学；它可能导致机械不稳定、异常力学和形状变形能力，可以在工程系统中利用。在这里，我们报告了一种新的几何挫折，一种与高斯挫折一样普遍的类型。我们证明了它的起源是对Mainardi-Codazzi-Peterson相容方程的破坏，并且它出现在欧几里德单上。结合实验、模拟和理论，我们研究了具有径向曲率和测地线曲率的欧几里德带的具体情况。使用不同材料和技术进行的实验揭示了形状转变、对称性破坏和自发应力集中。使用解析解和几何论证，这些观测结果在数量上是合理的。我们预计这种挫折感将在自然和工程系统中发挥重要作用，特别是在细长的3D打印薄片中。
Geometrical frustration in thin sheets is ubiquitous across scales in biology and becomes increasingly relevant in technology. Previous research identified the origin of the frustration as the violation of Gauss's Theorema Egregium. Such "Gauss frustration" exhibits rich phenomenology; it may lead to mechanical instabilities, anomalous mechanics, and shape-morphing abilities that can be harnessed in engineering systems. Here we report a new type of geometrical frustration, one that is as general as Gauss frustration. We show that its origin is the violation of Mainardi-Codazzi-Peterson compatibility equations and that it appears in Euclidean sheets. Combining experiments, simulations, and theory, we study the specific case of a Euclidean ribbon with radial and geodesic curvatures. Experiments, conducted using different materials and techniques, reveal shape transitions, symmetry breaking, and spontaneous stress focusing. These observations are quantitatively rationalized using analytic solutions and geometrical arguments. We expect this frustration to play a significant role in natural and engineering systems, specifically in slender 3D printed sheets.