Folding paper along curves leads to spatial structures that have curved surfaces meeting at spatial creases, defined as curve-fold origami. In this work, we provide an Eulerian framework focusing on the mechanics of arbitrary curve-fold origami, especially for multi-curve-fold origami with vertices. We start with single-curve-fold origami that has wide panels. Wide panel leads to different domains of mechanical responses induced by various generator distributions of the curved surface. The theories are then extended to multi-curve-fold origami, involving additional geometric correlations between creases. As an illustrative example, the deformation and equilibrium configuration of origami with annular creases are studied both theoretically and numerically. Afterward, single-vertex curved origami theory is studied as a special type of multi-curve-fold origami. We find that the extra periodicity at the vertex strongly constrains the configuration space, leading to a region near the vertex that has a striking universal equilibrium configuration regardless of the mechanical properties. Both theories and numerics confirm the existence of the universality in the near-field region. In addition, the far-field deformation is obtained via energy minimization and validated by finite element analysis. Our generalized multi-curve-fold origami theory, including the vertex-contained universality, is anticipated to provide a new understanding and framework for the shape programming of the curve-fold origami system.

中文翻译：

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多曲线折叠折纸的广义几何力学理论：顶点约束通用配置


沿着曲线折叠纸张会形成具有在空间折痕处相遇的曲面的空间结构，定义为曲线折叠折纸。在这项工作中，我们提供了一个欧拉框架，重点关注任意曲线折叠折纸的机制，特别是具有顶点的多曲线折叠折纸。我们从具有宽面板的单曲线折叠折纸开始。宽面板会导致由曲面的各种发生器分布引起的不同机械响应域。然后，这些理论被扩展到多曲线折叠折纸，涉及折痕之间的额外几何相关性。作为一个说明性的例子，从理论上和数值上研究了带有环形折痕的折纸的变形和平衡配置。随后，将单顶点曲线折纸理论作为多曲线折叠折纸的一种特殊类型进行了研究。我们发现，顶点处的额外周期性强烈限制了构型空间，导致顶点附近的区域无论机械性能如何，都具有惊人的通用平衡构型。理论和数值都证实了近场区域普遍性的存在。此外，远场变形是通过能量最小化获得的，并通过有限元分析进行验证。我们的广义多曲线折叠折纸理论，包括顶点包含的普遍性，预计将为曲线折叠折纸系统的形状编程提供新的理解和框架。