Periodic origami patterns made with repeating unit cells of creases and panels bend and twist in complex ways. In principle, such soft modes of deformation admit a simplified asymptotic description in the limit of a large number of cells. Starting from a bar and hinge model for the elastic energy of a generic four parallelogram panel origami pattern, we derive a complete set of geometric compatibility conditions identifying the pattern’s soft modes in this limit. The compatibility equations form a system of partial differential equations constraining the actuation of the origami’s creases (a scalar angle field) and the relative rotations of its unit cells (a pair of skew tensor fields). We show that every solution of the compatibility equations is the limit of a sequence of soft modes — origami deformations with finite bending energy and negligible stretching. Using these sequences, we derive a plate-like theory for parallelogram origami patterns with an explicit coarse-grained quadratic energy depending on the gradient of the crease-actuation and the relative rotations of the cells. Finally, we illustrate our theory in the context of two well-known origami designs: the Miura and Eggbox patterns. Though these patterns are distinguished in their anticlastic and synclastic bending responses, they show a universal twisting response. General soft modes captured by our theory involve a rich nonlinear interplay between actuation, bending and twisting, determined by the underlying crease geometry.

中文翻译：

---

从杆和铰链弹性推导平行四边形折纸的有效板理论


由折痕和面板的重复单位单元制成的周期性折纸图案以复杂的方式弯曲和扭曲。原则上，这种软变形模式允许在大量单元的限制下进行简化的渐近描述。从通用四平行四边形面板折纸图案的弹性能的杆和铰链模型开始，我们推导出一套完整的几何兼容性条件，用于识别该极限下图案的软模式。兼容性方程形成偏微分方程组，约束折纸折痕（标量角场）的驱动及其晶胞（一对斜张量场）的相对旋转。我们证明了兼容性方程的每个解都是一系列软模式的极限——具有有限弯曲能和可忽略拉伸的折纸变形。使用这些序列，我们推导出平行四边形折纸图案的板状理论，该理论具有显式粗粒度二次能量，具体取决于折痕驱动的梯度和细胞的相对旋转。最后，我们在两种著名的折纸设计的背景下阐述我们的理论：Miura 和 Eggbox 图案。尽管这些图案在反弹性和同弹性弯曲响应方面有所区别，但它们表现出普遍的扭曲响应。我们的理论捕获的一般软模式涉及驱动、弯曲和扭曲之间丰富的非线性相互作用，这是由底层折痕几何形状决定的。