Walker M G
Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, United Kingdom.
Phys Rev E. 2020 Apr;101(4-1):043001. doi: 10.1103/PhysRevE.101.043001.
Folded structures are often idealized as a series of rigid faces connected by creases acting as revolute hinges. However, real folded structures can deform between creases. An example of particular interest is a disk decorated by multiple radial creases. Such disks are bistable, snapping between a "natural" and "inverted" shape. We investigate the mechanical behavior of these creased disks and propose a new analytical approach to describe their mechanics. Detailed experiments are performed which show that, when indented at the center, a localized dimple forms, precluding the conical shape assumed in previous studies. As the indentation depth increases this dimple expands radially until reaching the disk edge when it snaps to the inverted shape, which has a conical form. We develop an analytical model which approximates each face as a series of rigid facets connected by hinges that can both rotate and stretch. Energy expressions are derived relating hinge rotation and stretching to compatible shell deformations of the facets and equilibrium enforced by minimizing the total strain energy. By increasing the number of facets, the mechanics of the continuum shell is approached asymptotically. The analysis shows that membrane stretching of the faces is required when a conical form of deformation is enforced. However, in the limit of zero thickness, the forming and propagation of a localized dimple is inextensional. This new approach relates the kinematic analysis of rigid origami to the mechanics of thin shells, offering an efficient method to predict the behavior of folded structures.
折叠结构通常被理想化地视为一系列由充当旋转铰链的折痕连接的刚性面。然而，实际的折叠结构在折痕之间会发生变形。一个特别有趣的例子是由多个径向折痕装饰的圆盘。这种圆盘是双稳态的，会在“自然”形状和“倒置”形状之间突然转变。我们研究了这些有折痕圆盘的力学行为，并提出了一种新的分析方法来描述它们的力学原理。进行了详细的实验，结果表明，当在圆盘中心施加压痕时，会形成一个局部凹坑，这与先前研究中假设的圆锥形状不同。随着压痕深度的增加，这个凹坑会径向扩展，直到到达圆盘边缘，此时它会突然转变为倒置形状，该形状呈圆锥状。我们开发了一个分析模型，将每个面近似为一系列由既能旋转又能拉伸的铰链连接的刚性小平面。推导了能量表达式，将铰链的旋转和拉伸与小平面的兼容壳变形以及通过最小化总应变能来实现的平衡联系起来。通过增加小平面的数量，可以渐近地逼近连续壳的力学原理。分析表明，当强制形成圆锥状变形时，面的薄膜拉伸是必要的。然而，在厚度为零的极限情况下，局部凹坑的形成和扩展是不可拉伸的。这种新方法将刚性折纸的运动学分析与薄壳力学联系起来，提供了一种预测折叠结构行为的有效方法。
