The Mechanics of Self-Folding Structures

非技术概述该奖项支持理论和计算研究，以及有关机械结构的教育，这些结构由可编程和响应性材料制成，并从最初的平面形状折叠成3D形状。自折叠材料结构将给制造业带来革命性的变化，特别是通过制造从不到一毫米到人类规模的非常复杂的三维结构。因为它们是扁平的，所以可以廉价、快速和批量地制造自折叠结构。然而，实现这一技术的主要障碍之一是错误折叠结构的扩散。PI和他的团队将开发新的理论和计算工具，以设计能够牢固折叠的材料结构，从而消除这一制造障碍。这项研究还将通过结合离散几何领域的工具，在自折叠材料结构的力学和更一般的材料壳力学之间建立联系。这个项目将向研究生介绍材料分析的尖端技术，并促进理论和实验之间的合作。在这个项目中开发的方法将被合并到一个计算包中，该包将被记录下来并分发给更广泛的材料社区。将开发一个中学几何和折纸教育模块，向学生介绍数学思维。技术概述该奖项支持理论和计算研究，以及关于自动折叠折纸结构的教育。自折叠折纸结构将为制造业带来革命性的变化，因为它提供了一种简单的方法，可以在任何长度尺度上从最初平坦的基板上创建复杂的三维几何图形。最近，有大量的努力致力于创造材料系统，这些材料系统可以在施加外部刺激时被制造出来折叠或展开。虽然许多复杂的折叠结构可以牢固地折叠成目标形状，但许多折叠结构并非如此，即使折叠图案很简单。实现这种结构的主要障碍之一是配置空间的分叉性质，它显示出结构的复杂性呈指数级的多个分支。这是这种结构变形的一个关键和不可避免的特征，并导致在折叠中出现明显的玻璃样行为。PI将发展一种关于自折叠结构的力学理论，并设计出使这些结构牢固折叠的原理。具体地说，这个项目的科学目标是发展一种关于自折叠结构的构形空间的理论，该理论结合了褶皱的弯曲弹性，允许在顶点引入高斯曲率，并适当地解释了二阶及以上结构的几何刚度。该项目将通过引入新的分析技术来预测三维结构，从而对结构材料的设计产生影响。这将导致在数学中进一步开发和编制一个可用于对刚性机构进行复杂计算的软件包。它还资助了一些教育活动，包括对研究生进行尖端理论技术培训。将开发一个中学几何和折纸教育模块，向学生介绍数学思维。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Nontechnical Summary This award supports theoretical and computational research, and education on mechanical structures that are made from programmable and responsive materials, and fold themselves into 3D shapes from an initially flat shape. Self-folding material structures are poised to revolutionize manufacturing, particularly by allowing the fabrication of very complex three-dimensional structures from sizes less than a millimeter to human scale. Because they are fabricated flat, self-folding structures can be made cheaply, rapidly, and in bulk. However, one of the primary obstacles to realizing this technology is the proliferation of misfolded structures. The PI and his team will develop new theoretical and computation tools to design material structures that will fold robustly, thus removing this manufacturing obstacle. The research will also establish connections between the mechanics of self-folding materials structures and the mechanics of material shells more generally by incorporated tools from the field of discrete geometry. This project will introduce graduate students to cutting edge techniques in the analysis of materials, and foster collaborations between theory and experiment. The methods developed in this project will be incorporated into a computational package that will be documented and distributed to the broader materials community. An educational module on geometry and origami for middle school will be developed to introduce students to mathematical thinking. Technical Summary This award supports theoretical and computational research, and education on self-folding origami structures. Self-folding origami structures are poised to revolutionize manufacturing by providing an easy means of creating complex three-dimensional geometries from an initially flat substrate on any length scale. Recently, there has been a great deal of effort devoted to creating material systems that can be fabricated to fold or unfold upon application of an external stimulus. While many complex folding structures fold robustly into their target shape, many do not, even when the fold patterns are simple. One of the primary obstacles to realizing such structures is the bifurcated nature of the configuration space, which exhibits a multitude of branches that are exponential in the complexity of the structure. This is a key and unavoidable feature of the way such structures deform and leads to apparent glass-like behavior in folding. The PI will develop a theory for the mechanics of self-folding structures and devise principles by which those structures can be made to fold robustly. The scientific aims of this project, specifically, are to develop a theory for the configuration space of self-folding structures that incorporates the bending elasticity of folds, allows for the introduction of Gaussian curvature at vertices, and properly accounts for the geometrical rigidity of the structures at second-order and beyond. This project will have an impact on the design of structured materials by introducing new analytical techniques for the prediction of three-dimensional structures. It will lead to the further development and documentation of a software package in Mathematica that can be used to perform complex calculations on rigid mechanisms. It also funds a number of education activities, including the training of a graduate student in cutting edge theoretical techniques. An educational module on geometry and origami for middle school will be developed to introduce students to mathematical thinking.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.