CAREER: Geometric and topological mechanics of flexible structures

该职业奖支持柔性结构设计和表征方面的理论和计算研究以及教育活动。科学和日常生活的世界被分成不同阶段的系统。尽管它们的原子结构大不相同，但包括砖、骨头、金属、玻璃和木材在内的固相体系都有一个共同点，那就是它们保持着特定的形状，而不像流体那样随容器的形状变化。因此，如果处理不当，所有固体都表现出普遍的响应，例如形成能够支持载荷力和通过振动波传递能量的应力模式。然而，像纸、布、电线和许多有机组织（如花瓣）这样的柔性固体与这种看似普遍的行为不同，即使在轻轻地探测时，它们也会发生很大的变形。这种灵活性带来了大量的新现象，特别是通过改变形状来实现功能，例如展开的叶子，折叠的帐篷或鸟或飞机的翅膀根据变化的风况进行自我重塑。这个项目探讨了灵活系统的一个特定子集。这些系统由重复的几何图案组成，例如在铰链上相互旋转的刚性块或沿折痕折叠的折纸板。这种结构被称为柔性机械超材料，因为它们由于这些结构元素而显示出新的性能，而这些结构元素不是来自它们的化学成分。这项研究应用并扩展了科学和数学原理，允许设计可以变形成一系列形状的新结构。选择特定的结构为系统提供了新的扩展、剪切和曲线方式。这项研究计划辅以教育和外展活动。PI正在开发一门应用拓扑概念研究问题的高级课程。该团队还在开发K12课堂模块，这些模块使用固体、可操作的系统，如折纸和弦，来实现深层的拓扑和几何概念。最后，科学概念通过多种渠道向广大公众传播。该职业奖支持理论和计算研究以及灵活结构的教育活动，其合理设计导致分析可处理和通用行为。软物质系统，如液晶、颗粒物质、凝胶和各种各样的生物结构，在许多尺度上都具有很大的变形能力，同时仍然具有一些类似固体的抗张力能力。这些软结构具有丰富而有用的现象，包括屈曲、记忆、多稳定性、自适应、挫折和相变。然而，它们的多样性和复杂性限制了确定可以统一该领域的普遍原则的能力。该项目侧重于合理设计由刚性元素组成的系统，这些刚性元素通过相对灵活的铰链以几何互补的方式连接在一起，使结构能够在低能量下承受大变形。这些结构是非线性和复杂的，足以显示丰富的现象，同时又具有足够的理性和控制，可以分析处理，并受普遍理论的支配。该团队将开发识别新结构的规则，包括用离散微分几何、组合学和张量微积分的语言制定相容性条件。与此问题相辅相成的是，力学临界性表明，变形模态可以作为系统的对称性，在结构的不同部位，由于模态的激活程度不同，导致了一个机制场。这些现象与曲率、对称和拓扑的作用密切相关。该团队将确定在这种结构中可能存在哪些非线性、非均匀低能变形。最后，这些准静态几何性质定义了一个大的非线性变形的低能量空间，在这个空间上可以发生高频动力学。该团队将探索这些机制场如何以与传统结构根本不同的方式积极驱动和耗散能量。这项研究计划辅以教育和外展活动。PI正在开发一门应用拓扑概念的高级课程，包括拓扑绝缘体，晶体缺陷和规范领域来研究问题。该团队还将开发K12课堂模块，使用可靠的、可操作的系统来实现深层拓扑和几何概念，如多面体的欧拉公式和图上的欧拉路径。最后，科学概念将通过一些渠道传达给公众。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

NONTECHNICAL SUMMARYThis CAREER award supports theoretical and computation research and educational activities into the design and characterization of flexible structures. The worlds of science and of everyday life are divided into systems in different phases. Despite their vastly different atomic structures, systems in the solid phase, including brick, bone, metal, glass and wood all have in common that they retain a particular shape, unlike fluids which take the shape of their container. Consequently, solids all display universal response when not treated too harshly, such as developing patterns of stress that can support loading forces and transmitting energy through vibrational waves. However, flexible solids like paper, cloth, wires and many organic tissues such as flower petals differ from this seemingly universal behavior by undergoing large deformations even when gently probed. This flexibility leads to vast new phenomena, and especially to achieving function derived from changing shape, such as in unfurling leaves, collapsing tents or the wing of a bird or plane reshaping itself in response to changing wind conditions.This project explores a particular subset of flexible systems. These systems consist of repeating geometrical motifs, such as rigid pieces that rotate against one another at hinges or origami panels that fold along creases. Such structures are referred to as flexible mechanical metamaterials because they display new properties due to these structural elements that are not derived from their chemical composition. This research applies and extends scientific and mathematical principles that permit the design of new structures that can be deformed into sets of shapes. Choosing particular structures leads to new ways for the system to dilate, shear and curve as desired.This research program is complemented by educational and outreach activities. The PI is developing an advanced course on applying topological concepts to research problems. The team is also developing K12 classroom modules that use solid, manipulable systems such as origami and string to realize deep topological and geometrical concepts. Finally, scientific concepts are communicated to the public at large via a number of channels.TECHNICAL SUMMARYThis CAREER award supports theoretical and computational research and educational activities into flexible structures whose rational design leads to analytically tractable and universal behavior. A hallmark of soft matter systems as diverse as liquid crystals, granular matter, gels, and various living structures across many scales is the ability to undergo large deformations while still offering some solid-like resistance to strain. These soft structures host a rich and useful set of phenomena, including buckling, memory, multistability, adaptation, frustration and phase transitions. However, their diversity and complexity limits the ability to identify universal principles that can unite the field.This project focuses on rationally designed systems consisting of rigid elements joined via relatively flexible hinges in a geometrically complementary way which allows the structure to undergo a large deformation at low energies. These structures are nonlinear and complex enough to display rich phenomena while being sufficiently rational and controlled to be analytically tractable and governed by universal theories. The team will develop rules for identifying new structures involves formulating compatibility conditions in the language of discrete differential geometry, combinatorics, and tensor calculus. Complementary to this problem, mechanical criticality indicates that the deformation mode can serve as a symmetry of the system, leading to a mechanism field as the mode is activated to different degrees in different parts of the structure. These phenomena are closely informed by the role of curvature, symmetry and topology. The team will identify which nonlinear, non-uniform low-energy deformations are possible in such structures. Finally, these quasistatic geometric properties define a large low-energy space of nonlinear deformations on which high-frequency dynamics can occur. The team will explore how these mechanism fields can be actively driven and dissipate energy in ways that are fundamentally distinct from conventional structures.This research program is complemented by educational and outreach activities. The PI is developing an advanced course on applying topological concepts, including topological insulators, crystal defects and gauge fields to research problems. The team will also develop K12 classroom modules that use solid, manipulable systems to realize deep topological and geometrical concepts such as Euler's formula for polyhedra and Eulerian paths on graphs. Finally, scientific concepts will be communicated to the general public via a number of channels.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.