CAREER: Geometric and topological mechanics of flexible structures

非技术总结这个职业奖项支持柔性结构的设计和表征方面的理论和计算研究以及教育活动。科学世界和日常生活世界在不同的阶段被划分为系统。尽管它们的原子结构截然不同，但固相系统，包括砖、骨、金属、玻璃和木材，都有一个共同点，它们保持着特定的形状，而不是液体的容器形状。因此，当不太严厉地处理时，所有固体都表现出普遍的反应，例如形成能够支持负载力的应力模式，并通过振动波传递能量。然而，纸、布、电线等弹性固体和许多有机组织(如花瓣)与这种看似普遍的行为不同，即使在轻轻探测时也会发生巨大的变形。这种灵活性导致了大量的新现象，特别是通过改变形状来实现功能，例如展开树叶、折叠帐篷或鸟类或飞机的机翼，以适应不断变化的风条件。这个项目探索了柔性系统的一个特殊子集。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部件，或沿折痕折叠的折纸面板。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不是来自其化学成分而显示出新的性能。这项研究应用并扩展了科学和数学原理，这些原理允许设计出可以变形为一组形状的新结构。选择特定的结构为系统带来了新的扩张、剪切和弯曲的方式。这项研究计划得到了教育和推广活动的补充。PI正在开发一门高级课程，内容是将拓扑概念应用于研究问题。该团队还在开发K12课堂模块，这些模块使用坚实的、可操作的系统，如折纸和绳子，来实现深刻的拓扑和几何概念。最后，科学概念通过多种渠道向公众传达。技术总结这个职业奖项支持将理论和计算研究和教育活动转化为灵活的结构，这些结构的合理设计将导致分析上的易操控和普遍的行为。软物质系统如液晶、颗粒物质、凝胶和许多尺度上的各种生命结构的一个特点是能够经历大变形，同时仍然提供一些类似固体的应变抵抗。这些软结构包含一组丰富而有用的现象，包括屈曲、记忆、多稳定性、适应、挫折和相变。然而，它们的多样性和复杂性限制了确定能够统一场的普遍原则的能力。本项目专注于由刚性元件组成的合理设计的系统，这些刚性元件通过相对灵活的铰链以几何互补的方式连接，允许结构在低能量下经历大变形。这些结构是非线性和复杂的，足以展示丰富的现象，同时足够合理和受控制，可以分析处理并受普遍理论支配。该团队将开发识别新结构的规则，涉及用离散微分几何、组合学和张量微积分的语言来制定相容条件。作为对这一问题的补充，力学临界性表明，变形模式可以作为系统的对称性，当该模式在结构的不同部分被不同程度地激活时，导致机构场。这些现象与曲率、对称性和拓扑学的作用密切相关。该团队将确定在这种结构中可能发生哪些非线性、非均匀的低能量变形。最后，这些准静态几何属性定义了一个大的低能非线性变形空间，在该空间上可以发生高频动力学。该团队将探索如何以与传统结构根本不同的方式积极驱动这些机制场并分散能量。这项研究计划得到了教育和推广活动的补充。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、晶体缺陷和规范场来研究问题。该团队还将开发K12课堂模块，这些模块使用坚实的、可操作的系统来实现深层次的拓扑和几何概念，如图上多面体和欧拉路的欧拉公式。最后，科学概念将通过一些渠道传达给公众。这一奖项反映了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 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.