Exotic Continua: Geometry, Topology and Mechanics in Soft Matter

这笔资金由材料研究部的凝聚态物质和材料理论项目以及数学科学部的应用数学项目资助。当工程世界是由直线和刚性成分构建的时候，自然充满了美丽、柔软和起伏的形状。我们在地衣和珊瑚、羽衣甘蓝和海懒、仙人掌和扁虫中都看到了这些形状。一个自然的问题是为什么？为什么褶皱、褶皱的形状在自然界中无处不在，这些形状带来了什么潜在的进化好处？我们可以从大自然中学到什么，以帮助开发用于实际应用的软而灵活的机器人？这是这项研究将解决的一些问题。简短的答案是，起伏、褶皱的形状具有一些迷人的机械特性，这些特性影响着生物有机体的生长、移动和与环境相互作用的方式。为了理解这些力学性质，并进一步使用它们来模拟自然系统以及进行技术设计，我们必须求助于多个学科的想法和工具：数学(微分几何)、材料(弹性)和力学(力和运动)。该项目将探索这些不同领域之间的联系，并开发新的理论和实践工具，用于软/柔性材料的建模和设计。该项目包括在生长、植物、海洋无脊椎动物和软机器人方面的具体应用，因此将引起应用力学、物理学、生物学和工程学研究人员的兴趣。这项研究具有很强的跨学科性质，并为培训下一代科学家和数学家的各种技术技能以及他们为实际应用抽象、建模和设计复杂系统的能力提供了极好的机会。自然经常使用的TECHNICALA模板是一个褶皱、起伏的形状。我们在许多活着的有机体中都看到了它。一个自然的问题是为什么？更准确地说，这种褶皱起伏的形状的独特之处在于，它在自然界中如此普遍。答案似乎在于这种形状赋予生物体的一些迷人的机械特性，这些特性源于有趣的拓扑和几何考虑。研究拓扑学、几何学和力学之间的相互作用是这个项目的首要主题。褶皱的形状自然地出现在本质上是负弯曲的薄弹性物体中。这些是奇异连续体的例子，即自组织成集体状态的材料，显示几何和拓扑的局部签名。这种自组织表现为极强的柔韧性和强烈的非线性以及对外力的空间不均匀响应。这一现象具有重要的生物力学意义，也为仿生设计提供了潜在的技术应用。PI建议研究一类具有这些特征的材料的机械性能：非欧几里得弹性薄板。这个项目将三个不同的研究线索编织在一起，创造出新的理论和数值工具，用于建模和分析软材料的力学、生长和动力学。这些“以前不相关”的领域是：(1)离散微分几何，它发展了连续几何中概念的离散类比，并在图形学和计算机科学的背景下得到发展；(2)研究洛伦兹曲面，其根源是纯数学和对2D时空因果结构的研究；以及(3)不变变分双复合体，它起源于对称和变分演算之间的相互作用的研究，根源于分析、群论和几何。在这个项目中，理论问题和实际应用之间的相互作用为本科生和研究生提供了极好的教育和跨学科培训机会。PI将继续努力鼓励本科生参与科学研究，帮助初学研究生过渡到开始研究，指导研究生研究助理，并积极与数学科学中代表性不足的群体合作，帮助他们推进科学事业。所有这些努力都是为了在我们的社会中培养一支多样化、创新性和训练有素的STEM劳动力队伍。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。

This grant is being funded by the Condensed-Matter and Materials Theory program in the Division of Materials Research and by the Applied-Mathematics program in the Division of Mathematical Sciences.NONTECHNICALWhile the engineered world is built with straight lines and rigid components, nature is filled with beautiful, soft and undulating shapes. We see these shapes in lichens and in corals, in kale and in sea slugs, in cacti and in flatworms. A natural question is why? Why are frilly, crenelated forms ubiquitous in nature, and what potential evolutionary benefits arise from these shapes? What can we learn from nature to help develop soft and flexible robots for practical applications? These are some of the questions that will be addressed in this research.The short answer is that an undulating, ruffled shape possesses some fascinating mechanical properties that influence the ways in which living organisms grow, move, and otherwise interact with their environment. To understand these mechanical properties and, further, to use them to model natural systems as well as for technological design, we have to turn to ideas and tools from multiple disciplines: mathematics (differential geometry), materials (elasticity), and mechanics (forces and motion). This project will explore connections between these disparate fields and develop new theoretical and practical tools for modeling and designing with soft/flexible materials. This project includes specific applications to growth, plants, marine invertebrates, and soft robotics, and will, therefore, be of interest to researchers in applied mechanics, physics, biology, and engineering. This research is strongly interdisciplinary and offers excellent opportunities for training the next generation of scientists and mathematicians in a variety of technical skills, as well as in their ability to abstract, model, and design complex systems for practical applications.TECHNICALA template, that nature uses repeatedly, is that of a ruffled, undulating shape. We see it in a multitude of living organisms. A natural question is why? More precisely, what is unique about this ruffled, undulating shape that it is so prevalent in nature. The answer seems to lie in some fascinating mechanical properties this shape confers on organisms, properties that arise from interesting topological and geometric considerations. Studying this interplay between topology, geometry and mechanics is the overarching theme of this project.The ruffled shape arises naturally in thin elastic objects that are intrinsically negatively curved. These are examples of exotic continuua, i.e. materials that self-organize into collective states that display local signatures of geometry and topology. The self-organization manifests itself as extreme pliability and strongly nonlinear and spatially inhomogeneous response to external forces. This phenomenon has important biomechanical implications, as well as potential technological applications for biomimetic design. The PI proposes to investigate the mechanical properties of a class of materials that exhibit these features: non-Euclidean elastic thin sheets. This project weaves together three disparate strands of research to create new theoretical and numerical tools for modeling and analyzing the mechanics, growth, and dynamics of soft materials. These ``formerly unrelated" areas are (1) Discrete differential geometry, which develops discrete analogs of concepts in continuous geometry and has been developed in the context of graphics and computer science; (2) The study of Lorentz surfaces with roots in pure mathematics and investigations into the causal structure of 2D spacetimes; and (3) The invariant variational bicomplex, which arose from a study of the interplay between symmetries and calculus of variations, and has roots in analysis, group theory and geometry. The interplay between theoretical questions and practical applications in this project offers excellent opportunities for education and interdisciplinary training to undergraduate and graduate students. The PI will continue ongoing efforts to encourage undergraduates to participate in scientific research, help beginning graduate students transition into starting research, mentor graduate research associates, and actively work with people from groups that are under-represented in the mathematical sciences to help advance their scientific careers. All of these efforts are geared toward developing a diverse, innovative, and broadly trained STEM workforce in our society.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.