Exotic Continua: Geometry, Topology and Mechanics in Soft Matter

奇异的连续体：软物质中的几何、拓扑和力学

• 批准号: 1923922
• 负责人: Shankar Venkataramani
• 金额: $ 30万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1923922&HistoricalAwards=false
• 关键词: Exotic; Continua; Geometry; Topology; Mechanics

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.

这项资助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数学数学项目提供资金。非物质性工程世界是由直线和刚性部件构成的，而自然界则充满了美丽、柔软和起伏的形状。我们在地衣、珊瑚、羽衣甘蓝、海蛞蝓、仙人掌和扁形虫中看到这些形状。 一个自然的问题是为什么？ 为什么褶边、锯齿状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益处？ 我们可以从自然界中学到什么来帮助开发用于实际应用的柔软和灵活的机器人？ 这些都是本研究要解决的问题，简单的回答是，波浪形的褶皱形状具有一些迷人的机械特性，这些特性会影响生物体的生长、运动以及与环境的相互作用。 为了理解这些力学性质，并进一步利用它们来模拟自然系统以及技术设计，我们必须求助于多个学科的思想和工具：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。该项目包括生长，植物，海洋无脊椎动物和软机器人的具体应用，因此，应用力学，物理学，生物学和工程学的研究人员会感兴趣。 这项研究是强烈的跨学科和培训下一代的科学家和数学家在各种技术技能，以及在他们的抽象，建模和设计复杂系统的实际应用能力提供了极好的机会。 我们在许多生物体中都能看到它。 一个自然的问题是为什么？ 更确切地说，这种褶皱的起伏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，这些特性来自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该项目的首要主题。褶边形状自然出现在本质上呈负弯曲的薄弹性物体中。 这些都是奇异连续体的例子，即自组织成集体状态的材料，显示出局部几何和拓扑特征。 自组织表现为极端的柔韧性和强烈的非线性和空间不均匀的响应外力。 这种现象具有重要的生物力学意义，以及仿生设计的潜在技术应用。 PI建议研究一类具有这些特征的材料的力学性能：非欧几里得弹性薄板。该项目将三个不同的研究方向编织在一起，以创建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似物，并在图形学和计算机科学的背景下发展起来;（2）洛伦兹曲面的研究，其根源在于纯数学和对二维时空因果结构的调查;（3）不变变分双复形，它起源于对对称性和变分法之间相互作用的研究，并植根于分析、群论和几何学。 该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良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，帮助开始研究生过渡到开始研究，指导研究生研究助理，并积极与数学科学代表性不足的群体合作，以帮助推进他们的科学事业。 所有这些努力都是为了在我们的社会中培养多元化、创新和受过广泛培训的STEM劳动力。该奖项反映了NSF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。

项目成果

Disk galaxies and their dark halos as self-organized patterns

• DOI: 10.1016/j.physletb.2020.136060
• 发表时间: 2021-02
• 期刊: Physics Letters B
• 影响因子: 4.4
• 作者: S. Venkataramani;A. Newell

Distributed Branch Points and the Shape of Elastic Surfaces with Constant Negative Curvature

分布分支点与恒负曲率弹性曲面的形状

• DOI: 10.1007/s00332-020-09657-2
• 发表时间: 2021
• 期刊: Journal of Nonlinear Science
• 影响因子: 3
• 作者: Shearman, Toby L.;Venkataramani, Shankar C.
• 通讯作者: Venkataramani, Shankar C.

Mechanics of moving defects in growing sheets: 3-d, small deformation theory

生长板材中移动缺陷的力学：3-d、小变形理论

• DOI: 10.1186/s41313-020-00018-w
• 发表时间: 2020
• 期刊: Materials Theory
• 作者: Acharya, Amit;Venkataramani, Shankar C.
• 通讯作者: Venkataramani, Shankar C.

Nature’s forms are frilly, flexible, and functional

自然的形式是褶边、灵活且实用的

• DOI: 10.1140/epje/s10189-021-00099-6
• 发表时间: 2021
• 期刊: The European Physical Journal E
• 作者: Yamamoto, Kenneth K.;Shearman, Toby L.;Struckmeyer, Erik J.;Gemmer, John A.;Venkataramani, Shankar C.
• 通讯作者: Venkataramani, Shankar C.