CAREER: Morphologies of Tensed Sheets

技术总结这个职业奖项支持理论研究和教育，以开发一个全面的理论框架，研究加载和限制下薄板的形态，这是一个与生命系统连接的问题。薄固体薄板在一般的约束和无特征的力分布下表现出复杂的图案。这种形态的多样性反映了弹性薄板中几何和力学之间的耦合。最近，这一领域的研究活动激增，这是因为有研究表明，它与形态发生过程有关，例如动物上皮或植物叶片中出现的组织形成不稳定性，以及能够对微小结构进行机械控制的不断缩小的规模的材料应用。这一进展带来的一个主要理论挑战是发展一种形式主义，解释在不同载荷和约束下薄板中图案形成的基本机制。当薄板厚度变得非常小时，这里的障碍与高度非线性的性质和问题的几何复杂性有关。为了取得进展，需要超越传统使用的新概念和方法。发展这些是研究的重点。通常，薄板的形态被描述为由褶皱、皱折、折叠、折痕和水泡组成。这些描述性的词语可能看起来很清楚，但实际模式之间的数量差异远不明显；此外，这些不同类型的变形出现的条件也不清楚。这个项目的一个主要目标是帮助澄清这些问题。径向拉伸问题将被发展为一种模板，用于将各种变形类型分类为弹性薄板中形状和应力场的对称破坏不稳定性。这些研究项目将产生新的分析技术，并将改进研究薄板力学和几何的数值方法。拟议的项目应与新兴技术高度相关。两个潜在应用的代表性例子与纳米薄膜的计量学和新兴的“可拉伸电子学”技术有关。该项目对教育的更广泛影响包括对学生和博士后进行各种分析和数值方法的培训，以及为教师举办的暑期讲习班：“我们周围的图案”，其中选定的研究成果和自然界中图案形成的其他例子将被同化为实际操作的工具包，将交付给中学教师。这项活动将与马萨诸塞大学STEM教育研究所协调。非技术SUMMARY这个职业奖项支持理论研究和教育，以开发一个理论框架，以了解薄板在各种力的响应下所承担的结构模式。弹力片表现出高度复杂的形态，通常被描述为皱纹、皱纹、皱折、折叠和水泡。仔细检查糖果包装、人的皮肤或拉伸的塑料袋，就会发现这些图案通常以相同的形状共存，从微米甚至纳米到人体尺寸。为什么薄膜在拉伸或限制时会折叠，而相同材料的厚板却不会折叠？为什么纸往往会起皱，而橡皮板会平滑地起皱？为什么某些植物的叶子是弯曲的，而其他植物的叶子是平的？回答这些问题不仅对满足我们与生俱来的好奇心很重要。天然和合成薄膜中复杂形态的出现影响了它们的机械、光学和化学性能，并可能在材料科学和工程以及生物世界中产生深远的影响。了解如何在无特征的作用力下自发地形成板材中的复杂图案，可能会启发人们采用有效的方法来裁剪所需的表面图案，或者从一块均匀的物质中随意地自组装结构。本课题试图通过运用模式形成理论的概念和方法来解决这些问题。通过聚焦于一类问题，PI试图将不同的变形类型识别为薄板的不同形态阶段。薄片的这些相将类似于普通水中常见的冰、液体、蒸汽等物质的相。然而，这些“形态阶段”在概念上是不同的，因为它们出现在与水不同的系统中，远未达到人们所熟知的平衡平衡。开发一个概念框架将能够对不同图案在床单上出现和消失的力和几何约束进行量化分析。该奖项的教育部分支持由PI与马萨诸塞州大学STEM教育研究所合作开发的夏季研讨会《我们周围的图案》。该计划将使用常见的形状，如浴帘和贴在弯曲保险杠上的贴纸，目的是：(A)通过日常现象激发物理-数学思维。(B)向教师和学生传授模式形成理论的概念。

TECHNICAL SUMMARYThis CAREER award supports theoretical research and education to develop a comprehensive theoretical framework for studying morphologies of thin sheets under loading and confinement, a problem with connections to living systems. Thin solid sheets exhibit complex patterns under generic confinements and featureless distribution of forces. This morphological diversity reflects the coupling between geometry and mechanics in elastic sheets. This field has seen recently a surge of research activity, driven by studies that demonstrated its relevance for morphogenetic processes, such as the tissue-shaping instabilities occurring in animal epithelia or plant leaves, and by material applications at ever decreasing scales that enable the mechanical control of tiny structures. A major theoretical challenge posed by this progress is to develop a formalism that explains the basic mechanisms for pattern formation in thin sheets under various loadings and confinements. The hurdle here is associated with the highly nonlinear nature and the geometric complexity of the problem when the sheet thickness becomes very small. To make progress, new concepts and methods, beyond traditionally used ones are required. Developing these is a focus of the research.Quite generally, morphologies of thin sheets are described as composed of wrinkles, crumples, folds, creases, and blisters. These descriptive words may appear lucid, but a quantitative distinction between the actual patterns is far from being obvious; moreover, the conditions under which these various types of deformations emerge are unclear. A primary goal of this project is to help clarify these questions. The radial stretching problem will be developed as a template for classifying various deformation types as symmetry breaking instabilities of the shape and stress field in an elastic sheet. The research projects will yield new analytic techniques and will improve numerical methods for studying the mechanics and geometry of thin sheets. The proposed projects should be of high relevance for emerging technologies. Two representative examples for potential applications are related to the metrology of nanofilms, and to emerging "stretchable electronics" technologies.The educational broader impacts of this project include training students and postdocs in a variety of analytical and numerical methods, and a summer workshop for teachers: "Patterns around us" in which selected research results and other examples of pattern formation in nature will be assimilated into hands-on kits that will be delivered to middle-school teachers. This activity will be coordinated with the University of Massachusetts STEM Education Institute.NONTECHNICAL SUMMARYThis CAREER award supports theoretical research and education on developing a theoretical framework to understand the structural patterns thin sheets assume in response to various forces. Elastic sheets exhibit highly complex morphologies that are often described as wrinkles, creases, crumples, folds, and blisters. A close inspection of a candy wrap, a human skin, or a stretched plastic bag, reveals that these patterns often coexist in the same shape, ranging from microns or even nanometers, up to human body scales. Why do films become folded upon stretching or confinement whereas a thick slab of an identical material does not fold? Why does paper tend to crumple whereas rubber sheets would smoothly wrinkle? Why do certain plant leaves have a buckled shape, whereas others are flat? Answering these questions is important not only for satisfying our natural curiosity. The emergence of complex morphologies in natural and synthesized sheets affects their mechanical, optical and chemical properties, and may have far reaching consequences in material science and engineering, as well as in the bio-world. Understanding how complicated patterns in sheets emerge spontaneously under featureless forces may inspire efficient methods for tailoring a desired surface pattern, or the self-assembly of structures at will from a homogenous piece of matter. This project seeks to address these questions by applying the concepts and methods of pattern formation theory. By focusing on a class of problems, the PI seeks to identify various deformation types as different 'morphological phases' of thin sheets. These phases for thin sheets would be analogous to phases of materials like ordinary water's familiar phases of ice, liquid, steam. However, these 'morphological phases' are conceptually distinct as they arise in systems that, unlike water, are far from the well understood balance of equilibrium. Developing a conceptual framework will enable a quantitative analysis of the forces and geometric constraints under which distinct patterns emerge and disappear in sheets.The education component of this award supports a summer workshop "Patterns around us" that will be developed by the PI in coordination with the University of Massachusetts STEM Education Institute. The program will use common shapes, such as shower curtains and stickers attached to curved bumpers in order to: (a) Stimulate a physical-mathematical thinking through everyday phenomena. (b) Impart to teachers and students concepts of pattern formation theory.