CAREER: Morphologies of Tensed Sheets

该职业奖支持理论研究和教育，以开发一个全面的理论框架，用于研究加载和限制下的薄板形态，这是一个与生命系统相关的问题。在一般的约束和无特征的力分布下，固体薄板呈现出复杂的模式。这种形态多样性反映了弹性片材中几何和力学之间的耦合。这一领域最近出现了研究活动的激增，研究表明其与形态发生过程的相关性，例如动物上皮细胞或植物叶片中发生的组织成形不稳定性，以及材料应用在不断减少的规模上，使微小结构的机械控制成为可能。这一进展所带来的一个主要理论挑战是发展一种形式主义，解释各种载荷和约束下薄板图案形成的基本机制。这里的障碍是与高度非线性的性质和几何复杂性的问题时，板厚度变得非常小。要取得进展，就需要有新的概念和方法，而不是传统使用的概念和方法。开发这些是研究的焦点。通常，薄片的形态被描述为由皱纹，褶皱，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，但实际模式之间的定量区别远不明显;此外，这些各种类型的变形出现的条件也不清楚。这个项目的主要目标是帮助澄清这些问题。 径向拉伸问题将被开发为一个模板，用于分类各种变形类型的形状和应力场的对称破缺不稳定性的弹性片材。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形状的数值方法。拟议的项目应与新兴技术高度相关。潜在应用的两个代表性例子与纳米薄膜的计量学和新兴的“可拉伸电子”技术有关。该项目的教育影响更广泛，包括培训学生和博士后各种分析和数值方法，以及为教师举办夏季研讨会：“我们周围的模式”，其中选定的研究成果和其他自然界模式形成的例子将被吸收到手中-将分发给中学教师的工具包。这项活动将与马萨诸塞州大学STEM教育研究所协调进行。非技术性总结这项职业奖支持理论研究和教育，开发一个理论框架，以了解薄板在各种力的作用下所呈现的结构模式。 弹性片材表现出高度复杂的形态，其通常被描述为皱纹、折痕、褶皱、折叠和气泡。仔细观察糖果包装、人体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度。为什么薄膜在拉伸或限制时会折叠，而同样材料的厚板却不会折叠？为什么纸张容易起皱，而橡胶板却能平滑起皱？为什么有些植物的叶子是弯曲的，而有些是扁平的？思考这些问题不仅对满足我们天生的好奇心很重要。天然和合成片材中复杂形态的出现影响其机械、光学和化学性质，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。了解复杂的图案如何在无特征的力量下自发出现，可能会激发有效的方法来定制所需的表面图案，或者从同质物质中随意自组装结构。本项目旨在通过应用模式形成理论的概念和方法来解决这些问题。通过关注一类问题，PI试图将各种变形类型识别为薄板的不同“形态阶段”。薄片的这些相类似于物质的相，比如普通水熟悉的冰、液体、蒸汽相。然而，这些“形态相”在概念上是不同的，因为它们出现在与水不同的系统中，远离人们所熟知的平衡。 开发一个概念框架将能够定量分析的力量和几何约束下，不同的模式出现和消失在sheets.教育部分，这个奖项支持一个夏季研讨会“模式在我们身边”，将由PI与马萨诸塞州大学STEM教育研究所协调开发。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，以便：（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.