CAREER: Morphologies of Tensed Sheets

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

TECHNICAL SUMMARY This 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 SUMMARY This 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.