Asymptotically Isometric Mechanics

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NONTECHNICAL SUMMARYThis award supports theoretical research and education to advance understanding of the structure and mechanics of very thin sheets and shells. Experiments triggered by the discovery of two-dimensional, atomically-thin solids, most notably single atom thick carbon sheets known as Graphene and plastic sheets that are only nanometers-thick. These developments opened new platforms for designing novel types of extremely thin, highly foldable solid materials, and a host of possible applications that may revolutionize material science and industry, from stretchable electronic circuitry to origami-based design of meta-materials to the encapsulation of liquids by surfactants that impart to them solid-like properties to much more. This project focuses on developing theoretical tools to study the intricate mechanics of such ultrathin solid materials. Of particular interest is how the extreme bendability of these materials enables deformations that would otherwise require substantial input of elastic energy. The wrapping of a ball by a foil, for example a candy wrapper, provides an illustrative example. The foil deforms irreversibly from its original flat state, due to the high, unavoidable strains that are generated by the wrapping process. By contrast, the extreme bendability of ultrathin solids makes them highly efficient wrappers that can almost entirely avoid strain. Currently, it is not known how the level of energy that is wasted in straining the foil depends on its thickness, nor is it known how to control a wrapping process that will minimize such strains. This PI will advance a theoretical framework called Asymptotically Isometric Mechanics introduced by the PI and his collaborators. This theory shows that strains, and thereby elastic energy, may be eliminated almost entirely for wrappers made of an ultrathin, highly bendable solid sheet. The PI and his research group and collaborators will employ this framework to unravel the principles that dictate energetically efficient wrapping and other processes that impose strains on ultrathin solids, and will provide a theoretical platform for computing the emerging structures of highly bendable, ultrathin sheets. This project also supports in part an annual summer school for graduate students, Soft Solid and Complex Fluids, co-organized by the PI, and an annual outreach workshop, Patterns Around Us, for middle school science teachers. TECHNICAL SUMMARY This award supports theoretical research and education to advance understanding of the structure and mechanics of very thin sheets and shells through revisiting basic concepts and providing new analytical tools to study the mechanics of very thin solids. Experiments that study the confinement of ultrathin sheets through geometric constraints or the exertion of boundary loads reveal mechanical response and pattern formation phenomena that appear to be inconsistent with the predictions of classical elasticity theory. This prompts the need for a new approach which the PI has answered by the development of a framework called Asymptotically Isometric Mechanics (AIM) which addresses the effect of incompatible geometric confinement of solid sheets and shells.AIM underlies the approach the PI will use in the research; it refers to a variational approach that characterizes the mechanical response and morphology of thin solid bodies under geometrical constraints that impose strain. Examples include the wrapping of liquid volumes by ultrathin polymer sheets or "solid surfactants," the exertion of localized force on shells, and the effect of inclusions on the mechanics and shape of an otherwise pure crystalline sheet like graphene. Central to these problems is that geometrically incompatible constraints prevent a complete elimination of highly energetic strain, hence the mechanics is governed by the tendency to reduce strain to the minimal level allowed by the geometric constraints, exerted loads, and cost of bending. The PI will develop AIM theory by deriving a comprehensive set of effective variational principles, and corresponding analytical tools, that can be used to describe the mechanics and morphology of sheets and shells subjected to a broad range of geometric constraints and external loads. The outcome will be actual bridge between two classical theories: Euler's "elastica," which amounts to minimizing bending energy alone, and Foppl-von Karman equations, which express the simultaneous minimization of the full elastic energy. Through collaboration with several experimental and computational groups, the PI and his group will implement this new analytical approach to study the geometry and elasticity of thin solids.This project also supports in part an annual summer school for graduate students, Soft Solid and Complex Fluids, co-organized by the PI, and an annual outreach workshop, Patterns Around Us, for middle school science teachers.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.