Asymptotically Isometric Mechanics

非技术总结该奖项支持理论研究和教育，以促进对超薄薄板和壳的结构和力学的了解。由发现原子厚度的二维固体引发的实验，最著名的是被称为石墨烯的单原子厚度的碳片和只有纳米厚的塑料片。这些发展为设计新型极薄、高度可折叠的固体材料开辟了新的平台，并可能带来材料科学和工业的革命性应用，从可伸缩的电子电路到基于折纸的超材料设计，再到表面活性剂对液体的封装，赋予它们类似固体的特性，等等。该项目致力于开发理论工具来研究这种超薄固体材料的复杂机制。特别令人感兴趣的是，这些材料的极端弯曲能力如何使原本需要大量输入弹性能量的变形成为可能。用箔片(例如糖果包装纸)包裹球提供了一个说明性的例子。由于包裹过程中不可避免地产生高应变，铝箔从其原始平坦状态不可逆转地变形。相比之下，超薄固体极易弯曲，使它们成为几乎可以完全避免应变的高效包装物。目前，还不知道在使箔片应变时浪费的能量水平如何取决于其厚度，也不知道如何控制将使这种应变最小化的包裹过程。这个PI将推进由PI和他的合作者提出的渐近等距力学的理论框架。这一理论表明，对于由超薄、高度弯曲的固体薄板制成的包装纸来说，应变以及由此产生的弹性能量几乎可以完全消除。PI和他的研究小组及其合作者将利用这个框架来揭示指示能量高效包裹和其他对超薄固体施加应变的过程的原理，并将为计算高度可弯曲的超薄薄板的新兴结构提供一个理论平台。该项目还部分支持了由PI联合举办的研究生、软固体和复杂流体的年度暑期班，以及为中学科学教师举办的年度外展研讨会--我们周围的模式。该奖项支持理论研究和教育，通过重温基本概念和提供新的分析工具来研究超薄固体的力学，从而促进对超薄薄板和壳的结构和力学的理解。通过几何约束或施加边界载荷来研究超薄板的约束，发现了机械响应和图案形成现象，这似乎与经典弹性理论的预测不一致。这就需要一种新的方法，PI已经通过开发一种称为渐近等距力学(AIM)的框架来回应这种新方法，该框架解决了固体薄板和壳的不相容几何约束的影响。AIM是PI将在研究中使用的方法的基础；它指的是一种变分方法，它描述了施加应变的几何约束下薄实体的机械响应和形态。例如，用超薄的聚合物薄片或“固体表面活性剂”包裹液体体积，对壳层施加局部作用力，以及夹杂物对石墨烯等纯结晶薄片的力学和形状的影响。这些问题的核心是，几何不相容的约束阻止了高能应变的完全消除，因此，力学受将应变降低到几何约束、施加的载荷和弯曲成本允许的最小水平的趋势所支配。PI将通过推导一套有效的变分原理和相应的分析工具来发展AIM理论，这些原理和分析工具可以用于描述在广泛的几何约束和外部载荷作用下的板壳的力学和形态。其结果将是两个经典理论之间的实际桥梁：欧拉的“弹性”理论，它等于单独最小化弯曲能量，以及Foppl-von Karman方程，它表示同时最小化全部弹性能量。通过与几个实验和计算小组的合作，PI和他的团队将实施这种新的分析方法来研究薄固体的几何和弹性。该项目还部分支持由PI联合举办的研究生、软固体和复杂流体的年度暑期班，以及为中学科学教师举办的年度外联研讨会。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.