Structures, Composites, and Inhomogeneous Bodies

这个项目的重点是材料和机械系统的数学建模。这项工作旨在推进几个系统的数学基础，这些系统在技术发展的前沿应用。该项目的一部分涉及隔震器，即专门用于保护建筑物免受地震破坏性剪切和垂直运动影响的机械结构。这部分工作的目的是指导设计在大位移下性能更好的隔振器。项目的其他部分涉及复合材料和超材料。复合材料在现代技术和制造业中有着重要的应用。超材料是一种工程复合材料，将不同的材料组合在一起，产生与组件不同的结构，可以具有天然材料所没有的特性，并且在其他用途中可以在导波中发挥重要作用。项目的这些部分的目标是指导动态超材料的设计，增强对复合材料数学基础的理解，以及对其性质和行为的固有限制的理解。这项工作的一部分有可能应用于医学成像和其他非侵入性检测。该项目包括一名博士后研究助理的跨学科培训。本研究包括五个部分：(1)隔震器的研究工作利用板铰连接机制开发了一种新的概念方法来构建结构单元，使建筑物在地震事件中免受地面水平或垂直运动的影响；(2)设计能量局部守恒的时空超材料微结构，消除当前时空微结构与外部系统耦合的需要，或采用补充泵浦波，根据需要产生或吸收能量；(3)寻求对复合理论中精确关系和联系的基本理解，这有助于发现适用于介质的新关系，其中材料模量的局部张量场在非旋转不变的流形上取值；(4)获得粘弹性两相复合材料或体在时域内可能行为范围的估计，这有助于设计具有所需时间响应的材料和估计相所占体积分数；(5)确定复介电张量在两个各向异性相的各向异性复合材料中所能吸收的范围，这有助于设计最大限度地吸收电磁能或以理想方式引导电场和电位移场的结构。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project is focused on the mathematical modeling of materials and mechanical systems. The work aims to advance the mathematical underpinnings of several systems that have applications at the leading edge of technological development. Part of the project concerns seismic isolators, mechanical structures tailored to protect structures from the destructive shearing and vertical motions of earthquakes. The goal of this part of the work is to guide the design of isolators with improved performance under large displacements. Other parts of the project concern composite materials and metamaterials. Composite materials are important in applications throughout modern technology and manufacturing. Metamaterials, which are engineered composite materials where dissimilar materials are combined to produce a structure with properties unlike those of the components, can have properties not found in naturally occurring materials and among other uses can serve important functions in guiding waves. The goals of these parts of the project are to guide the design of dynamic metamaterials and to enhance understanding of the mathematical underpinnings for composite materials, as well as understanding inherent limits on their properties and behavior. Parts of this work have potential application to medical imaging and other noninvasive testing. The project includes the interdisciplinary training of a postdoctoral research associate.The research consists of five parts: (1) work on earthquake isolators utilizes mechanisms of plate-hinge linkages to develop a new conceptional approach to the construction of structural units that insulate a building from horizontal or vertical movement of the ground during seismic events; (2) design of space-time metamaterial microstructures in which energy is locally conserved, eliminating the need of current space-time microstructures to couple with outside systems, or employ supplementary pumping waves, to generate or absorb energy as demanded; (3) search for fundamental understanding of exact relations and links in the theory of composites, which could help in the discovery of new relations applicable to media where the local tensor field of material moduli take values on a manifold that is not rotationally invariant; (4) obtaining estimates of the range of possible behaviors in the time domain for viscoelastic two-phase composites or bodies, which could help in design of materials with desired temporal responses and in estimation of the volume fractions occupied by the phases; and (5) work on determining the range that the complex dielectric tensor can take in an anisotropic composite of two anisotropic phases, which could facilitate design of structures that absorb a maximum amount of electromagnetic energy or that guide electric and electric displacement fields in desired ways.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.