Dynamic and Nonlinear Static Problems in Periodic and Random Composites

这项研究针对强非均质材料，主要是复合材料的数学建模中出现的问题。复合材料是几种不同的组成材料或相的混合物，它结合了每种材料或相的最有用的特征。例如，高导热系数的复合材料由陶瓷填充珠在聚合物基质中组成。其目标是最大限度地提高热导率。陶瓷是很好的导热体，而聚合物不是。由于纯陶瓷的脆性，使用纯陶瓷是不现实的，但聚合物/陶瓷复合材料是一种良好的导热材料，具有良好的力学性能。这里的问题是通过正确选择填料的特性、尺寸、形状和随机尺寸分布来优化强度和导热系数。这项工作的三个问题领域是：优化环氧树脂/陶瓷复合材料的介电性能和力学性能，并将其应用于电容器的设计；研究换能器材料中的动力学问题和频率相关效应，并将其应用于声信号的传感器和发射器；以及研究来自超导、超流和液晶的问题，目的是了解由于涡旋、非线性和非标准边界条件的存在而引起的数学问题。计划中的工作位于数学和材料科学之间的前沿，并将与来自大学和私营行业的材料科学家密切合作。数学可以帮助开发满足各种工业需求的具有优异性能的新材料，验证实验数据的可靠性，并设计出计算设计复合材料性能的有效方法。三个领域中的第一个领域(环氧/陶瓷复合材料)的工作结果将为各种重要工业应用中的制造和材料开发的未来方向提供指导。两个典型的例子是为集成电路设计新的‘封装’，它可以更有效地从电子设备中散热，并显著增强用于各种电子产品的电容器的功能。计划中的换能器材料工作立即应用于水下声纳和医学成像。第三个领域(超导)的工作将有助于澄清现有的哪种物理模型提供了计算具有大量涡旋的超导薄膜有效性质的最佳方法，并将导致对这些现象的更好的模型。

This research deals with problems arising in the mathematicalmodeling of strongly heterogeneous materials, primarily composites.A composite is a mixture of several different constituent materials or phases so that it combines the most useful features of each.For example, a high thermal conductivity composite consists of ceramic filler beads in a polymeric matrix. The goal is to maximize thermal conductivity. Ceramic is a very good thermal conductor while the polymer is not. It is not practical to use pure ceramic since it is too brittle.However, the polymer/ceramic composite is a good thermal conductor and it has desirable mechanical properties. The problem in this context is to optimize strength and thermal conductivity by the correct selection of the filler identity, size, shape, and the random size distribution. Tools fromthe mathematical theory of homogenization will be used to attack these questions.The three problem areas of the proposed work are the optimization of dielectric and mechanical properties of epoxy/ceramic composites, with applications to thedesign of capacitors; an investigation of dynamical problems and frequency dependent effects in transducer materials, with applications to sensors and transmitters of acoustic signals; and a study of problems from superconductivity, superfluidity, and liquid crystals, with the goal of understanding mathematical issues caused by the presence of vortices, nonlinearity, and nonstandard boundary conditions.The planned work lies at the frontier between mathematics and materials science, and close collaborations with materials scientists from universities and private industry will be carried out. Mathematics can help to develop new materials with superior properties for various industrials needs, verify the reliability of experimental data, and devuise efficient ways to compute properties of designer composites. The results of the work in the first of the three areas (epoxy/ceramic composites) will provide guidance for future directions in manufacturing and materials development in various important industrial applications. Two typical examples are the design of new 'packages' for integrated circuits which remove heat from the electronics more efficiently and significant enhancement in functionality of capacitors which are used in various electronics products. The planned work on transducer materials has immediate applications in underwater sonar and medical imaging. The work in the third area (superconductivity) will help to clarify which of the existing physical models provide the best way of calculating effective properties of superconducting thin films with a large number of vortices, and it will lead to better models for these phenomena.