Dynamic and Nonlinear Static Problems in Periodic and Random Composites

本研究涉及在强非均相材料，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组成材料或相的混合物，因此它结合了每种材料或相的最有用的特性。例如，高导热复合材料由聚合物基体中的陶瓷填充珠组成。目标是最大化导热性。陶瓷是很好的导热体，而聚合物不是。使用纯陶瓷是不实际的，因为它太脆了。然而，聚合物/陶瓷复合材料是一种良好的导热体，具有理想的机械性能。在这种情况下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性。同质化数学理论中的工具将用于解决这些问题。提出的工作的三个问题领域是环氧/陶瓷复合材料的介电性能和力学性能的优化，并应用于电容器的设计；动态问题和频率依赖效应在换能器材料中的研究，并应用于声信号的传感器和发射器；研究超导性、超流动性和液晶问题，目的是理解由涡流、非线性和非标准边界条件引起的数学问题。计划中的工作处于数学和材料科学的前沿，并将与来自大学和私营企业的材料科学家密切合作。数学可以帮助开发具有各种工业需求的优越性能的新材料，验证实验数据的可靠性，并设计出有效的方法来计算设计复合材料的性能。在这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产品中去除热量，并显着增强各种电子产品中使用的电容器的功能。在换能器材料方面的计划工作将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物理模型中哪一种提供了计算具有大量涡旋的超导薄膜有效特性的最佳方法，并将为这些现象提供更好的模型。

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.