Towards a Complete Theory of Exact Relations

SageDMS-0606300 通常，复合材料的物理性能强烈依赖于微观结构。 然而，在特殊的情况下，确切的关系存在的微观结构无关。 这些表示在给定的物理环境中的基本不变性。 精确的关系已经被广泛研究，但经典的方法一直严重依赖于物理环境。 在20世纪90年代后期，精确关系的抽象理论由格拉博夫斯基首创，并由格拉博夫斯基、米尔顿和研究者进行了极大的扩展。它被证明是非常强大的。 事实上，通过将精确关系的推导简化为一个涉及群表示理论的纯代数问题，它已经导致了三维热压电复合材料的所有旋转不变精确关系的完整列表，其中包括弹性，热弹性和压电的所有精确关系作为特殊情况。 这一新方法是材料科学这一领域的一个巨大飞跃。 然而，这一理论并不完整。 这个项目的目的是通过解决剩下的三个主要的开放问题来完成精确关系理论。首先，以前的工作只在相对简单的物理环境中给出了精确关系的完整列表。 该程序研究高度耦合的问题，最终目标是找到一个明确的参数化的精确关系的物理问题与任何数量的温度，电力和elasticfields。 第二个项目是探索分层与均匀化之间的关系，以获得精确的关系。 基本的问题是是否存在精确的关系，是稳定的分层下，但不均匀。 第三，研究者对与正合关系相关的代数结构进行了详细的研究。 特别是，关于精确关系的问题可以用称为Jordan代数的代数对象来表述。 这种方法似乎很有前途，它被认为是对前面描述的两个问题的进展至关重要。 复合材料在现代世界中无处不在。它们被用于制造从滑雪板到飞机，从网球拍到手机的各种产品。 因此，了解复合材料的物理性质（如导电性和弹性）与其组分性质的关系具有重要的技术意义。这在一般情况下是困难的，因为组分放在一起的方式强烈影响最终结果。例如，取两种材料，一软一硬。 如果硬材料嵌入软物质中，则复合材料将是可压缩的。 另一方面，如果软质材料位于硬质材料的基体中，则复合材料将变硬。 然而，在某些情况下，这种典型的可变性大大降低。 这个项目的目标是理解和分类这些情况。 从实践的角度来看，该项目建立了工程中的“不可能性定理”，即结果表明，具有所需性能的复合材料不能从给定的一组起始材料中构建。因此，该建议对工业研究和发展具有重大意义。

SageDMS-0606300 Typically, physical properties of composite materials arestrongly dependent on microstructure. However, in exceptionalsituations, exact relations exist that aremicrostructure-independent. These express fundamentalinvariances in a given physical setting. Exact relations havebeen extensively studied, but the classical approach has beenheavily dependent on the physical context. In the late 1990's,an abstract theory of exact relations was originated by Grabovskyand greatly extended by Grabovsky, Milton, and the investigator. It has proved to be enormously powerful. Indeed, by reducing thesearch for exact relations to a purely algebraic probleminvolving group representation theory, it has led to completelists of all rotationally invariant exact relations forthree-dimensional thermopiezoelectric composites, which includeall exact relations for elasticity, thermoelasticity, andpiezoelectricity as special cases. This new approach has beenresponsible for a great leap forward in this area of materialscience. However, the theory is by no means complete. Thepurpose of this project is to complete the theory of exactrelations by addressing the three remaining major open questions. First, previous work has given complete lists of exact relationsonly in relatively simple physical contexts. The investigatorstudies highly coupled problems with the ultimate goal of findingan explicit parameterization of exact relations for the physicalproblem with any number of temperature, electric, and elasticfields. The second project is the exploration of therelationship between lamination and homogenization for exactrelations. The basic question is whether there exist exactrelations that are stable under lamination, but not underhomogenization. Third, the investigator undertakes a detailedstudy of the algebraic structures associated with exactrelations. In particular, questions about exact relations may bereformulated in terms of algebraic objects called Jordanalgebras. This approach seems very promising, and it isconsidered to be crucial for progress on the two problemsdescribed previously. Composite materials are everywhere in the modern world. They are used in the manufacture of products ranging from skis toairplanes and from tennis rackets to cell phones. It isaccordingly of great technological importance to understand howphysical properties (such as conductivity and elasticity) of acomposite are related to the properties of its constituents. This is difficult in general because the way in which theconstituents are put together strongly influences the end result. For example, take two materials, one soft and one hard. If thehard material is embedded in the soft substance, the compositewill be compressible. On the other hand, if the soft materiallies in a matrix of the hard material, the composite will berigid. However, in certain situations, this typical variabilityis greatly reduced. The goal of this project is to understandand classify these situations. From a practical point of view,the project establishes "impossibility theorems" in engineering,i.e. results showing that a composite with desired propertiescannot be constructed from a given set of starting materials. The proposal thus has significant implications for industrialresearch and development.