Towards a Complete Theory of Exact Relations

通常，复合材料的物理性能强烈依赖于微观结构。然而，在特殊情况下，存在与微观结构无关的精确关系。它们表达了给定物理环境中的基本不变性。精确的关系已经被广泛研究，但经典的方法严重依赖于物理环境。20世纪90年代末，格拉博夫斯基提出了一种抽象的精确关系理论，并经格拉博夫斯基、弥尔顿和研究者大力推广。它已被证明是非常强大的。事实上，通过将对精确关系的研究简化为涉及群表示理论的纯粹代数问题，它导致了三维热电复合材料的所有旋转不变精确关系的完备性，其中包括弹性，热弹性和压电性的所有精确关系作为特殊情况。这种新方法在材料科学领域取得了巨大的飞跃。然而，这个理论绝不是完整的。这个项目的目的是通过解决剩下的三个主要悬而未决的问题来完成精确关系理论。首先，以前的工作只给出了相对简单的物理环境中精确关系的完整列表。研究者研究高耦合问题的最终目标是找到一个明确的参数化精确关系的物理问题与任何数量的温度，电，和弹性场。第二个项目是探索层压和均质化之间的确切关系。基本问题是是否存在在层压下稳定而在欠均匀化下不稳定的精确关系。第三，研究者对与精确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Jordanalgebras的代数对象来重新表述。这种方法似乎很有希望，并且被认为是在前面描述的两个问题上取得进展的关键。复合材料在现代世界无处不在。它们被用于制造各种产品，从滑雪板到飞机，从网球拍到手机。因此，了解复合材料的物理性质（如导电性和弹性）与其成分的性质之间的关系在技术上具有重要意义。这通常是困难的，因为组成部分组合在一起的方式对最终结果有很大的影响。以两种材料为例，一种是软的，一种是硬的。如果硬材料嵌入软物质，复合材料将是可压缩的。另一方面，如果软材料在硬材料的基体中，则复合材料将发生贝氏性。然而，在某些情况下，这种典型的可变性会大大降低。这个项目的目标是理解和分类这些情况。从实际的角度来看，该项目建立了工程中的“不可能定理”，即：结果表明，从一组给定的起始材料不能构造出具有期望性能的复合材料。因此，该提案对工业研究和发展具有重要意义。

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.