Towards a Complete Theory of Exact Relations

通常情况下，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确切关系。它们表示给定物理环境中的基本不变性。精确关系已经得到了广泛的研究，但经典的方法严重依赖于物理环境。20世纪90年代末，S提出了一种抽象的精确关系理论，并由格拉博夫斯基、弥尔顿和研究者作了极大的扩展。事实证明，它具有巨大的威力。事实上，通过将精确关系的研究简化为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的精确关系的完备化，其中包括作为特例的弹性、热弹性和压电性的所有精确关系。这一新方法推动了材料科学这一领域的一大飞跃。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，以前的工作只在相对简单的物理环境中给出了确切关系的完整清单。研究人员研究高度耦合的问题，最终目标是找到具有任意数量的温度、电场和弹性场的物理问题的精确关系的显式参数化。第二个项目是探索层合与齐次化之间的关系，以求精确的关系。最基本的问题是，是否存在在层合条件下是稳定的，而不是欠均匀的精确关系。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。具体地说，关于精确关系的问题可以用称为乔丹止痛的代数对象来表示。这一方法似乎很有希望，被认为是在上述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)如何与其组分的性质相关具有重要的技术意义。这通常是困难的，因为组成成分的组合方式对最终结果有很大影响。例如，拿两种材料来说，一种是软的，一种是硬的。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，如果软材料在硬材料的基质中聚集，复合材料将呈棕黄色。然而，在某些情况下，这种典型的可变性会大大降低。本项目的目标是了解和分类这些情况。从实践的角度来看，该项目建立了工程上的“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构造出来。因此，该提案对工业研究和发展具有重大影响。

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.