Stability and macroscopic properties of heterogeneous media

GRABOVSKY DMS-1008092研究人员研究与复合材料、马氏体相变和材料的形态稳定性有关的问题。首先，他应用他和他的合作者发展的一般理论，考虑了纤维增强弹性复合材料有效张量的精确关系和联系。一般理论提供了一种寻找每一种关系和联系的策略。然而，这一战略的实际执行远非微不足道。纤维增强弹性复合材料不仅具有重要的应用价值，而且在技术上具有极大的挑战性，因为复合材料的微观结构是二维的，而复合材料的性能是由三维的完全各向异性弹性张量来表示的。这一主题建立在霍尔效应电导率背景下一个类似的、更简单的问题的成功解决上。第二个问题涉及能够进行马氏体相变的材料中相界的形态稳定性。研究人员研究了一种构型的可能性，在这种构型中，唯一的不稳定性模式是相边界的整体运动。从数学上讲，这对应于广义的第二种变化未能保持正值。困难之处在于，需要满足其他稳定性标准。这一问题的解决为完全理解具有光滑相边界的组态的所有不稳定性开辟了道路。在周期性复合体的背景下，形态不稳定性可能在周期细胞内局部发生。在结构和材料不稳定性的先前工作之后，研究了它们与复合材料的有效行为的相互作用。非均质介质，或具有内部结构的介质，在应用中具有重要意义。复合材料，现在已经变得无处不在，就是一个例子。另一个例子是形状记忆合金和其他智能材料。这两组材料非常不同，但有一些重要的共同特征。这两个理论之间的一个众所周知的联系可以用来解释人们不太理解的马氏体相变理论，该理论是导致形状记忆行为的主要原因。本项目的目的之一是了解纤维增强弹性复合材料的弹性性能。这种复合材料可能会将两种廉价但不完美的材料结合起来，产生一种稀有或根本不存在于自然界中的新材料。例如，现代滑雪板使用复合材料来创造一种材料，这种材料很容易弯曲，但在扭转方面却非常坚硬。然而，复合材料所能实现的功能是有局限性的。研究人员列出了由于“精确关系”的存在而大大减少了制造复合材料的好处的情况，即材料弹性系数之间的关系，无论人们如何努力，都不能通过制造复合材料来改变这些关系。本报告的重点是一类应用非常广泛的纤维增强复合材料。该项目的另一个目标是了解非线性弹性和形状记忆合金模型中的不稳定性。一种广为人知的弹性失稳类型是屈曲。引起形状记忆行为的马氏体相变是另一种不稳定性。这些例子说明了不稳定性在应用中有多么重要。研究人员以系统的方式对所有可能的不稳定性进行分类，次要目标是解决一个长期存在的数学问题：找到稳定的“正确”充分条件。

GrabovskyDMS-1008092 The investigator studies problems related to compositematerials, martensitic phase transformations, and morphologicalstability in materials. First, he considers exact relations andlinks for effective tensors of fiber-reinforced elasticcomposites, by applying the general theory developed by him andhis collaborators. The general theory provides a strategy forfinding every relation and link. However, the actual executionof that strategy is far from trivial. The case offiber-reinforced elastic composites is both important forapplications and incredibly challenging technically, because themicrostructure is two-dimensional while the properties of thecomposite are represented by three-dimensional fully anisotropicelastic tensors. This topic builds on the successful solution ofa similar, simpler problem in the context of Hall-effectconductivity. A second problem concerns morphological stabilityof phase boundaries in materials capable of undergoingmartensitic phase transformations. The investigator studies thepossibility of a configuration in which the only mode ofinstability is the global motion of the phase boundary. Mathematically this corresponds to the failure of the generalizedsecond variation to stay positive. The difficulty is that allother stability criteria are required to be satisfied. Theresolution of this issue opens the way to a completeunderstanding of all instabilities of configurations with smoothphase boundaries. In the context of periodic composites,morphological instabilities may occur locally within a periodcell. Their interaction with the effective behavior of acomposite is investigated, following prior work for structuraland material instabilities. Heterogeneous media, or media with internal structure, areof great importance in applications. Composite materials, whichhave by now become ubiquitous, are one example. Another exampleis shape memory alloys and other smart materials. These twogroups of materials are very different, yet have some importantcommon features. A well-known connection between the twotheories can be used to shed light on the less well-understoodtheory of martensitic phase transformations responsible for muchof the shape memory behavior. One aim of this project is tounderstand elastic properties of fiber-reinforced elasticcomposites. Such composites may combine two cheap but imperfectmaterials to produce a new material that is either rare or notfound in nature at all. For example, modern skis use compositesto create a material that bends easily but is incredibly stiffwith respect to torsion. However, there are limitations to whatcan be achieved by composites. The investigator maps out thoseinstances where the benefits of creating composites aredramatically reduced due to the presence of "exact relations,"i.e. relations between material moduli that cannot be altered bymaking composites, no matter how hard one tries. The presentwork focuses on a very widely used class of fiber-reinforcedcomposites. Another aim of the project is to understandinstabilities in models of non-linear elasticity and shape memoryalloys. One widely known type of elastic instability isbuckling. Martensitic phase transitions responsible for shapememory behavior are another kind of instability. These examplesshow how important instabilities are in applications. Theinvestigator classifies all possible instabilities in asystematic way, with the secondary goal of solving one of thelong-standing mathematical problems: finding "correct" sufficientconditions for stability.