Modeling of Multiscale Inhomogeneous Materials with Periodic and Random Microstructure

提案#0204637PI：Leonid Berlyand机构：宾夕法尼亚州立大学标题：具有周期性和随机微结构的多尺度非均匀材料的建模摘要该提案的科学核心围绕四个方面。同质化是所有四个领域的共同主题，预计拟议的工作将导致开发新的同质化技术和工具。第一个领域涉及高填充高对比度复合材料的离散网络近似。主要目的是在严格的数学框架下描述复合材料的有效输运性质与填充粒子的形状和随机位置的依赖关系，并给出可控的误差估计。我们在第二个领域的目标是获得复合材料有效性能的解析公式。这些公式揭示了有效性质与几何和物理参数的显式依赖关系，并提供了有价值的物理见解，可用于测试针对一般情况开发的数值算法。第一个领域和第二个领域将从不同的角度讨论多分散性这一重要的实际问题，并对结果进行比较。第三个领域涉及复杂流体的流变性，如聚合物复合材料、悬浮液和胶束流体。这里的主要特点是：(I)胶束管或胶束球之间的相互作用，导致有效本构方程与相的本构关系相比发生了巨大的变化；(Ii)流动的层状化和粘弹性流动的减阻(压力降低)。第四个领域是对超导体和液晶建模中出现的一些具有非标准边界条件的非线性问题的齐次化新特征的探索。主要目的是刻画均匀极限对区域尺寸的依赖关系，并探索这种尺寸效应在物理问题中的分支。复合材料具有关键的技术重要性。这些材料的建模和设计提出了物理、材料科学和数学的基本问题。其中许多问题还没有得到回答，而数学有很大的贡献。这个项目将通过主要研究人员、他的合作者和顾问与材料科学家的实验研究相协调的理论努力来促进我们对复合材料的理解。其长期目标是提高数学对当代技术问题的贡献。这项工作的重点将是促进跨相邻学科以及学术界、实验室和行业之间的跨学科联系。这项研究的成果将被用于开发各种工业需求的具有优异性能的新材料。主要应用包括为电子工业设计热保护组件，这将满足现代电子设备(例如手机)进一步小型化的需要；使用带有聚合物和胶束添加剂的流体来冷却各种设备(例如反应堆)和更高效地输送石油；以及优化多分散悬浮液的运输性能。

Proposal #0204637PI: Leonid BerlyandInstitution: Penn State UniversityTitle: Modeling of Multiscale Inhomogeneous Materials with Periodic and Random MicrostructureABSRACTThe scientific core of this proposal is centered around four areas. Homogenization is the common theme of all four areas, and it is expected that the proposed work will result in the development of new homogenization techniques and tools. The first area concerns the discrete network approximation for highly packed high-contrast composites. The main objective is to characterize the dependence of the effective transport properties of composites on the shapes and random locations of the filling particles in a rigorous mathematical framework with a controlled error estimate. Our objective in the second area is to obtain analytical formulas for the effective properties of composites. Such formulas reveal the explicit dependence of the effective properties on geometrical and physical parameters and provide a valuable physical insight, which can be used to test numerical algorithms developed for generic situations. The important practical issue of polydispersity will be addressed from different perspectives in the first as well as the second area, and the results will be compared. The third area concerns the rheology of complex fluids such as polymeric composites, suspensions and micellar fluids. The main features here are: (i) the interaction between micellar tubes or balls, which leads to a drastic change in the effective constitutive equations as compared with the constitutive law of the phases, (ii) the laminarization of the flow and drag reduction in viscoelastic flows (reduction in pressure). The fourth area is the exploration of novel features of homogenization for some nonlinear problems with nonstandard boundary conditions arising in modeling of superconductors and liquid crystals. The main objective is to characterize the dependence of the homogenization limit on the domain size and to explore the ramification of this size effect in physical problems.Composite materials are of critical technological importance. The modeling and design of these materials raises fundamental questions of physics, materials science, and mathematics. Many of these questions are not yet answered, and mathematics has much to contribute. This project will advance our understanding of composite materials through a theoretical effort of the principal investigator, his collaborators and advisees coordinated with experimental studies by materials scientists. The long-term goal is to enhance the contribution from mathematics to very contemporary technological problems. This will be done with an emphasis on fostering interdisciplinary connections across neighboring disciplines, as well as between academia, laboratories and industries. The results of this research will be used in developing new materials with superior properties for various industrial needs. The main applications include the design of thermal protection packages for electronic industries, which will address the need for further miniaturization of modern electronic devices (e.g., cell phones); the use of fluids with polymer and micellar additives for cooling of various devices (e.g., reactors) and more efficient transport of oil; and the optimization of transport properties of polydispersed suspensions.