Modeling of Multiscale Inhomogeneous Materials with Periodic and Random Microstructure

提案#0204637 PI：Leonid Berlyand机构：宾夕法尼亚州立大学标题：具有周期性和随机微观结构的多尺度非均匀材料的建模ABSRACT该提案的科学核心围绕四个领域。 均匀化是所有四个领域的共同主题，预计拟议的工作将导致新的均匀化技术和工具的发展。 第一个领域涉及高度堆积的高对比度复合材料的离散网络近似。 其主要目的是描述复合材料的有效输运性质对填充颗粒的形状和随机位置的依赖性，并在严格的数学框架内进行误差估计。 我们的目标在第二个领域是获得复合材料的有效性能的解析公式。 这些公式揭示了几何和物理参数的有效属性的显式依赖，并提供了一个有价值的物理见解，它可以用来测试数值算法开发的通用情况。 多分散性的重要实际问题将从不同的角度在第一和第二领域，并将结果进行比较。第三个领域涉及复杂流体的流变学，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）胶束管或球之间的相互作用，这导致与相的本构律相比有效本构方程的急剧变化，（ii）流动的层化和粘弹性流动中的减阻（压力降低）。 第四个领域是探索新的功能的均匀化的一些非线性问题的非标准边界条件所产生的超导体和液晶的建模。 其主要目的是描述均匀化极限对畴尺寸的依赖性，并探索这种尺寸效应在物理问题中的分支。复合材料具有重要的技术意义。 这些材料的建模和设计提出了物理学、材料科学和数学的基本问题。 这些问题中有许多还没有答案，数学可以做出很大的贡献。 该项目将通过主要研究者、他的合作者和被研究者的理论努力，与材料科学家的实验研究相协调，推进我们对复合材料的理解。 长期目标是提高数学对当代技术问题的贡献。 这将重点放在促进相邻学科之间以及学术界，实验室和行业之间的跨学科联系上。 这项研究的结果将用于开发具有上级性能的新材料，以满足各种工业需求。 主要应用包括电子工业的热保护封装设计，这将解决现代电子设备进一步小型化的需求（例如，蜂窝电话）;使用具有聚合物和胶束添加剂的流体来冷却各种装置（例如，反应器）和更有效的石油运输;和多分散悬浮液的运输性能的优化。

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.