Optimal structures of multimaterial composites

Cherkaev 0707974 研究人员确定了最佳的微观结构和有效的导电性或刚度的多材料复合材料的新边界。 在数学上，这个问题等价于一个多变量变分问题的松弛与逐点二次多阱拉格朗日。 本文研究了新的充分条件（界）和特殊的极小化序列（微结构）。 一种新的技术称为localizedpolyconvexification的探讨，细化knownpolyconvexification程序推导和占newpointwise约束极小。 本文提出了一种确定复杂极小化层基序列的互补技术。 给出了几个最佳导电和弹性多材料结构的例子，可用于多材料结构的优化设计。 研究人员研究如何将几种不同导电性或刚度的材料放置在一个周期性的复合结构中，以最大限度地提高其整体导电性或刚度。 从某种意义上说，这个问题仍然是一个拼图游戏，要求从不同的物理属性中构建一个最佳的马赛克。 马赛克可以是二维或三维的。 关于这个问题的工作开始于1963年由著名的文件Hashinand Shtrikman。 解决了两种混合材料的问题，所提出的新方法允许将结果扩展到两种以上的材料。 所获得的最佳微观结构是多样的和不可预测的。 由给定的组分得到的最佳复合材料可用于人造材料的制造和结构设计。 基础数学一旦发展起来，也应该有助于更好地理解和控制智能材料和固体相变以及异质材料中自组织的类似现象。

Cherkaev0707974 The investigator determines optimal microstructures and newbounds for effective conductivity or stiffness of multimaterialcomposites. Mathematically, the problem is equivalent to therelaxation of a multivariable variational problem with apiece-wise quadratic multiwell Lagrangian. The investigatorstudies new sufficient conditions (bounds) and special minimizingsequences (microstructures). A new technique called localizedpolyconvexification is explored that refines the knownpolyconvexification procedure by deriving and accounting for newpointwise constraints on minimizers. A complementary techniqueis considered for determining complicated minimizinglaminate-based sequences. Several examples of optimal conductingand elastic multimaterial structures are worked out; there couldbe used for optimal design of multimaterial structures. The investigator studies how to place several materials ofdifferent conductivity or stiffness in a periodic compositestructure in order to maximize its overall conductivity orstiffness. In a sense, the problem remains a jigsaw puzzle thatasks to build an optimal mosaic from pieces of different physicalproperties. The mosaic can be two- or three-dimensional. Workon this problem was started in 1963 by the famous paper by Hashinand Shtrikman. The problem is solved for two mixing materials,and the suggested novel approach allows to extend the results tomore than two materials. The obtained optimal microstructuresare diverse and unpredictable. The best possible composites fromgiven components can be used in manufacturing of artificialmaterials and structural design. The underlying mathematics,when developed, should also help to better understand and controlsmart materials and phase transitions in solids and similarphenomena of self-organization in heterogeneous materials.