New extremal inclusions and their applications to two-phase composites
New extremal inclusions and their applications to two-phase composites
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发表时间：
2021-07
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通讯作者：
Liping Liu;R. James;P. Leo
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作者：
Liping Liu;R. James;P. Leo
在本文中，我们找到了一类特殊的包含，它对于二阶线性偏微分方程具有与椭球相同的性质。也就是说，在最简单的情况下，在物理术语中，夹杂物的恒定磁化意味着夹杂物上的恒定磁场。这些特殊的包含被发现是一个简单的变分不等式的解。这个变分不等式允许我们规定的连通性和周期性的包含。例如，我们发现周期性阵列的夹杂物在两个和三个维度的恒定磁化的夹杂物意味着恒定的磁场的夹杂物。夹杂物的体积分数可以是零和一之间的任何数字。我们发现这样的包含与任何有限数量的组件和组件是多重连接。这些特殊的包裹体在均匀化和能量最小化方面具有许多有用的性质。例如，我们使用它们给出了新的结果a）两相复合材料的有效性能和B）两相复合材料的最佳界限和最佳微观结构。
In this paper, we find a class of special inclusions that have the same property with respect to second order linear partial differential equations as holds for ellipsoids. That is, in the simplest case and in physical terms, constant magnetization of the inclusion implies constant magnetic field on the inclusion. The special inclusions are found as solutions of a simple variational inequality. This variational inequality allows us to prescribe the connectivity and periodicity properties of the inclusions. For example we find periodic arrays of inclusions in two and three dimensions for which constant magnetization of the inclusions implies constant magnetic field on the inclusions. The volume fraction of the inclusions can be any number between zero and one. We find such inclusions with any finite number of components and components that are multiply connected. These special inclusions enjoy many useful properties with respect to homogenization and energy minimization. For example, we use them to give new results on a) the effective properties of two-phase composites and b) optimal bounds and optimal microstructures for two-phase composites.