最小弹性应力能的三相平面复合材料：高孔隙率结构
Three-phase plane composites of minimal elastic stress energy: High-porosity structures
摘要：
本文建立了二维三材料复合材料在均匀各向异性应力作用下有效弹性能的精确下限。假设材料以给定的体积分数混合，并且其中一个相退化为空隙，即有效复合材料是多孔的。利用附加不等式增强的平移法，得到了能量界的显式表达式。充分的最优性条件的能量界被用来设置的要求，必须满足的应力场在每个阶段的最佳有效材料，无论其微观结构的几何形状的复杂性。我们表明，这些要求得到满足的一类特殊的微观几何形状，所谓的层压板的一个排名。它们的最优性详细阐述了具有大量空隙的结构，也称为高孔隙率结构。结果表明，在不同的体积分数范围和外应力的各向异性水平下，最优多阶高孔隙率层合板的几何参数是不同的。本文还利用充分条件技术讨论了其他作者提出的非层状三相微结构及其在高孔隙率区的最优性。关于低孔隙度区域的猜测，但这个问题的全面处理被推迟到一个单独的出版物。相应的“G-封闭问题”的三相各向同性复合材料也解决和精确的边界上的有效各向同性性能明确确定在这些地区的应力能约束是最佳的。
英文摘要：
The paper establishes exact lower bound on the effective elastic energy of two-dimensional, three-material composite subjected to the homogeneous, anisotropic stress. It is assumed that the materials are mixed with given volume fractions and that one of the phases is degenerated to void, i.e. the effective composite is porous. Explicit formula for the energy bound is obtained using the translation method enhanced with additional inequality expressing certain property of stresses. Sufficient optimality conditions of the energy bound are used to set the requirements which have to be met by the stress fields in each phase of optimal effective material regardless of the complexity of its microstructural geometry. We show that these requirements are fulfilled in a special class of microgeometries, so-called laminates of a rank. Their optimality is elaborated in detail for structures with significant amount of void, also referred to as high-porosity structures. It is shown that geometrical parameters of optimal multi-rank, high-porosity laminates are different in various ranges of volume fractions and anisotropy level of external stress. Non-laminate, three-phase microstructures introduced by other authors and their optimality in high-porosity regions is also discussed by means of the sufficient conditions technique. Conjectures regarding low-porosity regions are presented, but full treatment of this issue is postponed to a separate publication. The corresponding "G-closure problem" of a three-phase isotropic composite is also addressed and exact bounds on effective isotropic properties are explicitly determined in these regions where the stress energy bound is optimal.
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