具有极值体模量和剪切模量的平面多晶体
Planar polycrystals with extremal bulk and shear moduli
摘要：
在这里，我们考虑可能的体积和剪切模量的平面多晶体从一个单一的晶体在不同的取向。以前的工作给出了正交各向异性对称晶体的完整表征。具体而言，边界分别推导出的有效体积和剪切模量，从而限制有效模量位于（体积，剪切）平面的矩形内。已经确定，这个矩形中的每个点都可以通过适当的分层层压微观几何结构来实现，晶体在层中具有不同的取向，并且层本身处于不同的取向。边界很容易扩展到没有特殊对称性的晶体，但是构建实现边界所定义的矩形中每个点的微观几何形状的路径要困难得多。我们表明，具有最小体积模量的盒子的两个角落总是通过分层层压板获得。对于其他两个角落，我们提出的算法生成层次层压板，达到他们。数值证据强烈表明，角落具有最大的体积和最大的剪切模量总是达到。对于具有最大体积模量和最小剪切模量的剩余拐角，尚不清楚算法是否总是成功，因此是否总是获得矩形中的所有点。我们使用的微结构是分层层压几何结构，其核心具有自相似的微结构，在某种意义上，一个长度尺度上的微结构是较小长度尺度上的微结构的旋转和重新缩放。
英文摘要：
Here we consider the possible bulk and shear moduli of planar polycrystals built from a single crystal in various orientations. Previous work gave a complete characterization for crystals with orthotropic symmetry. Specifically, bounds were derived separately on the effective bulk and shear moduli, thus confining the effective moduli to lie within a rectangle in the (bulk, shear) plane. It was established that every point in this rectangle could be realized by an appropriate hierarchical laminate microgeometry, with the crystal taking different orientations in the layers, and the layers themselves being in different orientations. The bounds are easily extended to crystals with no special symmetry, but the path to constructing microgeometries that achieve every point in the rectangle defined by the bounds is considerably more difficult. We show that the two corners of the box having minimum bulk modulus are always attained by hierarchical laminates. For the other two corners we present algorithms for generating hierarchical laminates that attain them. Numerical evidence strongly suggests that the corner having maximum bulk and maximum shear modulus is always attained. For the remaining corner, with maximum bulk modulus and minimum shear modulus, it is not yet clear whether the algorithm always succeeds, and hence whether all points in the rectangle are always attained. The microstructures we use are hierarchical laminate geometries that at their core have a self-similar microstructure, in the sense that the microstructure on one length scale is a rotation and rescaling of that on a smaller length scale.
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