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.

在这里，我们考虑了由各种方向的单晶构建的平面多晶的可能体积和剪切模量。先前的工作给出了具有正交各向异性对称性的晶体的完整表征。具体来说，边界是在有效体积和剪切模量上分别导出的，因此将有效模量限制在（体积，剪切）平面中的矩形内。已确定该矩形中的每个点都可以通过适当的分层层压微观几何形状来实现，晶体在层中具有不同的取向，层本身也处于不同的取向。边界很容易扩展到没有特殊对称性的晶体，但是构建微观几何结构以实现边界定义的矩形中的每个点的路径要困难得多。我们表明，具有最小体积模量的盒子的两个角总是通过分层层压板获得。对于另外两个角，我们提出了生成获得它们的分层层压板的算法。数值证据有力地表明，总能达到具有最大体积和最大剪切模量的拐角。对于剩余的角，具有最大体积模量和最小剪切模量，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。我们使用的微观结构是分层层压几何结构，其核心具有自相似的微观结构，从某种意义上说，一个长度尺度上的微观结构是较小长度尺度上的微观结构的旋转和重新缩放。