Planar polycrystals with extremal bulk and shear moduli

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.