  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.
