A continued fraction approximation for the effective elasticity tensor of two-dimensional polycrystals as a function of the crystal elasticity tensor

For two-dimensional polycrystals the effective elasticity tensor $C_*$ as a function $C_*(C_0)$ of the elasticity tensor $C_0$ of the constituent crystal is considered. It is shown that this function can be approximated by one with a continued fraction expansion resembling that associated with a class of microstructure known as sequential laminates. These are hierarchical microstructures defined inductively. Rank 0 sequential laminates are simply rotations of the pure crystal. Rank $j$ sequential laminates are obtained by laminating together, on a length scale much larger that the existing microstructure and with interfaces perpendicular to some direction $n_j$, rank $j-1$ sequential laminates with a rotation of the pure crystal. The continued fraction approximation for arbitrary polycrystal microstructures typically takes a more general form than that of sequential laminates, but has some free parameters. It is an open question as to whether these free parameters can always be adjusted so the continued fraction approximation matches exactly that of a sequential laminate. If so, one would have established that the elastic response of two-dimensional polycrystals can always be mimicked by that of sequential laminates. Our analysis carries over to the more general case where the strain is replaced by a field $E(x)$ that is the gradient of a vector potential $u(x)$, i.e. $E=\nabla u$ and the stress is replaced by a matrix valued field $J(x)$ that need not be symmetric but has zero divergence $\nabla\cdot J=0$. The tensor $L(x)$ entering the constitutive relation $J=L E$ is locally a rotation of the tensor $L_0$ of the pure crystal that need not have any special symmetries and has 16 independent tensor elements.