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

TL;DR

This work addresses the problem of characterizing the two-dimensional polycrystal effective elasticity tensor ${\bf C}_*( {\bf C}^{(0)} )$ as a function of the constituent crystal tensor by constructing a continued-fraction representation whose structure mirrors that of sequential laminates. Using a Hilbert-space homogenization framework (the Z-problem) and a careful truncation scheme, the authors derive a recursion that links successive effective tensors, yielding a continued-fraction approximation for ${\bf L}_*( {\bf L}^{(0)} )$ with analytic, homogeneous, and Herglotz properties. While this framework generalizes the sequential-laminate expansion to elasticity and accommodates non-symmetric local tensors, it remains an open question whether an exact sequential-laminate realization exists for arbitrary polycrystals; the paper also surveys alternative continued-fraction expansions and discusses conditions under which a sequential-laminate correspondence may hold. The resulting representation provides a practical, representation-theoretic route to bounds and design insights for the macroscopic elasticity of composites, with potential implications for three-dimensional columnar composites and frequency-dependent (viscoelastic) settings through analytic continuation in ${\bf L}^{(0)}$. Overall, the work advances a principled approach to approximate, and potentially bound, ${\bf C}_*({\bf C}^{(0)})$ via hierarchical laminates-inspired constructions, while clarifying the limits of exact replication by sequential laminates.

Abstract

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.

Figures (3)

• Figure 1: An example of a periodic two dimensional polycrystal showing the unit cell $Q$ of periodicity. The double arrowed blue lines denote the the direction of the non-trivial local eigenvector of the rank-one matrix valued function $\chi$ relative to which the crystal orientation is determined. We are free to rotate these eigenvectors by a common angle while keeping ${\bf L}({\bf x})$ unchanged: the coefficients $L^{(0)}_{ij}$ need to be adjusted accordingly.
• Figure 2: An example of a rank 3 two dimensional sequential laminate polycrystal. It is schematic in the sense that there should be a large separation of length scales at each stage of the lamination. Here the crystal orientation is controlled by the function $\chi$ which takes rank one values with the non-trivial eigenvector being denoted by the blue doubled headed arrows. The insert shows the details of the orientations in the first layering.
• Figure 3: (a) An example of a field ${\bf v}^\perp$ in a sequential laminate which is nonzero only in the last layers and takes a symmetric rank one value ${\bf n}_\perp\otimes{\bf n}_{\perp}/\sqrt{f}$ there. Here ${\bf n}$ is the direction of the last lamination, and ${\bf n}_\perp={\bf R}_{\perp}{\bf n}$. The double headed green arrows denote the non-trivial unit eigenvectors $\pm {\bf n}_\perp$ of ${\bf v}^\perp$ in those last layers, and we see that $\nabla \cdot{\bf v}^\perp=0$ implying $\hbox{\boldmath ${\Gamma}$}_1{\bf v}^\perp=0$. The blue double headed blue arrows denote the non-trivial unit eigenvectors $\pm {\bf n}$ of $\chi$ in the last layers, ensuring that $\hbox{\boldmath ${\chi}$}{\bf v}^\perp=0$. (b) More general crystal orientations, again with the blue double headed arrows denoting the non-trivial unit eigenvectors of $\chi$, are obtained by looking for an angle $\phi$ such that ${\bf v}^\perp= {\bf n}_\perp\otimes{\bf n}_{\perp}/\sqrt{f}$ satisfies $\hbox{\boldmath ${\chi}$}_\phi{\bf v}^\perp=0$. Thus the non-trivial unit eigenvectors of $\chi_\phi$ in the last layers are $\pm{\bf n}$, denoted by the magneta double headed arrows, and we have $\hbox{\boldmath ${\Gamma}$}_1{\bf v}^\perp=0$ and $\hbox{\boldmath ${\chi}$}_\phi{\bf v}_{\perp}=0$.