Rigorous link between the conductivity and elastic moduli of fibre-reinforced composite materials

We derive rigorous cross-property relations linking the effective transverse electrical conductivity $\sigma _{\ast}$ and the effective transverse elastic moduli of any transversely isotropic, two-phase `fibre-reinforced' composite whose phase boundaries are cylindrical surfaces with generators parallel to one axis. Specifically, upper and lower bounds are derived on the effective transverse bulk modulus $\kappa _{\ast}$ in terms of $\sigma _{\ast}$ and on the effective transverse shear modulus $\mu $ $_{\ast}$ in terms of $\sigma _{\ast}$. These bounds enclose certain regions in the $\sigma _{\ast}$-$\kappa _{\ast}$ and $\sigma _{\ast}$-$\mu _{\ast}$ planes, portions of which are attainable by certain microgeometries and thus optimal. Our bounds connecting the effective conductivity $\sigma _{\ast}$ to the effective bulk modulus $\kappa _{\ast}$ apply as well to anisotropic composites with square symmetry. The implications and utility of the bounds are explored for some general situations, as well as for specific microgeometries, including regular and random arrays of circular cylinders, hierarchical geometries corresponding to effective-medium theories, and checkerboard models. It is shown that knowledge of the effective conductivity can yield sharp estimates of the effective elastic moduli (and vice versa), even for infinite phase contrast.