Linear comparison estimates for the effective resistivity of three-dimensional nonlinear polycrystals

Abstract

Estimates for the effective resistivity of nonlinear polycrystals are obtained using the ‘linear comparison’ homogenization scheme of DeBotton and Ponte Castañeda (DeBotton & Ponte Castañeda 1995 Proc. R. Soc. A 448 , 121–142). Computing the effective properties of linear composites, with the same microstructure as the nonlinear composite, is an essential part of this scheme. The classical self-consistent method is employed for this purpose. An important characteristic of these estimates, for polycrystals with field thresholds, is that they satisfy the recent bound of Garroni and Kohn (Garroni & Kohn 2003 Proc. R. Soc. A 459 , 2613–2625), which dramatically improves upon the classical Taylor upper bound at large crystal anisotropy. In addition, the estimates also satisfy the Hashin–Shtrikman bounds, which are more restrictive than the Garroni–Kohn bound at small crystal anisotropy. Interestingly, the scaling exponents for the linear comparison estimates are found to be independent of the constitutive nonlinearity. This last observation provides an explanation for the relative weakness of an earlier linear comparison bound obtained by Garroni and Kohn.