Improved bounds on the conductivity of composites by interpolation

Different bounds on the conductivity of a composite material may improve on each other in different conductivity regimes. If so, the question arises of how to efficiently interpolate between the bounds. In this paper I show how to do an interpolation with a two-point Pade approximation method. For bounds on two-component composites the interpolation method is shown to be, in a sense, the best possible. The method is discussed in the context of equiaxed polycrystals where the classic Hashin-Shtrikman bounds and the more recent null-lagrangian bounds, partly improve on each other. Denoting the principal conductivities of the crystallite $\sigma _{1}\leq \sigma _{2}\leq \sigma _{3}$, the method gives improved lower bounds for equiaxed polycrystals which have $\sigma _{2}$ (0.77$\sigma _{1}$ + 0.23$\sigma _{3}$) $\geq \sigma _{1}\sigma _{3}$. The method also gives improved upper bounds.