Voigt-Reuss and Hashin-Shtrikman bounds revisited

Abstract

The Voight-Reuss-Hill (VRH) bounds have simple structure (“rules of mixture”); hence they can be extended, in a straightforward way, to cases of anisotropy. However, they are overly wide in cases of large property contrast between constituents. The Hashin-Shtrikman (HS) bounds are substantially narrower, but they have more complex structure and do not easily extend to cases of anisotropy (at least in the explicit way), without additional information on microstructure (that may not be available). We show that, in broad range of property contrast, the mid-line between the VRH bounds either lies inside the HS bounds or is close to one of them – and thus has the HS accuracy. The mid-line estimates readily extend to cases of anisotropy. They also yield simple and explicit cross-property connection that, however, has very approximate character.