On the Relationship Between the H-Tensor and the Concentration Tensor and Their Bounds

For composite materials, two quantities that are useful for characterizing the contribution of inhomogeneities in a matrix material to the overall properties are (1) the individual H-tensor, H _i , which describes the contribution of a single inhomogeneity and (2) the overall strain concentration tensor, which describes the relationship between the overall volumetric strain to the average strain of all of the inhomogeneities. In this paper, we develop a relationship expressing the overall H-tensor, ${\mathcal{H}}$ , in terms of the overall strain concentration tensor. An important feature of the derivation is that it allows for rigorous upper and lower bounds on the overall H-tensor. In the special case that the inhomogeneities are all the same, with the same orientation, then ${\mathcal{H} = {\bf H}_i}$ , and the results derived for ${\mathcal{H}}$ also hold for H _i .