The theoretical connection between Mori-Tanaka's theory and the Hashin-Shtrikman-Walpole bounds

Mori-Tanaka's theory with the general anisotropic constituents has been recast into a new form and it is shown that this form bears an identical structure to that developed by Walpole for the bounds. The equivalent polarization stress and strain in the former theory are exactly those chosen by Hashin-Shtrikman and Walpole and the average stress and strain of the matrix phase are equal to the image stress and strain imposed on the approximate fields by Walpole to meet the required boundary conditions. The consequence is that the effective moduli of the composite containing either aligned or randomly-oriented, identically shaped ellipsoidal inclusions always have the same expressions as those of the H-S-W bounds, only with the latter's comparison material identified as the matrix phase and Eshelby's tensor interpreted according to the appropriate inclusion shape. This connection allows one to draw a line of important conclusions regarding the predictions of the M-T theory, and it also points to the conditions where this theory can always be applied safely without ever violating the bounds and where such an application might be less reliable.