Explicit evaluation of Willis' bounds with ellipsoidal inclusions

Abstract The integral involved in Willis' bounds is now explicitly evaluated when the two-point correlation function assumes the ellipsoidal symmetry, and, in terms of Eshelby's S- tensor and the moduli L 0 of the comparison material, it is found to be simply given by P 0 = S 0 L 0 −1 . This simple outcome allows Willis' bounds to be calculated readily for the class of composites containing aligned ellipsoidal inclusions, and it also shows how his bounds lead to those of Hashin-Shtrikman, Hill and Hashin, and Walpole when the correlation function takes the spherical, cylindrical, and disc symmetries. The new result further helps place Mori-Tanaka's theory with aligned ellipsoidal inclusions on a firmer theoretical footing. In a multiphase composite if the matrix is the softest (or hardest) phase, the M-T moduli will coincide with Willis' lower (or upper) bounds; otherwise they will always lie inside the bounds. Numerical results for a glass/epoxy system indicate that the bounds and the corresponding moduli are indeed quite sensitive to the shape of inclusions.

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