Elastic moduli of heterogeneous solids with ellipsoidal inclusions and elliptic cracks

Abstract

The influence of ellipsoidal inclusions and elliptic cracks on the overall effective moduli of a two-phase composite and of a cracked body, respectively, is investigated by means of Mori-Tanaka's theory for three types of inclusion and four types of crack arrangements: monotonically aligned, 2-D randomly oriented (two kinds for cracks), and 3-D randomly oriented. The effective moduli of the composite in the aligned case are known to coincide with Willis' orthotropic lower (or upper) bounds with a two-point ellipsoidal correlation function if the matrix is the softer (or harder) phase. With 2-D randomly oriented inclusions, the effective moduli are examined under Willis' transversely isotropic bounds with a two-point spheroidal correlation function, and it is found that, as the cross-sectional aspect ratio of the ellipsoidal inclusions flattens from circular shape to disc-shape, the two effective shear moduli and the plane-strain bulk modulus all lie on or within the bounds. The effective bulk and shear moduli of an isotropic composite containing randomly oriented ellipsoidal inclusions also fall on or within Hashin-Shtrikman's bounds as the shape of the ellipsoids changes. The obtained moduli are then extended to a cracked body containing elliptic cracks, which are generated by compressing the thickness of ellipsoidal voids to zero. It is found that only selected components of the effective moduli are dependent upon the crack density parameter η. Their dependence on η and the crack shape γ are explicitly established.