On Eshelby’s tensor in transversely isotropic materials

Eshelby’s tensor of a general ellipsoidal inclusion in transversely isotropic materials is obtained by integrating the associated Green’s function over the surface of the ellipsoid. Unlike those derived via Fourier integral transform, the present mathematical expressions of the tensor involve some definite integrals that are related to those appearing in other branches of physics. Theoretical aspects concerning the real-valued integrals known in the past are generalized by introducing complex numbers to fit for the present context. Some divergent integrals encountered in the present study are discussed in detail. It is concluded that only the finite parts or the principal values of the integrals are needed for the evaluation of the Eshelby tensor.