Effective transport properties of arrays of multicoated or graded spheres with spherically transversely isotropic constituents

This work is concerned with the determination of the effective conductivity and potential fields of a periodic array of spherically transversely isotropic spheres in an isotropic matrix. We generalize Rayleigh’s method to account for the periodic arrangements of the inclusions. The inclusions considered in the formulation could be multicoated, generally graded, or exponentially graded. For the multicoated spheres, we derive a recurrence procedure valid for any number of coatings. We show that a $(2×2)$ array alone can mathematically represent the effect of the multiple coatings. For a graded inclusion, the method of Frobenius is adopted to obtain series solutions for the potential fields. For an exponentially graded sphere, we show that the admissible potential field in the inclusion admits a closed-form expression in terms of confluent hypergeometric functions. All these types of inclusions can be characterized by simple scalar coefficients $Tl$ in the estimate of effective conductivities. Simple orthorhombic, body-centered orthorhombic, and face-centered orthorhombic lattice structures are considered in the formulation. Numerical results are presented for selected systems with sufficient accuracy. We demonstrate that the anisotropy of the spheres can strongly influence the potential fields inside the inclusions. The effects of spherical anisotropy, multiple coatings, and the grading factor are also studied.