TY - JOUR

T1 - Effective transport properties of arrays of multicoated or graded spheres with spherically transversely isotropic constituents

AU - Kuo, Hsin Yi

AU - Chen, Tungyang

N1 - Funding Information:
This work was supported by the National Science Council, Taiwan, under Contract No. NSC 94-2211-E006-056. One of the authors (T.C.) would like to thank G. J. Weng for many valuable comments on the manuscript. Table I. Numerical constants a 1 and a 2 for Eq. (63) . sc bcc fcc a 1 − 1.304 71 − 0.129 30 − 0.075 29 a 2 0.405 41 0.764 20 − 0.741 54 Table II. Numerical constants for d 1 , ⋯ , d 15 in Eq. (65) . sc bcc fcc d 1 − 1.3046 − 0.1293 − 0.0753 d 2 − 0.0723 − 0.2570 − 0.2420 d 3 0.5289 0.0988 − 0.0558 d 4 0.3069 − 0.0534 0.0524 d 5 − 0.0428 0.1115 0.0013 d 6 − 8.9976 − 0.0312 − 0.0762 d 7 − 2.8355 − 0.2776 − 0.0056 d 8 − 0.2345 − 0.4479 − 6.88 × 10 − 5 d 9 9.1692 − 0.2402 − 8.75 × 10 − 4 d 10 0.5082 − 0.4775 − 0.0028 d 11 0.4054 0.7642 − 0.7415 d 12 − 0.5918 0.4339 0.0055 d 13 − 6.8967 − 0.2412 − 1.0125 d 14 − 3.2426 − 1.7431 − 2.84 × 10 − 4 d 15 7.0285 − 1.8584 − 0.0116 FIG. 1. A schematic representation of a unit cell. FIG. 2. Potential contours on the plane x 3 = 0 for a simple cubic system under a unit intensity along the x 1 direction with spherically transversely isotropic spheres ( f = 0.2 ) : (a) k r ∕ k m = 8 , k θ ∕ k m = 2 , and (b) k r ∕ k m = 2 , k θ ∕ k m = 8 . FIG. 3. Potential contours on the plane x 3 = 0 for a simple cubic array of doubly coated spheres under a unit intensity along the x 1 direction. Phase properties are k r ( 1 ) ∕ k 0 = 2 , k θ ( 1 ) ∕ k 0 = 8 , k r ( 2 ) ∕ k 0 = 8 , k θ ( 1 ) ∕ k 0 = 2 , k r ( 3 ) ∕ k 0 = 3 , k θ ( 3 ) ∕ k 0 = 9 , ρ 1 = 0.362 78 , ρ 2 = 0.272 08 , and ρ 3 = 0.163 24 . FIG. 4. The effective conductivity vs the grading parameter for a simple cubic array of composite with k r ∕ k m = k θ ∕ k m = 4 . FIG. 5. Potential contours on the x 3 = 0 plane for a simple cubic system ( k r ∕ k m = k θ ∕ k m = 4 , f = 0.2 ) under a unit intensity along the x 1 direction with exponentially graded spheres: (a) β = − 5 and (b) β = 5 .

PY - 2006/5/1

Y1 - 2006/5/1

N2 - 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.

AB - 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.

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U2 - 10.1063/1.2191668

DO - 10.1063/1.2191668

M3 - Article

AN - SCOPUS:33646893517

VL - 99

JO - Journal of Applied Physics

JF - Journal of Applied Physics

SN - 0021-8979

IS - 9

M1 - 093702

ER -