Anisotropy in effective conductivities of rectangular arrays of elliptic cylinders

Effective conductivities of inhomogeneous systems with rectangular arrays of elliptic cylinders are accurately evaluated with a boundary collocation scheme. The investigation is conducted from 0 to the close packing in inclusion volume fractions f, and from 0 to ∞ in conductivity ratios of the inclusion to matrix σ. Results of the calculation agree very well with a reciprocal theorem developed by Keller [J. Appl. Phys. 34, 991 (1963); J. Math. Phys. 5, 548 (1964)]. The effect of different array arrangement on effective conductivities is studied by changing the aspect ratio of the inclusion ra, from 2 to 10 and the array edge length ratio re, from 1 to 10. The anisotropy in effective conductivities of the array is also examined. It is found that, with unit ra/re, the enhancement in conductivity in the direction parallel to the major axis of the inclusion increases appreciably when compared to that of the square array of circular cylinders of equal inclusion volume fractions, while the enhancement in the perpendicular direction decreases only slightly.