Effective Conductivity and Critical Properties of a Hexagonal Array of Superconducting Cylinders

Abstract

Effective conductivity of a 2D composite corresponding to the regular hexagonal arrangement of superconducting disks is expressed in the form of a long series in the volume fraction of ideally conducting disks. According to our calculations based on various re-summation techniques, both the threshold and critical index are obtained in good agreement with expected values. The critical amplitude is in the interval (5.14, 5.24) that is close to the theoretical estimation 5.18. The next order (constant) term in the high concentration regime is calculated for the first time, and the best estimate is equal to (−6.229). Final formula is derived for the effective conductivity for arbitrary volume fraction.

Effective transport properties of composites with a doubly‐periodic array of fiber pairs and with a triangular array of fibers

“…Perrins et al [36] and McPhedran et al [27] modified and extended Rayleigh's solution [38] for square array and hexagonal array. Mityushev et al [11,13,32] applied the method of functional equations to the square array and hexagonal array, and discussed two extreme cases of porous materials and superconducting cylinders. Rylko [42] and Mityushev [33] also applied this method to the cases of rectangular array of cylinders.…”

“…This motivated us to extend the complex variable solution [50] for array of sole fibers to a general doubly-periodic array of fiber pairs. The present complex variable method [50] is expressed in a series form and based on the elliptic function theory, thus, which is related to other solutions based on the elliptic function theory, such as the exact method of functional equations by Mityushev et al [11,13,32,33,42], and the solutions by Jiang et al [21] and by Godin [15]. When the series is truncated to finite order, an approximate solution is obtained.…”