Physics
Scientific paper
Apr 1992
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1992rspsa.437..215t&link_type=abstract
Proceedings: Mathematical and Physical Sciences, Volume 437, Issue 1899, pp. 215-227
Physics
13
Scientific paper
I use conformal mapping techniques to determine the change in the conductivity of a sheet containing a few well-separated holes. The hole shapes studied are the equilateral triangle, square, pentagon and regular n-gons. I show that the conductivity can be written as σ /σ 0 = 1-α nf + o(f2), where f is the area fraction of the inclusions and the coefficient α n = tan (π /n)/2π nGamma 4(1/n)/Gamma 2(2/n), which is 2.5811, 2.1884, 2.0878 for triangles, squares and pentagons, and tends to the circle limit of 2 as n -> ∞ . The coefficient α n is proportional to the induced dipole moment around the polygonal hole which can be found using an appropriate conformal mapping. I have also examined and compared the results for long thin needle-like holes in the shape of diamonds, rectangles and ellipses. In all cases the conductivity parallel to the needles has the limiting form σ /σ 0 = 1 - f, while for the perpendicular conductivity, I find that σ /σ 0 = 1 - nπ a2, where 2a is the length of the needle, and n is the number of needles per unit area. For thicker needles, the shape becomes important and I compare the results with recent analog experiments and computer simulations. Because of the reciprocity theorem, all the results found here apply equally well to superconducting inclusions.
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