Effective Conductivity of 2D Ellipse – Elliptical Ring Composite Material

Abstract

For 2D composite material geometrically composed %presented
by an interior of ellipse $E_1$ (with semi-axes $a_1$, $b_1$) and an
outer elliptical ring with an outer ellipse $E_2$ (with semi-axes
$a_2$, $b_2$), where $E_1$ and $E_2$ are confocal ellipses; it is
determined in an analytic form the $x$-component of the effective
conductivity tensor as a sum of convergent power series with
coefficients depending on conductivities of the components of the
composite material and geometrical characteristics $a_1$, $b_1$ and
$a_2$, $b_2$.