An analytical formula for the effective conductivity of 2D domains with cracks of high density

We construct the conformal mapping of the square with circular non-overlapping holes onto the square with non-overlapping slits. It is constructed as a solution of the Riemann-Hilbert problem for a multiply connected domain in a class of double periodic functions. The Riemann-Hilbert problem is reduced to a system of functional equations which is solved with an arbitrary order of approximations. On the basis of this conformal mapping, an analytical formula for the effective conductivity of randomly distributed cracks in 2D media is deduced. This formula extends the known before formulae to high density fractures in 2D media.