  An analytic solution to a stationary heat conduction problem in 2D unbounded
doubly periodic composite materials with temperature dependent conductivities
of their components is given. Corresponding nonlinear boundary value problem is
reduced a Laplace equation with nonlinear transmission conditions. For special
relationships between the conductivity coefficients of the matrix and
inclusions, the problem is transformed to fully linear boundary value problem
for doubly periodic analytic functions. This allows to reconstruct the solution
of the originally nonlinear composite and to find its effective properties. The
results are illustrated by numerical examples.
