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Exact solution to a nonlinear heat conduction problem in doubly periodic 2D composite materials

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.