Relationship between the effective thermal properties of linear and nonlinear doubly periodic composites

The present paper is devoted to the study of the effective properties of 2D unbounded composite materials with temperature dependent conductivities. We consider a special case of nonlinear composites, when the conductivity coefficients of the matrix and the composite constituencies are proportional. This allows us to transform the problem for the nonlinear composite to a problem for an equivalent linear composite and then to find a solution of the nonlinear type. Analyzing the effective properties of the composites we derive relationships between their average properties. We show that, when computing the effective properties of the representative cell of the nonlinear composite, the result may depend not only on the temperature but also on its gradient.