Exact solution of a nonlinear heat conduction problem in a doubly periodic 2D composite material

Abstract

An analytic solution of a stationary heat conduction problem in an unbounded doubly periodic 2D composite whose matrix and inclusions consist of isotropic temperature-dependent materials is given. Each unit cell of the composite contains a finite number of circular non-overlapping inclusions. The corresponding nonlinear boundary value problem is reduced to a Laplace equation with nonlinear interface conditions. In the case when the conductive properties of the inclusions are proportional to that of the matrix, the problem is transformed into a fully linear boundary value problem for doubly periodic analytic functions. This allows one to solve the original nonlinear problem and reconstruct temperature and heat flux throughout the entire plane. The solution makes it possible to calculate the average properties over the unit cell and discuss the effective conductivity of the composite. We compare the outcomes of the present paper with a few results from literature and present numerical examples to indicate some peculiarities of the solution.