In this research, the electric potential equations and the boundary conditions for weakly nonlinear composites based on the third-order perturbation expansion are developed. These expressions are applied to solve for the electric potentials up to the third-order of a weakly nonlinear dielectric composite consisting of dilute linear cylindrical inclusions randomly dispersed in a nonlinear host medium. General formulae for calculating the effective nonlinear coefficients up to the ninth-order by using the perturbation expansion method, including terms up to the fifth-order nonlinear coefficient, are derived from the composite energy definition of the effective coefficients. These general formulae are applied to a case of nonlinear dielectric composite consisting of dilute weakly nonlinear cylindrical inclusions randomly dispersed in a linear medium. Our results, which include the fifth-order nonlinearity of the effective coefficients, reduce to the simple results of Gu and Yu (1992) considering only the third-order nonlinearity obtained by using the average electric displacement definition of the effective nonlinear coefficients. Moreover, we show that our method is more concise and gives more accurate results than those obtained by using the method of Gu and Yu (1992). Furthermore, we also apply our general formulae to a more complicated weakly nonlinear composite consisting of dilute linear cylindrical inclusions randomly dispersed in a nonlinear host medium.
