Abstract
This paper presents a unified method to predict the effective properties of the composite materials with periodic microstructure. First, based on the energy analysis, we use the multiple scale asymptotic analysis for a periodic unit cell to achieve a series of differential equations. Those equations include the first-order and the second-order strain gradient constitutive equations besides the conventional homogenized constitutive equations. Second, we further show that these equations can be converted into solving unified formal differential equations. Finally, we give the numerical method based upon Fast Fourier Transforms and taking the strain gradients into account and the solution is based on the explicit form of the periodic Green's function of the reference medium. This method is different from classically investigated by the Finite Element Method which need meshing.