Asymptotic justification of the three-phase composite model

The three-phase composite model (TPM) is analyzed from the viewpoint of the asymptotic homogenization method. It is shown that such an asymptotic process can be constructed that the three-phase model will represent a zero-order approximation. Therefore the present paper presents a mathematical asymptotic justification of the TPM technique. Further more, it is shown that the TPM approach provides a sufficiently accurate solution of the unit cell local problem arising in the asymptotic homogenization for periodic composite materials with cubic inclusions with arbitrary properties. The same conclusion is proved for the unidirectional fiber-reinforced composites in the cases of parallelepiped inclusions and rectangular periodic structure. The multi-point Padé approximants are used in the present paper in order to evaluate the accuracy of three-phase models in the cases of small and large inclusions, and to derive the estimations for the effective properties of composite materials.