Homogenization of a fibre/sphere with an inhomogeneous interphase for the effective elastic moduli of composites

An effective interphase model (EIM) and a uniform replacement model (URM) are proposed to study the effect of an inhomogeneous interphase with varying elastic properties in the radial direction on the effective elastic moduli of composites reinforced by fibres/spheres. The central idea of these models is to convert a fibre/sphere with its interphase into a two-phase or homogeneous fibre/sphere. Then, the strain energy changes can be obtained using the three-phase model or Eshelby's solution. Detailed comparisons with the finite-element method (FEM) results of the strain energy changes for various possible material combinations of fibre/sphere, interphase and matrix are carried out to check the validity of the two models. Moreover, the other two existing models, the uniform interphase model (UIM) and differential replacement model (DRM), are compared with the new ones. It is shown that the validity of these analytical models depends oil the material combinations. The EIM can be valid for general cases, while the simple URM is only valid for some cases. The validity ranges of the two existing models lie between those of the two new ones. Finally, the expressions of the effective elastic moduli of composites involving an inhomogeneous interphase are given by combining these models and the Mori-Tanaka method. The application of these expressions is illustrated through three examples and further comparisons with FEM results are also given.