A CLOSED SOLUTION AND ITS APPLICATIONS FOR THE 3-PHASE CONFOCAL ELLIPSE MODEL UNDER LONGITUDINAL SHEAR
纵向剪切三相共焦点椭圆模型的精确解及其应用

Extending the 3-phase concentric circle model, a 3-phase confocal ellipse model is proposed, which can account for a wide range of inclusion shape variations from a ribbon to a circle. By using the conformal mapping technique, a closed form solution is obtained for the model under longitudinal shear. The 3-phase model can serve as a fiber/interfacial layer/matrix model, which is useful in studying stress concentrations around the fiber with an interfacial layer between the fiber and matrix. It is found that serious stress concentrations occur as the aspect ratio (the ratio of the semiminor axis to the semimajor axis of an ellipse) of the fiber section is small. The 3-phase model can also serve as a fiber/matris/composite model, in terms of which a generalized self-consistent approach is developed and a new formula is derived for predicting the effective longitudinal shear modulus. A comparison of the new formula with the classical formula and experimental results is made. The reason is found that the classical formula gives a poor theoretical prediction for the effective longitudinal shear modulus, that is, the classical formula fails to account for the effect of the fiber section shape. The new formula shows that the effective longitudinal shear modulus is sensitive to the fiber section shape, and the theoretical predictions by using the new formula show reasonable agreement with experimental data.