纵向剪切三相共焦点椭圆模型的精确解及其应用

推广夹杂问题的三相同心圆模型，提出了三相共焦点椭圆模型，从而计及了夹杂形状变化的重要因素。利用保角变换结合罗朗级数展开技术获得了纵向剪切载荷下的封闭形式解。本模型和解答对夹杂、界面层附近的细观应力场分析、对复合材料和含缺陷材料的有效弹性模量预测有重要实用价值，获得了比经典公式更精确的结果。对特殊模型，可以得到许多有意义的问题（如二相模型）的解答。

A CLOSED SOLUTION AND ITS APPLICATIONS FOR THE 3-PHASE CONFOCAL ELLIPSE MODEL UNDER LONGITUDINAL SHEAR

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.