基于等效特征应变原理的复合材料有效模量细观力学分析

One of the main objectives of micro-mechanics is to predict the effective moduli of composites. Most micro-mechanics models are based on Eshelby's equivalent inclusion method under the hypothesis of ellipsoidal inclusion. Therefore, appropriate theoretical treatment for non-ellipsoidal inclusions in inhomogeneous media is still needed. In this work, we proposed a principle of averaged equivalent eigenstrain, and developed a new analytical micro-mechanics method for determining the effective elastic moduli of composites. By introducing the concept of averaged eigenstrain, the average stress and average strain in the matrix and inclusions of the representative volume element were analyzed. The principle of averaged equivalent eigenstrain was thus developed based on the principle of equivalent eigenstrain. Identical to the principle of equivalent eigenstrain which was built on the corner-stone of principle of virtual work, this new principle is also applicable to the inclusions with arbitrary shapes (either convex or concave), and multiple inclusions within a solid of finite volume. Then, focusing on the long-fiber-reinforced two-phase composites, we obtained the analytical relations between the average stresses and average strains in the matrix and fibers. The averaged equivalent eigenstrain was thus combined with the direct homogenization method of micro-mechanics to determine the effective elastic moduli of two-phase composites, by selecting different deformation models and the corresponding boundary conditions. Finally, the effective moduli of composites were predicted using the mechanical properties of fiber and matrix constituents as well as the volume fraction of fibers, through a MATLAB program. The engineering constants such as tensile moduli, shear moduli and Poisson's ratios of nine types of commonly-used composites were discussed in detail. After comparing the predicted values with the experimental results and the predictions from other theoretical models, it was found that the averaged relative errors of the new approach were less than those of the Halphin-Tsai model and the Mori-Tanaka model, indicating that the new analytical method is inherently suitable for the effective moduli of more general composites (either two-phase or multi-phase composites, containing inclusions/damage of either convex or concave shapes).