中文摘要：

在三维孔隙材料如泡沫材料中，胞孔的形状及大小并非完全均匀，因此借助二维均匀化方法来讨论胞孔形状及大小对多孔材料性能的影响。在线弹性范围内，根据均匀化理论，基于虚位移原理，结合有限元方法推导出二维周期性结构的均匀化的有限元格式。取具有不同孔洞形状的正方形胞元作为周期性结构的代表胞元，将三维孔隙材料简化为截面上具有规则孔洞的二维结构，来计算不同微孔形状及大小下的等效弹性常数；比较分析了微孔结构对多孔材料等效弹性常数的影响。计算结果分析表明，多孔材料的等效弹性参数不仅取决于微孔结构的数量，而且对微孔结构的形状也有一定程度的敏感，同时对基体材料的泊松比的变化敏感与否也与微孔结构有关。

英文摘要：

In 3-D porous materials, the cells are not the same in shape and dimension. 2-D homogenization method was applied to study the effects of the micro--structure （cell shape and size ）on the equivalent elastic constants of porous structures so as to simplified the problem. Starting from the weak form of virtual displacement equation, the homogenization method is introduced for linear elastic problems. Then a homogenization finite element method （homo FEM） is developed for the 2-D periodic structures and applied to the calculation of the equivalent elastic constants of thin regular 3-D periodic structure with various relative densities and with square as base-cell including different shapes of hole （circular and hexagonal section in-plane）. The numerical results obtained by homo FEM shows that both the cell volume fraction and cell shape influence the equivalent elastic constants of porous structures. Meanwhile the influence of the mechanical property on the micro-mechanical equivalent elastic moduli of cellular structures are also evaluated. In the range of analysis（0.1--0.4）, the elastic constant E of circular structure decreases with the increase of from 0. 1 to 0.4,but that of hexagonal structure doesn＇t change , this means that they has different influence on the equivalent elastic constants of cellular materials of different micro-structure.