Microscopic Buckling Analysis of Cellular Solids Based on a Homogenization Theory of Finite Deformation

研究了胞状固体在宏观均匀压缩下的微观对称分叉屈曲问题。首先，用更新的拉格朗日形式描述无限周期材料的虚功原理，建立了满足材料客观性原理的有限变形均匀化理论。然后，我们提出了一个假设，即在微观对称分叉开始时，微观速度成为自发的，但改变这种自发速度的符号对宏观状态的变化没有影响。将这一假设应用到齐次化理论中，得到了微观对称分叉开始时应满足的条件。利用齐次化理论和所建立的屈曲条件分析了弹性六角蜂窝的面内双向屈曲问题。由此得出以下结论：当最大压缩载荷分别沿细胞壁的一个、两个和三个方向传递时，可以发生简单、二次和三次分叉。在双分叉点和三分叉点，单轴屈曲模式在细胞壁的两个和三个方向同时发展，并被线性组合，产生了Gibson和Ashby(1997)报道的双轴模式II和Papka和Kyriakdes(1999a)观察到的花状模式III。换句话说，这些双轴模式是由于分叉的多重性而产生的，因此与发生在简单分叉点的单轴屈曲模式I相比，它们具有非常复杂的胞型。此外，我们证明了模式II和模式III以及模式I被归类为微观对称分叉，尽管它们的胞型非常复杂，因为它们不像假设中所描述的那样在宏观上受到自发扰动速度符号的影响。

The microscopic symmetric bifurcation buckling of cellular solids subjected to macroscopically uniform compression was studied. To begin with, describing the principle of virtual work for infinite periodic materials in the updated Lagrangian form, we built a homogenization theory of finite deformation, which satisfies the principle of material objectivity. Then, we stated a postulate that at the onset of microscopic symmetric bifurcation, microscopic velocity becomes spontaneous, yet changing the sign of such spontaneous velocity has no influence on the variation in macroscopic states. By applying the postulate to the homogenization theory, we derived the conditions to be satisfied at the onset of microscopic symmetric bifurcation. The homogenization theory and buckling conditions established were employed to analyze the in-plane biaxial buckling of an elastic hexagonal honeycomb. We thus found the following : Simple, double and triple bifurcations can take place, if the largest compressive load is transmitted in one, two and three directions of cell walls, respectively. At the double and triple bifurcation points, uniaxial buckling modes develop simultaneously in the two and three directions of cell walls and are linearly combined to generate a biaxial mode, Mode II, reported by Gibson and Ashby (1997) and a flower-like mode, Mode III, observed by Papka and Kyriakides (1999a). In other words, these biaxial modes result from the multiplicity of bifurcation, so that they have very complex cell-patterns in comparison with the uniaxial buckling mode, Mode I, occurring at the simple bifurcation points. Moreover, we showed that Modes II and III, as well as Mode I, are classified as microscopic symmetric bifurcation in spite of their very complex cell-patterns, since they are not macroscopically influenced by changing the sign of spontaneous perturbed velocity as described in the postulate.