Large-scale Multiscale Analysis for Microscopic Buckling and Macroscopic Instability of Periodic Cellular Solids Based on a Homogenization Theory

在本研究中，首先建立了一个分析周期性胞状固体微观分叉和分叉后行为的一般框架。该框架是建立在一个双尺度理论的基础上，称为均匀化理论，更新拉格朗日类型。由此导出了微观分岔的本征模问题和本征模所应满足的正交性。结果表明，正交性使得宏观增量与本征模无关，从而简化了基于比较体概念的弹塑性后分叉分析过程。其次，利用上述框架，对面内压缩弹塑性六边形蜂窝结构的胞元集合体进行了分叉和后分叉分析。由此，证明了单轴压缩下微观分叉的一个基本长波本征模，并表明本征模具有纵向分量的优势 关于我们 并因此导致微观屈曲定位在垂直于加载轴的单元行中。进一步表明，在等双轴压缩下，宏观稳定状态下的花状屈曲模式由于宏观失稳状态下的六重分叉而转变为非对称的长波模式，导致微观屈曲在三角洲区域的局部化。采用双尺度理论分析了平面内双向压缩下弹性方蜂窝结构的长波和短波屈曲。将细胞团视为周期单元，分析了细胞团的分叉和分叉后行为，讨论了屈曲应力与周期长度的关系。结果表明，屈曲应力随周期长度的增加而减小，如果周期长度足够长，则在宏观失稳发生后不久就会发生超长波屈曲。在此基础上，通过对宏观失稳条件的探讨，推导出了一个简单的计算平面内双向压缩条件下超长波屈曲应力的公式。用能量法对所得公式进行了验证。少

In this study, first, a general framework was developed to analyze microscopic bifurcation and post-bifurcation behavior of periodic cellular solids. The framework was built on the basis of a two-scale theory, called a homogenization theory, of the updated Lagrangian type. The eigenmode problem of microscopic bifurcation and the orthogonality to be satisfied by the eigenmodes were thus derived. It was shown that the orthogonality allows the macroscopic increments to be independent of the eigenmodes, resulting in a simple procedure of the elastoplastic post-bifurcation analysis based on the notion of comparison solids.Second, by use of the framework mentioned above, bifurcation and post-bifurcation analysis were performed for cell aggregates of an elastoplastic hexagonal honeycomb subject to in-plane compression. Thus, demonstrating a basic, long-wave eigenmode of microscopic bifurcation under uniaxial compression, it was shown that the eigenmode has the longitudinal component dominant … More to the transverse component and consequently causes microscopic buckling to localize in a cell row perpendicular to the loading axis. It was further shown that under equi-biaxial compression, the flower-like buckling mode having occurred in a macroscopically stable state changes into an asymmetric, long-wave mode due to the sextuple bifurcation in a macroscopically unstable state, leading to the localization of microscopic buckling in deltaic areas.Third, long-wave and short-wave buckling of elastic square honeycombs subject to in-plane biaxial compression were analyzed using the two-scale theory. By taking cell aggregates to be periodic units, the bifurcation and post-bifurcation behavior were analyzed to discuss the dependence of buckling stress on periodic length. It was shown that buckling stress decreases as periodic length increases, and that very-long-wave buckling occurs just after the onset of macroscopic instability if the periodic length is sufficiently long. Then, a simple formula to evaluate the very-long-wave buckling stress under in-plane biaxial compression was derived by exploring the macroscopic instability condition in the light of the two-scale analysis. The resulting formula was verified using an energy method. Less