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

在这项研究中，首先，建立了一个总体框架来分析周期性细胞固体的微观分岔和分岔后行为。该框架是建立在双尺度理论的基础上的，称为均匀化理论，更新的拉格朗日型。由此导出了微观分岔的本征模问题和本征模所满足的正交性。结果表明，正交性允许宏观增量独立于特征模态，从而使基于比较固体概念的弹塑性分岔后分析过程变得简单。其次，利用上述框架，对受面内压缩的弹塑性六边形蜂窝细胞聚集体进行了分岔和后分岔分析。结果表明，在单轴压缩条件下，微观分岔的基本长波特征模态以纵向分量为主，而横向分量占主导地位，从而导致微观屈曲在垂直于加载轴的单元排中局部化。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不稳定状态下的六分岔而转变为不对称的长波屈曲模式，导致三角洲地区微观屈曲局部化。第三，利用双尺度理论分析了弹性方蜂窝在平面内双轴压缩作用下的长波和短波屈曲。以细胞聚集体为周期单位，分析了细胞聚集体的分岔和分岔后行为，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，当周期长度足够长时，超长波屈曲发生在宏观失稳开始后。在此基础上，结合双尺度分析，探讨了材料的宏观失稳条件，推导出了单向面内双轴压缩下超长波屈曲应力的简单计算公式。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