论文标题
非线性周期性超材料的均质化和多尺度稳定性分析的计算框架
Computational frameworks for homogenization and multiscale stability analyses of nonlinear periodic metamaterials
论文作者
论文摘要
本文为多尺度的一阶有限菌株均质化和与周期性微观结构的速率无关固体的稳定性分析提供了一个一致的计算框架。基于多尺度虚拟功率的原理，均化公式是基于先验离散的微观结构的，并且用于计算同质应力和切线模量的矩阵表示的算法始终得出。均化结果在第一分叉开始时失去了其有效性，可以根据多尺度稳定性分析计算。多尺度不稳定性包括：a）通过Bloch波分析计算的显微镜结构不稳定性； b）宏观材料不稳定性，通过对均质的切线模量进行等级-1凸度检查计算。提供了有关BLOCH波分析的实现的详细信息，包括选择波矢量空间的选择以及从复杂价值的BLOCH波中检索实值的屈曲模式。详细详细介绍了三种方法，用于求解所得的约束特征值问题 - 两种冷凝方法和一种基于空空间的投影方法。均质化和稳定性分析的两种实现均使用包括超弹性和弹性塑料超材料在内的数值示例验证。还通过检查几个代表性的超材料实例来证明各种微观屈曲现象。与理论结果保持一致，数值结果表明，可以通过均质切线模量的等级1凸度的丢失来等效地检测到显微镜长波长屈曲。
This paper presents a consistent computational framework for multiscale 1st order finite strain homogenization and stability analyses of rate-independent solids with periodic microstructures. Based on the principle of multiscale virtual power, the homogenization formulation is built on a priori discretized microstructure, and algorithms for computing the matrix representations of the homogenized stresses and tangent moduli are consistently derived. The homogenization results lose their validity at the onset of 1st bifurcation, which can be computed from multiscale stability analysis. The multiscale instabilities include: a) microscale structural instability which is calculated by Bloch wave analysis; and b) macroscale material instability which is calculated by rank-1 convexity checks on the homogenized tangent moduli. Details on the implementation of the Bloch wave analysis are provided, including the selection of the wave vector space and the retrieval of the real-valued buckling mode from the complex-valued Bloch wave. Three methods are detailed for solving the resulted constrained eigenvalue problem - two condensation methods and a null-space based projection method. Both implementations of the homogenization and stability analyses are validated using numerical examples including hyperelastic and elastoplastic metamaterials. Various microscale buckling phenomena are also demonstrated by examining several representative metamaterial examples. Aligned with theoretical results, the numerical results show that the microscopic long wavelength buckling can be equivalently detected by the loss of rank-1 convexity of the homogenized tangent moduli.
