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A computational framework for homogenization and multiscale stability analyses of nonlinear periodic materials
International Journal for Numerical Methods in Engineering ( IF 2.9 ) Pub Date : 2021-08-01 , DOI: 10.1002/nme.6802
Guodong Zhang 1 , Nan Feng 1 , Kapil Khandelwal 1
Affiliation  

This article presents a consistent computational framework for multiscale first-order finite strain homogenization and stability analyses of rate-independent solids with periodic microstructures. 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 first bifurcation, which can be computed from multiscale stability analysis. The multiscale instabilities include: (a) microscale structural instability calculated by Bloch wave analysis; and (b) macroscale material instability calculated by rank-1 convexity checks on the homogenized tangent moduli. Implementation details of the Bloch wave analysis are provided, including the selection of wave vector space and the retrieval of real-valued buckling mode from complex-valued Bloch wave. Three methods are detailed for solving the resulted constrained eigenvalue problem—two condensation methods and a null-space projection method. Both the homogenization and stability analyses are verified using numerical examples including hyperelastic and elastoplastic periodic materials. Various microscale buckling phenomena are demonstrated. 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.

中文翻译:

非线性周期材料均质化和多尺度稳定性分析的计算框架

本文提出了一个一致的计算框架,用于对具有周期性微观结构的与速率无关的固体进行多尺度一阶有限应变均匀化和稳定性分析。均质化公式建立在先验离散的微观结构上,计算均质化应力和切线模量的矩阵表示的算法是一致推导出来的。同质化结果在第一次分叉开始时失去其有效性,这可以从多尺度稳定性分析中计算出来。多尺度不稳定性包括: (a) 通过布洛赫波分析计算的微尺度结构不稳定性;(b) 通过对均质切线模量进行 1 级凸度检查计算得出的宏观材料不稳定性。提供了布洛赫波分析的实现细节,包括波向量空间的选择和复值Bloch波实值屈曲模态的反演。详细介绍了解决约束特征值问题的三种方法——两种压缩方法和一种零空间投影方法。均质化和稳定性分析均使用数值示例进行验证,包括超弹性和弹塑性周期性材料。展示了各种微尺度屈曲现象。与理论结果一致,数值结果表明,微观长波长屈曲可以通过均质切线模量的 1 阶凸度损失等效检测。详细介绍了解决约束特征值问题的三种方法——两种压缩方法和一种零空间投影方法。均质化和稳定性分析均使用数值示例进行验证,包括超弹性和弹塑性周期性材料。展示了各种微尺度屈曲现象。与理论结果一致,数值结果表明,微观长波长屈曲可以通过均质切线模量的 1 阶凸度损失等效检测。详细介绍了解决约束特征值问题的三种方法——两种压缩方法和一种零空间投影方法。均质化和稳定性分析均使用数值示例进行验证,包括超弹性和弹塑性周期性材料。展示了各种微尺度屈曲现象。与理论结果一致,数值结果表明,微观长波长屈曲可以通过均质切线模量的 1 阶凸度损失等效检测。
更新日期:2021-08-01
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