石墨烯纳米复合材料的降阶均匀化方法及其数值实现

鞠晓喆; 朱加文; 梁利华; 许杨剑

doi:10.13801/j.cnki.fhclxb.20210202.002

石墨烯纳米复合材料的降阶均匀化方法及其数值实现

doi: 10.13801/j.cnki.fhclxb.20210202.002

浙江工业大学　机械工程学院，杭州 310023

基金项目: 国家自然科学基金青年科学基金(12002309)；国家自然科学基金面上项目(51875523)；浙江省自然科学基金探索项目(LQ21A020002)

详细信息

通讯作者:
许杨剑，博士，教授，研究方向为计算固体力学　 E-mail：xuyangjian571@163.com

中图分类号: TB332
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出版历程
- 收稿日期: 2020-12-23
- 录用日期: 2021-01-22
- 网络出版日期: 2021-02-02
- 刊出日期: 2021-12-01

Reduced order homogenization of graphene nanocomposites and its numerical implementation

College of Mechanical Engineering, Zhejiang University of Technology, Hangzhou 310023, China

摘要

摘要: 对石墨烯纳米复合材料进行三维有限元建模通常需极其精细的网格。在考虑塑性演变的情况下，细观代表性单元体模型的计算效率极其低下。为此，基于非均匀变换场分析理论，提出了石墨烯纳米复合材料的降阶均匀化方法。首先，针对不同加载路径进行预分析，提取细观塑性应变场信息；然后对这些信息进行本征正交分解，从而得到若干个塑性模态，用作降阶模型的基函数；基于宏、细观耗散功的等效原理，导出降阶变量的本构模型。该方法的离线分析部分通过MATLAB编程实现。为了便于工程计算，在线分析部分则由商业有限元软件ABAQUS的UMAT用户子程序接口实现。基于三维算例分析，验证了所提方法的有效性。结果显示，在保证较高精度的前提下，针对三维代表性单元体计算的加速率可达10³~10⁴量级。
- 石墨烯纳米复合材料 /
- 模型降阶 /
- 均匀化方法 /
- 非线性多尺度分析 /
- 代表性单元体
Abstract: A three-dimensional finite element modeling for graphene nanocomposites usually requires very fine meshes. In case of plasticity, the computational efficiency of a micro representative volume element model is extremely low. For a remedy, based on the theory of nonuniform transformation field analysis, a reduced order homogenization method for graphene nanocomposites was proposed. First, we performed a pre-analysis for different loading paths to extract micro plastic strain fields; then a proper orthogonal decomposition of those field information was performed to obtain several plastic modes, which were used as basis functions for model order reduction. A constitutive model of reduced variables was derived from the equivalence of macro and micro dissipation power. The offline analysis of this method was implemented in MATLAB. For engineering computations, the online analysis was implemented by the user subroutine interface UMAT of the commercial finite element software ABAQUS. The effectiveness of the proposed method was illustrated by three-dimensional numerical examples. The results show that the acceleration rate for 3D representative volume element computations is of the order of 10³~10⁴, while maintaining a sufficient accuracy level.
- graphene nanocomposites /
- model order reduction /
- homogenization method /
- nonlinear multiscale analysis /
- representative volume element

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图 1 宏-细观双尺度问题

Figure 1. Illustration of a two scale (macro-micro) problem

L—Macro characteristic length; l—Micro characteristic length;$\overline{\varOmega } $—Macro domain; ${\varOmega } $—Micro domain; $\overline\sigma $—Macro stress tensor;$\sigma $—Micro stress tensor; $\overline\varepsilon $—Macro strain tensor; $\varepsilon $—Micro strain tensor

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图 2 石墨烯纳米复合材料的降阶均匀化流程

Figure 2. Flowchart of reduced order homogenization of graphene nano-composites

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图 3 石墨烯纳米复合材料代表性体积单元(RVE)建模和分析流程

Figure 3. Flowchart of modeling and analyzing based on representative volume element (RVE) of graphene nanocomposites

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图 4 不同GPLs含量的石墨烯纳米复合材料RVE模型

Figure 4. RVE models of graphene nanocomposites with different volume fractions of GPLs

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图 5 石墨烯纳米复合材料双向拉伸试验宏-细观双尺度有限元模型(FEM)

Figure 5. A two scale (macro-micro) finite element model (FEM) for a biaxial tensile test of a graphene nanocomposte

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图 6 石墨烯纳米复合材料塑性模态$ {\boldsymbol{\mu }}^{1} $的不同分量

Figure 6. Different components of plastic mode $ {\boldsymbol{\mu }}^{1} $ for graphene nanocomposites

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图 7 石墨烯纳米复合材料双向拉伸试验的力(F) -位移 (u)曲线

Figure 7. Force (F)-displacement (u) curve of a biaxial tension test for graphene nanocomposites

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图 8 石墨烯纳米复合材料宏观模型Mises应力云图

Figure 8. Contour plot of von Mises stresses for the macro model of graphene nanocomposites

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图 9 石墨烯纳米复合材料积分点A上的宏观应力-应变曲线

Figure 9. Macro stress-strain curves at an integration point A of graphene nanocomposites

NTFA—Nonuniform transformation field analysis

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图 10 石墨烯纳米复合材料降阶模型和有限元模型的Mises应力比较

Figure 10. Comparison of Mises stress localization between reduced order model and FE model of graphene nanocomposites

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表 1 石墨烯纳米复合材料的材料属性

Table 1. Material properties of a graphene nanocomposite

Material	Young’s modulus/GPa	Poisson's ratio	Yield stress/MPa	H/MPa	b	c/MPa
Matrix	70	0.3	350	1500	25	360
GPLs	1050	0.186	—	—	—	—
Notes: H, b and c—Material parameters in Eq.(22); GPLs—Graphene nanoplates.

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