基于等参梯度单元分析界面层属性对石墨烯/环氧树脂复合材料弹性性能的影响

黄立新; 杨绍钊; 梁福安; 黄君

doi:10.13801/j.cnki.fhclxb.20210402.001

基于等参梯度单元分析界面层属性对石墨烯/环氧树脂复合材料弹性性能的影响

doi: 10.13801/j.cnki.fhclxb.20210402.001

广西大学　土木建筑工程学院，南宁 530004

基金项目: 国家自然科学基金(12002093)；广西自然科学基金(2018GXNSFBA281199)

详细信息

通讯作者:
黄君，硕士，讲师，研究方向为复合材料结构与力学分析　E-mail：jun_517561912@qq.com

中图分类号: TB332
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出版历程
- 收稿日期: 2021-01-30
- 修回日期: 2021-03-17
- 录用日期: 2021-03-29
- 网络出版日期: 2021-04-02
- 刊出日期: 2022-02-01

Effects of interfacial layer properties on elastic properties of graphene/epoxy composites based on isoparametric graded element

School of Civil Engineering and Architecture, Guangxi University, Nanning 530004, China

摘要

摘要: 针对石墨烯片在环氧树脂基体内定向连续和定向非连续分布的复合材料，通过分别构建三明治代表体单元和嵌入式代表体单元，进行了弹性性能预测的研究。代表体单元是三相复合结构，其中石墨烯和环氧树脂基体之间的界面层视为连续介质，其材料属性分别考虑均匀、线性和指数变化等3种情况。在代表体单元的有限元建模过程中，石墨烯和环氧树脂基体分别采用梁单元和实体单元进行离散，而界面层则采用等参梯度单元逼近。采用有限元软件ABAQUS 分析了小应变下的代表体单元的力学变形行为并提取其弹性性能，然后分析了界面层属性对石墨烯/环氧树脂复合材料弹性性能的影响。通过与混合率模型、修正的Halpin-Tsai模型及实验数据对比，验证了本文提出的基于等参梯度单元计算方法的有效性。数值算例表明，在处理界面层材料属性不均匀分布的问题方面，等参梯度单元具有计算量少、收敛快和精度高的优点。石墨烯/环氧树脂复合材料杨氏模量预测结果显示，界面层材料属性采用梯度变化的模型时，杨氏模量计算结果比均匀分布的模型、混合率模型和修正的Halpin-Tsai模型的结果偏大，但是更接近实验值。本文的研究结果说明界面层属性是影响复合材料力学性能的重要因素，并为寻求复合材料力学性能更精确地分析提供有效的途径。
- 等参梯度单元 /
- 界面层属性 /
- 石墨烯 /
- 复合材料 /
- 弹性性能 /
- 环氧树脂
Abstract: The elastic properties of the composites with graphene sheets distributed continuously and discontinuously in the epoxy matrix were investigated via sandwich representative volume element (RVE) and embedded RVE, respectively. The RVEs were considered as a three phase composite structure, in which the interphase between graphene and epoxy resin matrix was treated as continuum medium while its material properties were considered to vary uniformly, linearly and exponentially. In the finite element modeling process of the RVE, the graphene was discretized using beam element and the epoxy matrix was discretized via the use of solid element while the interfacial layer was approached using isoparametric graded element (IGE). The finite element software ABAQUS was used to analyze the mechanical deformation behavior of the RVE subjected to small strains and extract its elastic properties. The extracted results of elastic properties were then used to study the effects of interfacial layer properties on the elastic properties of graphene/epoxy composites. The validity of the proposed computational method based on the IGE was verified by comparing with rule of mixture (ROM), the modified Halpin-Tsai model and the experimental data. Numerical examples illustrate that the IGE has the advantages of less computation, fast convergence and high accuracy in dealing with the uneven distribution of material properties in the interfacial layer. The prediction results of Young's modulus of composites reveal that when the material properties of interfacial layer adopt the gradient model, the calculated results of Young's modulus are larger than those of uniform distribution model, ROM and the modified Halpin-Tsai model, but closer to the experimental value. The results of this research show that the property of interfacial layer is an important factor affecting the mechanical properties of composites, and provide an effective way to seek more accurate analysis of the mechanical properties of composites.
- isoparametric graded element /
- interfacial properties /
- graphene /
- composites /
- elastic properties /
- epoxy

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图 1 两种石墨烯/环氧树脂复合材料代表体单元(RVE)的选取方式

Figure 1. Selection of two representative volume elements (RVE) of graphene/epoxy composites

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图 2 两种石墨烯/环氧树脂复合材料RVE的尺寸示意图

Figure 2. Dimension diagram of two RVE of graphene/epoxy composites

L—Length of the RVE; W—Width of the RVE; t—Thickness of RVE; L_g—Length of graphene sheet; W_g—Width of the graphene sheet; t_m—Single layer thickness of substrate; t_in—Unilateral thickness of the interfacial layer; t'—Distance between the outer boundary of graphene sheet in each direction and the outer surface of RVE in embedded RVE

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图 3 石墨烯/环氧树脂复合材料模型单向拉伸受力图

Figure 3. Force diagram of model of graphene/epoxy composites under uniaxial tension

R_n—Support reaction in x direction of each node; ΔL—Displacement in x direction of surface, x=L; ΔW'—Displacement in y direction of surface, y=W; Δt'—Displacement in z direction of surface, z=t

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图 4 石墨烯有限元模型

Figure 4. Finite element model of graphene

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图 5 石墨烯/环氧树脂复合材料RVE的有限元网格

Figure 5. Finite element mesh of RVE of graphene/epoxy composites

IGE—Isoparmetric graded element

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图 6 石墨烯/环氧树脂复合材料界面层材料属性的3种分布形式

Figure 6. Three distribution forms of material properties of interfacial layer of graphene/epoxy composites

p—material attribute; Subscripts in, m and g—Interphase, matrix and graphene, respectively.

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图 7 等参梯度单元

Figure 7. Isoparametric graded element

ζ, η, ξ—Local coordinates of the element node

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图 8 石墨烯片的手性定义

Figure 8. Chirality definition of graphene sheet

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图 9 石墨烯/环氧树脂复合材料界面层的层数划分

Figure 9. Division of interfacial layer of graphene/epoxy composites

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图 10 石墨烯体积含量与石墨烯/环氧树脂三明治模型杨氏模量的关系

Figure 10. Relationship between volume fraction of graphene and Young's modulus of graphene/epoxy sandwich model

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图 11 界面层厚度与石墨烯/环氧树脂复合材料三明治模型弹性性能的关系

Figure 11. Relationship between the thickness of interfacial layer and the elastic properties of graphene/epoxy composite sandwich RVE model

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图 12 界面层厚度与石墨烯/环氧树脂复合材料嵌入式模型弹性性能的关系

Figure 12. Relationship between the thickness of interfacial layer and the elastic properties of graphene/epoxy composite embedded RVE model

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图 13 石墨烯体积含量与石墨烯/环氧树脂嵌入式模型杨氏模量的关系

Figure 13. Relationship between volume fraction of graphene and Young's modulus of graphene/epoxy embedded model

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表 1 分子力学常数

Table 1. Molecular mechanics constants

Stretching constant k_r/(N·nm⁻¹)	Bond angle bending constant k_θ/(N·nm·rad⁻²)	Torsion constant k_τ/(N·nm·rad⁻²)
6.52×10⁻⁷	8.76×10⁻¹⁰	2.78×10⁻¹⁰

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表 2 材料参数和几何参数

Table 2. Material and geometric parameters

Length of beam l/nm	Diameter d/nm	Sectional area A/nm²	Young’s modulus of beam E/GPa	Shear modulus G/GPa	Moment of inertia I/nm⁴	Polar moment of inertia J/nm⁴	Young’s modulus of epoxy E_m/GPa	Poisson’s ratio of epoxy ν_m
0.1421	0.1466	0.01688	5487.5	7627.6	2.267×10⁻⁵	4.534×10⁻⁵	3.8	0.4

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表 3 石墨烯片杨氏模量和泊松比计算结果

Table 3. Calculation results of Young's modulus and Poisson's ratio of graphene sheet

Size of graphene/(nm×nm)	Armchair		Zigzag
Size of graphene/(nm×nm)	Young’s modulus E_g/GPa	Poisson’s ratio ν_g	Young’s modulus E_g/GPa	Poisson’s ratio ν_g
1×1	1106.811	0.133	999.142	0.119
2×2	1084.729	0.096	1020.232	0.089
3×3	1073.213	0.083	1027.222	0.079
4×4	1062.424	0.077	1030.972	0.074
5×5	1059.296	0.073	1032.965	0.071
6×6	1056.941	0.071	1034.302	0.069
7×7	1053.812	0.069	1035.346	0.068
8×8	1052.836	0.068	1036.168	0.067
9×9	1051.802	0.067	1036.693	0.066
10×10	1050.294	0.066	1037.156	0.065
20×20	1045.813	0.063	1039.111	0.062
50×50	1042.975	0.061	1040.307	0.061
100×100	1042.042	0.060	1040.702	0.060

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表 4 界面层属性呈线性分布情况时不同体积含量的石墨烯/环氧树脂三明治模型的杨氏模量

Table 4. Young's modulus of graphene/epoxy sandwich model with different volume fractions under linear distribution of interfacial properties

Volume fraction V_fr/vol%	Number of layers in single side interface
	1		2		3		4
	Young’s modulus E_c_x/GPa	—	Young’s modulus E_c_x/GPa	${\varDelta _1}$/%	Young’s modulus E_c_x/GPa	${\varDelta _2}$/%	Young’s modulus E_c_x/GPa	${\varDelta _3}$/%
2.5	43.596	—	43.441	0.357	43.408	0.076	43.395	0.030
5.0	83.389	—	83.079	0.373	83.012	0.081	82.987	0.030
7.5	123.172	—	122.707	0.379	122.606	0.082	122.569	0.030
10.0	162.936	—	162.315	0.383	162.180	0.083	162.130	0.031
Note: Δ_i(i=1,2.3)—Deviation of Young's modulus when the interface is divided into i and i+1 layer.

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表 5 界面层属性呈指数分布情况时不同体积含量的石墨烯/环氧树脂三明治模型的杨氏模量

Table 5. Young's modulus of graphene/epoxy sandwich model with different volume fractions under exponential distribution of interfacial properties

Volume fraction V_fr/vol%	Number of layers in single side interface
	1		2		3		4
	E_c_x/GPa	—	E_c_x/GPa	${\varDelta _1}$/%	E_c_x/GPa	${\varDelta _2}$/%	E_c_x/GPa	${\varDelta _3}$/%
2.5	43.595	—	39.238	11.104	38.373	2.254	38.064	0.811
5.0	83.387	—	74.673	11.670	72.942	2.373	72.324	0.854
7.5	123.170	—	110.098	11.873	107.501	2.416	106.575	0.869
10.0	162.935	—	145.502	11.981	142.040	2.437	140.804	0.878

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表 6 等参梯度单元与分层法的计算量对比

Table 6. Comparison of computational cost between isoparametric graded element and layering method

Method	Interfacial property	Number of layers	Total number of elements	Total number of nodes
Layering method^[15]	Linear	5	—	—
Layering method^[16]	Linear	5	—	—
Layering method^[17]	Linear	17	—	—
Layering method^[14]	Exponential	5	62400	78864
IGEM	Linear	1	7790	15936
IGEM	Exponential	2	15580	23904

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表 7 不同体积含量和尺寸情况下石墨烯/环氧树脂嵌入式模型的杨氏模量

Table 7. Young’s modulus of graphene/epoxy embedded model with different volume contents and sizes

Volume fraction/ vol%	Size of graphene/(nm×nm)
	1×1			5×5			10×10
	E_c_x/GPa		Deviation	E_c_x/GPa		Deviation	E_c_x/GPa		Deviation
	H-T equation	Present study	Deviation	H-T equation	Present study	Deviation	H-T equation	Present study	Deviation
2.5	4.085	4.053	−0.80%	4.812	4.733	−1.65%	5.665	5.593	−1.27%
5.0	4.386	4.290	−2.19%	5.876	5.777	−1.68%	7.621	7.503	−1.54%
7.5	4.702	4.546	−3.32%	6.994	6.878	−1.66%	9.674	9.531	−1.48%
10.0	5.036	4.882	−3.06%	8.173	8.030	−1.75%	11.833	11.628	−1.73%

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表 8 体积含量为1.12vol%的石墨烯/环氧树脂嵌入式模型归一化杨氏模量值与实验值对比(E_c_x/E_m)

Table 8. Comparison of the normalized Young's modulus of 1.12vol% graphene/epoxy embedded model with the experimental result (E_c_x/E_m)

Model	Present study	Experiment^[25]	Deviation
Homogeneous	1.211	1.312	−7.70%
H-T equation^[24]	1.217		−7.24%
Exponential	1.247		−4.95%
Linear	1.254		−4.42%

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