应变率相关的橡胶本构模型研究

魏家威; 石霄鹏; 冯振宇

doi:10.11858/gywlxb.20210815

应变率相关的橡胶本构模型研究

doi: 10.11858/gywlxb.20210815

魏家威^{1, 2,},
石霄鹏^{1, 2},
冯振宇^{1, 2,*, ,}

1.
中国民航大学安全科学与工程学院, 天津　300300
2.
民航航空器适航审定技术重点实验室, 天津　300300

详细信息

作者简介:
魏家威（1996－），女，硕士研究生，主要从事材料动态力学行为研究. E-mail：13821971667@163.com

通讯作者:
冯振宇（1966－），男，博士，教授，主要从事飞机结构强度研究. E-mail：mhfzy@163.com

中图分类号: O345
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出版历程
- 收稿日期: 2021-06-18
- 修回日期: 2021-07-07
- 录用日期: 2021-07-16

Strain Rate Dependent Constitutive Model of Rubber

WEI Jiawei^{1, 2
,},
SHI Xiaopeng^{1, 2},
FENG Zhenyu^{1, 2,*
, ,}

1.
College of Safety Science and Engineering, Civil Aviation University of China, Tianjin 300300, China
2.
Key Laboratory of Civil Aircraft Airworthiness Technology, Tianjin 300300, China

摘要

摘要: 为研究橡胶在不同应变率下的响应特性，建立应变率相关的橡胶黏超弹性本构模型，分别采用超弹性本构模型和黏弹性本构模型表征其非线性弹性行为和应变率相关的弹性行为。首先，对于超弹性模型，基于最小二乘法，对比了Mooney-Rivlin模型、修正的Mooney-Rivlin模型、Yeoh模型、修正的Yeoh模型、Ogden模型和Arruda-Boyce模型等超弹性本构模型的拟合能力。结果表明，经修正的Mooney-Rivlin模型和Yeoh模型的拟合优度与Ogden模型和Arruda-Boyce模型接近。在此基础上，基于一种参数较少且拟合效果良好的修正Mooney-Rivlin模型和应变率相关的Maxwell模型，建立了橡胶黏超弹性本构模型，考察了该黏超弹性本构模型在单轴拉伸和单轴压缩情况下中高应变率时的拟合能力。结果表明，对于这两种受力情况下的应变率相关的实验数据，该黏超弹性本构模型的拟合优度均在0.95以上。研究结果为大应变率范围内单轴拉伸和单轴压缩下橡胶的本构模型选择提供了参考。
- 本构模型 /
- 应变率 /
- 橡胶 /
- 最小二乘法
Abstract: In order to study the mechanical response characteristics of rubber under large loading strain rates, a strain-rate-dependent visco-hyperelastic constitutive model is established. The nonlinear and the strain-rate-dependent elastic behavior are characterized by the hyperelastic constitutive model and the viscoelastic constitutive model, respectively. Firstly, based on least square method, the fitting abilities of several hyperelastic constitutive models, i.e., Mooney-Rivlin model, the modified Mooney-Rivlin model, Yeoh model, the modified Yeoh model, Ogden model and Arruda-Boyce model, are compared. The results show that the modified Mooney-Rivlin model and the modified Yeoh model have similar fitting goodness as Ogden model and Arruda-Boyce model. Furthermore, the visco-hyperelastic constitutive model based on a modified Mooney-Rivlin model with a few parameters leading to good fitting results and the Maxwell model reflecting the strain-rate-dependent property of rubber is described. The fitting ability of the proposed model under uni-axial tension and uni-axial compression loading conditions at medium and high strain rates is assessed. It is concluded that the fitting goodness of the model is above 0.95 under both loading conditions. This study can provide a reference for the selection of constitutive models to characterize the mechanical behavior of rubber under uni-axial tension and uni-axial compression tests at different strain rates.
- constitutive model /
- strain rate /
- rubber /
- least square method

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图 1 基于M-R模型拟合的工程应力-伸长比的结果

Figure 1. Fitting results of the principle stress versus the principle stretch using M-R model

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图 2 M-R模型和修正后的M-R模型拟合结果对比

Figure 2. Comparison of fitting results between M-R model and the modified M-R model

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图 3 基于Yeoh模型拟合的工程应力-伸长比的结果

Figure 3. Fitting results of the principle stress versus the principle stretch using Yeoh model

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图 4 Yeoh模型和修正后的Yeoh模型拟合结果对比

Figure 4. Comparison of fitting results between Yeoh model and the modified Yeoh model

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图 5 不同模型拟合Treloar3种实验数据的结果

Figure 5. Fitting results of different models of Treloar’s three kinds of experimental data

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图 6 不同实验类型下不同模型的拟合结果对比

Figure 6. Comparison of the fitting results of different models for different experiment types

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图 7 黏超弹性模型示意图

Figure 7. Schematic diagram of visco-hyperelastic model

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图 8 M-R黏超弹性本构模型拟合硅橡胶单轴拉伸实验数据^[29]的结果

Figure 8. Fitting results of M-R visco-hyperelastic constitutive model for the uni-axial tensile experimental data of silicone rubber^[29]

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图 9 M-R黏超弹性本构模型拟合硫化橡胶单轴压缩实验数据^[28]的结果

Figure 9. Fitting results of M-R visco-hyperelasticconstitutive model for the uni-axial compressionexperimental data of vulcanized rubber^[28]

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表 1 不同超弹性模型对 ST、PT 和 ET 实验数据的拟合效果比较

Table 1. Comparison of fitting results of different hyperelastic models on ST, PT and ET experimental data

Model	Equation	Parameters	R²
M-R model	Eq.(2)	C₁₀, C₀₁	0.8043
Modified M-R model	Eq.(15)	C₁₀, C₀₁, C₂₀	0.9704
Yeoh model	Eq.(6)	C₁₀, C₂₀, C₃₀	0.9897
Modified Yeoh model	Eq.(21)	C₁₀, C₂₀, C₃₀, C₀₁	0.9961
Ogden model (N=2)	Eq.(8)	$ {\,\mu } $₁, $ {\alpha } $₁, $ {\,\mu } $₂, $ {\alpha } $₂	0.9769
Ogden model (N=3)	Eq.(8)	${\,\mu }$₁, $ {\alpha } $₁, $ {\,\mu } $₂, $ {\alpha } $₂, ${\,\mu }$₃, $ {\alpha } $₃	0.9924
A-B model	Eq.(10)	$\,\mu$, $ {\lambda } $_m	0.9891

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表 2 单轴拉伸和单轴压缩实验的拟合参数值

Table 2. Fitting parameter values of the uni-axial tensile and the uni-axial compression experiment

Experiment	C₁₀	C₀₁	C₂₀	${E}$₁	$ {\theta } $₀	$ \,\beta $	R²
Uni-axial tensile experiment^[29]	0.73	1.00	1.0×10⁻¹⁴	7.98	21.90	0.82	0.9811
Uni-axial compression experiment^[28]	−0.86	−0.10	0.30	−4.40	9.30	0.70	0.9585

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