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橡胶材料弹性的一种新的螺旋管模型

魏志刚 陈海波 罗仲龙 胡文锋

魏志刚, 陈海波, 罗仲龙, 胡文锋. 橡胶材料弹性的一种新的螺旋管模型. 力学学报, 2023, 55(2): 470-485 doi: 10.6052/0459-1879-22-435
引用本文: 魏志刚, 陈海波, 罗仲龙, 胡文锋. 橡胶材料弹性的一种新的螺旋管模型. 力学学报, 2023, 55(2): 470-485 doi: 10.6052/0459-1879-22-435
Wei Zhigang, Chen Haibo, Luo Zhonglong, Hu Wenfeng. A new helical tube model for the elasticity of rubber-like materials. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(2): 470-485 doi: 10.6052/0459-1879-22-435
Citation: Wei Zhigang, Chen Haibo, Luo Zhonglong, Hu Wenfeng. A new helical tube model for the elasticity of rubber-like materials. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(2): 470-485 doi: 10.6052/0459-1879-22-435

橡胶材料弹性的一种新的螺旋管模型

doi: 10.6052/0459-1879-22-435
基金项目: 安徽省自然科学基金(2008085MA15)和安徽省重点研发计划(202004h07020005)资助项目
详细信息
    通讯作者:

    魏志刚, 副教授, 主要研究方向为橡胶本构关系. E-mail: zhigwei@163.com

  • 中图分类号: TQ330.1

A NEW HELICAL TUBE MODEL FOR THE ELASTICITY OF RUBBER-LIKE MATERIALS

  • 摘要: 建立统计力学模型正确描述材料微观结构与宏观力学特性之间的关系是软物质类材料的最大挑战之一, 已有的橡胶材料统计模型尚存在一些不足. 文章根据橡胶类材料宏观各向同性、连续均匀和不可压缩特性, 结合分子链的非高斯统计模型, 提出一种橡胶材料网络结构的力学特性模型. 该模型将代表体元上对应点之间的传力路径用一个类螺旋管区域约束的分子链子网络来描述, 螺旋管的表面随材料的宏观变形做仿射变形, 分子链子网络由方向和长度随机的分子链或链段首尾链接而成, 在此基础上由分子链的熵推导出描述材料宏观力学特性的本构关系. 通过大量的材料测试数据对本构模型进行拟合验证, 拟合结果表明该模型具有非常好的精度, 并且在采用两个参数时模型具有非常高的可靠性, 仅用单轴拉伸实验数据拟合模型就能准确预测全部3类实验数据. 该模型使用了仿射的弯曲管假设, 能从微观结构尺度上说明材料的不可压缩特性, 避免了直管模型的近似性, 为微观尺度的随机性和宏观的均匀性的联系提出一个新的模型.

     

  • 图  1  提出的管模型的原理示意图

    Figure  1.  Schematic diagram of the principle of the proposed tube model

    图  2  橡胶网络结构的等效力学模型

    Figure  2.  The equivalent mechanical model of the network structure of rubber-like materials

    图  3  螺旋管模型

    Figure  3.  The helical tube model

    图  4  螺旋管模型的参数示意图

    Figure  4.  The parameters of the helical tube model

    图  5  本模型拟合Treloar数据[39]

    Figure  5.  Predictions of the proposed model against Treloar’s data[39]

    图  6  本模型拟合Meunier数据[40]

    Figure  6.  Predictions of the proposed model against Meunier’s data[40]

    图  7  本模型拟合Zhao数据[41]

    Figure  7.  Predictions of the proposed model against Zhao’s data[41]

    图  8  本模型拟合Kawabata双轴实验数据[38]

    Figure  8.  Predictions of the proposed model against Kawabata’s biaxial[38]

    图  9  本模型拟合James数据[42]

    Figure  9.  Predictions of the proposed model against James’ data[42]

    图  10  本模型拟合Kawabata基本实验[43]

    Figure  10.  Predictions of the proposed model against Kawabata’s basic test data[43]

    图  11  Davidson模型[44]拟合Treloar数据[39]

    Figure  11.  Predictions of the Davidson model[44] against Treloar’s data[39]

    图  14  Davidson模型[44]拟合Kawabata双轴实验数据[38]

    Figure  14.  Predictions of the Davidson model[44] against Kawabata’s biaxial test data[38]

    图  12  Davidson模型[44]拟合Meunier数据[40]

    Figure  12.  Predictions of the Davidson model[44] against Meunier’s data[40]

    图  13  Davidson模型[44]拟合Zhao数据[41]

    Figure  13.  Predictions of the Davidson model[44] against Zhao’s data[41]

    图  15  模型[45]拟合Treloar数据[39]

    Figure  15.  Predictions of the Xiang model[45] against Treloar’s data[39]

    图  16  Xiang模型[45]拟合Meunier数据[40]

    Figure  16.  Predictions of the Xiang model[45] against Meunier’s data[40]

    图  17  Xiang模型[45]拟合Zhao数据[41]

    Figure  17.  Predictions of the Xiang model[45] against Zhao’s data[41]

    图  18  Xiang模型[45]拟合Kawabata双轴实验数据[38]

    Figure  18.  Predictions of the Xiang model[45] against Kawabata’s biaxial testdata[38]

    图  19  本模型用3个参数拟合Treloar数据[39]

    Figure  19.  Predictions of the proposed model of three parameters against Treloar’s data[39]

    图  20  本模型用3个参数拟合Meunier数据[40]

    Figure  20.  Predictions of the proposed model of three parameters against Meunier’s data[40]

    图  21  本模型用3个参数拟合Zhao数据[41]的情况

    Figure  21.  Predictions of the proposed model of three parameters against Zhao’s data[41]

    图  22  本模型用3个参数拟合Kawabata双轴实验数据[38]

    Figure  22.  Predictions of the proposed model of three parameters against Kawabata’s biaxial test data[38]

    图  23  本模型用两个参数拟合Treloar数据[39]

    Figure  23.  Predictions of the proposed model of two parameters against Treloar’s data[39]

    图  24  本模型用两个参数拟合Meunier数据[40]

    Figure  24.  Predictions of the proposed model of two parameters against Meunier’s data[40]

    图  25  本模型用两个参数拟合Zhao数据[41]

    Figure  25.  Predictions of the proposed model of two parameters against Zhao’s data[41]

    图  26  本模型用两个参数拟合Kawabata双轴实验数据[38]

    Figure  26.  Predictions of the proposed model of two parameters against Kawabata’s biaxial test data[38]

    表  1  本文所提模型拟合得到的参数

    Table  1.   Parameters of the proposed model derived from data fitting

    Data source$\overset{\lower0.5 em\hbox{$\smash{\scriptscriptstyle\frown}$}}{a} $$ \overset{\lower0.5 em\hbox{$\smash{\scriptscriptstyle\frown}$}}{b} $$ \overset{\lower0.5 em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} $$\overset{\lower0.5 em\hbox{$\smash{\scriptscriptstyle\frown}$}}{d} $
    Treloar[39]0.38280.13690.74380.4442
    Meunier et al.[40]0.08890.36360.65230.4126
    Zhao[41]0.38190.12150.86640.1910
    Kawabata et al.[38]0.30500.12650.89100.3050
    James et al.[42]0.62900.11221.26550.6290
    Kawabata et al.[43]1.04990.34810.70601.0500
    下载: 导出CSV

    表  2  Davidson模型[44]拟合得到的参数

    Table  2.   The parameters of the Davidson model[44] derived from data fitting

    Data source${G_{\rm{C}}}$$\lambda _{\max }^2$${G_{\rm{e}}}$
    Treloar[39]0.24728.3600.087
    Meunier et al.[40]0.2264.1170.044
    Zhao[41]0.15723.4690.074
    Kawabata et al.[38]0.26315363.3000.100
    下载: 导出CSV

    表  3  Xiang模型[45]拟合得到的参数

    Table  3.   The parameters of the Xiang model[45] derived from data fitting

    Data source${G_{\rm{C}}}$3N${G_{\rm{e}}}$
    Treloar[39]0.273690.35430.0960
    Meunier et al.[40]0.249012.59780.0487
    Zhao[41]0.176973.33150.0808
    Kawabata et al.[38]0.289925448.99840.1098
    下载: 导出CSV

    表  4  $ \overset{\lower0.5 em\hbox{$\smash{\scriptscriptstyle\frown}$}}{d} = 0.5\overset{\lower0.5 em\hbox{$\smash{\scriptscriptstyle\frown}$}}{a} $时本模型拟合得到参数

    Table  4.   The parameters of the proposed model with$ \overset{\lower0.5 em\hbox{$\smash{\scriptscriptstyle\frown}$}}{d} = 0.5\overset{\lower0.5 em\hbox{$\smash{\scriptscriptstyle\frown}$}}{a} $ derived from data fitting

    Data source$\overset{\lower0.5 em\hbox{$\smash{\scriptscriptstyle\frown}$}}{a} $$ \overset{\lower0.5 em\hbox{$\smash{\scriptscriptstyle\frown}$}}{b} $$ \overset{\lower0.5 em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} $
    Treloar[39]0.554770.121580.76541
    Meunier et al.[40]0.215250.315090.72497
    Zhao[41]0.381890.121510.86643
    Kawabata et al.[38]0.792260.078320.66172
    下载: 导出CSV

    表  5  $ \overset{\lower0.5 em\hbox{$\smash{\scriptscriptstyle\frown}$}}{d} = 0.5\overset{\lower0.5 em\hbox{$\smash{\scriptscriptstyle\frown}$}}{a} $, $ \overset{\lower0.5 em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} = 0.8 $时本模型的拟合参数

    Table  5.   The parameters of the proposed model with $ \overset{\lower0.5 em\hbox{$\smash{\scriptscriptstyle\frown}$}}{d} = 0.5\overset{\lower0.5 em\hbox{$\smash{\scriptscriptstyle\frown}$}}{a} $ and $ \overset{\lower0.5 em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c} = 0.8 $ derived from data fitting

    Data source$\overset{\lower0.5 em\hbox{$\smash{\scriptscriptstyle\frown}$}}{a} $$ \overset{\lower0.5 em\hbox{$\smash{\scriptscriptstyle\frown}$}}{b} $
    Treloar[39]0.6220.117
    Meunier et al.[40]0.2410.296
    Zhao[41]0.4050.120
    Kawabata et al.[38]0.8730.064
    下载: 导出CSV

    表  6  各模型拟合实验数据的平均相对误差对比

    Table  6.   Comparisons of the mean relative errors of data-fitting of models

    TypeModelData
    Test[39]Test[53]Test[40]Axel testTest[41]Test[54]Test[42]Test[55]Test[43]sum
    4-para-meterproposed0.0360.0480.0240.0420.0280.0370.0160.0420.0110.284
    Khiêm et al.[29]0.0350.0520.0310.0610.0250.0390.0150.0440.010.312
    3-para-meterproposed0.0360.080.0380.060.0290.04710.0260.0430.0220.381
    Xiang et al.[45]0.0390.0720.0350.0550.0350.0440.0250.0420.0330.38
    Davidson et al.[44]0.0330.0670.0340.0530.0280.0380.020.0420.0210.336
    Dal et al.[46]0.0240.0710.0330.050.0230.0520.0180.0420.0210.334
    straight tube0.0880.0540.070.0740.1050.1180.0590.0630.0370.668
    2-para-meterproposed0.0370.0770.0390.060.0340.0650.0320.0450.0230.412
    proposed UT fitted only0.0380.0770.0480.0700.0340.0750.0320.0530.0200.448
    Anssari-Benam et al.[49]0.0870.0920.040.0740.0940.1280.0680.0470.080.71
    James et al.[15]0.3050.1050.0940.1070.140.1280.0760.0860.0761.12
    下载: 导出CSV
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  • 收稿日期:  2022-09-19
  • 录用日期:  2022-12-17
  • 网络出版日期:  2022-12-21

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