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.Abstract: One of the biggest challenges for soft materials is to establish statistical mechanical models to correctly describe the relationship between its microstructure and macroscopic mechanical properties, and the statistical models for rubber-like materials still have some imperfections. Based on the macroscopically isotropic, continuous uniform and incompressible properties of rubber-like materials, combined with a non-Gaussian statistical model for molecular chains, a new elastic model for rubber material is proposed. The force transfer path between the corresponding points on the representative volume element is described by a subnetwork constrained to a region as a spiral helical tube, whose surfaces all deform affinely with the macroscopic deformation. The sub-network consists of molecular chains or chain segments linked end-to-end with random orientation and length. Hence, the constitutive model describing the macroscopic mechanical characteristics of the material is derived from the entropy of the subnetwork. A large number of test data were used to fit the constitutive model, which show that the model has very good accuracy. Especially, the proposed model with two parameters show very high reliability that it gives good predictions of the three basic test with the parameters derived from data-fitting with uniaxial tension data only. With the proposed curved affine tube confinement, this model can explain the incompressible properties of the material from the microstructure scale, overcome the shortcoming of straight tube model, and build a new model for the correlation between the stochastic at the micro scale and the uniform at the macro scale.
-
Key words:
- rubber elasticity /
- constitutive relation /
- network structure /
- tube model
-
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] 0.3828 0.1369 0.7438 0.4442 Meunier et al.[40] 0.0889 0.3636 0.6523 0.4126 Zhao[41] 0.3819 0.1215 0.8664 0.1910 Kawabata et al.[38] 0.3050 0.1265 0.8910 0.3050 James et al.[42] 0.6290 0.1122 1.2655 0.6290 Kawabata et al.[43] 1.0499 0.3481 0.7060 1.0500 表 4 在
$ \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 fittingData 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] 0.55477 0.12158 0.76541 Meunier et al.[40] 0.21525 0.31509 0.72497 Zhao[41] 0.38189 0.12151 0.86643 Kawabata et al.[38] 0.79226 0.07832 0.66172 表 5 在
$ \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
Type Model Data Test[39] Test[53] Test[40] Axel test Test[41] Test[54] Test[42] Test[55] Test[43] sum 4-para-meter proposed 0.036 0.048 0.024 0.042 0.028 0.037 0.016 0.042 0.011 0.284 Khiêm et al.[29] 0.035 0.052 0.031 0.061 0.025 0.039 0.015 0.044 0.01 0.312 3-para-meter proposed 0.036 0.08 0.038 0.06 0.029 0.0471 0.026 0.043 0.022 0.381 Xiang et al.[45] 0.039 0.072 0.035 0.055 0.035 0.044 0.025 0.042 0.033 0.38 Davidson et al.[44] 0.033 0.067 0.034 0.053 0.028 0.038 0.02 0.042 0.021 0.336 Dal et al.[46] 0.024 0.071 0.033 0.05 0.023 0.052 0.018 0.042 0.021 0.334 straight tube 0.088 0.054 0.07 0.074 0.105 0.118 0.059 0.063 0.037 0.668 2-para-meter proposed 0.037 0.077 0.039 0.06 0.034 0.065 0.032 0.045 0.023 0.412 proposed UT fitted only 0.038 0.077 0.048 0.070 0.034 0.075 0.032 0.053 0.020 0.448 Anssari-Benam et al.[49] 0.087 0.092 0.04 0.074 0.094 0.128 0.068 0.047 0.08 0.71 James et al.[15] 0.305 0.105 0.094 0.107 0.14 0.128 0.076 0.086 0.076 1.12 -
[1] Zhong M, Wang R, Kawamoto K, et al. Quantifying the impact of molecular defects on polymer network elasticity. Science, 2016, 353(6305): 1264 doi: 10.1126/science.aag0184 [2] Sussman T, Bathe KJ. A model of incompressible isotropic hyperelastic material behavior using spline interpolations of tension-compression test data. Communications in Numerical Methods in Engineering, 2009, 25: 53-63 doi: 10.1002/cnm.1105 [3] Treloar LRG. The elasticity of a network of long-chain molecules-I. Transactions of the Faraday Society, 1943, 39: 36-41 doi: 10.1039/tf9433900036 [4] Rivlin RS. Large elastic deformations of isotropic materials. IV. Further developments of the general theory. Philosophy of Transaction of Royal Society of Lond. Series A Mathematical Physical Science, 1948, 241(835): 379-397 [5] Ogden RW. Large deformation isotropic elasticity-on the correlation of theory and experiment for incompressible rubberlike solids. Proceeding of the Royal Society of Lond. A Mathmatical Physical & Engineering Science, 1972, 326(1567): 565-584 [6] Yeoh OH. Some forms of the strain energy function for rubber. Rubber Chemistry and Technology, 1993, 66(5): 754-771 doi: 10.5254/1.3538343 [7] Gent AN. A new constitutive relation for rubber. Rubber Chemistry and Technology, 1996, 69(1): 59-61 doi: 10.5254/1.3538357 [8] Shariff MHBM. Strain energy function for filled and unfilled rubberlike material. Rubber Chemistry and Technology, 2000, 73(1): 1-18 doi: 10.5254/1.3547576 [9] Carroll MM. A strain energy function for vulcanized rubbers. Journal of Elasticity, 2011, 103(2): 173-187 doi: 10.1007/s10659-010-9279-0 [10] Khajehsaeid H, Arghavani J, Naghdabadi R. A hyperelastic constitutive model for rubber-like materials. European Journal of Mechanics A/Solids, 2013, 38: 144-151 doi: 10.1016/j.euromechsol.2012.09.010 [11] Mansouri MR, Darijani HD. Constitutive modeling of isotropic hyperelastic materials in an exponential framework using a self-contained approach. International. Journal of Solids and Structures, 2014, 51: 4316-4326 doi: 10.1016/j.ijsolstr.2014.08.018 [12] Kuhn W, Molekülgröße BZ. statistischer molekülgestalt und elastischen eigenschaften hochpolymerer stoffe. Kolloid-Z, 1936, 76: 258-271 [13] 杨小震. 链分子的构象弹性理论. 中国科学: B辑, 2001, 31(1): 11 (Yang Xiaozhen. Conformational elastic theory of chain molecules. Science in China (Series B) , 2001, 31(1): 11 (in Chinese) [14] Kuhn W, Grün F. Beziehungen zwischen elastischen Konstanten und Dehnungsdoppelbrechung hochelastischer stoffe. Kolloid-Z, 1942, 101: 248-271 [15] James HM, Guth E. Theory of elastic properties of rubber. Journal of Chemical Physics, 1943, 11: 455-481 doi: 10.1063/1.1723785 [16] James, HM. Statistical properties of networks of flexible chains. Journal of Chemical Physics, 1947, 15: 651-668 doi: 10.1063/1.1746624 [17] Miehe C, Göktepe S. Lulei F. A micro–macro approach to rubber-like materials–part I: the non-affine micro-sphere model of rubber elasticity. Journal of the Mechanics and Physics of Solids, 2004, 52(11): 2617-2660 [18] Nishi K, Fujii K, Chung U, et al. Experimental observation of two features unexpected from the classical theories of rubber elasticity. Physical Review Letters, 2017, 119: 267801-1-5 doi: 10.1103/PhysRevLett.119.267801 [19] Gusev AA. Numerical estimates of the topological effects in the elasticity of gaussian polymer networks and their exact theoretical description. Macromolecules, 2019, 52: 3244-3251 doi: 10.1021/acs.macromol.9b00262 [20] Arruda EM, Boyce MC. A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. Journal of the Mechanics and Physics of Solids, 1993, 41(2): 389-412 doi: 10.1016/0022-5096(93)90013-6 [21] Fu B, Yang X, Li Q. A network decomposition model for rubber-like materials considering topological constraints. Acta Mechanica Solida Sinica, 2018, 31(6): 785-793 doi: 10.1007/s10338-018-0068-9 [22] Treloar LRG. The photoelastic properties of short-chain molecular networks. Transactions of the Faraday Society, 1954, 50: 881 doi: 10.1039/tf9545000881 [23] Wu PD, Van Der Giessen E. On improved network models for rubber elasticity and their applications to orientation hardening in glassy polymers. Journal of the Mechanics and Physics of Solids, 1993, 41: 427-456 doi: 10.1016/0022-5096(93)90043-F [24] Beatty MF. An average-stretch full-network model for rubber elasticity. Journal of Elasticity, 2003, 70: 65-86 doi: 10.1023/B:ELAS.0000005553.38563.91 [25] Aydogdu AB, Loos K, Johlitz M, et al. A new concept for the representative directions method: Directionalisation of first and second invariant based hyperelastic models. International Journal of Solids and Structures, 2021, 222-223: 111017 doi: 10.1016/j.ijsolstr.2021.03.004 [26] 陈晓红. 高聚物模糊随机网络统计力学. 中国科学(A辑), 1995, 25(5): 505-513 (Chen Xiaohong. Statistical mechanics of polymer fuzzy stochastic networks. Science in China (Series A) , 1995, 25(5): 505-513 (in Chinese) [27] Rubinstein M, Panyukov S. Elasticity of polymer networks. Macromolecules, 2002, 35: 6670-6686 doi: 10.1021/ma0203849 [28] Doi M, Edwards SF. The Theory of Polymer Dynamics. Oxford: Clarendon Press, 1986 [29] Khiêm VN, Itskov M. Analytical network-averaging of the tube model: Rubber elasticity. Journal of the Mechanics and Physics of Solids, 2016, 95: 254-269 doi: 10.1016/j.jmps.2016.05.030 [30] Diani J, Tallec PL. A fully equilibrated microsphere model with damage for rubberlike materials. Journal of the Mechanics and Physics of Solids, 2019, 124: 702-713 doi: 10.1016/j.jmps.2018.11.021 [31] Darabi E, Itskov M. A generalized tube model of rubber elasticity. Soft Matter, 2021, 17(6): 1675-1684 doi: 10.1039/D0SM02055A [32] Steinmann P, Hossain M, Possart G. Hyperelastic models for rubber-like materials: consistent tangent operators and suitability for Treloar’s data. Archive of Applied Mechanics, 2012, 82(9): 1183-1217 doi: 10.1007/s00419-012-0610-z [33] Ilg P, Karlin IV, Succi S. Super symmetry solution for finitely extensible dumbbell model. Europhysics Letters, 2000, 51(3): 355-360 [34] Edwards SF. The statistical mechanics of polymerized material. Proceding of Physics Society (London) , 1967, 92(1): 9 [35] Wei Z, Yang S. An elastic model for rubber-like materials based on a force-equivalent network. European Journal of Mechanics A/Solids, 2020, 84: 104078 doi: 10.1016/j.euromechsol.2020.104078 [36] Drucker DC. On the postulate of stability of material in the mechanics of continua. MeWtanika. Period. Sbornik Perevodov Invsts Srarei, 1964, 3: 115-128 [37] 高梦霓, 赵亚溥. 若干弹性力学问题解的唯一性定理. 中国科学:物理学、力学、天文学, 2020, 50(8): 084601 (Gao Mengnin, Zhao Yapu. Some uniqueness theorems of solutions for the problems of elasticity. Scientia Sinica Physica,Mechanica &Astronomica, 2020, 50(8): 084601 (in Chinese) [38] Kawabata S, Matsuda M, Tei K, et al. Experimental survey of the strain energy density function of isoprene rubber vulcanizate. Macromolecules, 1981, 14: 154-162 doi: 10.1021/ma50002a032 [39] Treloar LRG. Stress–strain data for vulcanised rubber under various types of deformation. Transactions of the Faraday Society, 1944, 40: 59-70 doi: 10.1039/tf9444000059 [40] Meunier L, Chagnon G, Favier D, et al. Mechanical experimental characterisation and numerical modelling of an unfilled silicone rubber. Polymer Test, 2008, 27(6): 765-777 doi: 10.1016/j.polymertesting.2008.05.011 [41] Zhao F. Continuum constitutive modeling for isotropic hyperelastic Materials. Advances in Pure Mathmatics, 2016, 6: 571-582 [42] James AG, Green A. Simpson GM, Strain energy functions of rubber I. characterization of gum vulcanizates. Journal of Applied Polymer Science, 1975, 19: 2033-2058 [43] Kawabata S, Kawai H. Strain energy density functions of rubber vulcanizates from biaxial extension. Advances in Polymer Science, 1977, 24: 89-124 [44] Davidson JD, Goulbourne NC. A nonaffine network model for elastomers undergoing finite deformations. Journal of the Mechanics and Physics of Solids, 2013, 61: 1784-1797 doi: 10.1016/j.jmps.2013.03.009 [45] Xiang Y, Zhong D, Wang P, et al. A general constitutive model of soft elastomers. Journal of the Mechanics and Physics of Solids, 2018, 117: 110-122 doi: 10.1016/j.jmps.2018.04.016 [46] Dal H, Gültekin O, Açıkgöz K. An extended eight-chain model for hyperelastic and fifinite viscoelastic response of rubberlike materials: Theory, experiments and numerical aspects. Journal of the Mechanics and Physics of Solids, 2020, 145: 104159 doi: 10.1016/j.jmps.2020.104159 [47] Hossain M, Steinmann P. More hyperelastic models for rubber-like materials: consistent tangent operators and comparative study. Journal of the Mechanical Behavior of Materials, 2013, 22(1-2): 27-50 doi: 10.1515/jmbm-2012-0007 [48] Xiang Y, Zhong D, Rudykh S, et al. A review of physically-based and thermodynamically-based constitutive models for soft materials. Journal of Applied Mechanics, 2020, 87: 110801 doi: 10.1115/1.4047776 [49] Anssari-Benam A, Bucchi A. A generalised neo-Hookean strain energy function for application to the finite deformation of elastomers. International Journal of Nonlinear Mechanics, 2021, 128: 103626 doi: 10.1016/j.ijnonlinmec.2020.103626 [50] He H, Zhang Q, Zhang Y, et al. A comparative study of 85 hyperelastic constitutive models for both unfilled rubber and highly filled rubber nanocomposite material. Nano Materials Science, 2022, 4(2): 64-82 [51] 魏志刚, 陈海波. 一种新的橡胶材料弹性本构模型. 力学学报, 2019, 51(2): 473-483 (Wei Zhigang, Chen Haibo. A new elastic model for rubber-like materials. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(2): 473-483 (in Chinese) doi: 10.6052/0459-1879-18-303 [52] Ogden RW, Saccomandi G, Sgura I. Fitting hyperelastic models to experimental data. Computational Mechanics, 2004, 34(6): 484-502 doi: 10.1007/s00466-004-0593-y [53] 丁智平, 杨荣华, 黄友剑等. 基于连续损伤模型橡胶弹性减振元件疲劳寿命分析. 机械工程学报, 2014, 50(10): 80-86Ding Zhiping, Yang Ronghua, Huang Youjian, et al. Fatigue life analysis of rubber vibration damper based on continuum damage model. Journal of Mechanical. Engineering, 2014, 50(10): 80-86 (in Chinese) [54] Heuillet P, Dugautier L. Modelisation du comportement hyper_elastique des caoutchoucs et elastomeres thermoplastiques, compacts on cellulaires. Genie Mecanique des Caoutchoucs et des _Elastomeres Thermoplastiques, 1997 [55] Fujikawa M, Maeda N, Yamabe J, et al. Determining stress–strain in rubber with in-plane biaxial tensile tester. Experimental Mechanics, 2014, 54: 1639-1649 doi: 10.1007/s11340-014-9942-7 -