考虑缠结效应的超弹性本构模型

肖锐; 向玉海; 钟旦明; 曲绍兴

doi:10.6052/0459-1879-21-008

考虑缠结效应的超弹性本构模型

doi: 10.6052/0459-1879-21-008

浙江大学工程力学系, 杭州 310027

基金项目: 1)国家自然科学基金(12022204);国家自然科学基金(11872170);国家自然科学基金(91748209)

详细信息

作者简介:
3)曲绍兴, 教授, 主要研究方向: 智能软材料和软机器. E-mail: squ@zju.edu.cn
2)肖锐, 研究员, 主要研究方向: 软材料本构关系. E-mail: rxiao@zju.edu.cn;

通讯作者:
肖锐

曲绍兴

中图分类号: O341
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出版历程
- 收稿日期: 2021-01-06
- 刊出日期: 2021-04-10

HYPERELASTIC MODEL WITH ENTANGLEMENT EFFECT

Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, China

摘要

摘要: 经典熵弹性模型, 如 Neo-Hookean模型和Arruda-Boyce八链模型, 被广泛应用于预测橡胶等软材料的超弹性力学行为. 然而, 大量实验结果也显示仅采用一套模型参数, 这类模型不能同时准确地描述橡胶在多种加载模式下的应力响应. 为了克服上述模型的不足, 本文在熵弹性的模型基础上引入缠结约束效应. 微观上, 采用Langevin统计模型来表征熵弹性变形自由能, 通过管模型(tube model)引入缠结约束自由能, 并基于仿射假设, 建立微观变形与宏观变形之间的映射关系. 在宏观上, 所建立的超弹性模型的Helmholtz自由能同时包含熵弹性和缠结约束两部分, 其中熵弹性自由能与经典的Arruda-Boyce八链模型一致, 依赖于柯西-格林应变张量的第一不变量, 而缠结约束自由能依赖于柯西-格林应变张量的第二不变量. 与文献中的实验结果对比发现, 该三参数模型能准确地预测实验中所测得的橡胶材料在单轴拉伸、纯剪切和等双轴拉伸变形条件下的应力响应, 也能较好地描述不同预拉伸比条件下双轴拉伸实验结果. 最后, 本文比较了所建立的基于应变不变量的缠结约束模型与文献中相关的缠结约束模型在多种加载模式下自由能的异同. 总的来说, 本文所建立的本构理论能准确模拟橡胶等软材料的大变形力学行为, 对其工程应用有促进作用.
- 超弹性模型 /
- 缠结效应 /
- 橡胶 /
- 软材料 /
- 管模型
Abstract: Classic hyperelastic models, such as the Neo-Hookean model and Arruda-Boyce eight-chain model, have been widely adopted to describe the mechanical response of rubbers. However, the experimental data has shown that using a single set of parameters these models have difficulty in accurately predicting the measured stress-strain relationship of rubbers under various loading modes. For example, the Arruda-Boyce model fails to describe the stress response in biaxial loading conditions of Treloar's classic experiments. To address this limitation, this work develops a hyperelastic theory incorporation the entanglement effect. At the microscale, the Langevin statistical model is adopted for the entropic part and the tube model is used for the entanglement part. The affine assumption is used to construct the micro-macro mapping. Macroscopically, the Helmholtz free energy of the model consists of both an entropic part and an entanglement part. The entropic part has the same form as the eight-chain model, depending on the first invariant of the Cauchy-Green deformation tensor. In contrast, the entanglement part is a function of the second invariant of the Cauchy-Green deformation tensor. Compared with the eight-chain model and Neo-Hookean model, the developed model with three parameters provides a greatly improved prediction on the experimentally measured stress response of rubbers in uniaxial, pure shear and equibiaxial loading conditions, as well as that of biaxial tension tests with different pre-stretch ratios. The model shows superior prediction ability compared with the classic models, such as the Neo-Hookean model, the eight-chain model, the Yeoh model and the generalized Rivlin model. Finally, the work also compares the free energy density of entanglement part developed in this work and those of the related models in the literature. The constitutive theory developed in this work can accurately predict the large deformation behaviors of rubbers and other related soft materials, which can potentially benefit their engineering applications.
- hyperelastic model /
- entanglement /
- rubber /
- soft material /
- tube model

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参考文献(36)

[1]	李晓芳, 杨晓翔. 橡胶材料的超弹性本构模型. 弹性体, 2005,15(1):50-58 (Li Xiaofang, Yang Xiaoxiang. A review of elastic constitutive model for rubber materials. China Elastomerics, 2005,15(1):50-58 (in Chinese))
[2]	Mihai LA, Budday S, Holzapfel GA, et al. A family of hyperelastic models for human brain tissue. Journal of the Mechanics and Physics of Solids, 2017,106:60-79
[3]	Holzapfel AG. Nonlinear Solid Mechanics. New York: Johns Wiley & Sons, 2000
[4]	Huang ZP. A novel constitutive formulation for rubberlike materials in thermoelasticity. Journal of Applied Mechanics, 2014,81(4):041013
[5]	张希润, 蔡力勋, 陈辉. 基于能量密度等效的超弹性压入模型与双压试验方法. 力学学报, 2020,52(3):787-796 (Zhang Xirun, Cai Lixun, Chen Hui. Hyperelastic indentation models and the dual-indentation method based on energy density equivalence. Chinese Journal of Theoretical and Applied Mechanics, 2020,52(3):787-796 (in Chinese))
[6]	Sasso M, Palmieri G, Chiappini G, et al. Characterization of hyperelastic rubber-like materials by biaxial and uniaxial stretching tests based on optical methods. Polymer Testing, 2008,27(8):995-1004
[7]	杨海波, 刘枫, 李凡珠等. 圆柱形橡胶试样压缩变形有限元分析的超弹性本构方程对比研究. 橡胶工业, 2018,10:1085-1093 (Yang Haibo, Liu Feng, Li Fanzhu, et al. Finite element analysis of compressive deformation for cylindrical rubber components based on hyperelastic constitutive models. China Rubber Industry, 2018,10:1085-1093 (in Chinese))
[8]	李雪冰, 危银涛. 一种改进的 Yeoh 超弹性材料本构模型. 工程力学, 2016,33(12):38-43 (Li Xuebing, Wei Yingtao. An improved Yeoh constitutive model for hyperelastic material. Engineering Mechanics, 2016,33(12):38-43 (in Chinese))
[9]	刘滢滢, 邢誉峰. 超弹性橡胶材料的改进 Rivlin 模型. 固体力学学报, 2012,33(4):408-414 (Liu Yingying, Xing Yufeng. An improved Rivlin model of hyperelastic rubber materials. Chinese Journal of Solid Mechanics, 2012,33(4):408-414 (in Chinese))
[10]	Lopez-Pamies O. A new I1-based hyperelastic model for rubber elastic materials. Comptes Rendus Mecanique, 2010,338(1):3-11
[11]	Mangan R, Destrade M, Saccomandi G. Strain energy function for isotropic non-linear elastic incompressible solids with linear finite strain response in shear and torsion. Extreme Mechanics Letters, 2016,9:204-206
[12]	罗文波, 谭江华. 橡胶弹性材料的一种混合本构模型. 固体力学学报, 2008,29(3):277-281 (Luo Wenbo, Tan Jianghua. A hybird hyperelastic constitutive model of rubber materials. Chinese Journal of Solid Mechanics, 2008,29(3):277-281 (in Chinese))
[13]	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))
[14]	Treloar LRG. The elasticity of a network of long-chain molecules-II. Transactions of the Faraday Society, 1943,39:241-246
[15]	Wang MC, Guth E. Statistical theory of networks of non-Gaussian flexible chains. The Journal of Chemical Physics, 1952,20(7):1144-1157
[16]	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
[17]	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(3):427-456
[18]	Tian C, Xiao R, Guo J. An experimental study on strain hardening of amorphous thermosets: effect of temperature, strain rate, and network density. Journal of Applied Mechanics, 2018,85(10):101012
[19]	Boyce MC, Arruda EM. Constitutive models of rubber elasticity: a review. Rubber Chemistry and Technology, 2000,73(3):504-523
[20]	Diani J, Le Tallec P. A fully equilibrated microsphere model with damage for rubberlike materials. Journal of the Mechanics and Physics of Solids, 2019,124:702-713
[21]	Hossain M, Amin AF, Kabir MN. Eight-chain and full-network models and their modified versions for rubber hyperelasticity: A comparative study. Journal of the Mechanical Behavior of Materials, 2015,24(1-2):11-24
[22]	Treloar LR. Stress-strain data for vulcanized rubber under various types of deformation. Rubber Chemistry and Technology, 1944,17(4):813-825
[23]	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
[24]	Klüppel M, Schramm J. A generalized tube model of rubber elasticity and stress softening of filler reinforced elastomer systems. Macromolecular Theory and Simulations, 2000,9(9):742-754
[25]	Miehe C, G?ktepe 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
[26]	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
[27]	Davidson JD, Goulbourne NC. A nonaffine network model for elastomers undergoing finite deformations. Journal of the Mechanics and Physics of Solids, 2013,61(8):1784-1797
[28]	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
[29]	Doi M, Edwards S. The Theory of Polymer Dynamics. Oxford: Oxford University Press, 1986
[30]	Kuhn W, Grün F. Beziehungen zwischen elastischen Konstanten und Dehnungsdoppelbrechung hochelastischer Stoffe. Kolloid-ZeitschrifT, 1942,101(3):248-271
[31]	Kearsley EA. Note: Strain invariants expressed as average stretches. Journal of Rheology, 1989,33(5):757-760
[32]	Zhao F. Continuum constitutive modeling for isotropic hyperelastic materials. Advances in Pure Mathematics, 2016,6(9):571-582
[33]	Thiel C, Voss J, Martin RJ, et al. Shear, pure and simple. International Journal of Non-Linear Mechanics, 2019,112:57-72
[34]	Treloar LR. The Physics of Rubber Elasticity. Oxford: Oxford University Press, 1975
[35]	Jones DF, Treloar LR. The properties of rubber in pure homogeneous strain. Journal of Physics D$:$ Applied Physics, 1975,8(11):1285-1304
[36]	Hong W, Liu Z, Suo Z. Inhomogeneous swelling of a gel in equilibrium with a solvent and mechanical load. International Journal of Solids and Structures, 2009,46(17):3282-328