﻿ 帘线/橡胶复合材料各向异性黏-超弹性本构模型
«上一篇
 文章快速检索 高级检索

 力学学报  2016, Vol. 48 Issue (1): 140-145  DOI: 10.6052/0459-1879-15-189 0

### 引用本文 [复制中英文]

[复制中文]
Huang Xiaoshuang, Peng Xiongqi, Zhang Bichao. AN ANISOTROPIC VISCO-HYPERELASTIC CONSTITUTIVE MODEL FOR CORD-RUBBER COMPOSITES[J]. Chinese Journal of Ship Research, 2016, 48(1): 140-145. DOI: 10.6052/0459-1879-15-189.
[复制英文]

### 文章历史

2015-05-27收稿
2015-09-16录用
2015-09-28网络版发表

 图 1 典型的子午线轮胎结构[1]：单层带束层帘线方向为${\pmb a}_0$ Fig.1 Typical radial tire construction[1]：${\pmb a}_0$ represents the direction of a single belt cord

1 各向异性黏--超弹性本构模型 1.1 本构模型的一般形式

 $W\left( {C,\dot C,{a_0}} \right) = {W^{\rm{e}}}\left( {C,{a_0}} \right) + {\psi ^{\rm{v}}}\left( {C,\dot C,{a_0}} \right){\rm{ }}$ (1)

 ${W^{\rm{e}}}\left( {C,{a_0}} \right) = {W^{\rm{e}}}\left( {{I_1},{I_2},{I_3},{I_4},{I_5}} \right)$ (2)

 $\left. \begin{array}{l} {I_1} = I:C = {\rm{tr}}\left( C \right)\\ {I_2} = \frac{1}{2}\left[{{{\left( {{\rm{tr}}C} \right)}^2} - tr{{\left( C \right)}^2}} \right]\\ {I_3} = \det \left( C \right)\\ {I_4} = {A_0}:C = {a_0} \cdot C \cdot {a_0} = \lambda _{\rm{a}}^2\\ {I_5} = {A_0}:{C^2} = {a_0} \cdot {C^2} \cdot {a_0} \end{array} \right\}$ (3)

 $W\left( {{ C},\dot{ C},{ a}_0 } \right) = \psi^{\rm v}\left( {J_1 ,J_2 ,J_3 ,J_4 ,J_5 } \right)$ (4)

 $\left. \begin{array}{l} {J_1} = I:\dot C = {\rm{tr}}\left( {\dot C} \right)\\ {J_2} = \frac{1}{2}\left( {I:{{\dot C}^2}} \right) = \frac{1}{2}{\rm{tr}}\left( {{{\dot C}^2}} \right)\\ {J_3} = \det \left( {\dot C} \right)\\ {J_4} = {A_0}:\dot C = {a_0} \cdot \dot C \cdot {a_0}\\ {J_5} = {A_0}:{{\dot C}^2} = {a_0} \cdot {{\dot C}^2} \cdot {a_0} \end{array} \right\}$ (5)

.

 $\begin{array}{l} S = {S^{\rm{e}}} + {S^{\rm{v}}} = 2\left[{\frac{{\partial {W^{\rm{e}}}}}{{\partial C}} + \frac{{\partial {\psi ^{\rm{v}}}}}{{\partial \dot C}}} \right] = \\ \qquad 2\left[{\sum\limits_{i = 1}^5 {\left( {\frac{{\partial {W^{\rm{e}}}}}{{\partial {I_i}}}\frac{{\partial {I_i}}}{{\partial C}}} \right)} + \sum\limits_{j = 1}^5 {\left( {\frac{{\partial {\psi ^{\rm{v}}}}}{{\partial {J_j}}}\frac{{\partial {J_j}}}{{\partial \dot C}}} \right)} } \right] \end{array}$ (6)
1.2 超弹性应变能的解耦

 ${W^{\rm{e}}} = W_{\rm{m}}^{\rm{e}} + W_{\rm{f}}^{\rm{e}} + {W_{{\rm{shear}}}} = {V_{\rm{m}}}{W_{\rm{m}}} + {V_{\rm{f}}}{W_{\rm{f}}} + {W_{{\rm{shear}}}}$ (7)

1.2.1 基体等容变形能函数

 ${W_{\rm{m}}} = {\rm{ }}\sum\limits_{i = 1}^2 {{C_{i0}}{{\left( {{I_1} - 3} \right)}^i}} + \sum\limits_{j = 1}^2 {{D_{j0}}{{\left( {{I_2} - 3} \right)}^j}}$ (8)

1.2.2 帘线拉伸应变能函数

 ${W_{\rm{f}}} = \left\{ \begin{array}{l} \sum\limits_{i = 2}^4 {{C_i}{{\left( {{I_4} - 1} \right)}^i}} ,\;{I_4}1\\ 0,\;\;\;\;\;\;\;\;\;\;\;\;\;{I_4} \le 1\; \end{array} \right.\;$ (9)

1.2.3 帘线--橡胶剪切应变能函数

 ${W_{{\rm{shear}}}} = {W_{{\rm{shear}}}}({I_4},\chi ) = f({I_4})(a{\chi ^2} + b\chi ){\rm{ }}$ (10)

 $f({I_4}) = \exp \left[{{c_1}\left( {{I_4} - 1} \right)} \right]{\rm{ }}$ (11)

1.3 黏性应变能函数的推导

 ${\psi ^{\rm{v}}} = {\psi ^{\rm{v}}}\left( {{I_1},{J_1}} \right) = g\left( {{J_1}} \right){\rm{ }}\sum\limits_{i = 0}^2 {{\eta _i}{{\left( {{I_1} - 3} \right)}^i}} {\rm{ }}$ (12)

 $g\left( {{J_1}} \right) = \ln \left( {{c_2}{J_1} + 1} \right){\rm{ }}$ (13)

1.4 帘线/橡胶黏--超弹性本构模型

 $\begin{array}{l} {S^{\rm{e}}} = 2[\left( {W_1^{\rm{e}} + {I_1}W_2^{\rm{e}}} \right)I - W_2^{\rm{e}}C + W_4^{\rm{e}}{a_0} \otimes {a_0} + \\ \qquad W_5^{\rm{e}}\left( {{a_0} \otimes C \cdot {a_0} + {a_0} \cdot C \otimes {a_0}} \right)] \end{array}$ (14)
 ${S^{\rm{v}}} = 2{c_2}\ln \left( {{c_2}{J_1} + 1} \right)\left( {\sum\limits_{i = 0}^2 {{\eta _i}{{\left( {{I_1} - 1} \right)}^i}} } \right)$ (15)

(1) 拟合纯橡胶的准静态单轴拉伸实验数据得到橡胶的材料参数$C_{i0}$和$D_{j0}$.

(2) 拟合纯帘线的准静态单轴拉伸实验数据得到帘线的材料参数${C_i}{\rm{ }}\left( {i = 2,3,4} \right)$.

(3) 由(1)和(2)两步得到$C_i$,$C_{i0}$和$D_{i0}$后，拟合 准静态偏轴拉伸实验数据得到角剪切应变能密度函数$W_{\rm shear}$的参数$a$,$b$和$c_1$.

(4) 根据前面3步得到的材料参数，拟合高应变率拉伸条件下的实验数据得到与率相关的黏性系数$\eta _0$,$\eta _1$和$\eta _2$ 和材料参数$c_2$.

2 本构模型的参数确定 2.1 模型应用于单轴拉伸

 ${a_0} = \left[{\cos \alpha \;\;\sin \alpha \;\;0} \right]{\rm{ }}$ (16)
 图 2 单层帘线/橡胶复合材料单轴拉伸变形 Fig.2 Uniaxial tensile deformation of one family of cord reinforced composites

 $\left. \begin{array}{l} F = {\rm{dig}}\left[{{\lambda _1}\;\;{\lambda _2}\;\;{\lambda _3}} \right]\\ C = {F^{\rm{T}}} \cdot F = {\rm{dig}}\left[{\lambda _1^2\;\;\lambda _2^2\;\;\lambda _3^2} \right]\\ \dot C = {{\dot F}^{\rm{T}}} \cdot F + {F^{\rm{T}}} \cdot \dot F = {\rm{dig}}\left[{2{\lambda _1}{{\dot \lambda }_1}\;\;2{\lambda _2}{{\dot \lambda }_2}\;\;2{\lambda _3}{{\dot \lambda }_3}} \right] \end{array} \right\}$ (17)

 $\left. \begin{array}{l} {I_1} = \lambda _1^2 + \lambda _2^2 + \lambda _3^2\\ {I_2} = \lambda _1^2\lambda _2^2 + \lambda _2^2\lambda _3^2 + \lambda _1^2\lambda _3^2\\ {I_3} = \lambda _1^2\lambda _2^2\lambda _3^2\\ {I_4} = \lambda _1^2{\cos ^2}\alpha + \lambda _2^2{\sin ^2}\alpha \\ {I_5} = \lambda _1^4{\cos ^2}\alpha + \lambda _2^4{\sin ^2}\alpha \\ {J_1} = 2\left( {{\lambda _1}{{\dot \lambda }_1} + {\lambda _2}{{\dot \lambda }_2} + {\lambda _3}{{\dot \lambda }_3}} \right) \end{array} \right\}$ (18)

 $\begin{array}{l} S_{11}^{\rm{e}} = 2[W_1^{\rm{e}}I + W_2^{\rm{e}}\left( {{I_1} - \lambda _1^2} \right) + \\ \qquad \left( {W_4^{\rm{e}} + 2W_5^{\rm{e}}\lambda _1^2} \right){\cos ^2}\alpha] \end{array}$ (19)
 $S_{11}^{\rm{v}} = 2{c_2}\ln \left( {{c_2}{J_1} + 1} \right)\sum\limits_{i = 0}^2 {{\eta _i}{{\left( {{I_1} - 3} \right)}^i}}$ (20)

 $P = S \cdot {F^{\rm{T}}}$ (21)
2.2 本构模型参数拟合 2.2.1 橡胶基体材料参数

 $\left. \begin{array}{l} {C_{10}} = 7.2449{\rm{MPa}},\quad {C_{20}} = 2.1952{\rm{MPa}}\\ {D_{10}} = - 3.6129{\rm{MPa}},\quad {D_{20}} = - 1.9871{\rm{MPa}} \end{array} \right\}$ (22)
 图 3 横向单轴拉伸应力-应变曲线 Fig.3 Stress-strain curve of transverse tension
2.2.2 帘线拉伸材料参数

 ${C_2} = 12434{\rm{MPa}},\;{C_3} = 148{\rm{MPa}},\;{C_4} = 3{\rm{MPa }}$ (23)
 图 4 帘线单轴拉伸应力-应变曲线 Fig.4 Uniaxial tensile stress-strain curve of pure cord
2.2.3 剪切应变能参数

 $a = 0.2743{\rm{MPa}},\;b = 0.3882{\rm{MPa}},\;{c_1} = - 0.6601{\rm{ }}$ (24)
 图 5 应变率为0.0033s$^{-1}$和26s$^{-1}$的偏轴拉伸应力-应变曲线 Fig.5 Stress-strain curve of 45$^\circ$ off-axis tension under strain rate 0.0033s$^{-1}$ and 26s$^{-1}$
2.2.4 黏性应变能参数

 $\left. \begin{array}{l} {\eta _0} = 0.0157{\rm{MPa}},\;\;{\eta _1} = 0.0138{\rm{MPa}}\\ {\eta _2} = - 0.0038{\rm{MPa}},\;\;{c_2} = 5.2368{\rm{s}} \end{array} \right\}$ (25)
3 本构模型验证及结果讨论

 图 6 不同应变率下的偏轴拉伸应力-应变曲线 Fig.6 Stress-strain curve of 45$^\circ$ off-axis tension under different strain rates
4 结论

(1) 基于连续介质力学理论，提出了一种帘线/橡胶复合材料各向异性黏-超弹性本构模型来描述轮胎胎体层在服役条件下的大变形、非线性和高应变率力学行为.

(2) 单位体积的应变能被解耦为便于参数识别的基体等容变形能、帘线拉伸应变能、剪切应变能 和黏性应变能四部分，给出了模型参数的确定方法.

(3) 拟合出了本构模型参数，并验证了模型的准确性和有效性. 这一本构模型具有简单实用、材料参数容易确定的优点，为整体轮胎的动力学有限 元分析奠定了基础.

 1 郭国栋, 彭雄奇, 赵宁. 一种考虑剪切作用的各向异性超弹性本 构模型. 力学学报, 2013, 45(3): 451-455 (Guo Guodong, Peng Xiongqi, Zhao Ning. An anisotropic hyperelastic constitutive model with shear interaction for cord-rubber tire composites. Chinese Journal of Theoretical and Applied Mechanics, 2013, 45(3): 451-455 (in Chinese)) 2 Akasaka T, Kabe K, Sako K. Bending sti ness of a tire-belt structure with steel cords. Composites Science and Technology, 1985, 24(3):215-230 3 Clark SK. Theory of the elastic net applied to cord-rubber composites. Rubber Chemistry and Technology, 1983, 56(2): 372-389 4 A dl J, Kardos J. The Halpin-Tsai equations: A review. Polymer Engineering & Science, 1976, 16(5): 344-352 5 Mansor M, Sapuan S, Zainudin E, et al. Rigidity analysis of kenaf thermoplastic composites using Halpin-Tsai equation. In: Proc. of Conference of in Applied Mechanics and Materials, 2014: 29-33 6 Giner E, Vercher A, Marco M, et al. Estimation of the reinforcement factor for calculating E2 with the Halpin-Tsai equations using the finite element method. Composite Structures, 2015, 124: 402-408 7 Helnwein P, Liu CH, Meschke G, et al. A new 3-D finite element model for cord-reinforced rubber composites—application to analysis of automobile tires. Finite Elements in Analysis and Design,1993, 14(1): 1-16 8 Meschke G, Helnwein P. Large-strain 3D-analysis of fibrereinforced composites using rebar elements: Hyperelastic formulations for cords. Computational Mechanics, 1994, 13(4): 241-254 9 Yanjin G, Guoqun Z, Gang C. 3-dimensional non-linear FEM modeling and analysis of steady-rolling of radial tires. Journal of Reinforced Plastics and Composites, 2011, 30(3): 229-240 10 Itskov M, Aksel N. A class of orthotropic and transversely isotropic hyperelastic constitutive models based on a polyconvex strain energy function. International Journal of Solids and Structures, 2004,41(14): 3833-3848 11 Brown LW, Smith LM. A simple transversely isotropic hyperelastic constitutive model suitable for finite element analysis of fiber reinforced elastomers. Journal of Engineering Materials and Technology,2011, 133(2): 021021 12 Ciarletta P, Izzo I, Micera S, et al. Sti ening by fiber reinforcement in soft materials: A hyperelastic theory at large strains and its application. Journal of the Mechanical Behavior of Biomedical Materials, 2011, 4(7): 1359-1368 13 王友善, 王锋, 王浩. 超弹性本构模型在轮胎有限元分析中的应 用. 轮胎工业, 2009, 29(5): 277-282 (Wang Youshan, Wang Feng, Wang Hao. Application of hyperelastic constitutive models in finite element analysis on tires. Tire Industry, 2009, 29(5): 277-282 (in Chinese)) 14 Peng X, Guo Z, Moran B. An anisotropic hyperelastic constitutive model with fiber-matrix shear interaction for the human annulus fibrosus. Journal of Applied Mechanics, 2006, 73(5): 815-824 15 Brieu M, Diani J, Bhatnagar N. A new biaxial tension test fixture for uniaxial testing machine-A validation for hyperelastic behavior of rubber-like materials. Journal of Testing and Evaluation, 2007,35(4): 364 16 Aboshio A, Green S, Ye J. New constitutive model for anisotropic hyperelastic biased woven fibre reinforced composite. Plastics, Rubber and Composites, 2014, 43(7): 225-234 17 Guerin HL, Elliott DM. Quantifying the contributions of structure to annulus fibrosus mechanical function using a nonlinear, anisotropic, hyperelastic model. Journal of Orthopaedic Research, 2007, 25(4):508-516 18 Eiamnipon N, Nimdum P, Renard J, et al. Experimental investigation on high strain rate tensile behaviors of steel cord-rubber composite. Composite Structures, 2013, 99: 1-7 19 Choi SS, Kim OB. Influence of specimen directions on recovery behaviors from circular deformation of polyester cord-inserted rubber composites. Journal of Industrial and Engineering Chemistry, 2014,20(1): 202-207 20 Rao S, Daniel IM, Gdoutos EE. Mechanical properties and failure behavior of cord/rubber composites. Applied Composite Materials,2004, 11(6): 353-375 21 Zhang F, Kuang Z, Wan Z, et al. Viscoelastic constitutive model of cord-rubber composite. Journal of Reinforced Plastics and Composites,2005, 24(12): 1311-1320 22 Shim V, Yang L, Lim C, et al. A visco-hyperelastic constitutive model to characterize both tensile and compressive behavior of rubber. Journal of Applied Polymer Science, 2004, 92(1): 523-531 23 Anani Y, Alizadeh Y. Visco-hyperelastic constitutive law for modeling of foam's behavior. Materials & Design, 2011, 32(5): 2940-2948 24 Zhurov AI, Evans SL, Holt CA, et al. A nonlinear compressible transversely-Isotropic viscohyperelastic constitutive model of the periodontal ligament. In: Proc. of ASME 2008 International Mechanical Engineering Congress and Exposition, 2008: 707-719 25 Spencer AJM. Continuum Theory of the Mechanics of Fibrereinforced Composites. New York: Springer, 1984 26 Jiang Y, Wang Y, Peng X. A visco-hyperelastic constitutive model for human spine ligaments. Cell Biochemistry and Biophysics, 2015,71(2): 1147-1156 27 彭雄奇, 王宇. 腰椎椎间植骨融合有限元分析. 力学学报, 2011,43(2): 381-389 (Peng Xiongqi, Wang Yu. Finite elment analysis on lumbar interbody fusion. Chinese Journal of Theoretical and Applied Mechanics, 2013, 45(3): 451-455 (in Chinese)) 28 Yeoh O. Characterization of elastic properties of carbon-black-filled rubber vulcanizates. Rubber Chemistry and Technology, 1990,63(5): 792-805 29 Holzapfel GA, Gasser TC. A viscoelastic model for fiber-reinforced composites at finite strains: continuum basis, computational aspects and applications. Computer Methods in Applied Mechanics and Engineering,2001, 190(34): 4379-4403 30 Peng X, Guo Z, Du T, et al. A simple anisotropic hyperelastic constitutive model for textile fabrics with application to forming simulation. Composites Part B: Engineering, 2013, 52: 275-281
AN ANISOTROPIC VISCO-HYPERELASTIC CONSTITUTIVE MODEL FOR CORD-RUBBER COMPOSITES
Huang Xiaoshuang, Peng Xiongqi, Zhang Bichao
School of Materials Science and Engineering, Shanghai JiaoTong University, Shanghai 200030, China
Abstract: Based on fiber reinforced continuum mechanics theory, an anisotropic visco-hyperelastic constitutive model for cord-rubber composites was developed to characterize their highly non-linear, strongly anisotropic and strain rate dependent mechanical behaviors under high speed impact or large deformation condition. The unit-volume strain energy function for the visco-hyperelastic model was decomposed into four parts, representing the strain energy from isochoric rubber, the tensile energy from cord elongation, shearing energy from interaction between cord and rubber and viscous potential energy due to viscous characteristics, respectively, which greatly facilitated and simplified the identification of material parameters. By introducing the so called viscous potential energy that could not be neglected under particular loading conditions, the computation accuracy of the model was significantly improved. A simple approach for fitting the parameters was given. Experimental data from literature was used to identify material parameters in the constitutive for a specific cord-rubber composite. The developed model was validated by comparing numerical results with experimental uniaxial tension and bias-tension data under di erent strain rates, demonstrating that the developed constitutive model is highly suitable for characterizing the anisotropic and viscous material behaviors of cord-rubber composites under large deformation. The proposed model is simple, useful and easy for material parameter determination. It provides a theoretical foundation for dynamic finite element analysis of tire in the future.
Key words: cord-rubber composites    anisotropic    visco-hyperelastic    constitutive model    parameter determination