曲线保持了较好的一致性，验证了超弹性模型和参数以及仿真过程的有效性.在此基础上，进一步仿真分析小形变单轴拉伸以及不同试件尺寸和加载边界条件下的单轴压缩试验，仿真与实测的应力-应变关系比较分析详见图5～7.由图5可知，仿真结果与单轴拉伸和压缩试验结果保持一致，由图6和图7可知，该方法还可以准确预测压缩试验对试件尺寸和边界条件的依赖性.对于边界条件②和③，试件上、下表面横向约束相对严格，此时的轴向承载过程元件内部产生两种响应应力：与简单压缩形变ε对应的均匀压缩应力σ，以及与接触表面剪切应变
c
对应的剪切应力σ.产生剪切应变是由于接触表面的约束作用使接触表面各质点
s
始终具有由变形后位置向初始位置恢复的倾向导致的，从而使表征压缩模量大于材料弹性模量.当横向约束力较大时，橡胶元件最大剪切应变接近于最大压缩应变；而当横向摩擦被消除时，最大剪切应变远远小于最大压缩应变，可忽略不计，试件被视为发生了纯压缩形变.综上所述，大形变力学试验结合超弹性理论和仿真分析，可有效获取橡胶材料的弹性特征参数，也可合理预测大形变状态下的非线性特性以及压缩特性易于受到元件尺寸和实际加载条件影响的特性.
a 单轴拉伸
b 单轴压缩
c 等轴拉伸 图8 仿真示意图 Fig.8 Simulation process
4 结论
橡胶材料弹性特征参数的掌握对于工程弹性体的设计和优化至关重要，根据弹性体的实际工程应用，确定其特征参数的试验手段和理论方法也有所不同.本文采用试验与数值分析相结合的方法，对一种橡胶材料进行了大、小形变试验及理论分析.试验及数值计算结果表明：
(1) 基于超弹性理论的单轴拉伸、单轴压缩和平面拉伸试验和分析结果，可相对完整地定义材料的回弹特性，定义初始剪切模量、杨氏模量、体积压缩模量和表征压缩模量，并描述大形变非线性特征.
(2) 小形变试验结合线性理论与大形变试验所确定的杨氏模量和剪切模量保持一致.
(3) 通过小形变力学试验及线弹性理论确定某橡胶元件力学特征的方法具有一定局限性，不仅无法获取大形变状态下的模量硬化现象，也无法准确预测试
件尺寸和边界条件等对压缩应力-应变关系的影响规律，从而预测杨氏模量与表征压缩模量的理论关系.
综上所述，当橡胶元件以拉伸或剪切形变为主且变形量属于线性形变范围时，小形变力学试验结合线性理论即可获取材料弹性特征参数；然而当弹性体形变量较大，或者当弹性体发生以压缩为主的复杂形变时，采用大形变力学试验结合非线性模型和仿真分析才是完成弹性体测试-设计-优化的最佳方法. 参考文献：
[1] Gent A N.橡胶工程——如何设计橡胶配件[M].张立群,等译. 北京：化学工业出版社，2001.
Gent A N. Engineering with rubber-how to design rubber components [M]. Translated by ZHANG Liqun, et al. Beijing: Chemical Industry Press, 2001.
[2] Charlton D J, Yang J. A review of methods to characterize rubber elastic behavior for use in finite element analysis [J]. Rubber Chemistry and Technology, 1994, 67(3): 481.
[3] Miller Kurt. Testing elastomers for hyperelastic material models in finite element analysis [R]. Ann Arbor: Axel Products, Inc, 2000.
[4] Horton J M, Tupholme G E, Gover M J C. Axial loading o bond rubber blocks [J]. Journal of Applied Mechanics, 2002, 69(6), 836.
[5] Gent A N, Lindley P B. The compression of bonded rubber blocks [C]∥Proceedings of the Institution of Mechanical Engineers. London: Institution of Mechanical Engineers, 1959: 111-117.
[6] 危银涛,张文飞.橡胶悬置材料与结构特性实验[R]. 北京: 清华大学, 2010.
WEI Yintao, ZHANG Wenfei. Rubber mount material and structural characteristics experiment investigation on the material and structural characteristics of automotive suspension [R]. Beijing: Tsinghua University, 2010.
[7] Williams J G, Gamonpilas C. Using the simple compression test to determine young’s modulus, Poisson’s ratio and coulomb friction coefficient [J]. International Journal of Solis and Structures, 2008, 45(16), 4448.
[8] Day J R, Miller K A. Equibiaxial stretching of elastomeric sheets, an analytical verification of experimental technique [C]∥Abaqus 2000 User’s Conference Proceedings. Newport: Sorensen, Inc, 2000: 205-220.
[9] 胡小玲. 炭黑填充橡胶黏超弹性力学行为的宏细观研究[D]. 湘潭：湘潭大学, 2013.
HU Xiaoling. Micro- and macro- viscohyperelastic behavior of carbon black filled rubbers [D]. Xiangtan: Xiangtan University, 2013.
[10] Axel Products, Inc. Nonlinear finite element analysis of elastomers[DB/OL]. com/pages/loads.html.
[11] Axel Products, Inc. Where do the “pure” and “shear” come from
in
the
pure
shear
test?
[DB/OL].
[2015-06-17].
[2015-06-17].
http://www.axelproducts.
http://www.axelproducts.com/pages/loads.html.
[12] Sang J L. Active, polymer-based composite material implementing simple shear [D]. [S.l.]: Texas A & M University, 2008.
[13] 3DS SIMULIA. Abaqus analysis user’s manual. Volume III: materials,
Version
6.12
[DB/OL].
[2015-01-17].
http://www.material.physik.uni-goettingen.de/abaqus/v6.12/index.html.
Determination of Resilient Coefficients of Engineering Rubber
Material
LIU Yan, ZHANG Jimin, LUO Yanyun, LI Qiutong
1,2
1,2
2
2
(1. School of Mechanical Engineering, Tongji University, Shanghai 201804; China; 2 Institute of Rail Transit, Tongji University, Shanghai
201804, China)
Abstract：Determination of resilient coefficients of rubber material is essential to the design and optimization of elastomers in most engineering fields. Based on the nonlinear elasticity theory, a set of experiment including uniaxial tension, uniaxial compression and planar tension tests under large deformation was conducted and coefficients of a hyperelastic model were fitted. The testing data and corresponding simulation results can be used, on the one hand, to integrallty define the initial shear modulus, Young’s modulus, bulk modulus and apparent compressive modulus of present material, and on the other hand, to accurately predict the material nonlinearity and strong dependence of compressive modulus under different test conditions. Then a set of material mechanics experiment within small deformations was performed and the corresponding linear modulus were calculated. Finally, several conclusions about small and large material mechanics experiment as well as linear and nonlinear elastic theory were proposed through comparative analysis.
Key words：engineering rubber material; shear modulus; Young’s modulus; bulk modulus; apparent compressive modulus; finite element analysis