COMPLETE CONSTITUTIVE RELATION OF HYPERELASTIC MATERIALS FOR TRELOAR’S EXPERIMENTAL DATA
-
摘要: 超弹性材料在航天航空、民用工业等多个领域已得到广泛应用, 但因具有的大变形非线性特性, 其本构行为异常复杂, 本构模型多种多样. 本文从超弹性材料的应变能函数出发, 在连续介质力学框内开展超弹性材料完全本构关系的理论与应用研究. 首先, 分析了Treloar针对某硫化橡胶超弹性材料开展的单轴拉伸、等双轴拉伸以及纯剪切3种基本变形模式实验数据的特点; 接着, 理论分析了这3种变形模式所具有的相同应力条件, 从而将3种模式的本构关系均表示为加载方向应力随伸长比的变化, 并就
$I_1^m$ 和$I_2^m$ 两类幂函数型应变能函数的本构特性进行了研究; 然后, 将实验曲线分为初始和剩余两个阶段, 对初始阶段采用neo-Hookean模型应变能函数、对剩余阶段采用非指定指数的幂函数型应变能函数, 在3种变形模式实验数据总体误差泛函最小条件下对模型参数进行识别, 最终建立了典型超弹性材料的完全本构关系. 对3种基本变形模式下的不同响应进行了重新预测, 其结果均优于文献上已发表的多个本构模型. 本文工作表明, 依据多种不同变形模式下全程变形范围的实验曲线, 可建立超弹性材料的完全本构关系, 从而为超弹性材料断裂等复杂问题的理论研究与实际应用提供支撑.Abstract: Hyperelastic material is a typical one widely used in many fields such as aerospace engineering and civil industrials. However, due to the property of nonlinear large deformation, the constitutive behavior of hyperelastic materials is extremely complex and the models are quite different in form. Starting from the strain energy function, a complete constitutive relation of hyperelastic materials is studied within the theoretical framework of continuum mechanics in this paper. Firstly, the feature is analyzed for the experimental curves under three essential deformation modes like uniaxial tension, equibiaxial tension and pure shear, which are conducted by Treloar for a vulcanized rubber hyperelastic material. Next, the same stress conditions of the three deformation modes are summarized in detail, based on which the constitutive relationship is derived in a same manner in terms of the stress and the principal stretch ratio in the loading direction for the three modes according to the hyperelastic constitutive theory. The constitutive behaviors of two typical power-law strain energy functions, namely$I_1^m$ and$I_2^m$ , are accordingly studied for the three essential modes. The experimental curves are divided into the initial regime and the remaining regime, and then the neo-Hookean model is adopted for the initial regime while the power-law functions with variable exponents are used for the remaining regime. The complete constitutive model is eventually established after the model parameters are identified by minimizing the overall error functional of the three modes. The responses are re-predicted for the three essential deformation modes, and the results agree better with the experimental than other models available in published literature. The present work indicates that a complete constitutive relation can be obtained for a hyperelastic material in light of the experimental curves with whole deformation range under multiple deformation modes, which is therefore instructive and meaningful to theoretical research and engineering application of complex practical problems such as fracture of hyperelastic materials. -
图 1 ST, ET和PS实验数据汇总
Figure 1. Collection of experimental data under ST, ET and PS states
图 3 典型m取值时
$ {W_1} = {I_1}^m $ 对3种变形模式的本构描述Figure 3. Constitutive behavior of three deformation modes described by
$ {W_1} = {I_1}^m $ for typical values of m
图 4 典型m取值时
$ {W_2} = {I_2}^m $ 所表征的三种变形模式本构特性Figure 4. Constitutive behavior of three deformation modes described by
$ {W_2} = {I_2}^m $ for typical values of m
图 6 对3种变形模式更新实验数据的预测效果
Figure 6. Predicted effect for the updated experimental data of three deformation modes
图 7 完全本构关系对3种不同模式实验曲线的预测效果
Figure 7. Predicted effect of complete constitutive relation for the three different modes
图 9 Mansouri模型对不同变形模式实验数据的预测效果
Figure 9. Predicted effect of the Mansouri model for experimental data of different deformation modes
9 Mansouri模型对不同变形模式实验数据的预测效果(续)
9. Predicted effect of the Mansouri model for experimental data of different deformation modes (continued)
表 1 单轴拉伸变形模式的实验数据
Table 1. Experimental data of the ST deformation mode
No. Stretch
ratio $\lambda $Engineering
stress T/MPaNo. Stretch
ratio $\lambda $Engineering
stress T/MPa1 1.02 0.0255 13 5.36 1.9502 2 1.12 0.1343 14 5.75 2.3128 3 1.24 0.2254 15 6.15 2.6852 4 1.39 0.3165 16 6.4 3.0380 5 1.58 0.4077 17 6.6 3.4104 6 1.90 0.4998 18 6.85 3.7730 7 2.18 0.5880 19 7.05 4.1258 8 2.42 0.6762 20 7.15 4.4884 9 3.02 0.8624 21 7.25 4.8608 10 3.57 1.0486 22 7.4 5.2234 11 4.03 1.2250 23 7.5 5.5860 12 4.76 1.5876 24 7.6 6.3112 
下载: 导出CSV
表 3 纯剪切变形模式的实验数据
Table 3. Experimental data of the PS deformation mode
No. Stretch
ratio
$\lambda $Engineering
stress T/MPaNo. Stretch
ratio
$\lambda $Engineering
stress T/MPa1 1.00 0.00 8 2.40 0.76 2 1.06 0.07 9 2.98 0.93 3 1.14 0.16 10 3.48 1.11 4 1.21 0.24 11 3.96 1.28 5 1.32 0.33 12 4.36 1.46 6 1.46 0.42 13 4.69 1.62 7 1.87 0.59 14 4.96 1.79
下载: 导出CSV
表 2 等双轴拉伸变形模式的实验数据
Table 2. Experimental data of the ET deformation mode
No. Stretch
ratio
$\lambda $Engineering
stress T/MPaNo. Stretch
ratio
$\lambda $Engineering
stress T/MPa1 1.00 0.00 10 1.94 0.77 2 1.04 0.09 11 2.49 0.96 3 1.08 0.16 12 3.03 1.24 4 1.12 0.24 13 3.43 1.45 5 1.14 0.26 14 3.75 1.72 6 1.20 0.33 15 4.03 1.96 7 1.31 0.44 16 4.26 2.22 8 1.42 0.51 17 4.44 2.43 9 1.69 0.65
下载: 导出CSV
表 4 初始阶段数据及其对应变形模式
Table 4. Data in the initial regime and the associated deformation modes
No. Deformation mode Stretch ratio $\lambda $ Engineering stress T/MPa 1 ST 1.02 0.0255 2 ET 1.04 0.09 3 PS 1.06 0.07 4 ET 1.08 0.16 5 ST 1.12 0.1343 6 ET 1.12 0.24
下载: 导出CSV
表 5 不同本构模型的预测精度比较
Table 5. Comparison of prediction accuracy of different constitutive models
Strain energy function Overall error functional ST ET PS Overall Eq. (19)[26] 0.2136 0.02925 0.01810 0.2609 Eq. (17) 0.1371 0.01824 0.007811 0.1632 Eq. (22) 0.1488 0.01351 0.007124 0.1694
下载: 导出CSV
1 单轴拉伸实验数据
1. Experimental data under the ST state
No. Stretch ratio$\lambda $ Engineering stress T/MPa No. Stretch ratio$\lambda $ Engineering stress T/MPa 1 1.052 0.074 11 2.299 0.688 2 1.116 0.134 12 2.502 0.749 3 1.191 0.198 13 2.685 0.809 4 1.267 0.259 14 2.873 0.870 5 1.363 0.319 15 3.052 0.933 6 1.474 0.383 16 3.223 0.994 7 1.610 0.443 17 3.382 1.058 8 1.765 0.504 18 3.522 1.115 9 1.936 0.564 19 3.677 1.178 10 2.116 0.624
下载: 导出CSV
2 纯剪切实验数据
2. Experimental under the PS state
No. Stretch ratio$\lambda $ Engineering stress T/MPa No. Stretch ratio$\lambda $ Engineering stress T/MPa 1 1.047 0.068 12 1.996 1.147 2 1.097 0.114 13 2.098 1.291 3 1.149 0.152 14 2.197 1.443 4 1.199 0.205 15 2.297 1.618 5 1.298 0.304 16 2.396 1.777 6 1.397 0.418 17 2.495 1.952 7 1.497 0.524 18 2.595 2.127 8 1.596 0.638 19 2.697 2.301 9 1.698 0.752 20 2.796 2.484 10 1.797 0.873 21 2.846 2.582 11 1.897 1.010
下载: 导出CSV
-
[1] Nkenfack AN, Beda T, Feng ZQ, et al. HIA: A hybrid integral approach to model incompressible isotropic hyperelastic materials—Part 1: Theory. International Journal of Non-Linear Mechanics, 2016, 84: 1-11 doi: 10.1016/j.ijnonlinmec.2016.04.005 [2] Lu Y, Gu ZX, Yin ZN, et al. New compressible hyper-elastic models for rubberlike materials. Acta Mechanica, 2015, 226(12): 4059-4072 doi: 10.1007/s00707-015-1475-3 [3] 郭辉, 胡文军, 陶俊林. 泡沫橡胶材料的超弹性本构模型. 计算力学, 2013, 30(4): 575-579 (Guo Hui, Hu Wenjun, Tao Junlin. The superelasticty constitutive model for foam rubber materials. Chinese Journal of Computational Mechanics, 2013, 30(4): 575-579 (in Chinese) [4] Boyce MC, Arruda EM. Constitutive models of rubber elasticity: A review. Rubber Chemistry and Technology, 2000, 73(3): 504-523 doi: 10.5254/1.3547602 [5] Dal H, Acikgoz K, Badienia Y. On the performance of isotropic hyperelastic constitutive models for rubber-like materials: A state of the art review. Applied Mechanics Reviews, 2021, 73(2): 020802 doi: 10.1115/1.4050978 [6] Steinmann P, Hossain M, Possart G. Hyperelastic materials behavior modeling using consistent strain energy density functions. Archive of Applied Mechanics, 2012, 82(9): 1183-1217 doi: 10.1007/s00419-012-0610-z [7] Bazkiaei AK, Shirazi KH, Shishesaz M. A framework for model base hyperelastic material simulation. Journal of Rubber Research, 2020, 23(4): 287-299 doi: 10.1007/s42464-020-00057-5 [8] Beda T. Reconciling the fundamental phenomenological expression of the strain energy of rubber with established experimental facts. Journal of Polymer Science Part B: Polymer Physics, 2005, 43(2): 125-134 doi: 10.1002/polb.20308 [9] Nunes LCS. Mechanical characterization of hyperelastic polydimethylsiloxane by simple shear test. Materials Science and Engineering A, 2011, 528(3): 1799-1804 doi: 10.1016/j.msea.2010.11.025 [10] 彭向峰, 李录贤. 超弹性材料本构关系的最新研究进展. 力学学报, 2020, 52(5): 1221-1234 (Peng Xiangfeng, Li Luxian. State of the art of constitutive relations of hyperelastic materials. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(5): 1221-1234 (in Chinese) doi: 10.6052/0459-1879-20-189 [11] Khodadadi A, Liaghat G, Ahmadi H, et al. Numerical and experimental study of impact on hyperelastic rubber panels. Iranian Polymer Journal, 2019, 28(2): 113-122 doi: 10.1007/s13726-018-0682-x [12] Shahzad M, Kamran A, Siddiqui MZ, et al. Mechanical characterization and FE modelling of a hyperelastic material. Materials Research, 2015, 18(5): 918-924 doi: 10.1590/1516-1439.320414 [13] Destrade M, Saccomandi G, Sgura I. Methodical fitting for mathematical models of rubber-like materials. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2017, 473(2198): 20160811 doi: 10.1098/rspa.2016.0811 [14] Han L, Peng XF, Li LX. Entire-region constitutive relation for Treloar’s data. Rubber Chemistry and Technology, 2022, 95(1): 119-127 doi: 10.5254/rct.21.78993 [15] 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 [16] 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 [17] 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 [18] Marckmann G, Verron E. Comparison of hyperelastic models for rubber-like materials. Rubber Chemistry and Technology, 2006, 79(5): 835-858 doi: 10.5254/1.3547969 [19] Bien-Aimé LKM, Blaise BB, Beda T. Characterization of hyperelastic deformation behavior of rubber-like materials. SN Applied Sciences, 2020, 2(4): 1-10 [20] Blaise BB, Bien-Aimé LKM, Betchewe G, et al. A phenomenological expression of strain energy in large elastic deformations of isotropic materials. Iranian Polymer Journal, 2020, 29(6): 525-533 doi: 10.1007/s13726-020-00816-6 [21] Darijani H, Naghdabadi R. Hyperelastic materials behavior modeling using consistent strain energy density functions. Acta Mechanica, 2010, 213(3): 235-254 [22] Mansouri MR, Darijani H. Constitutive modeling of isotropic hyperelastic materials in an exponential framework using a self-contained approach. International Journal of Solids and Structures, 2014, 51(25-26): 4316-4326 doi: 10.1016/j.ijsolstr.2014.08.018 [23] Doğan Kİ. A hyperelastic constitutive model for rubber-like materials. Archive of Applied Mechanics, 2020, 90(3): 615-622 doi: 10.1007/s00419-019-01629-7 [24] Bahreman M, Darijani H, Fooladi M. Constitutive modeling of isotropic hyperelastic materials using proposed phenomenological models in terms of strain invariants. Polymer Engineering and Science, 2016, 56(3): 299-308 doi: 10.1002/pen.24255 [25] 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 Yintao. An improved Yeoh constitutive model for hyperelastic material. Engineering Mechanics, 2016, 33(12): 38-43 (in Chinese) doi: 10.6052/j.issn.1000-4750.2015.05.0388 [26] Carroll MM. A strain energy function for vulcanized rubbers. Journal of Elasticity, 2011, 103(2): 173-187 doi: 10.1007/s10659-010-9279-0 [27] 徐中明, 袁泉, 张志飞等. 基于超静定方程的橡胶材料本构模型参数识别. 重庆大学学报, 2017, 40(2): 1-9 (Xu Zhongming, Yuan Quan, Zhang Zhifei, et al. Parameter identification of constitutive models of rubber material based on hyperstatic equations. Journal of Chongqing University, 2017, 40(2): 1-9 (in Chinese) doi: 10.11835/j.issn.1000-582X.2017.02.001 [28] 肖锐, 向玉海, 钟旦明等. 考虑缠结效应的超弹性本构模型. 力学学报, 2021, 53(4): 1028-1037 (Xiao Rui, Xiang Yuhai, Zhong Danming, et al. Hyperelastic model with entanglement effect. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(4): 1028-1037 (in Chinese) doi: 10.6052/0459-1879-21-008 [29] Rivlin RS. Large elastic deformations of isotropic materials IV. Further developments of the general theory. Philosophical Transactions of the Royal Society A - Mathematical Physical and Engineering Sciences, 1948, 241(835): 379-397 [30] 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 [31] Jones DF, Treloar LRG. The properties of rubber in pure homogeneous strain. Journal of Physics D: Applied Physics, 1975, 8(11): 1285-1304 doi: 10.1088/0022-3727/8/11/007 -
下载:
点击查看大图
计量
- 文章访问数: 286
- HTML全文浏览量: 95
- PDF下载量: 63
- 被引次数: 0
