Accuracy analysis of Young’s modulus and stress-strain curve in the elastic stage of materials using Hopkinson bar experimental method

ZHOU Xuan; WANG Botong; WU Yiding; LU Wencheng; MA Minghui; YU Yilei; GAO Guangfa

doi:10.11883/bzycj-2023-0380

Article Contents

Article Navigation > Explosion And Shock Waves > 2024 >

ZHOU Xuan, WANG Botong, WU Yiding, LU Wencheng, MA Minghui, YU Yilei, GAO Guangfa. Accuracy analysis of Young’s modulus and stress-strain curve in the elastic stage of materials using Hopkinson bar experimental method[J]. Explosion And Shock Waves. doi: 10.11883/bzycj-2023-0380

Citation:

ZHOU Xuan, WANG Botong, WU Yiding, LU Wencheng, MA Minghui, YU Yilei, GAO Guangfa. Accuracy analysis of Young’s modulus and stress-strain curve in the elastic stage of materials using Hopkinson bar experimental method[J]. Explosion And Shock Waves. doi: 10.11883/bzycj-2023-0380

Citation:

ZHOU Xuan, WANG Botong, WU Yiding, LU Wencheng, MA Minghui, YU Yilei, GAO Guangfa. Accuracy analysis of Young’s modulus and stress-strain curve in the elastic stage of materials using Hopkinson bar experimental method[J]. Explosion And Shock Waves. doi: 10.11883/bzycj-2023-0380

PDF( 3500 KB)

Accuracy analysis of Young’s modulus and stress-strain curve in the elastic stage of materials using Hopkinson bar experimental method

doi: 10.11883/bzycj-2023-0380

School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, China

Received Date: 2023-10-17
Rev Recd Date: 2024-01-24

Available Online: 2024-02-29

Abstract

Abstract

The stress-strain data obtained from split Hopkinson pressure bar (SHPB) tests include both strain rate effects and structural effects, where the structural effects result in non-uniform stress in the elastic phase of the stress-strain curve. The elastic phase is a critical focus of study for materials like concrete with low sound velocity or certain metals under high strain rate loading conditions. In this paper, we focus on one-dimensional rod systems and employ one-dimensional elastic incremental wave theory to derive analytical expressions for stress-strain curves and Young’s modulus under one-dimensional stress wave conditions with linear incident waves. We investigate the effects and mechanisms of stress difference and velocity difference at both ends of the specimen on the accuracy of stress-strain curves and Young’s modulus. Furthermore, we provide a method for determining stress-strain curves and tangent Young’s modulus during the elastic phase for arbitrary incident waveforms. We analyze the influence of the incident wave slope and shape characteristics on the stress uniformity in specimens and stress-strain curves. We establish the inherent relationship between stress uniformity and experimental stress-strain curves, and clarify the relative accuracy and applicability conditions of tangent modulus and secant modulus. The results indicate that stress uniformity is a key factor affecting the accuracy of stress-strain curves and Young’s modulus. However, the accuracy of Young’s modulus is not solely dependent on the change in stress difference at both ends of the specimen; it is also related to the factors such as the incident wave slope, shape characteristics, and the elastic segment range of the specimen. An increase in the linear wave slope leads to a greater difference between the tangent modulus and the secant modulus from the actual values. For larger slopes, the accuracy of the secant modulus is higher than that of the tangent modulus. When the incident wave shape is considered as a reference, curves with low initial slopes, such as sine waves, have higher accuracy for the tangent modulus compared to the secant modulus, whereas curves with high initial slopes show the opposite trend. For concrete specimens, we verify the influence of incident wave slope on Young’s modulus and evaluate the maximum incident wave slopes for concrete specimens to reach accurate values, which are 0.128 MPa/μs for the tangent modulus and 0.319 MPa/μs for the secant modulus.
- SHPB,
- Young’s modulus,
- stress uniformity,
- stress wave

FullText(HTML)

References(20)

References

[1]	KOLSKY H. An investigation of the mechanical properties of materials at very high rates of loading [J]. Proceedings of the Physical Society. Section B, 1949, 62(11): 676–700. DOI: 10.1088/0370-1301/62/11/302.
[2]	TAN J Q, ZHAN M, LIU S, et al. A modified Johnson-Cook model for tensile flow behaviors of 7050-T7451 aluminum alloy at high strain rates [J]. Materials Science and Engineering: A, 2015, 631: 214–219. DOI: 10.1016/j.msea.2015.02.010.
[3]	BARR A D, RIGBY S E, CLAYTON M. Correction of higher mode Pochhammer-Chree dispersion in experimental blast loading measurements [J]. International Journal of Impact Engineering, 2020, 139: 103526. DOI: 10.1016/j.ijimpeng.2020.103526.
[4]	TYAS A, WATSON A J. An investigation of frequency domain dispersion correction of pressure bar signals [J]. International Journal of Impact Engineering, 2001, 25(1): 87–101. DOI: 10.1016/S0734-743X(00)00025-7.
[5]	ZHANG D N, SHANGGUAN Q Q, XIE C J, et al. A modified Johnson-Cook model of dynamic tensile behaviors for 7075-T6 aluminum alloy [J]. Journal of Alloys and Compounds, 2015, 619: 186–194. DOI: 10.1016/j.jallcom.2014.09.002.
[6]	刘志杰, 朱志武, 谢东海, 等. 基于线性黏弹性模型的冻土动态本构关系 [J]. 西南科技大学学报, 2015, 30(4): 85–88. DOI: 10.3969/j.issn.1671-8755.2015.04.019. LIU Z J, ZHU Z W, XIE D H, et al. Dynamic constitutive relation of frozen soil based on liner viscoelastic model [J]. Journal of Southwest University of Science and Technology, 2015, 30(4): 85–88. DOI: 10.3969/j.issn.1671-8755.2015.04.019.
[7]	杜瑞锋, 裴向军, 贾俊, 等. 多次冲击下砂岩粘弹性损伤本构关系 [J]. 吉林大学学报(工学版), 2021, 51(2): 638–649. DOI: 10.13229/j.cnki.jdxbgxb20191171. DU R F, PEI X J, JIA J, et al. Viscoelastic damage constitutive relation of sandstone under multiple impact load [J]. Journal of Jilin University (Engineering and Technology Edition), 2021, 51(2): 638–649. DOI: 10.13229/j.cnki.jdxbgxb20191171.
[8]	黄锐宇, 于培师, 刘禹, 等. 聚硅氧烷硅胶的黏超弹性力学行为研究 [J]. 力学学报, 2021, 53(1): 184–193. DOI: 10.6052/0459-1879-20-287. HUANG R Y, YU P S, LIU Y, et al. Study on the visco-hyperelastic behavior of polysiloxane rubber [J]. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(1): 184–193. DOI: 10.6052/0459-1879-20-287.
[9]	周忠彬, 陈鹏万, 丁雁生. PBX代用材料非线性粘弹性本构模型研究 [J]. 兵器装备工程学报, 2021, 42(6): 276–281. DOI: 10.11809/bqzbgcxb2021.06.047. ZHOU Z B, CHEN P W, DING Y S. Study on nonlinear viscoelastic constitutive model of polymer-bonded explosive mock materials [J]. Journal of Ordnance Equipment Engineering, 2021, 42(6): 276–281. DOI: 10.11809/bqzbgcxb2021.06.047.
[10]	雷经发, 许孟, 刘涛, 等. 聚氯乙烯弹性体静动态力学性能及本构模型 [J]. 爆炸与冲击, 2020, 40(10): 103103. DOI: 10.11883/bzycj-2019-0249. LEI J F, XU M, LIU T, et al. Static/dynamic mechanical properties and a constitutive model of a polyvinyl chloride elastomer [J]. Explosion and Shock Waves, 2020, 40(10): 103103. DOI: 10.11883/bzycj-2019-0249.
[11]	毛勇建, 李玉龙, 史飞飞. 用经典Hopkinson杆测试弹性模量的初步探讨 [J]. 固体力学学报, 2009, 30(2): 170–176. DOI: 10.19636/j.cnki.cjsm42-1250/o3.2009.02.010. MAO J Y, LI Y L, SHI F F. A discussion on determining Young’s moduli by conventional split Hopkinson bar [J]. Chinese Journal of Solid Mechanics, 2009, 30(2): 170–176. DOI: 10.19636/j.cnki.cjsm42-1250/o3.2009.02.010.
[12]	YUAN P, MA Q Y, MA D D. Stress uniformity analyses on nonparallel end-surface rock specimen during loading process in SHPB tests [J]. Advances in Civil Engineering, 2018, 2018: 5406931. DOI: 10.1155/2018/5406931.
[13]	YANG L M, SHIM V P W. An analysis of stress uniformity in split Hopkinson bar test specimens [J]. International Journal of Impact Engineering, 2005, 31(2): 129–150. DOI: 10.1016/j.ijimpeng.2003.09.002.
[14]	MENG H, LI Q M. Correlation between the accuracy of a SHPB test and the stress uniformity based on numerical experiments [J]. International Journal of Impact Engineering, 2003, 28(5): 537–555. DOI: 10.1016/S0734-743X(02)00073-8.
[15]	HONG L, LI X B, LIU X L, et al. Stress uniformity process of specimens in SHPB test under different loading conditions of rectangular and half-sine input waves [J]. Transactions of Tianjin University, 2008, 14(6): 450–456. DOI: 10.1007/s12209-008-0077-8.
[16]	ZHOU Z L, LI X B, LIU A H, et al. Stress uniformity of split Hopkinson pressure bar under half-sine wave loads [J]. International Journal of Rock Mechanics and Mining Sciences, 2011, 48(4): 697–701. DOI: 10.1016/j.ijrmms.2010.09.006.
[17]	任文科, 李汶峰, 王江波, 等. 整形器对SHPB入射波形影响规律的定量研究 [J]. 北京理工大学学报, 2021, 41(9): 901–910. DOI: 10.15918/j.tbit1001-0645.2021.010. REN W K, LI W F, WANG J B, et al. Quantitative study on influence of pulse shaper on split Hopkinson pressure bar (SHPB) incident waveform [J]. Transactions of Beijing Institute of Technology, 2021, 41(9): 901–910. DOI: 10.15918/j.tbit1001-0645.2021.010.
[18]	高光发. 固体中的应力波导论 [M]. 北京: 科学出版社, 2022: 239–245. GAO G F. Introduction to stress waves in solid [M]. Beijing: Science Press, 2022: 239–245.
[19]	WANG W, YANG J, DENG G Q, et al. Theoretical analysis of stress equilibrium of linear hardening plastic specimen during SHPB tests [J]. Experimental Mechanics, 2023, 63(8): 1353–1369. DOI: 10.1007/s11340-023-00994-3.
[20]	王江波, 丁俊升, 王晓东, 等. 粗骨料粒径对混凝土动态压缩行为的影响研究 [J]. 爆炸与冲击, 2022, 42(2): 023101. DOI: 10.11883/bzycj-2021-0147. WANG J B, DING J S, WANG X D, et al. Effect of coarse aggregate size on the dynamic compression behavior of concrete [J]. Explosion and Shock Waves, 2022, 42(2): 023101. DOI: 10.11883/bzycj-2021-0147.