Accuracy analysis of Young’s modulus and stress-strain curve in the elastic stage of materials using Hopkinson bar experimental method
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摘要: 霍普金森压杆（split Hopkinson pressure bar，SHPB）实验中试件的应力不均匀对应力-应变曲线的弹性阶段有显著影响，而弹性阶段是研究混凝土等低声速材料或高应变率加载条件下某些金属材料的关键。针对一维杆系统，利用一维弹性增量波理论，推导了线性入射波作用时应力应变和杨氏模量的解析式，研究了试件两端应力差和速度差对试件弹性阶段曲线及杨氏模量准确性的影响；进一步给出了任意形状入射波作用下试件弹性阶段曲线和切线杨氏模量的求解方法，分析了入射波斜率和形状特征对试件应力均匀性及曲线的影响。结果表明：试件弹性阶段曲线及杨氏模量的准确性与试件两端应力差的变化趋势有关，但并不完全依赖试件两端应力差，与入射波斜率、形状特征以及试件屈服强度等因素耦合相关；线性加载波斜率增大，切线模量和割线模量与实际值的差异均增大，在斜率较大时，割线模量的准确性要高于切线模量；入射波形状以正弦波为参考，曲线的初始斜率低时，切线模量的准确性高于割线模量，曲线的初始斜率高时则相反。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.
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Key words:
- SHPB /
- Young’s modulus /
- stress uniformity /
- stress wave
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图 7 不同上升沿时长时应力-应变曲线的拟合模量和割线模量
Figure 7. Fitted modulus and secant modulus of stress-strain curves with different rise times
图 9 一维应力波条件下数值模拟与理论计算的应力-应变曲线
Figure 9. Stress-strain curves for numerical simulation and theoretical calculation under one-dimensional stress wave conditions
图 10 “三波法”与“二波法”计算结果
Figure 10. Calculation results of the three-wave method and two-wave method
图 12 采用“三波法”计算得到的三种模量无量纲时程曲线
Figure 12. Dimensionless time history curves for three moduli in the three-wave method
图 13 不同波长线性入射波计算结果
Figure 13. Calculation results by using different wavelength of the linear incident wave
图 14 不同波长时切线模量和无量纲应力差曲线
Figure 14. Tangent modulus and dimensionless stress difference curves at different wavelengths
图 15 不同波长时切线模量与质点速度差曲线
Figure 15. Tangent modulus and particle velocity difference curves at different wavelengths
图 17 无量纲时间分别为4和1的线性波和正弦波计算结果
Figure 17. Calculated results for linear and sinusoidal waves at dimensionless times 4 and 1
图 19 不同形状特征入射波得出的试件两端应力差
Figure 19. Stress difference at both ends of specimens resulting from incident waves of different shapes
图 20 入射波上升沿无量纲时间为4时，4种曲线的计算结果
Figure 20. Calculation results of four curves when dimensionless time of incident wave rise time is 4
图 21 入射波上升沿无量纲时间为1时，4种曲线的计算结果
Figure 21. Calculation results of four curves when dimensionless time of incident wave rise time is 1
图 22 无量纲时间为4时三种波形切线和割线杨氏模量曲线对比
Figure 22. Comparison of tangent and secant Young’s modulus curves for three waveforms at a dimensionless time of 4
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