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 引用本文: 杨洪升, 李玉龙, 周风华. 梯形应力脉冲在弹性杆中的传播过程和几何弥散[J]. 力学学报, 2019, 51(6): 1820-1829.
Yang Hongsheng, Li Yulong, Zhou Fenghua. THE PROPAGATION PROCESS AND THE GEOMETRIC DISPERSION OF A TRAPEZOIDAL STRESS PULSE IN AN ELASTIC ROD[J]. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(6): 1820-1829.
 Citation: Yang Hongsheng, Li Yulong, Zhou Fenghua. THE PROPAGATION PROCESS AND THE GEOMETRIC DISPERSION OF A TRAPEZOIDAL STRESS PULSE IN AN ELASTIC ROD[J]. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(6): 1820-1829.

## THE PROPAGATION PROCESS AND THE GEOMETRIC DISPERSION OF A TRAPEZOIDAL STRESS PULSE IN AN ELASTIC ROD

• 摘要: 在应力波传播过程中,几何弥散效应往往难以避免.对应力波在弹性杆中传播的几何弥散效应进行解析分析,对于基础波动问题研究以及材料动态力学行为表征等课题,显得至关重要.本文简单说明了弹性杆中考虑横向惯性修正的一维 Rayleigh-Love应力波理论,概述了其波动控制方程的变分法推导过程;针对 Hopkinson杆实验中常用的梯形应力加载脉冲,建立了相应的偏微分方程初边值问题的求解模型,并运用 Laplace变换方法研究了脉冲在杆中传播的几何弥散现象;根据留数定理进行 Laplace反变换,给出了杆中不同位置和时刻的应力波的级数形式解析解,分析了计算项数对结果收敛性的影响;将解析计算结果与采用三维有限元数值模拟的计算结果进行对比,两者吻合程度良好,从而证明 Rayleigh-Love横向惯性修正理论可以有效地表征典型 Hopkinson杆实验中的几何弥散效应.在此基础上围绕梯形加载脉冲的弥散效应进行参数研究,定量描述了传播距离、泊松比、脉冲斜率等参数的影响.本文给出的 Rayleigh-Love杆在梯形加载条件下的解析解,揭示了几何弥散效应的本质规律,可以用于实际实验的弥散修正过程.

Abstract: Geometric dispersion effects are often difficult to avoid during stress wave propagation. Analytical analysis of the geometric dispersion of stress wave propagation in elastic rods is crucial for the study of fundamental wave problems and the dynamic mechanical behavioral characterization of materials. This paper briefly describes the one-dimensional Rayleigh-Love stress wave theory considering the lateral inertia correction in the elastic rod, and summarizes the derivation process of the control equation by the variation method. Aiming at the trapezoidal stress loading pulse commonly used in Hopkinson rod experiments, the corresponding model of the initial boundary value problem (IBVP) of the partial differential equations is established. The geometric dispersion phenomenon of pulse propagation in the rod is studied by using the Laplace transform method. The inverse Laplace transform is carried out according to the residue theorem. The analytic solutions of the stress waves at different positions and times are given in the form of series representation. The influence of the number of calculation terms on the convergence of the results is analyzed. These analytical calculation results are in good agreement with the results using three-dimensional finite element numerical simulation, which proves that the Rayleigh-Love lateral inertia correction theory can effectively characterize the geometric dispersions in typical Hopkinson bar experiments. Based on the analytic solutions, the parametric study of the trapezoidal loading pulse is conducted. The influences of propagation distance, Poisson's ratio, and the pulse slope on the geometric dispersions are quantitatively described. The analytical solution of the Rayleigh-Love rod under trapezoidal pulse loading reveals the essential law of geometric dispersion effect and can be used for the dispersion correction process in the real experiments.

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