UNLOADING FLEXURAL STRESS WAVE IN A TIMOSHENKO BEAM AND THE SECONDARY FRACTURE OF THE BEAM
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摘要: 半无限长梁承受恒定弯矩作用后, 如果自由端的初始弯矩突然释放, 将在梁中激发出一列卸载弯曲应力波. 采用铁木辛柯梁和瑞利梁来研究突然卸载所激发出的弯曲波的传播特征. 利用拉普拉斯变换方法进行分析, 首先推导出铁木辛柯梁和瑞利梁中的卸载弯曲波的像函数解析解, 采用数值反变换方法给出了时域上波传播的响应解, 并研究了梁中各点的横向位移、弯矩和剪力随时间的变化规律. 计算结果表明: 与简化的欧拉梁不同, 旋转惯性的引入使铁木辛柯梁和瑞利梁中的弯曲波传播具有强烈的局部化效应, 特别是梁中各点经历的弯矩变化, 和其距离自由端的位置相关, 不同时刻的弯矩峰值大小不同;瑞利梁中离自由端不同距离各点的峰值弯矩先增大后降低, 最后达到一个渐近值;铁木辛柯梁中各点的峰值弯矩总体上随着时间单调增大到同一个渐近值, 该渐近值与欧拉梁中的峰值弯矩值相同, 均为1.43.切应力效应的引入进一步降低了铁木辛柯梁中卸载弯曲波的波速, 同时也使得铁木辛柯梁中弯矩峰值的最大值小于瑞利梁中的最大值. 对于脆性细长梁的纯弯曲断裂, 铁木辛柯梁可以较好地预测二次断裂的发生位置, 相应的碎片尺寸约为7倍梁横截面厚度.Abstract: When a half-infinite beam is subjected to a constant bending moment, if the initial bending moment at the free end is suddenly released, a series of unloading flexural stress waves will be excited. This paper studies the propagation characteristics of the excited flexural stress waves using Timoshenko and Rayleigh beam theories. The Laplacian transform method is used for derivation and analysis. The analytical image function solutions of the unloading flexural waves in Timoshenko and Rayleigh beams in the frequency domain are derived, the numerical inverse Laplacian transform method is used to give the quantitative solutions of wave propagation in the time domain, and the changes over time of the deflection, the shear force and the bending moment at each point in the beam are studied. The calculation results reveal that: Unlike the simple Euler-Bernoulli beam, the introduction of the rotary inertia effect leads to a strong localization effect during the propagation in both Timoshenko and Rayleigh beams. Especially the values of the bending moment at each point in the beam are different related to distance from the free end, and the peak values change over time. The peak values of the bending moment in a Rayleigh beam firstly increase with the distance from the free end, then decrease, and finally reach an asymptotic value; the peak values of the bending moment in a Timoshenko beam generally monotonously increase over time to the same asymptotic value, which is identical with the value of the peak bending moment in a Euler-Bernoulli beam, being 1.43.The introduction of the shear effect further reduces the flexural stress wave speed, and also makes the maximum value of the peak bending moment in a Timoshenko beam smaller than that in a Rayleigh beam. For studying the flexural fracture process of a brittle thin beam, the Timoshenko beam theory can better predict the location of the secondary fracture, and the corresponding fragmentation size is about 7 times beam cross section thickness.
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