UNLOADING FLEXURAL STRESS WAVE IN A TIMOSHENKO BEAM AND THE SECONDARY FRACTURE OF THE BEAM

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.