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

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.