Longitudinal Wave Propagation in a Circular Bar Loaded Suddenly by a Radially Distributed End Stress

Transient longitudinal wave propagation in a semi-infinite, circular, elastic bar loaded by a radially distributed pressure-step end stress is investigated on the basis of the exact equations of motion. The stress applied to the end of the bar has a radial dependence which can be continuously varied, by means of a loading parameter, from a uniform distribution to a load concentrated at the bar axis. Both analytical and numerical techniques are employed to obtain a complete description of the pulse head strain (ezz + eθθ), as a function of the nonuniformity of the loading, the radial coordinate, the distance from the bar end, and time. The analytic solution, which is valid asymptotically at large distances from the bar end, describes the first mode and shows only very small effects from even a high degree of radial nonuniformity in the applied stress. Near the bar end, solutions for (ezz + eθθ) and the axial stress τzz are obtained by direct numerical integration of the equations of motion. Good agreement between the numerical and analytic results at a propagation distance of 20 dia demonstrates the accuracy of the numerical technique. At distances less than 20 dia from the bar end, the effect of increasing the nonuniformity of the end loading is to greatly enhance the contributions of the higher modes, especially at the bar axis. With regard to a dynamic Saint Venant’s principle, differences in average dynamic stresses and strains resulting from statically equivalent but different radial end stress distributions are negligible at distances greater than 5 bar dia from the end. Differences in peak values are insignificant only at distances greater than 20 bar dia from the end.