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

Abstract

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.

Displacement correction for punching at a dynamically loaded bar end

“…The elementary 1-D theory of wave propagation in bars, and also some more advanced theories that consider wave dispersion due to lateral inertia effects [1,2,3], do not tackle this problem. Rather they investigate wave propagation at large distances from the bar end [4]. The determination of the local displacement at a bar end is of practical importance for the use of the SHPB device, as suggested by some authors [5].…”

Symmetrical Taylor impact studies of copper

“…However, the VISAR record is limited in duration by the space available for the rear of the rod to move before reaching the probe. The VISAR is sensitive to dispersion effects if the rod it is observing is too short, as elastic waves have to propagate a certain distance (above 10 diameters) before approximating to a longitudinal wave (same amplitude over a cross section) (Davies & Hunter 1963;Lindholm 1964;Kennedy & Jones 1969;Gorham 1980;Safford 1988;Field et al 1994). The copper rods used were 10 diameters long.…”

Fidelity of a Finite Element Model for Longitudinal Wave Propagation in Thick Cylindrical Wave Guides

“…Shook and Curtis (1957) showed experimentally that the mixed end condition 12 solution predicted accurately the main features of the signal in a semi-infinite bar for distances larger than 20 diameters. Kennedy and Jones (1969) used the method of Folk et al (1957) to investigate the axially symmetric longitudinal response of a semi-infinite elastic bar to a pressure step end loading. Goldberg and Folk (1993) extended the method of Folk et al (1957) to allow it to be used for the pure-end-condition problem.…”