Axially Symmetric Cross-Sectional Strain and Stress Distributions in Suddenly Loaded Cylindrical Elastic Bars

Abstract

Axially symmetric cross-sectional strain and stress distributions in semi-infinite cylindrical elastic bars subjected to pressure-step and velocity-impact loading are considered on the basis of the exact equations of motion. Asymptotic solutions are obtained as functions of time, radius, and distance from the end of the bar for the strains, ezz, err, eθθ, and erz, and for the stresses, τzz, τrr, τrz, and τθθ. These solutions, which are valid asymptotically at large distances from the end of the bar, describe the head of the pulse when only the first mode is operative. Previous Airy integral solutions based on plane sections remaining plane are contained in these solutions as first-order terms; radial dependencies resulting from the warping of plane sections appear as second-order correction terms involving derivatives of the Airy function. The magnitudes of the correction terms decrease as the distance from the end of the bar increases. The pressure-step solution accurately predicts the experimental results of Miklowitz and Nisewanger for the radial surface displacement. Quantitative nonzero values are obtained for the radial, tangential, and shear stresses associated with the dispersive lateral inertia mechanism. These values are compared with the amplitude of the applied longitudinal load; for example, the maximum radial stress is about 4 percent of the applied stress for a propagation distance of 20 dia. Finally, the solutions to the pressure-step and velocity-impact problems are compared to determine the effect of markedly different end conditions. The difference, which is found to be less than 1 percent at 20 dia from the end of the bar, is discussed in terms of a dynamic Saint-Venant’s principle.