Elastic Cylinder with Free Lateral Surface and Mixed Time-Dependent End Conditions

Abstract

A method is described for formulating an exact solution to any problem involving an elastic cylindrical bar with a free lateral surface and mixed time-dependent end conditions. To illustrate the method and to test the practicality of simulating pure end conditions by conditions given in mixed form, a solution is obtained for a particular problem involving both longitudinal and flexural strain. Asymptotic expressions valid at large distances from the end of the bar or for long times after the load is applied are developed for the integrals of the exact solution to the particular problem and predictions based on these expressions are compared with results of an experimental test in which pure end conditions were used. In this case and for large distances or long times, all of the main features of the observed behavior were correctly predicted.

Saint-Venant’s Principle in Dynamics of Structures

“…The cross section peak stress is insensitive to excitation profile only beyond 20D. Karal and Alterman [87] examined the possibly different strains in the far field of a bar due to pure and mixed shock end [80] excitations. They concluded that already beyond 2D from the excited end it is immaterial whether the data is pure or mixed (p. 10): "…the output response for distances equal or greater than two diameters from the end of the rod exhibits the same general features whether the boundary conditions are pure or mixed.…”

“…DeVault and Curtis [80] examined the relevance of asymptotic solutions obtained for mixed type end data for prediction of actual results in experiments performed with pure end conditions [47]. An example of the mixed data used therein for asymptotic evaluation is shown in Fig.…”

A hierarchy of dynamic equations for solid isotropic circular cylinders

“…The lateral boundary conditions (BCs) at r = a are obtained through (5) and (6). Consider the first relation in (5) for a prescribed tractiont r given on a circular ring at fixed z 0 .…”

“…Studies of the corresponding dispersion curves have been investigated in detail [3,4,5] where results have been presented for the modes that are of most importance to the present work: the transverse lowest flexural modes and the higher order flexural mode families. Considering semi-infinite and finite cylinders, analytical transient solutions have been developed for the flexural modes in the special case of mixed boundary conditions [6,7]. Due to the complexity of the three dimensional theory of elastodynamics in general, and dealing with other end boundary conditions in particular (e.g.…”

A Study of the Propagation of Stress Waves in Sand

“…In fact, recent theoretical studies (Ref. 4) show that the distortions of sharp-fronted pulses may be even worse than those indicated by the figure. Ihus the analysis of such results, based on plastic theory which presupposes plane sections, cannot be expected to show precise agreement.…”