A double transform method is used to solve the problem of determining the elastic strain in a semi-infinite cylindrical bar with a stress free lateral surface, subject to the end conditions that the stress applied normally to the end is uniform and has a step function time dependence and that the radial displacement at the end is always zero. The exact solution appears as a sum of Fourier integrals whose integrands have the form of Pochhammer-Chree waves. These integrals cannot be evaluated in general by simple means, but asymptotic solutions have been obtained which are valid for large distances of travel. The theoretical predictions are compared with the results of experiment in a companion report.
Measurements are presented which show that the elastic strain produced in a cylindrical bar by stepfunction end-loading corresponds closely to predictions of the theory presented in Part I. In particular, even at large distances from the end of the bar, there is an observable strain moving faster than the bar velocity determined by Young's modulus. The theoretical shape for the head of the pulse conforms to experiment, and the head is followed by oscillations having the correct periods and amplitudes. Second-mode oscillations appear at the times predicted and with the expected periods and amplitudes.
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.
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