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Solutions to Time-Dependent Pure-End-Condition Problems of Elasticity Pressure-Step Wave Propagation and End-Resonance Effects

Irwin S. Goldberg and Robert T. Folk
SIAM Journal on Applied Mathematics
Vol. 53, No. 5 (Oct., 1993), pp. 1264-1292
Stable URL: http://www.jstor.org/stable/2102154
Page Count: 29
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Solutions to Time-Dependent Pure-End-Condition Problems of Elasticity Pressure-Step Wave Propagation and End-Resonance Effects
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Abstract

A double-transform method is developed for finding exact solutions to axisymmetric nonmixed time-dependent problems in elasticity in which the applied stresses are specified on the end of semi-infinite circularly cylindrical bars with stress-free lateral surfaces. The response of the bar to the sudden application of pressure to the end is determined. In another application, the method is used to calculate the reflection of a continuous train of waves off a free end of a semi-infinite cylindrical bar. For the reflection problem, an end resonance is found when the incident waves have an angular frequency in the neighborhood of ω = 1.51 Vd/a, for a rod with Poisson's ratio of 1/3, where a is the radius of the bar and Vd is the dilatation wave velocity. The neighborhood of the end-resonance frequency determined for the free-end reflection problem is shown to correspond to a range of frequencies of oscillation at which large deviations exist between the solution to the pure-end-condition problem and the solution to the comparative mixed-end-condition problem describing the pressure-step response.

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