Research Papers

Longitudinal Impact of Semi-Infinite Elastic Bars: Plane Short Time Solution

[+] Author and Article Information

Russian Research Center
for Artificial Intelligence,
Moscow 111141, Russia
e-mail: mamalkov@gmail.com

Manuscript received December 6, 2011; final manuscript received July 7, 2012; accepted manuscript posted July 25, 2012; published online November 19, 2012. Assoc. Editor: Martin Ostoja-Starzewski.

J. Appl. Mech 80(1), 011023 (Nov 19, 2012) (5 pages) Paper No: JAM-11-1462; doi: 10.1115/1.4007227 History: Received December 06, 2011; Revised July 07, 2012; Accepted July 25, 2012

Using the Sobolev-Smirnov method we find the exact analytical solution of the longitudinal impact of semi-infinite plane elastic bars. The solution allows us to investigate wave propagations shortly after the impact. The obtained results allow us to compare the exact and approximate solutions of impact problems. We show that approximate solutions are wrong in a short time. According to the exact solution five waves are created in the first time after impact: one longitudinal loading wave and two unloading longitudinal waves from the lateral surfaces of the bars and two unloading transverse waves, also from the surfaces. We find singular points of the solution. It can be used for other applications such as any impact of two elastic bodies since the same waves arise and the same singular points exist at the impact. One of the singular points moves along surfaces with the speed of Rayleigh waves. At this point the bar surface is perpendicular to the axis of the bars. Such a point arises at any impact. As the earth surface is also perpendicular to the direction of propagation of seismic waves, Rayleigh waves are especially destructive at earthquakes.

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Skalak, R., 1957, ”Longitudinal Impact of a Semi-Infinite Circular Elastic Bar,” J. Appl. Mech., 24, pp. 59–64.
Smirnov, V. I., and Sobolev, S. L., 1932, “New Method of Solution for Plane Problem of Elastic Vibration,” Trudy Seismologic Institute, 20, pp. 1–31 (in Russian).
Malkov, M. A., 1963, “Two-Dimension Problem of Elastic Impact of Bars,” Dokl. Akad. Nauk SSSR, 148.4, pp. 782–785 (in Russian).
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Malkov, M. A., 1966, “Asymptotic of Source Function of Elastic Perturbations in Plane Layer,” International Congress of Mathematicians, abstracts of brief scientific communications, Moscow, August 16–26, 12, p. 15.
Miklowitz, J., 1978, The Theory of Elastic Waves and Waveguides, North-Holland Publishing Company, Amsterdam.


Grahic Jump Location
Fig. 2

Discontinuity lines near lateral surface of bars

Grahic Jump Location
Fig. 1

Breakup and propagation of discontinuity lines after bar collision



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