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.

References

References
1.
Skalak
,
R.
,
1957
, ”
Longitudinal Impact of a Semi-Infinite Circular Elastic Bar
,”
J. Appl. Mech.
,
24
, pp.
59
64
.
2.
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).
3.
Malkov
,
M. A.
,
1963
, “
Two-Dimension Problem of Elastic Impact of Bars
,”
Dokl. Akad. Nauk SSSR
,
148.4
, pp.
782
785
(in Russian).
4.
Spillers
,
W. R.
, and
Callegeri
,
A.
,
1969
, “
Impact of Two Elastic Cylinders: Short-Time Solution
,”
Int. J. Mech. Sci.
,
11.40
, pp.
845
851
.10.1016/0020-7403(69)90036-8
5.
Gabriel
,
D.
,
Plesec
,
J.
,
Kolman
,
R.
, and
Vales
,
F.
,
2010
, “
Dispersion of Elastic Waves in Contact-Impact Problem of a Long Cylinder
,”
J. Comput. Appl. Math.
,
234.6
, pp.
1930
1936
.10.1016/j.cam.2009.08.043
6.
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
.
7.
Miklowitz
,
J.
,
1978
,
The Theory of Elastic Waves and Waveguides
,
North-Holland Publishing Company
,
Amsterdam
.
You do not currently have access to this content.