  Mechanics of Solids A Journal of Russian Academy of Sciences   Founded
in January 1966
Issued 6 times a year
Print ISSN 00256544 Online ISSN 19347936 
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N. A. Veklich, "The propagation of elastic waves in a rightangled wedge after facet impact on a fixed plane obstacle," Mech. Solids. 40 (2), 5870 (2005) 
Year 
2005 
Volume 
40 
Number 
2 
Pages 
5870 
Title 
The propagation of elastic waves in a rightangled wedge after facet impact on a fixed plane obstacle 
Author(s) 
N. A. Veklich (Moscow) 
Abstract 
In this study, we present an exact analytical solution to the plane
problem of impact of an elastic rightangled wedge (a quarter of a plane)
on a fixed perfectly smooth obstacle. This problem is a constituent part
of a more complicated plane problem of impact of two elastic bars in the
case where the wave reflection on the lateral bar surfaces is not taken
into account. The problem under consideration can also be regarded as a
particular case of a more complicated problem of the wave motion of an
elastic halfplane with a free surface under appropriate initial
conditions. The general solution of the wave problem for a halfplane with
arbitrary initial conditions was given in [1].
The problem of impact of two elastic bars was first considered in [1]
using the functionallyinvariant solution method. In some important
features, the solution of the problem turned out to be imperfect. Critical
comments on this solution were given in [3, 4]. The analysis of the
solution [2] indicates that the inconsistency of wave patterns and some
graphs and conclusions given in the paper is due to the fact that the
boundary conditions for the tangential stresses on the lateral bar surface
are not satisfied ([2], p. 782).
In the present study, the system of motion equations written in terms of
displacements is solved for a wedge using integral transformations, which were
earlier used, in particular, in [5] to solve the problem of impact of an
acoustic band on an obstacle.
Within the linear formulation adopted, the solution obtained makes it
possible to describe quantitatively all characteristics of the propagation
of elastic waves in the wedge after the impact on an obstacle. Based on
this solution, the dynamic calculations taking into account the
propagation of elastic waves in solids can be performed. The solution can
be used as a benchmark which is necessary for the development of accurate
and reliable numerical methods for solving 2D dynamic problems of
elasticity. In addition, the solution enables one to validate
theoretically the conditions of applicability of the SaintVenant theory
for impact of elastic bars. 
References 
1.  S. Soboleff, "Sur les vibrations d'un demiplan at d`une
couche a conditions initiales arbitrares," Matematicheskii Sbornik,
Vol. 40, No. 2, pp. 236266, 1933. 
2.  M. A. Malkov, "Twodimesional problem of impact of elastic
bars," Doklady AN SSSR, Vol. 148, No. 4, pp. 125140, 1979. 
3.  W. R. Spillers and A. Callegari, "Impact of two elastic
cylinders: shorttime solution," Int. J. Mech. Sci., Vol. 11,
pp. 845851, 1969. 
4.  V. B. Poruchikov, Methods of Dynamic Elasticity Theory [in
Russian], Nauka, Moscow, 1986. 
5.  N. A. Veklich, "Impact of a band of compressible fluid on an
obstacle," Izv. AN SSSR. MZhG [Fluid Dynamics], No. 6, pp. 138145, 1990. 
6.  L. I. Sedov, Continuum Mechanics. Volume 2 [in Russian], Nauka,
Moscow, 1970. 
7.  W. Nowacki, Theory of Elasticity [in Russian], Mir, Moscow, 1975. 

Received 
11 April 2002 
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