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IssuesArchive of Issues2005-2pp.58-70

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N. A. Veklich, "The propagation of elastic waves in a right-angled wedge after facet impact on a fixed plane obstacle," Mech. Solids. 40 (2), 58-70 (2005)
Year 2005 Volume 40 Number 2 Pages 58-70
Title The propagation of elastic waves in a right-angled 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 right-angled 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 half-plane with a free surface under appropriate initial conditions. The general solution of the wave problem for a half-plane with arbitrary initial conditions was given in [1].

The problem of impact of two elastic bars was first considered in [1] using the functionally-invariant 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 Saint-Venant 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. 236-266, 1933.
2.  M. A. Malkov, "Two-dimesional problem of impact of elastic bars," Doklady AN SSSR, Vol. 148, No. 4, pp. 125-140, 1979.
3.  W. R. Spillers and A. Callegari, "Impact of two elastic cylinders: short-time solution," Int. J. Mech. Sci., Vol. 11, pp. 845-851, 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. 138-145, 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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