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IssuesArchive of Issues2002-4pp.31-36

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E. E. Pavlovskaya and Yu. V. Petrov, "On some specific features of solutions of dynamic problems of elasticity," Mech. Solids. 37 (4), 31-36 (2002)
Year 2002 Volume 37 Number 4 Pages 31-36
Title On some specific features of solutions of dynamic problems of elasticity
Author(s) E. E. Pavlovskaya (St. Petersburg)
Yu. V. Petrov (St. Petersburg)
Abstract Three problems of linear elasticity are considered to illustrate the importance of essential dynamic features- a specific behavior of the energy, the inertia, and time factor. The solutions of these problems (obtained in closed form) are analyzed. Some paradoxes of the behavior of dynamical systems appearing when treating the results obtained in quasistatic terms are indicated and explained.

We present an exact closed form solution for the problem of two-sided tension of a one-dimensional elastic rod by constant stresses. On the basis of this solution, we find the time history of the rod total mechanical energy and show that in the dynamic case, this energy essentially depends on time and periodically vanishes.

In the problem of dynamic cyclic loading of an elastic half-space with a semi-infinite crack, we constructed an analytical expression for the time history of the stress intensity factor at the crack tip. It is shown that the ideally elastic model allows for the fatigue effect that manifests itself in the fact that the failure determined by a prescribed critical value of the stress intensity factor does not occur at the first cycle of the loading but can occur at some Nth cycle.

For the problem of dynamic loading of a system of remote rigid punches in an ideally elastic half-space, we present model equations governing the dynamics of these punches with allowance for their interaction with the half-space and with one another. This model is shown to cover dissipative phenomena the existence of which in a conservative system looks paradoxically.

The dynamic processes of loading and failure of solids have a number of specific features that make these processes essentially different from the quasi-static analogues. In some cases, the stress-strain state of elastic media subjected to a dynamic loading demonstrates a paradoxical behavior that leads to fine effects that look somewhat unusually when interpreted in terms of classical principles. Such effects can be explained by taking into account the dynamic features, such as a specific behavior of the mechanical energy, the inertia, and time factor.
References
1.  O. P. Barsukov, L. A. Vaisberg, B. V. Semkin, and V. A. Tsukerman, "On the energy relations for an impact action on a rod," Izv. Vysh. Uchebn. Zavedenii, Fizika, No. 7, pp. 96-101, 1972.
2.  G. P. Cherepanov, Mechanics of Brittle Fracture [in Russian], Nauka, Moscow, 1974.
3.  P. A. Martynyuk, "On the diffraction of a plane wave by a penny-shaped crack," Dinamika Sploshnoi Sredy, No. 25, pp. 82-91, 1976.
4.  P. G. Richards, "Elementary solutions to Lamb's problem for a point source and their relevance to three-dimensional studies of spontaneous crack propagation," Bull. Seism. Soc. Amer., Vol. 69, No. 4, pp. 947-956, 1979.
5.  N. A. Lavrov and E. E. Pavlovskaya, "Dynamics of a system of remote punches on an elastic half-space," Zhurnal Prikl. Mekhaniki i Tekhn. Fiziki, No. 6, pp. 204-210, 1999.
6.  W. Nowacky, Elasticity [Russian translation], Mir, Moscow, 1975.
7.  L. A. Galin, Contact Problems of Elasticity and Viscoelasticity [in Russian], Nauka, Moscow, 1980.
Received 25 January 2000
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