Mechanics of Solids (about journal) Mechanics of Solids
A Journal of Russian Academy of Sciences
in January 1966
Issued 6 times a year
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IssuesArchive of Issues2002-6pp.48-56

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M. I. Chebakov, "3D contact problem with friction for a layer," Mech. Solids. 37 (6), 48-56 (2002)
Year 2002 Volume 37 Number 6 Pages 48-56
Title 3D contact problem with friction for a layer
Author(s) M. I. Chebakov (Rostov-on-Don)
Abstract A 3D contact problem of elasticity is considered to study the action of a punch of an arbitrary shape on the surface of a layer of thickness h. The contact region is not prescribed beforehand but depends on the magnitudes of the normal force, P, and the tangential force, T, acting on the punch. Coulomb friction forces are assumed to act between the punch and the layer, these forces being collinear to the tangential force T. The punch does not rotate during the interaction. The surface acted upon by the punch is stress-free outside the punch, while the other surface of the layer is subject to zero displacement conditions. We consider the limiting equilibrium case. The case of the quasi-static motion of the punch along the surface of the layer in the moving reference frame can be considered in a similar manner.

An integral equation of this contact problem is obtained. To solve this equation, we utilize the nonlinear boundary integral equation method [1, 2]. The influence of the Coulomb friction coefficient, the shape of the punch, the elastic constants, and the layer thickness on the magnitude of contact stresses, the dependence of the vertical displacement of the punch on the impressing force, the contact region size and shape, and the displacement of points of the layer surface outside the contact region is analyzed.

Plane problems [3, 4], 3D problems for a half-space [5, 6], and 3D problems for a wedge [7, 8], with friction force being taken into account, have been considered previously by other authors.
1.  B. A. Galanov, "Hammerstein-type boundary equation method for contact problems of elasticity in the case of unknown contact regions," PMM [Applied Mathematics and Mechanics], Vol. 49, No. 5, pp. 827-835, 1985.
2.  B. A. Galanov, "Nonlinear boundary equations for contact problems of elasticity," Doklady AN SSSR, Vol. 296, No. 4, pp. 812-815, 1987.
3.  L. A. Galin, Contact Problems of Elasticity and Viscoelasticity [in Russian], Nauka, Moscow, 1980.
4.  V. M. Alexandrov, "On the plane contact problems of elasticity with adhesion or friction," PMM [Applied Mathematics and Mechanics], Vol. 34, No. 2, pp. 246-257, 1970.
5.  L. A. Galin and I. G. Goryacheva, "3D contact problem of the motion of a punch with friction," PMM [Applied Mathematics and Mechanics], Vol. 46, No. 6, pp. 1016-1022, 1982.
6.  A. S. Kravchuk, "The solution of some 3D contact problems with friction on the contact surface," Trenie i Iznos, Vol. 2, No. 4, pp. 589-595, 1981.
7.  D. A. Pozharskii, "3D contact problem for an elastic wedge with friction in the unknown contact region," Doklady AN, Vol. 372, No. 3, pp. 333-336, 2000.
8.  D. A. Pozharskii, "On the 3D contact problem for an elastic wedge with friction forces," PMM [Applied Mathematics and Mechanics], Vol. 64, No. 1, pp. 151-159, 2000.
9.  I. I. Vorovich, V. M. Alexandrov, and V. A. Babeshko, Non-classical Mixed Problems of Elasticity [in Russian], Nauka, Moscow, 1974.
10.  A. I. Lur'e, Theory of Elasticity [in Russian], Nauka, Moscow, 1970.
Received 03 December 2001
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