 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
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| D. A. Pozharskii, "Contact problems for a two-layer elastic wedge," Mech. Solids. 44 (2), 222-231 (2009) |
| Year |
2009 |
Volume |
44 |
Number |
2 |
Pages |
222-231 |
| DOI |
10.3103/S0025654409020071 |
| Title |
Contact problems for a two-layer elastic wedge |
| Author(s) |
D. A. Pozharskii (Rostov State Agricultural Engineering Academy, pl. Strany Sovetov 1, Rostov-on-Don, 344023, Russia, tmm@rgashm.ru) |
| Abstract |
We obtain integral equations for plane contact problems for a two-layer wedge (composite) under three types of boundary conditions on one of its sides (absence of stresses, sliding, or rigid fixation). The composite consists of two wedges completely linked with each other, which have different opening angles and elasticity parameters. Using the symbols (Mellin transforms) of the kernels of integral equations for the two-layer wedge, one can derive the symbols of the kernels of integral equations for symmetric problems about a crack in a three-layer wedge or a three-layer strip and for contact problems for a two-layer strip (by passing to the limit in a special way). The complex zeros of the Mellin transform determine the asymptotics of the normal contact pressure at the corner point of the composite as the contact region approaches this point. It is important that this asymptotics is also preserved in three-dimensional contact problems as the die enters the edge of a two-layer wedge (outside the corner points of the die itself). Taking into account this asymptotics, we obtain solutions of the contact problems as the die enters the vertex of the composite. We show that by appropriately choosing the materials and the internal angle of the two-layer wedge one can avoid contact pressure oscillations at the vertex, which occur in the case of a homogeneous wedge and result in loss of contact. The contact pressure at the wedge vertex can be made zero for a composite, while for a homogeneous wedge with the same opening angle it increases unboundedly. We construct asymptotic solutions of the contact problems for a plane die located relatively close or to the vertex of a two-layer wedge or relatively far from the vertex.
The asymptotic and other methods were earlier used to solve similar plane contact problems for a homogeneous wedge [1, 2]. In the case of sliding fixation of one of the sides of a plane homogeneous wedge, the closed solution of the contact problem is known for a die entering the corner point [3, p. 131]. Two-dimensional contact problems were studied for a truncated wedge [4] and for a wedge supported by a rod of equal resistance [5]. The out-of-plane shear vibrations were studied for wedge-shaped composites [6, 7]. The spatial contact problems were considered for a homogeneous wedge [8]. The plane contact problem was analyzed for a continuously inhomogeneous wedge one of whose sides was rigidly fixed (the shear modulus continuously depends on the angular coordinate and the Poisson coefficient is constant). For a two-layer composite, which is studied in the present paper, the kernel symbol has different asymptotic properties, which are used in asymptotic methods for solving the problem. A similar distinction of the symbol properties takes place in contact problems for a continuously inhomogeneous layer and a layered packet. |
| References |
| 1. | V. M. Alexandrov and B. L. Romalis, Contact Problems in Mechanical Engineering (Mashinostroenie, Moscow, 1986) [in Russian]. |
| 2. | I. I. Vorovich, V. M. Alexandrov, and V. A. Babeshko, Nonclassical Mixed Problems of Elasticity (Nauka, Moscow, 1974) [in Russian]. |
| 3. | L. A. Galin (Editor) The Development of Theory of Contact Problems in USSR (Nauka, Moscow, 1976) [in Russian]. |
| 4. | V. M. Alexandrov and M. I. Chebakov, Analytical Methods in Contact Problems in Theory of Elasticity (Fizmatlit, Moscow, 2004) [in Russian]. |
| 5. | B. M. Nuller, "Contact Problem for an Elastic Wedge Supported by a Rod of Equal Resistance," Dokl. Akad. Nauk SSSR 225(3), 532–534 (1975). |
| 6. | V. N. Berkovich, "Theory of Mixed Problems of the Dynamics of Wedge-Shaped Composites," Dokl. Akad. Nauk SSSR 314(1), 172–175 (1990) [Soviet Phys. Dokl. (Engl. Transl.) 35, 784 (1990)]. |
| 7. | V. N. Berkovich, "Nonstationary Mixed Problem of Dynamics of Inhomogeneous Elastic Wedge-Shaped Medium," Ekolog. Vestn. Nauchn. Tsentrov Chernomorsk. Ekon. Sotrudn., No. 3, 14–20 (2005). |
| 8. | V. M. Alexandrov and D. A. Pozharskii, Nonclassical Spatial Problems in Mechanics of Contact Interactions between Elastic Bodies (Factorial, Moscow, 1998) [in Russian]. |
| 9. | V. M. Alexandrov, B. I. Smetanin, and B. V. Sobol’, Thin Stress Concentrators in Elastic Bodies (Nauka, Moscow, 1993) [in Russian]. |
| 10. | B. Noble, Methods Based on the Wiener-Hopf Technique (Pergamon, New York, 1958; Izd-vo Inostr. Lit., Moscow, 1962). |
| 11. | Ya. S. Ufland, Integral Transforms in Elasticity Problems (Nauka, Leningrad, 1967) [in Russian]. |
|
| Received |
02 May 2006 |
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