| Title: | Crack in a solid under Coulomb friction law (English) | 
| Author: | Kovtunenko, Victor A. | 
| Language: | English | 
| Journal: | Applications of Mathematics | 
| ISSN: | 0862-7940 (print) | 
| ISSN: | 1572-9109 (online) | 
| Volume: | 45 | 
| Issue: | 4 | 
| Year: | 2000 | 
| Pages: | 265-290 | 
| Summary lang: | English | 
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| Category: | math | 
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| Summary: | An equilibrium problem for a solid with a crack is considered. We assume that both the Coulomb friction law and a nonpenetration condition hold at the crack faces. The problem is formulated as a quasi-variational inequality. Existence of a solution is proved, and a complete system of boundary conditions fulfilled at the crack surface is obtained in suitable spaces. (English) | 
| Keyword: | variational and quasi-variational inequalities | 
| Keyword: | crack | 
| Keyword: | Coulomb friction | 
| MSC: | 35J85 | 
| MSC: | 35Q72 | 
| MSC: | 73T05 | 
| MSC: | 74M10 | 
| MSC: | 74M15 | 
| idZBL: | Zbl 1058.74064 | 
| idMR: | MR1763172 | 
| DOI: | 10.1023/A:1022319428441 | 
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| Date available: | 2009-09-22T18:03:53Z | 
| Last updated: | 2020-07-02 | 
| Stable URL: | http://hdl.handle.net/10338.dmlcz/134439 | 
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| Reference: | [1] V. V. Alekhin, B. D. Annin and S. N. Korobeinikov: Accounting of a friction in contact elastoplastic problems.In: Fund. Prob. Mat. Mekh. 2, Novosibirsk Univ., 1996. (Russian) | 
| Reference: | [2] C. Baiocchi, A. Capelo: Variational and Quasivariational Inequalities. Applications to Free Boundary Problems.Wiley, Chichester, 1984. MR 0745619 | 
| Reference: | [3] G. P. Cherepanov: On some mechanism of the development of cracks in the Earth solid shell.Izvestiya USSR Acad. Sci., Physics of the Earth 9 (1984), 3–12. (Russian) | 
| Reference: | [4] R. Duduchava, W. Wendland: The Wiener-Hopf method for system of pseudodifferential equations with applications to crack problems.Integral Equations Operator Theory 23 (1995), 294–335. MR 1356337, 10.1007/BF01198487 | 
| Reference: | [5] G. Duvaut, J.-L. Lions: Les Inéquations en Mécanique et en Physique.Dunod, Paris, 1972. MR 0464857 | 
| Reference: | [6] C. Eck, J. Jarušek: Existence results for the static contact problems with Coulomb friction.Math. Models Methods Appl. Sci. 8 (3) (1998), 445–468. MR 1624879, 10.1142/S0218202598000196 | 
| Reference: | [7] P. Grisvard: Singularities in Boundary Value Problems.Masson, Paris & Springer-Verlag, Berlin, 1992. Zbl 0778.93007, MR 1173209 | 
| Reference: | [8] I. Hlaváček, J. Haslinger, J. Nečas and J.  Lovíšek: Solution of Variational Inequalities in Mechanics.Springer-Verlag, New York, 1988. MR 0952855 | 
| Reference: | [9] J. Jarušek: Contact problems with bounded friction, coercive case.Czechoslovak Math. J. 33 (108) (1983), 237–261. MR 0699024 | 
| Reference: | [10] A. M. Khludnev, J. Sokolowski: Modelling and Control in Solid Mechanics.Birkhäuser, Basel-Boston-Berlin, 1997. MR 1433133 | 
| Reference: | [11] V. A. Kovtunenko: Analytical solution of a variational inequality for a cutted bar.Control Cybernet. 25 (1996), 801–808. Zbl 0863.73077, MR 1420072 | 
| Reference: | [12] V. A. Kovtunenko: Iterative penalty method for plate with a crack.Adv. Math. Sci. Appl. 7 (1997), 667–674. Zbl 0896.73079, MR 1476271 | 
| Reference: | [13] V. A. Kovtunenko: A variational and a boundary problems with friction on the interior boundary.Siberian Math. J. 39 (1998), 1060–1073. MR 1650748 | 
| Reference: | [14] A. S. Kravchuk: Variational and Quasivariational Inequalities in Mechanics.MGAPI, Moscow, 1997. (Russian) | 
| Reference: | [15] J.-L. Lions, E. Magenes: Problémes aux Limites non Homogénes et Applications 1.Dunod, Paris, 1968. | 
| Reference: | [16] V. G. Maz’ya: Spaces of S. L. Sobolev.Leningrad Univ., 1985. (Russian) MR 0807364 | 
| Reference: | [17] N. F. Morozov: Mathematical Foundations of the Crack Theory.Nauka, Moscow, 1984. (Russian) MR 0787610 | 
| Reference: | [18] S. A. Nazarov, B. A. Plamenevskiĭ: Elliptic Problems in Domains with Piecewise Smooth Boundaries.Nauka, Moscow, 1991. (Russian) | 
| Reference: | [19] J. Nečas, J. Jarušek and J. Haslinger: On the solution of the variational inequality to the Signorini problem with small friction.Boll. Um. Mat. Ital. 17-B (1980), 796–811. MR 0580559 | 
| Reference: | [20] J. J. Telega, T. Lewinski: Mathematical aspects of modelling the macroscopic behaviour of cross-ply laminates with intralaminar cracks.Control Cybernet. 23 (1994), 773–792. MR 1303383 | 
| Reference: | [21] E. Zeidler: Nonlinear Functional Analysis and its Applications. 1. Fixed-Point Theorems.Springer-Verlag, New York, 1986. MR 0816732 | 
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