Previous |  Up |  Next

Article

Keywords:
variational and quasi-variational inequalities; crack; Coulomb friction
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.
References:
[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)
[2] C. Baiocchi, A. Capelo: Variational and Quasivariational Inequalities. Applications to Free Boundary Problems. Wiley, Chichester, 1984. MR 0745619
[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)
[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. DOI 10.1007/BF01198487 | MR 1356337
[5] G. Duvaut, J.-L. Lions: Les Inéquations en Mécanique et en Physique. Dunod, Paris, 1972. MR 0464857
[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. DOI 10.1142/S0218202598000196 | MR 1624879
[7] P. Grisvard: Singularities in Boundary Value Problems. Masson, Paris & Springer-Verlag, Berlin, 1992. MR 1173209 | Zbl 0778.93007
[8] I. Hlaváček, J. Haslinger, J. Nečas and J.  Lovíšek: Solution of Variational Inequalities in Mechanics. Springer-Verlag, New York, 1988. MR 0952855
[9] J. Jarušek: Contact problems with bounded friction, coercive case. Czechoslovak Math. J. 33 (108) (1983), 237–261. MR 0699024
[10] A. M. Khludnev, J. Sokolowski: Modelling and Control in Solid Mechanics. Birkhäuser, Basel-Boston-Berlin, 1997. MR 1433133
[11] V. A. Kovtunenko: Analytical solution of a variational inequality for a cutted bar. Control Cybernet. 25 (1996), 801–808. MR 1420072 | Zbl 0863.73077
[12] V. A. Kovtunenko: Iterative penalty method for plate with a crack. Adv. Math. Sci. Appl. 7 (1997), 667–674. MR 1476271 | Zbl 0896.73079
[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
[14] A. S. Kravchuk: Variational and Quasivariational Inequalities in Mechanics. MGAPI, Moscow, 1997. (Russian)
[15] J.-L. Lions, E. Magenes: Problémes aux Limites non Homogénes et Applications 1. Dunod, Paris, 1968.
[16] V. G. Maz’ya: Spaces of S. L. Sobolev. Leningrad Univ., 1985. (Russian) MR 0807364
[17] N. F. Morozov: Mathematical Foundations of the Crack Theory. Nauka, Moscow, 1984. (Russian) MR 0787610
[18] S. A. Nazarov, B. A. Plamenevskiĭ: Elliptic Problems in Domains with Piecewise Smooth Boundaries. Nauka, Moscow, 1991. (Russian)
[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
[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
[21] E. Zeidler: Nonlinear Functional Analysis and its Applications. 1. Fixed-Point Theorems. Springer-Verlag, New York, 1986. MR 0816732
Partner of
EuDML logo