Abstract
Under study is the nonlinear equilibrium problem for an elastic Timoshenko-type shallow shell containing a through crack. Some boundary conditions in the form of inequalities are imposed on the curve defining the crack. We establish the unique solvability of the variational statement of the nonlinear problem of the equilibrium of a shell. We prove that, for sufficient smoothness of the solution, the initial variational statement is equivalent to the differential formulation of the problem. We deduce the boundary conditions on the inner boundary that describes the crack. In the case of the zero opening of the crack, we prove the local infinite differentiability of the solution function with additional assumptions on the functions defining the curvatures of the shell and the external loads.
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References
G. P. Cherepanov, Mechanics of Brittle Fracture (Nauka, Moscow, 1974; McGraw-Hill, New York, 1979).
Yu. I. Rabotnov, Mechanics of Deformable Solids (Nauka, Moscow, 1988) [in Russian].
V. A. Levin, E. M. Morozov, and Yu. G. Matvienko, Selected Nonlinear Problems of Mechanics of Fracture (Fizmatlit, Moscow, 2004) [in Russian].
L. I. Slepyan, Mechanics of Cracks (Sudostroenie, Leningrad, 1981) [in Russian].
N. F. Morozov, Mathematical Questions in the Theory of Cracks (Nauka, Moscow, 1984) [in Russian].
A. M. Khludnev and J. Sokolowski, Modelling and Control in SolidMechanics (Birkhauser, Basel, 1997).
A.M. Khludnev and V. A. Kovtunenko, Analysis of Cracks in Solids (WIT Press, Southampton, 2000).
A. M. Khludnev, Problems of Elasticity Theory in Nonsmooth Domains (Fizmatlit, Moscow, 2010) [in Russian].
E. M. Rudoy, “Asymptotics of the Energy Functional for a Fourth-Order Mixed Boundary Value Problem in a Domain with a Cut,” Sibirsk. Mat. Zh. 50(2), 430–445 (2009) [Siberian Math. J. 50 (2), 341–354 (2009)].
A.M. Khludnev, “The Contact Problem for a Shallow Shell with a Crack,” Prikl. Mat. Mekh. 59(2), 318–326 (1995) [J. Appl. Math. Mech. 59 (2), 299–306 (1995)].
A.M. Khludnev, “Extreme Crack Shapes in a Shallow Shell,” Adv. Math. Sci. Appl. 7(1), 213–221.
N. V. Neustroeva, “Unilateral Contact of Elastic Plates with Rigid Inclusions,” Vestnik Novosib. Gos. Univ. Ser. Mat. Mekh. Inform. 9(4), 51–64 (2009) [in Russian].
A. M. Khludnev, “Problem of a Crack on the Boundary of a Rigid Inclusion in an Elastic Plate,”, Izv. Ross. Akad. Nauk Mekh. Tverd. Tela No. 5, 98–110 (2010) [Mech. Solids 45 (5) 733–742 (2010)].
K. Z. Galimov, Bases of Nonlinear Theory of Thin Shells (Izd. Kazan. Gos. Univ., Kazan, 1975) [in Russian].
B. L. Pelekh, Theory of Shells with Finite Shear Rigidity (Naukova Dumka, Kiev, 1973) [in Russian].
A. S. Vol’mir, Nonlinear Dynamics of Plates and Shells (Nauka, Moscow, 1972) [in Russian].
V. Z. Vlasov, General Theory of Shells and Its Applications (Gostekhizdat, Moscow, 1949) [in Russian].
Yu. G. Reshetnyak, Stability Theorems in Geometry and Analysis (Nauka, Novosibirsk, 1982; Kluwer Academic, Dordrecht, 1994)
C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free Boundary Problems (Wiley, New York, 1984; Nauka, Moscow, 1988).
V. P. Mikhaĭlov, Partial Differential Equations (Nauka, Moscow, 1976;Mir, Moscow, 1978).
J. L. Lions and E. Magenes, Problèmes aux Limites Non Homogènes et Applications (Non-Homogeneous Boundary Value Problems and Applications) (Dunod, Paris, 1968; Springer, Berlin, 1972; Mir, Moscow, 1971).
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Original Russian Text © N.P. Lazarev, 2012, published in Sibirskii Zhurnal Industrial’noi Matematiki, 2012, Vol. XV, No. 3, pp. 58–69.
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Lazarev, N.P. The equilibrium problem for a Timoshenko-type shallow shell containing a through crack. J. Appl. Ind. Math. 7, 78–88 (2013). https://doi.org/10.1134/S1990478913010080
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DOI: https://doi.org/10.1134/S1990478913010080