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Russia Novosibirsk; Yakutsk
Section  Mathematics 
Title  The equilibrium problem for a Timoshenko plate containing a crack along a thin rigid inclusion 
Author(s)  Lazarev N.P.^{ab} 
Affiliations  Institute of Hydrodynamics, Siberian Branch of the Russian Academy of Sciences^{a}, NorthEastern Federal University^{b} 
Abstract  We study the equilibrium problem of a transversely isotropic plate with rigid inclusions. It is assumed that the plate deforms under hypotheses of classical elasticity. The problems are formulated as the minimization of the plate energy functional on the convex and closed subset of the Sobolev space. It is established that, as the geometric parameter (the size) of the volume inclusion tends to zero, the solutions converge to the solution of an equilibrium problem of a plate with a thin rigid inclusion. Also the case of the delamination of an inclusion is investigated when a crack in the plate is located along one of the inclusion edges. In the problem of a plate with a delaminated inclusion the nonlinear condition of nonpenetration is given. This condition takes the form of a Signorinitype inequality and describes the mutual nonpenetration of the crack edges. For the problem with a delaminated inclusion, the equivalence of variational and differential statements is proved provided a sufficiently smooth solution. For each considered variation problem, unique solvability is established. 
Keywords  crack, Timoshenkotype plate, rigid inclusion, energy functional, variational problem, nonpenetration condition 
UDC  539.311 
MSC  74B99 
DOI  10.20537/vm140103 
Received  2 September 2013 
Language  Russian 
Citation  Lazarev N.P. The equilibrium problem for a Timoshenko plate containing a crack along a thin rigid inclusion, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 1, pp. 3245. 
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