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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2014, Issue 1, Pages 32–45 (Mi vuu414)

MATHEMATICS

The equilibrium problem for a Timoshenko plate containing a crack along a thin rigid inclusion

N. P. Lazarevab

a Institute of Hydrodynamics, Siberian Branch of the Russian Academy of Sciences, pr. Lavrent'eva, 15, Novosibirsk, 630090, Russia
b Scientific-Research Institute of Mathematics, North-Eastern Federal University, ul. Belinskogo, 58, Yakutsk, 677891, Russia

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 Signorini-type 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, Timoshenko-type plate, rigid inclusion, energy functional, variational problem, nonpenetration condition.

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Document Type: Article
UDC: 539.311
MSC: 74B99

Citation: N. P. Lazarev, “The equilibrium problem for a Timoshenko plate containing a crack along a thin rigid inclusion”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2014, no. 1, 32–45

Citation in format AMSBIB
\Bibitem{Laz14} \by N.~P.~Lazarev \paper The equilibrium problem for a~Timoshenko plate containing a~crack along a~thin rigid inclusion \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2014 \issue 1 \pages 32--45 \mathnet{http://mi.mathnet.ru/vuu414} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. N. V. Neustroeva, “An equilibrium problem for an elastic plate with an inclined crack on the boundary of a rigid inclusion”, J. Appl. Industr. Math., 9:3 (2015), 402–411
2. N. P. Lazarev, “Optimal control of the size of rigid inclusion in equilibrium problem for inhomogeneous Timoshenko-type plate with crack”, J. Math. Sci., 228:4 (2018), 409–420
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