This article is cited in 7 papers

Variational Equilibrium Problem for a Plate with a Vertical Crack with a Geometrically Nonlinear Nonpenetration Condition

N. P. Lazarev^ab, T. S. Popova<sup>c

a</sup> M. A. Lavrent'ev Institute of Hydrodynamics, Novosibirsk
^b Research Institute of Mathematics of Yakut State University
^c North-Eastern Federal University named after M. K. Amosov

Abstract: We consider a variational problem of minimizing energy functional of an elastic isotropic plate with a crack. On the curve describing the crack we impose boundary conditions prohibiting mutual penetration of the crack edges. Boundary conditions introduced are inequalities which define a non-convex set of admissible displacement functions. We prove the existence of the solution for this variational problem and establish that sufficiently smooth solutions satisfy boundary condition on the crack curve. For the problem with a given angle of curving of one of edges, the equivalence of variational and differential forms of problem is established.

Keywords: Kirchhoff–Love plate, crack, nonlinear boundary conditions, variational inequality, contact problem.

UDC: 539.311

Received: 18.10.2010

 English version: Journal of Mathematical Sciences, 2013, 188:4, 398–409