The nonpenetration condition for a plate with an oblique cut is proposed. The variational formulation of the equilibrium problem and the equivalent formulation in the form of a boundaryvalue problem are obtained. The analytical solution is given for a one-dimensional case ( a beam with a cut), and the qualitative properties of this solution are studied.Introduction. The presence of a cut in a plate means that in addition to the outside edges, the plate has inside ones, which are called cut faces. In an undeformed state, the cut faces are in contact with each other everywhere along the two-dimensional surface, determining the shape of the cut. If, for the external faces, one can impose, for example, a jam condition, then for the cut faces, it is natural to assume the possibility of contact along the cut surface and to require their mutual nonpenetration. The restrictions that characterize this class of displacements of the points of the cut faces will be called a nonpenetration condition. This condition can incorporate the friction between the cut faces during their contact as well.One can use the term crack instead of the term cut, assuming that the crack has a zero opening in an undeformed state. However, attention in the existing theory of cracks is mainly focused on the problems of crack propagation and the determinations of the quantities that characterize the deformed state [1]. In this case, the boundary conditions considered at the crack sides usually imply violation of the nonpenetration condition [2].We consider the problem of finding the displacement field of the cut-containing plate's points with allowance for the nonpenetration condition, which leads to the variational and boundary-value formulations. As the analytical and numerical results obtained by Kovtunenko [3, 4] show, taking into account the nonpenetration condition changes significantly the qualitative character of the solution for thin plates (Kirchhoff model).The problems with cuts have wide applications not only in designing structures, but, for example, in geology: cuts can simulate faults of tectonic plateforms described by thin plates in tectonics [5]. The nonpenetration condition for thin plates with cuts and the variational formulation of the equilibrium problem with a cut were proposed and studied by Khludnev and Sokolowski for the first time [6]. Khludnev [7] considered a cracked shell and studied control in the problem in which a crack opening serves as the optimality criterion. The contact problem for a cracked plate with a rigid punch was studied in [8]. An analytical solution for the one-dimensional case (for the problem of a beam with a cut) was constructed in [3]. An algorithm for numerical solution of the problem of a plate with a cut was proposed in [4].In the present paper, a nonpenetration condition for a plate with an oblique cut that generalizes the case of a vertical cut is proposed. Variational and equivalent differential formulations of the problem are given. The problem is solved analytically for the one-dimensional...
