Fictitious Domain Method for Equilibrium Problems of the Kirchhoff-Love Plates with Nonpenetration Conditions for Known Configurations of Plate Edges

Abstract

New models are investigated in this paper, that describe equilibrium states of plates with Signorini type nonpenetration conditions. In these models, it is assumed that under appropriate loading, plates have special deformations with already known configurations of edges. For this case, we deal with new non-penetration conditions that allow us to describe more precisely the possibility of contact interaction of plate edges. Using the method of fictitious domains, it is proved that an original contact problem for a plate can be obtained by passing to the limit when a rigidity parameter tends to infinity from a family of auxiliary problems formulated in a wider domain. The mentioned family of problems model an equilibrium state of plates with a crack and depend on the positive rigidity parameter. For these problems, to prevent a mutual penetration of the opposite crack faces boundary conditions of inequality type are imposed on the inner boundary corresponding to the crack. For the problem, describing a plate with a crack that intersects the external boundary at zero angle (a case of a boundary having one cusp), the unique solvability is proved.

On a Limiting Passage as the Thickness of a Rigid Inclusions in an Equilibrium Problem for a Kirchhoff-Love Plate with a Crack

“…In the case of prior knowledge that a certain configuration of plate edges near a crack results in an equilibrium state (see Fig. 2), we impose following mutual nonpenetration condition of opposite crack faces [25]…”

Equilibrium problem for an thermoelastic Kirchhoff–Love plate with a nonpenetration condition for known configurations of crack edges

“…In order to describe the possible contact interaction of the crack's edges, for the case of prior knowledge of a certain equilibrium conguration of plate edges near the crack (see Fig. 1), we specify following mutual nonpenetration condition of opposite crack faces [29] (1)…”

“…In particular, if a solution of an equilibrium problem for this type of boundary conditions has nonzero jumps on the crack curve for vertical displacements (deections), then the solution, generally speaking, can have displacements that satisfy the general nonpenetration condition and, nevertheless, for which we have a physically unacceptable phenomenon since there is a mutual penetration of opposite crack faces, see [18]. Therefore, the abovementioned questions of the study of problems for special cases with rened modications of the nonpenetration condition is a justied branch of the development of the mechanics of deformable solids, see, for example, [28,29].…”

Equilibrium problem for a thermoelastic Kirchhoff–Love plate with a delaminated flat rigid inclusion

“…This approach of mathematical modelling implies methods of variational inequalities and has been actively developing, see [4][5][6][7][8][9][10][11][12]. Among this type of nonlinear mathematical models, a wide range of various problems for Kirchhoff-Love plates in the framework of elastic constitutive relations has been studied [4,6,[13][14][15][16]. Problems for elastic plates with rigid inclusions are investigated in [17][18][19][20][21][22][23].…”