Investigation of the shapes of cuts in a plate in contact with a rigid stamp

. A mathematical model of a contact interaction between a plate and rigid stamp is derived taking into account physical and design details. The plate is considered to have a crack, that changes its form. The problem of the contact is evaluated based on the theory of variational inequalities. The shape of the stamp is assumed to be perpendicular to the plate surface and the Poisson’s ratio is between 0 and 0.5. Analytical formulation of the study consists of transformation equation, boundary conditions and integral equation. The result is used in maximization and minimization problems for choosing extremal shape of the vertical break in the plate.


Introduction
Dynamic contact problems of the theory of elasticity and plasticity have a wide range of applications. It is associated with the study of impact and penetration of obstacles, explosive and hydroexplosive stamping and mechanical processing of materials. The high-strength materials are prone to brittle fracture, the presence of microdefects, structural cuts and pointed cavities significantly affects the strength of structures and can lead to their complete or local destruction. Therefore, studies of the stress-strain state near cuts in thin plates are of theoretical and practical interest.
From the recent studies, it is found that for the contact problems with unknown contact area, a variational approach is effective [1][2][3][4]. A variational approach to solving contact problems is based on the formulation of the boundary conditions of contact interaction in the form of variational inequalities with one-sided constraints. The expediency of using inequalities in these problems is explained by the fact that the required boundary is included in the variational statement only implicitly, as the boundary between the regions of "active" and "passive" constraints. Thus, no a priori assumptions are required regarding the topology of the unknown boundary determined after solving the problem.
The problem of determining the shape of a crack in a two-dimensional case based on crack sizes, material orthotropy and shell curvature was considered in [5]. With the development of information technologies, numerical methods, implemented through computer programs, have become relevant in contact problems. A huge number of works have been published on numerical modeling and its application in problems of dynamic contact, for example, [6][7][8][9][10][11][12].
The properties of the solution to the problem of contact of a plate without cuts with a rigid stamp were studied in [13]. Whereas, authors in [14] investigated the problems of equilibrium of plates with cuts.
In this paper, a plate with a limited area median plane is studied. The plate interacts with a rigid stamp and the set of contact points is unknown. It is assumed that the plate has a vertical break, the shape of which can change. It is required to find the form of cuts that delivers the maximum functionality defined for the solution. For each fixed shape of the cut, the plate deflection is uniquely determined. It is necessary to specify the shape at which the deflection of the plate differs from the predetermined one as much as possible. The cut shapes that are the maximum of this functional will be called extremal. The paper substantiates the fundamental possibility of finding extreme forms of sections.  and is the solution of inequality (9) for a given . It is required to find a solution to the problem It is required to estimate the value of the first derivatives of the solution. It can be shown that there is a function ∈ that minimizes the functional 1 on the set . There are other problems for which similar statements are valid.

Conclusion
As a result of this study, the existence of cut forms that give a maximum and minimum to the functional ( ) on the set is established. The main outcomes can be summarized as follows : a) The bilinear form and set of permissible deflections are determined by the space of Sobolev function and Poisson's ratio, 0 < < 1 2 b) The problem of the contact between the plate and the stamp is a variational inequality c) For each fixed parameter problem has a unique solution -plate deflection d) The solutions of the inequality is boundedness in the norm of space e) The dependence of a function on its variables is determined by the nature of the change of variables and the form of inequality f) for any element ̄∈ there exists a sequence ̄∈ such that ̄→̄ strongly in 2,0 ( ) and → weak in 2,0 ( 0 ) g) for any element ̅ ∈ there exists a sequence ̅ ∈ strongly converging in 0 2 ( ) to ̅ h) the extremal cut shapes are found by the function , which solves the maximization and minimization problems ∈ ( ) and ∈ 1 ( ) , respectively