O.A. Saltykova – Ph.D.(Phys.-Math.), Associate Professor, Department «Mathematics and Modeling», Yuri Gagarin State Technical University of Saratov; Senior Research Scientist, Laboratory 3D Modeling, National Research Tomsk Polytechnic University
O.A. Afonin – Ph.D.(Phys.-Math.), Associate Professor, Department «Mathematics and Modeling», Yuri Gagarin State Technical University of Saratov
E-mail: email email@example.com
A.V. Krysko – Dr.Sc.(Phys.-Math.), Professor, Department «Applied Mathematics and Systems Analysis», Yuri Gagarin State Technical University of Saratov; Engineer, Laboratory 3D Modeling, National Research Tomsk Polytechnic University
Currently, there is a large number of works on the study of beams, plates and shells, described by various kinematic hypotheses. However, an analysis of modern scientific literature on the problem area under investigation has shown that there are no papers devoted to the mathematical substantiation of the existence of a solution to the problem of the contact interaction of plates (beams). The aim of the paper is the mathematical substantiation of the existence of solutions for the contact interaction of geometrically nonlinear plates (beams) described by the kinematic hypothesis of the first approximation, as well as the creation of a new iterative method for solving the problems of contact interaction of geometrically nonlinear plates (beams). It is assumed that the lamellar (beam) structures are under the action of a pulse of infinite duration in time and there is a gap between the plates (beams). In this paper, a theory of the contact interaction of flexible plates with a gap between them described by the kinematic hypothesis of the first approximation (the Euler-Bernoulli-Kirchhoff hypothesis) is constructed. The contact interaction of the elements is described by the Cantor model. The existence and convergence theorems of the solution are formulated and proved, which made it possible to obtain a mathematical justification for obtaining solutions to the posed problem of contact interaction between plate and beam structures. An iterative approach is proposed for investigating the contact interaction of two plates and beams with a gap between them. The convergence of the iterative process extends to different types of boundary conditions. The proposed iterative algorithm can be used in the problems of contacting plates and beams, taking into account, besides the geometric nonlinearity, also the physical one. As a numerical example, a mathematical model of the contact interaction of two beams described by the kinematic hypothesis of the first approximation (the Euler-Bernoulli hypothesis) is studied. The method of establishment used is described in detail. With the help of the establishment method, stationary solutions are obtained that are compared with solutions obtained by the proposed iterative procedure. Numerical results of the solution of the posed problem on the contact interaction of two flexible beams coincided, when solving by the establishment method and using the iterative procedure.
The curves for the dependence of the maximum deflection of the beams on the value of the applied load for different values of the geometric parameter and for certain values of the gap between the elements of the structure are obtained. It is shown that for a studied beam structure the geometric parameter does not have a decisive influence on the solutions obtained. Also in this work, we plotted the distribution diagrams of the contact pressure arising when the beams contact each other for three values of the gap between the beams and for three values of the geometric parameter. For different values of the geometric parameter, the value of the contact pressure changes (the thinner the beam, the less the contact pressure), but the location of the maxima of the contact pressure along the length of the beams is the same for the same gaps between the beams. Increasing the gap between the elements of the structure leads to a decrease in the contact area of the beams.
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