О ЕСТЕСТВЕННЫХ ГРАНИЧНЫХ УСЛОВИЯХ В ЗАДАЧЕ О ПОТЕРЕ УСТОЙЧИВОСТИ ПЛАСТИНЫ С ЭЛЛИПТИЧЕСКОЙ ВСТАВКОЙ ПРИ РАСТЯЖЕНИИ

Abstract

This paper studies the issue of determination and influence of natural boundary conditions in the problem of buckling of a thin plate with an elliptical inclusion under tension. First, the naturalness of the boundary conditions of the “free edge” type for a plate with a hole is proved. Then a plate with welded inclusion is considered and natural boundary conditions are derived. The limit cases are checked for an absolutely soft inclusion and for an absolutely rigid one. It is shown that the first case leads to a problem with a hole and the corresponding natural boundary conditions, and in the second case, to the absence of natural conditions, since a problem with a clamped edge occurs. The authors conclude that the use of additional restrictions will lead to the construction of a basis that will rapid up the convergence of the method. Variational methods are widely used in all fields of mechanics, including in the field of machine, aircraft, and rocket engineering. An exact solution to the problems of elasticity theory and structural mechanics is not always possible to construct, therefore, in practice, great importance is attached to various approximate methods. Among them a special place is occupied by variational methods based on the direct minimization of the corresponding energy of the body and making it possible to build approximate analytical solutions in the form of a functional series. The goal of variational methods is to construct a partial sum of this series, which, with a sufficient number of terms, will be maximum close to the solution. However, the issue of convergence is influenced by many factors, and one of them is the natural boundary conditions, which are derived in this paper for the problem of the stability loss of the plate with the elliptical inclusion under tension.