Вариационно-структурный метод решения плоской контактной задачи теории упругости

Abstract

Дана вариационная и структурная постановка плоской контактной задачи теории упругости для однородных тел произвольной геометрической формы с известными и неизвестными областями контакта на основе функционала Рейсснера. На конкретном примере рассмотрена методика использования разработанных структур и ее численная реализация

This paper proposes a systematic description of a general method of variational and structural formulation of a plane contact problem in elasticity for a homogeneous body having arbitrary geometrical shape. Various boundary conditions with known and unknown contact areas are considered. The variational formulation is based on the Reissner functional. A general method for constructing solutions for the displacements and stresses which exactly satisfy all the boundary and contact conditions using R-functions is set forth. The solution structures obtained allow for specification and modification depending on the particular problem. The significant advantages of the proposed method are the possibility of independent approximation of displacements and stresses and the simplicity of constructing the solution structures. The problem of determining the contact areas is reduced to a sequence of mixed linear problems. The iteration algorithm applied uses two independent criteria. To illustrate a specific implementation of the proposed method a problem for an elastic trapezoid having lower base resting without gap on an absolutely smooth solid surface, lateral sides free from load, and upper base remaining under the pressure of a print is considered. The results of numerical studies are given. Validity and accuracy of the solutions are confirmed by the compliance of integral and local criteria