Мішана контактна задача для пластинки з квадратним отвором і жорсткого диска

Abstract

Розглянуто мішану контактну задачу для ізотропної пластинки з квадратним отвором і жорсткого диска за наявності на лінії розмежування матеріалів зон спаю, гладкого без тертя контакту і відставання. Використовуючи інтегральні подання компонент вектора зміщення контурних точок, побудовано систему сингулярних інтегральних рівнянь для визначення контактних зусиль. Напружений стан на контурі отвору пластинки досліджено методом механічних квадратур і колокації.

Composite plastic materials as elements of engineering structures are widely used in machinery, instrumentation, construction and other fields of modern technology. From the technological or structural pornts of view, they may consist of different materials with straight or curved boundary of their division, on which while manufacturing or operation may arise and develop cracks transverse sections of zero width. We consider the mixed contact problem for a composite plate structure, which contains an infinite isotropic plate with a square hole and totally hard drive, fused together along the separation line of materials. By the action of concentrated power load applied at the center of the drive, the banks of lamellar structure cut are partially in contact. We consider that there is no friction in the area of plate and drive contact. Solution of this problem involves identifying the components of stress state on the hole contour of the plate and setting the size and position of the contact zone. In the area of smooth frictionless contact boundary conditions of the problem are elected as equality of normal plate and drive displacement, as in the area of junction are in the form of displacement equality of contour points. Dependencies between the components of the displacement vector contour points of the plate and the hard drive and contact stresses are recorded in the form of integral relations with logarithmic kernels. By substituting these expressions into the boundary conditions, a system of four singular integral equations for determining the functions is obtained through which the contact force on the contact area and the junction are expressed . For approximate solution of this problem the structure of the desired functions in the vicinity of the ends of sections of smooth frictionless contact zone and junction is set. At the ends of the junction area contact efforts have root feature. The desired searched functions have the same feature. The area smooth frictionless contact solution is chosen in the class of functions bounded at its ends. Approximate solution of the problem is realized by the mechanical quadrature and collocation method. Dichotomy method determines the size and position of the contact zone. The results of numerical calculation of the stress state components are illustrated in the graphs.