Згин кусково-однорідної пластинки з криволінійним розрізом за умови контактування його берегів

Abstract

Побудовано систему сингулярних інтегрально-диференціальних рівнянь для задачі про згин нескінченної кусково-однорідної пластинки з криволінійним розрізом за умови контактування його берегів. Розрахунок напруженого стану здійснено числовими методами механічних квадратур і колокації.

Mixed contact problem for the piecewise homogeneous lamellar construction which contains the infinite isotropic plate with the curvilinear hole and the complete hard disk, which are partly soldered between each other on the line of delimiting, is considered. The edges of the cut between the plate and the disk are contacting along the line providing bending load in the plate which is acting on the infinity in the direction of the coordinates. As a result of it the plate is in the conditions of the generalized two-dimensional stress state and the cylindrical bending. The solving of the problem includes the determination of the components of stress state on the contour of the hole of the plate, the size and the position of the contact zone. The boundary conditions of the problem on the smooth contact zone of the plate and the disk are formulated as the equality of the normal displacements of contour points on one of the faces. Thus, it was considered that the contact stress is brought in the mean plane of the plate with the addition of the corresponding bending moment. In the zone of the soldered joint the boundary conditions of the problem are formulated as the equality of contact stresses and moments and also as the equality of the displacement vector components and the rotation angle of normal line up to the mean plane of the plate and the disk. The dependencies between the displacement vector components and the rotation angle of normal line in the plate contour points and the stresses contact are presented in the from of the integral relations with the logarithmic kernels. Their substitution in the boundary conditions leads to the system of singular integral-differential equations for the determination of the functions through which the contact stresses and the moments are determined. Finding an exact solution for this system makes great mathematical problems, so the solution is found approximately. The decision structure for this system at the contact zones and soldered joint was determined for the case of the non-full contact of cut banks. As the contact between the plate and the disk is smooth the normal stresses and the moments are limited in the contact zone and are equal to zero. At the ends of the zone of the soldered joint the contact stresses and the moments have the root feature on which the local oscillation is superposed. The numerical realization of this problem has been conducted by the methods of mechanical quadratures and the collocation by which the influence upon the size of contact zone and the shape of the hole on the stress state components allocation at the triangular contour hole of plate have been analyzed. The size and the position of the contact zone are determined by the method of dichotomy. The results of the numerical calculation of stress state components are illustrated on the graphs. At the top of the graphs was built distributions of the contact stresses and the hoop stresses and at the bottom of graphs – distributions of the contact moments, the hoop and the rotational moments were built. According to the results the size of the contact zone considerably depends on its curvature. The values of the contact zone for these examples are presented in the article. This problem is new for the formulation and the solution method. In scientific literatures there are only some published works in which by the reduction to the problem of the linear conjugation the stress state of the complete infinite isotropic plate with one or two cuts on the arc of the circle, the edges of which are contacting in the process of the flexible deformation, is researched.