BENDING OF THE PLATE WITH AN ELASTIC CIRCULAR WASHER AND A RADIAL CRACK INSIDE THE WASHER CONSIDERING THE STRIP CONTACT OF ITS EDGES

This work presents a solution of biaxial bending of an infinite isotropic plate with an elastic circular washer made of a different material and a through straight crack inside the washer. The edges of the crack are free from external loads and the conditions of ideal mechanical contact are fulfilled at the boundary of the elastic circular washer and the plate. It is assumed that under the action of uniformly distributed bending moments at infinity, the edges of the crack are in contact on area of constant width (strip contact) on the upper base of the plate along the entire length of the cracks. The problem is solved under the followingboundary conditions: Pr1 = Pr2 , Mr1 = Mr2 , w1 = w2 , ��wr1 � ��wr2 , �(r1r) � �(rr2) ,� �r� r�(1) � (2) , ur(P1) = ur(P2) , u�P u�P(1) � (2) , on L (conditions of ideal mechanical contact);P� � 0 , M hNy1� � � , ��Ïx1y1 � 0 , �Ïy y N h1� � �0,5 , � �v � �x � h ��� w �x �y �� �Ï 121 10�,� � 0.5�1 � �1 � ��2� , � � 1 � � 3 (conditions of strip contact of crack edges),� � h h1, x L1 1∈ , where Mr – bending moment, Pr – generalized Kirchhoff shear force, σÏr and �Ïr� – stress tensor components in the polar coordinate system, N – contact force between the edges of the crack, h − plate thickness, h1 − height of the contact zone of crack, w – plate deflection, σÏy1y1 andτÏxy1 1– stress tensor components in the Cartesian coordinate system, vÏ – displacement vector component; � f � � f � � f � , the “+” and “–” signs indicate the boundary values of the function when the point of the plane approaches the crack at y1 � �0.Due to the contact of the crack edges, the solution to the problem will be sought as a superposition of two interrelated problems: the problem of the plate bending, using the Kirchhoff–Love theory and the plane problem. Using the methods of the theory of functions of a complex variable and complex potentials of Kolosov–Muskhelishvili, linear conjugate problems were constructed. The obtained lineare conjugate problems are solved analytically and a system of singular integral equations was obtained for the displacement jumps in the plane problem and the jumps of rotation angles in the bending problem at the crack edges. The system of singular integral equations was reduced to a system of algebraic equations, using the method of mechanical quadrature and was solved numerically by the Gaussian method. A numerical analysis of the problem is conducted and graphic dependencies of contact forces and intensity coefficients of forces and moments were constructed for different geometric parameters of the problem and different values of uniformly distributed bending moments at infinity. In some cases, the results obtained coincided with the known results obtained in scientific works by other authors.