Elastic compound plane with an interfacial absolutely rigid thin inclusion partially detached form the matrix subject to slippage at the ends

This article discusses the stress state of an elastic composite plane with a crack of finite length on the joining line of the half-planes. An absolutely rigid thin inclusion of the same length is indented into one of the edges of an interfacial crack under the action of a concentrated force. It is assumed that for the contacting side of the inclusion, there is adhesion to the matrix in its middle part, and slippage occurs along the edges, which is described by the law of dry friction. The problem is mathematically formulated as a system of singular integral equations. The behavior of the unknown functions in the vicinity of the ends of the inclusion-crack and at the separation points of the adhesion and slip zones is studied. The governing system of integral equations is solved by the method of mechanical quadratures. The laws of distribution of contact stresses, as well as the lengths of the adhesion and slip zones, depending on the coefficient of friction, Poisson's ratios and the ratio of Young's moduli of the materials of half-planes, as well as the inclination angle of the external force, are found.