On a Contact Problem for a Homogeneous Plane with a Finite Crack under Friction

An exact solution to the contact problem of indentation of an absolutely rigid stamp with a straight base, taking into account friction, into one of the edges of a finite crack located in a homogeneous elastic plane is constructed. It is assumed that tangential contact stresses are directly proportional to normal contact pressure. In this case, it is believed that the friction coefficient is directly proportional to the coordinates of the contacting points of the contacting surfaces. The defining system of equations for the problem is derived in the form of an inhomogeneous Riemann problem for two functions with variable coefficients and its closed solution in quadratures is constructed. Simple formulas for contact stresses and the normal dislocation component of displacements of crack edge points are obtained. The patterns of changes in contact stresses and crack opening depending on the maximum value of the friction coefficient have been studied.