Time-dependent problems of the concentration of elastic stresses near a conical defect

Problems of the stress state of an unbounded elastic medium containing a conical defect (a crack or a fine inclusion) and loaded with an unsteady torsional elastic wave or an impact load in the form of a torsional moment are solved. Using the discontinuous solutions of the equations of dynamic elasticity for a conical defect constructed earlier, the problem is reduced to an integrodifferential equation (in the case of a crack) or an integral equation (in the case of an inclusion) in the space of Laplace transforms. A method is proposed for solving these last equations based on the combined use of the method of orthogonal polynomials and the discretization of the equation with respect to time. A formula is obtained and the time-dependence of the stress intensity factor near the edge of the crack and the angle of rotation of the inclusion is calculated.