In this paper, we are interested to analyse an axisymmetric torsion problem of an elastic medium by an annular rigid disc attached to its surface. The considered medium is weakened by a plane annular crack. The elastostatic mixed boundary value problem is reduced to a system of \(J_{1}\) coupled triple integral equations by using the Hankel integral transforms method. The unknown functions of the transformed problem are obtained with the help of the Bessel functions product series development. Knowing some integral formulas and Gegenbauer addition formula the above system can be reduced to an infinite system of algebraic equations. The angular displacement and stress components as well as the stress intensity and singularity factors at both crack tips and disc are given in closed forms and are shown graphically. Some special cases of the study are also considered. The validation of the results is given using the ANSYS FEM method and the comparison with the results of the medium without crack.
