<p>The purpose of this study is to explore the axisymmetric torsion of an elastic layer bonded to a rigid plate and twisted by an annular rigid disc. The medium is weakened by a stress-free annular crack. Using the Hankel integral transform method, the elastostatic mixed boundary-value problem is reduced to a system of coupled triple integral equations. The unknown functions are expressed by some series expansion of the product of Bessel functions. By applying specific integral formulas and the Gegenbauer addition formula, the system is further reduced to an infinite set of algebraic equations. Closed-form expressions are derived for the angular displacement, stress components, crack opening displacement, stress intensity factors, and energy release rates, as well as singularity factors at the crack tips and disc edges. These physical quantities are illustrated graphically. Additionally, special cases of the problem are considered. The validation of the results are justified through the ANSYS finite element method (FEM) and compared with those obtained for the case of the half-space medium without and with the annular crack.</p>