An axisymmetric torsion problem by an annular disc of a semi-infinite elastic medium weakened by an annular crack

The article presents a detailed analysis of an axisymmetric torsion problem involving an annular disc in a semi-infinite elastic medium weakened by an annular crack. It begins by discussing the importance of mixed boundary value problems in mechanical and engineering developments, particularly in areas such as heat conduction, elastostatic solid deformation, and fluid flow. The study builds upon pioneering works by researchers like Reissner, Srivastava, and others, who have contributed to the understanding of elastic media under torsion. The article introduces the Hankel integral transforms method to reduce the problem to a system of coupled triple integral equations, which are then solved using the Gegenbauer addition formula. This approach leads to an infinite system of algebraic equations, solved numerically via the truncation method. The results include the distribution of angular displacement, shear stress, and stress intensity factors, providing valuable insights into the behavior of elastic media under torsion. The article also validates its findings through comparisons with existing solutions and finite element method (FEM) simulations, ensuring the accuracy and reliability of the presented approach. The detailed mathematical formulations and numerical solutions make this article a significant contribution to the field of mechanical engineering and applied mathematics.