Elastic stress distributions for hyperbolic and parabolic notches in round shafts under torsion and uniform antiplane shear loadings

Closed-form solutions are developed for the stress fields induced by circumferential hyperbolic and parabolic notches in axisymmetric shafts under torsion and uniform antiplane shear loading. The boundary value problem is formulated by using complex potential functions and two different coordinate systems, providing two classes of solutions. It is also demonstrated that some solutions of linear elastic fracture and notch mechanics reported in the literature can be derived as special cases of the general solutions proposed herein.Finally the analytical frame is used to link the Mode III notch stress intensity factor to the maximum shear stress at the notch tip, as well as to give closed-form expressions for the strain energy averaged over a finite size volume surrounding the notch root.