Stress fields due to inclined notches and shoulder fillets in shafts under torsion

Closed-form expressions of the stress fields in notched rounded bars subjected to torsion are obtained. The notch profile is mathematically described according to Neuber’s conformal mapping z = (u + iv) ^q , which gives parabolic and hyperbolic profiles depending on q. The notch axis is inclined with respect to the rounded bar axis. This condition results in two eigenvalue functions: the former is associated with the antisymmetric stress field and gives a stress distribution of asymptotic nature when the notch radius is equal to zero. Conversely, the latter is associated with the symmetric stress field and results in a non-singular stress distribution. This specific condition has been noted in the literature only rarely, where only the antisymmetric part of the stress field is discussed in detail. Theoretical results are compared with numerical data as determined from two models weakened by a parabolic notch and a hyperbolic notch, both notches having the local axis inclined to 45° with respect to the bar longitudinal axis. The agreement is found to be satisfactory.