Stress properties at the tip of a conical notch

The singular behavior of a three-dimensional conical notch under torsional or torsionless loading was analyzed using the Papkovich-Neuber displacement potential method. Based on the analysis of the Legendre polynomial with complex index λ_n (Thompson, T. R. and Little, R. W. (1970). End effects in a truncated semi-infinite cone, Quart. J. Mech. Appl. Math. 23, 185-196), transcendental equations to the stress order λ_n of a torsional or torsionless notch at the cone tip are given for various notch angles. Analytical solutions for stress distributions around the cone tip under a concentrated torque T, and around the line notch tip under an expansion source E, are derived using the Legendre polynomial expression. These solutions can be applied for the study of penetration or perforation, where a penetrator is simulated as a concentrated force and as an expansion source. Curves of the first order eigenvalues show that there exist very weak singular stresses at the three-dimensional conical notch. According to the evidence observed in the conical notch experiment (Williams, M. L. (1966). Stress singular, adhesion, and fracture. In Proc. 5th Nat. Congress for Appl. Mech. pp. 451-464), when the base of a cone is loaded, fracture occurs along its side instead of at its tip. This observation suggests that there may exist a comparatively weak singularity at the cone tip.