Evaluation of stress singularity orders and stress intensity factors for a circumferential V-notch in a round bar

• Published in: Journal of engineering mathematics Vol. 152; no. 1
• Main Authors: Ur Rehman, Inam, Cheng, ChangZheng, Li, Tengyue, Wang, Mingzhi, Hu, Zongjun
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Nature B.V 01.06.2025
• Subjects: Asymptotic methods; Asymptotic series; Boundary conditions; Circumferences; Clamping; Eigenvalues; Equilibrium equations; Finite element method; Mathematics; Matrix methods; Nonlinear differential equations; Notches; Ordinary differential equations; Round bars; Singularity (mathematics); Stress intensity factors
• ISSN: 0022-0833; 1573-2703
• DOI: 10.1007/s10665-025-10450-x

The determination of stress singularity orders and notch stress intensity factors are crucial aspects of stress singularity analysis for circumferential V-notches. In this study, the asymptotic expansion technique is firstly employed to evaluate the stress singularity order. Fourier expansions with respect to the angular coordinate in the cross-section of the round bar are applied to the displacement components, and the Williams asymptotic expansions are then used to separate the radial and angular coordinates in the longitudinal section. The elasticity equilibrium equations in terms of displacement are transformed into non-linear ordinary differential equations with respect to the stress singular orders. Three transformation equations are introduced to convert the nonlinear ordinary differential eigenvalue equations into the linear ones. The free-free, free-clamped, and clamped–clamped boundary conditions are respectively considered, and they are transformed into a set of ordinary differential equations using the singularity asymptotic expansion technique. The interpolation matrix method is then employed to solve the established singularity eigenvalue equations, allowing for the simultaneous determination of singularity orders and corner functions for various singularities. The proposed method can obtain both real and complex singularity orders without the need to solve the complicated transcendental equation required in the conventional methods. Then, the notch stress intensity factors are calculated using finite element analysis assisted by the stress singularity orders and corner functions derived from the singularity eigen-analysis. Finally, the results are compared with those for plane and anti-plane V-notches, and the three-dimensional constraint effect of the circumferential V-notch is examined.