T-stress solutions for cracks at notches and in cylinders

The elastic T-stress is a parameter used to define the level of constraint at a crack-tip. It is important to provide T-stress solutions for practical geometries in order to apply the constraint-based fracture mechanics methodology. In the present work T-stress solutions are provided for cracks at a U-shaped notch, and circumferential through-wall cracks in cylinders. The U-shaped notch with length a and a notch-tip radius rho in a finite width plate is analyzed using the finite element method. A crack with length l is located at the notch-tip. Three notch geometries were considered; defined by the notch length (a) to specimen width (W) ratios: a/W = 0.3, 0.4, and 0.5. The ratio rho /W was fixed at 0.025 for all geometries. The T-stress was obtained at nine cracks lengths for remote tension and remote bending. A cylinder with a circumferential through-wall crack was analyzed using the finite element method and the weight function method. Three cylinder geometries were considered; defined by the pipe radius (R) to wall thickness (t) ratios: R/t = 5, 10, and 20. The T-stress was obtained at eight crack lengths for remote tension and remote bending loads. The T-stress was approximated using empirical formulas obtained from least-squares curve fitting. Weight functions for T-stress were then generated for general loading conditions. First the weight function method developed in two-dimensions was extended to a cylindrical surface. Then weight functions were obtained for the three geometries defined by R/t = 5, 10, and 20. The weight functions were validated by comparing the weight function based T-stress to finite element results for remote tension and bending, and good agreement was achieved. The weight functions are valid for complex loading conditions and for cracks depths in the range 0 ≤ theta/pi ≤ 0.5.