Sharp notch fracture strength characterized by critical energy density

Abstract

As the order of the stress singularity at a sharp notch tip varies with its vertex angle, the units for the coefficient that represents the amplitude of the local stress field changes accordingly. Only in the limit of a zero notch angle that the coefficient could be directly associated with the energy release concept infracture mechanics owing to the preservation of self-similar crack extension provided that the loads are applied symmetrically across the crack plane. For a finite notch angle, self-similarity would no longer be preserved before and after fracture even for symmetrical loading. The delineation of notch geometry from material behavior is a prerequisite for notch strength characterization and requires a more general consideration. By focusing attention on an element of material ahead of the notch, failure by yielding and/or fracture could be predicted from the stationary values of the volume energy density d W/d V regardless of the order of the notch tip stress singularity. Fracture initiation is associated with the critical value of d W/d V or (d W/d V) c being characteristics of the material. As the stress singularity increases with decreasing notch angle, the critical applied stress to initiate failure also decreases and the initial ligament of fracture becomes more localized. This effect is much more pronounced for skew-symmetrical loading where the stress singularity becomes diminishingly weak as the half notch angle reaches 60°, a result that is not unexpected. The direction of fracture initiation for in-plane shear load was found to be away from the line that bisects the notch. It varied from ± 74.5° to ± 74.8° and ± 86.2° to ± 85.8° for Poisson's ratios of 0.1 and 0.4, respectively as the half notch angle is increased from 0° (crack) to 60°. The Poisson's ratio had a more appreciable influence on the crack initiation direction than the notch angle. This is in contrast to the maximum normal stress criterion that showed marked changes of the fracture angle with notch angle. The fracture angle decreases from ± 70.5° to ± 52.0° as the half notch angle is increased from 0° to 60°. Numerical solutions are obtained for different notch angles under symmetrical and skew-symmetrical loading. The rate change of volume to surface ratios for elements near the notch tip are compared with those obtained analytically and used as a guide for developing the finite element grid patterns. The distance of the nearest notch tip element in the numerical analysis tends to decrease with decreasing stress singularity. Such a knowledge can benefit the selection of finite element mesh sizes and distribution near sharp notches.