On Crack-Tip Stresses as Crack-Tip Radii Decrease

In classical elasticity, when cracks are modeled with stress-free elliptical holes, stress singularities occur as crack-tip root radii go to zero. This raises the question of when crack-tip stresses first start to depart from physical reality as radii go to zero. To address this question, here, cohesive stress action is taken into account as radii go to zero. To obtain sufficient resolution of the key crack-tip fields, two highly focused numerical approaches are employed: finite elements with successive submodeling concentrated on the crack-tip and numerical analysis of a companion integral equation with considerable discretization refinement at the crack-tip. Both numerical approaches are verified with convergence checks and test problems. Results show that for visible cracks, classical elasticity analysis leads to physically sensible stresses, provided that crack-tip radii are accounted for properly. For microcracks with smaller crack-tip radii, however, cohesive stress action also needs to be included if accurate crack-tip stresses are to be obtained. For cracks with yet smaller crack-tip radii, cracks close and stresses throughout the crack plane become uniform.