Evaluation of the stress field around a notch tip using contour integrals

The mode I and mode II asymptotic stresses around a notch tip are in general governed by different orders of singularity. Direct computation of the mixed-mode near-tip stress field therefore appears to be difficult. In this paper, we propose a pair of contour integrals J_kR. The integrals are shown to be path-independent in a modified sense and so they can be accurately evaluated with finite element solutions. As an aside, by defining a pair of generalized stress intensity factors (SIFs) (K_I)_β and (K_II)_β, the relationship between J_kR and the SIFs is derived and expressed as functions of the notch angle β. Once the J_kR-integrals are accurately computed, the generalized SIFs and, consequently, the asymptotic mixed-mode stress field can then be properly determined. The feasibility of our formulation is demonstrated in two numerical examples, where various instances with different notch angles are considered. No particular singular elements are used in this study.