Formulation of the stress fields in power law solids ahead of three-dimensional tensile cracks

To accurately predict damage behavior in engineering applications, it is important to investigate the three-dimensional (3D) stress state near a real crack border. Introducing the out-of-plane stress constraint factor T _z, Guo and his colleagues derived out the 3D asymptotic fields near the tensile crack border in power law plastic (Guo, 1993a, 1993b, 1995) and creeping solids (Xiang et al.; 2011). However, these theoretical solutions are presented in curves and too complicated for application. Here we formulize the 3D theoretical solutions into a set of empirical explicit formulae in the whole range of out-of-plane stress constraint from T_z = 0 at plane stress state to T_z = 0.5 at plane strain state. At the two limits of T _z = 0 and 0.5, the empirical formulae degrade into the two dimensional (2D) HRR (Hutchinson, 1968; Rice and Rosengren, 1968) or RR (Riedel and Rice, 1980) solutions with high accuracy. Detailed finite element analyses are performed for cracked plates with finite thickness under power law plastic and creeping conditions to verify the formulation of the asymptotic crack border stress fields. It is shown that the in-plane stress components and stress triaxiality on the ligament ahead of the crack border can be efficiently predicted by the explicit formulae. We also investigate the dominance of the formulation of stress components in the whole forward sector to give a more convenient description for wide applications. Based on the formulation, we discuss the influence of both in-plane and out-of-plane constraints. Three-parameter descriptions, such as the J-T_z-Q^T description for plastic solids proposed by Guo (2000) and the C(t)-T _z-Q^- description for creeping solids proposed by Xiang et al. (2011) are evaluated based on comparison of the empirical formulae and 3D finite element results. The three-parameter descriptions are shown to be necessary and efficient under large scale yielding or extensive creeping conditions in the whole forward sector of cracked plates with finite thickness.