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

• We formulate stress fields near three-dimensional crack tips in power law plastic solids.
• We formulate stress fields near three-dimensional crack tips in power law creeping solids.
• The formulation spans the out-of-plane constraint from plane stress to plane strain states.
• Three-parameter descriptions J–Tz–QT and C(t)–Tz–Q∗ are evaluated for the two solids.
• The formulae are verified against three-dimensional finite element results.

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 Tz, Guo and his colleagues derived out the 3D asymptotic fields near the tensile crack border in power law plastic ( Guo, 1993a, Guo, 1993b and Guo, 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 Tz = 0 at plane stress state to Tz = 0.5 at plane strain state. At the two limits of Tz = 0 and 0.5, the empirical formulae degrade into the two dimensional (2D) HRR ( Hutchinson, 1968 and 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–Tz–QT description for plastic solids proposed by Guo (2000) and the C(t)–Tz–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.

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