Stress-strain state near the crack tip under mixed-mode loading: Asymptotic approach and numerical solutions of nonlinear eigenvalue problems

Abstract

Creep crack problems in damaged materials under mixed-mode loading in the creep-damage coupled formulation are considered. The class of self-similar solutions to the plane creep crack problems in a damaged medium under mixed-mode loading is given. With the similarity variable and the self-similar representation of the solution for a power-law creeping material and the Kachanov-Rabotnov power-law damage evolution equation, the near-crack-tip stresses, creep strain rates and continuity (integrity) distributions for plane stress conditions are obtained. The self-similar solutions are based on the hypothesis of the existence of a completely damaged zone near the crack tip. It is shown that the asymptotical analysis of the near crack-tip fields gives rise to nonlinear eigenvalue problems. A technique enabling all the eigenvalues to be found numerically is proposed, and numerical solutions to nonlinear eigenvalue problems arising from the mixed-mode crack problems in a power-law medium under plane stress conditions are obtained. Using the approach developed, eigenvalues different from the eigenvalues corresponding to the Hutchinson-Rice-Rosengren (HRR) problem have been found. Having obtained the eigenspectra and eigensolutions, we find the geometry of the completely damaged zone in the vicinity of the crack tip all values of the mixity parameter.