Intermediate self-similar asymptotic presentation of stress and damage fields in the vicinity of mixed mode crack tip under creep regime

Abstract

Higher order asymptotic fields of stress, creep strain rate and damage of a mixed mode I/II creep crack tip are obtained and analyzed on the basis of Continuum Damage Mechanics by employing a similarity variable and similarity solution. In the paper the class of the creep mixed mode crack problems in damaged power law materials under creep-damage coupled formulation for plane strain conditions is considered. With the similarity variable and the self-similar representation of the solution for a power-law creeping material and the classical Kachanov – Rabotnov power-law damage evolution equation the near crack-tip stresses, creep strain rates and damage distributions for plane strain conditions are obtained. The approximate solutions are based on the idea of the existence of the completely damaged zone near the crack tip. The multi-term asymptotic expansions of the stress and damage fields outside the completely damaged zone are found. It is shown that the asymptotical analysis of the near crack-tip fields results in nonlinear eigenvalue problems. The technique permitting to find all the eigenvalues numerically is proposed and numerical solutions to the nonlinear eigenvalue problems arising from the mixed-mode crack problems in a power-law medium under plane strain conditions are obtained. Using the approach developed the eigenvalues different from the eigenvalues corresponding to the Hutchinson-Rice-Rosengren (HRR) problem are found. For new eigenspectra and eigensolutions obtained the geometry of the completely damaged zone in the vicinity of the crack tip is found for all values of the mixity parameter. Effect of the higher order terms of the asymptotic expansions on the near crack tip field description is elucidated. Special attention is paid to angular distributions of the stress and damage fields for mixed mode loading.