Mixed-mode loading of the cracked plate under plane stress conditions

The paper is devoted to the stress-strain analysis near the crack tip in a power-law material un- der mixed-mode loading. In the paper by the use of the eigenfunction expansion method the stress- strain state near the crack tip under plane stress conditions is found. The type of the mixed - mode loading is specified by the mixity parameter which is varying from 0 to 1. The value of the mixity pa- rameter corresponding to Mode II crack loading is equal to 0 whereas the value corresponding to Mode I crack loading is equal to 1. It is shown that the eigenfunction expansion method results in the nonlinear eigenvalue problem. The numerical solution of the nonlinear eigenvalue problem for all the values of the mixity parameter and for all practically important values of the strain hardening (or creep) exponent is obtained. It is found that the mixed-mode loading of the cracked plate gives rise change of the stress singularity in the vicinity of the crack tip. The mixed - mode loading of the cracked plate results in the new asymptotics of the stress-strain fields which is different from the classical Hutchinson - Rice - Rosengren stress field. The approximate solution of the nonlinear eigenvalue problem is either obtained by the perturbation theory technique (small parameter method). In the framework of the small parameter method the small parameter presenting the difference between the eigenvalue of the nonlinear problem and the undisturbed linear problem is introduced. The analysis carried out shows clearly that the stress singularity in the vicinity of the crack tip is changing under mixed-mode loading in the case of plane stress conditions. The angular distributions of the stress and strain components (eigenvalue functions) in the full range of values of the mixity parameter are given.