Asymptotics of the stress field near a crack tip under mixed-mode loading: small parameter method

In the present paper approximate analytical and numeric solutions to nonlinear eigenvalue problems arising in nonlinear fracture mechanics in analysis of stress — strain fields near a crack tip under mixed mode loading are presented. Asymptotic solutions are obtained via perturbation method technique (small parameter method). The artificial small parameter is the difference between the eigenvalue corresponding to the nonlinear eigenvalue problem and the eigenvalue related to the linear "undisturbed” problem. It is shown that the perturbation technique gives an effective method of solving nonlinear eigenvalue problems in nonlinear fracture mechanics. Comparison results of numeric and asymptotic results for different value of the mixity parameter and hardening exponent shows good agreement. Thus the perturbation theory technique for study of nonlinear eigenvalue problems is offered and applied for eigenvalue problems arising from fracture mechanics analysis in the case of mixed mode loading.