Asymptotics of eigenvalues of the nonlinear eigenvalue problem arising from the near mixed-mode crack-tip stress-strain field problems

In the present paper, approximate analytical and numerical solutions to nonlinear eigenvalue problems arising in the nonlinear fracture mechanics in analysis of stress-strain fields near a crack tip under a mixed mode loading are presented. Asymptotic solutions are obtained by the perturbation method (the small artificial parameter method). The artificial small parameter is a 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 the nonlinear fracture mechanics. Comparison of numerical and asymptotic results for different values of the mixity parameter and hardening exponent shows good agreement. Thus, the perturbation theory technique for studying nonlinear eigenvalue problems is offered and applied to eigenvalue problems arising from the fracture mechanics analysis in the case of a mixed mode loading.