On the nonlinear eigenvalue problems arising in fracture mechanics

In the article the stress-strain state near a Mode II crack tip under plane stress conditions in power-law materials is considered. It is noted that nowadays the whole eigenspectrum and orders of stress singularity at the crack tip for a power-law medium are of prevailing interest. Additional eigenvalues for the stress field at a static mode II crack under plane stress condition are numerically obtained for different values of the exponent $n$ via the Runge–Kutta method in conjunction with the shooting method. However, in this case the shooting method is multi-parameter since it is necessary to select two parameters and, consequently, the results obtained require further verification and justification. For this purpose the technique developed in perturbation theory for study of nonlinear eigenvalue problems is offered and applied for eigenvalue problems arising from fracture mechanics analysis.