Nonlinear Eigenvalues Problems Engendered by Nonlinear Fracture Mechanics Boundary Value Problems

Abstract

Nonlinear eigenvalue boundary value problems occurring in nonlinear fracture mechanics due to determination of the stress-strain state in the vicinity of the crack tip in power law hardening or creep power law materials are discussed. The study is aimed at analytical determination and approximate analytical expressions of eigenfunctions and eigenvalues of the nonlinear eigenvalue problems following from the problem of determining the crack tip fields at crack in power law materials under Mode III, Mode I, Mode II and Mixed Mode loadings. The analytical approach is based on the perturbation theory technique (small artificial parameter method) allowing us to find the analytical solution for the eigenvalues in the approximate form. The method of analytical determination of eigenfunctions of the problem is presented.