Higher-order Asymptotic Solution for the Fatigue Crack Growth Problem based on Continuum Damage Mechanics

In the paper an analytical solution of the nonlinear eigenvalue problem arising from the fatigue crack growth problem in a damaged medium in coupled formulation is obtained. In order to evaluate the mechanical behavior in the vicinity of a growing fatigue crack for plane strain and plane stress conditions of mode I the asymptotic governing equations are derived and analyzed by the light of Continuum Damage Mechanics. It is shown that the growing fatigue crack problem can be reduced to the nonlinear eigenvalue problem. The perturbation technique for solving the nonlinear eigenvalue problem is used. The method allows to find the analytical formula expressing the eigenvalue as the function of parameters of the damage evolution law. The eigenvalues of the nonlinear eigenvalue problem are fully determined by the exponents of the damage evolution law. The higher-order asymptotic expansions of the angular functions determining the stress and continuity fields in the neighborhood of the crack tip are given. The asymptotic expansions of the angular functions permit to find the closed-form solution for the problem. The higher order stress, strain and continuity asymptotic fields in the vicinity of the fatigue growing crack are either obtained, in which analytical expressions of the higher order exponents and angular distribution functions (eigenfunctions) of the near tip stress and continuity fields are derived.