The perturbation method for studying nonlinear eigenvalue problems was proposed and developed. These problems emerge from 1) the problems of stress, strain and continuity fields definition in the vicinity of the crack peak in material with powered defining equations on use of kinetic equation driven by the powered law of accumulated damages and 2) the problems of fatigue growing of crack in linear elastic material in the damaged medium with the power law evolution of accumulating damages. The exact dependence of eigenvalue from non-linear characteristic of material in the case of a non-plane shift was achieved. It was demonstrated that the proposed method allows solving effectively the non-linear mathematical models of the processes of deformation, damage evolution and defect accumulation in deformed solid bodies.
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