Analysis of eigenvalues for problem on the transverse shear crack in material with power defining law

Abstract

We research the non-linear eigenvalues problem which appears from the problem of determining the stress-strain state near the transverse shear crack tip in material with power relationship between strains and stresses. For the eigenvalues finding we use the perturbation method based on expanding the eigenvalue, corresponding eigenfunction and the index of non-linearity of material in series in terms of powers of small parameter, which is the difference between the eigenvalues related to the linear and non-linear problems. The comparison between the found eigenvalues and the exact numerical solution of considered non-linear eigenvalues problem is given.