PSI - Issue 39

L.V. Stepanova et al. / Procedia Structural Integrity 39 (2022) 735–747 Author name / Structural Integrity Procedia 00 (2019) 000–000

745 11

p M =

p M =

0.3

0.4

0 3 / 2 λ = (left) and for

1 / 2

λ = −

Fig. 10. Eigensolutions for

and

and

(right (right) (plane strain)).

0

Function 0 ( ) f θ is shown by light blue color. The numerical solution of the problem (42), (43) is shown by red line. The two-term asymptotic solution 0 1 ( ) ( ) ( ) f f f θ θ ε θ = + is shown by green color, the three-term asymptotic expansion 2 0 1 2 ( ) ( ) ( ) ( ) f f f f θ θ ε θ ε θ = + + is shown by black color and the fourth-term asymptotic expansion 2 3 0 1 2 3 ( ) ( ) ( ) ( ) ( ) f f f f f θ θ ε θ ε θ ε θ = + + + is shown by blue color. One can see that the angular distributions tend to approach the numerical solution as the number of retained terms in the asymptotic expansion of the function ( ) f θ increases. It is seen from Figs. 9 – 11 that it is sufficient to retain four terms in the asymptotic expansion if the small parameter method is used, because the angular distribution of the function ( ) f θ obtained with the use of the four-term asymptotic expansion is close to the limiting numerical solution. Discussion and conclusions The present study was caused by the necessity to grasp distributions of eigenvalues of nonlinear eigenvalue problems arising from the mixed mode crack problems in power law materials. For the eigenspectra to be found it is necessary to study multi-parametric two-point boundary value problems which can be solved by the multi-parametric shooting techniques. However, the choice of the shooting parameters is shown to be one of difficult problems for mixed mode loadings. It can be clearly seen from figs. 11-13 that the radial stress is changing very abruptly. Figs. 11 13 illustrate difficulties of the shooting method and show that the shooting method can be applied very carefully and numerical results have to be verified additionally and thoroughly. The angular distributions obtained here will allow us to find accurately crack-tip fields corresponding new eigenvalues. It underlines once more the importance of a detailed and accurate asymptotic analysis such as this.