PSI - Issue 39

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

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crack tip. The coordinate perturbation technique is employed to study the eigenspectrum of creeping body. To attain eigensolutions a numerical scheme is worked out and the results obtained provide the information including the number of singularities, and their orders, as well as the angular distributions of stresses. In particular, additional eigenvalues of the HRR/RR problem are presented. In (Stepanova (2008), Stepanova (2009)) eigenspectra and orders of singularity of the stress field near a mode I crack tip in a power-law material are discussed. The perturbation theory technique is employed to pose the required asymptotic solution. The whole set of eigenvalues is obtained. It is shown that the eigenvalues of the nonlinear problem are fully determined by the corresponding eigenvalues of the linear problem and by the hardening exponent. The Airy stress function is sought in the form ( ) 0 1 2 1 1 1 (0) (1) (2) , ( ) ( ) ( ) ... r r f r f r f λ λ λ χ θ θ θ θ + + + = + + + (14) The new eigenvalues of the HRR problem different from / ( 1) n n λ = + can be obtained numerically by the help of the Runge-Kutta -Fehlberg method together with the shooting method. The results for a mode I crack for plane strain and plane stress conditions are given in Tables 1,2. The results for mode II crack for plane strain and plane stress conditions are given in Tables 3,4. New eigensolutions are shown in Figs. 1 and 2. The results for a mode I crack for plane strain and plane stress conditions are given in Tables 1,2. The results for mode II crack for plane strain and plane stress conditions are given in Tables 3,4. Table 1. The new computed eigenvalues λ and ( ) f θ ′′ for mode I crack (plane strain conditions). n λ ''(0) f 2 0 -0.5 3 0.228641 -0.436714

4 5 6 7 8 9 10 11

0.331913 0.382535 0.410325 0.427209 0.438319 0.446089 0.451786 0.456120

-0.408663 -0.397996 -0.394479 -0.393828 -0.394328 -0.395275 -0.396363 -0.397463

( ) f θ ′′ for mode I crack (plane stress conditions).

Table 2. The new computed eigenvalues λ and

n

λ

''(0) f

2 3

-0.154032

-0.568609

0.0

-0.5

4 5 6 7 8 9 10

0.086754 0.142129 0.180348 0.208191 0.229306 0.245829 0.259090

-0.465378 -0.442391 -0.425683 -0.419738 -0.403021 -0.395063 -0.388592

( ) f θ ′′ for mode II crack (plane strain conditions).

Table 3. The new computed eigenvalues λ and

n

λ

(0) f ′′′

2 3 4 5 6 7 8 9

-0.277975 -0.270329 -0.265673 -0.259298 -0.253728 -0.249344 -0.245930 -0.243240 -0.241043

-1.656070 -1.531775 -1.480068 -1.451846 -1.432646 -1.419161 -1.409279 -1.401774 -1.332168

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