PSI - Issue 37

L.V. Stepanova et al. / Procedia Structural Integrity 37 (2022) 908–919 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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0,   (11) defines a nonlinear eigenvalue problem in which the constant  is the eigenvalue and ( ) f  is the corresponding eigenfunction. For plane stress conditions the compatibility condition (5) results in the nonlinear ordinary differential equation for the function ( ) f  : ( )                  ( )   2 2 2 2 4 2 2 2 2 2 2 2 1 ( 1)(2 ) 2 /2 2 6 ( 1) 1 ( 1) ' ( 1)( 3) ( 1)(2 ) 2 ( 1) ( 1)( 2) 2 ( 1) ' ( 1) ( 1) ( 1) ' ( 1) / 2 ( 1) ' ( 1) IV e e e e e f f n f f f n n f hf f f n n h f f n f f f f f f f f f ff ff f f                    − + − + + + − + − + + − −        + − + + − + + + + + + + + + +    + + + − + − + + + ( ) ( )   ( )          4 2 2 2 4 4 1 2 ( 1) ( 1) 3 2( 1) ( 1)(2 ) 2 / 2 ( 1) ( 1)(2 ) 2 ( 1) 1 ( 1) ( 1)(2 1) 0, e e e e f f f f f f f f f n f h f f nf f f n nf f f                   + + − −        − + + + + + + − + − + −   − − + − + + − + − + − − = (12) where the following notations are adopted     ( )         2 2 2 2 2 2 2 2 2 ( 1) ( 1) ( 1) 1 3 , ( 1) ( 1) ( 1) ( 1) ( 1) / 2 ( 1) ( 1) / 2 3 . e f f f f f f f f h f f f f ff f f f f f f f f                       = + + + + − + + + + = + +            + + + + − + + + − + + + + The boundary conditions imposed on the function ( ) f  follow from the traction free boundary conditions on the crack surfaces (11). The order of the stress singularity is the eigenvalue and the angular variations of the field quantities correspond to the eigenfunctions. The solution of Hutchinson and Rice and Rosengren is the most important achievement of nonlinear fracture mechanics. Equations (10) and (12) give the angular distribution function of the solution. The function is governed by very complicated nonlinear differential equations of the fourth order. Thus, the eigenfunction expansion method results in the nonlinear eigenvalue problem: it is necessary to find eigenvalues leading to nontrivial solutions of Eqs. (10) and (12) satisfying the boundary conditions (11). The eigenvalue corresponding to the HRR problem is well known / ( 1). n n  = + (13) 3. Eigenspectrum of the nonlinear eigenvalue problem The further development of fracture mechanics required analysis of eigenspectra and orders of singularity at a crack tip for power-law materials (Meng and Lee (1998), Stepanova (2008), Stepanova (2009)). In (Meng and Lee (1998)) the necessity of introducing higher or lower order singular terms to more correctly describe the asymptotic fields of 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         + + + = + + + (14) The new eigenvalues of the HRR problem different from / ( 1) n n  = + 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.    =  = =  = 0. f f