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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2. Mathematical statement of the problem. Basic equations In either generalized plane stress or plane strain, equilibrium is ensured for all stresses derived from a stress function by 2 2 (4) where cylindrical coordinates , r  are centered at the right end of the crack. According to (Hutchinson (1968), Rice and Rosengren (1968)) the non-dimensional stress function and coordinate are given in terms of the dimensional quantities by ( ) 2 0 / , L    = / , r r L = where L is the half length of the crack. Condition of compatibility has the form ( ) 2 2 2 0. r rr rr r r r r r r r r                   + − − =               (5) The power-law constitutive relations for the plane strain conditions are described by ( ) 1 1 3 3 , , 4 2 n n rr e rr r e r             − − = − = − = (6) where the Mises equivalent stress is expressed by ( ) 2 2 2 3 / 4 3 e rr r       = − + . For plane stress conditions the constitutive equations have the form ( ) ( ) 1 1 1 1 1 3 2 , 2 , , 2 2 2 n n n rr e rr e rr r e r                 − − − = − = − = (7) where the Mises equivalent stress is expressed by 2 2 2 3 . e rr rr r          = + − + An asymptotic expansion of the solution is attempted in the separable form ( ) 1 1 1 2 , ( ) ( ) ... r r f r f       + + = + + (8) where, if the first term is to be singled out as the dominant one, .    In (Hutchinson (1968), Rice and Rosengren (1968)) one can see the two-term asymptotic expansion. However, the attention is restricted to only the dominant (leading) term in the asymptotic series expansion ( ) 1 , ( ). r r f     + = (9) The resulting ordinary differential equation following from (5) is homogeneous in ( ) f  and is associated with homogeneous boundary conditions has the form of the nonlinear eigenvalue equation for  : ( ) ( )   ( ) ( )   ( ) ( )  ( ) ( ) ( )  ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 ( 1)( 3) 1 1 4 1 ( 1) 1 1 1 4 1 2( 1) 1 1 IV e e e e f f n f f f n n f f f f f f f f n f f f f f f f f f f f n f f f f               − − + + + − −                 − + − + + − + +               + − − + + − + − +           + + − + + −         − + − +     ( ) ( ) ( )   ( ) ( ) 2 2 2 2 2 2 1 4 4 2 4 2 1 2 4 1 ( 1) 1 1 4 1 1 0, e e e e f f f f f C n f f f f f f f f C f f C f f f f f                 + − + +               + − − + − + + +          + − − + + − =   (10) where the following notations ( )     2 2 2 2 2 1 2 1 4 , 4 ( 1) 1 , ( 1) ( 1) 2 e f f f f C n C n n           = − + + = − + = − − +   are adopted. The equation is derived for plane strain conditions. The fourth order nonlinear differential equation (10) with boundary conditions 2 2 2 1 1 1     ( , ) r   , ( , ) r   , ( , ) r   , rr r r r  r r    r r             = + = = −    