Issue34

Y. Sumi, Frattura ed Integrità Strutturale, 34 (2015) 42-58; DOI: 10.3221/IGF-ESIS.34.04 44 ( ) ( ) ( ), 1 1 h        (3) Following the same method as used by Banichuk [1], Goldstein and Salganik [2, 3], and Cotterell and Rice [4], the perturbation solution of the stress field can be expressed by Muskhelishvili’s analytic functions [18] given by Figure 2 : Coordinate systems for a kinked and curved crack. 2[ '( ) '( )], 11 22 2 2[( ) ''( ) ( ) '( )], 22 11 12 z z i z z z z z                    (4) where z = x 1 + ix 2 . The analytic functions   ( z ) and  ( z ) are expanded in terms of  (  1 ) up to the second order in the following form: 3 0 1 2 3 0 1 2 '( ) ' ( ) ' ( ) ' ( ) ( ), ( ) ( ) ( ) ( ) ( ), z z z z O z z z z O                   (5) in which    z  and   ( z ) are of the zero-th order,    z  and   ( z ) are of the first order, and    z  and   ( z ) are of the second order in      . The boundary conditions on the crack surfaces, at z =  1 + i      become 2 '( ) '( ) [( ) ''( ) ( ) '( )] ( ), i n s Z z e z z z z z T iT                (6) where  is the angle of the slope on the crack line. It is assumed that the crack surfaces are subjected to normal and shear tractions T n and T s on the upper surface and - T n and - T s on the lower surface, respectively. We also assume that tractions on the crack surfaces, T n and T s are bounded and integrable in the defined range. Approximate Description of a Slightly Kinked and Curved Extension of a Straight Crack The second order approximation of the stress distribution in the vicinity of the original crack tip can be expressed by 1 2 3 ( , 0) ( , 0) ( , 0) ( ), 1 ,2 1 ,22 1 2 x x x O ij ij ij ij            (7) in which I 1 ( , 0) ( ), 11 1 I 1 2 2 1 I 1 ( , 0) ( ), 22 1 I 1 2 2 1 II 1 ( , 0) ( ), 12 1 II 1 2 2 1 k x x T b O x x k x x b O x x k x x b O x x                    (8) where k I and k II are the stress intensity factors, and the coefficients T , b I , and b II are also determined from the solution of the boundary-value problem prior to the crack extension. The surface tractions on the extended crack surfaces, T n and T s