Issue 32

I. Telichev, Frattura ed Integrità Strutturale, 32 (2015) 24-34; DOI: 10.3221/IGF-ESIS.32.03 28 T singular integral equation technique is a powerful alternative to the finite element method in the non-linear analysis of crack propagation which provides very rapid convergence of the numerical results [7-11]. Fig. 5 summarizes the procedure of non-linear fracture analysis which employs the method of singular integral equation and includes the following basic steps: Modules 1-2: The analysis starts with specifying the design and material characteristics of the pressure wall and determining the impact hole parameters. Module 3: The piecewise traction distribution p(x) is applied to the crack surface as it is shown in Fig. 4-b. It divides the contour into 5 portions (links) L 0 , L 1 , L 2 , L 3 and L 4 , where each piece of the traction function is differentiable throughout each individual link. The traction-free link L 0 corresponds to the hole, links L 1 and L 3 are radial cracks and links L 2 and L 4 represent the plastic zones. The solution of the singular integral Eq. (1) must satisfy the condition of single-valuedness of displacements for the crack contour: 0 0 ' ' 0 0 1 ( ) exp( ) ( ) 0 q q l l H q q q q q l l g t dt i g t dt                    where H reflects the total number of links within the crack; q is the current number of link;  is the angle of link inclination and l is half of the crack link. Also, the symmetry of the problem and link angular positions (  1 =  2 =0,  3 =  4 = π ) were taken into account. Module 4: Unlike the finite element method the method of singular integral equations is free of mesh generation and only nodes are needed. The Chebyshev’s nodes with normalized coordinates  and  changing from -1 to 1are generated on each link of the contour (Fig. 4c, d) where        [ (2 1) / (2 )],   [ / (2 )] k m cos k N cos m N , 1,  ,  k N  1, ( 1) m N   . The open circles indicate the points ξ 1 ,.., ξ N on the crack faces where displacements are calculated. The closed circles correspond to the traction nodes η 1 , .., η N-1 . The normalized coordinates  and  change from -1 to 1. Module 5: An efficient approach to account for the jump discontinuities of traction applied to the crack faces was proposed by Savruk [7]. Following [7] the Eq. (1) for the case of 5-link crack is replaced by the system of singular integral equations:                     1 0 0 1 3 1 1 2 4 2 1 0 1 , , , , , } ,    0,2      0 n n n n n n M M M M M n                                                  (3) where M 00 ,.., M 02 , M 10 ,.., M 12 , M 20 ,.., M 22 are normalized kernels,  0 ,..,  2 are normalized derivatives of the displacement discontinuity across the crack contour  0 (  )=g 0 ’ (l 0  ),..,  2 (  )=g 2 ’ (l 2  ), and  0 ,..,  2 are normalized tractions  0 (  )=p 0 (l 0  ),. .,  2 (  )=p 2 (l 2  ). The last equation of the system (3) represents the condition of single-valuedness of displacements for the crack contour. Also, the symmetry of the problem and link angular positions are taken into account. Module 6: The numerical solution of the system of singular integral Eq. (3) is obtained by the method of mechanical quadratures [7]. Functions  0 (  ),  1 (  ),  2 (  ) are sought in the class of functions unbounded at the ends of intervals 2 ( ) ( ) / 1 n n u       , ( n =0, 1, 2) expressing u 0 (  ), u 1 (  ), u 2 (  ) in terms of the Lagrange interpolation polynomials over the Chebyshev nodes:         1 1 1 1 1 2 ,  0,2 N N n n k r k r k r u u T T n N                         (4) where   [  ( )] r T cos r arccos    is a first kind Chebyshev polynomial. Boundary conditions   1 1 0 u  and   2 1 0 u   are applied to complete the system of Eq. (3). By applying the Gauss-Chebyshev quadrature expressions the system of singular integral equations is transformed to the closed system of linear algebraic equations with 3 N unknowns where N is number of the Chebyshev nodes. Module 7: The solution of closed normalized and linearized system of equations is obtained by Gauss elimination.