13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- traction fundamental solutions. GIJ d and GIJ s are fundamental solutions of higher order. These fundamental solutions have been derived by Wang and Zhang [3]. 2.3. Laplace-domain dynamic fundamental solutions The traction fundamental solutions GIJ p and GIJ d have a strong singularity of Cauchy type, while the higher-order traction fundamental solution GIJ s has a hypersingularity. To deal with the singularity, it is advantageous to split the fundamental solutions into a singular static part and a regular dynamic part. The singular static parts are given by 3 G(S) 1 2 0 0 1 ( ) ( ) 1 , Re M IJ JM MI M M M n n p L Q z z x x x x , (8) 3 G(S) 1 0 2 0 0 0 1 ( ) ( ) 1 , Re M IJ JM MI M M M n n d L Q z z x x x x , (9) 3 G(S) 1 2 0 0 2 1 ( ) ( ) 1 , Re ( ) M M IJ IJ M M M n n s T z z x x x x , (10) where M M z x y , 0 0 0 M M z x y , 1,2,3 M . (11) 0 0 0 ( , ) x y x is the source point while ( , ) x y x the observation point. JML , MI Q , M IJ T and M are determined by the anisotropic material constants [3]. 2.4. Time-stepping scheme To approximate the Riemann convolution integrals in BIEs, Lubich quadrature formula is used [1] 0 n n j j f t g t h t t h j t , (12) where time t is divided into N equal time-steps, and the weights n j t are determined by ( ) 1 2 ( ) / 0 ˆ / e n j N i n j m N n j m m r t g t N , (13) in which ˆg stands for the Laplace-transformation of the function g(t). After the temporal and spatial discretization, a system of linear equations for the discrete boundary quantities can be obtained. Leaving all the unknown boundary quantities on the left-hand side, an explicit time-stepping scheme 1 0 1 1 n n n n j j k kl l kl l j A A x y x , (14) can be obtained for computing the unknown boundary quantities at the n-th time-step. 3. Computation of singular integrals When a collocation point is on one element Γe, the BIEs possess strongly singular and hypersingular integrals. After discretization, the singular integrals correspond to the following integrals, 1 2 0 ( ) ( ) e M M q M M n n I d z z x x , 1 2 0 2 ( ) ( ) ( ) e M M q M M n n I d z z x x (15) where q(q=1,2,3) is the quadratic shape function and ( ) n x donates the outward unit normal
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