13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- equations as shown in Eq.(2). Besides, the interface between the matrix and inclusion is assumed perfectly bonded and it satisfies the following interface condition for stress fields and electric fields I Mw w and I z Mz on kB , (5) I M and I MD D on kB . (6) 2.2. Dual null-field boundary integral formulation 2.2.1 Conventional version The integral equation for the domain point can be derived from the third Green’s identity, we have D U t dB T w dB w B B s x s s x s x s s x ( , ) ( ) ( ), ( , ) ( ) ( ) ( ) , (7) D L t dB t M w dB B B s x s s x s x s s x ( , ) ( ) ( ), ( , ) ( ) ( ) ( ) , (8) where B is the boundary, s and x are the source and field points, respectively, x x n x ( ) ( ) t w , ( ) ( ) t w s s s n , sn and xn denote the outward normal vectors at the source point s and field point x, respectively, D is the domain of interest and the kernel function, r U ln 2 1 ( , ) s x ( | | x s r ), is the fundamental solution which satisfies ) ( , ) ( 2 x s x s U , (9) in which ) (x s denotes the Dirac-delta function. The other kernel functions, ( , ) s x T , ( , ) s x L , and ( , ) s x M , are defined by sn s x s x ( , ) ( , ) U T , xn s x s x ( , ) ( , ) U L , s x n n s x s x ( , ) ( , ) 2U M . (10) By moving the field point x to the boundary, the dual boundary integral equations for the boundary point can be obtained as follows: 1 ( ) . . . ( , ) ( ) ( ) ( , ) ( ) ( ), 2 B B w CPV T w dB U t dB B x s x s s s x s s x , (11) 1 ( ) . . . (,) () () ... ( , ) ( ) ( ), 2 B B t HPV M wdB CPV L tdB B x s x s s s x s s x , (12) where C.P.V. and H.P.V. denote the Cauchy principal value and Hadamard (or called Mangler) principal value, respectively. Besides, once the field point x locates outside the domain ( cDx ), we obtain the dual null-field integral equations as shown below c B B D U t dB T w dB s x s s x s s x s ( , ) ( ) ( ), ( , ) ( ) ( ) 0 , (13) c B B D L t dB M w dB s x s s x s s x s ( , ) ( ) ( ), ( , ) ( ) ( ) 0 , (14) where cD is the complementary domain. Equations (7), (8), (13) and (14) are conventional formulations where the point is not located on the real boundary. Singularity occurs and concept of principal values is required once Eqs.(11) and (12) are considered. The traction t(s) is the directional derivative of w(s) along the outer normal direction at s. In order to satisfy the interface condition, the collocation points are located on the boundary. For calculating the stress in the domain, the normal vector of an interior point is artificially given, e.g. t w x ( ) ( ) x x , if (1,0) xn and t w y ( ) ( ) x x , if (0,1) xn . In other words, the selection of n depends on the stress under consideration. 2.2.2 Present version By introducing the degenerate kernels, the collocation point can be located on the real
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