13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- boundary free of calculating principal value using a small circular bump. Therefore, the representations of integral equations including the boundary point for the interior problem can be written as w TwdB UtdB DB B i B i s x s s x s s x s x ( , ) ( ) ( ), ( , ) ( ) ( ) ( ) , (15) L t dB D B t M w dB B i B i s x s s x s s x s x ( , ) ( ) ( ), ( , ) ( ) ( ) ( ) , (16) and T wdB U tdB DB c B e B e s x s s x s s x s ( , ) ( ) ( ), ( , ) ( ) ( ) 0 , (17) L t dB D B M w dB c B e B e s x s s x s s x s ( , ) ( ) ( ), ( , ) ( ) ( ) 0 , (18) once the kernels are expressed in terms of an appropriate degenerate forms (denoted by subscripts i and e) instead of the closed-form fundamental solution. It is noted that x in Eqs.(15)-(18) can be exactly located on the real boundary. For the exterior problem, the domain of interest (D) is in the external region of the elliptical boundary and the complementary domain (Dc) is in the internal region of the ellipse. Therefore, the null-field boundary integral equations are represented as w TwdB UtdB DB B e B e s x s s x s x s s x ( , ) ( ) ( ), ( , ) ( ) ( ) ( ) , (19) L t dB D B t M w dB B e B e s x s s x s x s s x ( , ) ( ) ( ), ( , ) ( ) ( ) ( ) , (20) and T wdB U tdB DB c B i B i s x s s x s s x s ( , ) ( ) ( ), ( , ) ( ) ( ) 0 , (21) L t dB D B M w dB c B i B i s x s s x s s x s ( , ) ( ) ( ), ( , ) ( ) ( ) 0 , (22) Also, the observation point x in Eqs.(19)-(22) can be exactly located on the real boundary. For various problems (interior or exterior), we used different kernel functions (denoted by superscripts “i” and “e”) so that the jump behavior across boundary can be captured. Therefore, different expressions of the kernels for the interior and exterior observer points are used and they will be elaborated on later. 2.2.3 Expansions of the fundamental solution and the boundary density Based on the separable property, the kernel function ( , ) s x U can be expanded into degenerate form by employing the separating technique for source point and field point under the elliptical coordinates. The fundamental solution, ( , ) s x U , in terms of degenerate (separable) kernel is shown below: , , sin sin sinh 2 cos cos cosh 2 2 ln 2 1 ( , ; , ) , , sin sin sinh 2 cos cos cosh 2 2 ln 2 1 ( , ; , ) ( , ) 1 1 1 1 m m m m e m m m m i e m m m m e m m m m c U e m m m m m m m e m c U U s x (23) where the position of the source point is ( , ) s and the field point is ( , ) x , the superscripts “i” and “e” denote the interior ( ) and exterior ( ) cases, respectively. The other kernels in the boundary integral equation can be obtained by utilizing the operators of Eq.(10) with respect to the kernel ( , ) s x U . In the real computation, the degenerate kernel can be expressed as finite sums of products of functions of s alone and functions of x alone.
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