3 [ ] ', ( ) ) ln( ( ) ( ) 2 1 2 ) ( 2 1 ( ) ( ) 2 ( ) 12 1 2 + g þ ý ü î í ì - - - - p y =Y = s-s+s + ò ò +G - a ¥ ¥ ¥ S i t dt t z tH t t z H t q t e i i z z dz z (7) = + Î +G + k+ = - a for t y iy S q t w t e H t i t 2 1 ( ) ] ( ) [ ( ) 1 1 ( ) , (8) Before we continue to consider the boundary values of our complex functions, it is useful to note from (8) that q(t) can be expressed in terms of H(t) and w(t) as q t H t w t e for t y iy S i t ( ) ( ) ( ) ( ) ( ) = + - = + Î + - k a 1 1 2 G. (9) Figure 1: A sketch for an infinite elastic body containing n cracks j (j = 1,...,n) and m holes Si (i=1,..., m) subjected to far field stressess1 ¥ , s 2 ¥ and s 12 ¥ . In obtaining the above formula, we have let the outer boundary tends to infinity and the components of stress at infinity are given as s1 ¥ , s 2 ¥ and s 12 ¥ . This formulation bears a close resemblance with Muskhelishvili’s formalism (1975). In these formulas, S denotes the union of the holes S1 , S2 , ... , Sm , C G = + ¥ 2 1 e k / ( ) with e¥ being the rotation at infinity, and the outer boundary S0 , and G the union of the cracks G1 , G2 , ... , Gn . The shear modulus and Poisson’s ratio are denoted by G and respectively. The plane parameter equals 3 4 for plane strain or (3 )/(1+ ) for plane stress. The angle between the tangent at t on S+ and the global coordinate axis ox1 is denoted by (t). And ij and ui (i,j =1,2) are the components of stress and displacement in the Cartesian coordinate system ox1x2 , respectively. n and ns respectively are the normal and shear stresses on the boundary. The superscripts “+” and “ ” denote the upper and lower crack faces respectively. The complex integration constants and relate only to rigid displacements. It is obvious that the only unknown function in the boundary integral formulation for (t), (t), (t) and (t) is H(t). Therefore, only one variable is needed in this complex formulation and this is one of the main advantage of using the present complex formulation. MIXED BOUNDARY INTEGRAL EQUATIONS
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