lo = 1kZ L Re( z, k)+, Im(r r'k)} (A25)


 where

 g = cos + p sin 

 where Re and Im stand for real and imaginary. Since the material 1 is divided into

two parts by a crack, the displacements and the stresses in the Material 1 has superscript

of (+) or (-). A superscript (+) to denote properties of the Material 1 in the positive

region of the xl and (-) to denote properties of Material 1 in the negative region of the xl

axis. The stress free boundary conditions at the crack surface are

 S= 0 at = (A26)
 Olj = 0 at 0 = -f

 The continuity conditions at the interface are

 u, u =0 at ( = / 2 (A27)
 ,+ ', =0 at 0 = ;/2

 u' -u =0 at =-'/2 (A28)
 (7f u -2 = 0 at 0 = r /2

 Using the displacement and stress equations, the boundary and the continuity

equation results in 18 equations for the eighteen coefficients of AL, BL A', B', AL and

BL. The equations can be written in matrix form

 K(-)q = 0 (A29)

 where K is an 18 X 18 square matrix and the elements of q are A+, BL, AL, BL,

AL and BL. For a nontrivial solution of q, determinant of the matrix K must be zero when

singularity 2 satisfies the matrix K. The determinant was calculated numerically.