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.