1 0 21 22 21 0 23 22 23 0 24 24 33 1 1 ln , 1 1 1 , , 1 1 , , a a a u u v a a a a a a v a a a a a ext a a k k t x t dt k t x t dt t dt t x k k t x t dt t dt k t x t dt t x k t dt k t x t dt x t x , x a (9b) 1 0 31 32 31 0 33 32 33 0 34 34 3 1 1 ln , 1 1 1 , , 1 1 , 1 , a a a u u v a a a a a a v a a a a a e ext a a k k t x t dt k t x t dt t dt t x k k t x t dt t dt k t x t dt t x k t dt k t x t dt k D x t x , x a (9c) 1 0 41 42 41 0 43 42 43 0 44 44 3 1 1 ln , 1 1 1 , , 1 1 , 1 , a a a u u v a a a a a a v a a a a a m ext a a k k t x t dt k t x t dt t dt t x k k t x t dt t dt k t x t dt t x k t dt k t x t dt k B x t x , x a (9d) where the functions , , ij k t x where , 1...4 i j , are known continuous and bounded kernels that depend on the nonhomogeneity parameter . The solution of (9a-d) subject to the single-valuedness conditions may be expressed as , 1..4 i i t w t t i . In this solution, 2 1 1 w t t is the weight function which is obtained from the nature of the singularity at the crack tips and which is associated with the Chebyshev polynomial of the first kind t n T t n cos arccos . The functions , 1..4 i t i are continuous and bounded functions in the interval 1,1 which may be expressed as truncated series of Chebyshev polynomial of the first kind. Using a suitable collocation method, a linear algebraic system of the unknown coefficients of the density functions is obtained. As a result, the stress intensity factors, the electric displacement intensity factor and the magnetic induction intensity factor can be expressed as: 0 0 0 1 22 23 24 1 1 , N n n n n k k b k c k d 0 0 0 1 22 23 24 1 1 1 , N n n n n n k k b k c k d (10a,b) 0 2 11 1 1 , N n n k k a 0 2 11 1 1 1 , N n n n k k a (10c,d) 0 0 0 32 33 34 1 1 , N D n n n n k k b k c k d 0 0 0 32 33 34 1 1 1 , N n D n n n n k k b k c k d (10e,f) 0 0 0 42 43 44 1 1 , N B n n n n k k b k c k d 0 0 0 42 43 44 1 1 1 . N n B n n n n k k b k c k d (10g,h)
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