ICF13A

13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- following system of boundary-domain integral equations (BDIEs) for the mechanical and thermal fields at the boundary and interior points is obtained as                                       ( ) ( ) ( ) ( ) ( ) ( ) 0 ( ) ( ) ( ) 0 0 0 1 , , , , , , , ( ) ( ) , , , , , , , 0, ( ) 1 , , , , , , , ( ) n n n j ij i ij i n u n n j n n j j n i i i i u p T p u p U p t p d E k Z p p U p q p d F p E p p T p u p U p t p d E                                                 x x y y x y y y y y x y y x y y x y y x x y y x y y y y                      ( ) ( ) ( ) 0 ( ) , , , , , , , 0, ( ) n n n k F p p T p q p d F p E                  y x y y x y y x y y    (11) where x and y represent the source and observation points, ( , , ) ij U p x y , ( , , ) iU p x y , (,,) T p x y , ( , , ) ij T p x y , ( , , ) iT p x y , (,,) iZ p x y and (,,) F p x y are the fundamental solutions [6, 8, 9]. Here, a tilde denotes the ratio of the non-homogeneous quantity to the corresponding homogeneous quantity. The functions (1)( )u jF and (1)( ) F  describe the non-homogeneity of the FG layer. They vanish completely for the homogeneous substrate (0)  . The functions (1)( )u jF and (1)( ) F  are defined as [8, 9] (1)( ) 2 (1) (1) (2) (1) (1) , (1) (3) (1) (2) (4) ( , ) ( ) ( , , ) ( , ) ( ) ( , , ) ( ) ( , , ) ( , ) ( ) ( , , ) ( ) ( , , ) ( , ) ( ) ( , , ) ( ) ( , , ) u j ij i y k ij ijk i k y j j kk y ji i j F p p U p u p d U p V p u p d p U p U p p d E p U p                               x y x y y y x y y x y y y x y y x y y y x y y x y (1) , ( , ) , i y p d        y (12) (1)( ) 2 (1) (1) 0 0 0 (2) (1) (1) , (1) (3) (1) 0 0 0 (2) (4 ( , ) ( ) ( , , ) ( , ) ( ) ( , , ) ( ) ( , , ) ( , ) ( ) ( , , ) ( ) ( , , ) ( , ) ( ) ( , , ) i i y k i ik i k y j j y i i p F p p U p u p d U p W p u p d p p T p U p p d G p                                               x y x y y y x y y x y y y x y y x y y y x y ) (1) , ( ) ( , , ) ( , ) , i y T p p d        y x y y (13) where the fundamental solutions ( , , ) ij E p x y , ( , , ) iG p x y , ( , , ) ijk V p x y and ( , , ) ikW p x y are given in [8]. It should be noted that Eqs. (11) are no longer pure boundary integral formulations in the non-homogeneous sub-domain (1)  because they involve domain integrals containing unknown fields. The BDIEs (11) contain boundary and domain integrals with singular kernels. The strongly singular integrals are interpreted in the sense of the Cauchy principal value. Making use of the

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