13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- and 2. From the mathematical point of view, the governing equations are the coupled partial differential equations with variable coefficients for the FG sub-domain (1) . A measure of the thermo-mechanical coupling due to the dilatational term ( ) , n n k k u in Eq. (4) is defined by a dimensionless coupling parameter [6, 7] 2 0 (1 ) , (1 )(1 2 ) n n n n n n n n E T c ñ (5) that equals zero for an uncoupled problem. The following essential and natural boundary conditions for the mechanical and thermal quantities are prescribed as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( , ) ( , ), , ( , ) ( , ), , ( , ) ( , ), , ( , ) ), , ( , n n n n n n i i u n n n n n n i i t q u t u t t t t t t t q t q t x x x x x x x x x x x x (6) where ( )n it and ( )n q represent the traction vector and the heat flux defined by ( ) ( ) ( ) ( ) , ( , ) ( , ) ( ), ( , ) ( ) ( , ) ( ). n n n n i ij j n i i t t t n q t k t n x x x x x x x Here, () in x denotes the components of the outward unit normal vector, ( )n u and ( )n t are the parts of the external boundary ( ) ( ) ( ) n n n u t , ( ) ( ) n n u t , in which the displacements ( )n iu and the tractions ( )n it are given, respectively; ( )n and ( )n q are the parts of the boundary ( ) ( ) ( ) n n n q , ( ) ( ) n n q with the specified temperature ( )n and the heat flux ( )n q , respectively. The crack-faces are assumed to be free of mechanical and thermal loadings ( ) ( ) ( , ) 0, , ( , , ) 0, n n i c c t t q t x x x x (7) where c c c represents the upper and lower crack-faces. The continuity conditions on the interface are prescribed as (0) (1) (0) (1) (01) (0) (1) (0) (1) (01) ( , ) ( , ), ( , ) ( , ), , ( , ) ( , ), ( , ) ( , ), . i i i i u t u t t t t t t t q t q t x x x x x x x x x x (8) The initial conditions are given by ( , ) ( , ) 0, ( , ) 0 for 0. i i u t u t t t x x x (9) Applying the Laplace-transform to Eqs. (3) and (4) and substituting Eqs. (1) and (2) yield ( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) , , , , , , ( ) ( ) 2 ( ) ( ) , , ( )0 , 0, 0, n n n n n n n n k lj j k l n i n i i n ijkl n i n n n n ii i n n k k n c u u p u k pu k 駐 (10) where a superimposed bar over a quantity denotes the Laplace-transformed quantity, p is the Laplace-transform parameter, 牋n n n n k c ñ is the thermal conductivity and 2 / n n p . Integral representations of the displacements and the temperature at an arbitrary point of the domain are derived from the generalized Betti’s reciprocal theorem in conjunction with the fundamental solutions of the Laplace-transformed linear coupled thermoelasticity for a homogeneous solid [6, 8, 9]. By moving the observation point to the boundary ( )n x or keeping it in the domain ( )n x the
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