13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- homogeneous and linear thermoelastic solids. The boundary-domain integral equations (BDIEs) are obtained for the unknown mechanical and thermal fields and then applied to each sub-domain and the continuity conditions are employed on the interface boundary. The BDIEs for the FG layer contain domain integrals, which describe the material's non-homogeneity. A crucial point of the numerical solution procedure is how to evaluate the domain integrals without discretization of the non-homogeneous sub-domain into internal cells. In this analysis, the domain integrals are transformed into boundary integrals over the global boundary by using the radial integration method (RIM) [3, 4]. For the homogeneous and linear elastic substrate, only boundary integrals need to be considered in the boundary integral equations. A collocation method is implemented for the spatial discretization. The final time-dependent solutions are obtained by using the Stehfest’s algorithm [5] for the inverse Laplace-transform. Numerical results are presented and discussed to demonstrate the accuracy and efficiency of the proposed BEM as well as the effects of the material gradation and thermo-mechanical coupling on the dynamic stress intensity factors (SIFs). 2. Problem statement Let us consider isotropic and linear thermoelastic bimaterials in a 2-D domain, which are composed of an FG layer attached to a homogeneous substrate. The FG/homogeneous bimaterials are modeled by using a sub-domain technique [2]. The bimaterial system is divided into a homogeneous (0) and a non-homogeneous sub-domain (1) along the interface. The material properties of the FG layer such as the mass density 1( )x ñ , the Young’s modulus 1( ) E x , the thermal conductivity 1( ) k x , the specific heat 1( ) c x and the linear expansion coefficient 1( ) x are assumed to be continuous functions of the Cartesian coordinates, while the Poisson’s ratio 1 and the material parameters of the homogeneous substrate, which are denoted by a subscript zero, are taken as constant. In this case, the elasticity tensor can be written as (0) (1) 0 (0)0 (1)0 1 , ( ) ( ) ijkl ij ijkl kl ijkl c c c c x x (1) with ( )0 0 1 0 1 0 1 ( ) , ( ) , , 0,1, 2(1 ) 2(1 ) 1 2 n ij kl ki lj kj n ijk n l l i E E n c x x where n is the shear modulus and ij is the Kronecker delta symbol. The relation between the stresses ( ) ( , ) n ij t x and the displacements ( ) , ( , ) n k l u t x with the consideration of the temperature changes ( ) ( , ) n t x is defined by the Duhamel-Neumann constitutive equations ( ) ( ) ( ) ( ) , ( , ) ( ) ( , ) ( ) ( , ) , 0,1, n n n n ij ijkl k l n ij t c u t n t x x x x x (2) where the stress-temperature modulus is given by / (1 2 ). n n n n E In the absence of body forces and heat sources, the governing equations in transient linear coupled thermoelasticity are given by the equations of motion and the thermal balance equations [6] ( ) ( ) , ( , ) ( ) ( , ) 0, i n n n ij j t u t x x x ñ (3) ( ) ( ) ( ) , , , ( ) ( , ) ( ) ( ) ( , ) ( ) ( ) ( , ) 0, n n n n i n n n n k k i k t c t k u t x x x x x x x x ñ (4) where 0 / n nT k and 0T is the reference temperature. Unless otherwise stated, the conventional summation rule over double indices is implied, 0,1 n and other Latin indices take the values of 1
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