4 ) , (1 1 ( ) , 1 ( ) 1 2 2 1 2 2 1 R Y X (10) where X and Y are complex constants to be determined. Consequently it follows from Eqs. (7) and (10) that , ( 1) 1 2 ( 1) ) ( ( 1) ( ) 2 4 1 4 1 2 1 1 2 2 1 2 1 2 1 1 2 2 1 1 1 2 2 R R R Y R X X ) ( 2 1 R R (11) It is found that the necessary and sufficient condition for the validity of Eq. (10) for the internal stress field is that 2( ) should take the form of ( ) ( 1 ) 2 2 2 with being a complex constant. This condition can be easily arrived at from Eq. (2.14) in [2]. The necessity is true only when 1 2 2 2 1 R R and 1 3 . Consequently the following relationship between X and Y establishes , ( 1, 1, 1) 2 3 1 2 1 2 1 R R X Y (12) Once the above relationship is satisfied, and all the interface conditions in Eqs. (7) are enforced, 2( ) and 2( ) are found to have the following expressions , 1 ) ( 1)( ) ( ) (4 4 ( 1) ( 1) 2 ) ( 1)( ) ( ) (4 4 ( 1) ( 1) 2 ( ) , 1 ) ( 1)( ) 2 ( 1) )( ( ( ) 2 2 1 2 1 1 2 1 1 2 6 1 2 1 1 2 1 1 1 1 1 2 4 1 2 2 1 2 1 1 2 1 1 2 6 1 2 1 1 2 1 1 1 1 1 2 4 1 2 2 2 2 1 2 1 1 2 1 2 1 2 1 1 1 2 R R R X R X R R R R X R X R R R R R X X ) ( 2 1 R R (13) Similarly by enforcing the interface conditions in Eq. (8), we arrive at the following expression of 3( ) and 3( ) , ) ( 1)( 1)( 1 ) ( ) (4 4 ( 1) ( 1) (1 ) 2 ) ( ) (4 4 ( 1) ( 1) (1 ) 2 ) 2 ( 1) )( ) ( 2(1 1 ) 2 ( 1) )( 1) ( ( ( ) 2 1 2 1 3 1 2 2 1 1 2 6 1 2 1 1 2 1 1 1 1 1 2 4 1 3 4 2 2 1 1 2 6 1 2 1 1 2 1 1 1 1 1 2 4 1 3 4 2 2 2 2 2 2 2 1 2 1 2 1 1 1 3 2 2 1 2 1 2 1 1 1 3 2 3 R R R X R X R R R X R X R R R R R R X X R R X X ) ( 2R (14)
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