8 view of the fact that 1 0 2 for 2R ) and meet the asymptotic condition in Eq. (9). It can be easily proved that under a linearly distributed stress field at infinity, the internal stress field within an elliptical inclusion perfectly bonded to the surrounding matrix is always a linear function of the coordinates. The above obtained results demonstrate the interesting fact that the internal stress field within a three-phase elliptical inclusion can still keep a simple linear form if the remotely applied non-uniformly loadings satisfy the two conditions in Eq. (18). 3. Discussions In this section several typical examples will be presented to illustrate the application of Eq. (18) to the design of three-phase elliptical inclusions when the matrix is subjected to a linearly distributed stress field at infinity. 3.1. The materials comprising the interphase layer and the matrix are identical ( 3 2 and 3 2 ) In this case, Eqs. (18) becomes , ) 2( 1) )( ( ) ) 2( (4 4 ( 1) ( 1) ) 2 ( 1 2 1 2 1 1 1 1 1 2 1 1 2 1 1 1 2 4 1 1 1 2 6 1 1 1 R R R R R A B (27) . ) 2( 1) )( ( ) ) 2( (4 ) (4 ( 1) ( 1) ) 2 ( 1 2 1 2 1 1 1 1 1 1 2 1 2 1 1 1 2 4 1 1 1 2 6 1 2 2 R R R R R A B (28) In particular, if the inclusion is a hole ( 1 1 ), Eqs. (27) and (28) become , 1 2 2 1 4 1 1 1 A R R B (29) . 1 2 2 1 4 1 2 2 A R R B (30) On the other extreme end, if the inclusion is rigid ( 1 1 ), Eqs. (27) and (28) become , 2 2 1 4 1 2 1 1 A R R B (31) . 2 2 1 4 1 2 2 2 A R R B (32) It is deduced from Eqs. (13) and (14) that , ) ( 1)( ) 2 ( 1) )( ( ( ) ( ) 2 2 1 2 1 1 2 2 1 2 1 2 1 1 1 3 2 z R R R R R X X z z (33)
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