9 which implies that the mean stress in S2 and S3 is not disturbed. Thus the elliptical inclusion is harmonic when the remote non-uniform loadings satisfy the two restrictions given by Eqs. (27) and (28). It seems that the discussion on harmonic elliptical inclusions under non-uniform loadings has not been recorded in the literature. Previously harmonic elliptical inclusions were found only for remote uniform stresses [2, 7, 8]; whereas only non-elliptical harmonic inclusions were found for non-uniform loadings [9, 10]. Results similar to those derived in this subsection can also be obtained from Eq. (18) by letting =1, or by assuming that the materials comprising the inclusion and the interphase layer are identical ( 2 1 and 2 1 ). 3.2. Extremely compliant interface layer ( 1 1 , 1 3 and 1 ) In this case, Eq. (19) and (20) become , 2) ( ( 1) 2 2 ( , ) 4 2 1 2 2 1 2 1 2 1 2 2 1 1 R R R R R f R (34) . 2) ( ( 1) 2 2 ( , ) 4 2 1 2 2 1 2 1 2 1 2 2 1 1 R R R R R g R (35) The above two equations together with Eq. (18) give us very simple formulas for determining the restrictions on the remote non-uniform loadings when the interphase layer is much more compliant than both the inclusion and the matrix. 3.3. Relatively rigid interphase layer ( 1 1 , 1 3 and 1 ) In this case, Eqs. (19) and (20) become , ( 2) ( 1 2 ) 2 ( , ) 4 1 2 1 2 1 2 1 3 2 1 1 R R R R R f R (36) . ( 2) ( 1 2 ) 2 ( , ) 4 1 2 1 2 1 2 1 3 2 1 1 R R R R R g R (37) The above two equations together with Eq. (18) also give us very simple formulas for determining the restrictions on the remote non-uniform loadings when the interphase layer is much stiffer than both the inclusion and the matrix. 3.4. A three-phase circular inclusion ( 1R ) In this case, Eqs. (19) and (20) reduce to . 1) )( ) ( )( (1 ) )( 2(1 )( ( , ) 2 ( , ) 1 3 2 1 1 3 1 2 2 3 1 3 1 1 2 1 1 2 1 R f R R g R (38) 4. Conclusions Similar to the case of uniform loading [2], confocal elliptical interfaces are used in the present design to achieve an internal stress field of linear form under a linearly distributed stress field at
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