2 For plane deformations of an isotropic elastic material, the in-plane displacements u and v, the two resultant forces fx and fy, and the in-plane stresses xx, yy and xy can be expressed in terms of two analytic functions (z) and (z) of the complex variable z=x+iy as [6] ( ) , ( ) i ( ) i ( ), ( ) ( ) 2 ( i ) z z z z f f z z z z u v y x (1) ( ) , ( ) 2i 2 ( )], 2[ ( ) z z z z z xy xx yy yy xx (2) where 3 4 for plane strain, and (3 ) (1 ) for plane stress; and , where 0 and 0.5 0 , are the shear modulus and Poisson’s ratio, respectively. In the physical z-plane, the boundary value problem for the three-phase elliptical inclusion takes the form: ; ( ) , ( ) ( ) 2 1 ( ) ( ) ( ) 2 1 ( ), ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 2 2 2 2 2 1 1 1 2 2 2 z z z z z L z z z z z z z z z z z z (3) ; ( ) , ( ) ( ) 2 1 ( ) ( ) ( ) 2 1 ( ), ( ) ( ) ( ) ( ) ( ) 2 3 3 3 3 3 2 2 2 2 2 3 3 3 2 2 2 z z z z z L z z z z z z z z z z z z (4) (1), i ) ( ) ( (1), i ) ( ) ( 2 2 1 3 2 2 1 3 z B B z O z A A z O as z (5) where A1, A2, B1 and B2 are real numbers related to the applied linearly distributed stress field at infinity. Elliptical Inclusion S 1 Interphase Layer S 2 Matrix S 3 x y L 1 L 2 2R -2R Figure 1 Three-phase elliptical inclusion with an internal stress field of linear form
RkJQdWJsaXNoZXIy MjM0NDE=