3 Consider the following conformal mapping [2, 6] , ( 1) 4 1 1 2 , () 1 ( ) 2 2 z R R z z R z (6) which maps the segment [2R 2R] onto the unit circle in the -plane, and the two interfaces L1 and L2 are mapped onto two coaxial circles with radii R1 and R2, respectively. Thus the three regions S1, S2 and S3 are mapped onto the annuli 1 1 R , 2 1R R and 2R , respectively, as shown in Fig. 2. For convenience we will write )), ( ( 1,2,3 (()), () ( ) i i i i i . In the mapped -plane, the boundary value problem takes the form ( ), 1 ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 2 2 2 2 1 1 1 2 2 2 on 1R (7) ( ), ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 2 2 3 3 2 2 3 3 3 3 2 2 2 3 3 3 on 2R (8) (1), i ) ( ( ) (1), i ) ( ( ) 2 2 1 2 3 2 2 1 2 3 R B B O R A A O as (9) where 1= 1/ 2 and 3= 3/ 2 are two stiffness ratios. In order to ensure that the internal stress field is a linear function of the coordinates x and y, 1( ) and 1( ) must take the following forms Figure 2 The mapped -plane 1R 2R 1
RkJQdWJsaXNoZXIy MjM0NDE=