13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- } { } } [ ]{ [ ]{ 0 Ψ Ψ U Φ Φ T M M M M , (31) from Eq.(22). For the interior problem of each inclusion in Figure 2(b), we have [ ]{ }[ ]{} {} T w U t 0 I I I I , (32) [ ]{ }[ ]{ }{}0 Ψ U Φ T I I I I , (33) from Eq.(18), where the subscripts “M” and “I” denote the matrix and inclusion, respectively. The four influence matrices, [ ] MU , [ ] MT , [ ] I U and [ ] IT , are obtained from the degenerate kernels, while { }Mw , { }Mt , { }Iw , { }I t , { }MΦ , { }MΨ , { }IΦ , { }IΨ represent the coefficient vectors of eigenfunction expansions. Based on the continuity of displacement and equilibrium of traction between the interface of matrix and the kth inclusion as shown in Eqs.(6) and (7), we have { } { } { } w w 0 I M , (34) [ ]{ }[ ]{}[ ]{ }[ ]{ }{} 15 15 44 44 0 Ψ e Ψ c t e c t I I I M M I M M , (35) { } { } { }0 Φ Φ I M , (36) [ ]{ }[ ]{}[ ]{ }[ ]{ } {} 11 11 15 15 0 ε Ψ ε Ψ e t e t I I I M M I M M , (37) where [ ] 44 Mc , [ ] 15 Me , [ ] 11 Mε , [ ] 44 Ic , [ ] 15 Ie and [ ] 11 I ε are the diagonal matrices to the material parameters. According to Eqs.(30)-(37), we have a linear system as follows: 0 0 0 0 0 b 0 a Ψ Φ Ψ Φ t w t w ε 0 ε 0 0 e 0 e I 0 I 0 0 0 0 0 0 e 0 e 0 c 0 c I 0 0 0 0 0 I 0 0 0 0 0 0 0 T U 0 0 0 0 T U 0 0 0 0 T U 0 0 0 0 T U 0 0 0 0 0 0 I I M M I I M M I M I M I M I M I I M M I I M M 11 11 15 15 15 15 44 44 , (38) where [ ]I is the identity matrix and { }a and { }b are the forcing terms due to the remote shear stress as shown below {}[ ]{ }[ ]{} a T w U t M M , (39) {}[ ]{ }[ ]{ } Ψ U Φ b T M M . (40) From Eq. (38), the unknown Fourier coefficients can be easily determined. 3. Numerical examples and discussions To the authors’ knowledge, we don’t find any paper to discuss on the piezoelectricity with two or more than two elliptical inhomogeneities. Therefore, we consider the available case containing two circular inhomogeneities to demonstrate the validity of present approach for dealing with a problem containing two elliptical inhomogeneities. Besides, we also provide a numerical example for the case containing two elliptical inhomogeneities. Case1: An infinite medium with two circular inhomogeneities The first example considered here is an infinite medium with two elliptical inhomogeneities. Here, we used the limiting concept by numerically setting the semi-major and semi-minor axes to be near the same by using 1.000000 and 0.999999 for the first inclusion and 2.000000 and 1.999999 for the second inclusion to compare with the available results of two circular inclusions subject to remote shears and electric fields. The mechanical and electric parameters of medium and inhomogeneities are
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