13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- ( xE and yE ) as shown in Fig. 1. Bleustein [12] has pointed out that if one takes the plane normal to the poling direction as the plane of interest, only the anti-plane displacement (w) couples with the in-plane electric fields (Ex and Ey). Therefore, only the anti-plane displacement and in-plane electric field are considered in this article such as u, v and Ez are the vanishing components. In the absence of the body forces and body charges, the governing equations coupled by the displacement and electric potential can be obtained as follows: 0 2 15 2 44 c w e , 0 2 11 2 15 e w , (1) where 2 is the two-dimensional Laplacian operator, c 44 is the elastic modulus, e15 is the piezoelectric constant, 11 is the dielectric constant, w is the anti-plane displacement and is the in-plane electric potential. From Eq.(1), we can simplify the equaitons as 0 2 w and 0 2 . (2) The constitutive equations coupled between the elastic filed and electric field are x zx zx c e E 15 44 , y zy zy c e E 15 44 , (3) x zx x E D e 11 15 , y zy y E D e 11 15 , (4) where zx and zy are the anti-plane shear strains, and Dx and Dy are the in-plane electric displacements. By taking free body technique, the problem can be decomposed into two parts. One is an infinite piezoelectric medium with N elliptical holes (Fig.2(a)) and the other is only N-inclusions problem (Fig.2(b)). For the problem in Fig.2(a), it can be superimposed by two parts as shown in Fig.3(a) and Fig.3(b). Both the two parts in Figs. 2(b) and 3(b) satisfy the Lapalce Figure 2(a) An infinite plane containing elliptical holes subject to remote shears and far-field in-plane electric fields Figure 2(b) Multiple elliptical inclusions Figure 3(a) An infinite medium subject to remote shears and far-field in-plane electric fields Figure 3(b) An infinite medium containing elliptical holes
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