13th International Conference on Fracture June 16–21, 2013, Beijing, China -22 Formulation of the problem 2.1 Theoretical model The thermal effect of mode III crack in a functionally graded strip under the electric shock is investigated. This fracture analysis can be expressed through the superposition of two problem solutions. The first solution refers to the dynamic behaviors of a functionally graded piezoelectric material with central crack subjected to the electric shock. The second solution means the temperature field by calculating the power of point heat source around the crack tip. Illustrated in Fig.1 is the fracture model of a functionally graded piezoelectric strip which is assumed to contain a center crack. The crack of length and the thickness of strip are defined as 2cand 2h. In addition, the rectangular coordinate system is established as fig 1. Since the poling directions of piezoelectric materials are orientated along z-axis, the antiplane mechanical field and inplane electric field are coupled. Here, the fundamental solution of crack under a pair of the equivalent electric shock 0 ( ) DH t - and shear traction 0 ( ) H t t- acting on the crack surface is considered. ( )t H is Heaviside function, and 0τ , 0D are the range of the impact load of force field and electric field. Figure 1. Functionally graded piezoelectric strip with central crack subjected to the electric shock In fracture analysis of functionally graded piezoelectric strip, for the convenience of employing some standard methods such as Fourier transforms and integral equations, material properties are always assumed to be continuous functions of spatial coordinates, among which the most widely used one is exponential function [4]. Therefore, like many previous literatures, the properties of the functionally graded piezoelectric strip are assumed in power forms along yaxis as follows: y e c c β 0 44 = ， y e e e β 0 15 = ， y eβ κ κ 0 11 = ， y eβ ρ ρ 0 = (1) where 0 0 0 0 , , , c e k r are the coefficient of the functionally graded strip at y=0, such as shear modulus, piezoelectric coefficient, dielectric permittivity and density permeability, respectively. β is the non-homogeneity parameters controlling the material coefficient in the graded layer. 2.2 Governing equations Firstly, without considering the related effect of temperature field, a theoretical model is developed for the dynamic fracture analysis of a functionally graded piezoelectric material of a finite dimension with central crack subjected to the electric shock. Under axial shear deformation, the constitutive relations can be expressed in terms of polar coordinate system in the form
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