13th International Conference on Fracture June 16–21, 2013, Beijing, China -8- ] ( , ) ( , , , ) ( , ) ( , , , ) ( , , ) [ 1 1 2 1 * ∑ ∑ = = + = N l N l l l dy l l dy y N p u S p y x u K N p u R pyxu K c pyx D (56) where ∫+∞ + + + + − − − + − − = 0 4 3 4 3 2 1 1 2 0 0 1 )] ]cos[( [ ( , , , ) 4 3 4 3 3 4 2 1 1 2 2 1 ds x cu s e t e t e t e t e t e t et e t e p y x u K h t h t y t h t y t h t h t h t y t h t y t h t ex κ (57) ds x cu s e t e t e t e t p y x u K h t h t y t h t y t h t ex )] cos[( ( , , , ) 0 4 3 3 4 2 4 3 3 4 4 3 − − − =∫ +∞ + + (58) ∫+∞ + + + + − − − + − − = 0 4 3 3 4 3 4 2 1 1 2 1 2 0 0 1 )] ]sin[ ( [ ( , , , ) 4 3 3 4 4 3 2 1 2 1 1 2 ds x cu s e t e t e t t e t t e t e t e t t e t t e p y x u K h t h t y t h t y t h t h t h t y t h t y t h t ey κ (59) ds x cu s e t e t e t t e t t p y x u K h t h t y t h t y t h t ey )] sin[ ( ( , , , ) 0 4 3 3 4 3 4 2 4 3 4 3 3 4 − − − =∫ +∞ + + (60) ∫+∞ + + + + − − − + − − = − 0 4 3 4 3 11 2 1 2 1 0 0 11 15 1 )] ]cos[ ( [ ) ( ( , , , ) 4 3 4 3 3 4 2 1 2 1 1 2 ds x cu s e t e t e t e t e t e t e t et e e p y x u K h t h t y t h t y t h t h t h t y t h t y t h t dx κ κ κ (61) ds x cu s e t e t e t e t p y x u K h t h t y t h t y t h t dx )] cos[ ( ( , , , ) 0 4 3 3 4 11 2 4 3 3 4 4 3 − − − =∫ +∞ + + κ (62) ∫+∞ + + + + − − − − − − = − 0 4 3 3 4 3 4 11 2 1 1 2 1 2 0 0 11 15 1 )] ]sin[ ( [ ) ( ( , , , ) 4 3 4 3 3 4 2 1 1 2 2 1 ds x cu s e t e t e t t e t t e t et e t t e t t e e p y x u K h t h t y t h t y t h t h t h t y t h t y t h t dx κ κ κ (63) ds x cu s e t e t e t t e t t p y x u K h t h t y t h t y t h t dx )] sin[ ( ( , , , ) 0 4 3 3 4 3 4 11 2 4 3 3 4 4 3 − − − = − ∫ +∞ + + κ (64) The solution of electric field strength and electric displacement intensity in the time domain are calculated by Laplace numerical inversion method. Furthermore using Eq. (48), the temperature field of the crack tip is obtained. 4. Numerical Examples Examples and discussions discussions In the numerical computation, the functionally graded piezoelectric strip layer is assumed to be a non-homogeneous BaTiO3 composite and the material constants of 0=y are Gpa c 44 44 = ， 2 15 11.4 / m C e = ， Vm C/ 128.3 1010 11 = × κ ， 3 / 5700 m Kg =ρ (65) Here, the fundamental solution of crack under a pair of the equivalent electric shock 0 ( ) DH t - acting on the crack surface is considered strongly, so it assume that 0 0 = τ . Fig. 2 show the the temperature field around the crack tip versus the value of time. It is indicated that the effect of time on the temperature field is simple, i.e., when / 1/ 2,1/3,1/ 4 =h c are specified, the temperature field increases drastically as time increases from 0 to 3. However, when time is larger than 3, the temperature field gradually decrease with time, finally, stabilized. It is also observed that the temperature field increases with the h c/ increasing. Fig. 3 depict the variation of the the temperature field around the crack tip versus the non-homogeneity parameter β. When time is small (t<2), the temperature field decreases as β increases; but when time is larger than 2, the temperature field increases drastically as β increases. Meanwhile, it is obviously indicated that when time is larger (t>2), the effect is larger than that when time is small (t<2). The above results show that the crack tip will cause high temperature change under high electric shock load. In this case, the crack tip temperature effect cannot be ignored
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