13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- normal stress H(t) 0P and impact electric displacement H(t) 0D are applied on the edges of the strip, where H(t) is the Heaviside step function. As shown in Fig. 1, symmetry conditions are used to allow for consideration of only the region ( 1 2 0, h z h x ≥ − ≤ ≤ ). In this paper the impermeable electric boundary condition on the crack faces is employed. Application of Laplace transform leads to the governing equations in the Laplace domain as follows: 0 ) ( ) ( ) ( ) ( * , 33 * , 11 * , 33 * , 15 * , 15 31 2 * * , 33 * , 15 * , 33 *, 44 * , 44 13 2 * * , 15 31 * , 44 13 * , 44 * , 11 = − − + + + = + + + + + = + + + + + zz xx z zz z xx x xz z zz xx z zz z xx x xz x xz z xz x zz x xx e e u e u e u p u e C C u C u C u e p u e e C u C u C C u φ λ φ λ ρ φ φ ρ φ (2) where p is the Laplace transform parameter and the superscript * denotes the quantities in the Laplace transform domain. The corresponding boundary conditions and continuity conditions in the Laplace domain are: Dxhp Dxhp D p x h p x h p x h p P p x h p z z xz xz zz zz 0 2 * 1 * 2 * 1 * 0 2 * 1 * ( , , ) ( , , ) ( , , ) 0 ( , , ) , ( , , ) ( , , ) = − = − = = − = = σ σ σ σ ( 0≥x ) (3) ( ,0 , ) ( ,0 , ) ( ,0 , ) ( ,0 , ) ( ,0 , ), ( ,0 , ) * * * * * * D x p D x p x p x p x p x p z z xz xz zz zz − + − + − + = = = σ σ σ σ ( 0≥x ) (4) ( ,0, ) 0, ( ,0, ) 0 ( ,0, ) 0, * * * = = = x p D x p x p z xz zz σ σ ( x c ≤ < 0 ) (5) ( ,0 , ) ( ,0 , ) ( ,0 , ) ( ,0 , ) ( ,0 , ), ( ,0 , ) * * * * * * x p x p ux pux pux pux p x x z z − + − + − + = = = φ φ ( x c ≥ ) (6) Fourier transform is applied to Eq. (2) to obtain the solutions as [ ] ξ ξ ξ γ ξ ξ γ ξ γ x d z z B p a A p u x z p j j n j j n j j j n x ) ) sin( ( , )cosh( ) ( , )sinh( ( , , ) 3 1 0 ( ) ( ) *( ) ∑∫ = ∞ + =− (7) [ ] x d T z p z z B p A p u x z p j j n j j n j n z 1 3 1 0 ( ) ( ) *( ) ) ) cos( ( , )sinh( ) ( , )cosh( ( , , ) + + =∑∫ = ∞ ξ ξ ξ γ ξ ξ γ ξ (8) [ ] x d T z p z z B p b A p x z p j j n j j n j j n 2 3 1 0 ( ) ( ) *( ) ) ) cos( ( , )sinh( ) ( , )cosh( ( , , ) + + =−∑∫ = ∞ ξ ξ ξ γ ξ ξ γ ξ φ (9) where ( 1,2) Τ = j j are constants and j j a b , ( 1,2,3) = j are known functions defined in Appendix A, the superscripts ( ) (1), (2) = n denote the fields quantities in the upper ( 1 0 y h ≤ ≤ ) and lower ( 0 2 − ≤ ≤ h y ) parts of the piezoelectric strip, respectively, and ( , ) ( , ), ( ) ( ) A p B p n j n j ξ ξ , ( 1,2; 1,2,3) = = j n are unknowns to be determined; jγ ( 1 3 = − j ) are the roots of the following characteristic equation:
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