13th International Conference on Fracture June 16–21, 2013, Beijing, China -3y y y y e D x y x y e D y y e y y c x y e x y c y x yz xz ∂ ∂ − ∂ ∂ = ∂ ∂ − ∂ ∂ = ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ = φ κ ω φ κ ω φ ω σ φ ω σ ( ) ( ) , ( ) ( ) ( ) ( ) , ( ) ( ) 11 15 11 15 15 44 15 44 (2) where f and w denote the electric potential and anti-plane mechanical displacement, D and s are the electric displacement and anti-plane stresses, respectively. The geometric equation can be written as x xz ∂ ∂ = ω ε2 ， y yz ∂ ∂ = ω ε2 ， x Ex ∂ ∂ =− φ ， y Ey ∂ ∂ =− φ ， z Ez ∂ ∂ =− φ (3) Under the static condition, when body forces and body charges are omitted, the stress, electric displacement should satisfy the following equations 2 2 t y x yz xz ∂ ∂ = ∂ ∂ + ∂ ∂ ω ρ σ σ ， 0= ∂ ∂ + ∂ ∂ y D x D y x (4) Introduce an auxiliary function by ω κ φ ψ 0 0e = − (5) Substituting Eqs. (1-3) into Eq. (4) yields the decoupled governing equations in the piezoelectric strip 0 , 2 2 2 0 2 = ∂ ∂ ∇ + ∂ ∂ = ∂ ∂ ∇ + y t s y ψ β ψ ω ω β ω (6) where ) /( 0 2 0 0 0 0 κ ρ e c s + = (7) 2.3 Boundary conditions The upper region [0, ] y h Î and lower region [ ,0] y h Î - is symmetric with respect to the xaxis. Therefore, in the following we confine our attention to the upper region [0, ] y h Î . Then, based on the symmetry, the corresponding boundary conditions are imposed ( ,0, ) 0= t x ω ， ( ,0, ) 0= t x φ | ) (| c x > (8) Here, the fundamental solution of crack under a pair of the equivalent electric shock 0 ( ) D H t - and shear traction 0 ( ) H t t- acting on the crack surface is considered. The corresponding mechanical boundary conditions on the crack surface are imposed ( ) ( ,0, ) 0 t H t x yz τ σ =− ， ( ) ( ,0, ) 0 t H D t x D y =− ) ( c x c− < < (9) Assume that the upper surface h y = is free of loading. Therefore, the electric displacement and anti-plane stresses on the upper surface is taken as zero, ( , , ) 0= t h x yz σ ， ( , , ) 0= t h x D y ) (−∞< <+∞ x (10) 3. Solution 3.1 Basic solution expression Using the Laplace transform and Fourier transform technique, ones can transform Eqs. (6) into a system of decoupled differential equations
RkJQdWJsaXNoZXIy MjM0NDE=