13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- { } [ ]{ } { } ( ) (0) , n N n R h R = Π + Π (18) Where {Rn(0)} and {Rn(hN)} are the state vectors of top and bottom surfaces of the plate, respectively, and [ ]Π is the state transfer matrix. The non-homogeneous vector { }Π contains 1 2 1 N K K K + +⋅⋅⋅+ + boundary unknown coefficients , t i j v , , b i j v which can be determined by mechanical and electric boundary conditions at the free and electric open-circuited edges. 2.2. Boundary conditions of cross-ply piezoelectric plate For an electric load free surface, open-circuited and traction-free conditions are considered, the top and bottom surface conditions are obtained: [ ] [ ] (0) (0) (0) (0) 0 0 0 0 , T T n n n xz D Z Y τ = (19a) [ ] [ ] ( ) ( ) ( ) ( ) 0 0 0 0 . T T n N n N n N xz N D h Z h Y h h τ = (19b) The plate has free edges and electrical open-circuited conditions at y = 0, y = b as follows: 0, y xy zy yD σ τ τ = = = = (20) where τxy = 0, τzy = 0, Dy = 0 are satisfied automatically. The remaining boundary condition to be satisfied is σy = 0. Due to symmetry, we only impose the condition at y = 0, and substitute the second equation of Eq. (9) into the first expression of Eq. (20) yields (0) 13 14 15 16 0 13 0 2 [ ( ) ( ) ( )] ( ) 0. n n n n v z Z z D z v z b α η α α α ε α ∞ = ⋅ ⋅ + ⋅ + ⋅ + ⋅ − ⋅ ⋅ = ∑ (21) The following algebra equation system can be obtained by introducing Eqs. (18) and (19) and eventually the boundary unknown constants can be determined by considering Eqs. (21) and (22). 1 2 1 11 15 16 21 25 26 3 2 21 25 26 31 35 36 4 3 31 35 36 41 45 46 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) . ( ) ( ) ( ) ( ) ( ) n n n z v z z z z D z z z z z Z z z z z z − ⎧ ⎫ ⎧ ⎫ Π Π Π Π Π Π Π Π ⎧ ⎫ ⎡ ⎤⎡ ⎤ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥⎢ ⎥ ⎨ ⎬=−Π Π Π ΠΠΠ ⎨Π⎬+⎨Π⎬ ⎢ ⎥⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥⎢ ⎥ Π Π Π Π Π Π Π Π ⎩ ⎭ ⎣ ⎦⎣ ⎦ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ (22) 3. Numerical examples and results To validate the present method, numerical examples are presented for symmetric cross-ply piezoelectric laminated plates and comparisons are made between the current solution and work done by Artel and Becker [13], and Mirzababaee and Tahani [14]. The free-edge effect in the laminated plate with and without electromechanical coupling is investigated and two laminated layups [0 /90 ]s o o and [90 /0 ] s o o are considered. The material properties are given in Table 1 which comprises the mechanical properties of a T300/Epoxy and the piezoelectric and electrical properties of a PZT-5A. The uniaxial extension ε0 is 0.1% and the width b is ten times larger than the thickness h, in addition, the thickness of each ply in the laminate is identical.
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