13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- { } [ ] { } 19 20 21 2 17 18 0 2 13 15 14 5 7 6 8 1 3 2 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , . 0 0 0 0 0 0 0 0 0 0 z z yz v D R A B w α α β α β α β α β β σ ε α β α β α β τ α β α α α φ α β α α α ⎡ ⎤ ⎧ ⎫ ⎧ ⎫ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ − − ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ − = = = ⎢ ⎥ ⎨ ⎬ ⎨ ⎬ − − − ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ ⎣ ⎦ (8) The following in-plane stresses and electric displacements are also obtained: [ ] 9 10 11 12 0 13 14 15 16 17 18 , . x z y z yz y v D D σ α β α α α σ ε σ α β α α α τ α α β φ ⎧ ⎫ ⋅ ⎧ ⎫ ⎡ ⎤ ⎧ ⎫ ⎪ ⎪ = + ⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎢ ⎥ ⋅ ⎣ ⎦ ⎩ ⎭ ⎩ ⎭ ⎪ ⎪ ⎩ ⎭ ⎧ ⎫ = ⋅ ⎨ ⎬ ⎩ ⎭ (9) Constants related to material property can be expressed as follows: 32 33 23 33 1 1 e e C α − ⋅ − ⋅∈ = Δ , 33 2 1 α ∈ = Δ , 33 3 1 e α= Δ , 31 33 13 33 4 1 e e C α − ⋅ − ⋅∈ = Δ , 33 32 23 33 5 1 C e C e α ⋅ − ⋅ = Δ , 33 6 1 e α = Δ , 33 7 1 C α − = Δ , 33 31 13 33 8 1 C e C e α ⋅ − ⋅ = Δ , 9 12 1 13 5 31 C C e α α α = + ⋅ + ⋅ , 10 2 13 6 31 C e α α α = ⋅ + ⋅ , 11 3 13 7 31 C e α α α = ⋅ + ⋅ , 12 11 4 13 8 31 C C e α α α = + ⋅ + ⋅ , 13 22 1 23 5 32 C C e α α α = + ⋅ + ⋅ , (10) 14 2 23 6 32 C e α α α = ⋅ + ⋅ , 15 3 23 7 32 C e α α α = ⋅ + ⋅ , 16 12 4 23 8 32 C C e α α α = + ⋅ + ⋅ , 24 17 44 e C α = , 2 24 18 22 44 e C α =− −∈ , 19 44 1 C α = , 24 20 44 e C α =− , 21 1 α =− , 2 1 33 33 33 e C Δ = + ⋅∈ , y β ∂ = ∂ . Assuming that the displacements v can be expressed as: (0) 2 ( , ) ( , ) ( ) (1 ), y v y z v y z v z b = + ⋅ − (11) Where v(0)(z) is the unknown boundary displacement function that can be determined by imposing traction free conditions and open-circuit conditions on the free edges. The following Fourier series expansions are used: 0 0 0 ( , ) ( ) ( ) ( , ) ( ) ( ), ( , ) ( ) ( ) n n z n n z n n v y z v z Sin y D y z D z Cos y y z Z z Cos y η η σ η ∞ = ∞ = ∞ = ⎧ = ⋅ ⋅ ⎪ ⎪ ⎪ = ⋅ ⋅ ⎨ ⎪ ⎪ = ⋅ ⋅ ⎪ ⎩ ∑ ∑ ∑ 0 0 0 ( , ) ( ) ( ) ( , ) ( ) ( ) , ( , ) ( ) ( ) yz n n n n n n y z Y z Sin y y z z Cos y w y z w z Cos y τ η φ φ η η ∞ = ∞ = ∞ = ⎧ = ⋅ ⋅ ⎪ ⎪ ⎪ = ⋅ ⋅ ⎨ ⎪ ⎪ = ⋅ ⋅ ⎪ ⎩ ∑ ∑ ∑ (12) 0 2 ( ) ( ), n b Cos n y Sin y n π η π ∞ = =− ⋅ ⋅ ∑ (13) Where η = nπ/b, since a uniformly distributed extension is applied, displacement v is zero at y = b/2. By introducing Eqs. (11)-(13) into Eq. (7) the following non-homogeneous state space equation for an arbitrary value of n is obtained:
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