13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- 0 ) ( ) ( ) ( ) ( 2 33 11 15 2 33 15 31 15 2 33 44 2 2 2 33 44 13 15 31 44 13 2 44 2 2 11 = − − + − − − + + + − + γ λ λ γ γ γ ξ ρ γ γ γ γ γ ξ ρ e e e e e p C e C C C e e C C C p C (10) Note that the sixth-order characteristic equation (10) has six roots which occur in pairs with the same magnitude but opposite signs, and for complex roots, they always appear in conjugate pairs. In Eqs. (7-9), the roots jγ ( 1 3 = − j ) with Re( ) 0 > jγ are chosen by requiring a positive internal energy for the system to be in a steady state, as stated by Suo et al. [10]. The stress and electric displacement components can be expressed as follows [ ] ∑∫ = ∞ + =− 3 1 0 ( ) ( ) *( ) ) ) sin( ( , )sinh( ) ( , )cosh( j j n j j n j j n xz x d z z B p f A p ξ ξ ξ γ ξ ξ γ ξ ξ σ (11) [ ] ∑∫ = ∞ + = − 3 1 0 ( ) ( ) 0 *( ) ) ) cos( ( , )cosh( ) ( , )sinh( j j n j j n j j n zz x d z z B p g A p P p ξ ξ ξ γ ξ ξ γ ξ ξ σ (12) [ ] ∑∫ = ∞ + = − 3 1 0 ( ) ( ) 0 *( ) ) ) cos( ( , )cosh( ) ( , )sinh( j j n j j n j j n xx x d z z B p q A p p ξ ξ ξ γ ξ ξ γ ξ ξ σ σ (13) [ ] ∑∫ = ∞ + = − 3 1 0 ( ) ( ) 0 *( ) ) ) cos( ( , )cosh( ) ( , )sinh( j j n j j n j j n z x d z z B p D D p m A p ξ ξ ξ γ ξ ξ γ ξ ξ (14) where 31 2 13 1 0 C T e T = + σ and the coefficients j j j j f g q m , , , are defined in Appendix A. By applying the boundary conditions (3) and (4), the unknown functions ( , ), (1) B p j ξ ( , ), (2) A p j ξ ( , ) (2) B p j ξ ( 1 3) = − j can be expressed by the independent unknowns ( , ) ( 1 3) (1) = − A p j j ξ as ( , , , ) ( , ) ( , ) ( , , , ) ( , ) ( , ) ( , , ) ( , ), ( , ) 3 1 (1) 1 2 (2) 3 1 (1) 1 2 (2) 3 1 (1) 1 (1) (1) A p T h h p A p B p R h p A p B p Q h h p A p i i ji j i i ji j i i ji j ξ ξ ξ ξ ξ ξ ξ ξ ξ ∑ ∑ ∑ = = = = = = (15) where ( , , ) 1 (1) R h p ji ξ , ( , , , ) 1 2 T h h p ji ξ and ( , , , ) 1 2 Q h h p ji ξ are known functions. Introduce the auxiliary functions ( , ) ( 1 3) = − Φ x p i i such that ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ − − − ∂ ∂ = ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ Φ Φ Φ + − − + − + ( ,0 , ) ( ,0 , ) ( ,0 , ) ( ,0 , ) ( ,0 , ) ( ,0 , ) ( , ) ( , ) ( , ) *(1) *(2) *(2) *(1) *(2) *(1) 3 2 1 x p x p u x p u x p u x p u x p x x p x p x p z z x x φ φ (16) By applying the solutions (7-9) and using the Fourier inverse transform, the independent unknowns can be obtained as
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