13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- ( , , ) ( , , ) ( , , ) ( ) ( , , ) 2 0 2 2 2 pysFps pysF y p y s F y p y s F s = ∂ ∂ + ∂ ∂ + − β (11) ( , , ) 0 ( , , ) ( ) ( , , ) 2 2 2 = ∂ ∂ + ∂ ∂ + − p y s G y p y s G y p y s G s β (12) where ∫ ∫ = = +∞ − Br pt pt dp e p y x t y x dt e t y x p y x ( , , ) , ( , , ) ( , , ) ( , , ) * 0 * ψ ψ ψ ψ (13) ∫ ∫ = = +∞ − Br pt pt dp e p y x i t y x dt e t y x p y x ( , , ) 2 1 , ( , , ) ( , , ) ( , , ) * 0 * ω π ω ω ω (14) ∫ ∫ +∞ −∞ +∞ −∞ − = = dx e p y x pysF ds epysF p y x isx isx ( , , ) , ( , , ) ( , , ) 2 1 ( , , ) * * ω π ω (15) ∫ ∫ +∞ −∞ +∞ −∞ − = = dx e p y x pysG ds epysG p y x isx isx ( , , ) , ( , , ) ( , , ) 2 1 ( , , ) * * ψ π ψ (16) where ( , , ), ( , , ) * * p y x p y x ψ ω are the Laplace transform of ( , , ), ( , , )tyxtyx ψ ω , Br means path integral Bromwich formulation. And ( , , ), ( , , )p y s G p y s F are the Fourier transform of ( , , ), ( , , ) * * p y x p y x ψ ω . Afterwards, solving Eqs. (11-12), one can finally express the mechanical displacement, electric potential of the strip in the Laplace domains below dx e epsA epsA p y x isx y t y t − +∞ −∞∫ + = ( , ) ] [ ( , ) 2 1 ( , , ) 2 1 2 1 * π ω (17) dx e epsA epsA p y x isx y t y t − +∞ −∞ ∫ + = ( , ) ] [ ( , ) 2 1 ( , , ) 4 3 4 3 * π ψ (18) dx e epsA epsA e p y x isx y t y t − +∞ −∞ ∫ + = ( , ) ] [ ( , ) 2 1 ( , , ) 2 1 2 1 0 0 * κ π φ dx e epsA epsA isx y t y t − +∞ −∞ ∫ + + ( , ) ] [ ( , ) 2 1 4 3 4 3 π (19) There are four unknown functions ( , ), 1,2,3,4 =i p s Ai which can be solved by considering the boundary conditions and continuity conditions which are formulated by Eqs. (8-10). 2 0 2 2 1 ) 2 ( 2 p s s t =− + + + β β , 2 0 2 2 2 ) 2 ( 2 p s s t =− − + + β β , 2 2 3 ) 2 ( 2 s t + =− + β β , 2 2 4 ) 2 ( 2 s t =− + + β β (20) 3.2 Transformation to singular integral equation In order to derive the singular integral equation, two dislocation density functions are introduced as ⎪ ⎩ ⎪ ⎨ ⎧ ∂ ∂ = 0 ( ,0, ) ( , ) * * x p x p x f ω c x c x > ≤ | | | | (21) ⎪ ⎩ ⎪ ⎨ ⎧ ∂ ∂ = 0 ( ,0, ) ( , ) * * x p x p x g φ c x c x > ≤ | | | | (22) Substituting Eqs. (17-19) into Eq. (8), and considering Eqs. (21-22), one can obtain the following relation
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