13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- ( ) ( ) ( ) ( ) , ( 2,3,..., 1) 1 , ( 1, ); 2 1 , ( 1,2,..., 1) 2( 1) 2 1 cos ); , ( 1,2,..., 1 1 cos − = − = = − = − ⎥ = ⎦ ⎤ ⎢ ⎣ ⎡ − − = ⎥ = ⎦ ⎤ ⎢⎣ ⎡ − − = n i n i n A n A n k n k n x i n i s i i k i π π π π (24) 3. Asymptotic fields near the crack tip Once functions ( , ) H s p j ( 1,2,3 = j ) are obtained from solving the algebraic equations (21-23), following the procedure in Li and Lee [12], the asymptotic expressions of the electro-elastic fields near the crack tip can be determined by introducing a polar coordinate system ( θ,r ) with the origin at the right crack tip, as follows [ ]) ( tan , ) ( 1 2 2 z x c r x c z − = = − + − θ (25) The hoop and shear stresses at an angle θ near the right tip of the crack are obtained from the following relations in terms of the polar coordinates ( , )θ r [ ] θ θ σ θ σ θ σ θ θ σ θ θ σ θ θ σ θ θ σ θ σ θ θθ ( , , ) (,, )2 (,, )cos2 ( , , ) sin2 ( , , )sin2 ( , , )sin ( , , )cos ( , , ) * * * * * 2 * 2 * * r p r p r p r p r p r p r p r p xz xx zz r xz xx zz + − = − + = (26) Define the hoop stress intensity factor and shear stress intensity factor associated with the hoop and shear stresses at an arbitrary angle θ as [13] ( ) ( ) * 0 * * 0 * , lim 2 lim 2 θ θ θθ θθ σ σ r r r r r r K K → → = = (27) Substituting Eqs. (26) into (27), the hoop and shear stress intensity factors in the Laplace domain can be obtained as: ∑ ∑ ∑ = = = ⎪ ⎪ ⎭ ⎪⎪ ⎬ ⎫ ⎪ ⎪ ⎩ ⎪⎪ ⎨ ⎧ ⎥⎦ ⎤ ⎢⎣ ⎡ Λ Λ − − − ⎥⎦ ⎤ ⎢⎣ ⎡ Λ Λ + − + = 3 1 1 3 2 0 2 0 1 1 0 2 3 2 0 1 0 1 1 2 0 2 0 * ( ) (1, ) ( ) ( 1) (1, ) sin2 ( ) (1, ) ( ) (1, ) ) ( 1) sin ( cos j j k jk k n j j j j k jk k j j n j j H p Y H p Y f H p Y H p Y q g c K θ θ θ θ θ θ θ θθ (28) ∑ ∑ ∑ = = = ⎪ ⎪ ⎭ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ ⎥⎦ ⎤ ⎢⎣ ⎡ Λ Λ − − + ⎥⎦ ⎤ ⎢⎣ ⎡ Λ Λ + − − = 3 1 1 3 2 0 2 0 1 1 0 2 3 2 0 1 0 1 1 0 0 * ( ) (1, ) ( ) ( 1) (1, ) cos2 ( ) (1, ) ( ) (1, ) ( 1) 2 )sin2 ( j j k jk k n j j j j k jk k j j n j j r H p Y H p Y f H p Y H p Y g q c K θ θ θ θ θ θ θ (29) where π θ ≤ ≤ 0 when 1=n for the upper part and 0 − ≤ ≤θ π when 2 =n for the lower part of the cracked layer, respectively; the angular functions ( ) 1 θ j Λ and ( ) 2 θ j Λ ( 1,2,3) = j are given in the following form [ ] [ ] [ ]2 0 2 2 0 2 ) sin( 2cos ( ) ) ( 1) cos( ) sin( cos ( ) ( ) θ γ θ θ θ γ θ θ j n j nj + + − + Λ = ( 1,2) = k (30)
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