13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- 2 2 1 1 log log ( ) r z i i z i r θ θ + = + − − (34) where r1 and r2 are the modulus of the complex variables z+i and z -i respectively. θ1 and θ2 are the inclined angles of the complex variables z +i and z -i with respect to the negative imaginary axis. From Eqs. (29) and (31), the unknown complex function vector g(z) has been obtained. Then substituting the result into Eq.(26b), we can derive the unknown function vector ' ( )z f , which provides the full-field solutions of the problem using Eq. (15). 3.2 Electric saturation zone size From Eq. (15), we obtain the electric displacement ahead of the crack tip 2 4 44 4 4 ' 4 0 0 2 2 2 2 2 2 2 2 2 2 2 2 ( ( ) ( )) ( ( ) ( )) 2 ( ) ( ( ) ( )) 2 2 1 ( arccos( )) log 2 2 j j j b s s s D g x g x g x g x T f x D g x g x x b a i a a x x b D D D D b i x b x b a i a x b Λ Λ π π π + − + − + − ∞ ∞ = + + + = + + ⎧ ⎫ − + ⎪ ⎪ ⎪ ⎪ − = − − + ⎨ − ⎬ − − ⎪ − ⎪ ⎪ ⎪ − ⎩ ⎭ (35) In order to ensure the non-singularity of the electric displacement at x b= , Eq. (35) only has a solution if the coefficient of the singular term 2 2 x x b− vanishes. The following equations must be satisfied 2 cos( ) 2 s D a b D π ∞ = (36) From the above equation, we can calculate the size of the electric saturation zone 2 sec( ) 2 s D r b a a a D π ∞ = − = − (37) Under small-scale yielding conditions, r<<a, Eq. (37) can be approximately reduced to 2 2 ( ) 2 2 s a D r D π ∞ = . 3.3 Electroelastic fields near the crack tip From Eqs. (15) and (20), the stresses and electric displacements can be obtained
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