13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- ∫ ∫ ∫ − − − ⋅ + − + − 1 1 1 1 * 11 0 1 1 ( , , ) ( , ) ( , ) ( , ) 1 du p u F p u r K c du r u p u G e du r u p u F π π π ∫− = − − 1 1 0 * 12 0 ( , , ) ( , ) p du p u G p u r K c e τ π (35) ∫ ∫ ∫ − − − ⋅ − − + − − 1 1 1 1 * 21 0 0 1 1 0 ( , , ) ( , ) ( , ) ( , ) du p u F p u r K c e du r u p u G du r u p u F e π π κ π ∫− = − + 1 1 0 * 22 0 ( , , ) ( , ) p D du p u G p u r K c π κ (36) (0, ) 0= p F , (0, ) 0= p G (37) According to the theory of singular integral equation, the solution of ( , )p u F 和 ( , )p u G may be expressed as [6] 2 * 1 ( , ) ( , ) ( , ) u p u R p u F p t f − = = ， ∑ ∞ = = 0 ( , ) ( , ) n n n p u TC p u R (38) 2 * 1 ( , ) ( , ) ( , ) u p u S p u G p t g − = = ， ∑ ∞ = = 0 ( , ) ( , ) n n n p u TD p u S (39) Based on the Cauchy-Chebyshev collocation method, Eqs. (35-37) are reduced into a system of algebraic equations [6] p N p u S p u r cK e r u e N p u R p u r cK r u l N l l m m l l N l l m m l 0 1 * 0 12 0 1 * 11 ( , ) ( , , )] [ ( , ) ( , , )] 1 [ τ =− + − + + − ∑ ∑ = = (40) p D N p u S p u r cK r u N p u R p u r cK e r u e l N l l m m l l N l l m m l 0 1 * 0 22 0 1 * 0 21 0 ( , ) ( , , )] [ ( , ) ( , , )] [ =− + − + + − − ∑ ∑ = = κ κ (41) ∑ = = N l l N p u R 1 0 ( , ) , ∑ = = N l l N p u S 1 0 ( , ) (42) where ) 2 2 1 cos( π N l ul − = ( 1,2, , )N l ⋅⋅⋅ = , ) cos( π N m rm = ( 1,2, , 1) ⋅⋅⋅ − = N m (43) Solving Eqs. (40-41) and taking (42) into account, the numerical values of the function ( , )p u R l and ( , )p u S l can be obtained. Furthermore, the values of the function ( , )p u F and ( , )p u G can be obtained numerically. Thus, after obtaining the solutions f , g from the Eqs. (19), we may obtain the four unknown functions ( , ), 1,2,3,4 =i p s Ai from Eqs. (27-28), furthermore, the mechanical displacement, electric potential of the strip in the Laplace domains will be obtained from Eqs. (17-19). 3.4 Temperature field of the crack tip Research of Bilyk et al [5] revealed that applying high current on the conductor, the temperature of conductor is lager than the temperature just under force load. Then, this paper discusses the heating effect of the crack tip on piezoelectric medium under the high electric shock load. Secondly, within supposing that it is a heat insulation process in a short time, and thermal field and electromechanical filed is decoupled, the thermal effect is calculated. according to ref. [5] which shows that electromagnetic field diffusion time scale is far less than the heat conduction time scale under action current load, the this process can be approximated as adiabatic process, so this assumption is established.
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