13th International Conference on Fracture June 16–21, 2013, Beijing, China -9- 1 15 2 23 23 2 1 11 15 23 2 11 ( )(cos sin ) 2 ( )cos 2 2 e a D r e a D r σ σ θ μ θ ε θ σ ε − ∞ ∞ ∞ ∞ ⎡ ⎤ ⎢ ⎥ = ℜ + + ⎢ ⎥ ⎣ ⎦ = + (45) If the ferroelectric material is such that a crack propagates in a direction the maximum shear stress, it can be seen that the maximum shear stress 23σ occurs at 0 θ= for all the crack speeds. It means the crack remains in its straight line path for all the crack speeds. 5 Conclusions The transient response of a anti-plane Yoffe-type crack moving with constant velocity in ferroelectric materials is investigated in this paper. The dynamic intensity factors of stress, electric displacement are obtained in explicit forms. When the velocity of the crack v→0，the moving PS model will reduce to the static PS model. When the size of the electric saturation zone r→0, the moving PS model is in agreement with the moving linear piezoelectric model. For the case of anti-plane problem, it is concluded that the crack remains in its straight line path for all the crack speeds. Acknowledgements The authors are grateful for the support by National Natural Science Foundation of China under Grants #11090330, #11090331, #11072003 and #G2010CB832701. Appendix The matrices Q, R and T are 44 15 15 11 c e e ε ⎡ ⎤ = ⎢ ⎥ ⎣ − ⎦ Q , 0 0 0 0 ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ R , 44 15 15 11 c e e ε ⎡ ⎤ = ⎢ ⎥ ⎣ − ⎦ T (A.1) The eigenproblem given by Eq. (12) becomes 2 2 2 44 44 15 1 2 2 2 15 11 (1 ) 0 (1 ) (1 ) c v c e a a e ρ μ μ μ ε μ ⎡ ⎤ − + + ⎡ ⎤ = ⎢ ⎥ ⎢ ⎥ + − + ⎢ ⎥⎣ ⎦ ⎣ ⎦ (A.2) Similar to the static case, two characteristic roots are μ1=i, . μ2=i β. in which 2 2 1/2 (1 / ) v c β= − (A.3) where 1/2 44 ( / ) c c ρ = is the speed of the piezoelectric stiffened bulk shear wave, 2 44 44 15 11 / c c e ε = + is the piezoelectric stiffened elastic constant. The matrix H is then
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