13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- as follows. Section 2 introduces the governing equations and the boundary conditions. Section 3 proposes the moving PS model and solves the problem using the complex function method. And the explicit expressions of the electroelastic fields are obtained. Section 4 discusses the anti-plane case. Finally, Section 5 gives the conclusions. 2 Statement of the problem Consider an infinite ferroelectric medium containing a Yoffe-type crack of fixed length 2a,which moves through an otherwise unbounded ferroelectric materials at speed v, with the crack opening at the leading crack tip and closing at the trailing crack speed. The ferroelectric solids are considered as a class of mechanically brittle and electrically ductile solids. The electrical polarization is assumed to be saturated only in a line segment in front of the crack. And the medium is subjected to remote uniform electro-mechanical loads as shown in Fig.1. Fig.1. Schematic representation of the moving PS model 2.1 Basic equations Our earlier work [4] has proposed the governing equations of the piezoelectric material and given the solutions using the Stroh formalism method in details. For the reason of a self-contained presentation, the basic equations and the solution method are also summarized as follows. In a rectangular coordinate system xi (i =1, 2, 3), the momentum balance equations and quasi static Maxwell equation for quasi-electrostatic piezoelectricity are as follows 2 2 , / ij j iu t σ ρ = ∂ ∂ , , 0 i i D = (1) where ρ is the density of the material, ui, σij and Di are the elastic displacements, stresses, and electric displacements, respectively, and a subscript comma denotes partial differentiation with respect to one of the coordinates xi. The constitutive relations are , , ij ijkl k l ijk k c u e σ φ = + , , , i ikl k l ik k D e u ε φ = − (2) where φ is the electric potential, the electric fields Ei are related to φ as Ei = - φ,i, cijkl, ekij and εij are the elastic stiffness, piezoelectric and dielectric constants, respectively. x1 x x2 (pole) y vt a b -b -a 23σ∞ 2D∞ 22σ∞ 21σ∞ 2D∞ 23σ∞ 22σ∞ 21σ∞
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