13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- For a two-dimensional problem, all the variables are independent of x3, Eqs. (1) and (2) can be expressed in the following compact form: 1,1 2,2 ρ + = & t t gU (3) T 1 ,1 ,2 2 ,1 ,2 , = + = + tQU RU t R U TU (4) where T 1 2 3 [ , , , ] u u u φ = U , T 1 2 3 [ , , , ] D β β β β β σ σ σ = t ( β =1,2), and diag[1, 1, 1, 0] = g . The matrices Q, R and T are related to the material constants by 1 1 11 T 1 1 11 i k i k c e e ε ⎡ ⎤ = ⎢ ⎥ ⎢ − ⎥ ⎣ ⎦ Q , 1 2 1 2 T 2 1 12 i k i k c e e ε ⎡ ⎤ = ⎢ ⎥ − ⎢ ⎥ ⎣ ⎦ R , 2 2 2 2 T 2 2 22 i k i k c e e ε ⎡ ⎤ = ⎢ ⎥ − ⎢ ⎥ ⎣ ⎦ T (5) Substituting Eq. (4) into Eq. (3) leads to ( ) T 2 ,11 ,12 ,22 t ρ + + + = ∂ ∂ 2 QU RRU TU gU/ (6) 2.2 Yoffe-type crack As shown in Fig.1. (x, y, z) is a moving coordinate system fixed on the crack with the center as its origin. It has the relation with the fixed coordinate system (x1, x2, x3) as follows 1 2 3 , , x x vt y x z x = − = = (7) Then v t x ∂ ∂ =− ∂ ∂ (8) Thus, Eq. (6) can be written as ( ) ( ) 2 T , , , xx xy yy vρ − + + + =0 Q g U R R U TU (9) Eq. (9) is the governing differential equation for the steady-state electroelastic fields. Note that the structure is identical to that of the static case when (Q − ρv2g) is identified with Q. 2.3 Boundary conditions The medium is subjected to remote uniform electro-mechanical loads given by T 2 21 22 23 2 [ , , , ] D σ σ σ ∞ ∞ ∞ ∞ ∞ = t . The crack surfaces are traction-free and charge-free, with electrical yielding along strip a≤|x|≤b. The full set of boundary conditions for the moving PS model considered in this paper can be summarized as 2 + − ∞ = =− t t t , at x a< (10a) i i u u + − = , i=1, 2, 3, 2 2 2 s D D D D + − ∞ = =− + , at a x b ≤ ≤ (10b) 0= t at y →∞ (10c)
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