13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- where Ds is the electrical saturation limit. 3 Solution of the problem 3.1 Full field solution Adopting Stroh formalism for anisotropic elasticity, a general solution to Eq. (9) can be sought in the form ( ) f z = U a , z x yμ = + (11) where μ and a are a constant and a constant vector respectively; and f(z) is an arbitrary function of variable z subject to the twice-differentiable requirement. Substitution of Eq. (11) into Eq. (9) results in ( ) 2 T 2 0 vρ μ μ ⎡ ⎤ − + + + = ⎣ ⎦ Q g R R T a (12) This is a nonlinear eigenvalue problem. A nontrivial solution of a requires that the determinant of its coefficient matrix must be zero, i.e., ( ) 2 T 2 det 0 vρ μ μ ⎡ ⎤ ⎣ − + + + ⎦ = Q g R R T (13) This is a polynomial of degree 8 for μ. If μα ( α = 1, 2, 3, 4) are assumed to be the four distinct roots with positive imaginary parts, and aα are the associated eigenvectors, the general solution can then be expressed as ( ) 4 1 2 f z α α α α= = ℜ∑ U a (14) where ℜ denotes the real part and z x y α αμ = + . Substituting Eq. (14) into Eq. (4) and by using Eq. (3), the stress and electric displacement vectors can be expressed as 2 1 , , y x v Φ ρ =− + t gU , 2 ,xΦ = t (15) in which ( ) 4 1 2 f z α α α α Φ = = ℜ∑b (16) where Φ = [ φ 1, φ 2, φ 3, φ 4] T is called the generalized stress function vector, and bα can be determined from aα by the following relation: ( ) ( ) T 2 1 v α α α α α μ ρ μ− ⎡ ⎤ = + =− − + ⎣ ⎦ b R T a Q g R a (17) Introducing two 4×4 matrices, i.e., [ ] 1 2 3 4 , , A a a a a , = , [ ] 1 2 3 4 , , B b b b b , = (18) and a function vector, i.e.,
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