13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- 11 12 13 31 12 22 23 32 13 23 33 33 44 24 55 15 66 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x x y y z z yz yz xz xz xy xy C C C e C C C e C C C e C e C e C σ ε σ ε σ ε τ γ τ γ τ γ ⎧ ⎫ ⎧ ⎫ ⎡ ⎤ ⎡ ⎤ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎢ ⎥ = − ⎢ ⎥ ⎨ ⎬ ⎨ ⎬ ⎢ ⎥ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎢ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎢ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎣ ⎦ ⎣ ⎦ ⎩ ⎭ ⎩ ⎭ , x y z E E E ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎥ ⎩ ⎭ ⎥ (3) 15 11 24 22 31 32 33 33 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 x y x x z y y yz z z xz xy D e E D e E D e e e E ε ε ε γ γ γ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ∈ ⎧ ⎫ ⎡ ⎤ ⎡ ⎤⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎢ ⎥ = + ∈ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎢ ⎥ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎢ ∈ ⎥ ⎩ ⎭ ⎣ ⎦ ⎣ ⎦⎩ ⎭ ⎪ ⎪ ⎪ ⎪ ⎪⎩ ⎪⎭ (4) Due to the uniform extension ε0 and infinite length in the x direction, the state variables are independent from the longitudinal coordinate x, therefore, the linear strain-displacement relations of elasticity and the components of electric field vector can be written as 0 x u x ε ε ∂ = = ∂ , y v y ε ∂ = ∂ , z w z ε ∂ = ∂ , yz w v y z γ ∂ ∂ = + ∂ ∂ , 0 xz u w z x γ ∂ ∂ = + = ∂ ∂ , 0 xy u v y x γ ∂ ∂ = + = ∂ ∂ , (5) 0 xE x ϕ∂ =− = ∂ , yE y ϕ∂ =− ∂ , zE z ϕ∂ =− ∂ . Where u, v and w represent displacements in the x, y and z directions, respectively. Gaussian law and stress equilibrium equations of the elastic body are given below , , , , , , , , , , , , 0 0 . 0 0 x x y y z z x x xy y xz z x xy x y y yz z y xz x yz y z z z D D D q f f f σ τ τ τ σ τ τ τ σ ⎧ + + − = ⎪ ⎪ + + + = ⎨ ⎪ + + + = ⎪ + + + = ⎩ (6) By considering Eqs. (3)-(6) and assuming the electrical body charge q as well as body forces fi to be zero, we can get τxz = 0, τxy = 0, Dx = 0 and other 9 state variables v, w, σx, σy, σz, τyz, Dy, Dz, φ. They are all independent of x, and can be expressed as v(y,z), w(y,z), σy(y,z), σz(y,z), τyz(y,z), Dy(y,z), Dz(y,z), and φ(y,z). After a lengthy derivation process based on Eqs. (3)-(6) the following first-order partial differential equations are obtained: { } [ ]{ } { }, R A R B z ∂ = + ∂ (7)
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