13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- ( ) ( ) ( ) , 1 / 4 3 sin 2 2 2 4 3 / 4 3 sin 2 sin 2 2 sin cos 2 cos cos 2 2 sin cos sin cos 3 / exp 1 3 4 0 * T 0 T gb 0 * 0 * 0 2 2 2 2 0 2 0 * 2 2 2 2 0 * 0 * T 0 T gb 2 * gb * gbθ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ Ω ∂ ⎟⎟− ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ × ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − − Ω + Ω − + ⎥ ⎥ ⎦ ⎤ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ∂ ∂ ∂ Ω ∂ ⎟⎟− ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ − Δ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ Ω ∂ ⎟⎟− ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ ⎢ ⎣ ⎡ × Ω ⎟⎟ − ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ + ∂ Ω ∂ − ⎭ ⎬ ⎫ ∂ ∂ Ω ∂ − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ + ⎩ ⎨ ⎧ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ + ∂ Ω ∂ − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ − ∂ ∂ − Δ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ + ∂ Ω ∂ − + Δ × ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − Ω + Ω − − = y N y N x N x N x N N j y N j Z e kT N N N Q T d y N x N x N N j y N j Z e d y N y N x N x N y N N j x N j d Z e kT N y N x N N d x y N N Z e x j y j y N x N N Z e y j x j d y N x N N Z e θ j θ j kT N N N Q d T C N AFD y x y x y x y x y x y x κ ρ σ κ θ κ ρ ϕ κ ρ κ κ θ κ θ ρ θ θ θ κ θ ρ ϕ θ θ κ ρ ϕ σ κ where C* gb represents the product D0δ /k denoting the effective width of the grain boundary by δ, d the average grain size, and Δ ϕ a constant related to the relative angle between grain boundaries as shown in Fig. 2. The quantities jx and jy are components of the current density vector j in Cartesian coordinates, x and y. θ is angle between microstructure unit shown as rectangular in the figure and x axis. (2)
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