13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- where EiGΔ represents the deviation from an ideal system due to the presence of elements other than the impurity, i. In a binary system, such as Fe-C or Fe-P, the kinetics of segregation Cgb(t) is given by [6]: ( ) ( ) ( ) , 2 2 1 2 2 2 2 4 2 α α exp α δ α δ gb gb eq o Dt Dt C t C T C erfc ⎡ ⎤ ⎡ ⎤ = − − ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ (3) where , gb eq C is the concentration at equilibrium of the impurity, at temperature T2, given by Eq. 1, Co = CB (t = 0) is the impurity concentration in the matrix which is assumed to be constant, α1 = Cgb,eq (T1) / Co with T1 the temperature from which the material is cooled down, α2 = Cgb,eq (T2) / Co, D the coefficient of diffusion of the impurity at T2, t the time and δ, the thickness of the grain boundary. In a ternary system, Guttmann [7] assumed that seg iGΔ can be expressed as: 2 seg o gb gb i i ii i ij j G G C C Δ =Δ − α +α (4) where the coefficients αii and αij describe the interaction between the elements segregating at grain boundaries (αii > O or αij < O correspond to a repulsive interaction which is the situation met with C and P). The solution for the kinetics of segregation for non-isothermal conditions can be derived from Militzer’s work [12]. All terms in Eqs. 1-4 have been determined in the literature: 10 810 m d − = ( ) 2 -1 200000 0.25exp cm s R p D T ⎛ ⎞ = ⎜ − ⎟ ⎝ ⎠ [13, 14] (5) ( ) 2 1 76000 0.003exp R C D cm s T − ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ [15] (6) ( ) -1 64273 23.7 J mol o pG T Δ = − [13] (7) ( ) -1 16752 40.9 J mol o cG T Δ = + [13] (8) -1 -1 -1 4000 J mole ; 1500 J mole ; 4500Jmole cc pp cp α = α = α =− [10, 16] (9) The McLean-Guttmann-Militzer model has been applied to our steel with T1= 888 K, B CC = 80 ppm (Wt %) and B PC = 150 ppm (Wt %) for predicting the C and P grain boundary concentration in the
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