13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- 3. Three-dimensional Vacancy Diffusion Analysis The numerical analytical method of stress induced particle diffusion was proposed by Yokobori, et al. [13-15]. In this method, the stress distribution is calculated by finite element analysis (FEA) and stress induced particle migration is calculated by finite difference analysis (FDA). This analytical method was applied to the problems of hydrogen embrittlement [13, 14, 16], stressmigration and electromigration of LSI interconnect [8-10, 17] and high-temperature creep of heat-resistant steels [7]. All of these analyses were conducted in two dimensions. However, the actual plant has a complicated three-dimensional structure. Therefore, to apply this analytical method to the actual components, it is important to conduct the three-dimensional vacancy diffusion analysis. Thus, in this paper, the numerical analytical method of three-dimensional vacancy diffusion was developed and applied to the high-temperature creep of C(T) specimen. 3.1. Basic Equation of Vacancy Diffusion Based on a multiplication concept [13-15], the basic equation of three-dimensional vacancy diffusion is given by Eq. (1). ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ + ∂ Δ ∂ − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ Δ ∂ ⎟⎟− ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 2 2 2 2 2 3 2 2 2 2 2 2 2 1 z y x C RT V D z z C y y C x x C RT V D z C y C x C D t C p p p p p p σ σ σ α σ σ σ α α (1) where C is concentration of vacancies, t is time, D is the self diffusion coefficient of P92 steel, ΔV is a volume change due to accommodation of a vacancy, R is gas constant, T is absolute temperature, σp is hydrostatic stress, and αi are the weight coefficients. The self diffusion coefficient D is given by Eq. (2). ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = RT Q D D exp 0 (2) where D0 and Q are diffusion constant and activation energy of P92 steel, respectively. 3.2. Analytical Methods and Conditions Distribution of hydrostatic stress for C(T) specimen was calculated by three-dimensional FEA. This analysis was performed by MSC Marc / MSC Marc Mentat ver. 2009 (Cyber Science Center, Tohoku University). A finite element model of C(T) specimen is shown in Fig. 3. Due to symmetry, only one quarter of the specimen is meshed. The material properties of P92 steel used in the FEA were shown in Table 3. Work hardening due to the plastic deformation is given by Eq. (3). p cε σ = (3) where, σ is equivalent (Mises) stress, and pε is equivalent plastic strain. The distributions of vacancy concentration were obtained by solving Eq. (1) using three -dimensional FDA. Figure 4 shows a model of three-dimensional vacancy diffusion analysis. To match the finite difference lattice and finite element mesh, unequally-spaced finite difference lattice was used in the FDA. Equation (1) was converted into the non-dimensional equation using the non-dimensional values given by Eq. (4). ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = = − = = = = = + + + + + + RT Q D D D a t t D C C C a z z a y y a x x exp , , , , , 0 0 2 0 (4) The non-dimensional equation is discretized by the Crank–Nicolson implicit method and Eq. (5)
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