13th International Conference on Fracture June 16–21, 2013, Beijing, China -8- one partial-differential and a series of algebraic equations (17) and (18) with respect to partial concentrations CI (I = A,B,…). This system transforms into the sole differential equation (17) with respect to CA by substitution there CB according to Eq. (18). It can be verified that, labelling A ≡ L to be lattice sites and B ≡ T traps, at θL << 1 and γT << γL ≈ 1, and so, YT << 1, which are usually met in HAF of engineering alloys [6], and dBX/dAX ∼ exp(–βEb) < ∼10–4 at Eb > 0.2 eV [2,14,15] for X = L or T, Eqs. (17)-(18) convert into the equation of trapassisted diffusion [4] as implementation of Oriani’s [17] equilibrium hypothesis: ( ) [ ] L L L L L L T D C C S t C C C ln 1 ∇⋅ ∇ − ∇ = ∂ ∂ ∂ ∂ + at ( )L L L T TC C KN N KC + = / . (19) 4.3.2. Diffusion with Account for Lattice/Trap Exchange Kinetics The system of equations (9), which describes hydrogen diffusion affected by traps in rather general terms, is tough to solve. Taking advantage of circumstances usually met in HAF, it can be reduced to more suitable forms. Namely, θL << 1, γT << γL ≈ 1, and thus YT << YL ≈ 1, as well as dTX/dLX << 1 (X = T and L) usually hold in HAF, but the matter of local L/T-equilibrium is not ensured a priori. Then, taken for granted that T→T jumps are fairly improbable, the total flux J, Eq. (6), and the partial balance for CT, Eq. (9), can be assessed as follows: ( ) ( ) [ ] ∇ ∇ + + + + ∇ ∇ + = L T LL TL L T LL LT L T L T T LL TT L L C C O d d Y O oC d d C C O C C Y O d d 1 1 \ J J , (20) ( ) TL T LT T L L L T LL LT T d YC d C fl Y d d t C − + ∇⋅ = ∂ ∂ 2 \ 1 J . (21) Accordingly, if neither ratios of partial concentrations CT to CL and their gradients nor the divergence of lattice-hopping flux JL\L are too large to forbid the disregard of terms with dTX/dLX << 1 and YT << 1 in Eqs. (20) and (21), then Eqs. (17)-(18) convert into the system of partial- and ordinary-differential equations postulated by McNabb-Foster [7], which is now extended for variable solubility SL: ( ) ( ) [ ] L L L L T LC C D C C S t ln ∇⋅ ∇ − ∇ + = ∂ ∂ ; ( )( ) T L TL T T LT T k N N C C k C dt dC − − = . (22) Obviously, these specialised equations of trap-affected diffusion are additively extensible for the case of multiple trap kinds Ti (i = 1,…) as the antecedent general Eqs. (7) and (9) do. The system of equations (22) with respect to a number of concentrations Ci (i = 1,…,m) may complicate numerical simulations of diffusion making corresponding discrete approximations of the boundary-value problems oversized. However, the second of Eqs. (22) has the closed-form solution + ∫ − ∫ = ∫ 0 0 0 0 ( ) ( ) exp ( ) exp T t T t T T P d Q d C P d C ξ ξ ζ ζ ζ ζ ξ , (23)
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