13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- phenomenological account of measurable variables, but not relying on specification of atomic mechanisms, the latter approach extends the suitability of hydrogen diffusion description by means of Eq. (14) over much wider range of circumstances, such as non-uniformity of alloy composition, transient external fields (e.g., mechanical stresses), simultaneous (e.g., strain induced) phase transformations, cold working, non-isothermal diffusion (Soret effect), etc. [1,5,9,11]. Till now, these factors could not be incorporated satisfactorily in diffusion models via kinetics considerations. From another side, diffusion modelling accounting for the kinetics of H transitions between different microstructural entities (“sites”) is not feasible on the way of thermodynamics. 4.3. Multi-level System, Explicit Description of Trapping When conditions of system equilibrium are not fulfilled, it evolves to equilibrium. Various transitions can be discerned there. One of them is rather long-distance transfer that drives to global equilibrium via fluxes represented by respective equations of the previous section. Besides, localised processes of H exchange between the nearest sites of different kinds are there involved, too, which tend to local equilibrium expressed by the detailed balance (12). Corresponding process rates, which may depend on both intrinsic (e.g., vibration frequency and jump length) and extrinsic (such as diffusion distance) factors, may be so distinct, that in the time scale of interest (e.g., for membrane permeation or for hydrogen delivery to fracture locations) some processes may have attained equilibrium while others have not yet. 4.3.1. Diffusion under Traps-Lattice Equilibrium One possibility of the mentioned partial equilibrium is when detailed balance is quickly reached and maintained during diffusion, e.g., when relatively long diffusion distances xcr with corresponding times t ∼ xcr 2/(4D) [23] are of interest. Suggested by Oriani [17] to be kept during diffusion, the hypothesis of local equilibrium assumes that detailed balance (12) holds for all nearest neighbour A and B sites, so that partial balance (9) in the case of multiple B-sites yields the following: ( ) ( ) [ ] ( ) ( ) [ ] ∑ ∇⋅ ∇ + ∇⋅ ∇ = ∂ ∂ B B B U B B B A AB A A U A A AA A Y C e C Y C C d d Y C e C Y t C / ln / ln β β . (16) The global balance (7) for two-level system (for multi-level one the sum is to be taken over all site types) under local equilibrium is reduced to the next: ( ) ( ) ∇ + + ∇⋅ = ∂ + ∂ ≡ ∂ ∂ A A A AB B A BB B B AA A A A B S Y C d Y C d Y C d Y C t C C t C ln 2 . (17) Accompanying Eq. (17) with the detailed balance (12), which may be solved with respect to CB, ( ) ( ) A B b E A A A B A B K e E G G C K N N C K N C b − = = − + = , 1 / 1 / β , (18) the system of partial-differential equations of diffusion with trapping converts into the system of
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