13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- 4. Special Field Equations and Retrieval of the Known Models 4.1. One-level System In the case of diffusion over identical sites, say L-sites, Eqs. (5), (6), and (9) in the absence of any potential field U(x) render Fick’s laws for isothermal diffusion with lattice diffusivity DL = dLL = ΩLLlLL 2 [11,13,15]. As well, they lead to the equations of uphill diffusion under imposed potential U, e.g., due to lattice dilatation under hydrostatic stress σ, when U = – VHβσ, VH is the partial molar volume of H in metal, as they are known both for dilute solutions at θ << 1 [1,2,11,21] (hereafter the site type labelling is skipped wherever convenient) and for arbitrary degree of saturation θ = C/N [22]. For the latter more general case, corresponding reduction of Eq. (5) reads: ( ) [ ] D C C U L − ∇ + − ∇ = θ β 1 J . (13) The balance equation is established then by common continuity relation, cf. Eq. (7). 4.2. Multi-level System, Implicit Description of Trapping: The Variable Solubility Model Since the origin of potential U(x) was not relevant in derivations, it can incorporate the intrinsic own lattice potential relief with variable depths of wells at interstitial sites, which yields variable solubility S(x) according to the well bottom ( )x G at location x. This can be complemented with variable saddle point energy E(x). Then, skipping site labels that become irrelevant, both Eqs. (5) and (6) can be rewritten in terms of the non-uniform both solubility and diffusivity as follows: ( ) ( ) − − − ∇ = − − − ∇ = S C S D S C D C (1 ) 1 (1 ) ln 1 2 θ θ θ θ J , (14) where (cf. expressions (4), (5) and (11)) ( ) [ ] 2 0 ( ) ( ) ( ) ( )exp l E G D x x x x − − = β ω and ( ) ( ) ( ) exp x x G S β − = . (15) With these flux equations, the mass balance is established by usual continuity relation, cf. Eq. (7). This mode to describe diffusion, named [10] the non-uniform solubility model, deals with a system where neither interstitial positions nor saddle points are all identical, and so, it treats in effect a multilevel system. Though, specific sites are here indiscernible within each elementary volume d3x around a point x, where the values ( )x G and E(x) can be interpreted as effective averages over different sites with account for their amounts in regions, which are small in macroscopic sense, but large enough in microscopic one. Such averaging counts implicitly on equilibrium partition of a diffuser between sites of different kinds in d3x disregarding traps filling/emptying kinetics. Anyhow, excepting the term (1 – θ) that accounts for sites saturability, these equations render fairly the same as established for dilute solutions ( θ << 1) elsewhere booth within the frameworks of microscopic kinetics [10] and macroscopic thermodynamics [1,5,9]. To this end, built up upon
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