13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- where σ y εp are, respectively, the tensors of stress and plastic strain. Condition of contact between the concentration and the criterial surfaces, C(x,t) and Ccr(x,t), respectively, which reads ( ) x x ε x σ x x ∂ = ∂ ∂ ( , ), ( , ) ( , ) t t C t C p cr , (2) accompanies the fracture criterion (1) to form the system of equations to define the location xcr and time tcr of HAF event [12]. Hydrogen transport towards fracture sites is dominated by diffusion, which defines the left-hand parts of Eqs. (1) and (2). It is known that material damage is associated with crystal imperfections, and that they act as hydrogen traps (T-sites) for H atoms where their free energies GT are less than that for ordinary lattice (L-)sites GL (Fig. 1a). The ratio at.H/at.Me can there substantially exceed that in L-sites [2,3,4,6], as follows from the equilibrium partition of hydrogen between T- and L-sites [2,4,6,13] K L L T T θ θ θ θ − = − 1 1 ( ) bE K e β = , (3) where θX = CX/NX is hydrogen saturation of X-type sites (X = L or T) defined by volume concentrations of these sites in metal, NX, and of hydrogen allocated to them, CX, so that the total concentration C = ΣCX, Eb = GL – GT is the binding energy of hydrogen to trap, and β = (RT) –1 is the Boltzmann’s factor in terms of the gas constant R and temperature T. Then, e.g., for steels at usual HAF occurrence conditions T ≈ 300 K and θL ∼ 10 –6 at utmost [3,6], reported values of Eb, being approximately in the range from 0.25 to 1.5 eV [2,14,15] yield K ≥ ~ 104 and θT/ θL ≥ ~10 4. S lAA lBB x lAB= lBA JB\A JB\B JA\A JA\B Δx GL ,GA Potential GT,GB ELL,EAA ELT,EAB x Eb ( ) ( ) ( ) G x G x U x = + x U(x) Potential G(x) b c d a S B AJ B BJ AJ x A BJ A G(x) Figure 1. Schematics of (a) potential-position trace G(x) for H in a lattice with different type sites L and T (or A and B); (b) combination of diffusion jumps between sites of different kinds (A – circles, B – quads) to evaluate partial fluxes; (c) distortion of lattice potential G(x) by superposed field U; (d) combination of hops to establish partial balances. To this end, whenever L/T-partition of hydrogen in volume element d3x around a point x is in equilibrium, all partial concentrations Ci(x,t) (i = 1,2,…) are related one to another via Eq. (3), so that all them, including the one corresponding to crystal imperfections responsible for HAF micromechanism, are biunivocally related to the total one C(x,t). In this case, continuum description of local HAF event by Eqs. (1)-(2) holds, as well as it may be rewritten explicitly in terms of the responsible partial concentration CX merely by changing there the variable according to Eq. (3). Otherwise, HAF should be described taking in Eqs. (1)-(2) responsible concentration CX instead of C, and accounting for L/T-exchange kinetics in analysis of hydrogen transportation.
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