PSI - Issue 5

Jesús Toribio et al. / Procedia Structural Integrity 5 (2017) 1291–1298 Toribio and Kharin / Structural Integrity Procedia 00 (2017) 000 – 000

1297

7

For interstitial diffusion and trapping in solids, the mass conservation equation with respect to the species residing in the type A sites ( A   ) within infinitesimal material element d 3 x results from the balancing within arbitrary material volume of the outgo and income fluxes through its surface A J B and B J A , respectively, which should be taken for the exterior sites of all kinds B   . According to Toribio and Kharin (2015), this reads:





 k C N C k C N C d C Y Y U d Y C C U        2                 AB A B B BA B A A B B AB A B B BA A

    

    

 t C







   B A B

   AA d Y C C Y C Y U      A A A A   A A

(  A   ), (15)

A





, 

where k AB = 6  AB / N are the local A  B jump rates between dissimilar sites within d 3 x .

Now, ordinary differential equations (DEs) (10)-(12) stated on the body surface for the problem variables p CT , C  and C  , where the gas-phase flow g * should be represented by one of the expressions (2) to (4) which suites for a given HEAF case, together with partial DEs (15) for C A -s ( A   ) in the body volume form the closed system of equations describing the entire process of hydrogen transport to fracture sites. Both DE subsystems are coupled by the boundary conditions (14) for C A -s ( A   ). As a complementary equation, which can substitute whichever of the partial balances (15), the balance equation for total concentration C =  A   C A can be obtained either by summing up all the partial ones (15), or as customary continuity equation J      C t using the total flux (9). In the present model, hydrogen transport from the source gas to potential fracture sites in metals comprises several individual transitions of the species between various positions in a system. These movements are kinetic processes that seek respective equilibriums between hydrogen in definite states booth locally (within d 3 x in continuum physics terms) among the distinguishable nearest neighbour sites Y and Z ) ( ,   Y Z , and globally. Characteristic rates that particular movements come after respective equilibriums may be so different, that some processes within definite time frame of a particular HEAF case may have approached closely the equilibriums whereas others have not yet done so. In the limit cases, DEs of balance corresponding to the fastest interchanges degenerate into algebraic equality constraints between the rates at which the forward and the backward movements, Y  Z and Z  Y , of certain individual Y  Z interchanges of the species occur in local equilibrium (cf. Shanabarger (1985) and Toribio and Kharin (2015)). Counting on local equilibriums brings a great simplification to the transport modelling and streamlines computations. Assuming such equilibrium for a definite individual transition, virtually every known particularised model, which associates the kinetics of HEAF with the rate of certain environmental, interface of metal bulk transport process qualified as the rate controlling step, can be recovered as a limit case of the present theory. Comparison of the characteristic times of the individual transport steps – the time of diffusion over the distance x cr towards HAF locus in metal t x D cr dif / 2  , where D is some effective diffusivity, and the times  YZ of relaxation of Y  Z transitions towards the equilibrium partial concentrations ( ) Z eq Y eq Y C C C  along the long-time asymptotic solutions   ( / ) YZ eq Y Y t C C     1 exp of the balance equations for fast-fillable sites Y (see. Shanabarger (1985), Toribio and Kharin (2015)) – can justify the appropriateness of local equilibrium approximations in specific HEAF cases. This depends on the combination of both intrinsic (characteristic site energies E YZ and G Y , particle jump length l , etc.) and extrinsic (hydrogen pressure, stresses, strain induced trap multiplication N T (  p ), required transport distance x , etc.) factors. To exemplify, consider the environment-material-load combination when the in-crack molecular transport is irrelevant for hydrogen delivery to FPZ (i.e., when the Poiseuille flow condition º º / º / p T pb T  ~100  holds), and the involved diffusion distance to fracture loci implies t dif >>  YZ for all   Y Z , , i.e., all the local equilibriums can be virtually fulfilled during rather slow diffusion. The left-hand portions of balances (10)-(13) then become virtually nil, which renders the Sieverts-like equalities representing the equilibrium at gas-metal interface as the boundary condition for diffusion                G A A A A A A sz S p S C ) / /(    1 (  A   ) (16) 4. Closing commentaries