PSI - Issue 5

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

1296

Toribio and Kharin / Structural Integrity Procedia 00 (2017) 000 – 000

6

where d AB =  AB l 2 is the diffusivity associated with A  B jumps of the length l . The total diffusion flux is              A B A B U AB B A A B AB e C Y d Y C , , / ln J J . In contrast to other diffusion-trapping theories, which overview can be found elsewhere (see, e.g., Toribio and Kharin (2015)), this one considers simultaneously ( i ) multiple site types, ( ii ) arbitrary occupancies of all type sites up to their blocking at  A  1, ( iii ) non-uniform transient site populations N A ( x , t ), ( iv ) the alteration of lattice potential relief by a superposed field (x, ) U t , and ( v ) particle jumps between dissimilar sites A  B . For the purposes of HAF analysis, this allows to describe hydrogen diffusion under the effects of the accumulated plastic strain  p ( x , t ) via strain dependent trap multiplication N A = N A (  p ), and of the stress field, e.g., in linear isotropic approximation using U ( x , t ) =  v H  ( x , t ), where v H is the partial atomic volume of hydrogen in metal and  is the hydrostatic stress. (9) Although presented flux expressions describe individual hydrogen transportation steps in the course of HEAF, they do not form the closed system of equations to describe the overall hydrogen transport process from a source environment to prospective fracture sites in terms of specific hydrogen amounts – its concentrations – in each state and location in a system. To complete the task, the mass transfer has to comply with the conditions of partial mass conservation referring to hydrogen residing in the sites of each particular type   Z present in a system. For the gaseous state of hydrogen within the surface layer of characteristic thickness   near the CT, the mass balance has to account for hydrogen supply by the in-crack gas-phase flow g * of appropriate type (*) and for its outgoing into the physisorbed state by the net H 2 ( G )  H 2 (  ) exchange flux ( q G  – q  G ). With the use of the ideal gas law, corresponding mass conservation equation with respect to atom number volume concentration of hydrogen in the CT environment can be obtained in terms of the local gas pressure p CT as follows: 3. Balance equations

dp

( * g q q  G

)

.

(10)

CT

2

   2



G



dt

Similarly, the mass conservation conditions for hydrogen states at the gas-metal interface can be derived as follows (cf. Kompaniets and Kurdyumov (1984) or Shanabarger (1985)):

dt dC dt dC

) ( G G     (  q q   q q

)



,

(11)



  A

   q q ) (  

(

)

 A A q q 



,

(12)





dt dC

 B    

n J ( A B

)

q q

(  A   ),

A

(13)

 

 A A



where the flows g * and fluxes q * are defined by the former expressions (2) – (7), n is the unit external normal to the entry surface, and A J B are the net escape diffusion fluxes from the interstitial near-surface A -sites to the deeper interior B -sites ( A , B   ) due to outward (with respect to near-surface crystal layer) A  B and inward B  A jumps. These fluxes can be obtained similarly to the derivation of the fluxes (8) by Toribio and Kharin (2015) as continuum limit of the discrete description of random walks, which now yields       Y U d Y C C U C Y C Y d C Y B BA A B B AB A B BA B A AB A B A B               2 1 2 1 J . (14)