PSI - Issue 28

Alla V. Balueva et al. / Procedia Structural Integrity 28 (2020) 873–885 Author name / Structural Integrity Procedia 00 (2019) 000–000

878

6

1

c K



p

(1.4)



2

a

As we mentioned in the begin of this section, the primary advantage of the work described in this work is that the relationship between pressure, volume, and gas molar mas is described using the Equation for Real Gas, given thus:

pV

(1.5)

mRT



1

B p



2

where B 2 = 5.02×10 - 9 1/Pa for T = 300 ◦ K [ Eliaz et al., 2004]. Now we substitute the expression for gas pressure (1.4) and the crack volume (1.3) into the Real Gas Equation (1.5) to yield the following formula:

5 2

E RT 

3

a

(1.6)

m



2

4

K



B K



c

a



2

c

2

where m is a molar mass and it is described in term of the full gas flux into the crack, Q ( t ) as

m



( )

(1.7)

Q t



t



The total fluid mass Q ( t ), accumulated inside the crack by time t , is given:

a

 q r t rdr    , 0,

( )

(1.8)

Q t

0

where q ( r, z, t ) is the diffusion flux density. The flux density, q , can be found from the usual boundary value problem for the gas diffusion, where c is atomic hydrogen concentration in metal:

c



2 D c

(1.9)

 

t



with boundary conditions  ( , , ) , , c r z t c z 



 c z ,



0

( , , )

0,   t

0

,

,

(1.10)

t

c r z t

  



0

0

c z

 









( , 0, ) 0, 0 c r t

( ), r a t t    

0

( , 0, ) 0, r t

( ), r a t t

0

, and

.

 



 c q D z   



Thus, q can be expressed in terms of the rate of change in concentration, namely , where c is the solution of the boundary value problem (1.9-1.10). We believe that the crack is the ideal sink (Balueva and Dashevskiy, 2016). Also, taking into account that we are solving the problem in quasi-stationary approximation, 0, 0    ( ) r a t z 