PSI - Issue 59

Oksana Hembara et al. / Procedia Structural Integrity 59 (2024) 190–197 Oksana Hembara et al. / Structural Integrity Procedia 00 (2019) 000 – 000

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trapping sites (trapped hydrogen). The concentrations of diffusion-mobile and trapped hydrogen are denoted as C L and C T . According to experimental findings by Liang and Sofronis (2003), the concentration of trapped hydrogen C T is related to the equivalent plastic deformation e p . Considering the equilibrium relationship between diffusion-mobile and trapped hydrogen, the concentration C T is related to C L (Oriani (1970)). The hydrogen flux J is induced not only by the gradient of hydrogen concentration C L , but also by the gradient of hydrostatic stress σ h , and it is determined as follows:

DCV

(1)

L L J D C

  

 

L L H

h

RT

where D L is hydrogen diffusion coefficient in lattice sites of the material, V H is molar volume of hydrogen, R is gas constant, i.e., 8.314 J/(mol×K), and T is absolute temperature. Taking this into account, the hydrogen diffusion equation is expressed as follows:





C

DCV

N





  

  





p

*

(2)

0

D

L L    D C

 

 



L

L L H

T

h

T

t

RT

t

  



p

where θ T is hydrogen trapping efficiency, N T is the number of hydrogen trapping sites, and effective diffusion coefficient D * , which depends on C L , C T and θ T , is given by:     * 1 L T T L D C C C     (3) Applying the standard finite element procedure using Galerkin’s method, the finite element eq uation for hydrogen diffusion with an unknown vector for hydrogen concentration at nodes { C L } is obtained as Mehrer (2007):

  L C



            1 2 1 2 L K K C F F    

  M

(4)

t



where the corresponding matrices and vectors are computed according to the following relationships:       T * M N N V D dV  

(5)

DV

and  

  2 K B  

      B N 

T

    T 1 K B B L V  

  D dV

(6)

dV

L H

h

RT

V



T N     

    T N T 

and   F

    T N 

  1 F

S 

p

 

(7)

dV



dS

2

t

p

S

In the mentioned above equations [N] represents the shape function of the finite element, [B] is the gradient of [N], i.e.       B N ,   is the vector of hydrogen flux on the surface with an external unit normal vector n , i.e.   , h j n     is the vector of nodal hydrostatic stress. For the discretization of the time derivative of { C L } in equation (4) the backward Euler method was used. 3. Computational estimations of local hydrogen distribution in the vicinity of voids of various shapes and micro and macro cracks The analysis of hydrogen diffusion was conducted for a pipe made of low-alloyed steel AISI 1020, which is subjected to internal pressure from the transported hydrogen-containing product. The pipe has an inner diameter of