PSI - Issue 59

Jesús Toribio et al. / Procedia Structural Integrity 59 (2024) 206–213 Jesús Toribio / Procedia Structural Integrity 00 ( 2024) 000 – 000

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i.e., Boltzmann's distribution where co is the equilibrium concentration for the specific couple metal-environment when the former is free of stress, and s  the hydrostatic stress at the boundary. The stationary solution of the diffusion partial differential equation (2) is also a distribution of Boltzmann type:

V  s(x) RT

c o(x) = c o exp  

 

(4)

where x indicates any point of the domain. Following hypothesis D3 , the hydrogen-diffusion problem to be solved (Fig. 2) is that of a cylinder of radius a and c = c o * as the value of hydrogen concentration at the inner boundary (r=a, x=0, r and z being cylindrical coordinates and x the depth). Taking into account the cylindrical symmetry, the diffusion equation leads to:

∂ 2 c ∂r 2

∂ 2 s ∂r 2

1 r

DV* RT

DV* RT c   

1 r

  

   ∂c ∂ r –

  

∂c ∂r

∂s ∂r –

∂s ∂r

∂c ∂t = D

+

+

(5)

in the interval 0 ≤ r ≤ a, with the initial condition: c (r,0) = 0 ; 0 ≤ r ≤ a

(6)

and the boundary conditions: ∂c ∂r (0, t) = 0 ; t ≥ 0

(7)

c (a, t) = c o * ; t ≥ 0

(8)

The boundary condition (7) is due to the symmetry of the problem: the hydrogen flux just at the cylinder axis (r=0) is zero. Condition (8) yields c=co when the solid is free of stress.

4.2. Fracture criterion The following hypotheses were formulated regarding the fracture process:

F1. Hypothesis of damage localization. – The fracture phenomenon is spatially localized and the damage is assumed to be concentrated as a discrete part-through crack perpendicular to the bar axis, thus losing the cylindrical symmetry. Then, although hydrogen diffusion is considered to be axisymmetric (and thus hydrogen concentration is constant along circumferences) there is a certain time at which local damage in the form of a crack appears, in the same way as the necking phenomenon emerges in a standard tension test. This hypothesis seems to be more adequate for high-strength steels (brittle) in which damage localization appears as a discrete surface crack, whereas in austenitic stainless steels (ductile), hydrogen damage develops in volume as a sort of notch extension. F2. Hypothesis of initiation. – Fracture initiates when the hydrogen reaches a critical concentration cc over a distance xc (process zone, damaged area or critical size for initiation); then it is assumed that a crack of depth xc is created and that this is precisely the embrittled zone . The critical concentration cc depends on the environmental conditions (HE in this case) and thus is considered as a material property. F3. Hypothesis of propagation. – The propagation time, i.e., the time required to propagate the crack from the initial size xc up to a critical value to produce the final fracture, is neglected. Then it is assumed that the initiation time coincides with the time to fracture or critical time tc. In accordance with this hypothesis, the damage process is time – localized just at the instant previous to global catastrophic failure (feasible in high-strength steels).