PSI - Issue 59

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

209

4

critical concentration is reached over a critical distance. The model accounts for residual stress laws influencing the hydrogen concentration at the wire surface and the hydrogen diffusion towards the inner points, thus conditioning the life of the wire in the ATT solution.

H

H

H

H

H

H

H

H

(b)

(a)

Fig. 2. Statement of the environment-assisted fracture problem: (a) axisymmetric hydrogen diffusion; (b) damage localization.

The numerical model is able to predict the experimental results of the ATT (time to failure at a given applied stress). The hydrogen diffusion equations are based on three main hypotheses: D1. Hypothesis of absorption. – The absorption of adsorbed hydrogen at the metal surface is considered quasi instantaneous. HE tests on brittle materials such as high-strength prestressing steels confirmed this assumption, since the damaging effect of hydrogen is detectable even in very short tests. D2. Hypothesis of transport. – It is assumed that the main hydrogen transport mechanism is stress-assisted diffusion. This hypothesis was seen to be adequate for high-strength steels in aqueous environments promoting HE. The other important mechanism of hydrogen transport (dislocation movement) is not very important in specimens under quasi-static loading. An effective diffusion coefficient is considered to account for the hydrogen traps (static dislocations and others). D3. Hypothesis of cylindrical symmetry. – The hydrogen entry and diffusion towards the inner points are assumed to possess cylindrical symmetry, i.e., every point of a circumference with its centre at the cylinder axis is supposed to receive the same amount of hydrogen, the concentration depending only on the radial length (distance from the free surface). This hypothesis is consistent with the symmetrical nature of the process, so it is assumed that there is no difference between points located at the circumferences, i.e., no privileged points for diffusion are allowed. From hypothesis D2 , the equations for stress-assisted diffusion are modified Fick's Laws to include terms dependent on the hydrostatic stress:

V* RT c grad  s)

(1)

J = – D ( grad c –

V* RT grad c • grad s –

V* RT c ∆s )

∂c ∂t = D ( ∆c –

(2)

where c is the hydrogen concentration, J the hydrogen flux, t the time, s the hydrostatic stress (s = tr /3, being the   stress tensor), D the hydrogen diffusion coefficient, V* the molar partial volume of hydrogen, R the ideal gases constant and T the absolute temperature. The term ∆ indicates the Laplacian operator (∆= div grad). Equation (2) is a parabolic-type partial differential equation with non-conventional boundary conditions, since the concentration at the boundary (surface of the structural element) changes as the hydrostatic stress does. According to hypothesis D1 , the concentration of absorbed hydrogen at the inner sample surface (co*  is reached quasi-instantaneously, which means that it is a direct function of the hydrostatic stress:

  V  s  RT

c o*  = c o exp  

(3)