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A model for high temperature hydrogen attack in carbon steels under constrained void growth

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Petrochemical vessels exposed to high temperature and high pressure hydrogen gas may suffer from high temperature hydrogen attack (HTHA). HTHA is a hydrogen-induced degradation of carbon steels whereby internal hydrogen reacting with carbides forms methane gas bubbles, mainly on grain boundaries (GBs), with an associated loss in strength that can result in premature fracture of structural components. The design of equipment against HTHA is primarily based on the use of the empirical Nelson curves which are phenomenological and do not account for the underlying failure mechanisms and the material microstructure. Starting from the underlying deformation and fracture mechanisms, we present a simple constraint-based model for failure of steels by HTHA which involves growth of GB voids due to coupled diffusion of atoms along the GBs and creep of the matrix surrounding the voids. Since voids form only on some of the GBs, the uncavitated GBs geometrically constrain the growth of voids on the cavitated ones. The model is used to study void growth in HTHA of 21/4Cr–1Mo steel both in the presence and absence of externally applied stress. In the latter case, the model predictions are in good agreement with experimental results. Lastly, the model is used to develop a Nelson-curve type diagram in the presence of external stress in which the curves demarcating the safe/no-safe regimes are functions of the time to failure. This diagram though should be viewed as the result of the application of a new methodology toward devising mechanism-based Nelson curves and not as proposed new Nelson curves for the steel under investigation.

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  1. See “Appendix A” for the details of Stone’s model. In order to calculate the stress on the uncavitated GBs, Stone (1984) assumed that the void growth rate approaches zero. There was a minor mistake in the calculation of Stone and the correct form for the uncavitated GB stress is \(\sigma _A \approx \left( {1/(1-f_b )} \right) \sigma _\infty +\left( {f_b /(1-f_b )} \right) p\). This equation was used to plot the void growth predictions in the figures for Stone’s model.


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The authors would like to acknowledge the funding and technical support from BP through the BP International Centre for Advanced Materials (BP-ICAM) which made this research possible.

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Correspondence to Mohsen Dadfarnia.

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D. E. Moore: Retired from BP Jan, 2017.


Appendix A: Stone’s model (Stone 1984)

Stone proposed a constrained void growth model for bubble growth under applied stress (Stone 1984). He considered that when void growth becomes constrained the driving force that grows the void approaches zero. Neglecting elasticity and the effect of creep, Stone calculated the void growth after it becomes constrained as

$$\begin{aligned} a_S =a_S^0 \left( {1+K_S t} \right) ^{1/3}, \end{aligned}$$


$$\begin{aligned} a_S^0 =\left( {\frac{3\lambda L(\sigma _\infty +p)}{4\pi \rho E}} \right) ^{1/3},\quad K_S =\frac{E{\dot{\varepsilon }}_A ^{\mathrm {c}}}{\sigma _\infty +p} \end{aligned}$$

\(a_S^0 \) is the radius at which the constraint sets in, t is time, \(\lambda \) is the fraction of the grain that accommodates the void growth, L the grain size, \(\sigma _\infty \) the macroscopically applied stress, p the internal void pressure, \(\rho \) the bubble number density on cavitated GBs, E the Young’s modulus of elasticity, and \({\dot{\varepsilon }}_A ^{\mathrm {c}}\) the creep rate in uncavitated grain boundaries. There was a minor mistake in the calculation of the uncavitated GB stress \(\sigma _A \) and the correct form is

$$\begin{aligned} \sigma _A =\frac{1}{1-f_b }\sigma _\infty +\frac{f_b }{1-f_b }p. \end{aligned}$$

This equation was used to plot the void growth results shown in Figs. 4, 5, and 7 for Stone’s model. Stone considered in Eq. (5) that \(\lambda =f_b \) for the fraction of the grain which though seems to be rather small as we elaborate next in “Appendix B”.

Fig. 13
figure 13

Effect of cavitated GB area fraction \(f_b \) on the present model’s predictions of constrained void growth for a quenched and tempered 21/4Cr–1Mo steel and b normalized and tempered 21/4Cr–1Mo steel in 13.8 MPa \(\hbox {H}_{{2}}\) gas, \(600\,^{\circ }\hbox {C}\) environment, and 110 MPa external stress \(\sigma _\infty \). The grain size is L, the bubble number density on cavitated GBs \(\rho \), and the fraction of the grain that accommodates the void growth \(\lambda \) are showed on the figures. The initial void diameter was assumed \(2a_0 =0.02b = 0.0447\, \upmu \hbox {m}\) with 2b being the bubble spacing, and the calculated bubble pressure was \(p = 94.3\) MPa. The experimental data identified with error bars are also superposed for comparison (Stone 1984)

Appendix B: Effect of parameters \(\lambda \) and \(f_b \) on void growth

In order to investigate the effect of the fraction of the grain that accommodates the void growth, \(\lambda \), and the area fraction of the cavitated grains, \(f_b \), on the constrained void growth results, we simulated void growth for Q&T and N&T 21/4Cr–1Mo steels with a hydrogen gas pressure of 13.8 MPa, temperature \(600\,^{\circ }\hbox {C}\), and \(\sigma _\infty = 110\) MPa for various values of \(\lambda \) and \(f_b \). As mentioned in “Appendix A”, Stone (1984) assumed \(\lambda =f_b \) in his model. Figure 12a, b show the effect of this assumption on the void growth results respectively for Q&T and N&T 21/4Cr–1Mo steels. Superposed on the figure are the void diameter results for \(\lambda =1/3\) (Figs. 4a, 5a). Although the predictions of the present model and Stone’s model with \(\lambda =f_b \) are closer to the experimental data, we believe that the assumption \(\lambda =f_b \) is not accurate and hence it should not be adopted.

We also investigated the effect of the area fraction of the cavitated grains \(f_b \) on void growth under external load. As shown in Fig. 13, increasing \(f_b \) increases the void growth rate for both Q&T and N&T 21/4Cr–1Mo steel. However, the effect of an increasing \(f_b \) on N&T steel, Fig. 13b, is more pronounced as it also affects the initial stage of the constrained void growth process as well as the growth rate. This is attributed to the smaller value of the bubble number density on cavitated GBs \(\rho \) in the N&T steel in comparison to that in the Q&T steel which results in a larger bubble spacing 2b.

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Dadfarnia, M., Martin, M.L., Moore, D.E. et al. A model for high temperature hydrogen attack in carbon steels under constrained void growth. Int J Fract 219, 1–17 (2019).

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