A coupled diffusion and cohesive zone modelling approach for numerically assessing hydrogen embrittlement of steel structures

TL;DR: In this article, a review of coupled diffusion and cohesive zone modelling is presented as a method for numerically assessing hydrogen embrittlement of a steel structure, and the model is able to reproduce single experimental results by appropriate fitting of the cohesive parameters, but there appears to be limitations in transferring these results to other hydrogen systems.

About: This article is published in International Journal of Hydrogen Energy.The article was published on 2017-04-20 and is currently open access. It has received 62 citations till now. The article focuses on the topics: Hydrogen embrittlement & Hydrogen.

Summary

1. Introduction

• Hydrogen induced degradation of mechanical properties, often termed hydrogen embrittlement (HE), is a well recognized threat for structural steels.
• It manifests as loss in ductility, strength and toughness, which may result in unexpected and premature catastrophic failures.
• Rather it appears that different mechanisms apply to different systems, and that a combination of mechanisms is more likely in many cases.
• In recent years, cohesive zone modelling has gained increasing interest as suitable method for modelling hydrogen embrittlement [10, 11, 12, 14, 16], with the possibility of providing increased understanding of the involved process and their interactions combined with reduced time and costs compared to experimental programs.
• The coupling aspect between hydrogen transport and cohesive zone modelling is discussed and put in conjunction with hydrogen diffusion models in Section 2.

2. Hydrogen transport models

• The process that results in hydrogen embrittlement includes an important transport stage of hydrogen to the site of degradation.
• Atomic hydrogen is generally considered to reside either at normal interstitial lattice sites (NILS) or being trapped at microstructural defects like dislocations, carbides, grain boundaries and interfaces.
• Traps generally reduce the amount of mobile hydrogen, thus decreasing the apparent diffusivity and increasing the local solubility of the system.
• To date, models of transient hydrogen diffusion generally account for trapping by dislocations and hydrostatic drift.
• Recent approaches include capturing the effect of multiple trap sites and hydrogen transport by dislocations [13, 21, 22].

2.2. Hydrogen in traps

• Similarly as for lattice sites, the hydrogen concentra- tion in a specific trapping site can be expressed by [8].
• The ability of a trap site to hold hydrogen is associated with the trap binding energy, representing the attractive interaction of a trap site compared to a normal lattice site.
• There have been significant advances in theoretical approaches to capture the effect of traps on hydrogen transport, with models by McNabb and Foster [31] and Oriani [32] describing the process for steel.
• Generally, the trapped concentration increases with increasing lattice concentration and increasing trap binding energy, until saturation is reached.

2.3. Hydrogen diffusion

• The main mechanism for hydrogen diffusion in steel is lattice diffusion by interstitial jumps, where the hydrogen atom occupy interstitial sites and move by jumping from one interstitial site to a neighbouring one [38].
• Chemical potential gradients constitute the main driving force for hydrogen diffusion in steel; hydrogen will diffuse from regions where the chemical potential is high to regions where it is low, and the process ceases once the chemical potentials of all atoms are everywhere the same and the system is in equilibrium [39].
• Assuming that the diffusion flux is proportional to the concentration gradient, which often is the case, Fick’s laws are the governing equations describing the processes.
• These laws represent a continuum description and are purely phenomenological.
• The large scatter observed for ferritic steels is generally considered to be associated with trapping [41].

2.4. Implications of the hydrogen transport model

• In the following section, the effect of varying the hydrogen solubility, the trap binding energy and the trap density on the total hydrogen distribution, as controlled by the hydrogen transport model in Equation (14), is illustrated.
• Displacements are enforced on the circular boundary, controlled by the stress intensity factor KI .
• Figure 6a-d displays the resulting hydrogen profiles in front of the notch tip at the end of loading, for trap binding energies in the range 20-60 kJ/mol and two initial hydrogen concentrations of 0.00034 wppm and 1 wppm, representative of the theoretical solubility of hydrogen in ferrite and a 3 % NaCl aqueous solution [11], respectively.
• In the plots, CL is therefore included as one single line, representing all the cases considered.
• For the low trap density model, the maximum attainable trapped concentration is 0.033 wppm, 100 times an initial lattice concentration of 0.00034 wppm.

3. A cohesive zone modelling approach to hydro-

• Gen embrittlement Cohesive models were first formulated by Barenblatt [44] and Dugdale [45], who introduced finite non-linear cohesive tractions in front of an existing crack, as a mean to overcome the crack tip stress singularity.
• To date, the cohesive model is extensively applied for crack propagation analysis using the finite element method.
• Among the various approaches available, it is appealing in that it requires few parameters and in its universality of applicability [46].

3.1. The cohesive model

• The cohesive theory of fracture is a purely phenomenological continuum framework, not representative of any physical material.
• Common to most cohesive laws is that they can be described by two independent parameters out of the following three: the cohesive strength σC , the critical separation δC and the cohesive energy ΓC .
• The area embedded by the curve represents the cohesive energy.
• Alvaro et al. [15] points out the importance of this in relation to modelling hydrogen embrittlement.
• Yu et al. [57] have applied the viscosity term by Gao and Bower [56] in a three step, un-coupled, hydrogen in- formed cohesive zone model under constant displacement, and found the viscous regularization to be effective in solving the convergence problem with good accuracy.

3.2. Implementing hydrogen influence

• Most known attempts of implementing hydrogen influence into the cohesive model is through the HEDE principle [11, 15, 16, 58, 59, 60]; hydrogen reduction of the cohesive energy at fracture.
• The data fit is illustrated by the red line in Figure 8b.
• Using the coupling between hydrogen coverage and bulk concentration as supplied by the Langmuir-McLean isotherm, Serebrinsky et al. [11] suggested ∆G0b = 30 kJ/mol, which represents the trapping energy of hydrogen at a Fe grain boundary, yielding a threshold concentration of about 0.001 wppm and an embrittlement saturation level of about 5 wppm.
• Figure 10b illustrates hydrogen influence, in terms of hydrogen coverage, according to Equation (20), on both the cohesive strength and on the critical separation, for the polynomial cohesive law by Needleman [48].
• 64], no quantification of any effect of hydrogen on the critical separation is found to date.

3.3. Coupling of diffusion and mechanical models

• The Langmuir-McLean isotherm defines the necessary coupling between the hydrogen diffusion model in Section 2.3 and the hydrogen-dependent cohesive law described in the previous section.
• Choosing a ∆G0b level in the lower range may be justified, conforming to the findings by Novak et al. [13] and Ayas et al. [37] that the only possible trap sites associated with hydrogen embrittlement are low-binding energy traps.
• (Equation (14)) to account for hydrogen transport by dislocations.
• Brocks et al. [14, 68] have developed a model of hydrogen induced cracking, which in addition to the coupled interactions of hydrogen diffusion and reduced cohesive strength, also includes the effect of surface kinetics on hydrogen absorption and hydrogen induced softening of the local yield strength (HELP mechanism).

5. Conclusion

• A coupled mass transport and cohesive zone modelling approach for simulating hydrogen induced cracking is described and discussed.
• Based on calculations, the main findings are summarized as follows: .
• These levels have again significant influence on hydrogen induced reduction of the cohesive strength.
• New developments within modelling of mass transport may improve the agreement.
• Further, transferability may be improved by appropriately identifying the required input parameters for the particular system under study.

Figures

Figure 3: Plot of the trapped hydrogen concentration (CT ) as a function of the lattice hydrogen concentration (CL), equivalent plastic strain (εp) and trap binding energy (EB). (a) Trapping model by Kumnick and Johnson [2]. (b) Trapping model by Sofronis et al. [34, 35].

Figure 8: Hydrogen effect on decohesion by quantum-mechanical approaches: (a) Traction separation curves for decohesion along Al(111) planes with a hydrogen coverage between 0 and 1, by Van der Ven and Ceder [54]. (b) Cleavage energy for decohesion along Al(111) and Fe(110) as a function of hydrogen coverage, by Jiang and Carter [55].

Figure 13: Normalized threshold stress intensity factor and normalized hydrogen dependent cohesive stress as a function of hydrogen concentration, according to experimental data by Thomas et al. [4], the linear decohesion model and the exponential decohesion model by Serebrinsky et al. [11], with ∆G0b = 30 kJ/mol.

Figure 4: Reported diffusion coefficients for hydrogen in iron and steel. Adapted from Grong [41].

Figure 5: Hydrostatic stress and equivalent plastic strain as a function of the distance from the notch tip, plotted at the end of loading.

Figure 7: Cohesive laws by Hillerborg et al. [47], Needleman [48] and Scheider [49].

Figure 11: Relationship between critical cohesive energy at fracture and hydrogen coverage for the polynomial cohesive law by Needleman [48]. Single hydrogen influence denotes hydrogen reduction of the critical cohesive stress. Double hydrogen influence denotes hydrogen reduction of the critical cohesive stress and of the critical separation.

Figure 12: Hydrogen influenced cohesive laws from the decohesion model by Liang and Sofronis [10], T 0n is the normal traction and q is a non-dimensional separation parameter.

Figure 10: Reduction in cohesive energy at different levels of hydrogen coverage for the polynomial cohesive law by Needleman [48], where (a) illustrates hydrogen influence on the cohesive strength only (single) and (b) illustrates hydrogen influence on both the cohesive strength and the critical separation (double).

Table 2: Input parameters and boundary conditions for FE hydrogen diffusion analysis.

Figure 2: Dislocation trap densities according to the work by Kumnick and Johnson [2] and the model by Sofronis et al. [34, 35]. In calculating CT , it is assumed αθT = 1, which accordingly gives the maximum possible hydrogen concentration trapped at dislocations.

Figure 15: CTOD-R curves for various deformation rates, comparing experimental tests (symbol) and simulation results (lines) [14].

Figure 9: Hydrogen coverage as a function of hydrogen concentration, for various levels of Gibbs energy (kJ/mol). Plotted according to the Langmuir-McLean isotherm [63].

Figure 6: Normalized hydrogen concentration as a function of distance from the notch tip using the boundary layer approach, plotted at the end of loading. (a) Low trap density model, C0 = 0.00034 wppm, EB = 30 − 60 kJ/mol. (b) Low trap density model, C0 = 1 wppm, EB = 30 − 40, 60 kJ/mol. (c) High trap density model, C0 = 0.00034 wppm, EB = 20 − 60 kJ/mol. (d) High trap density model, C0 = 1 wppm, EB = 20 − 40, 60 kJ/mol.

Figure 14: Hydrogen coverage and reduction in cohesive strength as a function of distance from the notch tip, plotted at the end of loading. ∆G0b = 30 kJ/mol. (a) Low trap density model, C0 = 0.00034 wppm, EB = 30 (CTot), 40 − 60 kJ/mol. (b) Low trap density model, C0 = 1 wppm, EB = 30 − 40, 60 kJ/mol. (c) High trap density model, C0 = 0.00034 wppm, EB = 20 − 60 kJ/mol. (d) High trap density model, C0 = 1 wppm, EB = 20 − 40, 60 kJ/mol.

Frequently asked questions

Q1. What are the contributions in "A coupled diffusion and cohesive zone modelling approach for numerically assessing hydrogen embrittlement of steel structures" ?

The present study presents a review of coupled diffusion and cohesive zone modelling as a method for numerically assessing hydrogen embrittlement of a steel structure. 

Q2. How much hydrogen coverage is attainable in the low trap density model?

For the low trap density model, the maximum attainable trapped concentration of 0.033 wppm corresponds to a hydrogen coverage of 0.29 and a reduction in cohesive strength of 29 %. 

Q3. What is the importance of assessing the hydrogen distribution in the material?

In order to predict the degrading effect of hydrogen on the mechanical properties, it is of fundamental importance to correctly assess the hydrogen distribution in the material. 

Q4. What is the common method of implementing hydrogen influence into the cohesive model?

Most known attempts of implementing hydrogen influence into the cohesive model is through the HEDE principle [11, 15, 16, 58, 59, 60]; hydrogen reduction of the cohesive energy at fracture. 

Q5. How much is the effective diffusivity ratio at the notch tip?

Assuming EB = 60 kJ/mol, the effective diffusivity ratio at the notch tip yield 0.62 and 0.005 for the low and high trap density models, respectively, at an initial concentration of 0.00034 wppm. 

Q6. How can the authors determine the binding energies of trap sites?

Trap sites and trap binding energies can be established experimentally for a microstructure using varies approaches like electrochemical permeation or thermal desorption spectroscopy (TDS), with TDS considered best suited to provide detailed trap characteristics [5, 13, 23]. 

Q7. How many times an initial lattice concentration is 0.00034 wppm?

For the high trap density model, the maximum attainable trapped concentration is 10.1 wppm, 30000 times an initial lattice concentration of 0.00034 wppm. 

Q8. How is the dislocation trapped hydrogen concentration calculated?

Using Equation (1) - (5), the dislocation trapped hydrogen concentration, CT , is calculated as a function of the lattice hydrogen concentration, CL, in terms of the trapping models by Kumnick and Johnson [2] and Sofronis et al. [34, 35], assuming VM = 7.106 · 10−6 m3/mol, β = 6, α = 1 and room temperature. 

Q9. How can the authors estimate the hydrogen dependent cohesive stress for the fast separation case?

Using parameters representing of Fe (110); (2γint)0 = 4.86 J/m 2 and Γmax = 5.85 · 10−5 mol/m2 [55], assuming ∆g0i −∆g0s = 74.5 kJ/mol [13], the hydrogen dependent cohesive stress for the fast separation case can be estimated. 

Q10. Why is austenite more diffusive than ferrite?

The substantially higher diffusivity in ferrite compared to austenite is due to the lower packing density of bcc metals, reducing the potential energy barrier for jumps. 

Q11. What is the effect of the trap density on the effective diffusivity?

when the lattice concentration is increased from 0.00034 wppm to 1 wppm, maintaining a constant trap binding energy level, the effective diffusivity will increase. 

Q12. What is the cleavage energy of Al(111) and Fe(110)?

An almost linear decrease in cleavage energy with increasing hydrogen coverage is observed for both Al(111) and Fe(110), as displayed in Figure 8b.