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

## Summary (4 min read)

### 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].

### 2.1. Hydrogen in lattice

- Given a metal lattice, the hydrogen concentration in NILS, CL, can be expressed by [8] CL = βθLNL (1) with θL being the lattice site occupancy, NL the density of solvent atoms and β the number of NILS per solvent atom, usually assigned to be 6 under the assumption of tetrahedral site occupancy in iron.
- The density of solvent atoms, NL, can be calculated through NL = NA VM (2) where NA is the Avogadro constant equal to 6.022 · 1023 mol-1, and VM is the molar volume of the host lattice, which for iron is 7.106 ·10-6 m3/mol at room temperature.

### 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.
- The dislocation trap density is then expressed as a function of the dislocation density ρ and the lattice parameter a N (d) T = √ 2 ρ a (7) The dislocation density (measured in dislocation line length per cubic meter) is considered to vary linearly with the equivalent plastic strain according to ρ = ρ0 + γεp for εp < 0.51016 for εp ≥ 0.5 (8) where ρ0 = 10 10 line length/m3, denotes the dislocation density at zero plastic strain, and γ = 2.0·1016 line length/m3.
- 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.
- Olden et al. [12, 15] and Moriconi et al. [16] have applied the latter approach.
- (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.

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##### References

20,482 citations

6,292 citations

### "A coupled diffusion and cohesive zo..." refers methods in this paper

...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....

[...]

...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....

[...]

5,032 citations

### "A coupled diffusion and cohesive zo..." refers background in this paper

...[47], a polynomial law suggested by Needleman [48] for ductile materials and, more recently, a versatile trapezoidal law suggested by Scheider [49] also for ductile materials....

[...]

...[47], Needleman [48] and Scheider [49]....

[...]

4,900 citations

4,294 citations

### "A coupled diffusion and cohesive zo..." refers methods in this paper

...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....

[...]

...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....

[...]