Influence of hydrogen on the elastic properties and dislocation behavior in Fe–Cr and Fe–Ni alloys by DFT calculations

Abstract

The effect of interstitial hydrogen on the elastic properties of bcc Fe, bcc Fe–Cr, and bcc Fe–Ni was investigated using density functional theory calculations. Our results indicate that the elastic moduli decrease linearly with increasing hydrogen concentration. The consequences of hydrogen for the mechanical properties of bcc Fe, bcc Fe–Cr, and bcc Fe–Ni were analyzed, considering various factors such as the ideal shear stress, Peierls stress, number of dislocation pile-ups, and critical crack growth lengths. At the same hydrogen concentration, compared to the bcc Fe and bcc Fe–Ni systems, fewer dislocation pile-ups and shorter critical crack growth lengths can facilitate the nucleation and propagation of cracks in the bcc Fe–Cr system. Finally, we propose a mechanism to explain the influence of Cr and Ni on hydrogen embrittlement.

Introduction

Green and clean hydrogen energy is expected to become increasingly important in the future global energy structure. However, the issue of hydrogen embrittlement in steel materials used for hydrogen energy storage and transportation poses a significant threat to the safe application of hydrogen energy. Metallic elements such as Cr and Ni are widely used to enhance the comprehensive mechanical properties of steel materials¹. However, studies have shown that alloy steels with high Cr content, such as 13Cr martensitic stainless steel, exhibit higher hydrogen embrittlement sensitivity in environments containing hydrogen gas and solutions with hydrogen sulfide compared to alloy steels with low or no Cr content^2,3. Therefore, understanding the intrinsic mechanisms by which different alloying elements influence the hydrogen embrittlement sensitivity of steel is crucial for optimizing alloy design for materials used in hydrogen environments.

Traditionally, hydrogen embrittlement in metals is attributed to the reduction in metallic bond strength after the H atom enters the interior of the metal, leading to a decrease in macroscopic mechanical properties such as strength and ductility⁴. Specifically, the hydrogen-enhanced decohesion mechanism (HEDE) theory suggests that the charge transfer between the 1 s orbital electrons of the H atom and the unfilled d orbital electrons of the transition metal reduces the interatomic strength between metal atoms^5,6,7,8. This reduction is believed to be a critical factor in the embrittlement process.

Since the atomic binding force is directly proportional to the elastic modulus, variations in elastic modulus are a crucial parameter for studying hydrogen embrittlement susceptibility. Hydrogen charging experiments and density functional theory (DFT) calculations on metals such as Fe, Ni, Fe–Al, and Fe–Cr alloys have demonstrated that the elastic modulus decreases with increasing hydrogen solubility^{9,10,11,12,13}. This supports the theory that hydrogen embrittlement causes low-stress cracking due to the reduced strength of metallic bonds and a reduction in toughness during the cracking process. However, in metallic materials, plastic deformation occurs at the crack tip before crack propagation¹⁴. Consequently, the crack propagation behavior in the hydrogen embrittlement process is influenced by both the reduction in metallic bond strength and the plastic deformation at the crack tip.

Crack propagation in metals is often accompanied by local plastic deformation at the crack tip, with dislocations serving as the carriers of plastic deformation^15,16,17,18. Hydrogen embrittlement mechanisms such as hydrogen-enhanced local plasticity mechanisms (HELP) and adsorption-induced dislocation emission mechanisms (AIDE) suggest that hydrogen promotes dislocation emission and motion in metals, thereby making the crack tip prone to local plasticity deformation^19,20. Nanoindentation experiments with Fe and Ni indicate that the H atom promotes extensive dislocation multiplication and reduces the external stress required for plastic deformation^21,22. In situ transmission microscopy experiments conducted in a hydrogen environment reveal that the H atom can enhance the motion of dislocations in Fe^17,20. Molecular dynamics simulations show that the H atom accumulates around the edge dislocations of bcc Fe, thereby reducing the Peierls stress and increasing dislocations mobility^23,24. First-principles calculations have demonstrated that the Peierls stress decreases with increasing hydrogen concentration in Al, Fe, and Ni^25,26,27,28. Theoretical derivations by Frank, Peierls, and Nabarro indicate the critical stress intensity for Frank-Read dislocation source emission and the resistance to dislocation motion (Peierls stress) are directly proportional to the elastic modulus of the metal^29,30,31. Consequently, the enhancement of dislocation emission and motion by H atoms is likely related to the decrease in the elastic modulus of the metal material.

Dislocation emission and motion ultimately lead to dislocation pile-ups at obstacles such as grain boundaries and second phases, resulting in stress concentration. The degree of the stress concentration is proportional to the applied stress and the number of dislocation pile-ups. According to the dislocation pile-up theory derived by Eshelby, the number of dislocation pile-ups is inversely proportional to the elastic modulus^30,31. The Griffith criterion for crack propagation states that when the stress at the dislocation pile-up exceeds the theoretical fracture strength (metallic bond strength) of the material, the metallic bond there will break and form a microcrack. Additionally, the critical length of crack propagation is directly proportional to the elastic modulus of the materia^32,33. Therefore, both the dislocation pile-up and the propagation of critical microcracks during hydrogen embrittlement are influenced by changes in elastic modulus in the material.

The metallic bond strength of the Fe–Fe, Fe–Cr, and Fe–Ni bonds varies due to the difference in electronegativity among Cr, Ni, and Fe atoms. The DFT calculations have shown that alloying has a minimal effect on the elastic modulus of Fe^34,35, therefore exerting a limited influence on dislocation emission, motion, and pile-up behavior. Some researchers have indicated a significant difference in the extent of charge transfer between the H atom and Fe, Cr, Ni, Ni–Cr, Fe–Ni, and Fe–Cr metals^{25,36,37,38,39}, indicating that the impact of the H atom on the elastic modulus of these alloys is substantial. Consequently, there will be notable variations in how the H atom influences the hydrogen embrittlement sensitivity of steel containing different alloying elements.

In this paper, we used the DFT method to calculate the elastic modulus of the bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems under different hydrogen concentrations. Based on the Peierls-Nabarro framework and dislocation pile-up theory, we analyzed the implications for the dislocation emission, motion, and pile-up properties of hydrogen-charged Fe, Fe–Cr, and Fe–Ni alloys. This analysis considered multiple factors, including the ideal shear stress, Peierls stress, the critical stress strength, the number of dislocation pile-ups, the length of dislocation pile-ups, and the critical crack propagation length. Partial density of states, charge density difference analysis, and population analysis were used to investigate the bonding interactions between the Fe, Cr, Ni, and H atoms and their effect on dislocation emission, motion, and pile-up behavior. Finally, we discussed the mechanisms by which the alloying elements Cr and Ni affect the hydrogen embrittlement susceptibility of steel.

Methods

In this paper, we calculated the elastic modulus of bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems with varying hydrogen concentrations using the first principles method (DFT).

Further, the elastic modulus was used to calculate the critical stress for dislocation source emission and resistance to dislocation motion, including the ideal shear stress (${\tau }_{is}$) and the Peierls stress (${\sigma }_{p}$). Based on the Eshelby and Griffith theories^30,31,40, we calculated the dislocation pile-up and critical crack propagation characteristics, including the dislocation pile-up length, number, and critical crack size. Finally, we analyzed the partial density of states, charge density difference, and population distributions in bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems.

Calculation of elastic modulus

The elastic constants of crystals are characterized by ${C}_{ij}$. In cubic systems, there are three independent elastic constants, denoted as ${C}_{11}$, ${C}_{12}$ and ${C}_{44}$⁴¹.

The elastic modulus of each system was calculated using the Voigt-Reuss-Hill approximation method (VRH)^42,43,44. The elastic modulus included bulk modulus ($B$), shear modulus ($G$), Young’s modulus ($E$), and Poisson's ratio $\left( v \right)$. The relationship between elastic modulus and elastic constants in a cubic system is defined as follows:

$$B = B_{H} = B_{V} = B_{R} = \frac{{C_{11} + 2C_{12} }}{3}$$

(1)

$$G_{v} = \frac{{C_{11} - C_{12} + 3C_{44} }}{5}$$

(2)

$$G_{R} = \frac{{5(C_{11} - C_{12} )C_{44} }}{{4C_{44} + 3\left( {C_{11} - C_{12} } \right)}}$$

(3)

$$G = G_{H} = \frac{{G_{V} + G_{R} }}{2}$$

(4)

$$E = E_{H} = \frac{{9B_{H} G_{H} }}{{3B_{H} + G_{H} }}$$

(5)

$$v = \frac{{3B_{H} - 2G_{H} }}{{2\left( {3B_{H} + G_{H} } \right)}}$$

(6)

In a cubic system, according to Born-Huang’s lattice dynamics theory^45,46, the corresponding mechanical stability criterion can be expressed as:

$$C_{11} > 0; C_{11} - C_{12} > 0; C_{44} > 0; C_{11} + 2C_{12} > 0$$

(7)

Solution energy and differential charge density calculation

To calculate the stable solute interstitial site of the H atom, the method for determining the solute energy of the H atom in bulk bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems is as follows⁴⁷:

$$E_{sol} = E\left( {Fe/Fe_{n} Cr_{m} /Fe_{n} CrNi_{m} + H} \right) - E\left( {Fe/Fe_{n} Cr_{m} /Fe_{n} Ni_{m} } \right) - \frac{1}{2}E\left( {H_{2} } \right)$$

(8)

where the $E\left( {Fe/Fe_{n} Cr_{m} //Fe_{n} Ni_{m} + H} \right)$ was the total energy of Fe, $Fe_{n} Cr_{m}$ or $Fe_{n} Ni_{m}$ supercell with a single H atom, and $E\left( {Fe/Fe_{n} Cr_{m} /Fe_{n} Ni_{m} } \right){ }$ is the reference energy of perfect Fe, $Fe_{n} Cr_{m}$ or $Fe_{n} Ni_{m}$ supercell. $E\left( {H_{2} } \right)$ was the total energy of the H₂ molecule, calculated by putting the H₂ in a cubic box with 10 Å and carrying out a G-point calculation. It obtained for H₂ a bond length of 0.750 Å, a vibrational frequency of 4367 cm⁻¹, and a binding energy of 4.54 ${\text{eV}}$, which were almost identical to previous GGA results with the value of 0.741 Å, 4395  cm⁻¹, and 4.75 ${\text{eV}}$⁴⁸.

The method for calculating the differential charge density of the H atom in bulk bcc Fe and bcc Fe–Cr/Ni systems was as follows

$$\Delta \rho = \rho \left( {Fe/Fe_{n} Cr_{m} /Fe_{n} CrNi_{m} + H} \right) - \rho \left( {Fe/Fe_{n} Cr_{m} /Fe_{n} Ni_{m} } \right) - \rho \left( H \right)$$

(9)

where the $\rho \left( {Fe/Fe_{n} Cr_{m} //Fe_{n} Ni_{m} + H} \right)$ was the charge density of Fe, $Fe_{n} Cr_{m}$ or $Fe_{n} Ni_{m}$ supercell with a single H atom, and $E\left( {Fe/Fe_{n} Cr_{m} /Fe_{n} Ni_{m} } \right)$ was the reference charge density of perfect Fe, $Fe_{n} Cr_{m}$ or $Fe_{n} Ni_{m}$ supercell, $\rho \left( H \right)$ was the charge density of the H atom.

Calculation method

The elastic constant ($C_{ij}$), supercell energy, state density, and charge differential density were all calculated using the CASTEP Package in Materials Studio⁴⁹. All analyses were performed using the projector augmented wave (PAW) potentials and the generalized gradient approximation (GGA), with the exchange–correlation functional of Perdew–Burke–Ernzerhof (PBE)⁵⁰. The spin-polarized GGA has been shown to give reliable results for the ground state⁵¹ and elastic propertie^52,53 for Fe. Spin polarization was considered for the calculations of the bcc structure because of the ferromagnetism of bcc Fe, bcc Fe–Cr, and bcc Fe–Ni structures.

A plane wave cut-off energy was 400 eV. This energy cut-off was sufficient for highly precise energetics for all the structures considered in this paper. A $5 \times 5 \times 5$ k-point mesh was generated using the Monkhorst–Pack scheme to sample the Brillouin zone. All calculations were based on a ferromagnetic, $3 \times 3 \times 3$ cubic bcc supercell containing 54 atoms in a defect-free state. The bcc Fe₅₃Ni (Fe-1.95 wt% Ni) and bcc Fe₅₃Cr (Fe-1.72 wt% Cr) cells are constructed by replacing a Cr or Ni atom with a Fe atom in the top corner of the bcc Fe cell. The lattice constants of bcc Fe, bcc Fe–Cr, and bcc Fe–Ni cells are 2.83 Å, 2.86 Å, and 2.84 Å, respectively^34,37,54.

Before calculating the elastic constants, we first optimized the structure of bcc Fe, bcc Fe–Cr, and bcc Fe–Ni supercells as well as those containing H atoms, to obtain a stable structure. During the structure optimization processes, energy minimization was converged until the energy was less than 5 × 10^–6 eV/atom, and the forces on all the atoms were below 0.01 eV/A. Subsequently, the elastic constants $C_{ij}$ of each stable structure were calculated using the Elastic Constants module of the CASTEP package.

When modeling Fe (110), we use seven layers of Fe atoms, with the bottom four layers fixed at their bulk positions. We use a 15 Å thick vacuum layer for both Fe (110), Fe–Cr (110) and Fe–Ni (110). We allow the H layer, together with the top three layers of the Fe (110), Fe–Cr (110), and Fe–Ni (110) slab to relax.

Results

The tetrahedral and octahedral interstitial sites in bcc Fe are equivalent, meaning the solution energy of the H atom in any tetrahedral or octahedral interstitial site in bcc Fe is equal. However, in the bcc Fe–Cr/Ni system, the distance between a Cr/Ni atom and the center of the four nearest tetrahedral sites or the five nearest octahedral sites was different, as shown in Fig. 1. Therefore, when an H atom dissolves in various tetrahedral or octahedral interstitial sites in bcc Fe–Cr/Ni, its solute energy varies and must be calculated individually for each interstitial site. The most stable site for the solution of the H atom in the bcc Fe–Cr/Ni system was determined based on the minimum solution energy.

Fig. 1

(a) Different tetrahedral interstitial sites (T₁–T₄) in the bcc Fe–Cr/Ni system. (b) Different octahedral interstitial sites (O₁–O₅) in bcc Fe–Cr/Ni system.

Taking the Cr/Ni atom as the reference point, different interstitial sites were categorized according to the distance between the H atom and Cr/Ni atom from near to far: the octahedral interstitial sites from 1st to 5th nearest neighbor was labeled as O₁–O₅, and the tetrahedral interstitial sites from 1st to 4th nearest neighbor was labeled as T₁–T₄, as shown in Fig. 1.

The solution energy of H atoms in the T-site of the bcc Fe is 0.11 eV, while the solution energy in the O-site is 0.23 eV, which is consistent with the results from other studies^38,55. The lower solution energy of H atoms in the T-site compared to the O-site indicates that the T-site is the stable interstitial site for H atoms in bcc Fe. Further calculations of the solution energy for the H atom in the bcc Fe–Cr and bcc Fe–Ni system also show that the solution energy of the H atom in the T-sites is smaller than that in the O-sites, as shown in Fig. 2. It suggests that the most stable solution interstitial sites for the H atom in both the bcc Fe and bcc Fe–Cr and bcc Fe–Ni systems are the T-site. Figure 2 also shows that the interstitial site T₃ in the bcc Fe–Cr/Ni system has the lowest solution energy of the H atom. Therefore, for the subsequent calculations of the elastic constants for the H atom, the interstitial site T₃ in the bcc Fe–Cr and bcc Fe–Ni system was chosen as the stable solution interstitial site for H atoms.

Fig. 2

The solution energy ($E_{sol - Fe - Cr/Ni}$) of the H atom at different tetrahedral and octahedral interstitial sites in bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems. The circle represents the solution energy of the H atom in the tetrahedral interstitial site.

Calculation of elastic modulus of bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems

Figure 3a presents the calculated s ($C_{11}$, $C_{12}$, and $C_{44}$) for the bcc Fe, bcc Fe–Cr and bcc Fe–Ni systems. The elastic constant of the $C_{11}$, $C_{12}$ and $C_{44}$ for bcc Fe are 283.74, 155.71, and 120.60 GPa, respectively, which are consistent with the calculated results in other studies³⁴. As shown in Fig. 3a, the elastic constants $C_{11}$, $C_{12}$ and $C_{44}$ for bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems exhibit a decreasing trend with increasing hydrogen concentration.

Fig. 3

(a) Elastic constants ($C_{11}$, $C_{12}$, and $C_{44}$) for bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems at different hydrogen concentrations. (b) Bulk modulus ($B$), shear modulus ($G$), and Young’s modulus ($E$) for bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems at different hydrogen concentrations.

The elastic modulus of the bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems was calculated using the Voigt-Reuss-Hill (VRH) approximation, based on Eqs. (1)–(6), which includes the bulk modulus ($B$), shear modus ($G$) and Young’s modulus ($E$), as shown in Fig. 3b. The mechanical stability of these elastic constants was verified using the Born-Huang lattice dynamics theory (Eq. 7)^45,46. The results confirm that the calculated elastic constants for these systems satisfy the mechanical stability criteria.

Without the solution of the H atom, the value of the shear modulus for bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems are 93.5, 97.6, and 93.1 GPa, respectively. It indicates that the addition of the alloying element Cr in bcc Fe can slightly increase the strength of the metallic bond. In contrast, the influence of the alloying element Ni is minimal, which is consistent with reported computational results³⁴.

As shown in Fig. 3b, the presence of the H atom results in a significant decrease in the values of $B$, $G,$ and $E$ for all three alloy systems with increasing hydrogen concentration. Similarly, some research has shown that the elastic constants and elastic moduli decrease approximately linearly with increasing hydrogen concentration for bcc Fe and bcc Fe–Cr system^13,56. For the primary slip system [110] < 111 > in bcc Fe, the shear modulus decreases by approximately 1.6% per atomic percent of hydrogen¹³. At a hydrogen concentration of 3.6%, the shear modulus of the bcc Fe–Cr system (Fe-1.72 wt% Cr) is approximately 68 ${\text{GPa}}$¹². However, for the bcc Fe–Ni system, there are no reported studies on how the shear modulus varies with the concentration of solution hydrogen.

Specifically, the reduction of $B$, $G,$ and $E$. values in the bcc Fe–Cr system is significantly greater than in the bcc Fe and bcc Fe–Ni systems, while the bcc Fe and bcc Fe–Ni systems exhibit similar trends.t indicates that although the bcc Fe–Cr system exhibits the highest strength of metallic bond without the solution of the H atom, its metallic bond strength is significantly lower than that of the bcc Fe and bcc Fe–Ni systems once the H atom is present.

Effects of the H atom on emission and motion of dislocations in bcc Fe, bcc Fe–Cr, and bcc Fe–Ni system

Dislocations are the carriers of plastic deformation in bcc structured metals⁵⁷, and variation in the elastic modulus will also affect the emission and motion behavior of dislocation. In the absence of defects, the ideal shear strength ($\tau_{is}$) of a metal is expressed as^25,30:

$$\tau_{is} = \frac{GKd}{{2\pi b}}$$

(10)

where $G$ is the shear modulus, $b$ is the magnitude of the Burgers vector, $d$ is the interatomic spacing in the direction of dislocation motion, and $K$ is the elastic parameter. For the bcc structured metal, the (110) slip plane and < 111 > slip direction were chosen when calculating the $b$ and $d$ values. The elastic parameter $K$ is expressed as:

$$K = \frac{{sin^{2} \theta }}{1 - \upsilon } + \cos^{2} \theta$$

(11)

where $\upsilon$ is the Poisson’s ratio, and angle $\theta$ is the variation in the dislocation characteristic, where $\theta = 0^{^\circ }$ and $\theta = 90^{^\circ }$ for the screw and the edge dislocation, respectively.

The critical stress for Frank-Read dislocation source emission is given as³¹:

$$\tau_{Frank} = \frac{Gb}{{2\pi r}} = \tau_{is} \frac{d}{r}$$

(12)

where $G$ is the shear modulus, $b$ is the magnitude of the Burgers vector, and $r$ is the length of the dislocation line.

Figure 4 shows the value of $\tau_{is}$ at different hydrogen concentrations for bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems. The ideal shear strengths for screw dislocations ($\tau_{is - edge}$) along the (110) < 111 > slip system in bcc Fe, bcc Fe–Cr, and bcc Fe–Ni are12.16, 12.69, and 12.15 ${\text{GPa}}$, respectively, as shown in Fig. 4a; for edge dislocations, the ideal shear strengths are 17.28, 17.38, and 16.79 ${\text{GPa}}$. as shown in Fig. 4b. These values are in good agreement with previously reported values^25,58.

Fig. 4

(a) The ideal shear strength of screw dislocations in the bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems at different hydrogen concentrations ($\tau_{is - screw}$). (b) The ideal shear strength of the edge dislocations in the bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems at different hydrogen concentrations ($\tau_{is - edge}$).

When containing the solution H atom, it can be seen from Fig. 4a and b that the values of $\tau_{is - screw}$ and $\tau_{is - edge}$ decrease with increasing hydrogen concentration, which is also consistent with previously reported values in bcc Fe²⁵. Comparatively, the decrease in ideal shear strength $\tau_{is}$ of the bcc Fe–Cr system is significantly greater than that of the bcc Fe and bcc Fe–Ni systems, while the $\tau_{is}$ values of the bcc Fe and bcc Fe–Ni systems are relatively similar.

For the critical stress required for the emission of a Frank-Read dislocation source, according to Eq. (12), $\tau_{Frank}$ is $\frac{d}{r}$ times of $\tau_{is}$, where $\frac{d}{r}$ is a fixed value, and $\tau_{Frank}$ can also be approximated as a function of the elastic modulus. Consequently, the value of $\tau_{Frank}$ variation follows the same trend as $\tau_{is}$, indicating that after the solution of the H atom, the bcc Fe–Cr system exhibits the lowest shear stress required for dislocation emission due to the most significant decrease in elastic modulus within the Fe–Cr alloy.

The minimum critical shear stress (Peierls stress) required for the free motion of dislocations at 0 $K$ without thermal activation in a crystal is Peierls stress ($\sigma_{p}$), It represents an intrinsic resistance to dislocation motion in a metal lattice and is a fundamental quantity related to crystal strength^25,29. For the case of wide dislocations, approximately $\zeta /d > 0.5$, the expression for the Peierls stress ($\sigma_{p}$) is given by²⁹:

$$\sigma_{p} = \frac{GbK}{d}exp\left( { - \frac{2\pi \zeta }{d}} \right)$$

(13)

where $G$ is the shear modulus, $K$ is the elastic parameter, $b$ is the magnitude of the Burgers vector, and $d$ is the interatomic spacing along the direction of dislocation motion, ${\upzeta }$ is the half-width of the dislocation. The dislocation half-width ${\upzeta }$ can be expressed as:

$$\zeta = \frac{KbG}{{4\pi \tau_{is} }}$$

(14)

The calculated results of the Peierls stress along the (110) <111> slip system for the bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems under different hydrogen concentrations are shown in Fig. 5, where the Peierls stress for the screw dislocations ($\sigma_{p - screw}$) is significantly greater than that for the edge dislocation ($\sigma_{p - edge}$). Both simulations and experimental studies have demonstrated that the Peierls stress for screw dislocations is higher than that for edge dislocations in bcc metals, and that edge dislocations migrate faster than screw dislocations^26,59. Our calculation results are consistent with these observations and align with those of other researchers.

Fig. 5

(a) The Peierls stress of the screw dislocation ($\sigma_{p - screw}$) in bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems at different hydrogen concentrations. (b) The Peierls stress of the edge dislocation ($\sigma_{p - edge}$) in bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems at different hydrogen concentrations.

Without the presence of the H atom, as shown in Fig. 5a and b, the values of $\sigma_{p}$ along the (110) <111> slip system for both screw and edge dislocations are as follows: $\sigma_{p} \left( {bcc Fe - Cr} \right) > \sigma_{p} \left( {bcc Fe} \right) \approx \sigma_{p} \left( {bcc Fe - Ni} \right)$. The value of $\sigma_{p - screw}$ in bcc Fe, bcc Fe–Cr, and bcc Fe–Ni are 1031.71, 1076.15, and 1031.36 ${\text{GPa}}$, as shown in Fig. 5a; for edge dislocations, the value of $\sigma_{p - edge}$ are 246.37, 254.43, and 245.10 ${\text{GPa}}$. as shown in Fig. 5b. It is noteworthy that the value $\sigma_{p - screw}$ for the bcc Fe–Cr system is 3.5% larger than that for the bcc Fe system, indicating that the addition of the alloying element Cr slightly increases the resistance to dislocation motion in bcc Fe. In contrast, the influence of the alloying element Ni is minimal.

Comparing our results with previously reported values, we find that discrepancies in the calculated $\sigma_{p}$ for the same system arises due to differences in the Peierls stress models and expressions used by various researchers⁶⁰. For instance, the $\sigma_{p}$ for edge dislocations in bcc Fe (for the slip system (110) <111>) is 57 ${\text{MPa}}$⁵⁶, while for screw dislocations, the $\sigma_{p}$ is reported to be 900 MPa⁵⁶, 390 MPa⁶¹, or in the range of 80–100 ${\text{MPa}}$^62,63. Therefore, our $\sigma_{p}$ results are within a reasonable range.

When containing the H atom, the values of $\sigma_{p}$ for screw and edge dislocations in the bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems gradually decrease with increasing hydrogen concentration, which is also consistent with previously reported values in bcc Fe²⁵. It can indicate that the solution H atom reduces the resistance to dislocation motion in the bcc Fe, bcc Fe–Cr, and bcc Fe -Ni systems. It is similar to the trend observed in the variation of ideal shear strength (Fig. 4).

Comparatively, the value of $\sigma_{p}$ for the bcc Fe–Cr system decreases significantly more than that for the bcc Fe and bcc Fe–Ni systems. For example, when the solution hydrogen concentration is 6.9%, the $\sigma_{p}$ value for the screw dislocation in the bcc Fe–Cr system decreases by 9.13% compared with the value in the bcc Fe system, while the value for the bcc Fe–Ni system decreases by 2.5% compared to the bcc Fe system. This indicates that after the solution of the H atom, dislocations in the Fe–Cr alloy become more mobile due to the most pronounced decrease in Peierls stress ($\sigma_{p}$) in bcc Fe–Cr system. In contrast, the differences in the resistance to dislocation motion between Fe–Ni alloy and Fe are less apparent.

Effect of the H atom on dislocation pile-ups and critical crack propagation of bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems

The emission and motion of dislocations lead to the formation of dislocation pile-ups at obstacles (such as grain boundaries and second phases). When the density of dislocation pile-ups reaches a critical level, it results in localized stress concentration. Under applied stress ($\sigma_{a}$), the stress at the dislocation head ($\sigma_{1}$) generated by $n$ dislocation is given by^30,31:

$$\sigma_{1} = n\sigma_{a}$$

(15)

The length of dislocation pile-ups ($L$)³¹ is:

$$L = \frac{{n{\text{KG}}b}}{{2\sigma_{a} }}$$

(16)

where the $n$ is the number of dislocation pile-ups, $G$ is the shear modulus, $K$ is the elastic parameter, and $b$ is the magnitude of the Burgers vector.

According to the Griffith criterion of fracture mechanics³³, under applied stress $\sigma_{a}$, when the stress concentration at the dislocation pile-up position reaches the critical state for crack initiation, the stress at the dislocation head ($\sigma_{1}$) equal the theoretical fracture strength ($\sigma_{m}$).

The relationship between the theoretical fracture strength $\sigma_{m}$ and the concentration stress at the head of the dislocation pile-ups $\sigma_{1}$ is expressed as follows:

$$\sigma_{m} = \sigma_{1}$$

(17)

The theoretical breaking strength ($\sigma_{m}$) of a metal is:

$$\sigma_{m} = \left( {\frac{E\gamma }{d}} \right)^{0.5}$$

(18)

where $d$ is the interatomic spacing along the direction of dislocation motion, $E$ is Young's modulus, and $\gamma$ is the surface energy of each system. The surface energy of the (110) planes $\gamma$ for bcc Fe, Fe–Cr, and bcc Fe–Ni systems were calculated using the DFT method, and the result is shown in Table 1. The $E$ expressions are as follows:

$$E = \frac{G}{{2\left( {1 + \upsilon } \right)}}$$

(19)

where the $G$ is the shear modulus and $\upsilon$ is the Poisson’s ratio.

Table 1 The surface energies $\gamma$ (${\text{J/m}}^{{2}}$)of bcc Fe, bcc Fe–Cr, and bcc Fe–Ni (110) plane.

Therefore,

$$\sigma_{m} = \left[ {\frac{G\gamma }{{2d\left( {1 + \upsilon } \right)}}} \right]^{0.5}$$

(20)

The relationship between the theoretical fracture strength $\sigma_{m}$ and the concentration stress at the head of a dislocation pile-up position $\sigma_{1}$ is described as:

$$\sigma_{m} = \sigma_{1} = n\sigma_{a}$$

(21)

At the dislocation pile-ups position, the number of dislocations pile-ups $n$ is denoted as:

$$n = \left[ {\frac{G\gamma }{{2d\left( {1 + \upsilon } \right)}}} \right]^{0.5}$$

(22)

According to Eqs. (16) and (22), the length of dislocations pile-ups $L$ can be expressed as:

$$L = \frac{nKGb}{{2\sigma_{a} }} = \left[ {\frac{{K^{2} G^{3} b^{2} \gamma }}{{8d\sigma_{a}^{2} \left( {1 + \upsilon } \right)}}} \right]^{0.5}$$

(23)

According to the Griffith theory, taking the edge dislocation as an example, the critical half-crack length $a$ for crack propagation is given by

$$a = \frac{nbG}{{\pi \left( {1 - v} \right)\sigma_{a} }} = \left[ {\frac{{G^{3} b^{2} \gamma }}{{2\pi^{2} d\sigma_{a}^{2} \left( {1 - \upsilon^{2} } \right)}}} \right]^{0.5}$$

(24)

From Eqs. (22)–(24), it is evident that the number of dislocation pile-ups ($n$), the length of the dislocation pile-ups ($L$), and the critical half-crack length ($a$) can all be regarded as functions of the shear modulus $G$. If the elastic modulus of the metal decreases (indicating a decrease in the strength of the metallic bonds at the microscopic level), the theoretical fracture strength $\sigma_{m}$ will decrease in the metal material. This results in a reduction in both the number ($n$) and length ($L$) of dislocation pile-ups, ultimately leading to a reduction of the critical crack length ($a$).

Therefore, as indicated by Fig. 2b and Table 1, it can be inferred that the presence of dissolved H atom significantly decreases the shear modulus $G$ and surface energy of the bcc Fe–Cr system compared to bcc Fe and bcc Fe–Ni. Consequently, based on Eq. (21), the theoretical fracture strength ($\sigma_{m}$) of the bcc Fe–Cr alloy is expected to decrease significantly in the presence of dissolved H atoms.

Under the critical fracture state of the metal, according to Eq. (22), for the same applied stress ($\sigma_{a}$), the number of dislocation pile-ups ($n$) in the bcc Fe–Cr alloy will be notably smaller than that in bcc Fe and bcc Fe–Ni alloys, as shown in Fig. 6a. For instance, when a hydrogen concentration is 6.9%, the number of dislocation pile-ups ($n$) in the bcc Fe–Cr alloy is approximately 13% lower than in bcc Fe and bcc Fe–Ni alloys. This indicates that in the bcc Fe–Cr alloy, due to the presence of dissolved hydrogen, the reduced number of dislocation pile-ups ($n$) can lead to the fracture of metallic bonds at the stress concentration position and form micro-cracks. In contrast, the difference is not significant in the bcc Fe and bcc Fe–Ni alloys.

Fig. 6

(a) The number of dislocation pile-ups ($n$) in bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems at different hydrogen concentrations. (b) Critical crack growth lengths ($a$) in bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems at different hydrogen concentrations.

According to Eq. (24), the critical half-crack length ($a$) is proportional to the square of the shear modulus ($G$) and surface energy ($\gamma$). As shown in Fig. 6b, In the presence of dissolved H atom, the values of $a$ in both bcc Fe, bcc Fe–Cr, and bcc Fe–Ni alloys decrease to varying degrees with increasing hydrogen concentration. Other studies have similarly noted that under conditions of sufficient hydrogen concentration and applied stress, dislocation pile-ups are promoted, which facilitates crack formation. This hydrogen effect is also consistent with the experimental data obtained using transmission electron microscopy¹⁵ or internal friction^64,65 where the early start of dislocation sources, the increased mobility of dislocations, and the rearrangement of dislocations in the pile-ups under the solution H atom effect were observed.

In particular, the reduction in the value of $a$ is more noticeable in the bcc Fe–Cr alloy than in bcc Fe and bcc Fe–Ni alloys. For instance, when the hydrogen concentration is 6.9%, the value of $a$ in the bcc Fe–Cr alloy decreases by 17% compared with bcc Fe, while in the bcc Fe–Ni alloy, the decrease is only 4%. It indicates that in the presence of dissolved H atoms, the critical half-crack length in Fe–Cr alloy is significantly shorter than in pure Fe and Fe–Ni alloy.

In metals, plastic deformation is a prerequisite for crack propagation. The resistance to crack propagation consists of surface energy ($\gamma$) and plastic deformation work ($\gamma_{p}$), as assessed by the Orowan criterion¹⁴. According to this criterion, the actual fracture strength ($\sigma_{c}$) at the dislocation pile-ups can be expressed as:

$$\sigma_{c} = \left[ {\frac{{2E\left( {\gamma + \gamma_{p} } \right)}}{\pi a}} \right]^{0.5}$$

(25)

where the $\gamma$ and $\gamma_{p}$ are the surface energy and plastic deformation work of each system, respectively, $E$ is Young's modulus, and $a$ is the critical half-length of crack propagation. For the metal $\gamma_{p} \approx 10^{3} \gamma$.

Therefore,

$$\sigma_{c} \approx \left[ {\frac{{2 \times 10^{3} E\gamma }}{\pi a}} \right]^{0.5}$$

(26)

where the $E$ and $\gamma$ expressions are as follows:

$$E = \frac{G}{{2\left( {1 + \upsilon } \right)}}$$

(27)

$$\gamma \approx \frac{Gb}{{10}}$$

(28)

where the $G$ is the shear modulus, $b$ is the magnitude of the Burgers vector, and $\upsilon$ is the Poisson's ratio.

So, the $\sigma_{c}$ can be expressed as:

$$\sigma_{c} \approx 10\left[ {\frac{{G^{2} b}}{{\pi a1 + \upsilon}}} \right]^{0.5}$$

(29)

According to Eq. (29), the actual breaking strength ($\sigma_{c}$) can also be seen as a function of the elastic modulus. Thus, with increasing hydrogen concentration, the most pronounced decrease in the elastic modulus of the bcc Fe–Cr alloy leads to the lowest actual fracture strength among the three alloy systems.

Therefore, when comparing the bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems, it becomes apparent that, in the presence of dissolved H atoms, the considerable decrease in the shear modulus of the bcc Fe–Cr alloy leads to a significant reduction in its theoretical fracture strength. At the site of the dislocation pile-up position, given the same applied stress, fewer dislocations in the bcc Fe–Cr system can cause the fracture of metallic bonds at the stress concentration points, resulting in the formation of micro-cracks. Furthermore, only a shorter critical crack length is necessary for further crack propagation.

Discussion

Increasing hydrogen concentration leads to a significant reduction in both the ideal shear strength, Peierls stress, and critical crack length. This observed decrease indicates that the interstitial H atom is a key that reduces the energy required for dislocation nucleation, motion, and further crack propagation. However, this analysis attempts to capture the effect of interstitial H atom in the bulk metal, thus the competing hardening observed when hydrogen segregates at the dislocation core is not explained. Hydrogen interacts significantly with the dislocation core, affecting both the core radius and core energy. This interaction leads to variations in the ideal shear strength and Peierls stress, ultimately influencing the dislocation pile-up behavior within the material.

In the Peierls-Nabarro model, the expression for the dislocation core radius r_c can be interpreted as the dislocation half-width^29,66. The formula for the dislocation core radius is given by: $r_{c} = \zeta = \frac{KbG}{{4\pi \tau_{is} }}$.

The dislocation core radius r_c is a physical quantity related to lattice constants and elastic moduli. Figure 7a shows the relationship between the ratio of the expanded volume to the original volume and the concentration of H atoms in the bcc Fe and bcc Fe–Cr/Ni system. It is observed that the cell volume expands with the increasing of the dissolved H atoms in the bcc Fe, bcc Fe–Cr, and bcc Fe–Ni supercell. The volume expansion rate is approximately proportional to the amount of H atoms.

Fig. 7

(a) Variation curves of the expansion rate of the cell volume with the concentration of hydrogen atoms in the bcc Fe, bcc Fe–Cr, and bcc Fe–Ni system. (b) Dislocation core radius with the concentration of hydrogen atoms in bcc Fe, bcc Fe–Cr, and bcc Fe–Ni system.

Figure 7b shows the variation of dislocation core radius with the different H concentrations for the slip system (110) <111>. The value of $r_{c}$ along the (110) < 111 > slip system in bcc Fe, bcc Fe–Cr, and bcc Fe–Ni are 2.9926, 2.9964, and2.9893 Å, Studies have shown that the value of the $r_{c}$ in bcc Fe is 2b ~ 4b^66,67,68. Our results are in good agreement with previously reported values. When the H atoms dissolve and accumulate around the dislocation core, an increase in hydrogen concentration leads to a local volumetric expansion of the dislocation core region, which increases the lattice constant of the dislocation region, and ultimately results in an increase in the dislocation core radius. However, when comparing the dislocation core radius $r_{c}$ in the bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems at the same hydrogen concentration, the differences in core radius are not significant.

In the framework of elastic theory, the dislocation line energy $E_{disloc}$^69,70 is given by:

$$E_{disloc} = \frac{{KGb^{2} }}{4\pi }In\frac{R}{{r_{c} }} + E_{c}$$

(30)

where K is the elastic parameter, b is the magnitude of the Burgers vector, G is the shear modulus, and R and $r_{c}$ are outer and core radius. The first term on the right-hand side is the dislocation line energy stored in the elastic region bounded by two circles of radii $r_{c}$ and R, and the second term $E_{c}$ is the dislocation core energy stored in the region bounded by the circle of radius $r_{c}$.

The H-reduced dislocation line energy in the elastic region can be attributed to H reducing the G or increasing $r_{c}$. Among them, the bcc Fe–Cr system shows the greatest decrease, leading to the most significant reduction in the dislocation line energy stored in the elastic region. The core energy is difficult to compute analytically due to the limitations of elasticity theory in the core region. The influence of the atoms on the dislocation core energy $E_{c}$ in bcc Fe, bcc Fe–Cr and bcc Fe–Ni systems have not been discussed in our article. Research has demonstrated that solution H atom decreases the core energies and modifies the core structures of both edge and screw dislocations in bcc Fe⁵⁶.

The critical stress for dislocation emission ($\tau_{Frank}$), resistance to dislocation motion ($\sigma_{p}$), theoretical fracture strength ($\sigma_{m}$), number of dislocation pile-ups ($n$), length of dislocation pile-ups ($L$), critical crack propagation length ($a$), and the dislocation line energy are all related to the elastic modulus. In the bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems, the differences in shear modulus are mainly due to the addition of Cr and Ni atoms.

Firstly, the lattice constants of bcc Fe and bcc Fe–Cr/Ni systems vary due to the different atomic radii of Fe, Cr, and Ni atoms. Specifically, the lattice constants of bcc Fe, bcc Fe–Cr, and bcc Fe–Ni cells are 2.83 Å,2.86 Å, and 2.84 Å, respectively^34,37,54. This indicates that the lattice constants of bcc Fe increase with the addition of Cr and Ni atoms.

For the same dislocation, the Berger vector and the distance between slip planes are affected. Additionally, the difference in electronegativity among Cr, Ni, and Fe atoms leads to changes in the strength of the metal bonds between Fe–Fe, Cr–Fe, and Ni–Fe. After the incorporation of H atoms, the strengths of Fe–H, Cr–H, and Ni–H bonds will also vary. Consequently, the elastic constants and shear modulus of bcc Fe and bcc Fe–Cr/Ni differ, as shown in Fig. 3. Changes in shear modulus and lattice constants ultimately result in variations in the dislocation core structure, the theoretical shear strength of dislocation propagation, and the Peierls stress affecting dislocation migration.

From a microscopic perspective, the elastic modulus represents the strength of atomic bonding, which is directly proportional to the strength of atomic interactions. Therefore, analysis of the partial density of states (PDOS) and charge density difference can provide insights into the bonding interactions between the Fe, Cr/Ni, and the H atom.

In the bcc Fe, bcc Fe–Cr and bcc Fe–Ni systems, the presence of the H atoms significantly modifies the local atomic bonding. Analysis the PDOS, as shown in Fig. 8a–c, reveals that in the presence of the H atoms, the 1 s orbital of the H atom resonates with the d orbitals of the adjacent Fe, Cr, and Ni atoms at approximately − 9.0 ${\text{eV}}$. This resonance indicates strong electronic interactions between the H atom and the neighboring Fe, Cr, and Ni atoms, suggesting a charge transfer between the 1 s orbital of the H atom and the d orbitals of the Fe, Cr, or Ni atom.

Fig. 8

(a) The partial density of states (PDOS) of the bcc Fe system. (b) Partial density of states (PDOS) diagram of bcc Fe–Cr system. (c) Partial density of states (PDOS) diagram of bcc Fe–Ni system.

The amount of charge transfer between H atoms and the Fe, Cr, and Ni atoms in the bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems can be determined by calculating the differential charge density and Milligan charge distribution. The calculation results are shown in Fig. 9. The blue regions in the charge density difference indicate a decrease in charge density, whereas red regions indicate an increase in charge density.

Fig. 9

When the H atom dissolved in the tetrahedral interstices site, the differential charge density in bcc Fe (a), (d); bcc Fe–Cr (b), (e) and bcc Fe–Ni (c), (f). (a)–(c) Is the charge differential density diagram of the (001) slice. (g) Schematic diagram of the bcc Fe, bcc Fe–Cr, and bcc Fe–Ni lattice, where the gray shaded part is the (001) plane. The blue areas are where the charge density decreases, and the red areas are where the charge density increases.

It can be observed that the charge density increases near the H atom, suggesting that the H atom may acquire electrons from neighboring Fe, Cr, or Ni atoms. The extent of electron transfer varies significantly among Fe, Cr, and Ni atoms, as shown in Fig. 9. This indicates that the presence of the H atom in the lattice leads to a redistribution of the outermost shell charges of the metal atoms in the bcc Fe, bcc Fe–Cr and bcc Fe–Ni systems. Consequently, this redistribution results in varying degrees of changes in the charge distribution among different metal atoms, which ultimately affects the strength of the metallic bonds. The results demonstrate that the presence of the H atom induces a reorganization of the outermost charges of the metal atoms in bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems.

The strength of the metallic bonds in the bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems can be quantified by calculating the bond population of each metallic bond. Figure 10a–c shows the population numbers of each metallic bond in the bcc Fe, bcc Fe–Cr and bcc Fe Ni systems along the (110) slip plane and <111> slip direction. Additionally, the schematic diagram is shown in Fig. 10d.

Fig. 10

The bond population of each metallic bond in the (a) bcc Fe, (b) bcc Fe–Ni, and (c) bcc Fe–Cr system in the (110) slip plane and < 111 > slip direction. (d) Schematic diagram of slip plane and slip direction in bcc Fe and bcc Fe–Cr/Ni systems, where the gray shaded part indicates that the slip plane in bcc Fe and bcc Fe–Cr/Ni systems is (110), and the arrow indicates that the slip direction is <111>.

Following the solution of the H atom, this is a notable reduction in the bond population of each metallic bond on the (110) plane, with the bond population transitioning from positive to negative along the <111> slip direction. In the presence of the H atom, the bond population of the $Cr - Fe_{4}$ bond in the bcc Fe–Cr system experiences the most significant decrease, followed by the $Ni - Fe_{4}$ bond in bcc Fe–Ni system and $Fe - Fe_{4}$ bond in bcc Fe system. A positive bond population indicates an increase in metallic bond strength, while a negative bond population signifies a decrease in bond strength, often associated with the formation of inverse bonds. Consequently, the elastic modulus of the bcc Fe–Cr system exhibits the greatest decrease after hydrogen incorporation, consistent with the calculated results shown in Fig. 2b.

Our studies on dislocation behavior were conducted at 0 K. However, the temperature change can directly affect the lattice constant and shear modulus of metals and indirectly influence dislocation behavior and hydrogen atom solubility diffusion. These effects, in turn, impact the overall mechanical properties and hydrogen embrittlement resistance of the material. In bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems, as the temperature increases, the lattice constants of each system increase due to thermal expansion effects. Concurrently, the elevated temperature excites phonon vibrations within the lattice, subsequently reducing the shear modulus in each system⁷¹.

Since first-principles calculations can only determine the elastic modulus of each system at 0 K, we referenced the relationship between shear modulus and temperature for bcc Fe and Fe-based alloys given in the literature. The study found that with increasing temperature, the shear modulus in bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems decreases progressively^61,72. We derived that at 300 K, the shear moduli for bcc Fe, bcc Fe–Cr, and bcc Fe–Ni are 80.81, 82.93, and 81.19 ${\text{GPa}}$, respectively⁷². Further analysis revealed that when the temperature increases from 0 to 300 K, the shear modulus decreases by 13.60%, 15.05%, and 13.17% for bcc Fe, bcc Fe–Cr, and bcc Fe–Ni, respectively. Notably, the Fe–Cr system exhibits the largest decrease in shear modulus with increasing temperature.

​ After introducing H atoms, we are unable to calculate the shear modulus of each system at different temperatures using DFT methods. However, previous calculations indicate that at 0 K, the shear modulus of each system decreases with increasing hydrogen concentration²⁶, with the bcc Fe–Cr system showing the most significant decrease. Furthermore, an increase in temperature promotes the solubility and diffusion of hydrogen atoms in metals, leading to a higher concentration of solute hydrogen near dislocations. This significantly increases the probability of interactions between hydrogen atoms and dislocations. Therefore, we predict that even at elevated temperatures (T < curie temperature), the Fe–Cr system will still exhibit the largest decrease in shear modulus with increasing hydrogen concentration.

The analysis indicates that the presence of the H atom significantly reduces the metallic bond strength in the bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems along the <111> slip direction of the (110) plane. Among these systems, the reduction in metallic bond strength is most pronounced in the bcc Fe–Cr system, compared to the bcc Fe and bcc Fe–Ni systems. Consequently, the decrease in the elastic modulus of the bcc Fe–Cr system is significantly greater than that of the bcc Fe and bcc Fe–Ni systems. This reduction facilitates easier dislocation emission and motion in the bcc Fe–Cr system. At the location of stress concentration due to dislocation pile-ups, crack nucleation and propagation in the bcc Fe–Cr system can occur under lower applied stress and fewer dislocation pile-ups number, which significantly increases the probability of crack nucleation and propagation in the bcc Fe–Cr system. This suggest that the alloying element Cr enhances the hydrogen embrittlement sensitivity of bcc Fe. Conversely, the presence of hydrogen has a relatively minor effect on the elastic modulus of the bcc Fe–Ni system compared to the bcc Fe system, indicating that Ni does not significantly impact the hydrogen embrittlement sensitivity of bcc Fe.

Conclusions

The presence of the H atom influences the elastic modulus of metals and their alloys. Specifically, the critical shear strength required for dislocation emission and the resistance to dislocation motion can be approximated as a function of elasticity. Additionally, the critical fracture strength for crack initiation at dislocation pile-ups, the number of dislocation pile-ups, the length of dislocation pile-ups, and the Griffith critical crack propagation length can also be estimated based on the elastic modulus. To investigate how alloying elements such as Cr and Ni affect the hydrogen embrittlement susceptibility of steel, this study used the DFT method to calculate the elastic modulus of the bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems at varying hydrogen concentrations. The study further examined the impact of the solution H atom on the reduction of elastic modulus in the bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems, specifically along the (110) slip plane and the <111> slip direction. The main conclusions of the study are as follows:

• 1. The solution H atom significantly reduces the elastic modulus of the bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems. This reduction leads to a decrease in the critical stress for dislocation emission ($\tau_{Frank}$) and the resistance to dislocation motion ($\sigma_{p}$), which can promote dislocation emission and motion in these systems. Moreover, the solution H atom reduces the length of dislocation pile-ups ($L$) and lowers the theoretical fracture strength ($\sigma_{m}$) of the metals under critical fracture conditions. Additionally, it reduces the critical length for crack propagation ($a$) in the presence of microcracks, thus enabling crack nucleation and propagation at lower applied stress levels.

• 2. At the same hydrogen concentration, the elastic modulus of the bcc Fe–Cr system decreases significantly more than that of the bcc Fe and bcc Fe–Ni systems. This pronounced reduction in the elastic modulus makes dislocations more prone to emission and motion in the bcc Fe–Cr system. Consequently, at stress concentration points in dislocation pile-ups, cracks in the bcc Fe–Cr system are more likely to nucleate and propagate with lower external force and fewer dislocation pile-ups compared to the bcc Fe and bcc Fe–Ni systems

• 3. At the same hydrogen concentration, the presence of the H atoms significantly affects the bond population in the (110) slip plane and < 111 > slip direction of the bcc Fe, bcc Fe–Cr, and bcc Fe–Ni systems. Specifically, the bond population of Cr-Fe bonds in the bcc Fe–Cr system experiences the most substantial decreases, while the changes in the bond population of Ni–Fe bonds in the bcc Fe–Ni system and Fe–Fe bonds in the bcc Fe system are relatively minor. As a result, the strength of the Cr-Fe metallic bonds is greatly reduced, leading to a markedly larger reduction in the elastic modulus of the bcc Fe–Cr system compared to the bcc Fe and bcc Fe–Ni systems. Therefore, the alloying element Cr increases the hydrogen embrittlement susceptibility of bcc Fe, whereas the effect of the alloying element Ni on the hydrogen embrittlement susceptibility in bcc Fe is minimal.