Nanoindentation Study on the Local Evaluation of Hydrogen-Induced Hardening Performance of Ferrite and Austenite in 2205 Duplex Stainless Steel: Experiment and Finite Element Modeling

Abstract

In this study, a combined experimental and finite element modeling methodology (FEM) for a nanoindentation study is presented to quantitatively investigate the influence of hydrogen on the mechanical properties of ferrite and austenite in 2205 duplex stainless steel. The experimental results showed that, during hydrogen charging, the nano-hardness of ferrite and austenite gradually increased with time, showing a hydrogen-induced hardening phenomenon. After 3 h of hydrogen charging, the nano-hardness of both ferrite and austenite reached a saturation state, and the values of the nano-hardness of ferrite and austenite increased by 17.5% and 46.1%, respectively. FEM is employed by using a dual-phase microstructure-based model to reproduce nanoindentation load–displacement curves. To minimize the indentation size effect, an analytical correction model considering geometrically necessary dislocations (GNDs) was proposed. By considering GNDs, the errors between numerical predictions and experimental data reduced from about 50% to less than 5%.

1. Introduction

The presence of hydrogen in materials results in a reduction in ductility and the formation of cracks, known as hydrogen embrittlement (HE) [1,2,3] and hydrogen-induced cracking (HIC) [4,5,6,7]. HE indicates a loss of plasticity and strength reduction due to hydrogen, while HIC refers to the initiation and propagation of cracks caused by absorbed hydrogen around defects, as hydrogen reduces the surface interaction energy of metallic atoms. Thus far, two predominant mechanisms, hydrogen-enhanced decohesion (HEDE) and hydrogen-enhanced localized plasticity (HELP), have been used to explain the impacts of HE and HIC on metallic materials in most of the published research. HEDE assumes that regions containing defects (such as a notch or crack) exhibit high triaxial stresses and attract dissolved atomic hydrogen. The introduced hydrogen locally reduces the cohesive strength. When a critical hydrogen concentration is reached, small cracks form and propagate in regions of high hydrogen concentration. Further crack propagation requires a local increase in hydrogen concentration in front of the crack [8,9,10]. HELP proposes that solute hydrogen atoms can shield the elastic interaction between the obstacles and dislocations, leading to a reduction in the interaction energy of metallic atoms. This enhances dislocation mobility and increases the local plasticity of metallic materials [11,12,13].

Ferrite ($\mathsf{\alpha }$)–austenite ($\mathsf{\gamma }$) duplex stainless steels (DSSs) have been one of the key materials used in offshore and subsea components for decades. The phase volume fractions of ferrite and austenite are controlled to be nearly equal to achieve high strength and corrosion resistance. However, it has been found that the $\mathsf{\alpha }$-$\mathsf{\gamma }$ DSS is also sensitive to HE [14,15,16] and HIC [17,18,19]. This is due to the different crystal structures of ferrite (body-centered cubic, BCC) and austenite (face-centered cubic, FCC), which result in differences in hydrogen diffusivity and solubility. The hydrogen diffusivity in ferrite (~10⁻¹¹ m²/s) is about five orders higher than that in austenite (~10⁻¹⁶ m²/s), while the hydrogen solubility in ferrite is about two orders lower than in austenite [20,21]. As a result, ferrite acts as a highway for hydrogen diffusion, while austenite is a trap for storing hydrogen. From a comprehensive perspective, the focus of HE and HIC research has been drawn to characterizing the mechanical performance of ferrite and austenite, respectively, in a hydrogen-containing environment.

For DSS, the individual phase stress–strain properties cannot be directly obtained by using the macroscopic test method, as there is no steel with a similar chemical composition and the same grain size in the constituent phase. On this basis, nanoindentation (NI) has become an effective approach. The measurement displacement resolution of NI can reach the nano-scale, showing the advantages of the extraction of local elastoplastic properties while minimizing the influences of the sample substrate [22,23,24]. According to the Oliver–Pharr method [25,26], the nano-hardness and modulus are determined as the two basic characteristic parameters in the NI test. For homogeneous materials, the indentation hardness is observed to increase with decreasing displacement, a phenomenon known as the indentation size effect (ISE). This effect is attributed to the nucleation of geometrically necessary dislocations (GNDs) within the plastic zone during the NI test [27].

Combined with the NI test, finite element modelling (FEM) is usually performed as a complementary approach in NI studies [28,29,30,31]. Although elastoplastic properties obtained by uniaxial tensile tests can be employed to simulate and reproduce nanoindentation load–displacement curves on the macro-scale, the obtained macroscopic mechanical properties are not applicable to the micro-scale and nano-scale cases. Microstructure-based FEM (MB-FEM) is an increasingly relevant method for analyzing deformation behavior, stress–strain distribution and failure initiation in duplex or multiphase steel. Since the bulk behavior of steel is significantly dependent on the phase morphology, MB-FEM can help to further explain the stress–strain distribution in the local phase [32,33]. In order to couple the hydrogen-containing environment effects, Barnoush et al. [34,35] used a nanoindentation testing system in combination with an electrochemical setup to investigate the effect of hydrogen on the nano-mechanical properties of different materials. This methodology allows for the observation of the immediate effect of hydrogen on the local material properties and significantly reduces the hydrogen charging time, ensuring the quality of the specimen surface. According to the published research [36], it has been confirmed that ferrite and austenite exhibit different mechanical performances under the hydrogen effect. Although great efforts have been made to predict and prevent HE and HIC, differences in local phase properties are not yet clear.

In this study, an in situ nanoindentation hydrogen charging experiment was performed to quantitatively observe differences in the mechanical properties of ferrite and austenite 2205 DSSs under hydrogen interaction. A GNDs dislocation-based yield strength determination model is presented to consider the ISE effect. Validation of the proposed model was employed by using a microstructure-based FE model.

2. Methodology

2.1. Experiment

Nanoindentation tests were performed using the Bruker’s Hysitron TI-Premier Nano-indenter with a customized hydrogen-charging electrolytic cell at room temperature, as shown in Figure 1. A fluid cell probe was used here, which has an extended shaft with approximately 4 mm in length, allowing the end of the probe to be completely immersed in hydrogen charging fluid. It is important to note that the extended shaft has a diameter of about 700 μm, which can minimize the meniscus forces that occur when the probe penetrates the fluid.

The commercially available 2205 DSS was used and its chemical composition is shown in

Table 1. Chemical compositions of the 2205 duplex stainless steel (wt.%).

C	Cr	Ni	Mo	Mn	Si	Cu	Co	N	Fe
0.015	22.15	5.69	3.14	1.41	0.40	0.27	0.2	0.18	Bal.

. Circular specimens with a size of $\mathrm{\varnothing }20\times 2$ ${\mathrm{m}\mathrm{m}}^2$ were cut from the as-received 4 mm thick plate. The specimens were successively ground to 2000 grit and then mechanically polished with 3 $\mathsf{\mu }\mathrm{m}$ and 1 $\mathsf{\mu }\mathrm{m}$ water-based diamond suspensions. After that, the electro-polishing procedure was performed to remove the working hardening microscopic layer caused by mechanical grinding and polishing. The electrolyte solution was composed of (vol. %) 14% H₃PO₄, 18% H₂SO₄, 53% C₃H₅(OH)₃ and 15% H₂O [37]. The obtained microstructure graphs of the sample surface after the electro-polishing process observed by an optical microscope, scanning electron microscope and scanning probe microscope are shown in Figure 2. As presented, the phase structure can be identified after electro-polishing and the surface roughness of the sample is guaranteed.

Prepared smooth specimens were cathodically hydrogen charged in 0.5 mol/L H₂SO₄ solution, which is a frequently used and effective method for introducing hydrogen into steels [38,39]. To avoid the influence of sample surface quality changes on the obtained experimental data, the current density applied here is about one-tenth of the commonly used value, i.e., 1 mA/cm². The specimen was used as the cathode, and a platinum sheet was used as the anode. We added 1 g/L thiourea as a hydrogen recombination poison to increase the absorbed hydrogen in specimen.

Machine compliance and the contact area function of the indenter were previously calibrated before formal test by using a standard fused quartz sample. The method of automatic setting indentation was adopted. After reaching the expected hydrogen charging time, indentation tests were first performed on the ferrite and then performed on the austenite during hydrogen charging, as the hydrogen diffusivity in ferrite is much higher than that in austenite [20,21], and so the effect of the duration of charging on the hydrogen concentration in ferrite can be reduced. The load function contains 5 s loading, 2 s holding and 5 s unloading segments and the distance between adjacent indentations was set to be 7.5 μm to avoid mutual influence.

2.2. Finite Element Model

An axisymmetric microstructure-based two-dimensional (2D) model was created within the Abaqus software version 6.13-4, as shown in Figure 3. A representative optical micrograph of the microstructure (100 $\mathsf{\mu }\mathrm{m}\times$ 100 $\mathsf{\mu }\mathrm{m}$) obtained from the cross-section of 2205 DSS was converted to vector format and inserted into the model, to represent the ferrite-austenite dual-phase microstructure along depth direction during the indentation progress.

The geometry of the Berkovich indenter was simplified as a conical indenter with a half-cone angle of 70.3$°$ [40,41]. As the elastic modulus of the Berkovich indenter is 1140 GPa, it was employed as a rigid body in the 2D axisymmetric model. To save computational time and achieve sufficient computational accuracy, the mesh of a local region where contact exists between the indenter and specimen is refined to 10 nm, while a coarse mesh of 100 nm is performed away from the indentation region. The refined 10 nm mesh is 1/10 the size of the indenter. And the adaptive re-meshing procedure was also applied. Multiple sensitivity analyses were previously performed to make sure that the employed mesh was fine to achieve convergent and mesh-independent results. The FE model contains 458,977 linear triangular elements of type CAX3. The material parameters were determined by nanoindentation tests in our previous research [42,43]. This process involved three steps. First, we determined the optimal range of indentation load for nanoindentation measurements of the ferrite and austenite phases, as testing load cannot be arbitrary due to irregular phase structures. Then, the power–law stress–strain relationship was used to deduce the yield strength and work-hardening exponent of ferrite and austenite phases. Finally, the stress–strain curves obtained for the ferrite and austenite phases were individually input into the finite element model to reproduce the overall stress–strain curve, which was then verified with the experimental flow curve of 2205 DSS.

A static, general analysis step was carried out, and the displacement control was used to incrementally press the indenter into the specimen. The interaction between the indenter and the specimen was defined as a surface-to-surface interaction. For the tangential behavior, a frictionless condition was employed. The normal behavior (pressure–-overclosure) of the contact was defined as hard contact. A fixed boundary was applied to the bottom surface of the model, and the left surface along the symmetry axis was defined to only move in the y-direction. Displacement control was applied to a defined reference point of the indenter. Upon completion of the simulation, the load–displacement response was obtained by extracting the axial displacement and axial reaction force acting on the indenter.

3. Results and Discussion

3.1. Hydrogen Free

Figure 4 presents the obtained load–displacement curves of ferrite and austenite of the non-hydrogen charged 2205 DSS specimen. A large number of tests were conducted here to validate the testing stability of the designed hydrogen charging setup. According to the proposed framework by Oliver-Pharr [25,26], the values of nano-hardness and modulus are determined as follows:

$$H={\displaystyle \frac{P_{\mathrm{m}\mathrm{a}\mathrm{x}}}{A_c}}$$

(1)

$$E_r={\displaystyle \frac{\sqrt{\mathsf{\pi }}S}{2\sqrt{A_c}}}$$

(2)

$${\displaystyle \frac{1}{E_r}}={\displaystyle \frac{1-{\nu }^2}{E}}+{\displaystyle \frac{1-{\nu }_i^2}{E_{\mathrm{i}}}}$$

(3)

$$A_c=C_0{h_c}^2+C_1{h}_c+C_2{h_c}^{{\displaystyle \frac{1}{2}}}+C_3{h_c}^{{\displaystyle \frac{1}{4}}}+C_4{h_c}^{{\displaystyle \frac{1}{8}}}+C_5{h_c}^{{\displaystyle \frac{1}{16}}}$$

(4)

where $H$ is nano-hardness; $E_r$ is reduced modulus; $E$ is elastic modulus; $P_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is maximum force; $S$ is contact stiffness of the initial unloading curve defined by $\mathrm{d}P/\mathrm{d}h$, $P$ is load force and $h$ is displacement; $E_{\mathrm{i}}=1140 \mathrm{G}\mathrm{p}\mathrm{a}$ and ${\nu }_{\mathrm{i}}=0.07$ are the elastic modulus and Poisson’s ratio of the diamond Berkovich indenter, respectively; $A_c$ is contact area and $h_c$ is contact displacement.

The determined nano-hardness, reduced modulus, and elastic modulus of ferrite and austenite are listed in

Table 2. Mean values of the nano-hardness, reduced modulus and elastic modulus of ferrite and austenite before hydrogen charging.

Property	Ferrite	Austenite
Nano-hardness, $H$ (GPa)	3.94$\pm$ 0.19	4.06$\pm$ 0.23
Reduced modulus, $E_r$ (GPa)	182.29$\pm$ 5.69	177.52$\pm$ 10.86
Elastic modulus, $E$ (GPa)	197.27	191.17

. It is found that, although the elastic modulus of ferrite is slightly higher than austenite, the nano-hardness is lower than that of austenite. This mixed mechanical performance of 2205 DSS is consistent with previously published research [44]. These values are consistent with the results obtained by tests with specimens put on the original TI-Premier Nano-indenter testing platform [42].

3.2. Hydrogen Charged

Figure 5 shows the obtained load–displacement curves of ferrite and austenite with increasing hydrogen charging time. Our research found that the displacement of ferrite and austenite decreases with increasing hydrogen time, indicating hydrogen-induced hardening performance.

The values of the nano-hardness of ferrite and austenite varied with the time were plotted in Figure 6 and listed in

Table 3. Mean values of nano-hardness of ferrite and austenite varied with the hydrogen charging time.

Time, t (h)	Nano-hardness, H (GPa)
	Ferrite	$∆\mathit{H}$	Austenite	$∆\mathit{H}$
0	3.94$\pm$ 0.19	$-$	4.06$\pm$ 0.23	$-$
0.5	4.21$\pm$ 0.20	6.9%	4.72$\pm$ 0.17	16.3%
1	4.36$\pm$ 0.19	10.7%	5.09$\pm$ 0.26	25.4%
2	4.56$\pm$ 0.18	15.7%	5.89$\pm$ 0.15	45.1%
3	4.63$\pm$ 0.17	17.5%	5.93$\pm$ 0.13	46.1%
After 12 h hydrogen release	3.66$\pm$ 0.08	$-$7.1%	3.98$\pm$ 0.12	$-$2.0%

Note: $∆H$ is the percent of nano-hardness change compared with non-hydrogen charged value ($H_{t=0}$), i.e., $(H_t-H_{t=0})/H_{t=0}$, where is the nano-hardness of non-hydrogen charged specimen and $H_t$ is the nano-hardness with hydrogen charging time.

. As shown, the hydrogen-induced hardening performance in austenite is more obvious than in ferrite. After 3 h of charging, the values of nano-hardness tend to be stable. The nano-hardness of ferrite increased by 17.5%, while the nano-hardness of austenite increased by 46.1%. This is attributed to the higher hydrogen solubility in austenite [20,21]. The nano-hardness increase saturation phenomenon is probably attributed to the fact that the amount of absorbed hydrogen in the sample approaching a saturated level, and the desorption rate of the hydrogen reaches a local equilibrium with its absorption rate at the objective surface after a certain hydrogen charging time. After 12 h of ageing in an ambient environment (~20 °C), the nano-hardness of the ferrite and austenite returned to a value lower than the original state. This reduction in nano-harness is associated with the hydrogen-enhanced dislocations at low hydrogen contents [45,46], as the hydrogen is not completely desorbed under this condition. Another contributing factor is likely the presence of damaged surfaces with formed micro-cracks during the hydrogen-releasing process [44,47]. However, the crack formation could be reduced by a current density of 1 mA/cm² and 3 h of hydrogen charging in the experiment.

3.3. Dislocation-Based Yield Strength Determination Model

For metallic materials, hardening is commonly understood and described in terms of the accumulation of two types of dislocations: (i) statistically stored dislocations (SSDs) and (ii) geometrically necessary dislocations (GNDs), based on the strain gradient plasticity theory [48,49]. The SSDs are defined as dislocations generated by randomly trapping each other, while GNDs are dislocations that relieve the plastic deformation incompatibilities within the polycrystal caused by non-uniform dislocation slip [50]. For the NI test, the derived local stress-strain performance is termed as the representative stress and strain. During NI testing, the presence of more dislocations leads to an increase in the representative yield strength. According to the Taylor equation [51,52], the shear stress $\tau$ is defined as being related to the total dislocation density of SSDs and GNDs [53] as follows:

$$\tau =\alpha Gb\left(\sqrt{{\rho }_{\mathrm{S}\mathrm{S}\mathrm{D}}}+\sqrt{{\rho }_{\mathrm{G}\mathrm{N}\mathrm{D}}}\right)$$

(5)

where $\alpha$ is a material constant, called the Taylor coefficient; $G$ is the shear modulus; $G=E/2\left(1+\nu \right)$; $E$ is the elastic modulus; $\nu$ is Poisson’s ratio; $b$ is the Burgers vector; ${\rho }_{\mathrm{S}\mathrm{S}\mathrm{D}}$ is the statistically stored dislocation density; and ${\rho }_{\mathrm{G}\mathrm{N}\mathrm{D}}$ is the geometrically necessary dislocation density.

Nix and Gao [27] proposed that the number and density of GNDs (${\rho }_{\mathrm{G}\mathrm{N}\mathrm{D}}$) in the nanoindentation test with a conical indenter can be described as:

$${\rho }_{\mathrm{G}\mathrm{N}\mathrm{D}}={\displaystyle \frac{3}{2bh}}{\tan }^2\theta$$

(6)

where $b$ is the Burgers vector; $h$ is the indentation displacement; and $\theta$ is the angle of the cone made with the specimen free surface, for Berkovich indenter, $\tan \theta =$0.358.

The number and density of SSDs (${\rho }_{\mathrm{S}\mathrm{S}\mathrm{D}}$) are associated with the original hardness $H_0$ [50]:

$${\rho }_{\mathrm{S}\mathrm{S}\mathrm{D}}={\left({\displaystyle \frac{H_0}{3\sqrt{3}\alpha Gb}}\right)}^2$$

(7)

Combining Equations (5)–(7), as the value of ${\rho }_{\mathrm{S}\mathrm{S}\mathrm{D}}$ is independent of indentation depth, the $\alpha Gb\sqrt{{\rho }_{\mathrm{S}\mathrm{S}\mathrm{D}}}$ can be expressed as ${\tau }_0$; thus:

$$\tau ={\tau }_0+\alpha Gb\left(\sqrt{{\rho }_{\mathrm{G}\mathrm{N}\mathrm{D}}}\right)$$

(8)

By using the Taylor model of linking the macroscopic and microscopic scales, the yield strength is expressed as [52,53]:

$${\mathsf{\sigma }}_y=M\tau$$

(9)

where ${\mathsf{\sigma }}_y$ is the yield stress and $M$ is the Taylor factor. Then, we have:

$${\mathsf{\sigma }}_{\mathrm{y}}=M{\tau }_0+M\alpha Gb\sqrt{{\displaystyle \frac{3}{2bh}}{\tan }^2\theta }$$

(10)

where $M{\tau }_0$ can be considered as the initial yield strength, ${\sigma }_0$.

The related parameters of ferrite and austenite are listed in

Table 4. Related parameters of ferrite and austenite.

Parameter	BCC (Ferrite)	FCC (Austenite)
${\sigma }_{\mathrm{y}0}$ (MPa)	579.49	653.10
$M$	2.9	3.06
$\alpha$	0.23	0.23
$G$ (GPa)	79.59	76.86
$b$ (nm)	0.2482	0.2503

. The determined material properties are listed in

Table 5. Calculated material parameters of ferrite and austenite at 5000 $\mathsf{\mu }\mathrm{N}$ indentation load before and after being hydrogen charged.

Parameters	Ferrite (Uncharged)	Ferrite (Charged)	Austenite (Uncharged)	Austenite (Charged)
Depth, $h$ (nm)	222.28	203.37	221.14	191.11
Dislocation, ${\rho }_{\mathrm{G}}$ (${\mathrm{m}}^{-2}$)	3.48 × 1015	3.81 × 1015	3.47 × 1015	4.02 × 1015
Yield strength, ${\sigma }_{\mathrm{y}}$ (MPa)	1356.77	1392.79	1450.68	1511.57

.

Based on the power-law assumption, the stress–strain curves of ferrite and austenite are plotted, as presented in Figure 7.

$$\mathsf{\sigma }={\sigma }_{\mathrm{y}}{\left(1+{\displaystyle \frac{E}{{\sigma }_{\mathrm{y}}}}{\epsilon }_p\right)}^n$$

(11)

where ${\sigma }_{\mathrm{y}}$ is yield strength, $E$ is elastic modulus, ${\epsilon }_p$ is plastic strain and $n$ is the hardening exponent.

3.4. Finite Element Modelling Results

Figure 8 shows the numerically predicted results for ferrite and austenite both without and with considering the GND effect. It is found that without considering GND, the simulation results of ferrite and austenite display a significant difference ($~$50% error) when compared with experimental results. After considering GND, the error between modelling results and experimental tests was reduced to less than 5%.

Regarding the hydrogen charged condition, it is confirmed that the presence of hydrogen can increase the dislocation in the material. Figure 9 shows comparisons between modelling results and experimental results. It is found that, although the GND was considered here, the numerically predicted results show a significant difference compared with the experimental test. Based on that, the increased dislocation density caused by hydrogen also needs to be considered, thus the equation is defined as:

$${\mathsf{\sigma }}_{\mathrm{y}}=M{\tau }_0+M\alpha Gb\sqrt{{\rho }_{\mathrm{G}\mathrm{N}\mathrm{D}}+{\rho }_{\mathrm{H}}}$$

(12)

where ${\rho }_{\mathrm{H}}$ is the hydrogen-induced dislocations.

Figure 10 shows the results after adding the effect of hydrogen-induced dislocations. It is seen that the numerical predicted results are in accordance with the experimental results. The hydrogen-induced dislocation in ferrite (${\rho }_{\mathrm{H},\alpha }$) is determined to be equal to 4 times the GND of ferrite (${\rho }_{\mathrm{G}\mathrm{N}\mathrm{D},\alpha }$), and the hydrogen induced dislocation in austenite (${\rho }_{\mathrm{H},\mathsf{\gamma }}$) is determined to be equal to 5.3 times the GND of austenite (${\rho }_{\mathrm{G}\mathrm{N}\mathrm{D},\mathsf{\gamma }}$).

3.5. Discussion

This research focus on evaluating the hardening effects of hydrogen on ferrite and austenite in 2205 duplex stainless steel. In addition to conducting experiments, a complementary microstructure-based FE model was also developed to simulate the results with a specific emphasis on replicating the load–displacement curves. To predict the nanoindentation results more precisely, a yield strength determination analytical model considering dislocation was introduced. The results show that the nano-hardness of ferrite and austenite are not too different without hydrogen charging. However, after hydrogen charged, the nano-hardness of austenite increased more significantly when compared to ferrite. The following sections analyze and discuss the relevant factors leading to differences between ferrite and austenite during hydrogen charging, including three aspects: (i) hydrogen diffusion and distribution, (ii) dislocation nucleation and (iii) indentation loading rate.

(i) Hydrogen diffusion and distribution in ferrite and austenite

Due to the different crystal structure, the austenite phase has a higher solubility but a lower diffusivity than ferrite, leading to increased hydrogen content and a convoluted diffusion path. Additionally, residual stresses are present within DSSs due to the manufacturing processes, consisting of macroscopic and microscopic components. Macroscopic residual stresses result from temperature gradients, while microscopic residual stresses arise from misfit strains between grains with different thermal expansion coefficients. Although the macroscopic residual stress can be ignored due to the small difference in the elastic moduli between ferrite and austenite, the microscopic stresses vary between grains [54]. A quantitative equation involving the microscopic residual stress of a hydrogen partitioning coefficient in ferrite and austenite can be defined as [36]:

$$P_{\mathsf{\gamma }/\mathsf{\alpha }}^{\mathrm{H}}={\displaystyle \frac{S_{\mathsf{\gamma }}}{S_{\mathsf{\alpha }}}}\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{\left({\sigma }_{\mathsf{\alpha }}-{\sigma }_{\mathsf{\gamma }}\right){\overline{\mathrm{V}}}_{\mathrm{H}}}{RT}}\right)$$

(13)

where $S_{\mathsf{\alpha }}$ and $S_{\mathsf{\gamma }}$ are the hydrogen solubility in ferrite and austenite, ${\sigma }_{\mathsf{\alpha }}$ and ${\sigma }_{\mathsf{\gamma }}$ are the internal residual stress in ferrite and austenite, ${\overline{\mathrm{V}}}_{\mathrm{H}}$ is the partial molality volume of the hydrogen with a typical value of 2 ${\mathrm{c}\mathrm{m}}^3/\mathrm{m}\mathrm{o}\mathrm{l}$, $R$ is the gas constant and $T$ is the absolute temperature. According to Ref. [36], the calculated hydrogen partitioning coefficient $P_{\mathsf{\gamma }/\mathsf{\alpha }}^{\mathrm{H}}$ is 8330. This means that the hydrogen concentration is expected to be 8330 times higher compared to ferrite. Therefore, the austenitic phase is able to store more hydrogen than ferrite due to the intrinsic solubility difference and internal residual stresses.

(ii) Dislocation nucleation

The research on hydrogen influence on dislocation nucleation is a significant aspect of interpreting hydrogen-induced hardening performance in steels. It has been reported that when steel is tested under small indentation loads, the load–displacement curves show a series of pop-in phenomena. These pop-ins are largely due to the nucleation of dislocations. It has been discovered that hydrogen can decrease the load required for the first pop-in, which indicates the nucleation of dislocation. In an indentation test, the stress needed to nucleate a dislocation can be assumed to be the maximum shear stress beneath the indenter during purely elastic loading. The Hertzian elastic contact solution and maximum shear stress are defined as [55,56]:

$$P={\displaystyle \frac{4}{3}}{E}_r{R}^{{\displaystyle \frac{1}{2}}}{h}^{{\displaystyle \frac{3}{2}}}$$

(14)

$${\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}=0.31{\left({\displaystyle \frac{6{E_r}^2}{{\pi }^3{R}^2}}P\right)}^{{\displaystyle \frac{1}{3}}}$$

(15)

where $P$ is the applied load, $R$ is the radius of the indenter, $E_r$ is the effective indentation modulus, and $h$ is the indentation depth. Basa et al. [44] found that the initial load required for the first pop-in of ferrite and austenite decreased after hydrogen charging, meaning the maximum shear stress decreased as well. The variation range of shear stress in the austenitic phase is higher than that in ferrite phase, which indicates that hydrogen induces more dislocation accumulation in austenitic phase, making hydrogen hardening more prominent.

(iii) Indentation loading rate

The indentation test has been used as a useful approach to investigate the effects of hydrogen on dislocation activity, as indicated in the Ref. [57]. Our research found that hydrogen caused a 30% increase in hardness because hydrogen increased slip planarity. These results reveal that hydrogen enhances dislocation nucleation and induces slip planarity during loading. According to the Orowan’s equation [58]:

$$\dot{\epsilon }={\rho }_m\mathrm{b}v$$

(16)

where $\dot{\epsilon }$ is the strain rate, ${\rho }_m$ is the mobile dislocation, $\mathrm{b}$ is the magnitude of Burgers vector and $v$ is the dislocation velocity.

Equation (16) indicates that strain rate affects the density and velocity of mobile dislocation, as well as the accumulation of hydrogen in local sites. Previous studies have indicated that the susceptibility to hydrogen embrittlement increases with decreasing the strain rate at any given temperature [59]. The interaction between hydrogen and dislocation depends on the strain rate. Under a small strain rate, hydrogen can be easily transported by mobile dislocation due to the low dislocation mobility. The increase in hydrogen transport results in increased dislocation activity in some local sites because of hydrogen-enhanced plasticity. This, in turn, enhances the effects of hydrogen transport [60]. Therefore, the strain rate has an influence on the interaction between hydrogen and dislocation, as well as on the mobile dislocation velocity ($v$). The interaction between hydrogen and dislocation depends on strain rate. Hydrogen can only be transported when the dislocation velocity is below the smallest hydrogen diffusion velocity in the hydrogen environment $v_c$ (known as critical hydrogen diffusion velocity), only then can hydrogen be transported by mobile dislocation. Otherwise, hydrogen cannot keep up with the mobile dislocation.

According to dislocation theory, a critical dislocation velocity $v_{\mathrm{c}}$ is defined as [61,62]:

$$v_{\mathrm{c}}={\displaystyle \frac{D}{kT}}·{\displaystyle \frac{E}{30b}}$$

(17)

Combing Equations (16) and (17), then

$$\dot{\epsilon }={\rho }_m{\displaystyle \frac{DE}{30kT}}$$

(18)

where $D$ is the hydrogen diffusivity; $E$ is the dislocation trap biding energy; $k$ is the Boltzmann constant; $T$ is temperature. According to the parameters reported in the literature, the values of $\dot{\epsilon }$ for ferrite and austenite are determined as being 0.034 ${\mathrm{s}}^{-1}$ and 140 ${\mathrm{s}}^{-1}$, respectively. According to Mayo and Nix [63], the definition of the indentation strain rate ($\dot{\epsilon }$) can be deduced from the concept of true strain, which can be estimated as [64]:

$$\dot{\epsilon }={\displaystyle \frac{\dot{h}}{h}}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\dot{P}}{P}}-{\displaystyle \frac{\dot{H}}{H}}\right)\approx {\displaystyle \frac{1}{2}}{\displaystyle \frac{\dot{P}}{P}}$$

(19)

where $h$ is the displacement, $\dot{h}$ is the displacement rate, $P$ is the load, $\dot{P}$ is the loading rate, $H$ is the hardness, $\dot{H}$ is the hardness rate. Based on our experimental conditions, the strain rate was calculated to be 0.1 ${\mathrm{s}}^{-1}$. This indicates that the indentation testing rate of ferrite may also affect its low hydrogen-induced hardening performance when compared to austenite.

4. Conclusions

The influence of hydrogen on the mechanical properties of ferrite and austenite in a 2205 duplex stainless steel has been investigated in this paper. The following key conclusions are drawn:

(1)

In the in situ hydrogen charging tests, the nano-hardness values of ferrite and austenite gradually increased with time, showing the performance of hydrogen-induced hardening. The nano-hardness increases of austenite increased more than that of ferrite.

(2)

After 3 h of charging, the hardness of ferrite and austenite reached a saturated state, resulting in the respectively increased the nano-hardness of ferrite and austenite by 17.5% and 46.1%, when compared with the non-charged results.

(3)

Using the determined material parameters while considering the geometrically necessary dislocations effect in the microstructure-based finite element model, the numerically predicted results are much closer to the experimental data, reducing the errors from about 50% to less than 5%. Based on the modeling analyses in this study, the prediction results are in agreement with our experiment, showing that this is a suitable predictive methodology for reproducing nanoindentation curves.