# Study of hydrogen embrittlement in steels using modified pressurized disks

L.M. Santana, D. Lopes Pinto, N. Osipov, J. Furtado, F. Bourguignon, P.-J.

Marchais, Yazid Madi, J. Besson

► To cite this version:

L.M. Santana, D. Lopes Pinto, N. Osipov, J. Furtado, F. Bourguignon, et al., Study of hydrogen embrittlement in steels using modified pressurized disks. International Journal of Hydrogen Energy, 2024, 88, pp.498-514. 10.1016/j.ijhydene.2024.09.165. hal-05010087

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# Study of hydrogen embrittlement in steels using modified pressurized disks

L.M. Santana<sup>a,*</sup>, D. Lopes Pinto<sup>a,b</sup>, N. Osipov<sup>b</sup>, J. Furtado<sup>c</sup>, F. Bourguignon<sup>d</sup>, P.-J. Marchais<sup>d</sup>, Y. Madi<sup>a</sup>, J. Besson<sup>a</sup>

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

**Keywords:**

Gaseous hydrogen embrittlement

Disk pressure test

Diffusion

Trapping

Mixed finite elements

## ABSTRACT

The integration of hydrogen into natural gas pipelines presents challenges due to Hydrogen Embrittlement (HE), requiring critical material selection and testing methods. One such test is the Disk Pressure Test (DPT), which consists in pressurizing a clamped disk to failure. However, failure happens often at the clamping zone, making analysis difficult. This study aims to develop new disk geometries to control failure location while maintaining the test setup. Using two steel grades (a vintage X52 pipeline steel and a modern E355 modified steel with potential for pipeline use), new disk geometries were tested under helium and hydrogen at various pressure rise rates. Results show successful displacement of failure away from the clamping zones, demonstrating the effectiveness of the new geometries. Hydrogen embrittlement is demonstrated by comparing failure pressures under helium and hydrogen at various pressure rise rates. Using both material testing and simulation, this study provides insights into hydrogen embrittlement.

## 1. Introduction

The raising consciousness of the greenhouse effect and its resultant global warming has encouraged a pressing need for a transition in the energy system towards sustainable sources. A promising solution is the use of green hydrogen, a clean and renewable energy vector that can be produced from water using renewable energy sources such as solar and wind [1,2] or nuclear power plants [3]. In recent times, there has been a growing interest in the potential of green hydrogen to facilitate the decarbonization of the energy sector and achieve climate goals. An advantageous approach to introduce hydrogen into the energy matrix is through its transport via existing natural gas pipelines, which offers a feasible way for hydrogen integration [4–6].

The use of hydrogen as an energy carrier in pipeline systems presents significant challenges. Key issues include gas compression, leakage, and material compatibility [7,8]. Hydrogen requires high pressures for storage and transportation, necessitating materials that can endure harsh conditions. As a highly flammable gas, hydrogen poses risks related to potential leakage from pipelines [9,10]. In hydrogen pipeline systems, seals, gaskets, and joints must be designed to withstand the high pressures of hydrogen and prevent leakage over time [11]. The use of specialized seals and materials resistant to hydrogen embrittlement is crucial. Additionally, high-pressure pipelines transporting hydrogen are vulnerable to vibrations caused by factors

such as fluid flow and operational conditions [12]. These vibrations can induce fatigue in the pipeline materials, potentially compromising the structural integrity of the pipeline over time [13,14].

The conversion of existing natural gas pipelines to hydrogen pipelines represents a significant undertaking with the dual objective of decarbonizing energy infrastructure and developing the use of hydrogen as a clean energy carrier [15,16]. This represents a solution for the transportation of hydrogen, whereby existing systems for the transportation of natural gas can be utilized, thereby avoiding the costs associated with the construction of new infrastructure [8]. The primary challenge associated with the introduction of hydrogen into existing pipelines is the significant difference in specific volume between hydrogen and natural gas. As a consequence of this difference, the transportation of hydrogen requires higher pressures than natural gas, which in turn gives rise to challenges in fracture mechanics and embrittlement [7,8].

Hydrogen embrittlement represents a significant challenge in the safe transportation of hydrogen [17]. The interaction between hydrogen and pipeline materials can lead to a degradation of the materials themselves. While H₂ molecules are unable to diffuse through metal due to their considerable size, hydrogen atoms are sufficiently small to enter the metal lattice. This allows them to accumulate at high-stress

<footnote><sup>a</sup> Centre des Matériaux, MINES Paris, Université PSL, CNRS UMR, 7633, France</footnote>

<footnote><sup>b</sup> Transvalor S.A., France</footnote>

<footnote><sup>c</sup> Air Liquide, Innovation Campus Paris, France</footnote>

<footnote><sup>d</sup> Mannesmann Precision Tubes France SAS, France</footnote>

<footnote>* Corresponding author.
E-mail address: luciano.meirelles_santana@minesparis.psl.eu (L.M. Santana).</footnote>
locations, which in turn induces a decrease in ductility and toughness and the promotion of cracking [18,19]. This phenomenon, known as Hydrogen Embrittlement (HE), is a complex mechanism which combines mechanical and chemical loadings. Several mechanisms are accepted to explain the HE phenomena, such as Hydrogen Enhanced Localized Plasticity (HELP) [20-22], Hydrogen Enhanced Decohesion (HEDE) [23,24], Adsorption-Induced Dislocation-Emission (AIDE) [25] and Hydrogen Enhanced Strain-Induced Vacancy (HESIV) [26].

The HELP mechanism suggests that hydrogen increases the mobility of dislocations near crack tips, which in turn locally enhances plasticity. This increase in localized plasticity makes the material more susceptible to global brittle fracture. In contrast, the HEDE mechanism is related to the reduction of cohesive energy at grain boundaries and interfaces, making them more prone to decohesion and facilitating crack propagation. The AIDE mechanism states that the adsorption of hydrogen weakens the interatomic bonds, facilitating the emission of dislocations from crack tips. Conversely, the HESIV mechanism is characterized by the propensity of hydrogen atoms to promote the creation of lattice vacancies within a metal, leading to decreased strength and higher susceptibility to fracture.

Additionally, several researchers suggest that hydrogen embrittlement (HE) may result from the interaction of multiple mechanisms. Novak et al. [27] argue that hydrogen plays a significant role by being trapped at dislocations. This trapped hydrogen decreases the stress hindering dislocation movement and also reduces the energy needed for decohesion at dislocation pile-ups at the carbide/matrix interface. The proposed model indicates that the combined effects of the HELP and HEDE mechanisms account for the observed behavior. Martin et al. [28] investigated quasi-cleavage surfaces using scanning electron microscopy (SEM), atomic force microscopy (AFM), and transmission electron microscopy (TEM). Their observations revealed that these surfaces are not flat but instead exhibit large-scale undulations. TEM analysis identified a high density of dislocation lines and loops just beneath the surface. They found that hydrogen contributes to the formation of this dislocation substructure and increases the local hydrogen concentration, thereby enhancing the susceptibility to cracking in these areas. This evidence supports the conclusion that the HELP mechanism, rather than HEDE, is responsible for the observed flat quasi-cleavage fractures.

Testing materials for hydrogen compatibility and monitoring their degradation during the operational life of hydrogen-exposed pipelines have underscored the need for developing experimental methods to assess the mechanical properties of these materials [29]. Technological necessity led to the creation of tests to measure HE levels in different materials, as hydrogen often needs to be used with metallic components. One of these tests is established by the ISO/FDIS 11114-4 standard (method A): the Disk Pressure Test (DPT). The DPT consists of pressurizing a clamped disk until it ruptures. The test is performed under both hydrogen and helium, and a Hydrogen Embrittlement Index (HEI) is calculated as the ratio between the rupture pressure under helium ($P_r(He)$) and that under hydrogen ($P_r(H_2)$), according to the following equation:

$$HEI = \frac{P_r(He)}{P_r(H_2)} \qquad (1)$$

As it can depend on the pressurization rate, the test is conducted under several rates, and the worst ratio is adopted. The first authors to implement the disk pressure tests highlighted as advantages its sensitivity, rapidity, simplicity and economy [30]. Over the years, the DPT has been successfully utilized as an industrial qualification test for metallic alloys in hydrogen storage [31-35] always using the standard test geometry.

Despite its usefulness for materials ranking, the DPT provides limited insights into HE research. A notable limitation of the standard specimen is its tendency to fail in the clamping area. This issue arises because the highest maximum principal stresses in the disk are located

near the flange, as identified by Charles et al. [36]. Indeed, the DPT presents a significant challenge for numerical simulation due to, among others reasons, the large plastic strain fields that develop in the fracture zone [37]. The complex stress field in this zone, combined with the inherent experimental variations, can lead to a significant scatter in the results, as reported by Charles et al. [37] and Briottet et al. [34]. This complicated stress field is considered detrimental to a deeper understanding of the HE phenomena itself, as it adds complexity to the identification of a rupture criterion. This tendency of the disk to fail in the clamping area is comparable to a tensile test in which the specimen fails either in the clamping area or at the fillets. In a tensile test, such failures invalidate the results because they occur outside the gauge length. It can be reasonably assumed that this issue is also relevant for the disk test. Furthermore, failures in the standard disk specimen can occasionally initiate at the center, making the precise location of failure unpredictable. This variability complicates the comparison of results due to the differing stress states: the clamping zone experiences plane strain, while the center of the specimen undergoes plane stress, as shown by Arnould-Laurent and Fidelle [38].

Precise and reliable predictive modeling of HE at a continuum scale is crucial for advancing hydrogen infrastructure development [39] as HE presents a significant challenge in various engineering applications, such as high-pressure tanks, pipelines, and automotive steels, particularly in the context of hydrogen storage, transport, and utilization. However, simulating HE is inherently complex due to the intricate coupling between mechanics and diffusion. The plastic behavior of materials profoundly influences hydrogen diffusion, as hydrogen atoms tend to migrate towards regions of high hydrostatic stress. Conversely, the diffusion of hydrogen atoms can significantly impact the mechanical properties of the material. Thus, both mechanisms are intricately interdependent, and the simulation model must comprehensively account for both aspects to achieve accurate results. Using this strategy, several authors have made efforts to simulate the DPT [36-38,40-42].

The simplicity and relatively low cost of the DPT apparatus, in comparison to tensile and toughness tests conducted under gaseous hydrogen, indicate the potential for further exploration of this testing methodology. Introducing modified geometries for the specimen may help achieve distinct stress and strain states, which can be utilized to study the influence of hydrogen on fracture behavior. In this study, new designs for the Disk Pressure Test are proposed, adhering to the constraint that the standard experimental setup must remain unchanged. The primary objective is to prevent failure in the clamping area. Additionally, a computational framework is developed to simulate the effects of plasticity and its interaction with hydrogen in analyzing these tests.

The organization of this paper is as follows: Section 2 provides an overview of the two steels utilized in the research, along with their microstructural characterization. Section 3 details the mechanical characterization process and the determination of the constitutive material laws necessary for numerical simulations. Section 4 discusses the experimental results obtained from disk tests and highlights observed effects related to geometry, hydrogen presence, and pressurization rates. It also includes a fractographic analysis of the specimens to assess the extent of hydrogen embrittlement. Section 5 presents the finite element approach used in the simulation model, which describes mechanical behavior and hydrogen transport. The model is used to interpret the experimental results.

## 2. Materials

Two distinct steels were investigated: a X52 normalized steel grade (referred as X52 steel), extracted from a vintage pipeline with a thickness of 7.92 mm, and a modern E355 modified steel grade (referred as E355 mod. steel), received in the form of a hot rolled tube manufactured by Thermo-Mechanical Controlled Process (TMCP) with a
<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Microstructure of (a) the E355 mod. steel, consisting of ferrite and pearlite with an ASTM grain size number of 9.0, and (b) the X52 steel, consisting of ferrite and pearlite with an ASTM grain size number of 9.5. Inclusions in the E355 mod. steel are shown in (c), featuring globular inclusions containing Ca and S with an average diameter of 0.25 µm. Inclusions in the X52 steel are depicted in (d), featuring elongated MnS inclusions with an average length of 5.36 µm.</figcaption></figure>

thickness of 12.7 mm. Both materials are seamless, and their chemical compositions are detailed in Table 1.

The microstructures of both steels were examined using optical microscopy after grinding, polishing, and etching with Nital 3%. Fig. 1 illustrates the microstructure of both steels in three planes (T-S, L-S and T-L, being L the longitudinal direction, T the transverse direction and S the tube's thickness direction).

From Figs. 1(a) (E355 mod. steel) and 1(b) (X52 vintage steel), it is possible to observe that both steels exhibit a ferritic matrix with pearlite bands elongated in both the longitudinal and transverse direction. The average grain size for the two steel samples was determined using the intercept method according to the ASTM E112 standard. The grain size analysis indicates that E355 mod. has a grain size number of 9.0, corresponding to an average grain diameter of 15.9 µm, while X52 steel has a grain size number of 9.5, corresponding to an average grain diameter of 13.3 µm.

Scanning Electron Microscopy (SEM) observations and Energy Dispersive Spectroscopy (EDS) analyses were performed on the unetched specimens to assess their inclusion content, morphology, and composition. Figs. 1(c) and 1(d) show the T-S cross sections of E355 mod. and X52 steel grades, respectively. The vintage X52 steel presents inclusions elongated in the T-direction, as seen in Fig. 1(d). These inclusions were also found to be elongated in the L-direction. EDS analysis identifies them as MnS, with an average length of 5.36 µm in the T-direction

<table><caption>**Table 1**

Chemical compositions of the X52 and E355 mod. steels.</caption><thead><tr><th>Wt.%</th><th>C</th><th>Mn</th><th>Si</th><th>Cr</th><th>Ni</th><th>Mo</th><th>S</th><th>P</th><th>Fe</th></tr></thead><tbody><tr><td>X52</td><td>0.16</td><td>0.64</td><td>0.08</td><td>&lt;0.03</td><td>0.05</td><td>&lt;0.03</td><td>0.024</td><td>0.015</td><td>Bal.</td></tr><tr><td>E355 mod.</td><td>0.15</td><td>1.38</td><td>0.19</td><td>0.04</td><td>0.02</td><td>0.01</td><td>0.002</td><td>0.015</td><td>Bal.</td></tr></tbody></table>

and 33.2 µm in the L-direction. The samples also revealed the presence of globular oxides and clusters of oxides/MnS. The volume fraction of inclusions in the X52 steel is 0.27%.

In contrast, the E355 mod. steel grade does not display elongated inclusions, as shown in Fig. 1(c). Instead, few globular inclusions are observed, with an average diameter of 0.25 µm. These inclusions are identified as CaS through EDS analysis. Additionally, Al and Mg oxides were also found in the sample. The volume fraction of inclusions in the E355 mod. steel is 0.06%.

### 3. Mechanical characterization

This section focuses on the mechanical characterization of two distinct steel grades, analyzing their mechanical properties. It includes the methodology, presentation of results, and the fitting of an elastoplastic constitutive model.
<figure><img src="image_5.png" /><figcaption>(a) Geometry of the tensile specimen, indicating the extraction orthogonal planes with retro-lighting to capture the contour of the specimen during the test.
(b) Tensile test assembly: two cameras are positioned on the direction and marking.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 2. Tensile specimen geometry and test assembly.</figcaption></figure>

## 3.1. Methodology

Tensile tests were conducted using smooth axisymmetric specimens with a diameter ($\Phi_0$) of 4 mm and gauge length ($L_0$) of 20 mm. These specimens were extracted from the L-direction of the tubes. The specimen's geometry is illustrated in Fig. 2(a). The S-direction is marked in the head of the specimen, as shown in the figure.

The tests were conducted using an MTS testing machine with a maximum capacity of 100 kN. The test setup is presented in Fig. 2(b). These tests were conducted under several strain rates ($5 \times 10^{-4}$, $1 \times 10^{-5}$ and $1 \times 10^{-6}$ s$^{-1}$), controlled by a precise displacement sensor (0.004 µm of standard deviation).

During the tests, two digital cameras record images on orthogonal planes against a retro-lit white background created by two LED panels, as illustrated in Fig. 2(b). The captured images are post-processed to detect the samples' contours and estimate the minimal cross-section. This allows the calculation of the radial strain ($\Delta\Phi/\Phi_0$) along the test. One camera captures the radial strain in the S-direction ($\Delta\Phi_S/\Phi_0$), while the other records the radial strain perpendicular to this direction ($\Delta\Phi_{\perp}/\Phi_0$), as depicted in Fig. 2(b). This technique is employed to characterize the anisotropic elastoplastic behavior of both steels. For more detailed information on this technique, please refer to these previous publications [29,43].

## 3.2. Results

Fig. 3 displays the $F/S_0 - \Delta\Phi/\Phi_0$ curve for tensile specimens extracted from X52 and E355 mod. steels in the L-direction (radial contraction $\Delta\Phi$ is represented as positive). F represents the force and $S_0$ the initial cross section area. The solid line represents the radial strain in the S-direction ($\Delta\Phi_S/\Phi_0$), while the dotted line represents the radial strain in the perpendicular direction ($\Delta\Phi_{\perp}/\Phi_0$). Table 2 provides a summary of the results, including Yield Strength (YS), Ultimate Tensile Strength (UTS), $\Delta\Phi_S/\Phi_0$ at rupture ($Z_s$), $\Delta\Phi_{\perp}/\Phi_0$ at rupture ($Z_{\perp}$) and ($\Delta\Phi_{\text{eq}}/\Phi_0$) at rupture ($Z_{\text{eq}}$) for a strain rate of $5 \times 10^{-4}$ s$^{-1}$. Here, $\Delta\Phi_{\text{eq}}$ represents the equivalent radial deformation, calculated using the following equation:

$$ \Delta\Phi_{\text{eq}} = \Phi_0 - \sqrt{(\Phi_0 - \Delta\Phi_{\perp})(\Phi_0 - \Delta\Phi_S)} \quad (2) $$

$\Delta\Phi_{\text{eq}}$ is such that the minimum cross-section is preserved, assuming that the actual one is an ellipse. Examining the results, it is noticeable that E355 mod. steel has a lower YS of 335 MPa and UTS of 498 MPa compared to the X52 steel with values of 384 MPa and 544 MPa for YS and UTS, respectively. However, E355 mod. steel exhibits a higher $Z_{\text{eq}}$, measuring 55%, as opposed to X52 steel with 41%. The modern E355 mod. steel is, therefore, highly ductile, while the vintage X52 steel

<table><caption>Table 2
Summary of mechanical characterization of X52 and E355 mod. steel under a strain rate of $5 \times 10^{-4}$ s$^{-1}$.</caption><thead><tr><th></th><th>YS (MPa)</th><th>UTS (MPa)</th><th>Z<sub>s</sub> (%)</th><th>Z<sub>⊥</sub> (%)</th><th>Z<sub>eq</sub> (%)</th></tr></thead><tbody><tr><td>X52</td><td>384</td><td>544</td><td>0.46</td><td>0.37</td><td>0.41</td></tr><tr><td>E355 mod.</td><td>335</td><td>498</td><td>0.55</td><td>0.56</td><td>0.55</td></tr></tbody></table>

exhibits lower ductility, which can be attributed to a higher inclusion content. Moreover, it is observed that E355 mod. steel displays similar radial strain at rupture in both the S and $\perp$ directions and can be considered isotropic. In contrast, the X52 steel is anisotropic and shows lower radial strain in the $\perp$ direction compared to the S direction, however, for the sake of simplicity, an isotropic constitutive law was considered for this material.

## 3.3. Elastoplastic behavior

The coefficients of the constitutive laws for the X52 and E355 mod. steels were determined using optimization algorithms implemented in Z-set,<fnref n="1" /> a finite element software developed at Mines Paris and Onera, the French aerospace lab. The materials are modeled using the von Mises plasticity framework with isotropic hardening. Thus, the yield surface is expressed as:

$$ \Phi = \sigma_{\text{eq}} - \sigma_{F}(p) \quad (3) $$

where $\sigma_{\text{eq}}$ is the von Mises equivalent stress and the flow stress, $\sigma_F(p)$, is expressed as a function of the accumulated plastic strain, $p$:

$$ \sigma_F(p) = \max(\sigma_L, \sigma_0 + Q_1(1 - \exp(-b_1 p)) + Q_2(1 - \exp(-b_2 p)) + H p) \quad (4) $$

where $\sigma_L$ represents the Lüders stress, fixed at 395 MPa for the X52 steel and at 325 MPa for the E355 mod. steel. These values correspond to the region of nearly constant stress observed before work hardening begins, as illustrated in Fig. 3. The yield stress in absence of plateau is denoted by $\sigma_0$, while the parameters $Q_1, b_1, Q_2, b_2$ and $H$ describe the evolution of plastic hardening. From the experimental observations, the materials were considered to be slightly rate-dependent. In that case, the plastic strain rate is given by:

$$ \dot{p} = \dot{p}_0 \left\langle \frac{\sigma_{\text{eq}} - \sigma_F(p)}{K} \right\rangle^n \quad (5) $$

where $\dot{p}_0$, $K$ and $n$ are coefficients describing the viscoplastic properties of the material. These coefficients were originally calibrated for

<footnote n="1">¹ http://www.zset-software.com/.</footnote>
<figure><img src="image_7.png" /><figcaption>Fig. 3. Tensile tests performed for both steels using specimens extracted along the longitudinal direction. The curves display the nominal stress ($F/S_0$) against the radial deformation in the thickness direction ($\Phi_\perp$) and in the perpendicular direction ($\Phi_\perp$) for the experimental data and the numerical results. Tests conducted under strain rates of $1 \times 10^{-6}$ and $5 \times 10^{-4}$ s$^{-1}$. The curves of the strain rate of $5 \times 10^{-4}$ s$^{-1}$ where shifted by 100 MPa for clarity.</figcaption></figure>

<table><caption>**Table 3**
Coefficients of the elasto-visco-plastic constitutive law for the X52 vintage and E355 mod. steels.</caption><thead><tr><th></th><th>σ<sub>0</sub><br>(MPa)</th><th>Q<sub>1</sub><br>(MPa)</th><th>b<sub>1</sub><br>(-)</th><th>Q<sub>2</sub><br>(MPa)</th><th>b<sub>2</sub><br>(-)</th><th>H<br>(MPa)</th><th>ρ<sub>0</sub><br>(s<sup>-1</sup>)</th><th>K<br>(MPa)</th><th>n<br>(-)</th></tr></thead><tbody><tr><td>X52</td><td>269</td><td>191</td><td>41.45</td><td>358</td><td>3.15</td><td>10.23</td><td rowspan="2">1</td><td rowspan="2">213</td><td rowspan="2">4.43</td></tr><tr><td>E355 mod.</td><td>273</td><td>214</td><td>19.29</td><td>141</td><td>2.75</td><td>148</td></tr></tbody></table>

E355 mod. steel. However, due to the significant variability in the experimental data for the X52 steel, and for the sake of simplicity, the same coefficients were used for both steels.

The various model parameters were adjusted by minimizing the difference between the experimental and numerical $F/S_0 - \Delta\Phi/\Phi_0$ curves. Using this optimization strategy, the hardening law was successfully fitted beyond the point of necking and up to just before crack initiation. However, to accurately capture the development of necking, a finite element simulation of the tensile specimen with a finite strain formulation is necessary. Details of this procedure are provided in [43]. The simulations employed axisymmetric elements. In the case of the anisotropic X52 steel, $\Delta\Phi_{eq}/\Phi_0$ was used. The coefficients determined for both steels are listed in Table 3. These values were then applied in simulations across several strain rates ($5 \times 10^{-4}$, $1 \times 10^{-5}$ and $1 \times 10^{-6}$ s$^{-1}$). Fig. 3 presents a comparison between the numerical and experimental results. For clarity, the curve corresponding to the strain rate of $1 \times 10^{-5}$ s$^{-1}$ was omitted from the graph, as its results were nearly identical to those at $1 \times 10^{-6}$ s$^{-1}$. For both steels, Poisson's ratio and Young's modulus were set at 0.3 and 200 GPa, respectively.

Table 4 compares the experimental and simulated Ultimate Tensile Stress (UTS) values for the X52 and E355 mod. steels for several strain rates. As previously noted, both steels exhibit a slight rate dependence, resulting in higher stress values at higher strain rates. The simulations are in good agreement with the experimental results for all cases.

# 4. Tests on pressurized disks

## 4.1. Methodology

Standard ISO 11114 [44] specifies the Disk Pressure Test (DPT) method requirements. According to this standard, the test is performed using a 58 mm diameter disk with 0.75 mm thickness, as depicted in Fig. 4(a). However, this specimen geometry often fails in the clamping area, posing potential issues when analyzing the test. To overcome this

issue, two new geometries were proposed in this study: a cup and a notched disk.

The modified specimens were previously optimized with regard to a number of features, including specimen thickness, cup radius and position, notch radius and position. This was achieved through the use of finite element modeling, with the objective of attaining an optimal result in displacing the rupture from the clamping zone to the region of interest, which was identified as the center of the specimen for cup disks and the bottom of the notch for notched disks.

The new geometries maintain the same diameter as required by ISO 11114 to enable testing using the same device. The cup specimen is a 3 mm thick disk in which a truncated spherical cup is machined in the center, obtaining a minimum thickness of 0.75 mm in the center, as shown in Fig. 4(b). The notched specimen is a 2 mm thick disk with an axisymmetric machined notch having a radius of 1.25 mm, resulting in a minimum thickness of 0.75 mm at the root of the notch (Fig. 4(c)). Consequently, the minimum thickness of both novel geometries is identical to that of the standard disk.

After machining, the specimens were measured using a white light non-contact surface profilometer (AltiSurf 500). Three-dimensional maps of the entire specimens were created with a resolution of 1 point every 50 µm at a scanning speed of 40 mm/s. It was ensured that all desired dimensions were attained for each sample before testing. Special attention is given to the roughness, as it may impact the severity of hydrogen embrittlement. The standard requires a maximum roughness Ra of 1 µm. Roughness was analyzed in the region of interest: the bottom of the notch for notched specimens, the center of the cup for cup disks, and the entire specimen for the standard disk.

Furthermore, the ISO FDIS 11114 standard requires specimens to have a flatness of less than 1 mm over a 10 mm deflection. Achieving this requirement can be challenging when machining standard disks with a uniform thickness of 0.75 mm, as these disks are more prone to bending during machining due to residual stresses, which complicates accurate testing. In contrast, cup and notched specimens, despite their more complex geometries, do not face these issues due to their increased maximum thickness, which increases the bending stiffness.

Once the correct dimensions were confirmed, the specimens were tested in a dedicated device, as depicted in Fig. 5. Prior to testing, each specimen was cleaned using alcohol and acetone in an ultrasonic bath. The upper flange was carefully placed to avoid displacing the specimen, and the bolts were cross-tightened to ensure an evenly distributed clamping load. The applied torque on the bolts was set at 30 N m for standard specimens, in accordance with the guidelines of the
<table><caption>**Table 4**
Experimental and simulated ultimate tensile stresses (in MPa) for the X52 vintage and E355 mod. steels for three different strain rates.</caption><thead><tr><th rowspan="2"></th><th colspan="2">5 × 10<sup>-4</sup> (s<sup>-1</sup>)</th><th colspan="2">1 × 10<sup>-5</sup> (s<sup>-1</sup>)</th><th colspan="2">1 × 10<sup>-6</sup> (s<sup>-1</sup>)</th></tr><tr><th>Experiment</th><th>Simulation</th><th>Experiment</th><th>Simulation</th><th>Experiment</th><th>Simulation</th></tr></thead><tbody><tr><td>X52</td><td>544</td><td>546</td><td>527</td><td>528</td><td>530</td><td>523</td></tr><tr><td>E355 mod.</td><td>498</td><td>500</td><td>482</td><td>481</td><td>475</td><td>476</td></tr></tbody></table>

<figure><img src="image_8.png" /><figcaption>**Fig. 4.** Disk specimens geometries (in mm). All of the specimens have a nominal minimum thickness of 0.75 mm. Pressure is applied to the underside of the specimens.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Fig. 5.** Schematic view of the disk pressure test. Source: Adapted from [44].</figcaption></figure>

standard. For thicker specimens, the torque was increased to 70 N m. A Linear Variable Differential Transformer (LVDT) was positioned in the discharge port to record the displacement of the central point of the specimen during the test. A purging cycle with nitrogen and vacuum was performed ten times within the test cell to eliminate any traces of air and moisture. The flow of the test gas was then regulated using a manual valve to achieve the desired Pressure Rise Rate ($\dot{P}_a$). Once the desired $\dot{P}_a$ was achieved, the acquisition of pressure and displacement data was initiated, the cell was isolated, and the test was launched.

The tests were conducted using analytical-grade helium or hydrogen until the specimen either ruptured or leaked. A total of 46 tests were conducted using four distinct nominal Pressure Rise Rates ($\dot{P}_a$): 0.025

MPa/min, 0.250 MPa/min, 2.500 MPa/min, and 25.00 MPa/min. However, due to the manual adjustment of the pressurization rate with a valve, achieving precise control over the desired rate is challenging. As a result, the tests were performed at $\dot{P}_a$ ranging from 0.025 MPa/min to 35 MPa/min. The complete set of test parameters used in this study can be found in Appendix.

The rupture pressure was recorded for helium ($P_r(He)$) and hydrogen ($P_r(H_2)$). In accordance with the ISO 11114 standard, the rupture pressure must be adjusted to account for deviations from the “ideal” value, which can be caused by variations in thickness. Therefore, the corrected rupture pressure ($P'_r$) is calculated using the following equation:

$$P'_{r} = P_{r} \frac{0.75}{e_{m}} \qquad (6)$$

where $P'_r$ is the corrected pressure, $P_r$ is the recorded rupture pressure, 0.75 is the value of the nominal thickness (in mm) proposed by the standard and unchanged for the modified specimens, and $e_m$ is the measured thickness of the sample (also in mm). For the modified geometries, $e_m$ represents the minimum thickness, measured at the bottom of the notch for notched disks and at the center of the cup for cup disks.

**Table 5** summarizes the experimental results obtained for this study. It presents the average, minimum, and maximum $P'_r$ values for both He and $H_2$ gases across the three specimen geometries and both materials. The average $P'_r(He)$ is slightly higher for the E355 mod. steel grade compared to the X52 steel grade. The range between the minimum and maximum $P'_r$ values demonstrates the characteristic dispersion associated with the Disk Pressure Test (DPT). For brevity, the complete set of $P'_r$ values obtained in this study is provided separately in Appendix.
<table><caption>**Table 5**
Summary of the results obtained in DPT with E355 mod. and X52 steels. The presented values refer to the complete set of tests conducted over a Pressure Rise Rate ($P_r$) range from 0.025 to 35 MPa/min.</caption><thead><tr><th rowspan="2"></th><th colspan="2">Standard</th><th colspan="2">Cup</th><th colspan="2">Notched</th></tr><tr><th>P'<sub>r</sub> (He)</th><th>P'<sub>r</sub> (H<sub>2</sub>)</th><th>P'<sub>r</sub> (He)</th><th>P'<sub>r</sub> (H<sub>2</sub>)</th><th>P'<sub>r</sub> (He)</th><th>P'<sub>r</sub> (H<sub>2</sub>)</th></tr></thead><tbody><tr><td rowspan="3">E355 mod.</td><td>Avg.</td><td>Min.</td><td>Avg.</td><td>Min.</td><td>Avg.</td><td>Min.</td></tr><tr><td rowspan="2">45.9</td><td>45.6</td><td>19.0</td><td>70.1</td><td>63.4</td><td>52.4</td></tr><tr><td>Max.</td><td>21.6</td><td>Max.</td><td>66.8</td><td>Max.</td><td>29.8</td></tr><tr><td></td><td>46.1</td><td>25.7</td><td>75.7</td><td>69.6</td><td>53.3</td><td>33.9</td></tr><tr><td rowspan="3">X52</td><td>Avg.</td><td>Min.</td><td>Avg.</td><td>Min.</td><td>Avg.</td><td>Min.</td></tr><tr><td rowspan="2">39.1</td><td>35.8</td><td>19.7</td><td>70.6</td><td>63.8</td><td>45.8</td></tr><tr><td>Max.</td><td>21.8</td><td>Max.</td><td>65.6</td><td>Max.</td><td>29.8</td></tr><tr><td></td><td>42.4</td><td>24.3</td><td>73.6</td><td>67.6</td><td>48.4</td><td>32.3</td></tr></tbody></table>

<figure><img src="image_10.png" /><figcaption>**Fig. 6.** Simulation and experimental results for the deflection at the center of the disk against the corrected gas pressure for the three geometries (standard, cup, and notched) for the two steel alloys (E355 mod. and X52 vintage) tested under helium for a pressure rate of 0.25 MPa/min. The pressure in the case of the X52 steel is shifted by 20 MPa for clarity.</figcaption></figure>

## 4.2. Effect of specimen geometry (under helium)

Fig. 6 depicts numerical and experimental results obtained for different specimen geometries under helium charging at a nominal rate of 0.25 MPa/min for both steels. The tests were simulated using the elasto-plastic law presented in Section 3.3. The boundary conditions related to displacement and pressure are further explained in Section 5.3. The purpose of the simulation is to validate the tests and their interpretation.

Comparison of displacement-pressure curves reveals that thicker specimens – specifically, notched specimens with a thickness of 2 mm and cup disks with a thickness of 3 mm – exhibit a more prolonged linear displacement-pressure behavior compared to the standard specimens. This linear portion of the curve indicates a phase during the test when the failure zone of the specimens remains elastic, as confirmed by the simulations. Furthermore, it is observed that $P'_r$ for both standard and notched disks remains within the same range (between 40–50 MPa) for both materials. In contrast, the $P'_r$ values for cup disks are consistently higher, around 70 MPa, for both materials.

Additionally, the displacement at rupture is observable across the specimen types. The standard disk, due to its lower stiffness compared to the thicker specimens, exhibits a higher displacement at rupture, typically between 6 and 8 mm. In contrast, the notched disk shows a lower displacement at rupture, ranging from 3 to 4 mm. The cup disk demonstrates an intermediate displacement at rupture, falling within the 4 to 6 mm range.

$P'_r$ for each type of disk was observed to fall within a similar range for both steels. Notably, the displacement at rupture was higher for the E355 steel. These results align with the mechanical characterization obtained from tensile tests, where the E355 mod. steel demonstrated higher ductility compared to the X52 steel. Although the simulations do not account for the development of damage, they can predict the

point of failure, which occurs when pressure can no longer be increased due to changes in the geometry of the disk – specifically, the reduction in thickness. This situation corresponds to a limit load condition. The predicted failure points, indicated by vertical arrows in Fig. 6, closely match the experimental outcomes. This consistency suggests that disk failure is predominantly governed by plasticity and corresponds to a limit load scenario.

Additionally, it is important to note that fractures consistently occurred at the center of the cup disks. In notched specimens, fractures invariably occurred at the notch, indicating that fractures took place at the minimal cross-section of the specimens, as intended. In contrast, fractures in standard disk specimens primarily occurred in the clamping zone, although some fractures were also observed at the center of the disk in cases involving the highly ductile E355 mod. steel.

Additionally, experiments and simulations were conducted with notched disks, applying pressure to the flat side. The simulation indicated that when pressure is applied to the flat side, the highest level of plastic strain is achieved at the bottom of the notch, which, in this case, is not directly exposed to $H_2$. However, to obtain optimal HE sensitivity, it was reasoned that the most effective approach is to induce the maximum plastic strain in a region exposed to $H_2$. Therefore, the tests with pressure applied to the flat side did not offer a significant advantage and are not presented in this paper.

## 4.3. Effect of hydrogen

Fig. 7 illustrates the impact of hydrogen on the E355 mod. steel for the three different specimen geometries at a nominal $P_a$ of 0.25 MPa/min. These plots clearly demonstrate the embrittlement effect of $H_2$, characterized by a significant reduction in $P'_r$ and displacement at rupture for standard specimens. In this case, $P'_r(H_2)$ decreases by approximately 50% compared to $P'_r(He)$. In the case of the cup disk,
E355 mod.

<figure><img src="image_11.png" /><figcaption>Fig. 7. Effect of hydrogen in the E355 mod. steel for the three different specimen geometries for a pressure rate of 0.25 MPa/min.</figcaption></figure>

X52 Vintage

<figure><img src="image_12.png" /><figcaption>Fig. 8. Effect of hydrogen in the X52 steel for the three different specimen geometries for a pressure rate of 0.25 MPa/min.</figcaption></figure>

the difference between $P'_r(H_2)$ and $P'_r(\text{He})$ is less pronounced. The difference remains significant when considering displacements at failure. This phenomenon arises from the rapid increase in displacement when the applied pressure nears the limit pressure. In the case of the notched disk, the effect of $H_2$ is notably more pronounced, resulting in a drop of approximately 50% in the pressure at failure, with a significant impact on displacement at failure as well.

It can be observed that the pressure-displacement curves for all geometries remain unaltered by hydrogen up to the point of failure. This indicates that hydrogen does not affect the macroscopic behavior, allowing the hardening law derived above to be utilized in simulations of tests under hydrogen. Rupture occurs very rapidly, so one can assume that crack initiation immediately leads to the catastrophic failure of the disk. This is indeed consistent with a load-controlled test. It can be assumed that crack propagation is so rapid that hydrogen cannot diffuse into the material during failure. Consequently, the crack will propagate in a fixed hydrogen concentration field.

Fig. 8 depicts the effect of hydrogen on the X52 vintage steel for the three different disk specimen geometries. The conclusions drawn are analogous to those observed in the case of the E355 mod. material.

### 4.4. Effect of the pressurization rate

This section presents the influence of $\dot{P}_a$ on $P'_r$ of the different disk geometries and steel alloys under hydrogen and helium. The disks were

tested at different $\dot{P}_a$ values ranging from 0.025 to 35 MPa/min. Results are gathered in Fig. 9 where $P'_r(\text{He})$ is indicated by black triangles and $P'_r(H_2)$ by red circles. Under helium, very little dependence on the pressurization rate is expected as the material has a limited strain rate dependence. Such a trend is observed in all cases (specimens and materials).

Surprisingly, the effect of the $\dot{P}_a$ under hydrogen is also very limited, with a scatter similar to that observed under helium. This result is unexpected if compared to those of other studies. In a study conducted by Fidelle et al. [45], it was demonstrated that $\dot{P}_a$ has a significant influence on the hydrogen embrittlement (HE) of materials that are particularly susceptible to this phenomenon. Fidelle et al. [45] identified the relationship between $P_r(H_2)$ and $\log t$ (which is inversely proportional to $\dot{P}_a$) and divided it into 4 distinct zones: Zone I, for high $\dot{P}_a$, where $P_r(H_2)$ increases appreciably; Zone II, for moderate $\dot{P}_a$, where a plateau is identified and maximum HE can be observed across a range of $\dot{P}_a$; Zone III, for low $\dot{P}_a$, where $P_r(H_2)$ increases compared to Zone II, because dislocations do not move fast enough, allowing H to escape and redistribute in the metal lattice by regular diffusion; and Zone IV, for very low $\dot{P}_a$, which exhibits a higher plateau than Zone II, attributed to the fact that H entry is essentially controlled by the regular diffusion process. Moreover, more recent findings by Charles et al. [37] for commercial pure iron and by Briottet et al. [34] for X80 grade steel have shown the influence of $\dot{P}_a$ on $P_r(H_2)$. Despite some variability in the results, both studies observed a $\dot{P}_r(H_2)$ plateau for moderate
<figure><img src="image_13.png" /><figcaption>(a) Effect of pressurization rate in the hydrogen embrittlement of the E355 mod. grade steel.</figcaption></figure>

X52 Vintage

<caption>(b) Effect of pressurization rate in the hydrogen embrittlement of the X52 vintage steel.</caption>

Fig. 9. Effect of the pressurization rate. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

$\dot{P}_a$, and an increase in $P_r(H_2)$ with increasing $\dot{P}_a$. These results are within the Zones I and II described by Fidelle et al. [45]. The observed independence on the pressurization rate is further investigated through fracture surfaces observation in Section 4.6 and discussed in Section 5.4

## 4.5. Hydrogen Embrittlement Index

The Hydrogen Embrittlement Index (HEI) is calculated according to the ISO 11114 standard (see Eq. (1)), and the obtained values are shown in Table 6. Average values of the failure pressures at all loading rates are used. The HEI is quite similar for standard disks, equal to 2.41 for E355 mod. steel and 2.47 for X52 steel, exceeding the acceptable threshold of 2 according to ISO 11114. Notably, the HEI obtained for Notched disks is closer to the values obtained for the standard disk, with values of 1.96 and 1.77 for E355 mod. and X52. In contrast, the lowest HEI was observed for cup disks, 1.2 for E355 mod. and 1.12 for X52. These results clearly evidence that the HEI strongly depends on the specimen type. The application of different geometries results in distinct stress states at the fracture zone, a topic that will be discussed in more details in Section 5.4. Consequently, the interaction of hydrogen with the fracture zone is expected to vary, leading to variations in the HEI. For instance, in the case of the notched disk, a slightly sharper notch radius could lead to an HEI equivalent to that of the standard disk.

<caption>Table 6</caption>

Hydrogen Embrittlement Index (HEI) for the three disk specimens for the E355 mod. and X52 steels. HEI is calculated considering the worst ratio $P_r(He)/P_r(H_2)$ obtained for several tests under distinct Pressure Rise Rate ($\dot{P}_a$). The HEI values presented here corresponds to a nominal $\dot{P}_a$ of 2.5, 25 and 2.5 MPa/min for X52 Standard, Cup and Notched specimens, respectively; and to a nominal $\dot{P}_a$ of 0.25, 25 and 2.5 MPa/min for E355 mod. Standard, Cup and Notched specimens, respectively.

<table><thead><tr><th></th><th>Standard</th><th>Cup</th><th>Notched</th></tr></thead><tbody><tr><td>X52</td><td>2.47</td><td>1.12</td><td>1.77</td></tr><tr><td>E355 mod.</td><td>2.41</td><td>1.21</td><td>1.96</td></tr></tbody></table>

## 4.6. Fractography

The fracture surfaces of all specimens tested under helium exhibited nearly identical features, characterized by ductile fracture with voids elongated in the thickness direction. For conciseness, one cup disk for each steel is presented in Fig. 10. The surface fracture of the E355 mod. steel specimen is shown in Fig. 10(a), where the ductile aspect of the fracture can be seen. A similar ductile appearance is obtained for the X52 steel cup disk specimen, as shown in Fig. 10(b). However, in this sample, some voids elongated in the transverse and longitudinal directions can also be observed. They correspond to the long MnS inclusions present in this material.

The fracture surfaces for specimens extracted from the E355 mod. material tested under $H_2$ are illustrated in Fig. 11. In the left column,
<figure><img src="image_14.png" /><figcaption>Fig. 10. Fracture surface of cup disks tested under helium. Red arrows show elongated voids initiated on MnS inclusions. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 11. Surface fractures of E355 mod. steel disk specimens tested under hydrogen.</figcaption></figure>

the general aspect of the fracture can be seen. In all specimens, two zones can clearly be distinguished. An H₂ embrittled zone can be seen on the side exposed to the gas, corresponding to the initiation side. Subsequently, the crack propagates towards the second zone in which rupture is ductile. The H₂ embrittled zone is presented at a higher magnification in the right column. In these figures, a typical quasi-cleavage fracture aspect is observed.

Fig. 12 illustrates the fracture surfaces of different specimens from the X52 steel tested with H₂. As previously mentioned, the left column provides a general view of fracture, while the right column offers a

higher magnification view of the embrittled zone. Similar conclusions can be drawn for the X52 steel compared to the E355 mod. steel. A clear division between an embrittled and a ductile zone is observed except for the standard disk (Fig. 12(a)), which exhibits extensive shearing, complicating the identification of fracture mechanisms.

Disk specimens were sectioned and examined under a SEM to visualize the depth of the embrittled zone around the entire perimeter of the fracture. Fig. 13 illustrates the maximum observed embrittled depth, normalized relative to the local final thickness of the samples, as a function of the pressurization rate ($P_d$) for the three disk geometries.
<figure><img src="image_16.png" /><figcaption>Fig. 12. Surface fractures of X52 steel disk specimens tested with hydrogen.</figcaption></figure>

The maximum embrittled depth is calculated as the average of nine measurements taken from the area where embrittlement is most extensive. Blue circles indicate values for E355 mod. steel, while red circles represent those for X52 steel. Error bars show the standard deviation of the measurements. Some experimental points are missing due to the inability to analyze fractures in disks that leaked due to a through-wall crack rather than rupturing. Notably, for standard disk specimens, the normalized embrittled depth ranges between 0.67 and 0.93; for cup specimens, it ranges between 0.67 and 0.81; and for notched specimens, between 0.61 and 0.78. The pressurization rate ($\dot{P}_a$) does not seem to affect the size of the embrittled zone, which aligns with the observation that $P'_r(H_2)$ is not influenced by $\dot{P}_a$ for any specimen or material type. This aspect will be further explored through simulations in Section 5.4.

# 5. Simulation of hydrogen embrittlement during the disk tests

## 5.1. Modeling of hydrogen transport

The hydrogen transport model was initially proposed by [46] and lately corrected by [47], which is based on the equilibrium theory presented by [48]. This model states that hydrogen can be localized at Normal Interstitial Lattice Sites (NILS — concentration $C_L$) or trapping sites (concentration $C_T$). It is assumed that traps are isolated, i.e., do not form an extended network. Thus, hydrogen transport between trap sites occurs through the diffusion of lattice hydrogen. The model only considers reversible and saturable traps such as a dislocation core [47]. According to the principle of mass conservation, the rate of change of

the total hydrogen concentration ($C = C_L + C_T$) within a volume $\Omega$ is equal to the flux through a surface $\partial\Omega$:

$$ \frac{\partial}{\partial t} \int_{\Omega} (C_L + C_T) dv + \int_{\partial\Omega} \mathbf{J} \cdot \mathbf{n} \, ds = 0 \qquad (7) $$

where $\mathbf{J}$ is the hydrogen flux, and $n$ is the outward-pointing normal vector. The hydrogen flux can be defined as:

$$ \mathbf{J} = -D_L \nabla C_L + \frac{D_L C_L V_H}{RT} \nabla P \qquad (8) $$

where $D_L$ denotes the lattice diffusion coefficient, $V_H$ the partial molecular volume of hydrogen, $P$ is the pressure (hydrostatic stress: $P = \frac{1}{3}\text{trace}(\sigma)$), $R$ is the molar gas constant and $T$ is the absolute temperature. The hydrogen chemical potential in regions of tensile hydrostatic stress is lower, so a hydrogen flux is generated towards these regions to lower the chemical potential gradient. The opposite effect occurs in regions of compression hydrostatic stress. The diffusion coefficient in solids at different temperatures is generally found to be well predicted by the Arrhenius equation:

$$ D_L = D_0 \exp \left( -\frac{E_A}{RT} \right) \qquad (9) $$

where $D_0$ is a constant, and $E_A$ is the activation energy for diffusion. According to [46,49], a good estimation for the lattice diffusion coefficient $D_L$ in m$^2$/s for iron can be obtained by using $D_0 = 2.1 \times 10^{-7}$ m$^2$/s and $E_A = 6.88$ kJ/mol. The hydrogen concentration at the lattice sites, measured in hydrogen atoms per unit volume (H atoms/m$^3$), can be expressed as a function of the lattice occupancy ($\theta_L$):
<figure><img src="image_17.png" /><figcaption>Fig. 13. Maximum normalized embrittled depth different $\dot{P}_a$. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

$$C_L = \beta N_L \theta_L. \qquad (10)$$

Here $\beta$ represents the number of lattice sites per host atom, and $N_L$ is the number of lattice atoms per unit volume, expressed as: $N_L = N_A/V_M$ where $N_A$ is Avogadro's number, and $V_M$ is the molar volume of the host lattice measured in units of volume per lattice mole. In a similar manner, the hydrogen concentration at trapping sites can be expressed as a function of the occupancy of trapping sites $\theta_T$, and the trap density $N_T$:

$$C_T = N_T \theta_T \qquad (11)$$

The trap density, expressed in m⁻³, depends on the particular trapping site under consideration. In the case of traps such as grain boundaries or carbides, it is commonly assumed that the trap densities remain uniform through the material [50]. However, when dislocations are regarded as trapping sites, as in this work, the trap density varies and depends on the local accumulated plastic strain $p$, as dislocation density increases with plastic strain [51]. The equilibrium between lattice and trapped hydrogen atoms can be expressed as [48]:

$$\frac{1 - \theta_L}{\theta_L} \frac{\theta_T}{1 - \theta_T} = K = \exp \left( \frac{W_B}{RT} \right) \qquad (12)$$

where $W_B$ is the trapping binding energy. It is assumed that $\theta_L \ll 1$, so that Eq. (12) can be rewritten as follows:

$$\theta_T = \frac{K \theta_L}{1 + K \theta_L} \qquad (13)$$

Substituting Eq. (13) into (11), $C_T$ can be expressed as:

$$C_T = N_T \frac{\zeta C_L}{1 + \zeta C_L} \quad \text{with} \quad \zeta = \exp \left( \frac{W_B}{RT} \right) \frac{1}{N_L \beta} \qquad (14)$$

Taking the temporal derivative of $C_T$ for isothermal conditions, one gets:

$$C_T^t = \frac{1}{N_T} \frac{d N_T}{dp} C_T \dot{p} + \frac{\zeta N_T}{(1 + \zeta C_L)^2} C_L^t = \dot{\omega} + h \dot{C}_L^t \qquad (15)$$

## 5.2. Finite elements discretization and model parameters

It is imperative to consider a strong coupling between mechanics and diffusion to model the plastic behavior of materials with hydrogen accurately, as hydrogen diffusion can significantly impact the mechanical properties of the material. It is also important to account for finite strains to represent the disk deformation accurately. The algorithm used to handle this coupling is presented in [42]. This is only briefly recalled here. The proposed model was implemented in the Z-set software.

This study proposes an alternative approach to solving the coupled diffusion/mechanics problem using mixed finite elements. In the mixed formulation, the pressure is defined at nodes so that its gradient can be

<figure><img src="image_18.png" /><figcaption>Fig. 14. Representation of a square element with the mixed formulation with 4 degrees of freedom and reduced integration.</figcaption></figure>

calculated directly from the node values. This avoids the procedure of extrapolating the hydrostatic stresses determined at Gaussian points to nodes in order to compute the gradient.

This technique has also proven to be effective in solving problems characterized by volumetric locking occurring at high plastic strains, as reported in previous literature [52–55]. The problem's nodal unknowns are the displacement $[u]$, pressure $[P]$, and volume variation $[\theta]$. Displacements are interpolated using quadratic shape functions, while pressure and volume variation are interpolated using linear shape functions. Since pressure is known at the nodes, its gradient can be directly computed without extrapolation from values at the integration points to the nodes. An additional nodal field is also introduced to represent the lattice hydrogen concentration $[C_L]$, which is also represented by linear shape functions. Fig. 14 shows a representation of the degrees of freedom of an element, with reduced integration, using this mixed formulation.

In scenarios with high binding energies, integrated concentration values may oscillate and become unstable, leading to convergence difficulties. To address this issue, a “lumped” diffusivity matrix is used [56,57]. This technique leads to a diagonal diffusivity matrix, which avoids oscillations of the nodal concentration. In this study, the Gauss integration of the diffusivity matrix is performed by utilizing the values of the shape functions calculated at the node closest to the Gauss point as schematically shown in Fig. 14.

Table 7 presents the coefficients used in the simulations. These parameters are taken from a previous study [42]. Different values for the binding energy ($W_B$) and the trap density ($N_T$) are used in the simulations.

## 5.3. Boundary conditions for the hydrogen diffusion problem

In this study, axisymmetric elements with reduced integration are used to represent the disks. The same methodology could be applied in 3D cases at the cost of a much higher computation time. Fig. 15 illustrates the boundary conditions in the case of the standard disk.
<figure><img src="image_19.png" /><figcaption>Fig. 15. Representation of the boundary conditions applied to the standard specimen for the simulation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<table><caption>**Table 7**
Material parameters.</caption><thead><tr><td>Property</td><td>Value</td><td>Unit</td></tr></thead><tbody><tr><td>β</td><td>6</td><td>–</td></tr><tr><td>R</td><td>8.31</td><td>J/mol K</td></tr><tr><td>T</td><td>295</td><td>K</td></tr><tr><td>C<sub>S</sub></td><td>0.0129</td><td>appm</td></tr><tr><td>P<sub>s</sub></td><td>0.1013</td><td>MPa</td></tr><tr><td>D<sub>L</sub></td><td>1.27 × 10<sup>-8</sup></td><td>m<sup>2</sup>/s</td></tr><tr><td>N<sub>A</sub></td><td>6.022 × 10<sup>23</sup></td><td>mol<sup>-1</sup></td></tr><tr><td>V<sub>M</sub></td><td>7.09 × 10<sup>-6</sup></td><td>m<sup>3</sup>/mol</td></tr><tr><td>N<sub>L</sub></td><td>8.47 × 10<sup>28</sup></td><td>m<sup>-3</sup></td></tr><tr><td>V<sub>H</sub></td><td>2.0 × 10<sup>-6</sup></td><td>m<sup>3</sup>/mol</td></tr></tbody></table>

Sieverts' law can be used to determine the lattice hydrogen concentration at boundaries in contact with the gas [46]. For simplicity, it is assumed that hydrogen behaves as an ideal gas. Therefore, the lattice hydrogen concentration, $C_L$, is expressed as function of the hydrogen pressure, as follows [58]:

$$C_L = C_S \sqrt{\frac{P_a}{P_s}} \qquad (16)$$

where $C_S$ is a thermally activated term, $P_a$ is the hydrogen gas pressure, and $P_s$ is a reference pressure. Such a condition is applied to the lower part of the specimen (green line in Fig. 15). Therefore, the applied concentration increases with increasing applied pressure. Furthermore, based on the experimental findings of [59], it is reasonable to assume that the hydrogen content escapes rapidly from the top surface of the sample (blue contour in Fig. 15), so $C_L$ is set to zero for this boundary. The hydrogen flux through the other boundaries is assumed to be zero.

Radial displacement is blocked on the symmetry axis. To represent clamping, both axial and radial displacements are blocked at $r = 13.25$ mm (see Fig. 4). Contact with the upper part of the clamping device, with a radius of $R = 0.5$ mm, is also represented. This is necessary to prevent excessive deformation in this region. Frictionless contact is assumed. Pressure is applied to the lower part of the specimen, with a linear increase rate (corresponding to the experimental $\dot{P}_a$).

Simulations of tests performed under helium described above (Section 4.2) use the same technique without accounting for hydrogen diffusion.

## 5.4. Simulations of hydrogen embrittlement and discussion

In the present work, the development of damage due to hydrogen embrittlement (HE) is not accounted for in the simulations, which were conducted up to the experimentally observed rupture pressure for each corresponding material and geometry. Subsequent analyses of the simulations are focused on hydrogen concentration and the stress state. For instance, Fig. 16 displays an overlay of the cross-section of a notched disk after fracture with the plastic strain field derived from the numerical simulation for the E355 mod. steel. Notably, the simulation accurately predicts the location of fracture, as indicated by areas of higher plastic strain. The primary fracture zone and smaller secondary cracks, marked by red arrows, are situated within the zone

<figure><img src="image_20.png" /><figcaption>Fig. 16. Overlay of a fractography of the cross-section of the notched disk after rupture and the plastic strain field obtained by finite element modeling for the E355 mod.. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Fig. 17. Evolution of the trap density according to the accumulated plastic strain, following the evolution laws proposed by [51] and [60].</figcaption></figure>

of maximum plastic strain. However, the greater deformation observed in the experiment, shown on the left side of the figure, is attributed to the actual failure of the specimen, whereas the simulation model is limited to considering the elastoplastic behavior of the material.

Several models in the literature describe the evolution of trap density as a function of plastic strain and trap binding energies, with the latter ranging from 15 to 110 kJ/mol [61]. This study has retained two models that are frequently applied to model hydrogen trapping in pipeline steels. The first model, proposed by Moro et al. [60], focuses on a high-strength steel (API X80 grade) and assumes a trap binding energy of $W_B = 40$ kJ/mol. The trap density in this model is governed by the expression $\log_{10} N_T = 24.73 - 3.74 \exp(-60.17p)$, where $p$ represents the accumulated plastic strain. The second model, originally introduced by Kumnick et al. [51] and further utilized by Depraetere et al. [62],
<figure><img src="image_22.png" /><figcaption>Fig. 18. Total hydrogen concentration field in the rupture zone of the notched disk at rupture ($P_r = 29$ MPa for all cases). Simulations considering all the four possible models based on the binding energies $W_B$ and the trap density expressions $N_T$ for the E355 mod. steel.</figcaption></figure>

analyzes an X70 pipeline steel. This model assumes a trap binding energy of $W_B = 60$ kJ/mol, with $\log_{10} N_T = 23.26 – 2.33 \exp(-5.5p)$ governing the evolution of trap density relative to accumulated plastic strain.

Despite the similar materials analyzed by both models, significant differences are observed in the evolution of trap density as a function of plastic strain, as illustrated in Fig. 17. Particularly, the model proposed by Moro et al. [60] reaches its maximum value at remarkably low strains and exhibits a considerably higher maximum value than that of the model proposed by Kumnick et al. [51]. Consequently, employing the former model results in a significantly higher number of traps compared to the latter.

Based on the binding energies values and on the expressions for the trap density, four possible cases arise:

* Case 1: $W_B = 40$, $\log_{10} N_T = 24.73 – 3.74 \exp(-60.17p)$

* Case 2: $W_B = 40$, $\log_{10} N_T = 23.26 – 2.33 \exp(-5.5p)$

* Case 3: $W_B = 60$, $\log_{10} N_T = 24.73 – 3.74 \exp(-60.17p)$

* Case 4: $W_B = 60$, $\log_{10} N_T = 23.26 – 2.33 \exp(-5.5p)$

Fig. 18 summarizes the results for each of the four notched disk cases at pressure rise rates of $\dot{P}_a = 0.25$ and $\dot{P}_a = 25$ MPa/min for the E355 mod. steel. It compares the total hydrogen concentration fields for each model during slow and fast tests, along with the maximum concentration values achieved with each model. Notably, using the $N_T$ function proposed by Moro et al. [60] (cases 1 and 3), a significant difference is observed in the penetration profiles between the two pressurization rates, with the maximum concentration reaching much higher values as the trap density rapidly increases with accumulated plastic strain.

Conversely, the models employing the $N_T$ function proposed by Kumnick et al. [51] (cases 2 and 4) display very similar concentration profiles for both rates, but with smaller maximum concentration values. According to experimental observations shown in Fig. 13, the embrittled depth in the notched specimens is independent of the pressurization rate and represents about 80% of the total thickness. This suggests that hydrogen diffuses deeply into the material before failure occurs, with crack propagation happening so rapidly that it can be assumed to occur within a nearly static hydrogen concentration field, as previously discussed in Section 4.3.

Therefore, the model aligning more closely with experimental observations is the one proposed by Kumnick et al. [51], where $N_T$ increases moderately with plastic strain $p$. Since this model represents dislocations as traps, considered to be shallow, a lower binding energy value is more consistent, as supported by literature Refs. [61,63]. Consequently, case 2 is selected for all subsequent simulations.

Fig. 19 shows the total hydrogen concentration fields for both standard and cup specimens, providing detailed views of the fracture zones in the clamping area and at the center, respectively. Remarkably, the hydrogen fields appear similar for both pressurization rates. Particularly, the slower test may exhibit slightly higher concentration values due to the extended diffusion time of hydrogen; however, the results demonstrate that the concentration fields at both rates are comparable.

An additional analysis was conducted for the three disk geometries and both the X52 and E355 mod. steels, examining the evolution of the maximum principal stress ($\sigma_T$) in the fracture zone as a function of hydrogen gas pressure. Fig. 20 presents the results for each case. Distinct behaviors are observed for each geometry. For both the standard and notched specimens, a change in slope at approximately 300 MPa corresponds to the onset of plasticity. In contrast, at the onset of loading of
<figure><img src="image_23.png" /><figcaption>**Fig. 19.** Total hydrogen concentration field in the rupture zone for both standard and cup specimens. In the standard disk, failure occurs at the clamping zone, while in the cup, it occurs at the center. Simulations are based on the $W_B$ and $N_T$ coefficients from case 2 for E355 mod. steel.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>**Fig. 20.** Evolution of the maximum principal stress in the fracture zone with respect to the gaseous hydrogen pressure for $\dot{P}_a = 0.25$ MPa/min.</figcaption></figure>

the cup specimen, bending causes the surface in contact with air to be in tension and the surface in contact with hydrogen to be in compression, as shown in Fig. 20. As the specimen deforms, tensile stresses develop in the membrane. However, the stress differential between the air side and the hydrogen side remains, resulting in higher stress levels on the air side. According to Eq. (8), hydrogen diffuses towards regions of high tensile stress. Given that the outer surface experiences higher tensile stresses, failure would be expected to initiate there. However, failure typically begins on the side exposed to hydrogen, suggesting hydrogen leakage on the outer surface (the side exposed to air). This observation supports the use of the boundary condition ($C_L = 0$) for this surface. The stress response behavior is similar for both steels.

To successfully describe hydrogen-induced quasi-brittle failure (HEDE) with a GTN model, Pinto et al. [42] proposed that the critical stress needed to trigger fracture is depleted in function of the increased concentration of hydrogen. As shown in Fig. 20, specimens with geometries that develop higher principal stress in the fracture zone fail with lower $P_a(H_2)$, which correlates to lower hydrogen concentration. This observation aligns with the proposal of Pinto et al. [42] and suggests that the failure criterion for the DPT is a relationship between maximum principal stress and hydrogen concentration in the fracture zone, consistent with HEDE theory.

## 6. Conclusions

This study has successfully achieved its primary objectives through an integrated approach that combined numerical simulations with experimental analysis, adhering to the ISO 11114 standard without necessitating changes to the established DPT testing configuration. Key achievements include:

* The development and validation of a constitutive law derived from an innovative tensile test setup, which facilitated the simulation of the DPT for three distinct disk specimen geometries across two different materials.

* The design of novel disk specimen geometries effectively redirected rupture locations away from the clamping zones, with ruptures specifically occurring in areas of reduced thickness—namely, the central zone for the cup disk and near the bottom of the notch for the notched disk.

* The experimental observation that, despite a wide range of $\dot{P}_a$ values tested, there was no evident correlation between the severity of hydrogen embrittlement and $\dot{P}_a$. This is attributed to the consistent maximum embrittled depth across varying $\dot{P}_a$ values within the same specimen geometry.

* Numerical simulations using the Kumnick et al. model [51] for the evolution of hydrogen trap density with accumulated plastic strain corroborated experimental findings, showing similar hydrogen concentration profiles for significantly different pressurization rates. This supported the experimental evidence of $P'_r(H_2)$ independence from $\dot{P}_a$.

* Detailed analyses connecting the evolution of maximum principal stress ($\sigma_I$) in the fracture zone with hydrogen pressure and concentration levels demonstrated that increasing hydrogen concentrations decrease the $\sigma_I$ necessary to initiate fractures, providing insights into the mechanical vulnerability caused by hydrogen embrittlement.

In conclusion, this study confirms that the Disk Pressure Test (DPT) provides a simpler and more cost-effective alternative to traditional mechanical tests under gaseous hydrogen. By incorporating modified specimen geometries alongside standard ones, distinct stress and strain states can be achieved, potentially allowing for a more detailed investigation of hydrogen's impact on fracture behavior.

### CRediT authorship contribution statement

**L.M. Santana:** Writing – original draft. Visualization, Validation, Methodology, Investigation, Formal analysis, Data curation, Conceptualization, Writing – review & editing. **D. Lopes Pinto:** Writing – review
& editing, Writing – original draft, Visualization, Validation, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **N. Osipov:** Writing – review & editing, Software, Resources. **J. Furtado:** Writing – review & editing, Supervision, Resources, Methodology, Conceptualization. **F. Bourguignon:** Writing – review & editing, Resources, Conceptualization. **P.-J. Marchais:** Writing – review & editing, Resources, Conceptualization. **Y. Madi:** Writing – original draft, Supervision, Project administration, Methodology, Formal analysis, Conceptualization, Funding acquisition, Writing – review & editing. **J. Besson:** Writing – original draft, Validation, Supervision, Project administration, Methodology, Investigation, Formal analysis, Data curation, Conceptualization, Funding acquisition, Writing – review & editing.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

This work is funded by the ANR Chair program MESSIAH, France (ANR-20-CHIN-0003). The authors would like to express their gratitude to Mr. Frédéric Foy from CTE Air Liquide France for his assistance and guidance throughout the testing process.

## Appendix. List of the disk tests

See Table A.8.

<table><caption>**Table A.8**
Summary of the results obtained in DPT with E355mod. and X52 steels.</caption><thead><tr><th>MAT</th><th>TYPE</th><th>ENV</th><th>PRR<br>MPa/min</th><th>P'<sub>r</sub><br>MPa</th><th>Displacement<br>mm</th></tr></thead><tbody><tr><td rowspan="10">E355mod.</td><td rowspan="4">Standard</td><td>He</td><td>0.03<br>0.26</td><td>46.12<br>45.60</td><td>7.94<br>8.61</td></tr><tr><td rowspan="3">H<sub>2</sub></td><td>0.24<br>0.26<br>2.38</td><td>18.96<br>25.74<br>20.19</td><td>2.89<br>3.69<br>2.99</td></tr><tr><td>0.12<br>0.26<br>2.53<br>2.67</td><td>72.55<br>70.14<br>74.17<br>75.67</td><td>5.94<br>5.79<br>5.68<br>5.81</td></tr><tr><td>He</td><td>0.03<br>0.27<br>0.27<br>2.44<br>20.92</td><td>69.55<br>64.35<br>69.32<br>67.22<br>63.43</td><td>4.89<br>3.52<br>4.67<br>3.76<br>3.60</td></tr><tr><td rowspan="4">Cup</td><td rowspan="3">H<sub>2</sub></td><td>0.06<br>0.26</td><td>52.41<br>53.27</td><td>4.14<br>4.17</td></tr><tr><td>0.21<br>0.23<br>0.24<br>2.47<br>3.10<br>22.60</td><td>33.93<br>30.41<br>29.16<br>30.11<br>27.02<br>28.34</td><td>2.14<br>2.22<br>2.29<br>2.30<br>1.73<br>2.16</td></tr><tr><td>Notched</td><td></td><td></td><td></td></tr></tbody></table>

(continued on next page)

<table><caption>**Table A.8 (continued).**</caption><thead><tr><th>MAT</th><th>TYPE</th><th>ENV</th><th>PRR<br>MPa/min</th><th>P'<sub>r</sub><br>MPa</th><th>Displacement<br>mm</th></tr></thead><tbody><tr><td rowspan="10">X52</td><td rowspan="4">Standard</td><td>He</td><td>0.03<br>0.25</td><td>35.77<br>42.44</td><td>4.57<br>6.12</td></tr><tr><td rowspan="3">H<sub>2</sub></td><td>0.24<br>0.27<br>2.67<br>16.77</td><td>19.67<br>24.28<br>20.16<br>22.91</td><td>2.73<br>3.26<br>3.21<br>3.08</td></tr><tr><td>0.08<br>0.26<br>2.71</td><td>70.61<br>71.58<br>73.61</td><td>4.80<br>4.89<br>4.75</td></tr><tr><td rowspan="4">Cup</td><td>0.14<br>0.21<br>2.29<br>35.28</td><td>63.76<br>64.10<br>66.79<br>67.56</td><td>3.75<br>3.58<br>3.30<br>3.96</td></tr><tr><td rowspan="3">H<sub>2</sub></td><td>0.06<br>0.28<br>0.29<br>3.71</td><td>45.75<br>50.06<br>49.94<br>47.91</td><td>3.09<br>2.97<br>3.24<br>2.79</td></tr><tr><td>0.03<br>0.22<br>0.27<br>0.28</td><td>30.74<br>30.59<br>30.66<br>32.32</td><td>2.20<br>1.53<br>2.02<br>1.99</td></tr><tr><td>Notched</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

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