Low-cycle fatigue mechanical behavior of 30CrMo steel under hydrogen environment and numerical verification of chaboche model

In order to investigate the fatigue behavior of the hydrogen storage material, 30CrMo steel, in a hydrogen environment, an electrochemical hydrogen charging method was employed. Low-cycle fatigue experiments were conducted on the material to obtain half-life stress–strain hysteresis curves, cyclic response characteristics, and strain-life relationships under different hydrogen charging durations. The results indicate that the material exhibited an overall cyclic softening behavior, transitioning from ductile fracture to brittle fracture after hydrogen charging, resulting in a significant reduction in fatigue life. The Manson-Coffin formula was fitted based on material cyclic response characteristics and strain-life relationship curves. Additionally, fatigue toughness and Chaboche kinematic hardening models were fitted based on low-cycle fatigue test data. Finite element analysis was used to validate the accuracy and reliability of the Chaboche kinematic hardening model. The Chaboche kinematic hardening model showed minimal error compared to experimental data and accurately described the influence of hydrogen on the low-cycle fatigue mechanical behavior of 30CrMo steel.


Introduction
Hydrogen, as a crucial energy carrier, has found widespread applications in various fields such as aerospace and new energy vehicles [1].Currently, high-pressure hydrogen storage has become the mainstream form of hydrogen storage due to its simplicity and rapid charging and discharging capabilities [2][3][4].However, highpressure hydrogen storage containers are susceptible to hydrogen embrittlement, which occurs as they are exposed to hydrogen for extended periods, significantly affecting the mechanical properties and fatigue life of the materials.Additionally, these containers are subjected to alternating loads during the hydrogen charging and discharging processes, making them prone to fatigue failure.Research has shown that cyclic loading in a hydrogen environment accelerates fatigue crack propagation, shortens fatigue life, and often leads to changes in fracture surface morphology [5][6][7].Therefore, predicting the fatigue life of materials in a hydrogen environment is crucial for ensuring the safe operation of high-pressure hydrogen storage containers.

Experimental method
The experimental material used is 30CrMo steel in an annealed state, with its chemical composition as shown in table 1. Static tensile specimens and low-cycle fatigue specimens were designed according to the GB/ T15248-2008 standard, as illustrated in figure 1.

Electrochemical hydrogen charging test
The specimens were subjected to hydrogen charging using cathodic electrolysis.Surface grinding was performed to remove any surface machining marks, followed by ethanol cleaning of the specimens.A direct current power supply (PS-3002D-II) was employed for the experiment.The electrolyte solution was prepared by mixing 0.5 mol/L sulfuric acid with 1 g L −1 of thiourea.The cathode of the power supply was connected to the specimen, and the anode was connected to a carbon rod.To control the influence of hydrogen charging current on the test results, the hydrogen charging current was maintained at 1 mA cm −12 , with charging durations of 0 h, 2 h, and 6 h, respectively.After hydrogen charging was completed, in order to prevent the escape of hydrogen atoms from the metal, low-cycle fatigue tests were conducted immediately [13].

Low-cycle fatigue test
The low-cycle fatigue tests were conducted on an MTS-809 fatigue testing machine under standard laboratory conditions, including room temperature and atmospheric pressure.An extensometer with a specification of 632.13F-20 was used for axial strain control.The strain ratio (e), loading frequency was set at 2Hz, and the cyclic loading pattern was triangular.The experiments involved four levels of strain amplitudes (e a ): 1.0%, 1.2%, 1.4%, and 1.6%.For each strain level, three parallel specimens were tested, and their average values were taken as the test results.Fatigue failure of the material was defined as a 25% reduction in the maximum tensile stress within each cycle [14,15], at which point the test was terminated.Specimens that developed fatigue cracks but did not fracture were pulled apart using a quasi-static tensile method, with a pulling rate of 0.02 mm s −1 .

Static tensile test
The engineering stress-strain curves for the steel specimens under different hydrogen charging durations are depicted in figure 2. It is observed that the tensile strength and yield strength of the specimens experienced a slight decrease, while the elongation significantly decreased.Specifically, the elongation decreased by 22.6% and 39.0%, respectively.This indicates a reduction in the plasticity of the specimens after hydrogen charging.

Experimental results and analysis
The low-cycle fatigue test data for the three hydrogen charging conditions are presented in table 2, with the halflife cycle during the fatigue test chosen as the data point.Hydrogen significantly reduced the fatigue life of the test steel.At a low strain amplitude of 1%, the fatigue life of the steel charged with hydrogen for 2 h and 6 h decreased by 2.9 times and 5.67 times, respectively.At a high strain amplitude of 1.6%, the fatigue life of the steel  charged with hydrogen for 2 h and 6 h decreased by 3.5 times and 8.2 times, respectively.With increasing strain amplitude, the proportion of plastic strain amplitude in the total strain amplitude also increased, indicating that plastic deformation predominated in the material during high strain amplitude cycles.Comparing the three sets of tests, at different strain amplitude levels, the stress amplitude increased with increasing hydrogen charging duration.Tsuchida [16] investigated the effect of hydrogen on the microstructure of low-carbon steel and explained this phenomenon.Hydrogen stabilizes vacancies formed by the motion of screw dislocations, leading to an increase in stress amplitude.Additionally, at different strain amplitude levels, as the hydrogen charging duration increased, the elastic strain amplitude increased while the plastic strain amplitude decreased.This suggests that the increase in hydrogen-induced stress amplitude is due to the increase in elastic strain amplitude [17].In other words, the elastic phase of hydrogen-induced materials elongates during the cyclic process, while the plastic phase shortens, resulting in an increase in elastic strain amplitude and a decrease in plastic strain amplitude.

Stress-strain hysteresis loops
The half-life hysteresis loops under the same hydrogen charging duration but different strain amplitude conditions were superimposed onto a common point, as shown in figures 3(a)-(c).Under a strain amplitude of 1%, the hysteresis loop area is relatively small, indicating that the test steel undergoes minimal plastic deformation during the cyclic process.In contrast, under a strain amplitude of 1.6%, the hysteresis loop area is larger, indicating that the test steel primarily experiences plastic deformation during the cyclic process.Therefore, its fatigue life is significantly lower than that at a 1% strain amplitude.Furthermore, it can be clearly observed that the loading branches of the half-life hysteresis loops under different strain amplitudes form an envelope curve, indicating that the material exhibits Masing behavior.This suggests that the hydrogen embrittlement behavior of 30CrMo does not alter its Masing characteristics.To compare the effect of hydrogen on the half-life hysteresis loops, figure 3(d) displays the half-life hysteresis curves for a 1.6% strain amplitude under three different hydrogen charging durations.As the hydrogen charging duration increases, the stress amplitude of the half-life hysteresis loops also increases.The elastic phase largely overlaps, but zero plastic strain increases with the hydrogen charging duration.This indicates that after hydrogen charging, the elastic phase of the test steel elongates, while the plastic phase shortens.The hysteresis curve test results align with the low-cycle fatigue data in table 3, confirming the reliability of the experimental data.

Cycle response characteristics and cycle hardening/softening rate
Figures 4(a)-(c) represent the cycle response characteristics of specimens with different hydrogen charging durations at various strain amplitudes.The overall trend in the cycle response characteristics of the three hydrogen-charged specimens indicates cyclic softening.In the first stage of cycling, there is a rapid softening phenomenon at the beginning of the cycle, which accounts for approximately 10% of the fatigue life.In the second stage of cycling, the stress amplitude gradually decreases during mid-cycle loading, showing a mild softening effect.In the third stage of cycling, the stress amplitude rapidly decreases in the late stages of cyclic loading.This is primarily due to the rapid propagation of fatigue cracks after their initiation, leading to fatigue fracture failure in the experimental specimens as cyclic loading continues.Furthermore, specimens charged with hydrogen for 2 h and 6 h exhibited a rapid decrease in stress amplitude when subjected to cyclic loading at strain amplitudes of 1.4% and 1.6%, respectively, reaching 60% of their fatigue life.This phenomenon occurs because, after hydrogen charging, the reduced plasticity of the material due to hydrogen embrittlement makes the material more susceptible to fatigue crack initiation.Fatigue crack initiation becomes more sensitive and easier, resulting in the premature appearance of fatigue cracks during the cyclic process.This reduces the effective loadbearing area, thereby reducing the required stress amplitude to achieve the specified strain amplitude.
To more accurately describe the cyclic hardening/softening behavior of 30CrMo steel under different hydrogen charging durations, the concept of the cyclic hardening/softening ratio (H) is introduced.The definition of the cyclic hardening/softening ratio is as follows: Among these s a sat and s a 1 represent the half-life cyclic stress amplitudes for the first cycle and subsequent cycles, respectively.The linear fitting of the cyclic hardening/softening ratio under different hydrogen charging durations is illustrated in figure 5.
Analysis based on the definition of the cyclic hardening/softening ratio reveals two key aspects.On one hand, as the cyclic hardening/softening ratio approaches zero, it indicates a weaker degree of cyclic hardening/ softening in the material.On the other hand, when the slope of the fitted curve is larger, it suggests that the strain amplitude has a greater influence on the cyclic hardening/softening of the material.In figure 5, the cyclic hardening/softening ratio for different hydrogen charging durations is negative, indicating that 30CrMo steel    experiences cyclic softening during the cyclic process, where the half-life cyclic stress amplitude is smaller than that of the first cycle.This aligns with the conclusions drawn from the cyclic response characteristic curve.The cyclic softening ratio of the test steel approaches zero with increasing hydrogen charging duration, signifying a reduction in the softening effect of 30CrMo steel after hydrogen charging.Furthermore, the slope of the fitted curve decreases with increasing hydrogen charging duration, indicating that after hydrogen charging, the influence of strain amplitude on cyclic softening diminishes for 30CrMo steel.

Strain-life relationship
In low-cycle fatigue, material fatigue characteristics are often represented using the strain-life relationship.This approach involves analyzing the variation in fatigue life with respect to strain amplitude, elastic strain amplitude, and plastic strain amplitude using logarithmic coordinates, as shown in figure 6 [18,19].From the graph, it can be observed that all three strain amplitudes exhibit a linear relationship with fatigue life.As strain amplitude increases, the elastic strain amplitude remains constant while the plastic strain amplitude increases.This suggests that the material primarily dissipates energy due to plastic strain [20].
The fatigue life corresponding to the point where plastic strain amplitude equals elastic strain amplitude is known as the transition fatigue life, which is one of the indicators representing the low-cycle fatigue performance of the material.The transition fatigue life and its corresponding strain amplitude values are presented in table 3.
The cyclic lifetimes of the test steel for the three different hydrogen charging durations exhibit a decreasing trend.As the hydrogen charging duration increases, the strain amplitude corresponding to the transition fatigue life also increases.This is attributed to the fact that hydrogen charging increases the proportion of elastic strain amplitude in the material, 30CrMo steel, during the cyclic process.Furthermore, the transition fatigue life of the test steel decreases significantly after hydrogen charging.The transition fatigue life of the uncharged test steel is 5.2 times that of the test steel charged for 2 h and 22 times that of the test steel charged for 6 h.Two primary factors contribute to this outcome: Firstly, hydrogen embrittlement accelerates the rate of fatigue crack propagation in the test steel, and secondly, hydrogen charging increases the elastic strain amplitude of the test steel.Consequently, an increase in plastic strain amplitude is required to match the elastic strain amplitude, resulting in an overall increase in total strain amplitude.The larger the total strain amplitude, the shorter the transition fatigue life.According to the Basquin formula and Manson-Coffin formula [21], the strain-life relationship is as follows: In the equation, s ¢ f and e ¢ f represent the fatigue strength coefficient and fatigue ductility coefficient respectively; E stands for the elastic modulus; b and c denote the fatigue strength exponent and fatigue ductility exponent, as determined by fitting the material parameters using the least squares method, as shown in table 4. Table 4 shows that the fatigue strength coefficient and fatigue strength index increase slightly with the increase of hydrogen charging time, indicating that the elasticity phase of the material increases, in line with the above stress-strain hysteresis curve research law, in which the fatigue strength coefficient of the hydrogen-charged 2h specimen is larger, probably because of the data fitting error.The fatigue toughness coefficient and fatigue toughness index decreased rapidly with the increase of hydrogen charging time, indicating that the toughness of the material decreased after hydrogen charging.

Strain energy density
The Basquin formula and Manson-Coffin formula discuss the strain-life relationship primarily from the perspective of elastic and plastic strain amplitudes during cyclic loading.To comprehensively analyze the lowcycle fatigue life of materials, it's essential to assess the fatigue life before and after hydrogen charging from the perspective of energy dissipation, which accounts for the material's damage and fracture due to energy dissipation caused by both elastic and plastic deformation.This is manifested when the accumulated plastic strain energy density reaches a critical value, resulting in the occurrence of fatigue cracks or fatigue fracture [22].The area within the hysteresis loop represents the energy dissipated by the material during the cyclic loading process.This energy arises from the plastic work during the cycles and is referred to as plastic strain energy density, which can be calculated using equation (3) [23,24].Additionally, the elastic and total strain energy densities are represented by Equations (4) and (5), respectively: where s D is the stress range (2 s a ), e D p is the plastic strain range (2 e p ), ¢ n is the cyclic hardening exponent, e D e is the elastic strain range (2e e ), and E is the elastic modulus.The strain energy density-life curve for 30CrMo steel under different hydrogen charging durations is fitted, as shown in figure 7.In the figure, , , t e pc pa represents the total strain energy density, represents the elastic strain energy density, represents the calculated plastic strain energy density and represents the experimental plastic strain energy density (half-life hysteresis loop area).
As observed from the figures, the calculated plastic strain energy density closely matches the experimental plastic strain energy density curve.This indicates that the calculated plastic strain energy density is effective.With a decrease in fatigue life, there is a significant growth trend in plastic strain energy density, while elastic strain energy density remains relatively constant.This suggests that, in low-cycle fatigue tests with high strain amplitudes, the test steel primarily experiences plastic deformation.Furthermore, with an increase in hydrogen charging duration, the proportion of plastic strain energy density to total strain energy density decreases, primarily due to an increase in elastic strain energy density after hydrogen charging.The results of strain energy density are consistent with the conclusions drawn from the low-cycle fatigue data.To provide a more comprehensive analysis of material fatigue life, a fatigue toughness model is introduced to predict material fatigue life based on the accumulated plastic deformation energy within the failure cycles as a function of the number of cycles.This relationship is also referred to as a fatigue toughness model.Firstly, equation ( 6) is introduced to determine the material's fatigue toughness, which is calculated using half-life stabilized plastic strain energy.Then, equation ( 7) is used to fit the power-law relationship between fatigue toughness and the number of cycles, establishing a fatigue toughness model for 30CrMo steel with different hydrogen charging durations, as shown in figure 8.
where DW , pa the half-life plastic strain energy; N , f number of cycles to failure, A, fatigue toughness coefficient and a,fatigue toughness exponent.Fatigue toughness is the cumulative plastic strain energy of a material under fatigue failure.Figures 8(a)-(c) show the fatigue toughness models with hydrogen charging duration of 0 h, 2 h and 6 h, respectively.As can be seen from the figure, the fatigue toughness of the material decreases rapidly with the increase of hydrogen filling time.This is because hydrogen molecules penetrate into the lattice of the material and accumulate, resulting in increased brittleness and decreased toughness of the material.The cumulative fatigue toughness required for the fatigue failure of the sample with hydrogen charging duration of 6h under cyclic load is the cumulative plastic strain energy of the material under fatigue failure.Figures 8(a)-(c) show the fatigue toughness models with hydrogen charging duration of 0 h, 2 h and 6 h, respectively.As can be seen from the figure, the fatigue toughness of the material decreases rapidly with the increase of hydrogen filling time.This is because hydrogen molecules penetrate into the lattice of the material and accumulate, resulting in increased brittleness and decreased toughness of material.The cumulative plastic strain energy required for fatigue failure of the samples with hydrogen charging of 6 is reduced under cyclic load, resulting in fatigue cracks or fractures of the samples earlier.

Fracture surface analysis
To compare the fracture surfaces and microstructural features under different hydrogen charging durations, scanning electron microscopy (SEM) was used to observe the fracture surfaces of specimens subjected to a strain amplitude of 1.4% without hydrogen charging, with 2 h of hydrogen charging, and with 6 h of hydrogen charging, as shown in 9, 10, and 11.(In the figures, a∼d represent the overall fracture surface, crack initiation zone, crack propagation zone, and instantaneous fracture zone, respectively.)Typical fatigue crack initiation zones, crack propagation zones, and instantaneous fracture zones are observable in all the figures.In the early stages of fatigue, microcracks form in multiple locations on the specimen's surface.As the cyclic loading progresses, these microcracks intersect and propagate into the interior of the test steel, forming parallel crack steps.The effective load-bearing area gradually decreases until fatigue failure or the fatigue fracture limit is reached [25].
From figure 9(a), it can be observed that the fracture surface of the specimen without hydrogen charging is relatively smooth and regular.However, with an increase in hydrogen charging duration, the fracture surface becomes rougher, with irregular shear lip morphology in multiple areas.Notably, the fracture surface characteristics are most pronounced in figure 11(a) for the specimen with 6 h of hydrogen charging, with a rough stepped crack propagation zone and multiple shear lip features in the fracture area.By comparing figures 9(b), 10(b), and 11(b), it can be seen that hydrogen charging has some influence on the initiation of microcracks in the specimens.Figure11(b) exhibits crack initiation due to stress concentration.Comparing figures 9(c), 10(c), and 11(c), it can be observed that as the hydrogen charging duration increases, the spacing between cracks in the fracture propagation zone of hydrogen-charged specimens becomes larger.Additionally, when the specimen's fatigue life exceeds 300, tire track marks appear on the fracture surface, as shown in figure 11(c).This is mainly due to the enrichment of hydrogen in the lattice, dislocations, and inclusions, which reduces the bonding strength at grain boundaries or between the matrix and inclusions [26,27], resulting in larger crack spacing between each cycle, increased fatigue crack propagation rate, and reduced fatigue life [28].
Comparing figures 9(d), 10(d), and 11(d), it can be observed that the microstructural features in the instantaneous fracture zone exhibit a large number of dimples, indicating ductile fracture.However, as the hydrogen charging duration increases, the distribution of dimples becomes sparser, and their number decreases.This suggests a transition from ductile to brittle fracture tendencies in the fracture mode as the hydrogen charging duration lengthens.

Chaboche model
For ease of application in engineering, the constitutive model proposed by Chaboche is used to describe the stress-strain relationship of materials under cyclic loading by defining the yield surface, the flow rule, the hardening rule, and the critical state [29].The Chaboche constitutive model is based on the von yield criterion to describe the plastic yielding behavior of materials during the cyclic process [30].The pressureindependent yield surface used in the linear kinematic hardening model and the nonlinear isotropic/kinematic hardening model is defined by the function: where s 0 is the yield stress, a¢ is the back stress, a -¢ ( ) f s is the equivalent von Mises stress or Hill's potential with respect to a.The equivalent von Mises stress is defined as: where S is the deviatoric stress tensor, a dev is the deviatoric part of the back stress tensor.The isotropic hardening model in the Chaboche constitutive model is used to describe the hardening phenomenon of materials during the early stages of cyclic loading [31], as shown in figure 12(a).The evolution equation is given by equation (10), where s | 0 is the yield stress at zero plastic strain, ¥ Q and ¢ b are determined through nonlinear fitting of experimental data s e ( ̄) , , where s i 0 represents the peak tensile stress for each cycle, ēi p is the corresponding equivalent plastic strain, ¥ Q is the maximum variation in the size of the yield surface, and ¢ b is the rate of change of peak tensile stress with plastic strain development.
This study primarily investigates the influence of different hydrogen charging durations on the stable hysteresis loops of the tested steel, while ignoring the isotropic hardening effects and critical state of the experimental steel.The calibration of the kinematic hardening model parameters for the three different hydrogen charging durations of the tested steel is conducted, and the accuracy of the kinematic hardening model is verified using finite element analysis.The kinematic hardening model is typically formulated as a superposition of multiple independent nonlinear components to describe the movement of the yield surface center, represented by the back stress, in the stress space under cyclic loading [32], as illustrated in figure 12(b).where e i p and a i back represent the equivalent plastic strain of the ith cycle and the back stress of the ith cycle, respectively e = 0. p 1 e p 0 is the plastic strain value at the intersection of the curve with the strain axis, and s s mean is the average value of s 1 and s .
n Based on the above equation, Chaboche model parameters for stable hysteresis loops of 30CrMo steel at a 1.6% strain amplitude are fitted for different hydrogen charging durations.Using MATLAB software and the least squares regression method, the calibrated kinematic hardening parameters are determined.The values of the kinematic hardening parameters are shown in table 5.As shown in table 5, the zero plastic strain yield stress increases with an increase in hydrogen charging duration, which is consistent with the results and analysis conclusions of the low-cycle fatigue tests mentioned above.

Verification of model parameters
To verify the accuracy and reliability of the model parameters, this study conducted numerical simulations of cyclic loading tests on 30CrMo steel with different hydrogen charging durations using finite element software ANSYS.Establish a 3D model in Ansys software, replicating the standard specimen shown in figure 1. Employ 2mm hexahedral solid elements for meshing and select the Chaboche kinematic hardening model for the material.Furthermore, apply fixed boundary conditions and displacement loads ( = ( ) U time 0.6 sin 20 ) to the respective clamping ends.The 1.6% strain amplitude symmetrical cyclic loading hysteresis test curves were compared with the simulation curves.Figures 13(a)-(c) show the comparisons between experimental curves and finite element simulation curves for uncharged specimens, specimens charged for 2 h, and specimens charged for 6 h, respectively.
Under the influence of repeated cyclic loading, the area of the hysteresis loop is an important indicator of energy dissipation capacity.To quantitatively assess the relative error between the numerical simulation curves and experimental curves [33], the following equation is introduced: FEM test test where S FEM and S test represent the area of the hysteresis loop in numerical simulation and experimental data, respectively.A smaller absolute value of ¢ A indicates a more accurate model.Table 6 shows the relative errors in simulation for different hydrogen charging durations.From the table, it can be observed that all three sets of model parameters exhibit relatively high accuracy in simulating the cyclic loading tests, with ¢ A values all below 5%.This indicates that by fitting the parameters of the Chaboche cyclic constitutive model using a reasonable approach, it is possible to accurately simulate the elastic-plastic response of 30CrMo steel under cyclic loading conditions after hydrogen charging.

Conclusion
This paper conducted monotonic cyclic loading tests on 30CrMo steel with varying hydrogen charging durations and four strain amplitudes to investigate the influence of hydrogen on the material's mechanical behavior and fatigue fracture modes.The Chaboche cyclic constitutive model parameters for 30CrMo steel with different hydrogen charging durations were determined through a least-squares fitting process.After a comparative analysis, the following conclusions were drawn: (1) Hydrogen has a relatively small impact on the yield strength, tensile strength, and elastic modulus of the tested steel.However, it mainly reduces the material's ductility and plasticity, with reductions in elongation of 22.6% and 39.0% for specimens charged with hydrogen for 2 h and 6 h, respectively.Hydrogen does not alter the Masing behavior of the tested steel but extends the elastic stage during cyclic loading while reducing the plastic stage.
(2) Hydrogen significantly affects the material's fatigue life, primarily because the presence of hydrogen reduces the bonding strength between the crystal lattice or matrix and the inclusion layer, leading to an increased  fatigue crack propagation rate.There is a tendency for the fracture mode to transition from toughness fracture to brittle fracture.
(3) Based on the experimental data, the Chaboche cyclic constitutive model parameters fitted by the leastsquares method accurately describe the macroscopic low-cycle fatigue mechanical behavior influenced by hydrogen.This can serve as a reference for the engineering application of 30CrMo steel in high-pressure hydrogen storage containers and the prediction of low-cycle fatigue life under seismic conditions.
The hydrogen environment simulated by the electrochemical hydrogen charging method used in this paper is different from the high-pressure hydrogen environment used in actual engineering.Therefore, it does not completely and accurately describe the low-week fatigue behavior of 30CrMo steel in high-pressure hydrogen environment.In addition, this paper only controls the duration of hydrogen charging to determine the amount of hydrogen molecules diffusing into the interior of the metal, and does not quantitatively analyze the effect of the amount of hydrogen molecules in the metal on low-cycle fatigue behavior.

Figure 3 .
Figure 3. Hysteretic curve.Figure note: (a) Unhydrogenated sample; (b) Sample filled with hydrogen for 2 h; (c) Sample filled with hydrogen for 6 h; (d) Samples with different hydrogen charging durations with a strain amplitude of 1.6%.

Figure 4 .
Figure 4. Cyclic characteristic curve.Figure note: (a) Sample filled with hydrogen for 0 h; (b) Sample filled with hydrogen for 2 h; (c) Hydrogen filled sample for 6 h.

Figure 6 .
Figure 6.Low cycle fatigue strain life curve.Figure note: (a) Sample filled with hydrogen for 0 h; (b) Sample filled with hydrogen for 2 h; (c) Sample filled with hydrogen for 6 h.

Figure 7 .
Figure 7. Strain energy density life curve Figure note: (a) Sample filled with hydrogen for 0 h; (b) Sample filled with hydrogen for 2 h; (c) Sample filled with hydrogen for 6 h.

Figure 8 .
Figure 8.The relationship between accumulated plastic strain and life Figure note: (a) Sample filled with hydrogen for 0 h; (b) Sample filled with hydrogen for 2 h; (c) Sample filled with hydrogen for 6 h.

Figure 9 .
Figure 9. Unfilled hydrogen fatigue fracture image with a strain amplitude of 1.4% Figure note: (a) Overall view of the fracture surface; (b) Crack source; (c) Crack propagation zone; (d) Transient zone.

Figure 10 .
Figure 10.Image of fatigue fracture specimen with a strain amplitude of 1.4% after 2 h of hydrogen charging Figure note: (a) Overall view of the fracture surface; (b) Crack source; (c) Crack propagation zone; (d) Transient zone.

Figure 11 .
Figure 11.Fatigue fracture image of hydrogen charging for 6 h with a strain amplitude of 1.4% Figure note: (a) Overall view of the fracture surface; (b) Crack source; (c) Crack propagation zone; (d) Transient zone.

Figure 13 .
Figure 13.Experiment and simulation of stress-strain curves of steel under different hydrogen charging durations Figure note: (a) Sample filled with hydrogen for 0 h; (b) Sample filled with hydrogen for 2 h; (c) Sample filled with hydrogen for 6 h.

Table 1 .
Chemical composition of test steel.

Table 2 .
Low cycle fatigue test data.

Table 3 .
Critical values of steel tested for different hydrogen charging durations.

Table 4 .
Fitting results of strain life formula.

Table 5 .
Follow up strengthening parameters.
plastic strain value at the intersection of the curve with the strain axis s s average of the first and the last stresses data points ¢ A relative error between the numerical simulation curves and experimental curves ¥ Q maximum variation in the size of the yield surface ¢ b rate of change of peak tensile stress with plastic strain development