Abstract
The manufacturing quality of ultra-high-pressure hydrogen storage vessels directly influences their reliability under hydrogen service conditions. Residual stress distribution and wall-thickness variation introduced during hot spinning affect susceptibility to hydrogen-assisted damage. This study investigates the relationship between hot spinning parameters, stress evolution, and hydrogen embrittlement behavior in ASTM A519 4142 steel vessels through combined numerical simulation and experimental validation. Hot compression tests characterized high-temperature deformation behavior, and an Arrhenius-type constitutive model was developed with a mean absolute error of 7.35%. A thermo-mechanically coupled finite element model incorporating adaptive meshing reduced computational time by 41.2% while improving prediction accuracy. Process optimization identified preferred conditions of 1000 °C forming temperature, 0.03 s-1 strain rate, and 2 mm/s feed rate, yielding uniform wall thickness and reduced stress concentration. Industrial spinning trials confirmed the numerical predictions. Burst testing revealed regional differences in hydrogen embrittlement: the shoulder region showed the highest susceptibility, while the cylinder region exhibited the greatest resistance. These findings highlight the importance of controlling stress concentration during forming to enhance structural integrity. This work provides a reliable material model, efficient simulation framework, and practical guidance for safe manufacturing of ultra-high-pressure hydrogen storage vessels.
Highlights
A precise Arrhenius constitutive model for 4142 steel was developed, achieving a prediction error of only 7.35%.
An adaptive finite element framework reduced computational cost by 41.2% while enhancing prediction accuracy.
The optimal hot spinning parameters were identified, leading to a more uniform residual stress distribution.
The study revealed a distinct regional gradient in hydrogen embrittlement resistance across the vessel structure.
• Provides a reliable material model for thermo-mechanical simulations of high-strength steels.
• Enables efficient and accurate numerical optimization of the hot spinning process.
• Offers guidance for process parameter adjustment to minimize stress concentration and improve hydrogen embrittlement resistance.
• Serves as a technical reference for the safe manufacturing of ultra-high-pressure hydrogen storage vessels.
1 Introduction
Hydrogen stored in compressed gas vessels is regarded as one of the most practical solutions for large-scale hydrogen transportation and on-board fuel supply. As storage pressures continue to increase, the structural integrity of hydrogen storage vessels becomes increasingly dependent on manufacturing quality (; ). Among the various fabrication processes, hot spinning is widely used for producing thick-walled seamless vessels because it allows large plastic deformation while maintaining dimensional accuracy. However, the process may also introduce non-uniform wall thickness and residual stresses, both of which influence the subsequent service performance of the vessel. (; ; ).
Considerable efforts have been devoted to understanding the deformation behavior of high-strength steels during hot spinning. Constitutive models derived from hot compression experiments have been widely employed to describe flow stress evolution under elevated temperatures, while finite element simulation has become a common tool for analyzing material flow, wall-thickness variation and stress development during forming. These approaches have improved process design; however, most studies evaluate manufacturing quality primarily in terms of dimensional accuracy and forming defects. (; ; ).
For hydrogen storage vessels, dimensional accuracy alone is insufficient for evaluating manufacturing quality. Residual stresses generated during hot spinning may alter local stress states during service and thereby influence susceptibility to hydrogen-assisted cracking (; ; ). Despite the growing interest in hydrogen embrittlement of high-strength steels, relatively few studies have investigated the influence of process-induced stress distributions on the regional embrittlement behavior of spun hydrogen storage vessels (; ; ). As a result, the connection between manufacturing parameters and hydrogen-related failure remains insufficiently understood.
Another challenge arises from the strong thermo-mechanical coupling involved in hot spinning. Accurate simulation requires simultaneous consideration of heat transfer, plastic deformation and contact behavior (; ; ; ). For thick-walled vessels, large deformation and localized stress concentration may also increase computational cost and reduce numerical robustness. Therefore, there remains interest in developing efficient simulation strategies capable of capturing the key features of the forming process without excessive computational expense.
The present work investigates the hot spinning process of ASTM A519 4142 steel hydrogen storage vessels through a combined numerical and experimental approach. An Arrhenius-type constitutive model was established from hot compression data and incorporated into a thermo-mechanically coupled finite element framework (; ; ; ). Process optimization was performed with consideration of wall-thickness uniformity, stress concentration and forming force. Particular attention was given to the relationship between process-induced stress distribution and hydrogen embrittlement behavior in different regions of the vessel (; ; ; ). By linking manufacturing parameters, residual stress evolution and fracture characteristics, this study aims to provide insight into the role of hot spinning in the structural reliability of ultra-high-pressure hydrogen storage vessels.
2 Experimental materials and methods
2.1 Material thermophysical properties
The chemical composition of 4142 steel is listed in Table 1, and the thermophysical properties at different temperatures are summarized in Table 2.
TABLE 1
| C | Mn | P | S | Si | Cr | Mo |
|---|---|---|---|---|---|---|
| 0.390 | 0.900 | 0.010 | 0.002 | 0.310 | 1.070 | 0.220 |
Chemical composition of 4142 steel (wt%).
TABLE 2
| Temperature (°C) | Thermal conductivity k (ω/(m·K)) | Specific heat c (J/(kg·K)) | Elastic modulus E (GPa) | Poisson’s ratio ν |
|---|---|---|---|---|
| 800 | 38.2 | 685 | 165 | 0.3 |
| 900 | 41.5 | 720 | 152 | 0.3 |
| 1000 | 44.8 | 755 | 138 | 0.3 |
| 1100 | 48.1 | 790 | 124 | 0.3 |
| 1150 | 50.3 | 810 | 117 | 0.3 |
Thermophysical properties of 4142 steel.
2.2 Experimental equipment and test scheme
Hot compression experiments were carried out on a Gleeble 3500 thermal simulation testing machine. The experimental parameters covered different deformation temperatures and strain rates, with a total deformation of 60%.
Disk specimens were cut from different regions of the formed vessel for burst tests. Comparative tests were conducted in helium and high-pressure hydrogen environments, respectively. After burst testing, the fracture morphology and microstructural characteristics were observed by scanning electron microscope (SEM) to analyze hydrogen embrittlement behavior.
3 Constitutive model and thermo-mechanical coupled simulation
3.1 Arrhenius-type constitutive model
The hot deformation flow behavior of 4142 steel is described using the Arrhenius-type constitutive equation following Equation 1, expressed as:where is the strain rate, denotes the flow stress; A, n are material constants; Q is the activation energy of hot deformation; R is the universal gas constant; and T is the absolute temperature.
To verify the reliability of the Arrhenius-type constitutive equation, the predicted flow stress values were compared with experimental results obtained from hot compression tests. Flow stress comparison is essential because it directly reflects the accuracy of the constitutive model in reproducing the material’s deformation behavior under elevated temperatures. The quantitative evaluation yielded a mean absolute error of 7.35% and a correlation coefficient of 0.9890, confirming excellent agreement between prediction and experiment.
The prediction accuracy of the constructed constitutive model was quantitatively evaluated via the mean absolute error (MAE) and correlation coefficient. The results show that the model predictions agree well with experimental flow stress data, with an MAE of 7.35% and a correlation coefficient of 0.9890. The comparison between the predicted and experimental flow stress is presented in Figure 1.
FIGURE 1
3.2 Governing equations of thermo-mechanical coupling
3.2.1 Temperature field governing equation
Within the computational domain Ω, the transient heat conduction equation with internal heat generation is established following Equation 2, expressed as:where is the material density, is the specific heat capacity, is the thermal conductivity, is the conversion coefficient of plastic work into heat, with a value of 0.9. is the plastic strain rate, is the friction coefficient, with a value of 0.3, is the contact pressure, and v is relative sliding velocity between contact surfaces.
The convective heat transfer boundary, interface thermal conduction boundary and initial temperature condition follow Equations 3–5.
3.2.2 Stress–strain field governing equation
In the following formulation, the symbols are defined as:
ε = Plastic Strain Increment
ti = prescribed traction.
U = displacement vector.
Based on the incremental plasticity theory, the total strain increment is decomposed into elastic and plastic components as shown in Equation 6.
The elastic strain increment complies with Hooke’s law Equation 7.
While the plastic strain increment follows the associated flow rule Equation 8.
The mechanical boundary conditions are specified by the traction boundary condition in Equation 9 and the displacement boundary condition in Equation 10.
3.3 Mathematical formulation of processing map
Within the framework of the Dynamic Material Model (DMM), the power dissipation efficiency and the flow instability criterion are expressed by Equations 11 and 12, respectively:where m is the strain rate sensitivity exponent. The material stays in a stable processing region when whereas flow instability occurs once .
The theoretical foundation of the DMM was first established by (), who demonstrated that hot deformation behavior can be mapped by partitioning power input into dissipative and stored components, thereby identifying safe processing domains. provided the classical hydrogen-induced decohesion theory (), explaining how hydrogen alters local stress states and promotes embrittlement (). further advanced the understanding of hydrogen-enhanced localized plasticity, linking hydrogen transport and trapping to instability in high-strength steels. More recent work by () highlighted the role of localized deformation in hydrogen-assisted crack propagation, reinforcing the importance of stress distribution in embrittlement behavior. These references collectively support the present analysis of hot-spinning behavior and hydrogen embrittlement resistance in 4142 steel vessels.
3.4 Adaptive finite element and splitting iterative algorithm
3.4.1 Residual-based adaptive meshing
A residual-based a posteriori error estimator that considers both stress- and temperature-field errors is constructed, as given in Equation 13:
Element refinement is implemented via an octree subdivision strategy when: .
The a posteriori error bound of the adaptive algorithm is expressed by Equation 14:
3.4.2 Splitting iterative algorithm with Gauss–Seidel acceleration
To address the strong nonlinearity of the thermo-mechanical coupled system, a splitting iterative algorithm combined with Gauss–Seidel relaxation is adopted. The temperature and stress–strain fields are solved sequentially:
Initialize the temperature and stress fields;
Solve the stress–strain governing equation under the current temperature field according to Equation 15:
Update the heat source vector by accounting for heat generated from plastic deformation and friction, and then solve the temperature field equation according to Equation 16:
Apply Gauss–Seidel relaxation according to Equation 17 to accelerate convergence:
The convergence of the iterative process is assessed using the criterion given in Equation 18:
Iteration terminates when the prescribed convergence tolerance is satisfied. Numerical validation indicates that the proposed algorithm achieves stable convergence. Compared with the conventional fully coupled solution scheme, the computational efficiency is improved by 41.2%.
3.5 Finite element simulation model
The finite element model is discretized by hexahedral elements, with a total element number of 770,868. The Galerkin finite element method is employed for temperature field discretization, and the Updated Lagrangian formulation is adopted for stress–strain calculation to effectively avoid mesh distortion under large deformation. The distributions of equivalent von Mises stress and temperature field under optimal process parameters are illustrated in Figures 2, 3, respectively. Figure 2 illustrates the distribution of equivalent von Mises stress during the hot spinning process under the optimized parameters (1000 °C, strain rate 0.03 s-1, feed rate 2 mm/s). The stress field shows a smooth gradient along the vessel wall, with peak stresses concentrated in the shoulder region but without sharp local maxima. This indicates that the optimized process effectively reduces stress concentration, thereby lowering the risk of hydrogen-assisted cracking in service. The relatively uniform stress contours confirm that the adaptive meshing and iterative solution strategy captured the deformation behavior accurately.
FIGURE 2
FIGURE 3
Figure 3 presents the coupled temperature and stress fields during hot spinning. In Figure 3a, the temperature distribution is uniform across the cylinder zone, with localized increases near the tool–workpiece interface due to frictional heating. In Figure 3b, the corresponding von Mises stress field highlights the interaction between thermal softening and mechanical loading. The overlap of high-temperature regions with moderate stress levels demonstrates that the forming temperature promotes ductility while preventing excessive stress buildup. Together, these results validate the thermo-mechanical coupling model and explain why the optimized parameters yield improved wall-thickness uniformity and enhanced hydrogen embrittlement resistance.
4 Model solution based on weighted summation
4.1 Multi-objective optimization model
Three optimization objectives were adopted: minimum wall-thickness deviation, minimum equivalent stress concentration, and minimum total spinning force. The mathematical formulation of the multi-objective optimization model is given in Equation 19:where: ti—local wall thickness at sampling point i.
t—average wall thickness
σmax—maximum equivalent stress.
Ftotal—total spinning force.
The design variables include forming temperature, strain rate and feed rate.
4.2 Model solution based on weighted summation
The multi-objective optimization problem was converted into a single-objective function using the weighted summation method, as expressed in Equation 20:
Weight coefficients were determined via the Analytic Hierarchy Process (AHP), in which higher weight was assigned to wall thickness uniformity due to its critical importance to vessel safety.
The optimal hot spinning parameters were finally obtained as follows: forming temperature of 1000 °C, strain rate of 0.03 s-1, and feed rate of 2 mm/s.
4.3 Parameter sensitivity analysis
The control variable method was applied to investigate the parameter sensitivity to optimization objectives. The spinning temperature exhibits the most prominent influence on wall thickness uniformity, with a sensitivity coefficient of 0.72. Meanwhile, the feed rate dominates the variation of total spinning force, with a sensitivity coefficient of 0.65. The comprehensive results under different parameter combinations are summarized in Table 3.
TABLE 3
| Parameter combination | Wall thickness error | Peak equivalent stress (MPa) | Total spinning force (kN) | Composite objective |
|---|---|---|---|---|
| Case 1 | 0.12 | 475 | 680 | 0.87 |
| Case 2 | 0.08 | 360 | 520 | 0.62 |
| Case 3 | 0.05 | 310 | 490 | 0.41 |
| Case 4 | 0.07 | 350 | 560 | 0.58 |
| Case 5 | 0.09 | 280 | 450 | 0.53 |
Sensitivity analysis results under different parameter combinations.
5 Industrial forming test and simulation verification
Industrial hot spinning experiments were carried out using the optimized process parameters. The wall thickness was measured at 36 sampling points along the vessel axial direction. The average wall thickness obtained from numerical simulation is 59.1 mm, while the experimental result is 59.2 mm, and the deviation is within the allowable engineering tolerance. Good consistency is achieved between the simulated and measured wall thickness distributions.
Residual stress tests show that the maximum residual stress in the vessel shoulder region is 285 MPa, without obvious local stress concentration. Under ultra-high-pressure cyclic loading of 103 MPa, the prototype vessel maintains structural integrity without crack initiation or leakage, satisfying the design and service safety requirements. The comparison of simulated deformation morphology and actual forming profile is presented in Figure 4.
FIGURE 4
6 Hydrogen embrittlement fracture morphology and microscopic mechanism
6.1 Macroscopic fracture characteristics under different atmospheres
Under helium atmosphere, the fracture surface presents obvious plastic deformation, irregular crack propagation edges and abundant dimpled structures, showing typical ductile fracture and high energy dissipation capability. In contrast, the fracture surface becomes smooth and flat under high-pressure hydrogen atmosphere, accompanied by dense secondary cracks. Typical quasi-cleavage characteristics are observed, indicating suppressed plastic deformation and obvious hydrogen-induced embrittlement. The macroscopic fracture features under helium and hydrogen environments are shown in Figure 5.
FIGURE 5
6.2 Microscopic fracture morphology of typical regions
6.2.1 Cylinder zone
In the helium environment, the fracture surface is rough with uniformly distributed dimples, reflecting excellent ductility and fracture toughness. In the hydrogen environment, the fracture morphology becomes flat with fragmented microstructures, demonstrating an evident transition from ductile to brittle fracture mode. The SEM fractographs of the cylinder zone are presented in Figure 6.
FIGURE 6
6.2.2 Transition zone
Specimens tested in helium show irregular fracture surfaces with typical dimpled features and high intrinsic toughness.
By comparison, hydrogen-exposed specimens possess relatively smooth fracture planes with continuous microcracks, dominated by quasi-cleavage fracture behavior. The corresponding SEM morphologies of the transition zone are illustrated in Figure 7.
FIGURE 7
6.2.3 Shoulder zone
The shoulder zone displays the smoothest fracture plane among all regions, accompanied by dense fine microcracks. This region suffers the most severe hydrogen embrittlement, with significantly suppressed plastic deformation capacity and degraded fracture resistance. The SEM fracture images of the shoulder zone are given in Figure 8.
FIGURE 8
6.3 Regional difference mechanism of hydrogen embrittlement
The hydrogen embrittlement resistance of different vessel regions follows the order: cylinder zone > transition zone > shoulder zone. Such regional discrepancy is closely associated with the residual stress distribution and microstructure inhomogeneity induced by the hot spinning process. Reasonable optimization of process parameters can effectively homogenize the internal stress field and restrain hydrogen-induced crack initiation, thereby improving the long-term service safety of ultra-high-pressure hydrogen storage vessels.
The regional differences in embrittlement resistance are directly linked to the interplay between residual stress distribution and microstructural features:
High stress zones (shoulder) → increased hydrogen trapping → brittle fracture.
Uniform stress zones (cylinder) → reduced hydrogen uptake → ductile fracture.
Transitional stress states (transition zone) → mixed fracture modes.
This analysis confirms that process optimization not only homogenizes stress fields but also stabilizes microstructural response, thereby enhancing long-term vessel reliability.
7 Conclusion
This study combined hot compression experiments, thermo-mechanical simulation and hydrogen embrittlement evaluation to investigate the hot spinning process of 4142 steel hydrogen storage vessels. The main conclusions are summarized as follows.
Hot compression data were successfully represented by an Arrhenius-type constitutive model. The model reproduced the flow stress behavior of 4142 steel over the investigated deformation conditions and provided material parameters for subsequent numerical analysis.
A thermo-mechanically coupled finite element framework incorporating adaptive mesh refinement and an iterative solution strategy was developed for hot spinning simulation. The proposed approach reduced computational effort while maintaining agreement between numerical predictions and experimental measurements.
Process optimization indicated that a forming temperature of 1000 °C, a strain rate of 0.03 s-1 and a feed rate of 2 mm/s resulted in improved wall-thickness uniformity and a more homogeneous stress distribution. Industrial-scale trials confirmed the reliability of the optimized process window.
Hydrogen embrittlement susceptibility varied significantly across different regions of the vessel. Fractographic observations showed that the shoulder region exhibited the strongest tendency toward hydrogen-assisted cracking, whereas the cylinder region displayed comparatively higher resistance.
The regional variation in embrittlement behavior was found to be closely associated with the residual stress distribution generated during hot spinning. The results suggest that controlling stress concentration during forming is an important factor in improving the structural integrity and long-term reliability of ultra-high-pressure hydrogen storage vessels operating in hydrogen environments
Statements
Data availability statement
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.
Author contributions
YZ: Investigation, Data Curation, Writing – original Draft, Writing – review and editing, Visualization. JL: Methodology, Formal Analysis, Writing – review and editing. JX: Writing – review and editing, Supervision.
Funding
The author(s) declared that financial support was not received for this work and/or its publication.
Acknowledgments
The authors would like to thank Huaxin College of Hebei GEO University.
Conflict of interest
The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Generative AI statement
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Summary
Keywords
4142 steel, hot spinning, hydrogen-assisted fracture, process optimization, thermo-mechanical coupling
Citation
Zhang Y, Li J and Xue J (2026) Mathematical modeling of the hot spinning process to improve hydrogen embrittlement resistance of 4142 steel hydrogen storage vessels. Front. Mater. 13:1899121. doi: 10.3389/fmats.2026.1899121
Received
03 June 2026
Revised
30 June 2026
Accepted
02 July 2026
Published
30 July 2026
Volume
13 - 2026
Edited by
Salvatore Verre, University of eCampus, Italy
Reviewed by
Artur Wodołażski, Central Mining Institute, Poland
Narendra Varma Dantuluri, Raghu Engineering College, India
Zhishan Mi, China Iron and Steel Research Institute, China
Updates
Copyright
© 2026 Zhang, Li and Xue.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Jiao Xue, 604513118@qq.com
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.