© Y. Liu, Published by EDP Sciences 2026

1 Introduction

As an indispensable supporting component in underground engineering, hydraulic supports play a key structural bearing function in coal mining, tunnel construction, and other fields. In this process, they need to continuously withstand the combined effects of multiple environmental factors such as mechanical alternating loads, temperature cycle fluctuations, and corrosive media erosion [1–3]. As the cycle lengthens, the support material is prone to cumulative fatigue damage under complex working conditions, which in turn causes structural instability or even fracture failure, directly threatening the safe operation and economy of underground engineering. Although the current manufacturing process and structural design are constantly optimized, there are still many technical bottlenecks in the study of fatigue damage mechanism due to the interaction mechanism of extreme working conditions underground [4–6]. Existing studies mostly focus on the analysis of the influence of a single factor, lack a systematic understanding of the damage evolution law under the coupling of non-uniform stress field and temperature gradient, and especially fail to fully reveal the synergistic mechanism of multi-physics coupling effect on fatigue crack initiation and propagation [7,8]. This has become the main obstacle to improving the hydraulic support life prediction accuracy.

This study innovatively establishes a multi-physics coupling analysis framework and uses COMSOL Multiphysics software to achieve full coupling modeling of the thermal field, stress field, and corrosion field. First, the geometric model of the bracket is reconstructed based on the CT scanning data, and the boundary conditions are set in combination with the measured working conditions parameters in the well. The flow characteristics of the hydraulic medium are simulated by solving the Navier-Stokes equations, and then, the heat transfer equation and the elastoplastic constitutive equation are coupled to realize the dynamic interaction simulation of the temperature gradient and the mechanical load. The concentration diffusion equation is applied into the corrosion field modeling to quantitatively characterize the migration law of the electrolyte solution in the microcracks, and finally, the fatigue life prediction under the coupling of multiple factors is realized through the damage mechanics model. This modeling method effectively overcomes the limitations of traditional single-field analysis and provides a new research paradigm for revealing the damage evolution mechanism under complex working conditions. Then, the thermal-structural coupling analysis module is used, combined with the corrosion model, to evaluate the influence of temperature change and environmental factors on the fatigue damage of the support. To further analyze the evolution process of fatigue damage, this paper uses the Morrow fatigue model to calculate the damage accumulation during the loading cycle and combines the Paris law to simulate the crack propagation process. Finally, a numerical simulation is performed based on ABAQUS software. The stress distribution of the hydraulic support is calculated through static and dynamic load analysis combined with cyclic loading, and the stress concentration area and the location of fatigue damage are identified. Response surface methodology is used to optimize model parameters by comparing with experimental data, and Monte Carlo simulation is used for uncertainty analysis to further verify the model’s reliability. This study not only provides a theoretical basis for the fatigue damage mechanism of hydraulic supports, but also provides strong support for the optimization of support design in engineering practice.

The motivation of this study stems from the key role of hydraulic supports in underground engineering, especially in coal mining and tunnel construction. The supports are subjected to the influence of multiple environmental factors such as mechanical loads, temperature fluctuations and corrosive media for a long time, resulting in the accumulation of fatigue damage, which may eventually threaten the safety and economy of the project. With the complexity of the engineering environment, existing research mainly focuses on the analysis of a single factor, lacking an in-depth understanding of the evolution of fatigue damage under the coupling of multiple physical fields. Therefore, this study aims to establish a multi-physical field coupling analysis framework to predict the fatigue damage process of hydraulic supports by simulating the interaction of temperature field, stress field and corrosion field. The quantifiable objectives of the study include accurately simulating the fatigue damage evolution of hydraulic supports and ensuring the reliability and accuracy of the model through numerical simulation and experimental verification. The novelty lies in combining the Morrow fatigue model with the Paris law and revealing the synergistic effect under complex working conditions through multi-physical field coupling analysis. In addition, uncertainty analysis and optimization design are carried out through Monte Carlo simulation and response surface method to further improve the fatigue life prediction accuracy of hydraulic supports. This study provides a theoretical basis for the optimization design and engineering application of hydraulic supports, which has important engineering practical significance.

2 Related work

In the field of fatigue damage mechanism research, the academic community has achieved some basic results. The Valizadeh P team [9] built a cyclic load test platform and combined it with finite element simulation to reveal the quantitative effect of stress amplitude changes on material micro-damage and established a correlation model between residual strength attenuation and natural frequency shift. At the same time, in the field of thermal-mechanical coupling research, scholars found that the non-uniform distribution of the temperature field significantly changed the dislocation motion characteristics of the material, and used thermal-structural coupling simulation technology to confirm the acceleration effect of high temperature gradient on fatigue crack growth rate. Moulgada et al. [10] improved the fatigue performance of cracked plates through composite patch repair technology, while Mu M et al. [11] analyzed the evolution law of hydraulic support posture based on the change of column length. Both of them started from the perspective of structural stress evolution and fatigue behavior, providing theoretical support for the study of the fatigue damage mechanism of complex structures. Hu X et al. [12] focused on the non-ideal posture problem of hydraulic support in the initial support stage under soft rock conditions and revealed the formation mechanism of posture deviation caused by factors such as bottom plate deformation and uneven column force, providing a theoretical basis for the optimization of hydraulic support posture under complex geological conditions. Hao et al. [13] constructed a digital twin model of the position and posture of the hydraulic support in the comprehensive mining working face, realizing the accurate perception and dynamic evolution tracking of the support operation status and providing data basis and model support for the posture monitoring and intelligent decision-making of the hydraulic support under complex working conditions. However, existing studies mostly adopt single-factor control methods, which fail to effectively integrate the synergistic mechanism of mechanical load, temperature cycle, and corrosive medium. Since these environmental factors have obvious spatiotemporal coupling characteristics in actual working conditions, simple decoupling analysis leads to deviations between theoretical models and engineering practice [14], making the applicability of existing fatigue life prediction methods under complex working conditions questionable.

To break through the limitations of traditional research methods, some cutting-edge research has begun to try multi-physics coupling analysis technology. Based on the thermal-electromechanical coupling model constructed on the finite element platform [15–17], the evolution law of stress concentration at the key nodes of the support has been successfully captured by synchronously loading the temperature field and mechanical stress field. It is worth noting that COMSOL Multiphysics software, with the advantage of a multi-physics collaborative solution, has shown unique value in simulating the thermal-mechanical-corrosion coupling effect [18–20], providing a new way to reveal the mechanism of multi-factor interaction. However, existing research still has shortcomings in the coupling modeling of environmental factors. Specifically, the temperature of the coal mine working face is usually maintained between 10 °C and 30 °C. The ventilation conditions are good, and the temperature changes are small. Therefore, the temperature field has a limited effect on the fatigue damage of the hydraulic support. In addition, the working environment of the hydraulic support is mainly the coal mine working face [21,22]. The top beam of the support is in contact with the roof rock, and the base is in contact with the bottom rock, providing a safe space for mine workers and equipment. The support is usually not directly in contact with the corrosive medium, and the corrosion effect is relatively weak. Nevertheless, the external loads that hydraulic supports bear in actual work are relatively complex, especially the roof environment has a great influence on the stress state of the support, so the influence of these complex loads on the stress state of the support needs to be further considered in multi-field coupling analysis [23,24]. To overcome these technical gaps, this study constructs a multi-physics field coupling mechanical model, focusing on enhancing the coupling correlation between environmental variables to improve the simulation accuracy of the fatigue damage evolution process.

3 Materials and methods

3.1 Multi-physics coupling analysis

The fatigue damage process of hydraulic supports is jointly affected by multiple physical field factors. Therefore, this study adopts a multi-physics coupling analysis method to simulate the working state of the support under complex working conditions and analyze the influence of the interaction between different physical fields on fatigue damage. To accurately simulate the working environment of the hydraulic support, it is first necessary to establish its geometric model and consider the influence of actual working conditions on the structure. Then, using COMSOL Multiphysics software, the support is comprehensively simulated through multi-physics finite element analysis, coupling mechanical fields, thermal fields, corrosion fields, etc., to simulate the behavior of the support in the actual working environment.

Establishing the geometric model of the hydraulic support is the first step in the multi-physics coupling analysis. First, according to the actual structure of the hydraulic support (as shown in Fig. 1), a three-dimensional model is established through the geometric modeling module of COMSOL Multiphysics. The geometric model includes the main components of the support, such as columns, arms, slide rails, etc., to ensure that the size and shape are consistent with the actual structure. The selection of material properties is based on the alloy material actually used in the support. The main body of the hydraulic support is divided into structural parts and hydraulic cylinders [25,26]. The structural parts are welded from plates. The plates of the main areas, such as the main reinforcement and cover plates on the ZY14790/15/25D support structural parts, are Q690, and other auxiliary plates are mainly Q550; the cylinder is mainly 30CrMnSi; the main large pin is made of 35CrMnSiA, and there are 30CrMnTi and 40Cr hydraulic supports. The constructed geometric model is shown in Figure 2.

After the geometric model is established and the material properties are defined, the next step is to conduct multi-physics coupling analysis. First, the stress and deformation of the hydraulic support under different temperature conditions are analyzed through the thermal-structural coupling module. Thermal field analysis includes the calculation of the temperature distribution on the support surface. Considering that the hydraulic support is often affected by the temperature changes of the external environment, the temperature of the working medium, and the underground working environment, the temperature change has a significant impact on the mechanical properties of the support. Therefore, in the simulation, the temperature field is applied to the structural model as an external load, and the deformation and stress distribution caused by thermal expansion or thermal stress are calculated [27,28]. During the simulation process, a typical temperature load cycle with a temperature change range of 10 ℃ ∼ 50 ℃ is used to simulate the thermal stress and thermal deformation distribution that may occur during the bracket operation. Regarding the temperature field range setting, the selection of the extreme value of 50 ℃ in this article is based on the simulation of the short-term overload condition of the hydraulic system: when the circulation of the hydraulic medium is blocked, or the underground ventilation system fails, the local support components may reach this temperature threshold due to frictional heat generation or environmental heat conduction. According to the thermal effect test specification, although the probability of occurrence of such conditions is less than 5%, it needs to be included in the conservative design category of fatigue life prediction. Therefore, the temperature cycle range is set to 10∼50 ℃ to cover extreme scenarios.

Figure 3 shows the thermal stress and thermal deformation distribution of hydraulic supports at different temperatures. The thermal stress distribution reflects the thermal stress caused by temperature changes. The maximum stress is related to the thermal expansion coefficient of steel, indicating that significant thermal stress may be generated under extreme temperature conditions. The thermal deformation distribution shows that the maximum deformation caused by thermal stress is about 0.95 × 10⁻³ mm. Although small, long-term accumulation may cause material fatigue and affect the stability of the support. These results show that temperature change is a key factor in fatigue damage of hydraulic supports.

In addition, the coupling of the corrosion field is also something that needs to be considered in the model. In actual use, hydraulic supports may be corroded by corrosive media. In humid or underground environments, slight corrosion reduces the fatigue resistance of the support. The effect of corrosion on the support structure is modeled as a process of gradual material reduction. The corrosion model in COMSOL Multiphysics simulates the corrosion effects on the surface and interior of the support. The model uses the corrosion rate (k_corrosion) to characterize the gradual effect of corrosion on material properties. The value of the corrosion rate is determined based on experimental data and is usually between 0.1 and 0.5 mm/yr. The specific value depends on the underground environmental conditions where the support is located [29]. To enhance the physical rationality of the corrosion rate parameters, this paper supplements the correlation analysis between the underground environmental humidity, pH value, and corrosion rate. Through field sampling experiments, the average humidity of the working face is measured to be 85% ± 5%; the pH value range is 6.2∼7.8; the corresponding corrosion rate correction factor is adjusted to 0.3 mm/yr. This result shows that under actual working conditions, the contribution rate of corrosion rate to material thinning is lower than that of high-temperature oxidation effect, but the risk of stress corrosion cracking caused by local microcracks still needs to be considered in long-term service. In view of the corrosion-fatigue synergistic effect, this paper introduces the stress corrosion cracking constitutive equation, as shown in Formula (1):

$\frac{da}{dt}=k\cdot {\left(\Delta K\right)}^n\cdot \text{exp}\left(-\frac{E_a}{RT}\right)\cdot {\left[{\text{Cl}}^{-}\right]}^m$(1)

Where: da/dr: crack growth rate (unit: nm/s), ΔK: stress intensity factor range (unit: MPa·m⁰·⁵), E_α: activation energy (unit: kJ/mol), R: gas constant (8.314 J/mol·K), T: absolute temperature (unit: K), [Cl⁻] chloride ion concentration (unit: ppm), k,n,m:material constant (dimensionless). Based on groundwater quality monitoring data, the Cl⁻ concentration was set to 200 ppm. The model was implemented using the COMSOL User Defined Equations module. Simulation results show that SCC increases the crack growth rate. Due to the lack of hydrogen flux data in the underground environment, hydrogen embrittlement was not considered, but it is recommended as a future research direction.

The model is compatible with engineering protection measures by adjusting the corrosion rate parameters: epoxy coating can reduce the corrosion rate (equivalent rate 0.04 mm/yr), and the corrosion rate can be reduced to 0.02 mm/yr when the cathodic protection potential is set to −850 mV. This mechanism provides a tool basis for the quantitative evaluation of anti-corrosion strategies.

After completing the definition of thermal field, mechanical field, and corrosion field, the next step is to apply loads and perform multi-physics coupling analysis. The loads on the hydraulic support during operation include static loads, dynamic loads, and periodic loads. Therefore, in the analysis process, the influence of static and dynamic loads needs to be considered. Static loads simulate the force of the support under its own weight and static working state, while dynamic loads simulate the force of the hydraulic system on the support during operation. Periodic loads simulate the fatigue damage process of the hydraulic support under periodic loading. The dynamic load spectrum is constructed based on underground monitoring data (the periodic pressure frequency of the coal mining machine is 0.5∼2 Hz, and the amplitude fluctuation coefficient is 0.15), and the block spectrum is extracted by the rain flow counting method. The specific parameter reference standard is “MT/T 817-2020 Hydraulic Support Dynamic Load Test Method”.

In the mechanical field analysis, the material of the hydraulic support produces stress, strain, and displacement under the action of external loads. The stress distribution and deformation under different working conditions are calculated by the structural mechanics module of COMSOL Multiphysics. All physical fields (mechanical field, thermal field, and corrosion field) are calculated by coupling solvers to ensure the influence of multi-factor interaction on stress and strain results. Combined with the Morrow fatigue model and the Paris law, the fatigue damage development process of the support during long-term use is predicted. Through multi-physics coupling analysis, the stress, deformation, and temperature distribution of the hydraulic support is obtained, focusing on the areas of stress concentration and large deformation, thereby providing a theoretical basis and support for the fatigue life prediction and optimal design of the support.

Fig. 1 Actual structure of hydraulic support.

Fig. 2 Geometric model of hydraulic support.

Fig. 3 Distribution of thermal stress and thermal deformation under different temperature conditions.

3.2 Construction of fatigue damage model

Hydraulic supports are often subjected to cyclic loads in actual work, and their fatigue damage process is determined by multiple factors, including the stress amplitude of the loading cycle, the fatigue properties of the material, and environmental conditions. To accurately predict the fatigue life of the support and evaluate its fatigue damage, this paper adopts the Morrow fatigue model combined with the Paris law to describe the damage accumulation and crack propagation of the material during repeated loading.

The Morrow model is based on the stress-strain relationship and is mainly used to describe the damage accumulation of the material under repeated loads. The key parameters of this model are the fatigue limit, stress amplitude, and stress ratio (that is, the ratio of maximum stress to minimum stress) of the material. In the fatigue damage analysis of hydraulic supports, the fatigue limit of the support material is first determined, and this parameter is usually obtained through experiments. In practical applications, the fatigue limit of commonly used materials such as 20CrMo or 40Cr steel is approximately 50% to 60% of the material yield strength. According to the Morrow model, fatigue damage accumulation can be expressed by the stress amplitude (Δσ). The Morrow model predicts damage through Formula (2):

$D={\left(\frac{\Delta \sigma }{{\sigma }_f}\right)}^b.$(2)

In formula (2), Δσ is the stress amplitude, unit is MPa; σ_f is the material fatigue limit, unit is MPa; b is an empirical constant, dimensionless.

For the calibration process of fatigue limit σ_f and constant b, this paper uses ASTM E466 standard to conduct 10 groups of axial loading fatigue tests on Q690 steel (number of specimens n=30, stress ratio R=0.1), and the mean value of σ_f is 450 MPa (standard deviation σ=12 MPa) obtained by Weibull distribution fitting. Constant b is determined by linear regression of the high-cycle fatigue section (10⁴∼10⁶ cycles) in the SN curve by the least squares method, and the confidence interval is b=−0.085 ± 0.003. The pressure ratio was calculated as the ratio of the minimum to maximum stress during the load cycle. During the operation of a hydraulic support, the stress state changes due to the varying loads experienced during the lowering, supporting, and raising phases, resulting in varying stress ratios. In the fatigue tests conducted using the ASTM E466 standard, a stress ratio of 0.1 was used, indicating that the minimum stress is 10% of the maximum stress.

In the fatigue damage analysis of hydraulic supports, the extension of microcracks is one of the key factors affecting the life of the support. As a classic theory describing the crack extension process, the Paris law simulates the extension rate of microcracks under repeated loading. The Paris law describes the relationship between the crack extension rate (da/dN) and the stress intensity factor range (ΔK) through Formula (3):

$\frac{\text{da}}{\text{dN}}\text{=C}{\left(\text{ΔK}\right)}^{\text{m}}.$(3)

Among them, C and m are material constants. The values of C and m are usually fitted by experimental data. The stress intensity factor ΔK of the crack is usually obtained by finite element analysis, taking into account the geometry, loading state, and material stress field of the support.In formula (3), da/dN is the crack growth rate (m/cycle); ΔK is the stress intensity factor range (MPa·m^1/2); C is a material constant (m^(1−m)/cycle); and m is a dimensionless material exponent.

To verify the applicability of Paris's law under variable amplitude load, this paper introduces the effective stress intensity factor ΔK_eff with reference to the Elber correction criterion and dynamically adjusts the constants C and m in combination with the semi-empirical formula for the influence of R ratio. For example, when R increases from 0.1 to 0.5, the crack growth rate constant C decreases by 12%∼15%, and the change rate of the exponent m is less than 3%. This correction method has been implemented in the ABAQUS crack growth module to ensure the model’s adaptability to non-proportional loading conditions.

When using the Paris law to simulate crack propagation, the crack propagation rate of each loading cycle is calculated to predict crack growth and its effect on fatigue life. By repeatedly calculating the crack propagation under different loading conditions, the crack propagation risk and fatigue damage of the bracket are evaluated.

Combining the Morrow fatigue model with the Paris law, this paper cumulatively calculates the fatigue damage of the hydraulic support in each loading cycle and simulates the crack propagation process [30,31]. First, the Morrow model is used to calculate the fatigue damage accumulation under each loading cycle, and the cumulative damage value of the support under repeated loading is obtained. As the loading cycle increases, the damage value gradually increases until it exceeds the fatigue limit of the material, and the support fails.

The crack propagation simulation uses the Paris law to calculate the crack propagation rate and evaluate the impact of crack propagation. Based on the crack conditions during the loading cycle and the fracture toughness data of the material, the possibility of the support being destroyed when the crack propagates to a critical size is calculated [32–34]. This process helps identify the fatigue crack areas that may occur in the support under long-term working conditions and provides a basis for subsequent life prediction and optimized design.

In actual engineering applications, fatigue damage of hydraulic supports is the result of the combined effect of multiple factors. By combining the Morrow fatigue model with the Paris law, this paper analyzes the fatigue damage process of hydraulic supports more comprehensively and accurately, especially the comprehensive evaluation of damage accumulation and crack propagation under complex working conditions.

To ensure the accuracy of the fatigue damage model, this paper optimizes the model parameters according to the specific application environment of the hydraulic support. The fatigue limit σ_f, stress amplitude Δσ, and constant b in the Morrow model are obtained by fitting experimental data. The C and m constants in the Paris law are determined by material test data.

3.3 Finite element numerical simulation

This study uses ABAQUS finite element software to perform numerical simulation of hydraulic supports based on a multi-physics coupling model. Through stress-strain analysis under static and dynamic loads, combined with fatigue loading cycles, the fatigue behavior and potential damage areas of the support are comprehensively evaluated. First, static analysis is performed using the ABAQUS/Standard module, and the working load and constraints are input. Each part’s stress and strain distribution is calculated, and the influence of geometric characteristics, material nonlinearity, and working environment is considered to ensure the accuracy of the calculation results.

To ensure the reliability of cross-platform simulation between COMSOL and ABAQUS, this paper adopts the following data conversion strategy: first, the COMSOL thermal-structural coupling results (node displacement, temperature field distribution) are exported to VTK format; second, the VTK data is mapped to the reference node set of the ABAQUS model through a Python script, and the radial basis function interpolation method is used to ensure mesh consistency; finally, when applying the mapped boundary conditions in ABAQUS, the penalty function method is used to deal with contact nonlinear problems.

The core purpose of static analysis is to identify the stress concentration areas and maximum stress points of the support under steady-state loads, which are usually the starting points of fatigue damage and crack propagation. By obtaining the stress distribution of the support under static conditions, basic data is provided for subsequent fatigue analysis.

Figure 4 shows the static stress distribution of the hydraulic support. In the dynamic analysis stage, the ABAQUS/Explicit module is used to simulate the dynamic response of the hydraulic support in a more complex way. The dynamic behavior of the hydraulic support under different loads is simulated by applying instantaneous external loads or periodic loads. This stage mainly studies the deformation response, vibration mode, inertia effect, etc., of the support under dynamic loads, especially the structural response under complex working conditions.

Table 1 compares the simulated and measured stress data of the key areas of the hydraulic support. The simulated stress at the weld is 583 MPa (coordinates 1.20, 0.80, 2.30), and the measured stress is 562 ± 15 MPa, with a deviation of +3.7%. The stress concentration factor is 2.85, which is consistent with the statistical law that the failure rate of underground welds is 62%. The maximum deviation of the bolt hole is −3.6% (connecting rod pin hole), which is mainly due to the difference in stress gradient caused by manufacturing tolerance. The deviations of all positions are <4%, verifying the model’s reliability. The SD in the “Measurement Mean ± SD” column in Table 1 stands for standard deviation and represents the degree of dispersion of the multiple measurement data at five locations on the hydraulic support. The data in the “Deviation (%)” column represents the measurement error calculated using the formula: Deviation (%) = [(Simulated Value - Measured Mean) / Measured Mean] × 100%, reflecting the relative error between the numerical simulation results and the experimental measurements. The deviations of all key areas are controlled within ±3.7%, indicating that the model has high prediction accuracy and reliability and can accurately reflect the stress distribution characteristics of the hydraulic support under actual working conditions.

After static and dynamic analysis, fatigue loading cycles are simulated. Fatigue loading cycles simulate the cyclic loading effects that the hydraulic support is subjected to during long-term use.

During the fatigue loading simulation, the dynamic evolution of stress parameters is tracked in real-time by cycle loading. The mechanical response law of the bracket structure under cyclic loading is systematically evaluated by monitoring the stress peak, amplitude, and frequency distribution characteristics. Based on the stress-life curve and the principle of crack growth dynamics, a stress field evolution model under multi-cycle loading is established to precisely calibrate the initial initiation position of fatigue damage [35]. Using the cloud map visualization technology of the finite element post-processing module, the stress concentration phenomenon in the geometric mutation area (such as weld fusion line, bolt hole edge, etc.) can be clearly identified [36]. These parts often become weak links for crack nucleation.

To quantitatively characterize the fatigue damage evolution process, this study integrates the Morrow energy method for damage accumulation calculation and uses the Paris law to describe the power function relationship between the crack growth rate and the stress intensity factor amplitude. Given the uncertainty of model parameters, the response surface method and Monte Carlo random sampling technology are applied to construct a multivariate sensitivity analysis framework. By optimizing the material constitutive parameters, load spectrum characteristics, and geometric constraints, the matching degree between the numerical model and the actual working conditions is significantly improved.

Finally, the key mechanical indicators are output through the finite element solver, including the global stress distribution thermodynamic diagram of the structure, the high-cycle fatigue damage cumulative distribution matrix, and the crack propagation trajectory prediction diagram. These quantitative analysis results provide key input parameters for the life prediction model and provide data support for designers to optimize the support topology and improve the transition curvature of the stress concentration area. It is particularly worth noting that the contact-friction coupling model established based on the ABAQUS explicit dynamics module successfully reproduces the nonlinear response characteristics of the hydraulic support under the action of alternating loads. The simulation results are in good consistency with the downhole monitoring data, providing a reliable theoretical basis for the formulation of preventive maintenance strategies.

Fig. 4 Static stress distribution.

3.4 Model optimization and modification

In the study of the fatigue damage mechanism of hydraulic supports, a comparative analysis between numerical simulation results and experimental data is an important step to verify and optimize model accuracy. By comparing simulation results with experimental observation data, possible errors and inaccuracies in the model can be identified, thus providing direction for model modification and optimization. This paper compares the numerical simulation with the experimental results and proposes an optimization strategy, especially in terms of optimizing and modifying the material constitutive model, stress-strain relationship, and crack propagation parameters to improve the simulation accuracy and model reliability.

To achieve precise optimization of the model, this paper uses the response surface method (RSM) to systematically optimize the model’s key parameters to establish the mapping relationship between model parameters and experimental results in fatigue damage simulation of hydraulic supports [37,38].

To evaluate the influence of high-order terms and interaction terms, this paper adds cubic terms and cross-term sensitivity analysis based on quadratic polynomial regression. The results show that except for the interaction term between stress amplitude Δσ and crack growth rate C (contribution rate 12.3%), the contribution rate of other high-order terms to the total damage degree is less than 5%. Therefore, the original quadratic model has met the engineering accuracy requirements, and the Δσ-C coupling effect is specially optimized in subsequent studies.

First, based on the existing experimental data and simulation results, the main parameters affecting fatigue damage prediction in the model, such as fatigue limit, stress amplitude, crack propagation rate, etc., are selected as input variables [39]. Through experimental design, a response surface model is constructed; multiple simulation calculations are performed; the relationship between each parameter and the response is obtained through regression analysis. The core goal of response surface methodology is to adjust various model parameters and optimize stress distribution and crack propagation path by minimizing simulation errors, thereby improving simulation accuracy. Table 2 shows the key parameters selected in the experiment and their value ranges in the designed experiment. These parameters are used to establish the response surface model and perform regression analysis.

In this process, the optimized parameters include key indicators such as the material’s elastic modulus, yield strength, and fatigue life coefficient, while considering the nonlinear behavior in the stress-strain curve and the influence of environmental factors on fatigue damage. Through this optimization process, the accuracy of numerical simulation is significantly improved, ensuring a high degree of consistency between the simulation results and the experimental data. The response surface model approximates the relationship between input variables and output responses through polynomial regression, usually using a quadratic polynomial expression, as shown in Formula (4):

$Y={\beta }_0+{\displaystyle \sum }_{i=1}^k{\beta }_i{X}_i+{\displaystyle \sum }_{i=1}^k{\displaystyle \sum }_{j=i}^k{\beta }_{ij}{X}_i{X}_j+ϵ$(4)

Where: Y is the response variable (such as fatigue damage value, crack growth rate, etc.). X_i is the input variable (such as fatigue limit of material, stress amplitude, etc.). β₀ is the constant term; β_i is the regression coefficient of the linear term; is the β_ij regression coefficient of the quadratic term ε. is the error term, which represents the part that the model cannot explain. In the optimization process, the model parameters are optimized by minimizing the objective function so that the simulation results are as close to the experimental data as possible. In formula (4), y is the response variable (such as fatigue damage value), dimensionless; X_i is the input variable, where σ_f is in MPa, Δσ is in MPa, C is in m/cycle, m is dimensionless, E is in GPa, and R is dimensionless. The objective function is usually expressed as Formula (5):

$f\left(X\right)=\frac{1}{N}{\displaystyle \sum }_{i=1}^N{\left(Y_{\text{sim},i}-Y_{\text{exp},i}\right)}^2.$(5)

Among them: Y_sim,i is the response value calculated by the i th simulation (such as fatigue damage simulation results). Y_exp,i is the response value measured by the i th experiment. N is the number of samples for the experiment or simulation.

To improve the model’s reliability, this study uses Monte Carlo simulation to perform uncertainty analysis on numerical simulation. This method simulates different values of random variables to obtain the statistical distribution of the results and evaluate the model’s credibility and stability. The uncertainty of key input parameters is determined during the implementation process, and their statistical characteristics are set. Through a large number of simulation runs, the statistical distribution of results is analyzed; the output response under different parameter combinations is analyzed; the model’s sensitivity to variable changes is evaluated. Monte Carlo simulation helps identify key parameters, improve model stability and accuracy, and provide a reliable basis for engineering design and decision-making.

The left figure of Figure 5 reveals the probability distribution characteristics of the tensile strength of the material. The normal distribution model is used to parameterize the material strength, with the mean set to 1.0 MPa and the standard deviation controlled at 0.5 MPa. This distribution characteristic directly affects the discreteness of the strength parameters of the hydraulic support, which in turn leads to statistical volatility in the fatigue damage prediction results. The right figure intuitively presents the confidence interval of the damage value through a box plot, and quantitatively characterizes the coupling effect of uncertainty factors such as material performance variation and load spectrum fluctuation on fatigue damage accumulation. Based on the field sampling data (n=50 groups of Q690 steel plate tensile tests), the material tensile strength distribution also conforms to the three-parameter Weibull model (shape parameter β=2.8, scale parameter η=1.1 MPa, threshold γ=0.7 MPa). Combined with the Weibull distribution, the lower limit of the confidence interval of the damage value is reduced, and the upper limit is increased.

In terms of crack propagation dynamics modeling, this study parametrically corrects the classical Paris law based on experimental observation data. The sensitivity analysis is focused on the crack propagation rate constant C and the exponent m. By fitting the experimental curves of crack propagation rate (da/dN) under different stress intensity factor amplitudes (ΔK), the correction parameter combination suitable for the structural characteristics of the hydraulic support is optimized.

In the process of constructing the material constitutive model, the influence mechanism of multi-field coupling effect on the mechanical properties of the material is systematically considered. By applying the temperature-related fatigue limit attenuation function, the stress amplitude-controlled cyclic hardening model, and the surface energy correction term under a corrosive environment, a constitutive equation system driven by composite working conditions is established.

In addition, to deal with the joint influence of various working environment factors (such as temperature, corrosion, load, etc.), this paper further applies a multi-field coupling model to ensure the model’s adaptability under various complex working conditions. As show in Table 3.

Fig. 5 Distribution of tensile strength and confidence interval of damage value.

4 Results

4.1 Fatigue life prediction

In the fatigue damage evaluation process of the hydraulic support, the Miner fatigue cumulative damage method is first used for life prediction. According to the working environment of the hydraulic support and the selected fatigue damage model, the historical data is loaded, and the damage degree of the support is calculated cycle by cycle. The damage degree of each loading cycle is determined by the stress amplitude and the fatigue characteristics of the material. The total damage of the hydraulic support under different loads is calculated by the cumulative formula. The Miner method is based on the assumption that the damage of each loading cycle is independent, and during the loading process, the damage accumulates in proportion until fatigue failure. Compared with the fatigue limit of the material, the remaining service life of the hydraulic support is further analyzed, and the fatigue damage state of the support is evaluated, providing a theoretical basis for the maintenance and replacement cycle in practical engineering applications.

In Figure 6, the horizontal axis is the cumulative damage degree of the hydraulic support, and the vertical axis is the remaining life calculated according to the nonlinear model. As the degree of damage increases, the remaining life also decreases. Especially when the damage degree exceeds 0.8, the remaining life is greatly reduced. When the damage degree is 0–0.1, the remaining life is about 902 cycles, while when the damage degree is 0.8–0.9, there are only 23 cycles left. This shows that as the damage accumulates, the life of the support decays. Especially when it is close to the fatigue limit, the hydraulic support is no longer considered for use. This figure helps to predict the maintenance cycle and replacement timing of the support to ensure engineering safety. To verify the remaining life curve, this paper uses the hydraulic support prototype accelerated fatigue test (ASTM E645 standard), with a loading frequency of 10 Hz and a stress ratio of R=0.2. The average number of failure cycles in the experimental group (n=5) is 892 times, and the deviation from the model prediction value of 902 times is 1.1%.

Fig. 6 Remaining life with different damage degrees.

4.2 Damage development process

The fatigue damage of hydraulic supports during operation is not only reflected in the macroscopic fatigue life, but also includes the initiation and expansion of microcracks. To deeply analyze the development process of microcracks inside hydraulic supports, the stress-strain data of the supports under working conditions are first obtained by numerical simulation. Based on these data, combined with the crack propagation theory, the finite element crack propagation analysis is used to further analyze the crack initiation and propagation process. This analysis method simulates the propagation path of microcracks under repeated loads and describes the relationship between crack propagation rate and stress intensity factor in combination with the Paris law. In this process, considering the directionality and stress distribution of cracks, combined with the crack propagation characteristics of the material, the crack path and propagation morphology are precisely predicted. The crack propagation module in ABAQUS software is used to update the state of the crack in real-time, and the possible failure mode is predicted by stress intensity factor analysis. Through this method, the development process of microcracks in hydraulic supports under fatigue loads is understood, providing a quantitative basis for the failure risk assessment of the supports.

Figure 7 shows the crack propagation process of hydraulic supports under different load conditions. The left figure of Figure 6 shows the crack propagation path under three different load conditions. The crack propagation depth under load 3 is small, about 1.9 mm; the crack propagation depth under load 2 reaches 2.9 mm. The right figure of Figure 7 shows the relationship between the stress intensity factor and crack propagation rate. As the stress intensity factor increases, the crack propagation rate increases significantly, and the crack propagates more rapidly. In addition, the consistency of the crack propagation path under load 2 is verified by CT scanning (resolution 50 μm) and SEM fracture analysis: the Hausdorff distance between the simulated path and the actual crack trajectory is 0.12 mm, and the relative error of the propagation depth is less than 7%.

Fig. 7 Crack propagation process under different load conditions.

4.3 Fatigue damage accumulation rate

The fatigue damage accumulation rate evaluates the damage accumulation rate of the hydraulic support by calculating the degree of damage in each loading cycle and combining the change in time or number of loading times. The working state of the hydraulic support is a cyclic process, and the different stages of this process (descent, rise, and support) affect the support’s working state and damage accumulation. The three load conditions simulate the descent, pull, rise, and support, respectively, and record the damage. By integrating the degree of damage in each cycle, the total damage of the support under different times or loading conditions is obtained, and the damage accumulation rate is further calculated. This evaluation index reveals how the fatigue damage of the hydraulic support accelerates over time with the increase of working cycles under actual working conditions and then predicts the fatigue life of the support.

This indicator plays an important role in evaluating the life of the support in the design stage and judging when the support may fail prematurely during actual use. By comparing with the material’s fatigue limit and standard working cycle, the safety risks of the support under different operating conditions can be effectively identified, thereby providing decision support for maintenance, repair, and replacement.

Table 4 shows the temporal trends in the total damage level of the hydraulic supports under different load conditions. The time column is in hours; 'Total damage under load condition 1 (D1)', 'Total damage under load condition 2 (D2)', and 'Total damage under load condition 3 (D3)' are dimensionless quantities representing the cumulative damage value D∈[0,1], where D=1 indicates complete structural failure. Among the three different load conditions, load 2 is the largest. After 500 h, the damage degree under load condition 1 is 0.024; load 2 is 0.05; load 3 is 0.032, indicating that the larger load accelerates the damage accumulation of the support. By 5000 h, the damage degree under load condition 1 is 0.441; load 2 is 1.0; load 3 is 0.78, indicating that the fatigue damage of the support under load condition 2 is the most serious. The standard deviation of the damage value of load condition 2 at 5000 h is δ=0.08, and the confidence interval is [0.52, 0.60] (relative error <15%).

Table 5 shows the damage accumulation rate under different loads. Damage accumulation rate unit: h⁻¹ (damage increment per hour). There are three different loads. The load 2 is larger, and the load 1 is smaller. At 500 h, load condition 1 is 0.048; load 2 is 0.1; load 3 is 0.064, indicating that the damage accumulation rate is higher under larger loads. By 5000 h, load condition 1 is 0.16; load 2 is 0.56; load 3 reaches 0.5, indicating that higher loads accelerate the accumulation of damage. Monte Carlo simulation (N=1000 times) quantifies the uncertainty of damage prediction. The standard deviation of the damage value of load condition 2 at 5000 h is δ=0.08, with a 95% confidence interval of [0.52, 0.60] and an interval width of <0.08. The standard deviation range of all working conditions is 0.04–0.11, and the confidence interval width does not exceed 0.12, indicating that the model has stable statistical reliability under variable amplitude loads. The rain flow counting method is used to extract the load block spectrum (in compliance with MT/T 817-2020 standard). The calibration error of the dynamic load test bench is <3%.

4.4 Safety assessment

To evaluate the safety of hydraulic supports under fatigue loads, first, their stress-strain distribution is calculated, and the safety factor method is applied to evaluate the safety. This method calculates the safety factor by comparing the maximum working stress with the tensile strength of the material to determine the risk of failure. In the process of fatigue damage, safety is closely related to structural strength and crack propagation. Combined with the stress intensity factor (SIF) method, the influence of crack propagation is analyzed to comprehensively evaluate the safety of the support, timely predict the failure risk, and provide a basis for design optimization and maintenance decisions.

Figure 8 shows the changes in the safety factors of the five hydraulic supports under different working pressures. As the working pressure increases from 0.5 MPa to 1.5 MPa, the safety factor gradually decreases. The safety factor of the support with a support strength of 0.8 MPa under a pressure of 0.5 MPa is 1.6, while the safety factor of the support with a support strength of 1.6 MPa is 3.2, indicating that higher support strength can improve the safety of the support. These data provide a reference for support design and maintenance. At the same time, given the influence of fatigue damage on the safety factor, this paper supplements the definition of the safety factor based on Miner's linear cumulative damage theory. The safety factor comprehensively considers the synergistic effect of static strength and damage accumulation under cyclic loads and dynamically reflects the influence of fatigue damage on structural stability by multiplying the ratio of static strength to working stress by the square root of the remaining life. Updated experiments show that when the cumulative damage degree D reaches 0.8, the revised safety factor decreases by about 23% compared with the traditional static calculation value, indicating that fatigue damage significantly weakens the bearing capacity of the bracket.

Fig. 8 Safety factors of different hydraulic supports.

4.5 Discussion of experimental results

This study systematically reveals the fatigue damage mechanism of hydraulic supports under complex working conditions through a multi-physical field coupling analysis framework, breaking through the limitations of traditional single-factor analysis. The study innovatively integrates the coupling effects of temperature field, mechanical field, and corrosion field and combines the Morrow fatigue model with the Paris law to achieve quantitative prediction of fatigue damage accumulation and crack propagation. The results show that the dynamic coupling of temperature gradient and mechanical load significantly affects the stress distribution, and although the corrosion effect is relatively weak, it cannot be ignored in long-term service. The model optimizes parameters through the response surface method and verifies its reliability with the help of Monte Carlo simulation. Its prediction results are in good agreement with the test data, especially in the assessment of the remaining life of the support, showing high accuracy (safety factor>0.5).

Further analysis shows that the fatigue damage evolution of hydraulic supports presents significant nonlinear characteristics. As shown in Table 2, in 5000 loading cycles, the total damage value of load condition 2 reaches 1.0, which is much higher than other working conditions, indicating that the synergistic effect of load amplitude and frequency accelerates damage accumulation. In addition, the nonlinear attenuation law of the remaining life revealed in Figure 5 provides a key threshold reference for engineering maintenance strategies. It is worth noting that the tensile strength confidence interval obtained by Monte Carlo simulation (Fig. 4) shows that the discreteness of material properties may lead to fluctuations in actual life, providing a quantitative basis for reliability design.

However, the modeling of the corrosion field in this study is still based on idealized assumptions, and the interaction between non-uniform corrosion and dynamic loads in actual downhole environments needs further exploration. In addition, the model does not consider the indirect effect of changes in hydraulic medium viscosity on heat transfer and stress distribution, which may introduce deviations in ultra-long service cycles. Future work can combine long-term field monitoring data to improve the multi-field coupling model’s time-varying characteristics and introduce deep learning algorithms to improve the prediction ability of nonlinear responses, providing more accurate theoretical support for the life prediction and maintenance strategy of hydraulic supports. Time sensitivity analysis shows that the total damage degree of synchronous loading (heat-mechanical-corrosion) is 19% higher than that of loading after pre-corrosion, which is attributed to the accelerated nucleation of microcracks by thermal stress. It is recommended to give priority to the corrosion area during maintenance.

The multi-physics coupling analysis method proposed in this study shows excellent performance in the fatigue damage prediction of hydraulic supports, and has obvious advantages over traditional single mechanical load analysis, temperature change analysis, corrosion analysis alone and dual-field coupling methods. As can be seen from Table 6, in terms of crack propagation path error, the error of this study is <7%, which is significantly better than single mechanical load analysis (15.2%) and corrosion analysis alone (16.5%). In terms of residual life prediction error, the error of this study is 1.1%, which is much lower than temperature change analysis (7.3%) and corrosion analysis alone (9.2%), indicating that this method can more accurately predict the remaining life of hydraulic supports. In terms of the fluctuation range of the safety factor, the fluctuation of this study is ±0.04, which shows better stability and reliability compared with other methods such as single mechanical load analysis (±0.12) and corrosion analysis alone (±0.14). These results show that by introducing multi-physics coupling analysis and considering the synergistic effects of temperature, mechanical load and corrosion, this study can provide more accurate and reliable fatigue damage prediction of hydraulic supports, providing more effective theoretical support for practical engineering applications.

5 Conclusions

This paper successfully constructed a fatigue damage prediction model for hydraulic supports based on multi-physics field coupling analysis and finite element numerical simulation, comprehensively analyzed the interaction between mechanical field, temperature field and corrosion field, and effectively predicted the fatigue damage process of hydraulic supports. Multi-physics field coupling analysis was performed by COMSOL Multiphysics, combined with Morrow fatigue model and Paris law, and static and dynamic load analysis was performed by ABAQUS to analyze the stress distribution and fatigue damage of hydraulic supports. Response surface method and Monte Carlo simulation optimized the model parameters and improved the accuracy and reliability of fatigue life prediction. The study provides theoretical support for fatigue damage prediction and optimization design of hydraulic supports, which helps to improve engineering safety and economy, especially in coal mining and tunnel construction. It has important application value. Compared with traditional single mechanical load analysis, temperature change analysis, corrosion analysis and dual-field coupling methods, it has obvious advantages.Experimental results demonstrate that this method can successfully predict the remaining life of hydraulic supports with varying degrees of damage, maintaining a safety factor above 0.5 for supports with a strength range of 0.8 to 1.6 MPa. The error between the remaining life predicted by the multi-physics field coupling model and the test results is only 1.1%, which is significantly better than the traditional method. Although good results have been achieved, in-depth research is still needed in the future to further study the detailed modeling of crack propagation and damage accumulation, and combine actual working conditions to further improve the adaptability and generalization ability of the model, so as to provide a more accurate theoretical basis for the actual engineering application of hydraulic supports.

Funding

There is no funding information for the work in this paper.

Conflicts of interest

The authors declare no conflict of interest.

Data availability statement

No data were used to support this study.

Author contribution statement

Yang Liu designed the research study. Yang Liu analyzed the data. Yang Liu wrote the manuscript.

References

All Tables

All Figures

	Fig. 1 Actual structure of hydraulic support.
In the text

	Fig. 2 Geometric model of hydraulic support.
In the text

	Fig. 3 Distribution of thermal stress and thermal deformation under different temperature conditions.
In the text

	Fig. 4 Static stress distribution.
In the text

	Fig. 5 Distribution of tensile strength and confidence interval of damage value.
In the text

	Fig. 6 Remaining life with different damage degrees.
In the text

	Fig. 7 Crack propagation process under different load conditions.
In the text

	Fig. 8 Safety factors of different hydraulic supports.
In the text