A novel analytic model for sealing performance of static metallic joint considering the yield hardening effect

The static metallic joints’ sealing performance is deeply impacted by the plastic deformation and the interfacial separation of the contact surfaces with self-affine roughness. The yield hardening effect, unavoidable at the contact interface, is of vital importance to the plastic deformation and the distribution of the interfacial separation. However, most of the previous research ignores the effect of yield hardening, assuming that the contact surfaces are elastic-perfectly plastic. To address the problem, a novel analytic model for investigating the sealing performance under the effect of yield hardening has been developed in this paper. The upper boundary of contact stress in the conventional Persson model has been enlarged from a constant nominal yield stress to a maximum yield stress that varies with contact strain, making the model adaptable to actual scenarios with yield hardening effects. Utilizing the measured data of contact surfaces as input, the corresponding leakage rates are calculated. Besides, the contact stress distribution as well as the real contact area at the interface is also discussed. The sealing experiments are carried out accordingly, verifying that the proposed model owns the ability to predict the leakage rate under the effect of yield hardening.


A r
Real contact area with yield hardening Half of the flare angle Half of the thread angle C(q) Surface roughness PSD function C y (q) PSD function of (yield-hardened) deformed surface E 1  Cutoff wave vector q 1 Longest wave vector Q Leakage rate r 1 Pitch radius of thread r 2 Corresponding equivalent radius of the sealing area Contact stress Tightening torque T 1 Thread friction torque T 2 Friction torque Magnification u Interfaces average separation The modified factor for elastic energy 1 Introduction The hydraulic system is commonly used in engineering devices, such as aircraft. The sealing performance of the hydraulic system plays a critical role in guaranteeing the safety and the regular work of an aircraft [1]. The sealing performance of the hydraulic system is highly related to the contact status of the sealing joints in the hydraulic pipeline, such as the interfacial separations [2,3]. Figure 1 is an illustration of the static metallic sealing joint. The two metallic surfaces get into direct contact with each other forming the sealing circle. At the macroscale, the contact surface of the sealing joint is smooth and the contact is fine [4,5]. However, due to the existence of the roughness, as the magnification increases, the interfacial separation appears and only a few points are in contact at the microscale [6,7]. As a result, the stress distribution of the rough contact interface is uneven and the contact surfaces strain unevenly. In the high-stress contact region, the unavoidable yield hardening effect occurs, while in the noncontact region, where the contact stress is low, the leakage channels form. It is clear that the yield hardening effect is an indispensable factor in the sealing interfacial contact status, which has a vital impact on the leakage rate of the sealing joints [8].
In the last decades, many theoretical models were developed to predict the leakage rate, by means of characterizing the interface micromorphology and revealing the contact mechanism [9][10][11]. The most popular model is developed by Persson [12]; it mainly focuses on the sliding contact mechanics of the metal-rubber contact pair interface under low stresses. His model was based on the G-W statistical contact model [13] and detected the interface contact status. To investigate the influence of roughness and sliding speed on friction, a method to characterize self-affine surfaces was developed using power spectrum density obtained with the FFT approach [14][15][16]. By assuming the interfacial maximum contact stress is infinite, the theory for stress distribution and the real contact area of an elastic material (rubber) was proposed by Persson et al. [10]. Also, on the rubbermetal contact pair, Müser et al. [17] presented the mean to predict average interface separation and, in conjunction with percolation theory, investigated the influence of both the interfacial elastic strain and average separation on the sealing performance. These researchers mainly focused on the sliding contact at the rubber-metal interface with small elastic strain.
When it comes to static metallic sealing, plastic strain occurs. By substituting the infinite maximum contact stress with the constant nominal yield stress, Persson [10] redefined the range of stress distribution. This made the classical theory suitable for the investigation of metal-metal contact mechanics, including not only elastic strain but also plastic strain. Combined with the plastic deformation mechanism revealed by Pei et al. [18], contact mechanics under low load and minor strain have been studied in various terms. Campañá et al. [19] investigated the contact stress between an elastic manifold and a rigid substrate with a self-affine fractal surface. Yastrebov et al. [20] studied the evolution of the real contact area between rough contact surfaces. The link between load and plastic deformation was established by Martínez et al. [21,22]. In terms of the interfacial average separation, Müser et al. [23] found the pressure dependence of the average separation and derived the distribution of separations between surfaces near the contact edge. It is easy to notice that the metallic contact mechanisms investigated above are all under low contact pressure and with little yield hardening effect.
As the contact pressure increases, the degree of yield hardening effect reaches a critical level. As a result, the nature of the plastically deformed surfaces as well as the related contact parameters was impacted strongly [24]. However, few of the existing methods considering the yield hardening effect, most of them adopted the elastic-perfectly plastic material assumption instead when studying plastic  [25][26][27]. Although several studies considering the yield hardening effect were conducted to reveal surface properties, such as stiffness and indentation depth [28,29], it is still not clear how to include yield hardening in the Persson contact mechanic approach and the support for leakage rate prediction remains limited. All above make the results that the existing models are not suitable to predict the leakage rate under the effect of yield hardening.
Thus, to tackle this problem, a statistical model able to calculate the leakage rate under the effect of yield hardening was developed. The remainder of this paper is organized as follows. In Sect. 2 the leakage prediction model of the hardened sealing interface has been established, which consists of the method to estimate the maximum local stress in the contact interface and the calculation of the leakage rate. The experiments of the leakage measurement are conducted on the designed experiment rig to validate the model, in Sect. 3. The key parameters related to leakage prediction are discussed in Sect. 4. Section 5 presents the conclusion. The developed prediction model in this research could effectively calculate the leakage and can be of considerable help in analyzing joint sealing performance and optimizing assembly process parameters.

Analytic model for the leakage prediction
The calculation of fluid leakage in metallic seals could be divided into three steps. First, two surfaces are approached by applying a certain torque, and the contact pressure distribution at the interface of the two metallic bodies should be obtained. Then, the key contact parameters such as contact area, plastic deformation, and interfacial average separations are calculated using contact mechanics theory. Finally, the fluid flow through the leakage channels will be calculated using the effective medium theory.

Normal contact force
The sealing interface is formed when two contact surfaces approached each other under the act of tightening torque. The effect of normal force on the average separation is much larger than the tangential force [29]. Thus, in this paper, only the normal contact force is taken into account. By analyzing the tightening process, the normal contact force F N on the sealing contact surface was deduced [30]. The tightening torque T is composed of two torques: the thread friction torque T 1 and the friction torque T 2 on the interface between the flat nuzzle and the outer sleeve nut, shown in Fig. 2. Through the Eqs. (1,2,3,4), the relationships among the normal contact force F N , the axial force F, and the torque are obtained: where, for the joints with a flare angle of 74° and a size of M16 mm × 1.5 mm, α = 37° is half of the flare angle, P t = 1.5 mm is the thread pitch, μ 1 = 0.15 is the friction coefficient of the threaded surface, β = 30° is the half of the thread angle, r 1 = 7.68 mm is the pitch radius of thread, μ 2 = 0.15 is the friction coefficient of the contact surface between the flat nozzle and the outer sleeve nut, and r 2 = 6.43 mm is the corresponding equivalent radius of the contact area.
Once the normal contact force is obtained via Eqs. (1,2,3,4), the nominal contact stress σ 0 , an important factor in Eq. (15), can be gained by: where where illustrated in Fig. 3, L w is the width of the sealing circle obtained by measuring the plastic deformation surface of the sealing joint, L 1 is the conical busbar of the tube inside surface, R 1 is the outer radius of the tube edge, R 2 is the corresponding radius of L 2 , and S is the nominal contact area. (1)

Contact status with yield hardening
The two roughness contact surfaces can be equivalently expressed, without loss of generality, as an elastoplastic body with a roughness profile h(x) = h 1 (x) + h 2 (x) in contact with a rigid body with a flat surface [21,31], as shown in Fig. 4. And the equivalent elastic modulus E * could be obtained: where ν 1, ν 2 is the Poisson ratio, and E 1 and E 2 are the elastic moduli of the contact bodies respectively. z = h 1 (x) and h 2 (x) describe the surface profiles.
The modified Persson contact mechanics theory was used to determine the normal contact profile. It is no doubt that when getting into contact, metal will deform plastically with the effect of yield hardening. Thus, for taking the yield hardening effect into consideration, the approach employed here assumes that yield stresses are no longer as constant as suggested by Persson's contact theory, but increase with the increasing plastic strain. The solution of the maximum contact stress was conducted in two scales: the macroscope and the microscope.
On the macroscopic scale, the contact states may be divided into elastic deformation and yield hardened plastic deformation. It may be stated using the macroscopic force conservation: In Eq. (8), F e is the elastic contact force and F p is the plastic contact force.
While, on the microscopic scale, contact stresses will surpass the nominal yield stress due to yield hardening. Utilizing the nominal yield stress σ y to differentiate the spectrum of plastic and elastic stress, the surface deforms elastically in the interval σ e ∈ (0, σ y ), and plastically in the interval σ p ∈ [σ y , σ f ), corresponding to the F e and F p in macroscope. Through Eq. (9) and Eq. (10), the contact stress and contact force could be linked.
where S is the nominal contact area, and the yield strength σ f is related to the hardness of an ideal elastic-plastic solid as measured by indentation hardness [31]. Beneath a particular magnification ξ for a given load F N , it can be obtained by Eq. (11) [21,32]: where σ y is the nominal yield stress, and σ f is the stress when hardening yielding occurs. And ε p is the strain of contact surfaces and N = 0.2 is the strain hardening exponent. The iterative analytical model for solving the maximum yield stress was developed. First, with the input values, the associated yield stress is computed. The ε p 0 = 0 denotes the beginning of the plastic stress. Second, via the stress distribution function, the normal contact force is obtained. The results are then compared with the preset values, and iteration (8)   continues until the error is acceptable. Finally, the maximum yield stress at the contact interface is calculated.
It is worth noting that as the nominal contact pressure at the sealing interface rises, so do the local contact stress and plastic deformation rate, resulting in a faster yield hardening rate (Fig. 5). The original contact stress is therefore enlarged from a constant σ y to σ f , which is associated with the amount of strain. At the same time, the contact stress distribution in the interface area will redistribute.
Based on Persson's classical contact theory [10], combined with the maximum yield stress obtained by the aforementioned iterative model, the distribution of the contact stress σ overall length scales can be expressed as: where σ(x, ξ) denotes the nominal contact stress at a certain magnification ξ that changes with position x inside the interface, and 〈…〉 is the ensemble average of multiple rough surfaces. As described in Eq. (12), more microscopic surface topography characteristics will be obtained as the magnification ξ increases, and the contact stress distribution law will be revised at this time. This is extended to the linear order of ξ and yields: By applying the following boundary conditions, the above partial differential equation Eq. (13) and Eq. (14) can be solved: When the local contact stress equals zero, the contact surfaces are separated; at the minimum magnification ξ, that is, at the minimum resolution, the contact stress distribution is the delta equation; the maximum contact stress will exceed the nominal yield stress σ y , but will not exceed the yield stress σ f (ε p ). At the same time, it implies that the modified Persson contact theory is not only suitable for ideal elastic-plastic solids without work hardening, but also contact surfaces with work hardening effect.
Following the solution, the analytical formula of the contact interface stress distribution is obtained as Eq. (15) and Eq. (16): where notably, where α n = (nπσ 0 )/σ f , q L = 2π/L, L = 600a is the width of the unit domain along the circumferential direction, and a = 0.35 μm is the sampling pitch of the microscope.
As the maximum yield stress rises, the surface plastic deformation rate of the contact interface can be given by [10]: where q 0 is the cutoff wave vector, and H = 0.8 is the Hurst number. The measurement and characterization of roughness contact surfaces using the power spectrum are shown in the next section in detail.
The real contact area of the interface at a certain magnification ξ is derived by integrating across the (0 + , σ f ) stress interval based on the stress distribution. It may be obtained by [33]: Among them, P p (q) is the real contact area of the interface without the yield hardening effective, P non is the uncontacted area of the interface, and P pl is the plastic deformation rate in the non-yield hardening state and can be calculated by substituting α n = (nπσ 0 )/σ f with α y = (nπσ 0 )/σ y in Eq. (17). And P h is the plastic deformation rate including the yield hardening effect. Furthermore, the contact mechanism here ignores the yield hardening effect's minor impact on the elastic contact area; the effect primarily effects the plastic strain of the contact interface.

Calculation of leakage
To calculate the leakage, the interfacial average separation should be obtained first. Here, it was assumed to be unchanged with the fluid pressure increasing, since the normal force acting on the interface is large enough to stand the fluid pressure. For elastic-plastic inviscid contact, Persson's contact theory [21,33] is applied to calculate the average contact separation u of the interfaces, which is described as: where A r (q) denotes the relative contact area under the effect of yield hardening when the contact interface is studied at a magnification ξ. According to the related research the γ is set to be 0.4 [31,34]. Although the nominal contact pressure σ 0 is smaller than the yield strength σ y , as the magnification increases, it can be found that several asperities are already deformed plastically. The heights of the asperities are changed if only they deform plastically. The surface roughness power spectrum C y (q) of the plastic deformation surface can be obtained from the following Eq. (24) [32]: where P pl = σ 0 /σ y is the contact area when the whole contact area has yielded plastically. Based on the interface average separation u obtained by Eq. (23), the leakage of the incompressible and Newtonian fluid (hydraulic oil) can be obtained by the Reynolds equation. Adopting the assumption that all the fluids, in the studied domain, go through from the percolation channel and the pressure drop ΔP occurs at the critical constriction, the leakage rate per unit width in the radial direction for the studied domain Q w can be obtained by Eqs. (25,26,27) [33]: where Figure 6 depicts the overall leakage rate calculation process. Table 1 shows the essential parameters for the calculation, and an example of the leakage calculation is shown in Appendix.
The method for analyzing the effect of yield hardening on leakage rate prediction is established. An improvement to the classical plastic contact theory is proposed to include the factors of physical properties of the contact surface materials, which could restore the contact state of the sealing interface more realistically. Simultaneously, the techniques for predicting the average interfacial gap, leakage rate, and other parameters under the corresponding contact state are upgraded, making them more appropriate for the prediction of leakage with the effect of yield hardening.

Model validation
In order to validate the present model, the experiment was conducted on the experimental rig as shown in Fig. 9. Firstly, the surface topography was scanned and the micro-topography was characterized. Secondly, the hydraulic joints were assembled with a certain torque. Thirdly, the pipeline was pressured and the leakage was measured. Finally, the results of the theory were validated by the experiment.

Surface topography characterization
The morphology of the metal surface in the sealing region of the contact surface was captured utilizing Alicona's InfiniteFocus optical depth-of-field microscope, and the microstructure characteristics were modeled to investigate the microcontact state. To detect the surface roughness, a 50 × optical magnification lens with a sampling pitch of 0.35 μm and a vertical resolution of up to 10 nm is used to observe the microscopic morphological properties of the actual surface of the joint. It can cover the surface "peaks" and "troughs" while portraying microscopic characteristics by producing 500,192 data points across a 215.6 × 284.2 μm 2 area. The measurement scene is shown in Fig. 7.
The joint was fixed to keep the surface parallel to the horizontal plane of the instrument. Then, it was scanned and the point cloud data was obtained. As demonstrated in Fig. 7, the 3D reconstruction was carried out. The  surfaces of the sealing circle were scanned at 6 surfaces (60 degrees per surface) to represent the overall roughness of the microscopic surface features and the point cloud data were averaged. Then, the root-mean-square roughness values of the contact surfaces could be obtained, and the power spectral density (PSD) was derived by the fast Fourier transform method [33], as shown in Fig. 8. For a randomly rough surface, when h(x) is a Gaussian random variable, the statistical properties of the surface are completely defined by the power spectrum C(q) [10]. It was defined by [35]: In Eq. (28) above, q is the length of the wavevector, and h(x) is the surface roughness height at the point x = (x, y), 〈h(x)〉 = 0. The long distance roll-off wavevector q 0 = 2π/λ 0 = 5.98 × 10 4 , and the short distance roll-off wavevector q 1 = 2π/λ L = 2.99 × 10 6 .

Leakage experiment
An experimental setup was established to measure the sealing performance of the hydraulic system joints. The hydraulic pump station maintains a constant hydraulic pressure, with a control precision of 0.01 MPa. The fixed torque wrench has an accuracy of 0.01 N m and is able to manage the joint assembly torque. The measurer has a mass accuracy of 0.01 g and a volume precision of 0.2 mL, and the monitor is utilized to record the precise moment each drop of oil falls.
In order to guarantee the macroscopic assembly precision such as concentricity and step difference, the optical support platform, with a displacement accuracy of 0.1 mm and the rotation accuracy of 1°, was applied to fix the hydraulic pipeline. To limit the influence of mesoscopic form tolerances, joints with contact surface circle run-out within 30 μm (better than the manufacturing standard (of 50 μm)) are selected for experiments. At the same time, each group of experiments was repeated three times with the same joint.
During the experiment, the pipeline at both ends of the joint was firstly fixed by the optical support platform (Fig. 9). The joint to be tested was preassembled manually, fixing its spatial location, and the outer nuts are tightened by a constant torque wrench. The pipeline is then filled with the No. 15 aviation hydraulic fluids with a low pressure to reach a steady state. The pressure was achieved by controlling the pumps and shutting off one valve. Once the fluid leaks uniformly and stably, each dropping moment is recorded and both the mass and the volume of droplets were measured. Table 2, and the measurement results are clearly shown in Fig. 10. The results of the experiments are mainly separated into three parts: the top part with huge leakage rate, the middle leakage with uniformed droplet and at the bottom of the figure is the seepage which can hardly be observed. In fact, only the middle part, with a reasonable leakage rate, is predictable, while at the bottom part, for a certain torque, the fluid pressure inside is not high enough for the fluid to fulfill the interfacial separation, forming a leakage channel. In this so-called critical state, the leakage rate and contact state cannot be measured precisely. At the top of the figure, corresponding to the flowing fluid instead of the fluid droplet, the interfacial separation at contact surface is enlarged by the high fluid pressure. In another word, for a given fluid pressure, the corresponding tightening toque is not big enough for the two contact surfaces forming a fine attach. Thus, these two situations of leakage cannot be predicted precisely by the existing method and this paper mainly focuses on the middle part, the one with a reasonable leakage rate. From Fig. 10, it can be seen that the inflection point of the leakage curve delays as the torques increase. This means that each torque can hold the sealing performance well until the fluid pressure reach a certain value. And the critical values depend on the torque. What's more, with the aid of Fig. 15, it is clear that the torques used in the experiment, falling in the range of 5-10 N m, will cause an obvious changing of the interfacial separation, influencing the leakage rate. Thus, it is reasonable to choose these torques to analyze the effect of torques on the leakage rate.

Other important experimental parameters are shown in
It was also found experimentally that during each measurement, the leakage rate is not a time-independent value, but decreases over time. The interval between each drop increases with the time going on, when the joint was tightened with torque of 6.95 N m and under the fluid pressure of 8 MPa. This may be explained by the fact that at the beginning of the leak, the oil flows through the leak channel to form a leak; due to the presence of impurity particles in the oil, along with the oil flowing through the leak channel, the impurity particles block the tiny leak channels, reducing the cross-sectional area of them and lowering the leak rate. As time passes, the flow of oil through the leakage channel rises, so do the impurity particles. Thus, the leakage rate measured in actual will drop as the time goes by. The average value of the measured leakage over 11 min is about 0.9 of the value with pure fluids. Figure 11 indicates the trend of the leakage rate over time.

Comparison of results
In this section, the results of the newly developed model were compared to the experimental results. The studies used preset torque to evaluate the leakage rate law, specifically the micro leakage rate, as liquid pressures rose. For a clearer representation of the micro leakage rate, the results are represented in Fig. 12.
Taking the yield hardening effect into account, the leakage was calculated and compared to the measured results. The torque used to build the joint was 6.69 N m, and the pressure drop varied from 8.5 to 10.5 MPa. The largest error occurs at a pressure of 10.5 MPa, and the error value can reach 5.98%. As expected, the leakage rate increased linearly with the rising fluid pressure. It is easy to find that the theory results fit well with the experimental.
The newly developed model is capable of properly predicting leakage volume. The sealing joint was assembled with a torque of 7.55 N m and a pressure drop of 11 MPa in this example. The result of the calculated leakage is 7.68 × 10 −4 mL and the experimental result is 7.76 × 10 −4 . As shown in Fig. 13, the volume of leakage fluid steadily increases with time, but when the time reaches 400 s the slope of the curve drops a bit, meaning that the rate of leakage starts to decrease. As mentioned above, the phenomenon is induced by the existence of invisible partials that reduce the area of the flow channels.

Effect on parameters
In this section, the influences of the yield hardening on the contact state parameters will be discussed. Under a certain load, the yield hardening effect is visible in the overall real contact area and the plastic deformation rate. Furthermore, the average interface separation is the primary cause of leakage effects.

The surface contact status
The present model demonstrates an increasement in the interface stress. Meanwhile, the interface stress distribution is rearranged. When the torque (nominal contact pressure) is small, the percentage of yield-hardening effects is relatively small. Most of the asperities deform elastically. As the torque increases, so does the amount of yield hardened asperity clusters.
By considering the yield hardened effect, the trends of the contact status with the torque were investigated, shown in Fig. 14. The elastic and yield hardened plastic contact area increases sharply with the torque raising at the beginning of the assembly. This means that when two surfaces start getting in contact, surfaces are flat as seen in macroscale, and the number of contact asperities increase immediately. Then, when the torques fall in the range of 5-10 N m, the inflection point appears in the curves of both plastic and elastic deformed area. This is the key stages of the contact area forming process, because the value of the torques here may determine whether the rate of contact area is enough to block the pressure fluid from leakage.
Usually, a large normal force will cause a non-negligible yield-hardening effect on the actual contact surface, increasing the asperity body stiffness to the point where it does not require much strain to balance the work done by the normal force. Meanwhile, the plastic deformation rate grows with the increasing torque. It is worth noting that the yield hardening influence is mostly represented in two aspects: the asperities strain in normal direction and the contact area in the in-plane direction. In more detail, the yield hardened asperities on the base surface will lower down with the base surface when the highest come into touch. In another word, the hardened asperities will slow down the increasement of the plastic deformed area. By the same way, the number of asperities in contact reduces, and the real contact area reduces.

The average interfacial separation
The average contact separation is a critical factor which is directly related to the leakage rate, influencing the sealing performance of the sealing contact interface. A tiny average contact separation u as well as a large real contact area is essential for a fine sealing performance. Figure 15 indicates the average interfacial separation changes with various of torques. It is clear that the yield hardening effect has an influence on the interfacial separation. The rate of interfacial separation reduction will be slower than that of non-yield hardening. The average drop rate of the gap at the interface with yield hardening effect is 4.48 × 10 −4 μm/(N m) when the torque is between 10 and 50 N m, whereas the drop rate at the interface without yield hardening effect is 7.28 × 10 −4 μm/(N m). The yield hardened separation is approximately 7.69 of the surfaces without yield hardened plastic deformation. The average gap at the yield hardening interface is 7.33 times that of the nonyield hardening interface when the torque is 10 N m, and the ratio climbs to 8.02 when the torque is 50 N m. The separation's change rate reduces as the torques increase. It is worthwhile to investigate the region around the turning point, which is in the 5-10 N m range. Prior to the turning point, it rapidly declined in the 0-5 N m range. For each 1 N m increase in torque, the interfacial separation decreases by 0.348 μm. This may refer to the formation process of the sealing surfaces, in which two rough surfaces are initially approached to each other and the sealing circle is constructed. The rate of fall slows down when the torque reaches 5 N m. Most of the asperities in this region are in touch, and leakage channels are visible. After then, it becomes harder for the increasing torque to modify the contact states at the interface. The rate of interfacial separation drop stabilized after the torque reached 10 N m, and the influence of torque increase on modifying the interfacial separation diminished. The distance only narrows by 0.004 μm for every 1 N m increment.

Conclusion
In this paper, a novel model was developed to calculate the leakage rate under the effect of yield hardening. By creating an iterative approach, the stress distribution of contact interface was investigated. The results of the novel proposed model were compared with experiments, which was carried out on the rig designed for leakage measurement. Through the study above, the follow conclusion can be obtained: 1. The modified contact mechanics theory is able to determine the contact status of metallic interface with yield hardening effects, and furthermore, it has been confirmed that the developed model owns the ability to calculate the leakage as well. 2. It was found that there exists a critical range of torque corresponding to a given sealing interface. For the joint surface in this paper, the critical range is 5-10 N m, which may contribute to the guidance of the assembly process. 3. It was investigated that the yield hardening effect will influence the metallic contact status and the extent of the influence grows with the increasing torque. The effect will slow down the increasement of the plastic deformation.
This research demonstrates that the current model can accurately calculate leakage. However, other potentially affecting aspects, such as the influence of fluid pressure on interfacial separation, were not taken into account. As a response, in future study, we will continue to investigate the impact of the aforementioned elements in order to widen the model's application.

Appendix
Under the torque T of 7.55 N m, the calculation process of leakage of the sealing joint is as follows: The normal contact force F N = 1.79 × 10 3 N can be obtained via Eqs. (1,2,3,4,5,6); when the R 1 = 9.55 mm, L 1 = 15.87 mm, and L = 0.5 mm, the nominal contact stress of the sealing interface can be expressed as Eq. (29): According to the analysis above, the leakage rate of the sealing interface under a pressure drops of 11 MPa can be obtained as Eq. (30):