A novel finite element model for the flare joint contact status at the seal ring with multiscale factors during assembly

doi:10.21203/rs.3.rs-3084764/v1

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A novel finite element model for the flare joint contact status at the seal ring with multiscale factors during assembly

https://doi.org/10.21203/rs.3.rs-3084764/v1

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The sealing performance of a hydraulic system is determined by the joint’s interfacial contact status, which is impacted by unavoidable assembly errors and joint manufacturing deviations on multiple scales. A novel geometric analysis model was heuristically developed for incorporating the macroscopic and mesoscopic topographies into the accurate estimation of the contact surfaces' initial relative assembly position. The microscopic contact model is subsequently constructed using the reverse reconstruction method based on the measured characteristics of rough surfaces. In conjunction with the relative contact position, a multiscale finite element contact model of the interface in the assembly was generated. The simulated indentation results are then compared to experimental data for selected typical circumferential angles, while the impact of topography at each scale is investigated. The findings show the variation in the seal formation tendencies and illustrate how effectively the model can predict the contact status at the seal ring. The influence of the macroscopic mating angle is mostly apparent in the distribution of plastic deformation along the circumferential direction. The mesoscopic flare angle is mostly reflected in the width and depth of the interfacial depression, and the radial circular runout impacts the seal ring formation process by altering the contact phase.

Contact mechanics

Flare joints

Seal ring

FEM

Assembly

The interfacial contact status of metallic rough surfaces is of vital importance in many fields, such as sealing performance, heat transformation, and contact electric resistance [1–3]. As a crucial component of the aviation hydraulic system pipeline, the sealing performance of flared pipe joints plays a vital role in the security and reliability of aircraft [4, 5]. A fine contact status at the seal ring is required to prevent the pressure fluid from leaking, whose formation is influenced by multiscale factors during assembly [6, 7]. To study the interfacial contact status at the seal ring, not only microscopic roughness but also macroscopic and mesoscopic impact factors with larger scales should be incorporated in the modeling [8–10].

The study of contact status was highly related to contact mechanics [11, 12]. To date, numerous attempts have been made by researchers to investigate the contact state using microscopic models due to the complexity of the contact mechanics of rough surfaces [13, 14]. Persson et al. [15, 16] heuristically introduced the concept of magnification into the research of contact mechanics and found that roughness appears and the contact status changes with increasing magnification. By comparing the results of the geometric model with numerical results, the interfacial contact stress distribution was investigated microscopically by Persson et al. [17]. Then, the finite element model was utilized by Hyun et al. [18] and Yong et al. [19, 20] to investigate the real contact area and the elastoplastic deformation of the contact with self-affine rough surfaces. Next, in regard to the leakage rate, Yu et al. [21, 22] investigated the leakage channel within the mechanical seal interface to calculate the leakage rate. Cui et al. [23, 24] proposed a porous medium leakage model to explore the influence of microscopic surface topography on sealing performance. Multiscale models were created when researchers discovered that microscopic models were unable to accurately predict the leakage rate in a multiscale environment when applied to practical problems with multiscale factors due to the influence of mesoscopic and macroscopic factors [25, 26].

However, only single-scale models are unable to adequately describe the actual problem; hence, multiscale models began to develop [27]. An early attempt to establish a multiscale model was accomplished by Hol et al. [28, 29] to describe the multiscale interfacial friction during metal forming. Guo et al. [30] proposed a multiscale theoretical model to formulate the contact status of rough surfaces. Wanger et al. [31, 32] developed a multiscale FEM approach to study the hysteresis friction of rubber on roughness. The coupling between the scales is performed by passing the friction coefficients. Furthermore, Yan et al. [33, 34] proposed a simplified multiscale model describing deviation-free joints with perfect macroscopic geometry, including microscopic roughness, to simulate the assembly process and sealing performance of pipeline fittings. Kang et al. [35] developed a multiscale model to describe the contact status of a rough interface. The multiscale topographies of each contact part were considered by simplifying the macroscopic simple geometric deviations into linear equations. Wang et al. [36] developed an approach to describe the leakage of seal joints, including the roughness characteristics and the ideally macroscopic geometry without macroscopic assembly errors and mesoscopic manufacturing deviations. To date, the current research on multiscale topographic descriptions is well developed; however, in regard to contact status, macroscopic and mesoscopic factors have a greater impact than microscopic morphology, which should not be ignored in modeling analysis [37–39]. Lacking a model that combines macroscopic, mesoscopic, and microscopic roughness, practical assembly of piping joints containing multiscale factors is unable to be accurately described, and as a result, the interface contact state generated through the assembly process cannot be accurately described.

To address this issue, a finite element model based on geometric characteristics is developed to effectively investigate the interface contact status of the seal ring under the impact of multiscale factors in the assembly process. The effects of multiscale errors and deviations at each level on the initial relative positions of contact points are analyzed, and an inverse reconstruction method is utilized to develop a geometry-based multiscale finite element model. The multiscale factors are incorporated through the initial relative position of each contact body derived from analysis to effectively describe the contact state of the interface obtained by the assembly. Then, the corresponding experiments were carried out, proving the enhanced predictive capabilities of the proposed FE model. Finally, the model was used to analyze the influence of various factors on the contact status, and the results can contribute to the optimization of the assembly process.

An uneven and incomplete seal ring is the main factor causing leakage, which is composed of three main scale factors. The absence of any element on any scale will result in invalid predictions of seal joint leakage. Thus, in this section, an effective geometric-based multiscale FE model was established to evaluate the impact of factors at each scale on the contact status at the seal ring.

2.1 The relative position of two parts during assembly

The main factors causing the change in relative position are the mating angle α followed by the circular runout and the flare angle γ. In this section, the initial relative position is investigated geometrically, and the schematic diagram is shown in Fig. 1.

To define the relative position of the two contact parts, the geometric model was built taking both the macroscopic error and mesoscopic deviation into consideration. There are three commonly occurring macroscopic errors in the assembly: the mating angle α between the two axes 1 of cylinders, the gap h in the z direction, and the gap between the two axes of cylinders. The last two errors will be reduced to the first one when two parts approach each other under the preload. Thus, only the error of angle α will be considered. As shown in Fig. 1a), the mating angle α during assembly was obtained by fixing the joint part and rotating the tube part around the axis of ϕ = 0. In Fig. 1c), the position of the tube was defined, and point A(r_A,θ_A,ϕ_Α), used as the positioning point for the FE model, could be described through spherical coordinates:

$${r_{\text{A}}}=a$$

$${\theta _{\text{A}}}=\frac{\pi }{2}+\delta \left( {\alpha ,\varphi } \right),\varphi \in \left. {\left[ { - \pi } \right.,\pi } \right)$$

where θ_A is the angle between the straight-line OA and the z-axis, a is the length of line OA, and ϕ is the circumferential angle ranging from − 90° to 90°. The point B(r_B,θ_B,ϕ_Β) is the top contact point of the tube, whose position can be defined by the following equation:

$${r_{\text{B}}}=\sqrt {{{(a+b\cos \beta )}^2}+{{(b\sin \beta )}^2}}$$

$${\theta _{\text{B}}}=\frac{\pi }{2} - \omega +\delta \left( {\alpha ,\varphi } \right),\varphi \in \left. {\left[ { - \pi } \right.,\pi } \right)$$

where b is the length of line AB, angle ω is the angle between straight lines OA and OB, and β is the angle related to the manufacturing deviation.

In Fig. 1c), the angle θ_B is defined by two parts: the constant angle ω related to the structure of the tube and angle δ, which changes nonlinearly with the growth of mating angle α. As shown in Fig. 1d), a new model was built to investigate the relationship between the mating angle α and the corresponding angle δ.

$$\left\{ \begin{gathered} l= - {r_B}' \cdot \sin \delta =n \cdot \tan \alpha \hfill \\ n=m \cdot \sin \varphi \hfill \\ m={r_B}' \cdot \cos \delta \hfill \\ \end{gathered} \right.$$

$$\delta =\arctan (\sin \varphi \cdot \tan \alpha )$$

$$OO'=b \cdot \sin \beta {\prime }$$

The mesoscopic error is the so-called circular runout of the joint ranging from − 50 µm to + 50 µm, and it is presented by the position of point E on generating line CD. As shown in Fig. 1b) (the profile at ϕ = 90°), e is the distance of E from endpoint C, and γ is the angle between the line CE and the vertical coordinate z, which is normally equal to the flare angle of the joints.

To define the relative position of the two parts, the contact area of the joints should also be determined formulaically. Point E can be defined by the following equations:

$${r_E}=\sqrt {{{(e \cdot \sin \gamma +c)}^2}+{{(e\cos \gamma +h)}^2}}$$

$\left\{ \begin{gathered} {\theta _E}=\arctan \frac{{c+e\sin {\gamma _j}}}{{h+e\cos {\gamma _j}}} \hfill \\ e=\frac{{{r_B}\cos {\theta _B} - h}}{{\cos {\gamma _j}}} \hfill \\ \end{gathered} \right.$ , ${\varphi _B} \in ( - \pi ,\pi ]$ (9)

where h equals the height of point C and length c represents the circular run-out of the joint, which is the distance of point C from the z-axis.

To date, the contact points of the joint and tube have been normalized to a unified spherical coordinate system that includes both macroscopic and mesoscopic features. In this paper, for the tube with a diameter of 16 mm, the angle β'= arctan (3.6/1.7), angle ω = arctan (3.6/9.7), and for the lengths a = 8 mm, b = 4.04 mm, c = 7.6 mm, and h = 10 mm.

2.2 The structural parameters

Based on the initial relative position obtained by the geometric model, the contact status of the seal ring could be simulated by reconstructing the FE model with the measured surface point cloud data.

2.2.1 Model generation

Through the procedure shown in Fig. 2, the two contact parts with rough surfaces were reconstructed. First, the surface characteristics were passed in the form of point clouds gained by the measured topography. The point clouds were filtered before fitting by NURBS, and then the fitted surfaces were imported into 3D design software [40]. After some stretching and combination with the fitted surfaces, solid CAE models were built and imported into the commercial finite element geometric software.

The accuracy of the reconstructed model relies on the density of the surface point cloud. The in-plane precision produced by 3D super-deep field microscope sweeping could reach as high as 0.57 µm, and the vertical accuracy can approach 0.05 µm. The in-plane accuracy of the reconstructed model used in this paper is 7.09 µm, and the vertical accuracy is 0.05 µm, counting the time consumption of subsequent computation. This balances the need to properly characterize roughness features with the need to speed up calculation. The lateral length of the tube and the joint was long enough to cover the whole conical rough seal surface, enabling the multiscale model to include the seal ring during assembly under mesoscopic geometric deviations. In addition, the width of the tube is larger than the joints to guarantee that the typical part will contact the tube during the simulation.

The multiscale solid CAE models containing both mesoscopic structural characteristics and microscopic roughness were then imported into the commercial finite element analysis software. First, the two contact parts were assembled with the prescribed relative position gain by the geometric model, and the relative positions at several typical circumferential angles were picked up.

The initial relative position shown above consists of not only the mating angle and flare angle but also the circular runout. Both the tube and the joint have circular runouts, but because the joint is made by turning rather than stamping, its manufacturing variation is substantially larger than the pipe's circular runout. Therefore, while utilizing the model developed in this work to determine the initial position of the two contact parts, only the circular runout of the joint is explored. A circular runout means that the radius of curvature varies with the circumferential angle, as illustrated in Fig. 3.

As shown in Fig. 4a) and b), to reduce the time consumption of the simulation, the parts were meshed unevenly, and unnecessary structures of the joints were ignored, such as the threaded surface. Meshes near the contact interfaces were much smaller than those far away from them. The element type of the joint and tube parts was hexahedral C3D8R elements with eight nodes. The joint part contains 143533 elements and 129600 nodes, while the tube part contains 344006 elements and 325000 nodes. The joint's mesh in-plane accuracy could reach 9.91*10.58 µm, while the tube's could reach 6.36*3.54 µm, both of which are high enough to accurately depict the degree of roughness on the contact surfaces.

2.2.2 FE material properties and boundary conditions

To investigate the contact behavior of the seal interface during the assembly process, a contact pair was established. The contact surface of the tube is the master surface, while the joint surface is the slave surface. The penalty function was adopted to describe the friction at the seal interface, and the friction coefficient was set to 0.15. The Newton-Raphson method was applied to the contact analysis. The joint and tube were made of stainless steel, whose mechanical properties are listed in Table 1.

Table 1

Mechanical properties of the tube and joint parts
Material	Young’s modulus (MPa)	Poisson ratio	Yield strength (MPa)
0Cr18Ni9	204000	0.285	205

In addition, Fig. 5 shows the boundary conditions of the model. To simulate the contact behavior during the assembly process, the bottom and side surfaces in the x direction of the joint were fixed. As the typical part was selected from the whole joint, the side surfaces were restricted with symmetry constraints from moving along the circumference. To simulate the preload provided by the sleeve nut, an axial displacement was applied on the tube part through a reference point coupled to it.

The procedure for constructing the model is illustrated in detail in Fig. 6. The characteristics of the seal joints were divided into three scales: the macroscopic assembly error (mating angle α), the mesoscopic manufacturing deviations (flare angle γ and the circular runout) and the microscopic roughness. The assembly errors are on the order of 10³ µm, the manufacturing deviations are on the scale of 10² µm, and the roughness is limited to the range of 10 µm. By establishing the geometric model including the mating angle α, the flare angle γ and the circular runout, the initial relative position of the two contact parts can be described. Based on the analyzed relative position, the FE model was modified, including the surface characteristics, such as the flare angle, surface material and roughness. In this way, the contact status of the seal ring could be accurately simulated.

The geometric-based multiscale finite element model developed in this paper could precisely predict the position of the plastic indentation, or the so-called seal ring. Considering the effect of the parameters on three scales, the macroscopic errors, the mesoscopic deviation and the microscopic roughness, the proposed model is reliable enough to predict the contact state of the seal interface. To validate the proposed model, assembly experiments were conducted, and the results were fitted well with the simulation in both the position of the seal ring and the surface topography of the interface.

3.1 Experiments

3.1.1 Joint assembly

An experimental setup was designed to validate the contact model proposed above. As shown in Fig. 7, first, one side of the tube was fixed on the optical platform with a displacement accuracy of 0.1 mm and a rotation accuracy of 1°, and the joint was preassassembled manually. Then, by turning the outer nut with a torque wrench with an accuracy of 0.01 N•m, the joint was assembled tightly by the axial force. In this way, macroscopic assembly precisions such as concentricity and step difference are guaranteed.

It is worth noting that there existed a mating angle α between the axial lines of the tube and joint, which was fairly common in the actual assembly process. To measure the angle α accurately, a cross-shaped laser was fixed parallelly on both the tube and the joint after assembly. On the other side, a ruler was set horizontally to receive the projection of the laser, which was 550 mm away from the laser. By calculating the tangent value of the measured value on the ruler to the distance between the ruler and laser, the angle α could be obtained. In this paper, the mating angle is α = 2°. The joint used in the experiment had a flare angle γ_j of 39°, and the flare angle γ_t of the tube was 37°. The roughness was limited to 10 µm, and the circular runout was limited to ± 50 µm.

3.1.2 Contact surface scanning

It is possible to establish a full-scale multiscale finite model logically by scanning the whole topography of the tubes and joints. However, considering the unacceptable time consumption of both scanning the full-scale microtopography and computation for tens of millions of elements, the model was conducted only based on several typical surfaces picked from each contact joint and tube.

As illustrated in Fig. 8, the rough surface characteristics of the joint and tube were measured with a Hirox RX-100 3D microscope whose longitudinal accuracy can reach up to 0.07 µm and horizontal accuracy is 0.50 µm. In addition, optical auto zoom technology was utilized so that the topography of surfaces could be measured accurately. To offset the measurement error induced by the inherent flare angle of the tube, a fixture was designed to support the specimens, guaranteeing that the generating line of the measured surface was parallel to the horizon of the microscope.

3.2 Results comparison

The seal ring position could be observed through the experiments mentioned above. In addition, taking the results of the scanning experiments as input, the contact seal ring will be simulated via the model developed in this paper. The experimental results of both the position and the deformed surface topography of the seal ring are in high agreement with the simulation.

The initial relative position of the two contact parts is impacted by the mating angle α as well as the circular runout, the flare angle γ, which is determined by the manufacturing technology. After the joints were disassembled, the plastic indentation on the joint surface could be observed clearly with the help of the 3D microscope experimentally, as shown in Fig. 10. Considering all the parameters above, the geometric model providing the position of the two contact parts makes the FE model able to simulate the position of the seal ring. Via the assembly process simulation, the seal ring will be obtained on the surface. Then, since the seal contact surface is symmetrical, typical angles ranging from ϕ = − 90° to 90° were picked up to show the differences in the seal ring (plastic indentation) position.

As illustrated in Fig. 10, the simulation results are in good agreement with the experimental results. The contact surfaces with plastic indentation obtained by both experiments and FEM show that with the increase in the circumferential angle, the position distribution of the seal ring (plastic indentation) moves away from the tube edge. The width and depth of the plastic indentation are also related to the circumferential angle ϕ. In Fig. 10, the size of the indentation's normal deformation variable is represented by the indentation's color shade. When the indentation is close to ϕ = − 90°, it is evidently deeper and narrower. For ϕ = 90°, however, its area increased, but the indentation became lighter and difficult to observe. This can be seen in the simulated results, where the plastic deformation is more concentrated on the ϕ = -90° side and more dispersed on the ϕ = 90° side. This could be explained by the contact stress concentration induced by the state of contact between two surfaces shown in Fig. 13. Under the combined influence of the mating angle and the flare angle, there exists a wedge that shows an uneven distribution along the circumferential angle, as shown in Fig. 9. When ϕ = − 90°, the angle between the wedge angle is larger, resulting in a smaller contact area. As ϕ approaches ϕ = 90°, more area comes to contact, the local stress is reduced, and the indentation becomes inconspicuous. Listed in Fig. 10, the plastic indentation at ϕ = 0° is located at approximately 2254.38 µm for the experiment and 2281.94 µm for the FEM. The errors of the FEM prediction of the seal ring position are limited to 8%. The model is credible in predicting the position of the seal ring.

In addition, the distribution of the seal ring is not parallel to the circumferential direction but rather at a certain angle to it, which further illustrates the impact of the mating angle on the distribution of the seal ring. It is most obvious at ϕ = 0° and nearly not in the direction of ϕ = ± 90°. The discontinuity of the plastic deformation distribution on the surface of the pipe simultaneously illustrates the correlation between the simulation and roughness. Above all, the model created in this work is able to precisely predict not only the initial relative position of the sealing ring but also the microscopic contact status at the sealing ring, which is crucial for assessing the effectiveness of the flare joint's seal.

Simulations were carried out via the developed geometric-based multiscale FE model. The effects on the seal ring on multiple scales were taken into account and discussed in this section.

4.1 Assembly error at macroscale

During the assembly process, the assembly error was presented in the form of mating angle α. It can first influence the initial position of the two contact parts and then impact the contact status at the interface along the whole seal ring.

4.1.1 Initial position for various mating angles α

The position of the seal ring at the interface changes with increasing mating angle α. Taking the contact position (at φ = 0°) as a reference, the positions of the seal ring along the joint’s generating line are shown in Fig. 11a). It is clear that each contact point is located in a nearly symmetrical pattern on both sides of the reference point. As the circumferential angle ϕ increases, the height of the initial seal ring position grows nonlinearly for a certain mating angle α. Meanwhile, its growth rate decreases as the mating angle increases. The radial contact angle and the contact area also vary with position, which makes the contact situation more complex. The larger the distance between the contact pairs, the greater the possibility of leakage. This indicates that the rate of change in the contact position increases as the circumferential angle's absolute value decreases; however, the contact position of the seal ring stabilizes once the circumferential angle approaches 60°. Through this phenomenon, we can determine that the seal ring is more equally distributed and more likely to make reliable sealing contact when the circumferential angle is between 60° and 90° in absolute terms. However, a stronger uncertainty to contact results from a larger angle between the pipe and the joint's cross-section perpendicular to the axis when the angle is closer to 0°.

When the assembly angle's effect on the position of the seal ring in Fig. 11b) is calculated and observed, it is clear that the seal ring's position grows almost linearly with the growth of the assembly angle and that its slope gradually decreases as the circumferential angle rises. This suggests that the influence of the mating angle on the position varies depending on the corresponding circumferential angle. The position of the seal ring is sensitive to the mating angle when the circumferential angle is small in absolute value, and vice versa when the circumferential angle is large. The position at the greater absolute value of the circumferential angle, however, varies more considerably and defines the ultimate span of the seal ring distribution when taking the absolute value of the seal ring position into account. The distribution of the seal rings in the center will be involved, which will have an impact on the total sealing state even if it has a more stable contact state in this condition. Therefore, it is best to minimize the possibility of leakage brought on by an excessive assembly angle.

4.1.2 Contact status along the seal ring

Not only the initial relative position mentioned above but also the contact states of the two contact parts are impacted by the mating angle α. Here, two states of contact on the seal ring will be analyzed separately: the critical contact state and the complete contact state that forms the complete seal ring. The displacement needed to achieve the critical contact status grows steadily with increasing mating angle α when the first point of contact is taken as the reference. This phenomenon is consistent with the results shown in Fig. 11, meaning that the longer the distance, the larger the displacement needed. In addition, the displacement, which is essential for the formation of a complete seal ring, is also highly related to the mating angle α. It influences the assembly posture of the joints, which is particularly evident at φ = ±90°. Because the initial gap between the joint and the tube varies with the circumferential angle φ, the contact status for each circumferential angle is different at each time.

Here, when α = 2°, we set the first point of the tube getting in contact as the reference point where the displacement is zero and pick φ = 0° as an example. Due to the existence of the mating angle, the contact edge of the tube has a certain angle to the circumference of the joint; thus, the contact starts from one side. The initial contact state is point contact, rather than line contact, confirming the existence of the asperities on the surfaces. As the tube approaches the joint driven by the axial force, more area comes into contact. When the displacement reaches 106.19 µm, a complete contact seal line is formed the first time. In the following, we call this state “the critical contact status”. The contact stresses are not uniformly distributed in the circumferential direction. This is due to the relative initial position of the wedge slit formed by the inherent radius of curvature of the two contact boundaries under the influence of the mating angle. To obtain a more stable seal performance and to resist higher fluid pressure from leakage, a larger displacement (113.27 µm) is adopted to increase the real contact area at the seal interface. During this process, both the contact stress and the plastic deformation of the contact surface increase steeply. Surface plastic deformation develops from local to global along the circumferential direction, ensuring good contact at the interface.

As shown in Fig. 12 and Fig. 13, for the whole seal ring, it did not take much feed to reach the critical state, from 1.25 to 3.62 µm. When the displacement equals 70.67 µm, the contact pair at φ = -90° has already formed a fine seal ring. At the same time, the tube at φ = 0° is just getting in touch with the joint, and the tube at φ = 90° is still 46.00 µm away from the joint. Moreover, when the displacement reaches 117.92 µm, all the other parts have been fully contacted, only where φ = 90° is still in critical contact.

The influence of the mating angle on the contact state is reflected not only in the angle of the contact boundary and the difference in the contact sequence but also in the interfacial real contact area. The mating angle makes it distribute unevenly and is mainly embodied in the area close to φ = 90°. The contact area at the critical state increases dramatically, from 0.83–6.67%. At φ = 90°, when the feed is equal to 10 µm, the real contact area is 12.33%, which is 3.29 times that at φ = -90° (3.75%). Moreover, the seal ring at φ = 90° exhibits a surface contact much wider than a contact line at φ = -90°. As a consequence, the maximum plastic indentation at φ = -90° is deeper, 1.91 times that at φ = 90°.

In addition, for α = 2°, it can be seen from Fig. 13b) that the initial contact is a surface contact rather than a line contact, which indicates that the two contact surfaces have reached a state of almost parallelism. It could be considered a critical value for the mating angle. Continuing to increase the angle will convert the initial contact from the opposite direction to linear contact, reduce the contact area, and seriously affect the contact state.

On the one hand, the mating angle could make one side of the seal ring more stable. However, on the other hand, the uneven distribution of the seal ring induced by the asynchronous contact will make it easy for the pressure fluid to escape through the micro leakage channel at the seal ring. Therefore, for a better seal performance, the mating angle should be limited to 2° for the case studied in this paper.

4.2 Manufacturing deviation at mesoscopic

The manufacturing deviation is mainly divided into the manufacturing deviation of the flare angle of the tube end and the radial runout of the joint. This section investigates how the interfacial contact state at the seal ring is affected by the flare angle and the joint's radial runout.

4.2.1 Tube flare angle γ

The tube flare angle γ impacts the contact status of the seal ring by defining the initial relative position when two parts approach. To investigate the influence of the flare angle, 3 typical tube flare angles γ (37, 38, 39°) are applied in the FE simulation by rotating the tube part model around the circumferential axis through point C.

An asynchronous contact phenomenon induced by the flare angle is reflected in the relationship of the displacement and axial force during the assembly process, which can be seen in Fig. 14. The amount of feed required for contact increases as the flare angle increases, ranging from 97.67 µm to 312.60 µm. Unlike the other two curves, for γ = 37°, there are two fluctuations during contact. This is because at 37 degrees, the contact state is closer to a line contact than a surface contact, making the contact process more influenced by anisotropic asperities. As seen from Fig. 15, the contact region corresponding to γ = 37° is significantly less than that corresponding to γ = 39° when contact has just occurred. As the feed increases, the former increases the contact area by extending the contact region, in other words, by contacting more asperities of the same grade, while the latter increases the contact area by making smaller asperities contact in the existing contact region. Thus, the latter curve does not fluctuate, while the former does.

In addition, the same axial force will cause different feeds (deformation). When the axial force increases to 100 N, the displacement at γ = 37° of 18.01 µm occurs, while those at γ = 38 and 39° are only 8.14 µm and 4.33 µm, respectively. This phenomenon could be explained by the fact that the former contact area is smaller, the stress is more concentrated and easily deformed, while the latter two contact areas are relatively larger, the pressure distribution is dispersed, and the local stress is smaller and hard to deform.

The influence is also reflected in the differences in the contact position and the plastic deformation of the joint surface. As shown in the first line of Fig. 15, the larger the flare angle is, the closer the contact point is to the outer edge of the joint. Under the same loading F = 100 N, the tube with a smaller flare angle has a higher rate of plastic deformation. As shown in the third line, the maximum plastic deformation at γ = 37° is larger than the other two. On the one hand, the appearance of plastic deformation indicates that the two bodies come into contact deeply. But on the other hand, once there is relative movement between two contact bodies, making the initial indentation staggered, the plastically deformed surface will reduce the sealing capability of the interface. Therefore, a lower angle is suitable for a low number of assemblies, and vice versa for larger angles. For high-pressure environments requiring a large contact area, the flaring angle should be selected to a larger value.

4.2.2 Radial circular runout

The radial circular runout of the joint and tube, which is of the mesoscopic scale, are also included in the manufacturing deviation in addition to the tube flare angle. The circular runout of the joint is limited to ± 50 µm per the manufacturing standard. Several typical runout values are chosen for a better understanding of the impact of runout on the contact state of the seal ring, which are 0, ± 25, and ± 50 µm. The flare angle γ_t is chosen to be 37°, γ_j equals 39° and the mating angle α is 0°. Taking the tube edge as a reference, the initial distance is introduced by the circular runout, and then the displacement needed to get into contact will vary.

The radial circular runout mainly causes asynchronous contact during assembly. As shown in Fig. 16, it is easy to find that the beginning points for contact increase linearly with increasing runout. The displacement ranges from approximately 39.23 µm to 162.15 µm, corresponding to a circular runout from − 50 µm to 50 µm. From the force-displacement curve, we can see that depending on the corresponding contact morphology, the entire contact process for each runout value could be divided into several stages, according to which the curves are divided into different regions. Regions 1 and 3 represent the contact surfaces that are easily pressed down, while regions 2 and 4 indicate the stage where the contact area increases more slowly. The stages of adjacent curves are similar only when the phases are different, which is caused precisely by the different positions of the asynchronous contact point on roughness.

Two sets of curves, runout = -25 µm and runout = 0 µm, were selected for comparison to illustrate the characteristics between the different stages. Shown in Fig. 17, state 1 represents the starting point of region 2, state 2 represents the transition point to region 2, and state 3 is the end point of region 3. In terms of the growth rate of the interfacial real contact area under unit displacement, region 2 is larger than region 3. For runout = -25 µm, the rates of real contact area growth are 1.83 and 0.64%/µm. For runout = 0 µm, they are 2.59 and 0.84%/µm. The reason for the various growth rates may be related to the asperities contacting during the two contact processes. From state 1 to 2, corresponding to the low growth rate, it can be found that the contact region hardly grows, and the increased area mainly comes from the fact that as the normal displacement increases, more asperities included in the inherent contact region are generated by the contact. In regard to the high growth rate part, the increased area mainly comes from the additional contact region, where the smaller asperities in fresh contact are more easily pressed down, and thus, the roughness growth rate is greater at this point compared to the previous stages. Since the turned joint has an anisotropic periodic roughness distribution, the two stages cycle through the contact process.

It can be seen that the radial circular runout affects the contact sequence so that the contact points have different roughness phases, resulting in different contact stages. To make fine contact between contact pairs at various values of circular runout, it is necessary to provide sufficient axial force so that they all cover at least two contact processes. The minimum axial force required here is approximately 112.65 N.

The radial circular runout affects the contact sequence and has a smaller effect on the seal ring contact condition than the assembly angle and flare angle. Reducing the circular runout value is only meaningful if the first two error values are controlled. As long as sufficient torque is provided, the impact induced by circular runout can be served.

In this paper, a novel FE model has been developed to map the relationship between the multiscale contact parameters and the contact status of the seal ring during assembly. The developed model was validated by assembly experiments, further demonstrating that the model can accurately describe the seal ring contact state under the influence of multiscale assembly parameters.

It has been discovered that the macroscopic mating angle α, through changing the initial relative position of the two contact pieces, dictates the actual formation position and shape of the seal ring. The influence of α is more obvious in areas close to φ = 90°. The mating angle can cause an uneven distribution of the real contact area; the contact area at φ = 90° is more than 8 times that at φ = -90°, and the maximum plastic deformation at φ = 90° is 1.91 times that at φ = -90°.

For the mesoscopic parameters, the flare angle γ_t has less influence on the contact status of the seal ring compared to the mating angle. The most obvious effect is mainly on the difficulty and width of plastic deformation at the seal ring. For each radial circular runout value, the phase of the contact point on the anisotropy roughness varied, resulting in asynchrony of the micro-interface contact processes. Although the circular runout has the least effect on the seal ring formation results, the analysis of runout can be very helpful in understanding the micro contact process.

In conclusion, variables of all scales have varied degrees of influence on the position and contact status of the seal ring creation throughout the assembly process, which in turn influences the sealing effect of the contact surface of the joint. As a result, the model created in this work is capable of precisely predicting the contact state of the seal ring and evaluating the impact of each parameter on it.

Authors' contributions Lichuan Deng: Methodology, Software, Writing - Original Draft. Bin Luo: Conceptualization, Methodology, Funding acquisition. Kaifu Zhang: Supervision, Funding acquisition. Hui Cheng: Resources, Supervision. Langkun Lin: Validation, Visualization. Biao Liang: Conceptualization, Writing - Review & Editing.

Funding This research is funded by the National Natural Science Foundation of China (Grant No. 52035011) and the Natural Science Basic Research Program of Shaanxi Province (Grant No. 2022KJXX-74).

Availability of data and material All data generated or mentioned during this study are included in this published article.

Code availability No code was provided in this manuscript.

Ethics approval This study does not involve ethical issues.

Consent to participate This study is not a study of life science, so the “Consent to Participate” was not applied.

Consent for publication All of the authors have reviewed the final version of the manuscript and approve it for publication. We claim that none of the material in the paper has been published or is under consideration for publication elsewhere.

Competing interests The authors declare no competing interests.

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No competing interests reported.

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A novel finite element model for the flare joint contact status at the seal ring with multiscale factors during assembly

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Abstract

Figures

1. Introduction

2. Modeling for the contact status

2.1 The relative position of two parts during assembly

2.2 The structural parameters

2.2.1 Model generation

2.2.2 FE material properties and boundary conditions

3. Model validation

3.1 Experiments

3.1.1 Joint assembly

3.1.2 Contact surface scanning

3.2 Results comparison

4. Effects of parameters on the contact status at the seal ring

4.1 Assembly error at macroscale

4.1.1 Initial position for various mating angles α

4.1.2 Contact status along the seal ring

4.2 Manufacturing deviation at mesoscopic

4.2.1 Tube flare angle γ

4.2.2 Radial circular runout

5. Conclusions

Declarations

References

Additional Declarations

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