Analysis of static sealing rules of foamed silicone rubber based on a porous media model

Abstract

We established a method for calculating and analyzing the static leakage rate based on a porous media model for foamed silicone rubber materials. The mechanical properties of the foamed silicone rubber material under macroscopic compression were described by the Ogden third (foam) model in the finite-element hyperelastic model. It solved the problem of difficult convergence of large compressible and volume compressible cell materials. The size and distribution of the cells on the surface of the foamed material were obtained by a white-light interferometer and mathematical fitting. The boundary conditions for solving the porous medium model were obtained by the coupling of the macroscopic contact pressure and the microscopic cell contact pressure. For the unique cell structure and contact state of the surface of the foamed material, the flow state of the fluid at the sealing interface was described by a porous medium model, and the leak rate was obtained. In addition, this article analyzed the effect of different compression and the relative pressure of the sealing end face on the leakage.

Keywords

Leakage foamed silicone rubber finite element simulation porous media model static sealing

Introduction

Foamed silicone rubber was prepared by the foaming of a silicone rubber matrix, which acts as a skeleton, and bubble holes were distributed evenly between the skeleton. The rubber exhibited excellent characteristics of the foamed materials and the silicone rubber materials, such as strong compressibility, good resilience, antiaging, and corrosion resistance, so it is used widely as a static sealing member. The gap in the sealing area is one of the most important factors that affect the sealing behavior. Larger gaps can lead to an increase in leakage. The gaps are determined by the external load and surface topography. Although the static sealing system appears simple, the sealing behavior is affected by many factors. If the gaps and leakage condition of the sealing system can be predicted theoretically according to the external working condition and the surface morphology of the rubber, the risk of leakage can be avoided and mechanical equipment failure can be prevented.

Gap distribution rules and leakage rate have been the focus of extensive research by scholars on static sealing systems. In the field of percolation theory, when two rough faces come into contact, the contact between the surfaces is incomplete. Noncontact areas form the gaps. The gaps communicate with each other and cause leakage. The leakage threshold is termed the percolation threshold and the leakage trend is determined by the percolation threshold. Lorenz and Persson¹ studied the problem of the static sealing of solid rubber and metal plates. The influence of channel clearance below the percolation threshold was ignored. By combining the percolation theory with the single-hinge sealing theory, the leakage rate at the hub was calculated^2,3 Bottiglione et al. believed that the pressure drop was caused by the smallest gap area. The leakage density function and the average separation rate of the contact faces were obtained.^4,5 In the field of homogenization theory, by considering the influence of rough peaks on the fluid in the gap, the flow transport equation was modified by introducing flow factors. Vallet et al. described the microscopic flow field by the Stokes and Fick equations and calculated the macroscopic permeability and diffusion by the volume-average method.⁶ Etsion and Front calculated the leak rate of the static sealing surface by introducing a flow factor method.⁷ In the field of porous media theory, the contact interface was treated as a porous medium space. Tripp et al. discussed that with sufficient rough peak contact, the porous medium model could better describe the flow variation of the fluid.⁸ Lo constructed a porous medium model through porous medium theory, percolation theory, and flow factor theory.⁹ Zheng et al.,¹⁰ Lesinigo et al.,¹¹ and Shou et al.¹² used fractal theory and a porous media model to analyze the leakage of fluid medium between two contact surfaces.

However, the above theoretical model was insufficient to solve the static sealing problem of foamed silicone rubber materials. First, the existing percolation model only gave the leakage situation at the seal interface junction. In fact, the distribution of the entire flow channel gap of the foamed material influenced the fluid pressure drop and leakage. Second, the gaps and leakage were small in the static sealing system, which resulted in failure to form a complete flow state. Therefore, the calculation by the introduction of the flow factor was inaccurate. Third, the existing theoretical model of porous media was often combined with fractal theory to obtain the leak rate. Because the foamed silicone rubber surface is composed of a bubble hole and rubber matrix, it was not verified that the characteristics of the material conform to the requirements of self-correlation and scale invariance in fractal theory. Finally, the contact pressure affects the leakage rate in the static seal systems significantly, but there is a lack of a theoretical explanation for the intrinsic relationship between the contact pressure and the leakage rate.

By focusing on static-seal surface leakage, the sealing surface is regarded as porous media processing. Combined with a finite-element simulation, the microcontact mechanics model, and a flow continuous equation, a numerical simulation model for foamed materials is constructed.

This article addressed the following issues: First, the establishment of a finite-element model of foamed silicone rubber sealing material solved the problems of stress-strain nonlinearity, volume compressibility, and large compression of the foamed silicone rubber material during the simulation process. Second, the foamed silicone rubber microscopic surface cell structure treatment. The surface of the foamed silicone rubber was composed of a rubber substrate and cell. The cell geometry was much larger than the rubber substrate rough peak. At this time, the rough peak on the rubber substrate and the rough peak on the platen could be neglected, and the rubber substrate protruding between the cell was regarded as a new rough peak. The size and distribution of the cell substrate of the cell could be better described by means of a white-light interferometer and mathematical processing. As a result of the white-light interferometer test, the sealing interface leakage channel originated from the gap between the cell and the rubber substrate rather than the rough peak on the rubber substrate. Third, the boundary conditions of the porous medium model were coupled by macroscopic contact pressure and microscopic rubber matrix contact pressure coupling. The porous medium theory was used to simulate the state of the fluid medium passing through the sealing interface. The correspondence between macroscopic contact pressure and leak rate at the sealing interface was established. Finally, the leakage rate on the sealed interface was solved.

Experimental

Material mechanical properties acquisition

During foamed silicone rubber compression, rubber deformation originated from bubble hole skeleton deformation and rubber solid deformation. At the beginning of compression, the rubber skeleton changed, the volume of the bubble hole decreased, and the bearing object was mainly the bubble hole skeleton. Later, during the compression process, the bubble hole structure was compacted and the silicone rubber entity became the main bearing object. During intermediate compression, the stress state of the foamed silicone rubber was affected by both, which resulted in a high degree of nonlinearity of the mechanical parameters of the foamed material, such as the Poisson’s ratio and the elastic modulus. Therefore, the uniaxial compressive stress and strain and transverse strain data need to be obtained before finite-element simulation. The complex mechanical state of the foamed silicone rubber compression process was described by fitting the experimental data to the constitutive model.

The stress–strain data that were used in the finite-element simulation of the foamed silicone rubber were obtained by a uniaxial compression test of the material-grade sample. This article used the standard closed-cell foamed silicone rubber material that was produced by Shenzhen Fuchengwei Technology Co., Ltd (Shenzhen City, Guangdong Province, China). The material-grade sample size was 40 mm × 40 mm × 9.5 ± 0.1 mm. The stress–strain measurement was performed by using the ZCGW-W10KN universal testing machine supplied by Jinan Zhongchuang Industrial Testing System Co., Ltd (Jinan City, Shandong Province, China). The test procedure was carried out in accordance with GB/T 18942.2-2003. The specific measurement method was as follows: the sample was compressed at 5 ± 1 mm min⁻¹ with a compression amount of 80%. The above steps were repeated four times. The first three times used mechanical zero adjustment, and the fourth compressive stress and strain data were recorded. A camera was used to record the compression of the fourth foamed silicone rubber. The key position picture in the compression process was determined, the transverse strain data were read in the image processing software, and the transverse strain data were obtained by interpolation.

Micromorphology parameter acquisition

The acquisition of the microscopic surface-morphology parameters was the basis for microscopic leakage calculations. Key parameters of the surface morphology of the foamed silicone rubber were obtained by using a ZYGO-Nexview three-dimensional white-light interferometer. Figure 1(a) shows the surface-topography distribution of the foamed silicone rubber under a 10× magnification. In the 834.370 × 834.370 µm² area, the red parts were the silicone rubber matrix, the blue parts were the bubble holes, and the arithmetic average height of the entire surface was Sa = 10.515 µm. The longitudinal section shows that there were 16 prominent protrusions. If we consider that there were approximately 16 × 16 rough peaks in the entire square area, the roughness peak density was 3.23 × 10⁸ m⁻². Figure 1(b) shows the rough peak distribution of the rubber matrix with the roughness of Ra = 0.734 µm. The surface roughness of the rubber was caused by the main bubble hole structure. The effect of the rough peak of the rubber matrix was not of the same order of magnitude compared with the pore structure. If we consider subsequent microscopic contact mechanics calculations, the rough surface of the rubber matrix was smooth. On this basis, the rough surface height of the foamed silicone rubber was analyzed statistically with a skewness = 0.395 and Kurtosis = –0.526. Both parameters were nearly close to 0, so it is considered that the roughness peak height distribution of the foamed silicone rubber approximated a normal distribution, and the standard deviation of the roughness peak height was σ = 7.49 µm. To obtain the peak radius parameter in the subsequent contact model, combined with Figure 1(b), the rubber matrix was regarded as the spherical crown. A circle with radius R was found on the sectional plane so that as many experimental test points fell on the circle as possible. The data were fitted to obtain the results of Figure 1(c), and the radius of the rough peak was R = 40.6 µm.

Figure 1.

White-light interferometer test results and single rubber base fitting results. (a) The overall distribution of foaming silicone rubber surfaces Sa = 10.515 µm, Sq = 11.871 µm, and Sz = 56.789 µm. (b) Rubber substrate surface rough peak distribution status Sa = 521 µm, Sq = 0.626 µm, and Sz = 4.066 µm. (c) Single rubber base fitting result, single peak radius is R = 40.6039 µm.

Verification experiment

Pressure-sensitive paper was used to verify the subsequent finite-element simulation results. The pressure-sensitive paper consisted of a microparticle film of a chromogenic material and a color-forming film. When the pressure-sensitive paper was subjected to pressure, the microparticles ruptured and reacted with the color-forming film to form a red mark. The magnitude of the contact pressure was characterized by the degree of microparticle breakage. We used the Japanese Fuji two-piece ultra-low-pressure type pressure-sensitive paper in the experiments.

During the experiment, the microparticle and color-forming films were first cut according to requirements. The film was 40 × 40 mm², and the two film layers were stacked for use. The rubber sample (18.3 × 40 × 9 mm³) was placed on a ZCGW-W10KN universal testing machine. The pressure-sensitive paper was placed between the sample and the indenter of the universal testing machine. The compression distance was calculated from the thickness of the rubber sample. The compression amounts were 40%, 50%, and 60%, respectively. The sample was compressed at 5 ± 1 mm min⁻¹. Compression was completed after 2 min. The color film was removed, and the standard color chart was selected according to the ambient temperature and humidity at the time of measurement. The measured value was compared with a standard colorimetric card to determine the surface pressure distribution of the rubber sample at a certain compression.

Method

Finite-element simulation

Commercial software ANSYS Release 15.0 was used to simulate the three-dimensional statics of the sealed structure. The sealing system consisted of three parts: the pressure plate, seal, and limit ring. The structure is shown in Figure 2. The left side is the complete sealing structure and the right side is the sectional view. In terms of computational efficiency, and considering the symmetry of the seal structure, a seal of one-eighth was taken for three-dimensional geometric modeling. The geometric model dimensions were 45-mm length, 9.265-mm width, and 2.66-mm height. The limit ring was a one-fourth ring with a height of 1 mm, an outer diameter of 8 mm, and an inner diameter of 6 mm. For the convenience of description, the width of the seal is the X direction, and a positive direction exists in the free face. The length direction is the Y direction, and the positive direction is away from the limit block. The thickness direction is the Z direction, and the positive direction is the compression direction.

Figure 2.

Diagram of the sealing system structure, the sealing system consists of a metal pressure plate, seal, and limit ring.

The Ogden foam third-order model can be selected for high-compressibility elastic foamed materials. The stress–strain data are imported into the constitutive model for fitting. The fitted constitutive model parameters are shown in Table 1. The following is the constitutive relation of the model:

W = \sum_{i = 1}^{N} \frac{μ_{i}}{α_{i}} [J^{\frac{α_{i}}{3}} (\bar{λ_{1}^{α_{i}}} + \bar{λ_{2}^{α_{i}}} + \bar{λ_{3}^{α_{i}}}) - 3] + \sum_{i = 1}^{N} \frac{μ_{i}}{α_{i} β_{i}} (J^{- α_{i} β_{i}} - 1)

Table 1.

Ogden foam third-order fitting parameters.

Name	Value
Mu1	−0.07629 MPa
A1	−0.33568
Mu2	0.65095 MPa
A2	0.37536
Mu3	0.51325 MPa
A3	−0.14753
B1	3.6612
B2	1.0116
B3	−2.4899

where W is the strain energy;

\bar{λ_{k}^{α_{i}}}

(k = 1, 2, 3) represents the principal stress offset; J represents the elastic deformation gradient determinant; and N,

μ_{i}

α_{i}

, and

β_{i}

are the material constants.

Preprocessor: The material property definition, pressure plate, and limit ring are defined as rigid bodies, and the foamed silicone rubber is defined according to the parameters of the Ogden foam third-order model fitting. Face-to-face contact pairs are used for possible contact locations between the various components. A fixed displacement constraint was applied to the symmetry plane. No load was applied to the outer boundary of the seal, which was in a free state. A displacement load was applied to the surface of the platen and the amount of compression was specified. The finite-element mesh was divided by a tetrahedral mesh, which had a good adaptability. To ensure the efficiency and calculation accuracy, the method of local refinement of the grid was used to refine the mesh between the seal and the pressure plate.

Postprocessing: After the simulation calculation had been completed, the pressure distribution option in the contact tool was set. The grid coordinates and the corresponding contact pressure were derived. The descending order was arranged in the Y direction, and the leak rate analysis section was intercepted. Then, the X direction was arranged in descending order, and the contact pressure of the same X-coordinate was averaged. Finally, an array of distributions of contact pressures in the X direction was obtained.

Contact mechanics analysis

The rough peak contact state and flow channel distribution on the microsealing surface are the premises of the calculation of the subsequent leak rate. This section gives the flow-channel distribution by coupling the macroscopic contact pressure with the contact pressure of the microrough peak. According to the test results from the white-light interferometer, the rubber surface was formed by the staggered distribution of a bubble hole structure and a silicone rubber matrix. From Section “Micromorphology parameter acquisition,” it is concluded that the silicone rubber-matrix protrusion can be regarded as a spherical rough peak of radius R. The height z of the rough peak obeys the Gaussian distribution. The single rough peak contact is Hertz contact under pressure deformation. For the entire rough peak distribution, the microcompression state of the foamed silicone rubber will be described by the Greenwood–Williamson contact model.¹³ Figure 3 provides a schematic diagram of the contact model.

Figure 3.

Schematic diagram of the contact model, h is the reference height of the neutral plane of the rough peak, and the nominal gap d refers to the distance from the contact surface to the neutral plane of the normal distribution of the rubber rough peak.

Because the surface roughness of the foamed silicone rubber was much larger than the roughness of the metal plate, the metal plate can be used as a smooth surface. According to the Greenwood–Williamson contact model, the rough peak contact pressure is calculated as follows:

p = \frac{F_{tol}}{A_{n}} = \frac{4}{3} ξ R^{\frac{1}{2}} E^{'} \int_{d}^{\infty} Γ (z) {(z - d)}^{3 / 2} d z

where ξ represents the roughness peak density on the nominal contact area, $F_{tol}$ represents the total contact pressure of the rough peak, h is the reference height of the neutral plane of the rough peak, and the nominal gap d refers to the distance from the contact surface to the neutral plane of the normal distribution of the rubber rough peak. The gap value is negative, which indicates the left branch of the normal distribution curve, and the gap value is positive, which indicates the right branch of the normal distribution curve. R is the rough peak radius and $E^{'}$ is the equivalent elastic modulus:

\frac{1}{E^{'}} = \frac{1 - v_{1}^{2}}{E_{1}} + \frac{1 - v_{2}^{2}}{E_{2}}

Because the elastic modulus of the metal platen is much larger than the modulus of elasticity of the seal, equation (3) can be simplified as follows:

\frac{1}{E^{'}} = \frac{1 - v_{1}^{2}}{E_{1}}

where E ₁ and v ₁ represent the silicone rubber elastic modulus and Poisson’s ratio, respectively. The mechanical parameters of the microscopic material differed from the macroscopic parameters. The contact pressure of the rough peak originated from the function of the rubber matrix, and the rough peak itself did not contain the bubble hole, which is composed of the rubber matrix. Therefore, when the microrough peak contact was calculated, the rough peak was treated as an incompressible silicone rubber. The height z of the rough peak is Gaussian, and the expression of the probability density function Γ(z) is

Γ (z) = \frac{1}{σ \sqrt{2 π}} e^{\frac{- z^{2}}{2 σ^{2}}}

If the fluid medium convection channel does not have an effect, the macroscopic contact pressure that is obtained by the finite-element simulation is equal to the microscopic rough peak contact pressure, so the nominal gap d is obtained using equation (2).

Porous media model

According to the classical porous medium model⁶ proposed by Bear and Bachmat, the convection field is solved on the characterization element. The characterization element is composed of a solid skeleton and fluid medium. The internal pore distribution has a certain statistical significance, which represents the basic characteristics of the porous media. When the characterization element is sufficiently thin, parameters, such as the internal porosity, permeability, and water radius, remain unchanged.

The following are differential equations:

η \frac{\partial^{2} q_{i}}{\partial x_{j} \partial x_{j}} - n T_{j i} \frac{\partial p}{\partial x_{j}} - η α_{i j} \frac{C_{f}}{△_{f}^{2}} q_{j} = 0

where η is the fluid viscosity, C_f is the shape factor, $△_{f}$ is the water radius, and n is the porosity. $T_{j i}$ and $α_{i j}$ are two transport coefficients. If we consider that rubber can be regarded as isotropic, the flow in the X and Y directions is equivalent $q_{i} = q_{j}$ . Therefore, only the X direction needs to be considered in the calculation. The equation is simplified as follows, where the first term is the resistance term and indicates the viscosity of the fluid itself and the solid skeleton creates resistance. The second and third items are power items, which represent the power that is generated by the infiltration and the power that is generated by the pressure difference:

\begin{matrix} \tilde{η_{x}} \frac{\partial^{2} q_{i}}{\partial z^{2}} - \frac{η}{k_{x}} q_{i} - \frac{\partial p}{\partial x} = 0 \\ \tilde{η_{x}} = \frac{η}{n T {(v_{x}^{2})}_{f f}} = \frac{η}{n T_{x}} \\ k_{x} = \frac{n T_{x} △_{f}^{2}}{C_{f} (1 - (v_{y}^{2})_{f s})} = \frac{n T △_{f}^{2}}{C_{f} α_{x}} \end{matrix}

The porosity of the current section is related closely to the contact state of the rough peak of the section. According to the distribution of the rough peaks, the porosity also obeys the Gaussian distribution in the height z direction:

\begin{array}{l} n (z) = \int_{- \infty}^{z} Γ (z) d z \\ Γ (z) = \frac{1}{σ \sqrt{2 π}} e^{\frac{- z^{2}}{2 σ^{2}}} \end{array}

The angle function $φ$ = $({(cos θ_{x} cos θ_{y})}^{- 1})_{f s}^{- 1}$ is introduced.

The permeability in the X direction can be written as follows:

k_{x} = \frac{n^{3} (z) T_{x} φ^{3}}{Γ^{2} (z) C_{f} α_{x}}

The X direction differential equation can be reduced to

\frac{\partial^{2} q_{i}}{\partial z^{2}} = \frac{n (z) T_{x}}{η} \frac{\partial p}{\partial x} + g_{x} {[\frac{Γ (z)}{n (z)}]}^{2} q_{i} g_{x} = C_{f} α_{x} / φ^{2}

Q = \int_{z_{1}}^{d} q_{i} d z

In equation (10), the upper limit of the integral is the nominal gap d, and the lower limit of the integral z ₁ is given according to percolation theory.⁴ Percolation theory holds that a percolation threshold exists at a certain magnification. No medium flow occurs in the pore flow channel below the threshold. For the Tripp⁸ view, z ₁ may take – $σ$ , where $σ$ is the rubber standard deviation of surface roughness. The leak rate is calculated and the height above – $σ$ is integrated. Because the static seal housing and the seal are both in a static state, there is no fluid flow near – $σ$ and the nominal gap d, that is, q(– $σ$ ) = 0 and q(d) = 0. The above relationships are considered as equation (9) boundary conditions for solving differential equations. g_x and T_x are parameters that are independent of height z. The change in height does not cause a change in these two parameters. The parameter determination refers to the Tripp⁸ flow-factor-related article, $I_{x} = T_{x}^{- 1} = 0.714$ , $g_{x} = 12.6953$ . The ∂p/∂x is the medium pressure gradient in the X direction, which needs to be obtained by reverse solution of the flow continuous equation.

After a clear declaration of the elemental flow description, the fluid flow in the X direction of the contact surface of the foamed silicone rubber is found. From the finite-element simulation results, the contact pressure changes and the flow path heights of the fluid are no longer the same. However, the fluid flow between the characterization elements is equal. The seal is divided into N sections along the X direction. When N approaches infinity, the pressure gradient $\partial$ p/ $\partial$ x can be regarded as a constant, then the flow rate of each segment is a function of (∂p_i )/∂x. The flow of each section is as follows:

Q_{i} = f (\frac{\partial p_{i}}{\partial x})

\frac{\partial p_{i}}{\partial x} = f^{- 1} (Q_{i})

If the N segments are summed:

\begin{array}{l} \sum_{i = 1}^{N} \frac{\partial p_{i}}{\partial x} = \sum_{i = 1}^{N} f^{- 1} (Q_{i}) \\ \frac{Δ p}{l / N} = \sum_{i = 1}^{N} f^{- 1} (Q_{i}) \end{array}

where Δp is the difference in the medium pressure between the ends of the sealing surface, and l is the length in the X direction of the sealing member. The leakage rate on the contact surface of the selected section can be obtained by programming in the MATLAB R2018b environment according to equation (11).

Solution process

The calculation process is shown in Figure 4.

Figure 4.

Leak rate calculation flow chart.

The material stress and strain data, structural parameters, and external working conditions were imported into the finite-element model to calculate the contact pressure distribution. The contact pressure was subjected to a piecewise averaging process. The key parameters of the microscopic morphology were inserted into the contact model, and after coupling with the macroscopic contact pressure, a microscopic nominal gap was obtained. The boundary conditions of the porous medium were determined by combining the permeability theory with the nominal gap. The differential equation was solved in the height direction of the characterization element. Combined with the flow continuity equation, the pressure distribution and leakage rate of the fluid medium were obtained.

Foamed silicone rubber seals are used commonly in lithium battery enclosure seals to prevent outside rain from entering the interior. In practice, water pressure tests have been used to verify the sealing performance of the sealing system. The water depth tends to be selected as 1, 1.6, and 2 m. The above water depth corresponds to a relative pressure of 5 × $10^{3}$ , 8 × $10^{3}$ , and 1 × $10^{4}$ Pa. To explore the variation in sealing performance for different compression amounts and water depths and to verify the applicability of the model, parameters such as the nominal clearance and leakage rate will be calculated for a 40%, 50%, and 60% compression and a water depth of 1, 1.6, and 2 m. Table 2 presents the specific calculation parameters.

Table 2.

Calculation parameters.

Parameter name	Numerical value	Parameter name	Numerical value
Rough peak density ξ	3.23 × 10⁸ m⁻²	g_x	12.6953
Micro Poisson’s ratio ν	0.48	Pressure difference on both sides p ₀	5 × 10³ to 1 × 10⁴ Pa
Micro elastic modulus E	20 MPa	T_x	1.371742
Rough peak height standard deviation σ	7.49 µm	Dynamic viscosity	0.8937 × 10⁻³ Pa × s
Seal width 1	9.15 mm	Rough peak radius R	41.3 µm

Results and discussion

Figure 5 shows the finite-element simulation results of foamed silicone rubber. Figure 5(a) to (c) corresponds to the sealing surface contact pressure distribution at 40%, 50%, and 60%, respectively. The overall contact pressure decreases from the centerline to the free boundary. When the compression is 40%, the average contact pressure is 0.417 MPa, and the maximum contact pressure is 0.567 MPa. When the compression amount is 50%, the average contact pressure is 0.665 MPa and the maximum contact pressure is 0.811 MPa. When the compression is 60%, the average contact pressure is 1.14 MPa and the maximum contact pressure is 1.33 MPa. Figure 6(a) shows the results of the pressure-sensitive paper experiments, which correspond to the experimental results of 40%, 50%, and 60% compression from right to left. The contact pressure at the edge of the sample is large. The average contact pressure was 1.2 MPa when the compression was 60%. When the amount of compression was 50%, the average contact pressure was 0.7 MPa. When the amount of compression was 40%, the average contact pressure was 0.3 MPa. The experimental results are consistent with the simulated calculated values, as shown in Figure 6(b). Because of the limited range and accuracy of the pressure-sensitive paper, the variation of the pressure gradient cannot be given clearly. Therefore, the comparison between the experimental results and the experimental test results is carried out by obtaining the average value of the contact pressure.

Figure 5.

Finite-element simulation results: (a) compression 40%, maximum contact pressure of 0.56377 MPa; (b) compression 50%, maximum contact pressure 0.81153 MPa; and (c) compression 60%, maximum contact pressure of 1.3353 MPa.

Figure 6.

Pressure-sensitive paper test results: (a) pressure-sensitive paper color rendering result and (b) comparison of theoretical calculation values with experimental values.

The nominal gaps represent the range of flow paths through which the medium can flow, which reflects the effect of the macroscopic contact pressure on the microscopic flow paths. The contact pressure distribution array is included in the Greenwood–Williamson contact model to calculate the nominal gaps of the flow channels, and Figure 7 shows the calculated results. The three curves represent the distribution of the gap for compressions of 40%, 50%, and 60%, respectively. The shrinkage distribution of the gap occurs from the free boundary to the centerline. The size of the gap decreases with an increase in compression amount. When the amount of compression was 40%, the average gap was 2.1 µm. When the amount of compression was 50%, the average gap was –0.228 µm. When the amount of compression was 60%, the average gap was –3.30 µm. For the position of the contact area above 35 mm, the contact state was complicated, and the contact pressure distribution was discontinuous with the result that the size of the gaps fluctuates greatly. The fluctuation of the gaps will lead to a loss of fluid momentum of the medium, which is helpful to prevent media leakage to a certain extent.

Figure 7.

The three curves represent the distribution of the gap for compressions of 40%, 50%, and 60%, respectively.

Figure 8 shows the leakage rate of the medium for different amounts of compression of 40%, 50%, and 60%, and the pressure difference of 5 × 10³ Pa, 8 × 10³ Pa, and 1 × 10³ Pa on both sides of the sealing system. As the amount of compression increases, the leak rate decreases. As the pressure difference between the two sides of the sealing end increases, the leakage rate increases. As the amount of compression increases, the rate of change in leakage rate decreases. When the amount of compression increases to a certain extent, the impact on the leak rate will not be significant. In fact, the amount of compression represents the different contact pressure of the sealing surface, and the amount of leakage can be controlled by preapplying the appropriate contact pressure.

Figure 8.

The leakage rate of the medium for different amounts of compression of 40%, 50%, and 60%, and the pressure difference of 5 × 10³, 8 × 10³, and 1 × 10³ Pa, respectively, on both sides of the sealing system.

Conclusion

In this article, the stress–strain curve of foamed silicone rubber was obtained by mechanical property test. The finite element simulation model was established by combining the Ogden third (foam) model. Through the simulation model, the contact pressure distribution on the sealing surface at 40%, 50%, and 60% compression was obtained. The microscopic cell structure and the cell distribution of the surface of the foamed silicone rubber were obtained by means of a white-light interferometer. The nominal gap of the sealing interface under different compression states was obtained by the coupling of microscopic contact mechanics and macroscopic contact pressure. With the nominal gap as the boundary condition, the porous medium theory and the flow continuous theory were used to simulate the flow state of the medium in the cell structure. The fluid leakage under different medium pressures was obtained by numerical calculation. The calculation results showed that the compression amount increased, the contact area on the sealing surface increased, the contact pressure increased, and the corresponding nominal gap decreased, resulting in a decrease in leakage. In addition, an increase in the relative pressure of the fluid medium at the sealing end face also caused an increase in the amount of leakage. Therefore, increasing the amount of compression of the seal, providing a larger preload, and reducing the pressure difference at the seal end face could effectively reduce leakage. The model provided a theoretical reference for static seal structure design, material selection, preload load application, and the control of leakage rate.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work described in this article has been supported by the National Key R&D Program of China (Grant No. 2018YFB2001001) and the National Natural Science Foundation of China (Grant No. 51575300 and 51735006).

ORCID iD

Fei Guo

References

Lorenz

Persson

BNJ

. Leak rate of seals: effective-medium theory and comparison with experiment. Eur Phys J E 2010; 31(2): 159–167.

Persson

BNJ

. Theory of rubber friction and contact mechanics. J Chem Phys 2001; 115(8): 3840–3861.

Persson

BNJ

Yang

. Theory of the leak-rate of seals. J Phys Condensed Matt 2008; 20(31): 315011.

Bottiglione

Carbone

Mantriota

Fluid leakage in seals: An approach based on percolation theory. Tribol Int 2009; 42(5): 731–737.

Bottiglione

Carbone

Mangialardi

, et al. Leakage mechanism in flat seals. J Appl Phys 2009; 106(10): 104902–104907.

Vallet

Lasseux

Sainsot

, et al. Real versus synthesized fractal surfaces: Contact mechanics and transport properties. Tribol Int 2009; 42(2): 250–259.

Etsion

Front

. A model for static sealing performance of end face seals. Tribol Trans 1994; 37(1): 111–119.

Tripp

JH.

Surface roughness in effects in hydrodynamic lubrication: the flow factor method. J Lub Technol 1983; 105(3): 458–465.

. A study on flow phenomena in mixed lubrication regime by porous medium model. J Tribol 1994; 116(3): 640–647.

10.

Zheng

Wang

, et al. A diffusivity model for gas diffusion through fractal porous media. Chem Eng Sci 2012; 68(1): 650–655.

11.

Lesinigo

Carlo

D'Angelo

Quarteroni

. A multiscale Darcy–Brinkman model for fluid flow in fractured porous media. Numerische Mathematic 2011; 117(4): 717–752.

12.

Shou

Fan

Ding

. A difference-fractal model for the permeability of fibrous porous media. Phys Lett A 2010; 374(10): 1201–1204.

13.

Guo

Jia

Suo

, et al. A mixed lubrication model of a rotary lip seal using flow factors. Tribol Int 2013; 57: 195–201.