Fluid flows from the high-pressure side at the inner diameter to the low-pressure side at the outer diameter of the O-ring seal, with pressure showing a continuous gradient attenuation along the radial direction. For any two adjacent radially discrete micro-elements i and i +1, the condition “outlet pressure of micro-element i = inlet pressure of micro-element i +1“ is satisfied, thereby forming an uninterrupted radial pressure distribution field.

Based on the incompressible fluid assumption, the total leakage rate Q passing through the sealing clearance is globally constant. For any single micro-element divided along the radial direction, its local leakage rate dQ _i is equal to the total leakage rate Q, i.e., dQ _i = Q.

The solution procedure of the model follows the approach of micro-element discretization→ coupling relationship establishment→integration and combination of total pressure difference, which transforms the complex micro-element coupling problem into a single-variable solution problem targeting the total leakage rate Q. The specific steps are as follows:

Step 1: Maintain the discretization of annular micro-elements.

Divide N equally wide micro-elements along the radial direction of the sealing contact (from r ₁ to r ₂, where (N ≥ 20) to ensure the pressure gradient is approximately linear. The key parameters of the micro-elements are defined as follows. -

Micro-element width: dr=(r ₂ - r ₁)/N;

Step 2: Establish the coupling relationship between annular micro-elements.

Based on the flow rate conservation constraint (dQ _i = Q) and the viscous flow formula, the coupling relationship between the pressure difference across each micro-element and the total leakage rate Q is derived as follows.

For the i-th micro-element, the fluid flow obeys the Hagen-Poiseuille law (viscous laminar flow), and its local leakage rate is expressed as: d Q i = μ 1 r i · 2 π r i · h r i 3 · d Δ p i 12 μ d r (4)

Based on the flow rate conservation constraint (dQ _i = Q)), the pressure difference across the i-th micro-element can be rearranged to obtain: d Δ p i = Q · 12 μ · d r μ 1 r i · 2 π r i · h r i 3 (5)

In the formula: μ ₁-Leakage rate correction factor, a coefficient accounting for the blockage/contraction effects of surface roughness on leakage channels; μ-Dynamic viscosity of the fluid. The stronger the viscous resistance of fluid flow, the smaller the leakage rate.

μ ₁(r _i ) and h(r _i ) denote the correction factor and sealing clearance of the i-th micro-element, respectively. Owing to the arched distribution of contact pressure p(r), both parameters vary with r _i .

The total pressure difference Δp (i.e., high pressure at the inner diameter minus low pressure at the outer diameter) equals the sum of the pressure differences across all micro-elements, which arises from the continuous radial superposition of pressure: Δ p = ∑ i = 1 N d Δ p i = ∫ r 1 r 2 Q · 12 μ · d r μ 1 r · 2 π r · h r 3 (6)

In the formula: r 1 is the inner diameter of the ring at the sealing contact; r 2 is the outer diameter of the ring at the sealing contact.

By rearranging and combining (Equations 4–6), the coupled solution formula for the total leakage rate Q is obtained as Equation 7. Q = Δ P · 2 π 12 · μ · 1 ∫ r 1 r 2 d r μ 1 r · r · h r 3 (7)

Step 3: Substitute Coupling Parameters (Address the Radial Variation of h(r) and μ ₁(r))

The coupling between micro-elements is also reflected in the radial variation of h(r) and μ ₁(r) with the radial position r (owing to the arched distribution of contact pressure p(r)), which requires accurate calculation according to the following method.

Yang (2016) used 39 different geometric models to analyze the influence of surface roughness and gap height on the leakage rate, and fitted the expression of the leakage rate correction factor μ ₁: μ 1 r = 1 − exp − 0.6 h r T 0.33 δ T − 0.96 (8)

In the formula: δ is the root mean square deviation of the surface profile; T is the surface autocorrelation length, reflecting the density of peaks on the actual engineering surface (Yang, 2016). Engineering value range of T: Precision-ground surface: T = 0.5–1.0 μm; Conventional turned surface: T = 1.0–2.0 μm; Rough-machined surface: T = 2.0–3.0 μm. As T increases, the peak-valley structures on the surface become sparser, and the rate of change of μ ₁ with respect to δ/T slows down.

The surface roughness of the O-ring is determined by the morphology of the mold surface during its vulcanization. For metal surface profiles processed by precision turning, precision grinding, and lapping, the ratio of the root mean square deviation δ to the arithmetic mean deviation Ra of the profile is 1.22∼1.27. In this study, the intermediate value is adopted (Yasuo et al., 2017), namely: δ = 1.25 R a (9)

When considering the influence of roughness on the leakage channel for the leakage rate of the annular flat - plate sealing model, when the sealing surface of rubber seal is not subjected to pressure, the initial contact height is taken as h 0 = 3 δ ; when the pressure is Δ p , the sealing gap is h, and its calculation formula is as follows (Roth, 1972): h r = h 0 e − Δ p λ = 3 δ e − Δ p λ (10) where λ is the sealing coefficient of the O-type rubber seal material.

Since the roughness value of the metal contact surface is far smaller than that of the rubber surface, the real contact state between the O-ring and the rough interface can be regarded as the contact between an elastic rough surface and a rigid smooth surface. Therefore, the rough peaks on the contact surface of the O-ring are simplified into a cone with a cone angle of 120° and a height of h ₀. An axisymmetric plane model is adopted, as shown in Figure 2. The upper cylinder is a metal rigid body, and the lower cone is a rubber deformable body. The rubber material uses silicone rubber consistent with the O-ring material, with a Shore hardness of 50 HA. It is assumed that the mechanical properties of the rubber material are incompressible, so the Mooney-Rivlin model is adopted.

The relationship between the sealing coefficient λ and medium pressure p depends on the hyperelastic sealing behavior of the O-ring, material properties, and surface contact conditions. As a hyperelastic material, the contact stress of the O-ring exhibits a nonlinear increase with p; the higher the pressure, the tighter the O-ring is compressed, resulting in a higher contact stress.

For silicone rubber with Shore Hardness 50 HA, which features low hardness and high elasticity, the deformation saturation effect is more pronounced. Its relationship with p can be expressed in the form of exponential decay: λ = λ 0 · e − α p + λ ∞ (11)

In the formula: λ ₀ - Initial sealing coefficient, with λ ₀ = 2.8 as p→0; α - Exponential decay coefficient, taken as α = 0.52 MPa⁻¹; λ ∞ - Sealing coefficient at pressure saturation, with λ ∞ = 0.35 as p→ ∞ .or typical operating conditions with p = 0.3 MPa and p = 0.6 MPa, the corresponding sealing coefficients are 2.747 and 2.399, respectively.

Step 4: Numerically Solve the Total Leakage Rate Q.

Since h(r) and μ ₁(r) vary nonlinearly with r, Equation 7 has no analytical solution and is thus solved using the trapezoidal numerical integration method.

When the radial sealing range of the O-ring (from r ₁ to r ₂) is divided into N annular micro-elements, N+1 micro-element endpoints are obtained (from the start point of the first micro-element to the end point of the last micro-element), denoted as r ₀, r ₁, r ₂, …, r _N . For the central radius r _i of each micro-element, the integral term is calculated as follows: f r i = 1 μ 1 r i · r i · h r i 3 (12)

I = ∫ r 1 r 2 f r d r ≈ d r 2 f r 0 + 2 ∑ i = 1 N − 1 f r i + f r N (13)

Herein, r ₀ denotes the start point of the integral interval, which corresponds to the inner diameter r ₁ of the O-ring sealing contact; r _N denotes the end point of the integral interval, corresponding to the outer diameter r ₂ of the O-ring sealing contact.

Parameters including pressure p(r), r ₁, r ₂, and h(r) need to be obtained via finite element simulation of the O-ring compressed by the medium pressure in the groove, which yields the radial contact length and pressure distribution on the sealing contact surface. By substituting the relevant parameters derived from Equations 8–13 into Equation 7, the total leakage rate Q can be calculated.

Rubber specimens are made into Type I dumbbell-shaped specimens in accordance with the national standard GB/T 528−2009. The specimen thickness is (2 ± 0.2) mm, and the length of the test section is (25 ± 0.5) mm. Three dumbbell specimens are prepared from the same rubber compound, and the experimental results are averaged. The experimental equipment is a WDW-20 Micro-controlled Electronic universal testing machine, as shown in Figure 3. The prepared dumbbell specimens are installed on the electronic universal tensile testing machine, and stretching begins at a constant rate of (500 ± 50) mm/min until the specimen fails, recording the load-displacement curve.

As illustrated in Figure 4a, the experimentally recorded load-displacement curve is normalized into an engineering stress-strain profile, where tensile stress (ordinate) and tensile strain (abscissa) characterize the material’s mechanical response. The derived curve exhibits pronounced nonlinear hyperelastic behavior, a hallmark of rubber materials. Notably, at a strain level of 0.8, the material demonstrates significant strain accommodation with minimal stress increment, indicative of its molecular chain reorientation under load. To further quantify multiaxial deformation characteristics, equibiaxial tensile testing was conducted using an ERBI-300 inflatable testing system, compliant with the GB/T 528-2009 international standard for tensile property evaluation of vulcanized and thermoplastic rubbers. The resulting equibiaxial strain-stress relationship, shown in Figure 4b, reveals distinct anisotropic stiffening behavior compared to uniaxial loading conditions.

For the rubber material of the O-ring seal, reliable stress-strain curves are derived through experimental data from uniaxial tension, equibiaxial, and planar shear tests, based on appropriate constitutive models like Mooney-Rivlin, Yeoh, Ogden, etc. The most widely used is the Mooney-Rivlin model, capable of simulating the mechanical properties of most rubber materials. This model typically includes four equation types with 2, 3, 5, or 9 parameters—more parameters yielding higher fitting accuracy. The compression ratio of the O-ring adopted in this study falls within the moderate deformation range, and the strain energy density function of the two-parameter Mooney-Rivlin model, expressed as W=C ₁₀(I ₁-3) + C ₀₁(I ₂-3), can accurately characterize the hyperelastic behavior of the O-ring within this deformation interval.

This study selects the 2-parameter Mooney-Rivlin model, widely applied in laboratories for rubber material analysis. The rubber’s stress-strain curve shows nonlinearity, with its mechanical behavior closely linked to factors such as experimental environment, working conditions, and loading rate. According to the uniaxial tension engineering stress-strain data measured by the electronic universal tensile testing machine and the equibiaxial tension engineering stress-strain curve, the 2-parameter Mooney-Rivlin model is used. As shown in Figure 5, fitting the experimental curve with the Mooney-Rivlin model yields two parameters: C₁₀ = 0.27, C₀₁ = 0.03.

For the simulation model, one end is fixed, and the other end uses the RBE element to grasp partial nodes for applying a forced displacement of 50 mm. The reaction force of RBE nodes is extracted and compared with the experimental tension under the same tensile displacement. As shown in Figure 6, the numerical values of the two are highly close, thereby verifying the correctness of the constitutive model.

As shown in Figure 7, when one end of the specimen is stretched by 50 mm, the equivalent stress of the specimen reaches a maximum of 10.92 MP, and the width of the narrow part changes from 6 mm to 4.4 mm. The maximum tensile stress during the specimen’s stretching-to-fracture process is the tensile strength. Experiments show that the specimen’s tensile strength is approximately 18.5 MP.

Taking a certain sealing device as an example, the dimensions of the O-ring and the groove structure are shown in Figure 8. The sealing medium of this sealing device is hydraulic oil, with its dynamic viscosity μ = 0.0087 Pa · s and working temperature 300 K. The inner diameter of the O-ring is ϕ 40 ± 0.3 mm, and the cross-sectional diameter is ϕ 1.8 ± 0.08 mm. The compression ratio of the O-ring reaches 26.11% after installation. After sealing, both the force inside the groove and the internal medium pressure are axially symmetrically distributed. Therefore, the sealing part of the O-ring is simplified into a two-dimensional axisymmetric model.

Given the significant deformation of the O-ring, the nonlinear finite element software MSC. Marc was employed. To eliminate errors induced by unreasonable mesh sizes, three mesh seed spacing schemes were designed: the coarse mesh had an average element size of 0.04 mm with a total number of elements of approximately 1,500; the medium mesh featured an average element size of 0.03 mm with around 2,800 total elements; and the fine mesh adopted an average element size of 0.02 mm with a total element count of roughly 6,300. The maximum contact pressure at the O-ring-metal interface was selected as the key verification index. When the mesh was refined from the coarse to the medium scheme, the relative variation in the maximum contact pressure reached 8.5%; whereas when further refined from the medium to the fine scheme, this relative variation decreased to less than 2.5%, indicating that the calculation results had tended to stabilize. Consequently, the medium mesh scheme was finally selected as the formal mesh for subsequent numerical simulations. For the nonlinear contact analysis involving hyperelasticity and large deformations, three core convergence criteria were implemented in MSC. Marc: force, displacement, and energy convergence. A single criterion is inadequate to address both rubber material nonlinearity and geometric nonlinearity, whereas the synergistic application of the three criteria guarantees the accuracy and robustness of simulation results.

The flange is assumed to be a rigid body, and the O-ring rubber seal is regarded as a deformable body. The boundary conditions are set so that the contact area between the workpiece and the O-ring does not bear pressure—ensuring the real pressure-bearing surface is only the unit edge where the O-ring deformable body and the rigid body do not contact. Meanwhile, according to the actual sealing pressure, the applied load is a surface force, with the pressure always acting perpendicularly on the unit boundary. The simulation model is shown in Figure 9.

During the initial compression process, the sealing groove remains stationary. As the upper cover plate slowly moves rightward, the O-ring undergoes compression deformation and makes contact with both the upper cover plate and the groove. The resilience generated by the deformation acts on the upper cover plate and the groove bottom surface, thereby generating initial compressive stress—i.e., the sealing pressure. The ratio of the O-ring’s compression amount to its initial cross - sectional diameter is defined as the O-ring compression rate. As shown in Figure 10, there is nearly a linear relationship between the O-ring compression rate ε and the maximum sealing pressure p max . The greater the compression rate, the higher the sealing pressure. However, a relatively high compression rate can cause the O-ring to experience permanent deformation, lose elasticity, and thereby fail in sealing.

In the working state, the simulation slowly applies liquid pressure differences of 0.3 MPa and 0.6 MPa to one side of the O-ring through the gap. At this point, the maximum sealing pressure p max increases to 1.216 MPa and 1.502 MPa respectively, and then remains unchanged, as shown in Figure 11. This is because during operation, the greater the liquid pressure difference on both sides of the O-ring seal, the greater the deformation of the O-ring, and the greater the pressure transmitted to the contact surface—thereby enhancing the sealing effect. This is termed the self - tightening sealing action.

The contact line between the O-ring and the groove is relatively long. Due to the self - tightening sealing effect, leakage often occurs between the O-ring and the upper cover plate. Therefore, in the calculation of leakage volume, the sealing contact length between the O-ring and the upper cover plate (referred to as “contact length” hereinafter) is the difference between the outer diameter and inner diameter at the sealing contact area. As shown in Figures 12a,b respectively present the magnitude of sealing pressure at each position of the O-ring and the location of sealing contact length under internal and external liquid pressure differences of 0.3 MPa and 0.6 MPa.

As illustrated in Figure 13, along the contact length of the O-ring, the sealing pressure differs across distinct contact positions. The sealing pressure at both ends is relatively lower, whereas the pressure in the middle is higher, exhibiting an arch - shaped distribution. When the sealed liquid pressure differences reach 0.3 MPa and 0.6 MPa, the sealing contact length w between the O-ring and the upper cover plate extends from 1.13 mm to 1.24 mm. The contact width varies under different pressure differences, and the outer contact radius r ₂ of the seal satisfies the relationship: r ₂ = r ₁ + w (where r ₁ is the inner contact radius and w is the contact width). The inner side of the O-ring is in contact with the high-pressure medium, and its inward deformation is constrained by the medium pressure. Therefore, the inner contact radius of the seal remains basically stable under different pressure differences, which is consistent with the original inner radius. Based on the fitting of the sealing self-tightening effect law, a higher pressure difference results in a stronger extrusion force pushing the O-ring toward the sealing surface, and thus the contact length increases with the corresponding increase in pressure difference. Concurrently, the sealing pressure p at the contact area increases accordingly, with the maximum sealing pressure values under these two conditions differing by approximately 0.3 MPa.

The number of annular micro-elements was set as N = 20. When Ra = 0.8 μm (classified as a low-to-moderate roughness surface), T = 1 was adopted; when Ra = 1.6 μm (classified as a moderate roughness surface), T = 2 μm was adopted; and when Ra = 3.2 μm (classified as a high roughness surface), T = 3 was adopted. The sealing clearance h (μm) and correction factor μ ₁ were calculated by substituting the above simulation data into the formula, respectively.As illustrated in Figures 14, 15, the sealing clearance h and correction factor μ ₁ of the O-ring seal exhibit a consistent and physically interpretable trend with the variation of surface roughness and pressure difference (Δp): when Ra is fixed, both h and μ ₁ decrease monotonically as Δp increases from 0.4 MPa to 1.6 MPa; conversely, when Δp is kept constant, larger Ra values lead to larger magnitudes of h and μ ₁. These trends are dominated by the coupling effect between pressure-driven elastic deformation and surface topography. Specifically, a higher Δp increases the contact pressure acting on the O-ring, prompting elastic deformation to fill the microscopic gaps on the sealing surface, thereby reducing the effective sealing clearance h and the proportion of active leakage channels. In contrast, a larger Ra results in more pronounced peak-valley uneven structures on the sealing surface, which form wider and more persistent microscopic gaps that cannot be completely eliminated by the elastic deformation of the O-ring. This consequently leads to an increase in h and a higher density of unclosed leakage channels.

By substituting these parameters into Equation 7, the leakage rates Q corresponding to the three surface roughness levels (Ra = 0.8 μm, Ra = 1.6 μm, and Ra = 3.2 μm) under different sealing pressures Δp were obtained, as illustrated in Figure 16.

When the differential pressure Δp is relatively low, the leakage rate remains at a low level, yet as Δp increases, the leakage rate exhibits a near-linear upward trend across all roughness groups. For identical Δp, a greater surface roughness value consistently leads to a higher leakage rate: for example, at Δp = 0.4 MPa, the leakage rate is 0.37 × 10⁻¹² m³/s for Ra = 0.8 μm, rising to 0.50 × 10⁻¹² m³/s for Ra = 1.6 μm and 0.67 × 10⁻¹² m³/s for Ra = 3.2 μm. The underlying mechanism lies in two coupled effects: first, an increase in Δp elevates the pressure gradient driving fluid flow through micro-leakage channels, directly enhancing the leakage flux. Second, larger Ra values reflect more pronounced surface asperities, which create wider and more persistent leakage pathways that cannot be fully eliminated by the elastic deformation of the O-ring under contact pressure. Even as Δp increases and contact pressure rises to compress the sealing interface, the rough surface topography retains unclosed gaps, sustaining fluid flow. This observation aligns with the principle that surface roughness dominates the initial leakage channel formation, while differential pressure governs the subsequent flux through these channels.

Before the test, the samples shall be placed at standard temperature for no less than 3 h. Subsequently, the cross-sectional diameter ϕ was measured, and the samples were installed into the test device according to the specified compression amount and duration for testing, with the compressed cross-sectional diameter ϕ ₁ measured thereafter. After the test, the samples were removed from the test device, allowed to undergo natural recovery at standard temperature for 3 h, and then their recovered cross-sectional diameter ϕ ₂ was measured. The test rig for the compression-rebound performance test is shown in Figure 17.

The compression rebound rate is calculated using Equation 14: ε = ϕ 2 − ϕ 1 / ϕ 0 − ϕ 1 × 100 % (14)

Parameters in the formula: ε —Compression rebound rate (%);

ϕ 0 —Original cross-sectional diameter of the sample before compression (mm);

ϕ 1 —Cross-sectional diameter of the sample after compression as specified (mm);

ϕ 2 —Effective cross-sectional diameter of the sample after recovery from compression (mm).

Based on the experimental data, as shown in Table 1, the compression rebound rates of the O-ring all fall within a reasonable range. In addition, the overall rebound rate is relatively high, with a minimum value of 97.83%, and most values are close to or reach 100%, indicating that the O-ring exhibits excellent rebound recovery performance after compression.

The static sealing structure system mainly consists of a grooved flange, a baffle plate, an oil collecting cup, and a precision pressure regulation system. The hydraulic system comprises a pressure transmitter, a relief valve, and oil circuit connecting pipes. The collection system includes an oil pan for capturing leaked oil, oil pipes for guiding fluid flow, and a dry beaker for collecting the leaked oil. A constant room temperature was maintained throughout the experiments to avoid the influence of oil temperature fluctuations on oil viscosity, and no additional vibration was applied to the sealing system. The test rig is capable of providing adjustable pressures ranging from 0 to 2.0 MPa to the sealing test cavity, thereby establishing the required operating conditions for O-ring sealing tests. The experimental test rig and experimental flow chart are depicted in Figure 18.

This study experimentally investigates the leakage characteristics of O-ring seals under differential pressures of 0.3 MPa and 0.6 MPa, with surface roughness values of 0.8 μm, 1.6 μm and 3.2 μm.

The metal sealing surface in contact with the O-ring adopts 316L austenitic stainless steel as the base material, as shown in the top-left corner of Figure 18. A flat basic surface is first obtained via CNC turning, followed by grinding with grinding wheels of different grit sizes to control the surface roughness: grinding with an 80-grit coarse grinding wheel yields a surface with a roughness Ra = 3.2 μm; replacing it with a 120-grit medium grinding wheel produces a surface with Ra = 1.6 μm; fine grinding with grinding wheels of 240 grit or finer, coupled with polishing using a polishing cloth, enables the preparation of a surface with Ra = 0.8 μm.

To determine the Ra parameter, a surface roughness tester was used to select at least 3 different measurement points along the circumferential and radial directions of the sealing surface. The Ra value at each measurement point was recorded and the average value was calculated, so as to ensure the reliability of the measured data.

The cumulative leakage volume is measured over a 24-h period using a gravimetric method, where the leaked hydraulic oil is collected via an oil pan, oil pipe and dry beaker with a hydraulic oil density of 0.85 mg/μL. The theoretical cumulative leakage volumes and experimental cumulative leakage volumes under the differential pressures of 0.3 MPa and 0.6 MPa are denoted as Q _V1, Q _V 2, Q _V 3 and Q _V 4, respectively. As shown in Figure 19, the theoretical cumulative leakage volume of the O-ring seals ranges from 33.60 μL to 85.44 μL under the test conditions. The relative error between experimental and theoretical values is controlled within 15% with random deviation directions. Specifically, a higher experimental leakage volume than the theoretical value is observed in the case of 0.3 MPa differential pressure and 0.8 μm surface roughness, which is attributed to micro-burrs on the sealing surface that widen the local leakage channels beyond the scope of theoretical assumptions. In contrast, a lower experimental value appears under 0.3 MPa differential pressure and 1.6 μm surface roughness, resulting from the elastic rebound of the O-ring material after long-term compression, which reduces the actual sealing gap. For the condition of 0.6 MPa differential pressure and 1.6 μm surface roughness, the experimental leakage volume exceeds the theoretical value due to the random distribution of micro-leakage paths on the rough sealing surface that cannot be fully characterized by the idealized theoretical model. Meanwhile, the experimental value is lower than the theoretical value under 0.6 MPa differential pressure and 3.2 μm surface roughness because of slight oil adhesion on the inner wall of the oil pipe during the collection process. The remaining test cases show a relative error below 10%. These results demonstrate that the experimental data meet the precision requirements of mechanical seal tests, validating the accuracy of the circular plate micro-element integral model for O-ring leakage prediction. This confirms that the overall uncertainties of the proposed model are controllable under specific conditions, i.e., room temperature, low pressure and short-term service.

Meanwhile, the random deviations between experimental and theoretical values are closely related to the micro-topography of the sealing surface, elastic deformation of the O-ring material, and oil adhesion during the measurement process, rather than temperature-related factors.