Hysteresis Heat Generation in Polyurethane O-Rings: Thermo-Mechanical Coupling Mechanism and Its Quantified Effect on Reciprocating Sealing Performance

Abstract

Polyurethane O-ring seals are vital for the service life and sealing reliability of hydraulic systems, yet internal hysteresis heat generation under reciprocating motion causes localized temperature rise, altering contact pressure distribution and impairing sealing performance. This study aimed to clarify the coupled effects of reciprocating motion parameters on O-ring hysteresis heat generation and sealing performance. A unified hysteresis heat generation rate expression was derived by combining the time–temperature superposition principle with the Maier–Göritz model, and the heat source model was integrated into a thermo-mechanically coupled finite element analysis (FEA) framework, validated by matching simulated and experimental temperature rise histories. Under baseline conditions, hysteresis heating causes the O-ring’s peak contact pressure to decrease by approximately 0.4 MPa during the outward stroke. Parametric analysis revealed that elevated operating parameters increase contact pressure to maintain effective sealing, but simultaneously intensify hysteresis heating. Quantitatively, the maximum O-ring temperature was highly sensitive to operating conditions, reaching 63.6 °C at 8 MPa hydraulic pressure, 60.0 °C at a 90 Hz reciprocating frequency, and up to 81.5 °C for a friction coefficient of 0.2. Although the current framework is limited by the exclusion of interfacial frictional heating, it enables the reliable quantitative prediction of thermal loads. Ultimately, this study provides a robust method for assessing sealing safety margins and offers theoretical guidance for the structural optimization of hydraulic sealing systems.

1. Introduction

Polyurethane O-ring seals are extensively used in hydraulic and pneumatic sealing systems due to their excellent elasticity, abrasion resistance and mechanical durability [1,2]. During the reciprocating movement of hydraulic cylinder pistons, the embedded O-rings play a pivotal role in sealing and pressure maintaining. However, under dynamic operating conditions, cyclic interfacial deformation of the seals induces a prominent viscoelastic hysteresis effect, which continuously converts mechanical work into thermal energy and thus elevates the material temperature [3]. This hysteresis heat is concentrated at the seals interface where deformation is the most severe, resulting in localized temperature rise with multiple detrimental effects: rapid depletion of interfacial lubrication, increased frictional resistance and adhesive wear, as well as accelerated thermal softening and material aging. These issues are the primary drivers for premature failure of the hydraulic sealing systems [4,5,6,7], leading to oil leakage, pressure loss and even system breakdown. Therefore, it is essential to investigate the heat generation characteristics of O-ring seals and their influence on sealing performance for mitigating deformation, wear and leakage, and further improve the reliability of hydraulic systems.

The sealing performance of O-rings has been extensively studied with respect to structural and operating parameters, including hydraulic pressure, pre-compression ratio, groove geometry, and material hardness [8,9]. Cheng et al. [10] emphasized the critical importance of structural parameter optimization for composite seal assemblies, demonstrating that the O-ring compression ratio must be matched to the operating pressure, while the chamfer and width of the wear ring exert a significant influence on the distribution of contact stress and frictional behavior. Mao et al. [11] combined theoretical models with experimental methods to evaluate the sealing performance of composite seals and identified rod velocity as a critical factor influencing seal effectiveness. On the numerical side, Wu and Li [12] investigated the impact of dimensions and material properties on O-ring sealing under deep-sea conditions, suggesting that the bulk modulus of the piston should exceed that of the bore, and increasing the filet radius at the groove bottom can further enhance sealing performance. Similarly, Li [13] indicated that optimizing the compression ratio and static pressure improves sealing efficiency by maintaining sufficient contact pressure under dynamic loads. Furthermore, Zhang et al. [14] analyzed the effects of pre-compression, fluid pressure, friction coefficient, and hardness on reciprocating D-rings, and emphasized that optimizing the pre-compression ratio and friction coefficient significantly enhances system reliability by mitigating friction and wear.

Although these structural optimization strategies have improved the sealing reliability of O-rings, most numerical studies assume constant material properties and neglect the influence of the thermal environment on seal performance. Numerous studies have confirmed that temperature exerts a profound effect on rubber-like materials, causing thermal softening which in turn degrade the mechanical properties and sealing performance of O-rings [15,16]. Consequently, predicting the internal temperature field of O-rings under dynamic conditions is critical for understanding the temperature-dependent evolution of sealing performance.

In recent years, thermal considerations have received increasing attention in seal research, but existing studies still have notable limitations and research gaps. Nikas [17] developed a theoretical model to examine the effects of various operating parameters on the performance of rectangular elastomeric seals over a wide temperature range. Zhao et al. [18] investigated the friction and wear performance of dynamic seal structures based on the Archard model and analyzed the effects of compression ratio, medium pressure, relative sliding speed, and temperature on O-ring seals. Xiang et al. [19] established a transient thermo-elastic hydrodynamic lubrication simulation model for reciprocating rod seals, and corrected the viscosity of the fluid film based on the simulated temperature results. Li et al. [20] developed a finite element method to simulate the thermal–mechanical deformation of O-rings, introduced dimensionless contact stress as a sealing degradation indicator, and established an aging kinetics model dependent on temperature and time. Yin et al. [21] and Zhang et al. [22] examined the influence of frictional heat sources on reciprocating sealing performance. Despite these advances, several limitations remain. Most existing seal-level studies focus mainly on frictional heating, lubrication, thermal aging, or purely mechanical sealing behavior, whereas the role of internal hysteresis heating as an intrinsic volumetric heat source is less explicitly quantified. In addition, material-level studies on hysteresis heat generation are seldom integrated into a seal-level finite element framework capable of resolving the coupled evolution of temperature field and contact response under reciprocating operating conditions. Consequently, the quantitative link between material-level viscoelastic dissipation and seal-level thermo-mechanical response remains insufficiently established for polyurethane-reciprocating O-rings.

Motivated by the above-mentioned research gap, this study develops a thermo-mechanically coupled finite element framework to predict hysteresis heat generation in polyurethane O-rings under reciprocating motion and quantify its influence on sealing response. The specific objectives are: (1) to derive a unified hysteresis heat-generation model coupling temperature, frequency, and dynamic strain amplitude; (2) to implement this model in ABAQUS 2021 as an internal volumetric heat source within a sequential thermo-mechanical framework; and (3) to quantify the effects of key operating parameters, including pre-compression ratio, hydraulic pressure, friction coefficient, and reciprocating frequency, on both the thermal load and the contact–pressure response of the O-ring.

The remainder of this paper is organized as follows: Section 2 formulates the hysteresis heat-generation model by combining the Maier–Göritz model with the time–temperature superposition principle. Section 3 describes the material characterization experiments and parameter identification for the proposed model. Section 4 details the establishment of the finite-element model for O-ring seals, including the constitutive model, geometrical model, mesh generation, and thermo-mechanical coupling analysis steps. Section 5 presents and discusses the simulation results, including the deformation, contact pressure, temperature distribution, and the effects of key operating parameters. Finally, Section 6 summarizes the main conclusions and highlights the practical implications of this study.

2. Hysteresis Heat-Generation Model

To link viscoelastic energy dissipation with thermal analysis, a quantitative model for hysteresis heat generation of polyurethane O-rings is formulated in this section. In polymer viscoelasticity, the deformation behavior can be represented as the combined response of an elastic spring and a viscous dashpot [23]. Due to this inherent viscoelasticity, hysteresis heat generation arises from energy dissipation associated with internal friction within the polymer network and at the filler–polymer/filler–filler interfaces [24].

Assuming a sinusoidal strain input in a viscoelastic body, as expressed by Equation (1):

$$\epsilon \left(t\right)={\epsilon }_0+{\epsilon }_{\Delta }\sin \left(\omega t\right)$$

(1)

where $\omega =2\pi f$, and f is the frequency (Hz), ${\epsilon }_0$ is the pre-strain, and ${\epsilon }_{\Delta }$ is the dynamic strain amplitude. The corresponding stress response can be written as:

$$\sigma \left(t\right)={\epsilon }_0{E}^{\prime }+{\epsilon }_{\Delta }\sqrt{{E^{\prime }}^2+{E^{″}}^2}\sin \left(\omega t+\phi \right)$$

(2)

where E′ is the storage modulus, E″ is the loss modulus, and φ = arctan(E″/E′) is the phase angle between stress and strain. The total strain energy of the material under one loading cycle is given by Equation (3) [25]:

$$W={\displaystyle \oint \sigma \left(t\right)}\mathrm{d}\epsilon \left(t\right)$$

(3)

Under cyclic loading, the total strain energy consists of recoverable stored energy and irreversible dissipated energy. Since the recoverable energy term does not contribute to hysteresis loss, the hysteresis heat generation rate density (per unit volume) can be expressed as the dissipated energy per unit time:

$$Q=fD=\pi f{\epsilon }_{\Delta }^2{E}^{″}$$

(4)

where D is the hysteresis loss per unit volume in one cycle of deformation, which is the direct cause of the material temperature rise.

In filled elastomers such as polyurethane, stress-induced adsorption/desorption of weakly bound molecular chains at filler surfaces alters the effective network density. The associated breakdown and reformation processes produce the well-known Payne effect, which is the typical viscoelastic phenomenon of filled elastomers. Maier and Göritz proposed a phenomenological model to describe this effect [26], as shown in Equations (5) and (6):

$$E^{\prime }\left({\epsilon }_{\Delta }\right)=E_{\mathrm{st}}^{\prime }+E_{\mathrm{i}}^{\prime }{\displaystyle \frac{1}{1+c{\epsilon }_{\Delta }}}$$

(5)

$$E^{″}\left({\epsilon }_{\Delta }\right)=E_{\mathrm{st}}^{″}+E_{\mathrm{i}}^{″}{\displaystyle \frac{c{\epsilon }_{\Delta }}{{\left(1+c{\epsilon }_{\Delta }\right)}^2}}$$

(6)

where $E_{\mathrm{st}}^{\prime }$ is the storage modulus at large dynamic strain amplitudes; $E_{\mathrm{i}}^{\prime }$ is the variation amplitude of the storage modulus; $E_{\mathrm{st}}^{″}$ represents the loss modulus at extremely high or low dynamic strain amplitudes; $E_{\mathrm{i}}^{″}$ relates to the variation amplitude of the loss modulus; and the value of c depends on the position of the maximum loss modulus on the dynamic strain amplitude axis.

In practical engineering applications, the parameters of the Maier–Göritz model vary with temperature, frequency, and dynamic strain amplitude. To capture the combined dependence on frequency and strain amplitude, the Maier–Göritz model was extended as follows:

$$E^{\prime }\left({\epsilon }_{\Delta },f\right)=E_{\mathrm{st}}^{\prime }\left(f\right)+E_{\mathrm{i}}^{\prime }\left(f\right){\displaystyle \frac{1}{1+c{\epsilon }_{\Delta }}}$$

(7)

$$E^{″}\left({\epsilon }_{\Delta },f\right)=E_{\mathrm{st}}^{″}\left(f\right)+E_{\mathrm{i}}^{″}\left(f\right){\displaystyle \frac{c{\epsilon }_{\Delta }}{{\left(1+c{\epsilon }_{\Delta }\right)}^2}}$$

(8)

Above the glass transition temperature of polymers, the temperature dependence of viscoelastic properties is commonly described using the WLF equation [27,28]. For WLF-type time–temperature superposition, the shift factor ${\varphi }_{\mathrm{T}}$ is defined as:

$$\log {\varphi }_{\mathrm{T}}=-C_1{\displaystyle \frac{T-T_{\mathrm{ref}}}{C_2+(T-T_{\mathrm{ref}})}}$$

(9)

where C₁ is a dimensionless empirical constant, C₂ is an empirical material constant with the unit of Kelvin (K), T_ref is the reference temperature, and T is the actual temperature.

By incorporating the WLF shift factor into the extended Maier–Göritz model, the loss modulus can be expressed as a function of temperature, frequency and dynamic strain amplitude:

$$E^{″}\left({\epsilon }_{\Delta },f,T\right)=E^{″}\left({\epsilon }_{\Delta },{\varphi }_{\mathrm{T}}f,T_{\mathrm{ref}}\right)=E_{\mathrm{st}}^{″}\left({\varphi }_{\mathrm{T}}f\right)+E_{\mathrm{i}}^{″}\left({\varphi }_{\mathrm{T}}f\right){\displaystyle \frac{c{\epsilon }_{\Delta }}{{\left(1+c{\epsilon }_{\Delta }\right)}^2}}$$

(10)

Combining Equations (4) and (10), the hysteresis heat generation rate density is finally derived as an explicit function of temperature, frequency, and strain amplitude:

$$Q\left({\epsilon }_{\Delta },f,T\right)=\pi f{\epsilon }_{\Delta }^2\left[E_{\mathrm{st}}^{″}\left({\varphi }_{\mathrm{T}}f\right)+E_{\mathrm{i}}^{″}\left({\varphi }_{\mathrm{T}}f\right){\displaystyle \frac{c{\epsilon }_{\Delta }}{{\left(1+c{\epsilon }_{\Delta }\right)}^2}}\right]$$

(11)

3. Material Characterization and Parameter Identification

To establish the quantitative relationship between the loss modulus $E^{″}$, temperature, and frequency as well as dynamic strain amplitude, dynamic mechanical analysis (DMA) tests were conducted to characterize the viscoelastic properties of the polyurethane material. The primary objective of these tests was to determine the model parameters for both the extended Maier–Göritz model (frequency–strain dependence) and the WLF equation (temperature dependence).

3.1. Material Testing Procedures

Rectangular polyurethane specimens with dimensions of 30 mm × 10 mm × 2 mm were prepared for the DMA tests. All tests were conducted on a Gabo Eplexor 500 N dynamic mechanical analyzer (NETZSCH-Gerätebau GmbH, Ahlden, Germany). Prior to data acquisition, all specimens underwent mechanical conditioning with at least six pre-loading cycles to eliminate the Mullins effect and ensure a stable and repeatable viscoelastic response.

To fully characterize the temperature, frequency and strain-amplitude dependence of the polyurethane material, two distinct testing protocols were designed and conducted:

(1)

Strain sweep tests: To investigate the combined influence of frequency and dynamic strain amplitude, strain sweep tests were conducted at a fixed environmental temperature of 27 °C. The dynamic strain amplitude was swept from 0.1% to 5% (with ten data points per decade) under a static pre-strain of 10%. The lower bound of 0.1% ensures a stable DMA response while remaining within the small-amplitude regime, whereas the upper bound of 5% is sufficient to capture the strain-amplitude dependence of the loss modulus and the associated Payne effect. This strain sweep procedure was repeated at discrete frequencies of 3, 10, 20, 30, 40, 50, 60, and 70 Hz, respectively.

(2)

Frequency sweep tests: To establish the TTS master curve for temperature dependence, frequency sweep tests were performed over a frequency range of 0.1 Hz to 100 Hz. These tests were conducted under isothermal conditions at −20 °C, −10 °C, 0 °C, 10 °C, 20 °C, 30 °C, 40 °C, 50 °C, and 60 °C, respectively. A static pre-strain of 5% and a dynamic strain amplitude of 2% were applied for all frequency sweep tests.

The strain sweep and frequency sweep tests were performed at static pre-strains of 10% and 5%, respectively, primarily for experimental practicality and signal stability under the corresponding DMA protocols. Since both values represent a finite pre-deformed state within a relatively narrow range, this difference is not expected to materially affect parameter identification.

3.2. Results and Parameter Fitting

The dependence of the loss modulus on frequency and dynamic strain amplitude was fitted and analyzed using the extended Maier–Göritz model. First, the standard Maier–Göritz model (Equation (6)) was fitted to the strain sweep experimental data at each discrete frequency to extract the basic material parameters ($E_{\mathrm{st}}^{″}$, $E_{\mathrm{i}}^{″}$), with the parameter c determined as 2.2 from the fitting results. The fitting curve is shown in Figure 1a; this initial step demonstrated excellent robustness, with the coefficient of determination ($R^2$) values exceeding 0.92 for all individual frequencies. The frequency dependence of the parameters $E_{\mathrm{st}}^{″}$ And $E_{\mathrm{i}}^{″}$ is mathematically expressed by Equations (12) and (13) (with $R^2$ > 0.99). Subsequently, substituting these frequency-dependent parameter expressions back into the model generated the unified theoretical curves, as shown in Figure 1c. The extended model accurately captures the Payne effect of the polyurethane materials across all tested conditions, with a global $R^2$ value exceeding 0.90. Accordingly, the frequency- and strain amplitude-dependent loss modulus at the reference temperature of 27 °C is given by Equation (14):

$$E_{\mathrm{st}}^{″}\left(f\right)=2.38f^{0.13}$$

(12)

$$E_{\mathrm{i}}^{″}\left(f\right)=5.78+0.074f$$

(13)

$$E^{″}\left({\epsilon }_{\Delta },f,T_{\mathrm{ref}}\right)=2.38f^{0.13}+\left(5.78+0.074f\right){\displaystyle \frac{2.2{\epsilon }_{\Delta }}{{(1+2.2{\epsilon }_{\Delta })}^2}}$$

(14)

The experimental data from the frequency sweep tests at different temperatures are presented in Figure 2a. Using 20 °C as the reference temperature T_ref, the isothermal curves at different temperatures were horizontally shifted along the frequency axis to construct the master curve, as shown in Figure 2b. The variation in the shift factor with temperature is plotted in Figure 2c, and the experimental points were fitted using the WLF equation to derive the empirical material constants in Equation (9) for the polyurethane material:

$$C_1=10.79, C_2=137.76 \mathrm{K}$$

(15)

The reduced frequency f_r is then computed as:

$$f_{\mathrm{r}}={\varphi }_{\mathrm{T}}f={10}^{\left[{\displaystyle \frac{-10.79\left(T-20\right)}{137.76+\left(T-20\right)}}\right]}f$$

(16)

With Equations (14)–(16), the loss modulus can be evaluated as a function of temperature, frequency, and dynamic strain amplitude according to Equation (10):

$$\begin{array}{ll}{E}^{″}\left({\epsilon }_{\Delta },f,T\right)& =E^{″}\left({\epsilon }_{\Delta },{\varphi }_{\mathrm{T}}f,T_{\mathrm{ref}}\right)=E_{\mathrm{st}}^{″}\left({\varphi }_{\mathrm{T}}f\right)+E_{\mathrm{i}}^{″}\left({\varphi }_{\mathrm{T}}f\right){\displaystyle \frac{c{\epsilon }_{\Delta }}{{\left(1+c{\epsilon }_{\Delta }\right)}^2}}\\ & =2.38{\left({\varphi }_{\mathrm{T}}f\right)}^{0.13}+\left[5.78+0.074\left({\varphi }_{\mathrm{T}}f\right)\right]{\displaystyle \frac{2.2{\epsilon }_{\Delta }}{{(1+2.2{\epsilon }_{\Delta })}^2}}\end{array}$$

(17)

The corresponding hysteresis heat generation rate density is further obtained by substituting Equation (17) into Equation (4):

$$\mathrm{Q}\left({\epsilon }_{\Delta },f,T\right)=\pi f{\epsilon }_{\Delta }^2\left\{2.38{\left({\varphi }_{\mathrm{T}}f\right)}^{0.13}+\left[5.78+0.074\left({\varphi }_{\mathrm{T}}f\right)\right]{\displaystyle \frac{2.2{\epsilon }_{\Delta }}{{(1+2.2{\epsilon }_{\Delta })}^2}}\right\}$$

(18)

This unified expression for the hysteresis heat generation rate density was implemented in the finite element software ABAQUS 2021 using the HETVAL user subroutine, which serves as the core heat source model for the subsequent thermo-mechanically coupled simulations.

4. Finite Element Calculation Model of O-Ring Seal

4.1. Constitutive Model

To accurately capture large deformation behavior of the polyurethane O-ring under cyclic loading, the Mooney–Rivlin hyperelastic model was adopted to describe the stress–strain relationship of the polyurethane material [29]. The strain energy density function of the Mooney–Rivlin model is expressed as:

$$W=C_{10}(I_1-3)+C_{01}(I_2-3)$$

(19)

where I₁ and I₂ are the first and the second deviatoric strain invariants, respectively. C₁₀ and C₀₁ are material constants, which were determined as 0.371 MPa and 0.864 MPa, respectively, based on uniaxial tensile tests of the polyurethane material. Notably, a fully visco-hyperelastic model was not adopted in this study. Simulating high-frequency dynamic contact over thousands of cycles using a visco-hyperelastic model is computationally prohibitive and prone to severe convergence issues. Therefore, to balance computational efficiency and accuracy, a decoupled approach was employed: the Mooney–Rivlin model was selected to capture the large bulk deformation and baseline stress, while the viscoelastic hysteresis heating was separately calculated and introduced as a heat source via the HETVAL user subroutine.

4.2. Geometrical Model and Mesh Generation

A typical hydraulic O-ring sealing system consists of four key components: the piston rod, the O-ring seal, the seal groove, and the hydraulic medium. Figure 3a presents the unstressed and undeformed state of the O-ring before assembly. Figure 3b,c show the deformation state of the O-ring under radial interference assembly (pre-compression) and hydraulic pressure loading, respectively. Figure 3d illustrates the reciprocating motion process of the piston rod with inward and outward strokes, which is the typical dynamic working condition of the O-ring seal. The sealing function is achieved through the contact pressure generated by radial pre-compression and the elastic recovery deformation that resists the hydraulic medium pressure.

To evaluate the stresses, strains, and contact pressure distribution of the O-ring under actual operating conditions, an axisymmetric finite-element model was established in ABAQUS 2021. The key geometric and physical parameters of the O-ring sealing system are summarized in

Table 1. Geometric and physical parameters of the O-ring seal system.

Property	Symbol	Unit	Value
O-ring cross-section diameter	a	mm	5.33
Groove width	b	mm	7.3
Groove depth	h	mm	4.24
Groove bottom radius	r1	mm	0.4
Groove edge radius	r2	mm	0.2
Density	ρ	kg·m−3	1200
Specific heat capacity	c	J·kg−1·K−1	2800
Coefficient of heat conduction	λ	W·m−1·K−1	0.25
Convective heat transfer coefficient from specimen to air	hc	W·m−2·K−1	15
coefficient of thermal expansion	α	K−1	1.48 × 10−4

. The following reasonable assumptions were made for the finite-element model to balance the simulation accuracy and computational efficiency:

(1)

The polyurethane material is considered an incompressible hyperelastic material with a Poisson’s ratio of 0.5.

(2)

The piston rod and hydraulic cylinder (including the seal groove) are made of metal with a much higher elastic modulus than polyurethane; thus, they were set as rigid bodies in the simulation to eliminate their deformation and reduce computational cost.

(3)

The O-ring, seal groove and piston rod exhibit perfect axisymmetric structural characteristics, and the applied loads, such as pre-compression and hydraulic pressure, are also axisymmetric; thus, the 3-D sealing system can be simplified to a 2-D axisymmetric model without losing simulation accuracy.

(4)

The heat exchange between the O-ring and the surrounding environment is only considered as convective heat transfer.

Mesh quality strongly influences the accuracy and convergence of contact simulations. The O-ring cross-section was carefully partitioned to enable structured quadrilateral meshing, which is generally more accurate and convergent than triangular elements for large-deformation contact problems. The final finite element mesh contains 3137 nodes and 3072 elements. The O-ring was discretized using CAX4HT elements (four-node axisymmetric, hybrid, and thermally coupled quadrilateral elements) which ensure stable convergence for the present coupled thermo-mechanical contact analysis. A three-level mesh sensitivity study was performed to evaluate the influence of mesh density on the numerical results. The differences among the three meshes were minimal, and the results from the medium and fine meshes were nearly identical. In particular, the differences in peak contact pressure and maximum temperature rise between the adopted mesh and the fine mesh were only 0.05% and 0.9%, respectively, indicating that the key results are weakly sensitive to further mesh refinement. Therefore, the adopted mesh was deemed adequate for the present study. The simplified finite element geometric model and mesh division are shown in Figure 4.

4.3. Thermo-Mechanical Coupling Analysis Step Setting

A sequential thermo-mechanical coupling procedure was adopted because a fully coupled analysis of high-frequency reciprocating contact with large deformation involves substantially higher computational cost and greater convergence difficulty. In the present framework, the polyurethane O-ring is described by a temperature-independent Mooney–Rivlin model; therefore, temperature affects the contact pressure only through thermal expansion. Under the operating conditions considered, the influence of the coupling strategy on contact–pressure prediction is therefore expected to be limited. As illustrated in Figure 5, the coupling analysis workflow comprises three modules:

(1)

Deformation module: This module consists of four sequential steps for mechanical deformation analysis: interference assembly (pre-compression), hydraulic pressure loading, inward stroke, and outward stroke. The hyperelastic constitutive model, geometric structure, boundary conditions and mesh generation were applied in this module to calculate the strain distribution and strain history at each integration point in the O-ring.

(2)

Dissipation module: The strain history of the O-ring was extracted via the USDFLD user subroutine, and the dynamic strain amplitude ${\epsilon }_{\Delta }$ at each integration point was determined by comparing the strain values of the inward and outward strokes. These strain amplitude values were then coupled with the temperature-dependent loss modulus in the HETVAL user subroutine to quantify the hysteretic heat generation rate density at each integration point according to Equation (17).

(3)

Thermal analysis module: The convective heat boundary conditions were set for the thermal analysis, and the transient temperature field of the O-ring was calculated based on the hysteresis heat generation rate density from the dissipation module. The calculated temperature data were then fed back into the dissipation module to update temperature-dependent loss modulus and heat generation rate density.

The three modules were iteratively executed until a steady-state thermal response was reached, with the updated temperature field transferred to the subsequent mechanical analysis via thermal expansion. Automatic time incrementation and the default nonlinear convergence controls in ABAQUS 2021 were adopted to improve numerical robustness in the coupled analysis.

4.4. Contact and Boundary Conditions

In the mechanical deformation analysis, the contact interaction between the O-ring and metal surfaces (piston rod and seal groove) was defined as hard contact in the normal direction and Coulomb friction in the tangential direction. Unless otherwise specified, a friction coefficient of 0.1 was adopted as the baseline condition, and in the parametric study, this value was varied from 0.01 to 0.2 to evaluate its influence on sealing performance and hysteresis heat generation. Lubrication was not modeled explicitly but was represented indirectly via an effective interfacial friction coefficient. Under this assumption, the friction coefficient was treated as constant, and the effects of material degradation and wear were neglected. As a result, the present model is primarily intended to provide a mechanistic and parametric description of hysteresis-induced thermo-mechanical behavior, and its quantitative predictions for long-term service conditions should be interpreted with caution.

The hydraulic pressure loading was incorporated into the deformation analysis using a pressure-penetration algorithm [30], which can autonomously detect the lip separation and contact transitions of the O-ring under hydraulic pressure, and is more accurate than the traditional uniform pressure loading method for sealing problems. As shown in Figure 6, when the hydraulic pressure at a certain node (node 1) of the O-ring lip exceeds the contact pressure at that node, the hydraulic medium will penetrate along the sealing interface to the next node (node 2) and apply pressure. This process continues until the hydraulic pressure is less than the contact pressure at the node, at which point the forward penetration of the hydraulic medium stops.

To simulate the reciprocating motion of the piston rod, upward and downward displacement loads were applied to the reference point of the piston rod to simulate the outward and inward strokes of the O-ring, respectively. The stroke length of the piston rod was set to 1 mm for both the outward and inward strokes. The dynamic strain amplitude (${\epsilon }_{\Delta }$) at each integration point of the O-ring was calculated by extracting the steady-state strain field of the inward (${\epsilon }_{\mathrm{recip}1}$) and outward (${\epsilon }_{\mathrm{recip}2}$) strokes using the USDFLD subroutine:

$${\epsilon }_{\Delta }={\displaystyle \frac{{\epsilon }_{\mathrm{recip}1}-{\epsilon }_{\mathrm{recip}2}}{2}}$$

(20)

The calculated dynamic strain amplitude was then passed to the HETVAL user subroutine to define the hysteresis heat generation rate as a volumetric heat source (Equation (18)), which drives the transient thermo-mechanical coupled analysis. For the thermal analysis, convective heat transfer boundary conditions with a convective heat transfer coefficient (h_c) were applied to all out surfaces of the O-ring to simulate the heat exchange between the O-ring and the surrounding air.

4.5. Model Validation

Direct measurement of the internal temperature field in an operating O-ring is extremely challenging due to its small dimensions and dynamic working condition. Therefore, an indirect experimental validation based on cylindrical compression tests was adopted in this study. The polyurethane specimen used for validation had a cylindrical geometry of 10 mm in diameter and 10 mm in height. The surface temperature was measured using a FOTRIC 628C infrared camera (FOTRIC, Shanghai, China), which has a thermal sensitivity of 30 mK at 30 °C, an accuracy of 2% for temperatures from −20 °C to 650 °C. The experimental setup is shown in Figure 7.

The calibration test conditions were as follows: initial temperature of 24 °C, reciprocating frequency of 50 Hz, static compressive strain of 20%, dynamic strain amplitude of 5%, and test duration of 1800 s. In addition to the maximum temperature deviation, the agreement between the simulated and measured steady-state temperature was quantified using the root mean square error (RMSE) and normalized root mean square error (NRMSE), which were determined as 0.6 °C and 1.39%, respectively. Figure 8 shows that the model captures both the transient heating stage and the subsequent stabilization trend with satisfactory accuracy, thereby validating the predictive capability of the proposed thermo-mechanical coupling framework.

5. Results and Discussions

To quantify the effects of key operating parameters on the sealing performance and hysteresis heating of polyurethane O-rings, a series of numerical simulations were performed under different reciprocating operation conditions. The baseline operating conditions for the simulation are: 10% pre-compression ratio, 5 MPa hydraulic pressure, 0.1 friction coefficient, 50 Hz reciprocating frequency, 1 mm stroke length, and 24 °C ambient temperature. Notably, the selected frequency of 50 Hz was not intended to represent a universal operating frequency for all hydraulic systems. Rather, when combined with the 1 mm stroke length, it corresponds to a peak rod velocity of approximately 0.16 m/s, which is within the typical range reported for reciprocating O-ring sealing applications [31].

5.1. Deformation Analysis

The stress and strain distribution of the O-ring under baseline operating conditions was analyzed for four key stages: interference assembly, hydraulic loading, inward stroke and outward stroke. Figure 9a shows the von Mises stress distribution of the O-ring after interference assembly with 10% pre-compression, with an approximately symmetric stress field and the peak stress located near the mid-right region of the O-ring cross-section. After hydraulic pressure loading of 5 MPa, as shown in Figure 9b, the stress field of the O-ring changes significantly, and the highest stress is concentrated in the rod–cylinder clearance where extrusion deformation occurs. This is because the hydraulic pressure drives the O-ring to extrude into the small clearance between the piston rod and the cylinder, resulting in localized stress concentration. Figure 9c,d compare the von Mises stress distribution of the O-ring during the inward and outward strokes under reciprocating motion. The stress level during the outward stroke is significantly higher than that during the inward stroke, and the extrusion deformation in the rod–cylinder clearance becomes more pronounced, highlighting the asymmetry of the loading cycle under reciprocating motion. Repeated high-stress extrusion events in this localized region are the main cause of seal damage and premature failure [32], as they lead to cumulative plastic deformation and material fatigue of the O-ring.

Figure 10a,b show the maximum in-plane principal strain distribution of the O-ring during the inward and outward strokes. Consistent with the stress distribution, the outward stroke produces a larger strain amplitude than the inward stroke, and the maximum strain is also concentrated in the extrusion zone of the rod–cylinder clearance. The steady-state strain field of the O-ring was extracted using the USDFLD user subroutine, and the dynamic strain amplitude at each integration point was calculated using Equation (20). This strain amplitude was then substituted into Equation (18) to compute the hysteresis heat generation rate density, which was implemented in the FEA model via the HETVAL user subroutine. The thermo-mechanical coupling analysis was iterated until convergence to obtain the steady-state temperature rise distribution of the polyurethane O-ring. The extrusion-dominated region therefore governs not only the peak mechanical response of the O-ring, but also the local dynamic strain amplitude that drives hysteresis heat generation in the subsequent thermal analysis.

5.2. Contact Pressure Distribution Analysis

The contact pressure at the polyurethane O-ring-metal interface is the most critical criterion for assessing seal failure [33]. The O-ring can maintain a reliable sealing function if its peak contact pressure exceeds the system hydraulic pressure. Conversely, if the peak contact pressure falls below the hydraulic pressure, seal failure occurs, leading to hydraulic oil leakage and pressure loss [34]. From Figure 11a,b, the peak contact pressure of the O-ring during the inward and outward strokes under baseline conditions is 6.965 MPa and 7.054 MPa, respectively, both of which are significantly higher than the applied pressure of 5 MPa. This indicates that the O-ring can maintain an effective sealing performance under the baseline operating conditions, with a sufficient contact pressure margin.

5.3. Effect of Hysteresis Heat Generation on Sealing Performance

Figure 12 shows the evolution of the temperature distribution of the O-ring at different representative time points under baseline operating conditions. The temperature rise in the O-ring is highly localized, and the highest temperature occurs near the extrusion zone of the rod–cylinder clearance. This is because the dynamic strain amplitude is the largest in this region, leading to the most intense hysteresis heat generation. The temperature rise in other regions of the O-ring is relatively small, which further confirms that hysteresis heat generation is directly correlated with the dynamic strain amplitude of the material.

Figure 13 compares the contact–pressure distributions along the sealing interface obtained with and without hysteresis heating. Under the baseline operating conditions, inclusion of the temperature field leads to a slight reduction in the peak contact pressure during the outward stroke. Notably, as a limitation of the present framework, the polyurethane O-ring is described by a temperature-independent Mooney–Rivlin constitutive model. Therefore, this result should not be interpreted as direct thermally induced material softening. Rather, it reflects the thermo-mechanical effect captured by the current model, namely the nonuniform thermal strain generated by localized hysteresis heating in a geometrically constrained seal. Because the temperature rise is concentrated near the extrusion region, the resulting thermal expansion slightly perturbs the local contact configuration and redistributes the interfacial normal traction, leading to a modest decrease in the local peak contact pressure. The limited magnitude of this effect is consistent with the fact that only bulk hysteresis heating is considered here, whereas interfacial frictional heating and temperature-dependent mechanical degradation are not included.

5.4. Parametric Study on Sealing Performance and Hysteresis Heat Generation

A systematic parametric study was conducted to investigate the effects of four key operating parameters (i.e., pre-compression ratio, hydraulic pressure, friction coefficient, and reciprocating frequency) on the stress, contact pressure and steady-state temperature rise in the O-ring. The range of each parameter was selected based on actual engineering applications of hydraulic sealing systems, to ensure the practical significance of the simulation results.

5.4.1. Effect of Pre-Compression Ratio

The pre-compression ratio is a key assembly parameter of the O-ring, which directly determines the initial contact pressure at the sealing interface. Four different pre-compression ratios (7%, 10%, 15%, and 20%) were simulated to investigate its effect on sealing performance and hysteresis heat generation.

As shown in Figure 14, the peak contact pressure of the O-ring during the outward stroke increases with the increase in pre-compression ratio. The peak contact pressure at 20% pre-compression is approximately 20% higher than that at 7% pre-compression. This is because a higher pre-compression provides a larger baseline sealing force, which combines with the elastic recovery force caused by hydraulic pressure to increase the total contact pressure at the sealing interface. Across the studied pre-compression ratio range, the peak contact pressure remains well above the hydraulic pressure of 5 MPa, indicating that the sealing safety margin increases with the increase in pre-compression ratio under these conditions.

Figure 15 shows the steady-state temperature distribution of the O-ring at different pre-compression ratios. Notably, the pre-compression ratio has a negligible effect on the steady-state temperature rise. This is because the pre-compression ratio only affects the initial static strain of the O-ring, and does not substantially alter the dynamic strain amplitude imposed by the hydraulic pressure and reciprocating stroke. Therefore, changing the pre-compression ratio does not significantly affect the hysteresis heat generation rate and the resulting temperature field of the O-ring under dynamic operating conditions. The pre-compression ratio therefore mainly governs the static sealing preload, whereas the thermal load is more directly controlled by the cyclic deformation amplitude during reciprocation.

5.4.2. Effect of Hydraulic Pressure

Hydraulic pressure is the most important operating parameter of hydraulic sealing systems, which directly drives the deformation and sealing behavior of the O-ring. Five different hydraulic pressures (2 MPa, 4 MPa, 5 MPa, 6 MPa, and 8 MPa) were simulated to investigate its effect on sealing performance and hysteresis heat generation.

As shown in Figure 16, the contact pressure of the O-ring shows a substantial increase with the increase in hydraulic pressure. This is a direct consequence of the hydraulic pressure acting on the cross-section of the O-ring, which transmits the pressure to the sealing interface and increase the contact pressure between the O-ring and the metal surface. Crucially, the peak contact pressure consistently exceeds the applied hydraulic pressure across the entire studied range, confirming that the O-ring can maintain its fundamental sealing function even at high hydraulic pressure. Furthermore, Figure 17 shows that the steady-state temperature rise in the O-ring exhibits a significant increase with the increase in hydraulic pressure. The maximum temperature at 8 MPa hydraulic pressure reaches 63.6 °C, while it is only 27.34 °C at 2 MPa hydraulic pressure. This thermal rise is primarily attributed to the increased dynamic strain amplitude in the extrusion zone with the increase in hydraulic pressure; according to the hysteresis heat-generation model, the heat generation rate density is proportional to the square of the dynamic strain amplitude, leading to a rapid increase in hysteresis heat generation and thus the temperature rise in the O-ring. These results reveal an intrinsic trade-off under high-pressure operation: a higher hydraulic pressure increases the sealing pressure margin but simultaneously intensifies the local deformation and the associated hysteresis thermal load.

5.4.3. Effect of Friction Coefficient

The friction coefficient between the O-ring and the metal surface affects the frictional behavior and deformation mode of the O-ring under reciprocating motion. Five different friction coefficients (0.01, 0.05, 0.1, 0.15, and 0.2) were simulated to investigate its effect on sealing performance and hysteresis heat generation.

As shown in Figure 18, the peak contact pressure of the O-ring exhibits a clear inverse relationship with the friction coefficient. The peak contact pressure decreases from 7.216 MPa to 6.972 MPa when the friction coefficient is increased from 0.01 to 0.2. As the friction coefficient increases, the interfacial shear traction becomes larger and tangential slip is more strongly constrained. This additional tangential constraint restricts the local readjustment of the O-ring material along the contact interface during reciprocation. As a result, the contact state is redistributed, and the local concentration of normal compressive traction is slightly weakened, leading to a reduction in the peak contact pressure. This interpretation is consistent with previous studies showing that friction significantly affects the mechanical response of dynamic O-ring seals, and that tangential loading in soft elastomer contacts can induce shear-driven contact-area reduction or redistribution [9,35]. Crucially, the peak contact pressure consistently exceeds the applied hydraulic pressure of 5 MPa across all studied friction coefficients, confirming that sealing integrity of the O-ring is maintained even at high friction coefficients. Nevertheless, a high friction coefficient poses a risk to the durability due to accelerated adhesive wear and excessive thermal buildup.

Figure 19 demonstrates a strong positive correlation between the friction coefficient and the steady-state temperature rise in the O-ring. To clarify the mechanism by which friction affects the temperature rise, the strain fields of the O-ring during the inward and outward strokes were further examined for three representative friction coefficients, $\mu =0.01$, 0.1, and 0.2, corresponding to low, intermediate, and relatively high friction levels. The results show that the strain during the outward stroke increases with increasing friction coefficient, whereas the strain during the inward stroke decreases. According to the definition of the dynamic strain amplitude, ε_Δ = (ε_recip1 − ε_recip2), these opposite trends lead to an increase in the overall strain amplitude of the O-ring as the friction coefficient increases. As a result, the volumetric hysteresis heat source defined by Equation (18) also increases. Since interfacial frictional heat generation is not included in the present model, the temperature rise observed here should be interpreted as an indirect consequence of friction-induced changes in the cyclic deformation state, rather than direct surface frictional heating. While the precise quantitative contribution of friction to hysteresis heating requires further experimental investigation, the simulation results clearly confirm that higher friction coefficients substantially elevate the seal’s steady-state temperature and thus increase the risk of thermal softening and premature failure.

5.4.4. Effect of Reciprocating Frequency

Reciprocating frequency determines the loading rate of the O-ring under cyclic motion, and is a key parameter for high-speed hydraulic sealing systems. Five different reciprocating frequencies (10 Hz, 30 Hz, 50 Hz, 70 Hz, and 90 Hz) were simulated to investigate its effect on sealing performance and hysteresis heat generation.

Notably, since the Mooney–Rivlin constitutive model adopted in this study does not account for viscoelastic effects or rate-dependent mechanical behavior, the reciprocating frequency is solely utilized within the HETVAL user subroutine to determine the heat source intensity. Consequently, the reciprocating frequency has no direct impact on the stress field and contact pressure of the O-ring during the mechanical deformation analysis.

As illustrated in Figure 20, the steady-state temperature of the polyurethane O-ring exhibits a marked increase with the rise in reciprocating frequency. The maximum temperature at 90 Hz is 60 °C, while it is 27.45 °C at 10 Hz. This indicates that reciprocating frequency is another key parameter governing the thermal load of the O-ring. Although the energy dissipated per loading cycle remains constant for a given dynamic strain amplitude, an increase in frequency proportionally elevates the energy dissipation rate per unit time. Thus, a higher frequency amplifies the heat generation rate, driving the system to a higher equilibrium temperature even while the mechanical deformation and strain amplitude per cycle remain unchanged. Consequently, the reciprocating frequency is identified as a governing parameter that determines the thermal load in high-speed reciprocating sealing applications.

The parametric study in this work was designed as a one-factor-at-a-time analysis, in which one operating parameter was varied while the remaining parameters were held at their baseline values. This strategy was adopted deliberately to isolate the first-order effect of each parameter on the contact pressure and steady-state temperature rise in the O-ring. the present results suggest that hydraulic pressure and friction coefficient affect the thermo-mechanical response mainly by altering the local deformation mode and contact state of the O-ring, whereas reciprocating frequency primarily acts as a thermal amplification factor by increasing the rate of hysteresis energy dissipation. By contrast, the pre-compression ratio mainly determines the initial sealing preload and has only a limited influence on the cyclic strain amplitude and the resulting temperature rise. Interaction effects between parameters were not resolved in the present study and should be addressed in future work using a more systematic multi-parameter sensitivity framework.

6. Conclusions

A thermo-mechanically coupled finite element framework was developed to quantify the hysteresis heat generation in polyurethane O-ring seals under reciprocating motion and its effect on sealing performance. This framework integrates a unified hysteresis heat generation rate model with a Mooney–Rivlin hyperelastic constitutive model, and was validated using cylindrical compression tests. A systematic parametric study was conducted to investigate the effects of key operating parameters, including pre-compression ratio, hydraulic pressure, friction coefficient, and reciprocating frequency, on the stress, contact pressure and temperature rise in the O-ring. The main findings and conclusions of this study can be drawn as follows:

(1)

Hysteresis-induced temperature rise leads to a modest but systematic redistribution of contact pressure in polyurethane O-rings under reciprocating motion. Under the baseline operating conditions (5 MPa hydraulic pressure, 10% pre-compression ratio, friction coefficient 0.1, and 50 Hz), the hysteresis heat generation causes a localized temperature rise in the O-ring, which reduces the peak contact pressure by approximately 0.4 MPa during the outward stroke. Although a positive sealing pressure margin is still maintained, this thermal-induced degradation highlights the necessity of thermo-mechanical coupling analysis for accurate assessment of the sealing safety margin, pure mechanical analysis will overestimate the sealing performance.

(2)

The contact pressure of the O-ring is significantly influenced by operating parameters, with distinct trends for different parameters. Contact pressure increases with pre-compression ratio and hydraulic pressure, but decreases with increasing friction coefficient. Crucially, the contact pressure remains above the hydraulic pressure across all studied parameter ranges, ensuring basic sealing integrity.

(3)

Hydraulic pressure, friction coefficient, and reciprocating frequency are the predominant drivers of the steady-state temperature rise in the O-ring, while the pre-compression ratio exerts a negligible influence. The pre-compression ratio only affects the initial static strain and does not alter the dynamic strain amplitude.

(4)

Thermal management strategies for reciprocating hydraulic sealing systems should prioritize controlling the dynamic operational limits (e.g., hydraulic pressure, friction coefficient, and reciprocating frequency) rather than the initial assembly parameters (i.e., pre-compression ratio). Reducing the friction coefficient and limiting the maximum operating frequency and hydraulic pressure are effective measures to mitigate hysteresis heat generation, reduce thermal softening, and improve the sealing reliability and service life of polyurethane O-rings.

The proposed thermo-mechanical coupling framework provides a robust tool for quantitatively predicting the thermal load of reciprocating O-ring seals, and the identified key parameters and their effects offer important theoretical guidance for the structural optimization, performance improvement and reliability design of hydraulic sealing systems. In addition, the present predictions should be interpreted as representative of the short-term thermo-mechanical response of the polyurethane O-ring under the specified operating conditions. Further extension of the model to incorporate temperature-dependent hyperelastic properties, temperature-dependent friction, interfacial frictional heating, and progressive material evolution will constitute an important direction for future work.