Elastohydrodynamic lubrication and finite configuration changes in reciprocating elastomeric seals

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Abstract

A computational framework has been developed for a fully coupled analysis of elastohydrodynamic lubrication and finite deformations of elastomeric reciprocating seals in hydraulic actuators. The relevant formulation is provided, which consistently treats finite configuration changes of the seal modelled as a hyperelastic (Mooney-Rivlin) solid. The steady-state hydrodynamic lubrication is modelled using the classical Reynolds equation. Coupling of the solid and fluid parts is fully accounted for, including friction due to shear stresses in the lubricant film. Detailed results of finite element simulations are provided for two benchmark problems of O-ring and rectangular rod seals in a wide range of process parameters.

Introduction

Hydrodynamic lubrication in reciprocating hydraulic seals is a classical topic studied experimentally and theoretically for more than 50 years now, cf. Nau [1]. It is thus rather surprising that detailed solutions of the corresponding elastohydrodynamic lubrication problems are not easily found in the literature, particularly in view of the substantial progress in computational techniques and increase of computer power observed in the last decades. This paper aims at filling this gap by providing the relevant formulation, which consistently treats finite configuration changes, along with a computational scheme and detailed results of numerical simulations.

Compared to the more classical hard EHL problems [2], in the soft EHL problems of elastomeric seals, the lubricant pressures are relatively low so that effects such as variation of viscosity with pressure and compressibility of the fluid are not essential. At the same time, in the case of hard EHL problems, the maximum pressure is typically two orders of magnitude smaller than the elastic modulus and thus the linear elasticity is an appropriate model for contacting members. This is not the case of elastomeric seals which are characterized by a very low elastic stiffness so that the lubricant pressures may easily exceed the shear modulus by one order of magnitude. Accordingly, finite deformations with finite configuration changes are expected to occur, at least locally, and these require appropriate theoretical and numerical treatment.

Simulation of the hydrodynamic lubrication in the seal–rod system requires determination of the flow of the lubricant (hydraulic fluid) in the thin film between the seal and the rod and, in parallel, determination of the deformation of the seal. Clearly, the two phenomena are coupled, and several available solution methods differ in the way in which this coupling is treated.

The inverse hydrodynamic theory (cf. [3]) is based on the assumption that the contact pressure is not affected by the lubricating film developing between the seal and the rod (as the film thickness is much smaller than the elastic deflections of the seal). Accordingly, the distribution of the contact pressure, obtained from a purely mechanical contact analysis, is used to estimate the film thickness in characteristic points along the contact interface. Here, and in all the other relevant models, the fluid part is conveniently described by the Reynolds equation.

In the elastohydrodynamic lubrication problem, the elastic deflections of the seal are solved simultaneously with the Reynolds equation. In the early works, the linear elasticity together with the finite element method were used for that purpose, typically combined with the static condensation, cf. Ruskell [4], Yang and Hughes [5]. Prati and Strozzi [6] developed a model based on the finite-deformation theory with a hyperelastic material model adopted for the seal. The corresponding finite element model was used to compute the contact pressures while the (linear) influence coefficient matrix obtained through the nodal perturbation technique was used in the EHL analysis. Influence coefficients were also used by Salant et al. [7].

Simplified analytical models have been developed for rectangular seals, which have particularly simple geometry, cf. Field and Nau [8]. Similar approach has been used in a recent model of Nikas [9] which has also been extended to account for nonlinear elasticity of elastomeric seals, cf. Nikas and Sayles [10]. However, these simplified models are not able to represent some features of the solution, for instance, the pressure peaks at the rounded corners which were observed experimentally and also were predicted numerically by Prati and Strozzi [6], see also Section 4.2.

In the present approach, the nonlinearities associated with the finite configuration changes and hyperelasticity of the seal are fully accounted for. The focus of this work is on the coupling of the hydrodynamic lubrication and finite deformations of the seal. Accordingly, several effects, such as the influence of surface roughness (e.g. [7], [9]) and extrusion of the seal at the air side, which are known to be important in some situations, are not addressed here. Also, to fix the attention, the rod seals are only referred to throughout the paper, although, the approach is obviously more general.

The formulation of the corresponding EHL problem is introduced in Section 2 and the finite element discretization along with the adopted solution strategy are discussed in Section 3. As an application, the analysis of the steady-state hydrodynamic lubrication and dynamic sealing performance of reciprocating O-ring and rectangular seals is carried out in Section 4. Due to the simple geometry, these two types of seals are particularly suitable for benchmark and verification examples. Based on the study of convergence of the solution with mesh refinement, relatively fine discretization has been used for the computations. Accordingly, fine features of the solutions could be captured, such as sharp minima and maxima of the film thickness and pressure at the outlet and inlet zones. Results of similar scope could not be found in the literature.

Section snippets

Finite deformations of hyperelastic seal

The present model of the seal–rod system accounts for the deformations of the seal due to the action of the hydrostatic sealed pressure as well as contact interactions with the housing and with the rod, the latter in the hydrodynamic lubrication regime. The housing and the rod are assumed rigid, while the elastomeric seal will typically undergo finite deformations, at least locally. Accordingly, two configurations are introduced, the stress-free initial (reference) configuration Ω and the

Finite element discretization

The finite element method is used to solve the elastohydrodynamic lubrication problem defined in the previous section. Recall that the problem is specified by the mechanical equilibrium equation (the solid part) and the Reynolds equation (lubrication), expressed by the respective variational weak forms (1), (14). The virtual work of surface tractions, i.e. the second term in Eq. (1), is further split into three parts corresponding to the hydrostatic pressure on pω, contact with the housing on

O-ring seal

The dynamic sealing performance of an O-ring seal in steady-state lubrication conditions has been studied as the first example. The basic geometrical, material and process parameters used in the computations are provided in Table 1. The elastic properties3

Conclusion and discussion

A solution method for the soft EHL problems of reciprocating elastomeric seals has been developed. The formulation takes full account of the finite deformations of the seal and of the coupling of the solid and fluid parts, including friction due to shear stresses in the lubricant film. The corresponding computational framework employs the finite element method to solve the equilibrium equation for the seal as well as the Reynolds equation describing the flow of the lubricant. The resulting

Acknowledgments

This work has been financially supported by the European Commission through the PROHIPP Project (NMP 2–CT–2004–505466).

References (18)

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