Finite Element Modeling of Elastohydrodynamic Lubrication Problems

By Wassim Habchi

Covers the latest developments in modeling elastohydrodynamic lubrication (EHL) problems using the finite element method (FEM)

This comprehensive guide introduces readers to a powerful technology being used today in the modeling of elastohydrodynamic lubrication (EHL) problems. It provides a general framework based on the finite element method (FEM) for dealing with multi-physical problems of complex nature (such as the EHL problem) and is accompanied by a website hosting a user-friendly FEM software for the treatment of EHL problems, based on the methodology described in the book. Finite Element Modeling of Elastohydrodynamic Lubrication Problems begins with an introduction to both the EHL and FEM fields. It then covers Standard FEM modeling of EHL problems, before going over more advanced techniques that employ model order reduction to allow significant savings in computational overhead. Finally, the book looks at applications that show how the developed modeling framework could be used to accurately predict the performance of EHL contacts in terms of lubricant film thickness, pressure build-up and friction coefficients under different configurations.

Finite Element Modeling of Elastohydrodynamic Lubrication Problems offers in-depth chapter coverage of Elastohydrodynamic Lubrication and its FEM Modeling, under Isothermal Newtonian and Generalized-Newtonian conditions with the inclusion of Thermal Effects; Standard FEM Modeling; Advanced FEM Modeling, including Model Order Reduction techniques; and Applications, including Pressure, Film Thickness and Friction Predictions, and Coated EHL.

This book:

• Comprehensively covers the latest technology in modeling EHL problems
• Focuses on the FEM modeling of EHL problems
• Incorporates advanced techniques based on model order reduction
• Covers applications of the method to complex EHL problems
• Accompanied by a website hosting a user-friendly FEM-based EHL software

Finite Element Modeling of Elastohydrodynamic Lubrication Problems is an ideal book for researchers and graduate students in the field of Tribology.

• Language: English
• Publisher: Wiley
• Release date: Mar 21, 2018
• ISBN: 9781119225140

Book preview

Preface

This book is intended for graduate students and/or researchers interested in modeling the elastohydrodynamic lubrication (EHL) problem using the finite element method (FEM). The level of details provided would allow readers to build their own in-house FEM-based EHL codes from scratch or to use any of the variety of available commercial FEM software to implement them. This latter option is probably the most attractive advantage of FEM modeling of the EHL problem. In fact, though FEM has grown over the years to become the dominant methodology used in scientific computing, for the EHL problem, the most widespread techniques today are based on the finite difference method. The lack of available commercial software allowing the implementation of finite difference codes is a major obstacle. The wide availability of commercial FEM software and handbooks is a clear testimony to the widespread use of the methodology in a variety of scientific areas. It is also a clear sign of maturity of the technology, which has been developed and carefully improved over the years to meet the complex requirements imposed by different fields of science and industries. Another advantage of FEM is its object-oriented nature, which allows a significant flexibility in extending models to include new features. Also, features such as non-structured meshing, high-order elements, and model order reduction (MOR) offer FEM a major advantage in terms of computational performance (memory and speed). The methodology detailed in this book also enables a straightforward incorporation of effects that are difficult (if not impossible) to include in traditional modeling approaches, for example, non-homogenous and/or anisotropic and/or nonlinear solid material properties, and plastic deformations.

EHL is a lubrication regime in which contacting surfaces in relative motion are fully separated by a lubricant film. It is a sub-field of tribology, the science of interacting surfaces in relative motion. Within EHL films, hydrodynamic pressures of several gigapascals may develop. This leads to an elastic deformation of the contacting solids, thus the name elastohydrodynamic. Also, film thicknesses may be as low as a few nanometers, and shear stresses within the lubricant film may reach hundreds of megapascals. Under such conditions, the Newtonian limit of most lubricants is exceeded, and significant heat generation by shear may occur within EHL conjunctions. Lubricant temperature rise within the film may exceed 100°C in some extreme cases. This makes EHL modeling a rather complex process involving some strong coupling between various physics: hydrodynamics, linear elasticity, heat transfer, rheology, and so on.

The author has gained extensive experience in FEM modeling of the EHL problem over the last decade. He started working on this topic as a PhD student at Institut National des Sciences Appliqées (INSA) de Lyon (France), back in 2005. Immediately after his PhD, he moved back to his home country, Lebanon, where he held an Assistant Professor of Mechanical Engineering position at the Lebanese American University (LAU), School of Engineering, between 2008 and 2014. In 2014, he became an Associate Professor of Mechanical Engineering at LAU and still holds this position to date. In his positions at LAU, the author continued his work on FEM modeling of the EHL problem, extending the methodology to incorporate some advanced and EHL-specific MOR techniques, but also using it to study a variety of EHL configurations: coated surfaces, complex rheology, and so on. The author also used the developed tools to help advance the fundamental understanding of friction generation and film-forming capability in EHL contacts. With this book, the author wishes to share his experience with graduate students and researchers interested in the topic. The author also hopes it will assist readers in building their own EHL solvers and using them to further advance the field of EHL.

The book is divided into three distinct parts. The first part is an introductory one in which both the EHL and FEM fields are introduced. Chapter 1 offers a complete and general overview of the EHL problem and lays down the equations governing the different coupled physics that are involved. In Chapter 2, the FEM method is introduced with enough details for non-familiar readers to be able to grasp the different modeling techniques introduced in subsequent chapters. Chapter 2 should not be viewed as a comprehensive coverage of the FEM method, but rather as a just enough coverage for the book to be complete and for readers to be able to go through it, without the need for further readings. Obviously, a full coverage of the FEM method would require several handbooks and is beyond the scope of the current book. Readers who are interested in a deeper treatment of FEM and its mathematical foundations may refer to the wide variety of available handbooks on the topic.

In the second part, the FEM techniques used to model the EHL problem are described in detail, under a variety of configurations. These techniques are implemented with in-house codes written in the standard C++ programming language. In Chapter 3, the modeling of the steady-state isothermal Newtonian line contact problem is described. Chapter 4 offers an extension to the more general case of a point contact. Chapters 5 and 6 describe the incorporation of non-Newtonian and thermal effects into the previously described models for line and point contacts, respectively. Chapter 7 details the incorporation of transient effects into the modeling of the EHL problem. Finally, Chapter 8 describes some advanced MOR techniques, specifically developed for the EHL problem to boost the computational performance of its corresponding FEM models in terms of both computational speed and memory requirements.

The last part covers some areas of application of the numerical tools developed in the second part and showcases how these could be used to establish a proper quantitative and fundamental understanding of the EHL problem. The author has used these tools over the years for a variety of applications. A complete coverage would be beyond the scope of this book. However, the author has selected the applications he thought would be most representative and interesting for readers willing to gain a deeper insight into the EHL problem. In Chapter 9, the developed tools are used to accurately predict pressure and film thickness in EHL contacts. Also, a proper understanding of the physical mechanisms behind film-forming capability is established. Chapter 10 covers the accurate prediction of friction in EHL contacts, which is a far more complex goal to achieve. Further, an interesting discussion on the delineation of EHL friction regimes using dimensionless groups is offered. Finally, Chapter 11 describes how surface coatings can be incorporated in the FEM analysis of EHL contacts, in a rather straightforward manner. It also offers an interesting discussion on how surface coatings may be selected on the basis of their thermo-mechanical properties to significantly enhance the frictional response of EHL contacts, without affecting the fatigue life of their corresponding machine components.

To conclude, the author wishes to express his extreme gratitude to Philippe Vergne from INSA de Lyon (France), his PhD advisor, for introducing him to the field of EHL, and to his co-advisor, Dominique Eyheramendy from Ecole Centrale de Marseille (France), for sharing his FEM knowledge and expertise. It was their scientific and social skills as well as their trust and insightful vision that allowed the author to start his research journey in the field of EHL on solid ground. The author is extremely grateful to his colleague and long-term friend Jimmy Issa for his careful reading of the second chapter of the book. The author also wishes to thank his many collaborators with whom he has had the pleasure of working and exchanging ideas on the EHL problem over the years. A special appreciation goes to Scott Bair from the Georgia Institute of Technology (United States), who has been more than a collaborator, an inspiration, and a research soul mate. Finally, the author wishes to dedicate this book to his family, his wife Maya and his baby girl Leah, for their endless love and support.

Nomenclature

α = Lubricant viscosity-pressure coefficient (Pa −1 ) c0x-math-0001 Diffusion tensor αi Generalized coordinate i for reduced FEM model αx, αy, αz Diffusion tensor components in the x , y , z -directions β Lubricant viscosity-temperature coefficient (K −1 ) βK Tait EoS isothermal bulk modulus temperature coefficient (K −1 ) c0x-math-0002 Convection vector βx, βy, βz Convection vector components in the x , y , z -directions δ Elastic deformation of equivalent solid (m) c0x-math-0003 Dimensionless elastic deformation of equivalent solid δ1, δ2 Elastic deformation of solids 1 and 2 (m) η Lubricant generalized-Newtonian viscosity (Pa·s) c0x-math-0004 Lubricant dimensionless generalized-Newtonian viscosity ηe First-order cross-film lubricant viscosity integral (Pa·s/m) c0x-math-0005 Dimensionless first-order cross-film lubricant viscosity integral c0x-math-0006 Second-order cross-film lubricant viscosity integral (Pa·s/m ² ) c0x-math-0007 Dimensionless second-order cross-film lubricant viscosity integral ηR Lubricant viscosity at reference state (Pa·s) ϵ, ω Damped-Newton parameters ϵc Doolittle model occupied volume thermal expansivity (K −1 ) ϵii Solid normal strain in the i -direction within a plane having i as normal γ Dowson and Higginson EoS density-temperature coefficient (K −1 ) γij Solid shear strain in the j -direction within a plane having i as normal γs Relaxation factor for SCS algorithm c0x-math-0008 Lubricant resultant shear rate (s −1 ) c0x-math-0009 Lubricant shear rate in the j -direction within a plane having i as normal (s −1 ) c0x-math-0010 , c0x-math-0011 Lamé constants of equivalent solid (Pa) c0x-math-0012 , c0x-math-0013 Lamé constants of solid 1 (Pa) c0x-math-0014 , c0x-math-0015 Lamé constants of solid 2 (Pa) λ(k) Damping factor at iteration k for damped-Newton method λinit Initial damping factor for damped-Newton method λmin Minimum damping factor for damped-Newton method λrecovery Recovery damping factor for damped-Newton method μ Lubricant low-shear/Newtonian viscosity (Pa·s) c0x-math-0016 Dimensionless lubricant low-shear/Newtonian viscosity μg Lubricant viscosity at glass transition temperature (Pa·s) μR Lubricant low-shear/Newtonian viscosity at reference state (Pa·s) c0x-math-0017 Lubricant low-shear/Newtonian viscosity at infinite temperature (Pa·s) μ0 Ambient lubricant low-shear/Newtonian viscosity (Pa·s) μ1, μ2 Lubricant first and second Newtonian viscosities (Pa·s) c0x-math-0018 , c0x-math-0019 Dimensionless lubricant first and second Newtonian viscosities μ1,R, μ2,R Lubricant first and second Newtonian viscosities at reference state (Pa·s) υ Equivalent solid Poisson coefficient υc Coating material Poisson coefficient υs Substrate material Poisson coefficient υ1, υ2 Poisson coefficient of solids 1 and 2 Ω Equivalent solid computational domain Ωe Computational domain of element e within Ω Ωc Contact computational domain Ωc,e Computational domain of element e within Ωc Ωf Lubricant film computational domain within thermal part Ωf,e Computational domain of element e within Ωf Ω1, Ω2 Computational domain of solids 1 and 2 within thermal part Ω1,e, Ω2,e Computational domain of element e within Ω1, Ω2 c0x-math-0020 Boundaries of Ω c0x-math-0021 Boundaries of Ωc c0x-math-0022 Symmetry boundary of Ωc c0x-math-0023 Fixed boundary of Ω c0x-math-0024 Symmetry boundary of Ω Λ Lubricant limiting shear stress-pressure coefficient Ψ1 , Ψ2 Complete elliptic integral of the first and second kind ϕ Correction factor for double-Newtonian film thickness formulas ϕi Basis function i in reduced FEM solution space c0x-math-0025 FEM polynomial approximation of ϕi within element e c0x-math-0026 Nodal value of ϕi at node j within element e ρ Lubricant density (kg/m³) ρID Constant term for Isotropic Diffusion stabilized FEM formulation ρc Coating material density (kg/m³) ρs Substrate material density (kg/m³) ρ1, ρ2 Density of solids 1 and 2 (kg/m³) c0x-math-0027 Lubricant dimensionless density ρe First-order cross-film lubricant density integral (kg/m²) c0x-math-0028 Dimensionless first-order cross-film lubricant density integral ρ′ First-order cross-film density-to-viscosity double-integral (s) c0x-math-0029 Dimensionless first-order cross-film density-to-viscosity double-integral ρ″ Second-order cross-film density-to-viscosity double-integral (m·s) c0x-math-0030 Dimensionless second-order cross-film density-to-viscosity double-integral ρR Lubricant density at reference state (kg/m³) σii Normal stress in the i-direction within a plane having i as normal (Pa) σn Normal component of 2D or 3D stress tensor (Pa) σt Tangential component of 2D stress tensor (Pa) {σt} Vector of tangential components of 3D stress tensor (Pa) θ Contact ellipticity ratio τ Lubricant resultant shear stress (Pa) c0x-math-0031 Lubricant dimensionless resultant shear stress τL Lubricant limiting shear stress (Pa) τR Reference shear stress (Pa) τij Shear stress in the j-direction within a plane having i as normal (Pa) c0x-math-0032 Dimensionless shear stress in the j-direction within a plane having i as normal c0x-math-0033 Lubricant shear stress τij over plane surface (Pa) c0x-math-0034 Dimensionless lubricant shear stress c0x-math-0035 over plane surface c0x-math-0036 FEM polynomial approximation of c0x-math-0037 within element e c0x-math-0038 Nodal value of c0x-math-0039 at node k within element e c0x-math-0040 Nodal value of c0x-math-0041 at global node k τu Unbounded lubricant shear stress (Pa) τ0 Eyring stress (Pa) τe Tuning parameter for stabilized FEM formulation within element e Γ Inlet Weissenberg dimensionless number ξηζ Reference FEM rectangular Cartesian coordinate system Θ Heaviside function φ Modified coated contact dimensionless Hertzian pressure parameter [Φ] Transformation matrix from full to reduced FEM solution space ξ Penalty term parameter ξ0 Penalty term constant a Hertzian circular contact radius (m) ac, nc Double-Newtonian modified Carreau model parameters ax, ay Hertzian elliptical contact semi-axes in the x, y-directions (m) aV Tait EoS volume-temperature coefficient (K−1) Ak, Bk, Ck, s Lubricant thermal conductivity scaling rule parameters Ae Area of a 2D element e in dimensionless space Af, ωf Dimensionless amplitude and wavelength of surface feature A1, C2 Modified WLF viscosity model parameters (°C) A2, B2 Modified WLF viscosity model parameters (Pa−1) B, R0 Doolittle model parameters BF, c0x-math-0042 , g Vogel-like thermodynamic scaling model parameters B1, C1 Modified WLF viscosity model parameters [Be] Connectivity matrix of elastic part [Bh] Connectivity matrix of hydrodynamic part [Bs] Nodal connectivity matrix of shear stress part c0x-math-0043 , c0x-math-0044 Connectivity matrices for solids 1 and 2 of thermal part c Lubricant specific heat (J/kg·K) cc Coating material specific heat (J/kg·K) cs Substrate material specific heat (J/kg·K) c1, c2 Specific heat of solids 1 and 2 (J/kg·K) c0x-math-0045 Mutual approach of contacting solids (m) C Lubricant volumetric heat capacity (J/m³·K) C0 Lubricant ambient volumetric heat capacity (J/m³·K) C′, m Lubricant volumetric heat capacity scaling rule parameters (J/m³·K) D Ratio of contact equivalent radii of curvature Rx and Ry E Equivalent solid Young's modulus of elasticity (Pa) Ec Coating material Young's modulus of elasticity (Pa) Es Substrate material Young's modulus of elasticity (Pa) E1, E2 Young's moduli of elasticity of solids 1 and 2 (Pa) Er Roller compliance dimensionless number f Friction coefficient F Contact external applied load (N/m: line contacts or N: point contacts) {f} FEM analysis source vector Gc Lubricant critical shear stress (Pa) Gs Shear modulus of roller material (Pa) GHD, UHD,WHD Hamrock and Dowson material, speed and load dimensionless groups h Lubricant film thickness (m) he Characteristic length of element e in dimensionless space h0 Rigid-body separation (m) hc Central film thickness (m) hc,Newtonian Newtonian central film thickness (m) hm Minimum film thickness (m) hNewtonian Newtonian film thickness (m) c0x-math-0046 Non-Newtonian film thickness (m) Hc Dimensionless central film thickness Hm Dimensionless minimum film thickness Hc,m Dimensionless minimum film thickness over contact central line in x-direction H0 Dimensionless rigid-body separation c0x-math-0047 , P(0), c0x-math-0048 Initial guesses for H0, P and c0x-math-0049 within damped-Newton procedure H Dimensionless lubricant film thickness I Material thermal inertia (J/m²·K·s¹/²) [J] Jacobian transformation matrix from actual to reference FEM space J Determinant of [J] k Lubricant thermal conductivity (W/m·K) k0 Lubricant ambient thermal conductivity (W/m·K) kc Coating material thermal conductivity (W/m·K) ks Substrate material thermal conductivity (W/m·K) k1, k2 Thermal conductivities of solids 1 and 2 (W/m·K) K00 Tait EoS isothermal bulk modulus at zero absolute temperature (Pa) c0x-math-0050 Tait EoS pressure rate of change of isothermal bulk modulus at zero pressure [K] Linear FEM analysis stiffness matrix [Kn], [Kf] Near and far sub-matrices of [K] L, M Moes dimensionless material properties and load parameters Li Limiting shear stress dimensionless number c0x-math-0051 Molecular weight (kg/kmol) c0x-math-0052 Number of elements in elastic domain Ω c0x-math-0053 Number of elements in hydrodynamic domain Ωc c0x-math-0054 , c0x-math-0055 Numbers of elements in solid domains Ω1 and Ω2 of thermal part c0x-math-0056 Number of elements across film thickness in Ωf c0x-math-0057 Number of nodes in elastic domain Ω c0x-math-0058 Number of nodes in hydrodynamic domain Ωc ndof Total number of degrees of freedom of FEM model c0x-math-0059 Total number of degrees of freedom of reduced FEM model c0x-math-0060 Number of degrees of freedom of elastic part c0x-math-0061 Number of degrees of freedom of hydrodynamic part c0x-math-0062 Number of degrees of freedom of shear stress part c0x-math-0063 Number of degrees of freedom of thermal part nm, ns Numbers of master and slave dofs in reduced FEM model c0x-math-0064 Normal outward unit vector nx, ny, nz Components of c0x-math-0065 in the x, y, z-directions c0x-math-0066 Normal outward unit vector to cavitation boundary N FEM shape function over actual element c0x-math-0067 FEM shape function over master element Ne, Nh, Nt FEM shape functions for elements of elastic, hydrodynamic and thermal parts NGP Number of Gauss points in quadrature formulas c0x-math-0068 FEM shape function by extrusion of Nh in the z-direction Ñt FEM shape function by integration of Nt in the z-direction Nmax _iter Maximum number of iterations for damped-Newton method Na Nahme–Griffith dimensionless number p Pressure (Pa) ph Hertzian contact pressure (Pa) pR Reference pressure (Pa) c0x-math-0069 Polynomial order of FEM shape functions P Dimensionless pressure c0x-math-0070 Nodal value of P at node i within element e Pi Nodal value of P at global node i c0x-math-0071 FEM polynomial approximation of P within element e Pmin Minimum dimensionless pressure Pee Local Peclet number within element e qx, qy, qz Heat fluxes in the x, y, z-directions (W/m²) Q Total heat generation per unit volume (W/m³) Qcomp Compressive heat generation per unit volume (W/m³) Qshear Shear heat generation per unit volume (W/m³) {q} FEM analysis vector of nodal secondary variables R Circular contact equivalent ball radius (m) Rg Universal gas constant (J/mol·K) R1x, R2x Principal radii of curvature of solids 1 and 2 in the xz-plane (m) R1y, R2y Principal radii of curvature of solids 1 and 2 in the yz-plane (m) Rx Radius of curvature of equivalent elastic solid in the xz-plane (m) Ry Radius of curvature of equivalent elastic solid in the yz-plane (m) c0x-math-0072 Equivalent radius of curvature of reduced contact geometry (m) {R} Nonlinear FEM analysis residual vector S Overall contact surface feature profile (m) c0x-math-0073 Dimensionless overall contact surface feature profile S1, S2 Surface feature profiles of solids 1 and 2 (m) S0, Z0 Roelands viscosity model parameters c0x-math-0074 , c0x-math-0075 Standard and reduced FEM solution spaces for c0x-math-0076 SRRx, SRRy Contact slide-to-roll ratio in the x, y-directions t Time (s) c0x-math-0077 Dimensionless time tc Coating thickness (m) c0x-math-0078 Dimensionless time step s t0, c0x-math-0079 Initial and final dimensionless times T Temperature (K) c0x-math-0080 Dimensionless temperature c0x-math-0081 Nodal value of c0x-math-0082 at global node i c0x-math-0083 Nodal value of c0x-math-0084 at node i within element e c0x-math-0085 FEM polynomial approximation of c0x-math-0086 within element e Tg Glass transition temperature (K) Tg0 Glass transition temperature at zero pressure (K) TR Reference temperature (K) T0 Ambient temperature (K) Ti Thermoviscous indicator dimensionless number [T] Nonlinear FEM analysis Jacobian/tangent matrix u1, v1, w1 Surface velocity components of solid 1 in the x, y, z-directions (m/s) u2, v2, w2 Surface velocity components of solid 2 in the x, y, z-directions (m/s) uf, vf, wf Lubricant velocity field components in the x, y, z-directions (m/s) u, v, w Equivalent solid deformation components in the x, y, z-directions (m) um, vm Contact mean entrainment speeds in the x, y-directions (m/s) us, vs Contact sliding speeds in the x, y-directions (m/s) c0x-math-0087 , c0x-math-0088 Surface velocity vectors of solids 1 and 2 (m/s) c0x-math-0089 Lubricant film velocity vector (m/s) U, V, W Equivalent solid dimensionless deformation components in x, y, z-directions Ui, Vi, Wi Nodal values of U, V, W at global node i c0x-math-0090 , c0x-math-0091 , c0x-math-0092 Nodal values of U, V, W at node i within element e c0x-math-0093 , c0x-math-0094 , c0x-math-0095 FEM polynomial approximations of U, V, W within element e c0x-math-0096 Equivalent solid dimensionless deformation vector c0x-math-0097 Nodal value of c0x-math-0098 at global node i c0x-math-0099 Nodal value of c0x-math-0100 at node i within element e c0x-math-0101 FEM polynomial approximation of c0x-math-0102 within element e Ve Volume of a 3D element e in dimensionless space we, wh, wt FEM trial functions for the elastic, hydrodynamic, and thermal parts Wi Weissenberg dimensionless number Ŵ Reduced dimensionless elastic deformation field over contact domain x, y, z Space coordinates (m) X, Y, Z Dimensionless space coordinates Xcav X-location of cavitation boundary Xinlet, Yinlet X and Y limits of contact domain inlet Xf X-location of surface feature center Xoutlet, Youtlet X and Y limits of contact domain outlet Xs X-location of surface feature center at initial time

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Part I

Introduction

Chapter 1

Elastohydrodynamic Lubrication (EHL)

1.1 EHL Regime

Fluid film lubrication is an essential mechanism for the safe operation of many machine elements/components in relative motion, for example, gears and roller-element bearings. It consists in separating contacting components in relative motion by inserting a high-viscosity fluid, known as a lubricant, between their corresponding surfaces. In a lubricated contact, the lubricant generally serves two distinct purposes. Primarily, it separates the contacting surfaces (partially or fully) and prevents direct solid-to-solid contact between surface asperities. On the one hand, this prevents wear in the corresponding components, providing a longer fatigue life. On the other hand, it leads to reduced friction and energy dissipation. A secondary purpose is that of cooling the lubricated components. In fact, the lubricant separating the contacting surfaces acts as an energy carrier. It enters the contact, extracts much of the heat generated by the relative motion of the surfaces, and carries it away from the contact. This prevents overheating and thermal damage of the contacting solids.

In general, three fluid lubrication regimes are defined, which are distinguishable by their range of friction coefficients on a Stribeck [1] curve, as illustrated in Figure 1.1. These are as follows:

Boundary lubrication: A major part of the contact load is supported by the direct contact of the surface asperities. This regime is characterized by high friction coefficients, governed by the properties of the contacting solids.

Mixed lubrication: The contact load is supported by both the direct contact of the surface asperities and the lubricant film. Friction coefficients for this regime are lower than for boundary lubrication and are governed by the properties of the solids as well as those of the lubricant.

Full film lubrication: Contacting surfaces are fully separated by a lubricant film. Friction coefficients are relatively low and are governed by lubricant properties.

Illustration of Stribeck curve delineating different fluid lubrication regimes.

Figure 1.1 Stribeck curve delineating different fluid lubrication regimes.

Under full film lubrication, two sub-regimes may be distinguished:

Hydrodynamic lubrication (HL): Pressures generated within the lubricating film are relatively low and do not induce any significant elastic deformation of the contacting solids. This is typical of conformal contacts, for which centers of curvature of the two contacting surfaces are located on the same side of the contact, as shown in Figure 1.2a. Such contacts are characterized by large contact areas and therefore low pressures. Journal bearings (see Figure 1.3a) are typical mechanical devices subject to hydrodynamic lubrication.

Elastohydrodynamic lubrication (EHL): Pressures generated within the lubricating film are high enough to induce elastic deformation of the contacting solids. Elastic deformation of the solid surfaces is typically orders of magnitude greater than the lubricant film thickness. This is typical of non-conformal contacts, for which centers of curvature of the two contacting surfaces are located on opposite sides of the contact, as shown in Figure 1.2b. Such contacts are characterized by small contact areas and therefore relatively high pressures. Roller-element bearings (see Figure 1.3b) and gears (see Figure 1.3c) are typical mechanical devices subject to elastohydrodynamic lubrication.

Illustration of Geometries of (a) conformal and (b) non-conformal contacts.

Figure 1.2 Geometries of (a) conformal and (b) non-conformal contacts.

Geometrical illustration of (a) Journal bearing, (b) roller-element bearing, and (c) gears.

Figure 1.3 (a) Journal bearing, (b) roller-element bearing, and (c) gears.

In general, given that the size of the contact (or wet area) in an elastohydrodynamic (EHD) contact is orders of magnitude smaller than the size of the contacting solids, the two solid surfaces may be approximated by elliptic paraboloids. The principal (maximum and minimum) radii of curvature of these surfaces in the vicinity of the contact point, Rx and Ry, are assumed to lie in the xz and yz planes, respectively, with the centers of curvature located along the z-axis, as shown in Figure 1.4. Two types of EHL contacts can be encountered in machine elements:

Line Contact: Contacting elements are infinitely long in one of the principal space directions compared to other directions. In other words, one of the principal radii of curvature of the paraboloids approximating the surfaces, Rx or Ry, is infinitely large. In the unloaded dry contact situation, the surfaces touch along a straight line, and hence the name line contact. If a load is applied, a rectangular shaped contact region is formed due to elastic deformation of the solids. Such contacts are found in gears or cylindrical roller-element bearings, for example.

Point Contact: This is one of the most general types of EHL contacts encountered in a wide variety of applications. Contacting elements have finite principal radii of curvature in both the xz and yz planes. In the dry contact situation, both surfaces nominally touch at one point in the unloaded situation, and hence the name point contact. When a load is applied, the shape of the contact region depends on the ratio of the radii of curvature, Rx/Ry, of the two contacting solids. In general, if the ratio is different from unity for any of the solids, the shape of the contact region is an ellipse, and therefore this type of contact is also referred to as an elliptical contact. A special case of an elliptical contact is the circular contact, which corresponds to the case where the ratio is equal to unity for both contacting solids; that is, c01-math-0001 and c01-math-0002 . Point contacts are found between the ball and the inner or outer raceway of a roller-element ball bearing, for example.

Geometrical illustration of contacting solids in a general EHL point contact.

Figure 1.4 Geometry of the contacting solids in a general EHL point contact.

Remark

In the above definition of a general EHL point contact (adopted throughout this book), it was assumed that the principal radii of curvature of the two contacting paraboloids lie within orthogonal planes. This is true for most EHL applications. A more general case may arise though, as discussed in [2], if these planes are not orthogonal, such as in Novikov gears. The footprint, under dry contact situation, remains elliptical in this case. However, the orientation of the principal axes of the ellipse with respect to the planes of principal radii of curvature can no longer be determined analytically, and a numerical solution is required [3].

Historically speaking, the EHL regime was discovered toward the mid-twentieth century, after the failure of HL theory to predict the safe operation of many moving machine elements with non-conformal contacts (e.g., roller-element bearings and gears). At that time, the existing HL theory could not explain the absence of wear in such contacts where the theoretical lubricant film thickness predictions fell well below the combined surface roughness of the contacting surfaces. Two key features led to the discovery of EHL: first, the realization that contact surfaces are not rigid and do actually deform under the effect of the high pressures generated in such contacts; and second, a proper understanding of the dependence of lubricant transport properties on pressure.

In fact, the first steps toward establishing a fundamental understanding of lubrication date back to the nineteenth century with the work of Hirn [4] in 1854. Then, in 1883, two experimental investigations lead by Beauchamp Tower [5] in England and Nicoli Petrov [6] in Russia made it clear that the rigid surfaces of the contacting bodies in a hydrodynamic journal bearing were fully separated by a fluid film. Thus, it was demonstrated that the friction forces in such contacts are governed by hydrodynamics rather than by the direct contact between the solids, and the fundamentals of HL were established. In 1886, Osbourne Reynolds [7] established the Reynolds equation, which is the basis of all actual lubrication theories. It expresses the relationship between the pressure in the lubricant film, the geometry of the conjunction, and the kinematics of the moving parts. The solution of this equation confirmed the observations made by Tower and Petrov. At the beginning of the twentieth century, Michell [8] and Kingsburry [9] took the first step toward understanding the phenomenon of lubrication in hydrodynamic journal bearings. A few years later, Martin [10] and Gümbel [11] applied hydrodynamic theory to the case of rigid gears. Surprisingly, the results they obtained predicted very small film thicknesses compared to the surface roughness. Nevertheless, the contact was well protected by a full lubricant film that separated the surfaces. It took two additional decades for the fundamentals of EHL to appear with the works of Ertel [12] and Grubin [13]. Introducing Hertz [14] theory for the deformation of semi-infinite elastic bodies under dry contact conditions along with the Barus [15] law for lubricant viscosity-pressure dependence, they calculated larger film thicknesses compared to those obtained by Martin and Gümbel for the same operating conditions. Thus, the fundamental features of EHL were revealed. The second part of the twentieth century witnessed an increasing interest on the part of the scientific community in EHL problems. At the same time, the development of experimental technology based on optical interferometry techniques along with the progress in numerical resolution of partial differential equations due to more powerful computers and better-performing algorithms allowed a more profound understanding of EHL contacts.

The topic of interest in this book is EHL contacts and their modeling using finite element techniques. Within EHD conjunctions, pressures may be as high as several gigapascals, lubricant film thicknesses as low as a few nanometers, and shear stresses within the lubricant film may reach hundreds of megapascals. Under such severe conditions, the Newtonian limit of most lubricants is exceeded, and significant heat generation by shear may occur, leading to a temperature rise that may exceed 100°C in extreme cases. As such, it becomes inevitable for any numerical model that simulates EHL contacts to account for the non-Newtonian response of the lubricant as well as the generation of heat within the lubricant film and its dissipation through the film and bounding solids. This is essential for an accurate prediction of lubricant film thickness and friction generation within these contacts, as will be thoroughly discussed in later chapters. Clearly, EHL contacts are multi-physical in nature, involving a strong coupling between several physics: hydrodynamics, which governs lubricant flow within the conjunction; linear elasticity, which governs the elastic deformation of the contacting solids; heat transfer, which governs heat flow within the lubricant layer and surrounding solids; and rheology, which describes the constitutive behavior of lubricants under the extreme conditions that are encountered in these contacts. This strong coupling makes modeling of EHL contacts rather complex and challenging, often requiring the development of sophisticated special techniques and tools.

1.2 Governing Equations in Dimensional Form

In this section, the different equations governing EHL are derived for the most general case of a transient point contact operating under a thermal non-Newtonian regime. That is, the operating conditions (load, speed, geometry, etc.) may vary in time, and both heat generation within the lubricant film as well as lubricant viscosity dependence on shear are considered. The flow of lubricant within the EHL conjunction is governed by the generalized Reynolds equation, while the geometry of the contact is defined by the film thickness equation. The deformation of the contacting solids is governed by the linear elasticity equations. The load balance equation describes the equilibrium of forces over the lubricated contact. Finally, the energy equation describes the generation of heat within the lubricant film and its dissipation through the film and bounding solids. In the derivation of the different equations, the usual thin-film simplifying assumptions are adopted:

Body forces are negligible.

Pressure is constant through the lubricant film thickness.

There is no slip at the fluid–solid interfaces.

Lubricant flow is laminar (low Reynolds number).

Inertia and surface tension forces are negligible compared to viscous forces.

Lubricant film thickness is small compared to the dimensions of the contact.

Lubricant film is continuous and fully separates the solid surfaces.

Elastic deformations of the solid components are small compared to their actual size.

The contact size is very small compared to the size of the contacting solids.

Some of these assumptions may be relaxed under certain configurations, leading to more generalized forms of the different equations described in this section. However, for most common applications involving EHL contacts, the above assumptions are completely justified.

In order to derive the EHL equations, the general case of a point contact represented by two paraboloid surfaces that are fully separated by a lubricant film is considered, as shown in Figure 1.5. The lubricant film thickness is denoted as h, and the contacting solids are pressed against each other by an external applied force F. The lower and upper surfaces are denoted by the subscripts 1 and 2, respectively. Given that the elastic deformations of the solid surfaces are relatively small compared to the actual size of the corresponding rotating components, the surface velocity vectors c01-math-0003 and c01-math-0004 are assumed to be constant in space in the vicinity of the contact (these can vary in time), where u, v, and w correspond to the x, y, and z components of the surface velocity vector c01-math-0005 , respectively. Let the contact mean entrainment speeds um and vm and sliding speeds us and vs in the x- and y- directions, respectively, be defined as follows:

1.1

equation

A Cartesian coordinate system xyz is adopted with its origin O located at the center of the contact, on the lower non-deformed surface. The center of the contact actually corresponds to the center of the ellipse-shaped contact area that would arise in a dry Hertzian contact configuration.

Geometrical illustration of Kinematics of a general EHL point contact.

Figure 1.5 Kinematics of a general EHL point contact.

1.2.1 Generalized Reynolds Equation

The original Reynolds [7] equation was derived to describe pressure variations within lubricated contacts assuming constant lubricant viscosity and density across the film thickness (z-direction). Therefore, the lubricant was assumed to have a Newtonian behavior, whereby its viscosity is not shear dependent, because shear stresses vary in the thickness direction. Also, isothermal operation was assumed, as temperature variations across the film thickness would result in both viscosity and density variations. A more general form that relaxes the assumption of constant viscosity and density in the film thickness direction was derived by Yang and Wen [16]. The generalized Reynolds equation allows for the incorporation of generalized Newtonian lubricant behavior as well as thermal effects. Like the original Reynolds equation, it describes the pressure distribution p within the lubricant film as a function of the geometry of the gap between the contacting surfaces, contact kinematics, and lubricant properties. The starting point for the derivation of the generalized Reynolds equation consists in isolating an infinitesimal volume dV of fluid within the lubricant film of side dimensions dx, dy, and dz in the x-, y-, and z-directions, respectively, as shown in Figure 1.5, at a location (x, y) where the film thickness is c01-math-0007 and the lubricant velocity field vector is c01-math-0008 . The forces acting on this volume of fluid are shown in Figure 1.6, neglecting inertia and body forces, as stated in the simplifying assumptions.

Illustration of Free body diagrams in the (a) x-direction and (b) y-direction for an infinitesimal volume of fluid within the lubricant film.

Figure 1.6 Free body diagrams in the (a) x-direction and (b) y-direction for an infinitesimal volume of fluid within the lubricant film.

The usual notation for shear stresses is employed here. That is, τzx corresponds to the shear stress in the x-direction in a plane having z as normal. Because the lubricant film thickness (in the z-direction) is small compared to the dimensions of the contact in the x- and y-directions, velocity gradients are only significant in the z-direction. As a consequence, shear stresses are only significant within planes having z as normal; that is, τxy, τxz, τyx, and τyz are negligible. Given that an infinitesimal volume of fluid is in equilibrium, and that inertia and body forces are negligible, the equilibrium of forces on this volume in the x-direction gives

equation

Dividing all terms by dx dy dz, one obtains

equation

Taking the limit when dx and dz tend toward zero, one obtains

1.2 equation

Similarly, the equilibrium of forces in the y-direction gives

equation

Dividing all terms by dx dy dz yields

equation

Taking the limit when dy and dz tend toward zero:

1.3 equation

Applying an equilibrium of forces in the z-direction would be of little importance as pressure is assumed constant in that direction, and shear stresses are negligible. The resulting equation would simply read

1.4 equation

which is the direct consequence of p being constant in the z-direction. Equations (1.2), (1.3), and (1.4) correspond to the Navier–Stokes [17] equations under thin-film simplifying assumptions. Let η(p, T, τ) be the generalized Newtonian lubricant viscosity which, in the most general case considered here, is a function of pressure p, temperature T, and the resultant shear stress c01-math-0012 . By the definition of the generalized Newtonian viscosity, the shear stresses τzx and τzy are expressed as a function of η and the shear rates c01-math-0013 and c01-math-0014 as follows:

1.5

equation

Substituting the above expressions for τzx and τzy into Equations (1.2) and (1.3):

1.6 equation

Integrating the above equations with respect to z and taking into consideration the fact that c01-math-0017 and c01-math-0018 do not vary in the z-direction (because p itself was assumed constant in this direction):

1.7 equation

Moving the generalized viscosity term to the right-hand-side and integrating with respect to z gives the variations of uf and vf with respect to z at (x, y), as follows:

1.8

equation

where z′ is a dummy integration variable representing z, and c1, c2, c3, and c4 are integration constants that are determined by applying the velocity field boundary conditions. Given that a no-slip condition was assumed at the solid–fluid interfaces, these boundary conditions are

equation

Applying the conditions at c01-math-0021 to Equations (1.8) gives

equation

Similarly, the conditions at c01-math-0022 give

equation

Let

1.9

equation

Then, given that by definition c01-math-0024 and c01-math-0025 , the expressions for c1 and c2 simplify to

equation

Replacing c1, c2, c3, and c4 in Equation (1.8) by their above expressions, the lubricant velocity field components uf and vf become

1.10

equation

Similarly, substituting c1 and c2 into Equations (1.7) gives the following expressions for the lubricant shear rate components c01-math-0027 and c01-math-0028 :

1.11

equation

Note that the lubricant flow within an EHL contact is the combination of a Poiseuille flow and a Couette flow, as shown in Figure 1.7. A Poiseuille flow is one that is driven by a pressure difference and usually entails a parabolic velocity profile, with the maximum or minimum velocity located within the middle layer of the flow. A Couette flow is one that is driven by the surface velocity of the bounding solids and usually entails a linear velocity profile. The fluid velocity at each solid–fluid interface corresponds to the surface velocity of the solid at that interface, assuming a no-slip condition. Within an EHL contact, owing to the curvature of the bounding solid surfaces and variations in lubricant properties as it moves through the contact, a hydrodynamic pressure builds up within the film. Pressure variations lead to a pressure-induced flow. Simultaneously, owing to the relative motion of the bounding surfaces, a surface-velocity-induced flow arises. The combination of the two flows results in a flow profile that is the superposition of a parabolic and a linear velocity profile. An example of such a profile is shown in Figure 1.7. Obviously, depending on the sign of the pressure gradient and that of the surface velocity difference between the two solids, the resulting velocity profile might have a relatively different shape, but in all cases it corresponds to the superposition of a parabolic and a linear profile. The middle terms on the right-hand-side of Equations (1.10) correspond to the Poiseuille components of the velocity field, whereas the remaining terms correspond to the Couette component. Similarly, the left terms on the right-hand-side of Equations (1.11) correspond to the Poiseuille components of the corresponding shear rates, whereas the right terms correspond to the Couette component.

Geometrical illustration of Velocity field within an EHL contact.

Figure 1.7 Velocity field within an EHL contact.

The next step in deriving the generalized Reynolds equation consists in applying the conservation of mass principle to the infinitesimal volume of fluid in order to derive the continuity equation. The conservation of mass principle states that the net mass flow rate c01-math-0030 entering a volume, that is, the difference between the incoming and outgoing mass flow rates, should equal the net rate of change of mass m within the volume with time. Or, in mathematical terms:

1.12 equation

But the mass flow rate of a fluid with a density c01-math-0032 flowing through a surface with an area A at a velocity V is simply expressed as c01-math-0033 . Thus, the mass flow rates into and out of the infinitesimal fluid volume, through its different faces, are as shown in Figure 1.8.

Geometrical illustration of Mass flow rates into and out of an infinitesimal volume of fluid.

Figure 1.8 Mass flow rates into and out of an infinitesimal volume of fluid.

Given that the mass of this infinitesimal volume c01-math-0034 , Equation (1.12) becomes

equation

Dividing all terms by dx dy dz, one obtains

equation

Taking the limit when dx, dy, and dz tend toward zero yields the continuity equation for a compressible flow:

1.13

equation

Integrating Equation (1.13) with respect to z between the limits c01-math-0036 and c01-math-0037 gives

1.14

equation

because at c01-math-0039 , c01-math-0040 and at c01-math-0041 , c01-math-0042 , on the basis of the no-slip boundary condition assumed at the fluid–solid interfaces. At this point, it is essential to recall the Leibniz integral rule, which states that for any continuous differentiable and integrable function f(x, y, z):

1.15

equation

where z1, z2 are also continuous and differentiable. Applying this rule to the integral terms of Equation (1.14) yields

1.16

equation

Keeping in mind that c01-math-0045 and c01-math-0046 at (x, y, z1) and similarly c01-math-0047 and c01-math-0048 at (x, y, z2), then:

equation

Substituting the above expressions for w1 and w2 into Equation (1.16) simplifies the equation to

1.17

equation

Replacing uf and vf by their expressions derived in equation (1.10) yields

1.18

equationequation

Given that u1, u2, v1, v2, c01-math-0051 , c01-math-0052 , ηe and c01-math-0053 are independent of z, these can be moved out of their corresponding integrals with respect to z, and Equation (1.18) becomes

1.19

equationequation

Equation (1.19) can be written in a more compact form as follows:

1.20

equation

Equation (1.20) is the generalized Reynolds equation for a thermal non-Newtonian point contact. It governs the pressure distribution within the EHL conjunction for a given film geometry, operating conditions, and lubricant properties. The integral terms in this equation allow the incorporation of viscosity variations across the film thickness with both shear stress and temperature as well as density variations with temperature. The left-hand-side terms are the Poiseuille components, whereas the right-hand-side terms are the Couette components. The latter are split into wedge (the space-dependent terms) and squeeze (the time-dependent term) components. Note that, even though the starting point for deriving the generalized Reynolds equation is the three-dimensional simplified Navier–Stokes equations, the resulting equation is two-dimensional in the xy-plane. This is a consequence of the thin-film simplifying assumptions according to which the pressure across the lubricant film thickness is assumed constant. Thus, the dimension of the problem has been reduced by one, and the solution of Equation (1.20) gives rise to a two-dimensional pressure distribution in the xy-plane.

Note that the solution of the generalized Reynolds equation requires knowledge of the geometry of the lubricant-filled gap, the lubricant viscosity dependence on pressure, temperature, and shear stress – c01-math-0056 – as well as its density dependence on pressure and temperature – c01-math-0057 . The rheological models that are most commonly employed in the EHL literature to describe these dependencies will be detailed in Section 1.4. It is important to mention that these dependencies make the generalized Reynolds equation highly nonlinear and particularly difficult to solve, as will be explained in later chapters. Finally, note that boundary and initial conditions also need to be specified in order to complete the generalized Reynolds equation. These will be detailed in later chapters, according to the configuration at hand.

1.2.2 Film Thickness Equation

The solution of the generalized Reynolds equation requires knowledge of the geometry of the lubricant-filled gap separating the two solids in an EHL contact at any instant t in time. The geometry is described by the film thickness equation. Figure 1.9 shows an xz-view of the gap geometry. Consider a point (x, y) of the contact where the lubricant film thickness is h(x, y, t). It is composed of the non-deformed shape of the gap hu and the elastic deformation of the solid components δ1 and δ2:

1.21 equation

The non-deformed shape of the film geometry is composed of the rigid body separation term h0, which may vary in time, and the original non-deformed shapes of the solids h1 and h2:

1.22

equation

The rigid body separation term h0 corresponds to the distance between the original non-deformed surfaces at the contact center. It is often negative (as discussed in later chapters), indicating inter-penetration between the non-deformed solids. This is why rigid body HL analysis failed to properly predict the safe operation of non-conformal contacts, leading to the discovery of the EHL regime, as discussed earlier. In fact, in most cases where the contact load F is relatively high, without the elastic deformation of the contacting solids, their surfaces would be in direct contact. For such cases, the lubricant film thickness is smaller than the overall elastic deformation of the solids. It is actually often orders of magnitude smaller, as will be discussed in later chapters.

Illustration of EHL film geometry (xz-view).

Figure 1.9 EHL film geometry (xz-view).

Recall that the contacting surfaces are approximated by elliptic paraboloids whose principal radii of curvature lie in the xz and yz planes. Thus, at any instant t in time, the terms h1 and h2 can be expressed as follows:

equation

where a1, b1, a2, and b2 are positive constants to be determined. Also recall that for any given surface f(x, y), the local radii of curvature Rx and Ry in the x and y-directions respectively, at a given point (x0, y0) of the surface are expressed as [18]

equation

Applying the above relationships to R1x, R1y, R2x, and R2y, the local principal radii of curvature of the contacting solids at the center of the contact ( c01-math-0060 ), one obtains:

equation

Thus, at any instant t in time, the constants a1, b1, a2, and b2 are expressed as a function of the local principal radii of curvature of the contacting solids as follows:

equation

Substituting the expression for hu defined in equation (1.22) in the film thickness equation, Equation (1.21), it becomes

1.23

equation

Let Rx and Ry be the equivalent or reduced radii of curvature of the contacting solids at the center of the contact in the x- and y-directions, respectively, defined as follows at any instant t in time:

1.24

equation

and let δ be the overall elastic deformation of the two solids ( c01-math-0063 ), then Equation (1.23) reduces to

1.25

equation

Note that, with the above definition of the reduced/equivalent radii of curvature Rx and Ry, the geometry of the contact may be reduced to that of an equivalent contact between an elastic elliptic paraboloid with principal radii of curvature Rx and Ry