8. Elastohydrodynamic, Mixed and Boundary Lubrication

Introduction and context

Lubricated machine elements rarely operate in a single, perfectly defined lubrication regime. A rolling bearing, gear tooth, cam–follower pair, or seal may pass repeatedly through full-film elastohydrodynamic lubrication (EHL), mixed lubrication, and boundary lubrication during one operating cycle. The transitions are governed by speed, load, temperature, lubricant properties, surface roughness, conformity, material elasticity, and the chemical condition of the contacting surfaces.

The central feature of EHL is that the lubricant is not merely a fluid separating two rigid surfaces. At the high pressures generated in concentrated contacts, its viscosity increases dramatically, while the contacting solids deform elastically. These two effects cooperate to generate a load-carrying film. In mixed and boundary lubrication, however, the film becomes comparable with or smaller than the combined surface roughness, and some load is transferred through asperities. Friction and wear then depend increasingly on surface chemistry, additive-derived films, and the mechanical response of the near-surface material.

Understanding these regimes is essential for predicting fatigue life, frictional losses, scuffing, wear, efficiency, and reliability. It also provides a basis for selecting lubricants, specifying surface finish, designing lubrication systems, and interpreting failures.

Learning objectives

After studying this chapter, the reader should be able to:

1. Explain pressure–viscosity coupling and elastic deformation in concentrated contacts.

2. Distinguish isothermal, thermal, fully flooded, and starved EHL conditions.

3. Estimate and interpret central and minimum film thickness using established dimensionless relationships.

4. Explain traction, rolling–sliding effects, inlet heating, and lubricant starvation.

5. Use Stribeck behaviour and the lambda ratio to identify lubrication regimes.

6. Describe asperity-contact models and the transition from mixed to boundary lubrication.

7. Explain the chemical and mechanical functions of boundary and tribofilm layers.

8. Apply these concepts to engineering design, diagnosis, and lubricant selection.

From hydrodynamic films to elastohydrodynamic films

In classical hydrodynamic lubrication, two surfaces are separated by a converging wedge of lubricant. The Reynolds equation describes the pressure generated by viscous flow. This model works well for conformal bearings, where the contact area is relatively large and pressures are moderate. In gears and rolling bearings, by contrast, the nominal contact is small and highly loaded. The undeformed geometry would produce excessive pressure and an extremely thin film.

Two additional effects make separation possible. First, the lubricant’s viscosity rises with pressure. A common engineering representation is the Barus relation,

μ(p) = μ₀ exp(αp),

where μ₀ is the ambient-pressure viscosity and α is the pressure–viscosity coefficient. Real lubricants may depart from this exponential relationship at very high pressure, so more advanced expressions, such as the Roelands or modified pressure–viscosity equations, are often used in numerical analysis. Nevertheless, the Barus form clearly illustrates the feedback: pressure increases viscosity, increased viscosity restricts flow, and restricted flow permits a larger pressure gradient to support the load.

Second, the contacting solids deform elastically. The pressure distribution flattens the surfaces and enlarges the contact region. Hertzian contact theory provides the first approximation for the pressure and contact dimensions before lubricant-film effects are included. For two curved bodies, the reduced elastic modulus is

1/E′ = (1 − ν₁²)/E₁ + (1 − ν₂²)/E₂,

where E is Young’s modulus and ν is Poisson’s ratio. A lower reduced modulus generally produces greater elastic deformation and a larger contact area. EHL therefore results from a coupled fluid–solid problem: pressure determines viscosity and deformation, while viscosity and geometry determine the pressure field.

Contact geometry and film formation

The reduced radius of curvature combines the principal curvatures of the two bodies. In a line contact, as in an idealized gear or roller, the contact is long in one direction and narrow in the other. In a point or elliptical contact, as in a ball bearing, curvature exists in two principal directions. The film is usually thickest near the inlet and becomes thinner toward the central contact region. The minimum film thickness often occurs just downstream of the point of maximum Hertzian pressure because of pressure-induced viscosity and elastic displacement.

The film thickness is controlled by several dimensionless groups. A widely used set is:

U = η₀u/(E′R), the speed or entrainment parameter;

W = w/(E′R²), the load parameter for a point contact;

G = αE′, the material or pressure–viscosity parameter.

Here η₀ is lubricant viscosity, u is mean entrainment speed, w is load, and R is an effective radius. For elliptical contacts, an ellipticity parameter is also required. In general, film thickness increases with entrainment speed, viscosity, and pressure–viscosity response, and decreases with load and curvature severity.

For engineering estimates, central film thickness is often expressed in the form

h_c/R = C U^m G^n W^p,

with exponents and constant C selected for the contact geometry and correlation used. For a representative point-contact correlation, the exponents on U and G are positive, whereas the exponent on W is small and negative. Minimum film thickness has a similar structure but is more sensitive to load, ellipticity, and the detailed pressure distribution. Such equations are valuable for preliminary design, but they should not be treated as universal laws. Surface roughness, thermal effects, starvation, lubricant non-Newtonian behaviour, and transient operation can make the actual film substantially different from an isothermal, fully flooded prediction.

Thermal effects, traction, and rolling–sliding contact

At sufficiently high speed, viscous shearing heats the lubricant. The resulting temperature rise lowers viscosity and can reduce film thickness. This is the thermal EHL effect. The importance of thermal reduction depends on the slide-to-roll ratio, contact size, lubricant viscosity, speed, and heat dissipation through the solids. A calculation based only on ambient viscosity may therefore overestimate film thickness.

The slide-to-roll ratio is commonly written as

SRR = 2(u₁ − u₂)/(u₁ + u₂),

where u₁ and u₂ are the surface velocities. Pure rolling has zero SRR, whereas gear contacts normally combine rolling and sliding. Sliding generates tangential stress and frictional heat. The ratio of traction force to normal load is called the traction coefficient. In EHL contacts, traction is influenced by lubricant viscosity, pressure, shear-thinning, molecular structure, temperature, and the limiting shear strength of the confined lubricant.

A lubricant does not necessarily behave as a simple Newtonian liquid under EHL pressure. Some traction fluids exhibit strong pressure-induced solid-like behaviour, while many modern lubricants shear-thin at high shear rates. Lower traction can improve gear efficiency, but excessive reduction in viscosity or pressure response may compromise film thickness. Design is therefore a compromise between fatigue protection and frictional power loss.

Fully flooded and starved EHL

Classical film-thickness formulas generally assume that the inlet is fully flooded: an adequate supply of lubricant enters the contact and replenishes the inlet meniscus. In practice, the inlet may receive insufficient lubricant because of low flow rate, high speed, seal drag, lubricant churning, cage dynamics, orientation, or poor oil distribution. This condition is called starvation.

Under starvation, the inlet meniscus moves closer to the contact, reducing the available lubricant reservoir. Central film thickness can fall substantially even when speed, load, and lubricant viscosity remain unchanged. Starvation is especially important in high-speed rolling bearings, grease-lubricated contacts, and systems where lubricant delivery is intermittent. Starvation should not be confused with lubricant starvation caused by complete oil depletion; a contact can be partially starved while still appearing wet.

Engineering responses include improving oil jets or nozzles, controlling grease quantity, using suitable separators, reducing churning, improving lubricant distribution, and allowing adequate relubrication. Monitoring temperature and torque can help identify inadequate supply, but such measurements must be interpreted alongside speed and load because both also influence these signals.

Stribeck behaviour and the regime map

The Stribeck curve relates friction coefficient to a dimensionless lubrication parameter, often approximated by ηu/P, where η is viscosity, u is characteristic speed, and P is unit load. At low values, boundary lubrication dominates and friction is relatively high. As the parameter increases, a mixed regime develops: a fluid film carries an increasing share of the load, asperity interactions decline, and friction generally falls. At still higher values, full-film lubrication is established. Friction then consists mainly of viscous shear and may rise again with increasing viscosity or speed.

The curve is a useful conceptual framework, not a universal numerical plot. The positions of the transitions depend on roughness, surface chemistry, geometry, lubricant additives, temperature, and transient history. A modern interpretation distinguishes at least three load-carrying contributions: hydrodynamic or EHL pressure in the fluid film, direct asperity contact, and solid or tribofilm interactions at the surfaces.

A practical indicator is the lambda ratio,

λ = h_min/σ,

where h_min is the minimum film thickness and σ is the composite root-mean-square roughness. For statistically independent surfaces,

σ = (σ₁² + σ₂²)^(1/2).

Approximate interpretation is as follows:

• λ greater than about 3: predominantly full-film separation;
• λ between roughly 1 and 3: mixed lubrication is likely;
• λ below about 1: significant asperity interaction is likely.

These limits are guidelines rather than strict boundaries. Two surfaces with the same lambda ratio may show different friction and wear because roughness texture, hardness, skewness, wavelength, coating, and additive chemistry differ. A high lambda ratio also does not guarantee safety against micropitting or fatigue if stress concentrations and traction are excessive.

Mixed lubrication: sharing the load

Mixed lubrication is not simply a failed form of hydrodynamic lubrication. It is a coupled state in which part of the load is carried by the lubricant film and part by contacting asperities. The total load may be represented conceptually as

W = W_f + W_a,

where W_f is the fluid-film contribution and W_a is the asperity contribution. Friction similarly contains fluid shear, asperity adhesion and deformation, ploughing, and boundary-film contributions.

Asperity models commonly assume a statistical distribution of summit heights. The Greenwood–Williamson model represents asperities as spherical summits with a distribution of heights and predicts the number and load of contacting asperities as a function of separation. More advanced models incorporate measured three-dimensional topography, non-Gaussian height distributions, plasticity, and interaction between neighbouring asperities. The model is useful because it links film thickness and roughness to real contact area, but it depends strongly on how roughness is measured and on the validity of the assumed summit geometry.

In a mixed regime, increasing viscosity or speed may increase film thickness and reduce asperity load. Increasing load, temperature, roughness, or misalignment generally has the opposite effect. Surface finishing can be more effective than simply increasing viscosity when the contact is already near full film, whereas chemical protection becomes increasingly important as asperity interaction grows.

Boundary lubrication and tribofilm chemistry

Boundary lubrication occurs when the fluid film is too thin to prevent frequent interaction of the highest asperities. The surfaces are then protected by adsorbed molecules, reaction products, oxides, and mechanically formed transfer layers. The boundary layer may be only a few nanometres thick, but it can determine whether the contact survives or rapidly scuffs.

Boundary-film formation involves several mechanisms. Polar molecules may adsorb physically or chemically onto a surface. Anti-wear additives can react under pressure and temperature to form a sacrificial film. Extreme-pressure additives may produce reaction layers during severe asperity heating and stress. Friction modifiers, often containing polar functional groups, can reduce shear strength at the interface. Detergents and dispersants may influence deposit formation and surface cleanliness, while antioxidant chemistry controls lubricant degradation and indirectly affects film persistence.

Zinc dialkyldithiophosphate, or ZDDP, is a familiar anti-wear additive in many lubricants. It forms a phosphate-rich tribofilm under suitable temperature, load, and surface conditions. Its effectiveness is not universal: concentration, base oil, temperature, competing additives, surface material, and operating history all matter. Modern additive systems increasingly seek lower ash, lower toxicity, compatibility with catalysts and electrical components, and reduced interference with friction modifiers.

Boundary films are sacrificial and condition-dependent. They may be removed by sliding, replenished by chemical reaction, or altered by water, oxidation, contaminants, and surface potential. Thus a lubricant’s performance cannot be inferred solely from its viscosity grade. Base-oil properties, additive compatibility, replenishment rate, and material chemistry must be considered together.

Engineering application and design decisions

In a rolling bearing, the designer estimates film thickness under representative speed, load, temperature, and lubricant supply. The lambda ratio is then compared with the required reliability level. If the predicted ratio is low, possible remedies include increasing viscosity within acceptable churning limits, improving surface finish, reducing contamination, lowering operating temperature, changing material or coating, or improving lubricant delivery.

For gears, the calculation must account for varying rolling and sliding velocities, contact stress, transient mesh conditions, and possible micropitting or scuffing. A lubricant that provides adequate film at the pitch point may be inadequate near the tip or root, where sliding and temperature are higher. Gear oils therefore require a balance of viscosity, anti-scuffing chemistry, oxidation resistance, and low-foam behaviour.

In piston rings and cylinder liners, the contact moves repeatedly between mixed and boundary regimes near top dead centre, where speed approaches zero. Oil-control design, surface texturing, coating selection, and low-friction additive chemistry become important. In seals, a deliberately thin film may be needed to control leakage; excessive film thickness can increase leakage, while insufficient film promotes wear and heat.

Digital methods now support these decisions. Numerical EHL solvers couple Reynolds flow, elastic deformation, energy equations, and sometimes non-Newtonian constitutive laws. Roughness measurements can be incorporated through deterministic or statistical contact models. Digital twins may combine speed, load, temperature, and vibration data to estimate changing lubrication state. Machine-learning models can assist fault detection, but they require physically meaningful training data and should not replace verification through tribological analysis.

Illustrative case study: premature micropitting in a wind-turbine gearbox

Background

A wind-turbine gearbox uses case-carburized steel gears and a high-quality industrial gear lubricant. The gearbox was designed for a predominantly full-film condition under rated operation.

Situation and problem

Inspection after a relatively short service period revealed grey staining and micropitting near the pitch lines of several high-speed-stage gears. Oil analysis showed no unusual concentration of large abrasive particles. Operators also reported elevated gearbox temperature during cold starts and during periods of low wind speed with frequent speed variation.

Analysis

A review of operating data showed that the contact frequently experienced low entrainment speed while load remained substantial. The estimated minimum film thickness during these transients was much lower than the rated-condition value. Surface roughness had increased modestly during running-in, and the lambda ratio periodically entered the mixed regime. The gear contact also had significant sliding, so local temperature and traction stress were higher than predicted by an isothermal calculation.

The lubricant supply system was functioning, but low-temperature viscosity caused delayed circulation during start-up. Consequently, the problem was not simply a lack of lubricant quantity. It was a combination of transient starvation, thermal viscosity loss during loaded operation, and repeated mixed-lubrication cycles. Micropitting was promoted by repeated asperity and near-surface stress interactions rather than by one isolated overload.

Approach and solution

The engineering team adopted a combined strategy: start-up procedures were modified to limit load until oil temperature reached an acceptable level; oil distribution near the high-speed mesh was improved; gear surface finishing was reviewed; and an alternative lubricant was evaluated for pressure–viscosity response, traction, additive compatibility, and low-temperature flow. Film-thickness calculations were repeated over the complete duty cycle rather than only at rated speed.

Outcome

The revised assessment predicted fewer periods with lambda below the mixed-lubrication threshold. Subsequent inspections showed a reduction in the growth of micropitted areas. The case illustrates that nominal viscosity and rated-load film thickness are inadequate indicators when a machine experiences frequent transients.

Learning points

The important lesson is diagnostic integration. Film calculations, thermal behaviour, surface topography, lubricant supply, and duty-cycle data must be considered together. A lubricant change alone may not solve a problem caused by starvation or operating procedure, and a surface finish improvement may be ineffective if the lubricant chemistry is incompatible with the materials.

Limitations, risks, and common errors

Several errors recur in EHL and mixed-lubrication assessment. Using ambient viscosity instead of operating-temperature viscosity can substantially overpredict film thickness. Applying a fully flooded equation to a grease-lubricated or poorly supplied contact ignores starvation. Treating the lambda ratio as an exact regime boundary overlooks surface texture and tribofilm effects. Assuming that a harder surface eliminates wear is also incorrect: hard surfaces may still suffer fatigue, micropitting, or chemical attack.

Another error is optimizing friction alone. A low-friction additive or low-viscosity lubricant may reduce power loss while lowering film thickness or increasing fatigue risk. Conversely, a very viscous lubricant may improve separation but cause churning losses and excessive temperature. Compatibility must also be checked when changing lubricants, particularly where seals, coatings, additive packages, or electrical components are involved.

Environmental and professional considerations are increasingly important. Lubricant leakage can contaminate soil and water, while certain additive elements can complicate recycling or emissions-control systems. Selection should consider service life, safe handling, biodegradability where appropriate, and disposal. Tribological decisions should be documented with assumptions, operating envelopes, uncertainty, and validation evidence rather than presented as precise predictions without qualification.

Current developments and best practice

Current research is directed toward low-viscosity lubricants with strong pressure response, ionic and bio-derived lubricants, nanostructured additives, diamond-like carbon and other hard coatings, engineered surface textures, and real-time lubrication monitoring. Electrification introduces additional concerns: stray currents, electrical discharge damage, lubricant ageing under electrical fields, and compatibility with polymeric or ceramic components.

A robust engineering workflow is to:

1. Define the full speed–load–temperature duty cycle.

2. Determine material properties, geometry, roughness, and surface treatments.

3. Estimate film thickness using an appropriate EHL model.

4. Apply thermal, starvation, and transient corrections where necessary.

5. Calculate the lambda ratio using measured composite roughness.

6. Assess traction, asperity stress, wear, fatigue, and additive-film requirements.

7. Validate predictions through test-rig measurements, oil analysis, temperature data, and inspection.

8. Review the result after changes in lubricant, surface finish, or operating practice.

The most reliable design is not the one that maximizes a single parameter. It is the one that maintains acceptable film separation, traction, temperature, chemical protection, and supply throughout the actual duty cycle.

Chapter Summary

EHL contacts develop load-carrying films through the combined effects of pressure-induced viscosity increase and elastic deformation of the contacting solids. Film thickness depends on speed, viscosity, pressure–viscosity response, load, geometry, temperature, and lubricant supply. Thermal effects, rolling–sliding motion, traction, and starvation can make real contacts differ substantially from ideal fully flooded predictions.

The Stribeck framework describes the progression from boundary through mixed to full-film lubrication. The lambda ratio, defined as minimum film thickness divided by composite roughness, provides a useful but approximate indicator of asperity interaction. In mixed lubrication, fluid-film and asperity loads are shared; in boundary lubrication, adsorbed and reaction-derived tribofilms become essential. Asperity models help estimate real contact, while additive chemistry controls surface protection under severe conditions.

Professional design requires analysis of the entire duty cycle, not merely nominal speed and load. Surface finish, lubricant chemistry, thermal behaviour, starvation, coatings, contamination, and environmental considerations must be integrated and validated by measurement.

Key Terms

Pressure–viscosity coefficient: A parameter describing the increase of lubricant viscosity with pressure.

Elastohydrodynamic lubrication: Lubrication in which fluid pressure changes viscosity and elastically deforms the contacting solids.

Reduced elastic modulus: Combined elastic property of two contacting materials.

Hertzian contact: Elastic contact solution used to estimate pressure and deformation in concentrated contacts.

Entrainment speed: Mean surface velocity that carries lubricant into a contact.

Minimum film thickness: Smallest lubricant-film thickness in a contact.

Central film thickness: Film thickness near the central region of an EHL contact.

Thermal EHL: EHL in which viscous heating and temperature-dependent viscosity are included.

Traction coefficient: Ratio of tangential traction force to normal load.

Slide-to-roll ratio: Dimensionless measure of sliding relative to mean rolling speed.

Starvation: Reduction in film thickness caused by insufficient lubricant replenishment at the inlet.

Stribeck curve: Relationship between friction and a speed–viscosity–load parameter.

Lambda ratio: Ratio of minimum film thickness to composite surface roughness.

Mixed lubrication: Regime in which fluid film and asperity contacts share the load.

Boundary lubrication: Regime in which surface films and asperity interactions dominate because the fluid film is very thin.

Asperity: Individual microscopic summit on a nominally rough surface.

Tribofilm: Protective surface layer formed by adsorption, chemical reaction, transfer, or mechanical action.

Anti-wear additive: Lubricant additive that forms a protective layer and reduces wear.

Extreme-pressure additive: Additive that reacts under severe pressure and temperature to prevent welding or scuffing.

Review Questions

Short Answer Questions

1. Why does lubricant viscosity increase in an EHL contact?

2. Define the lambda ratio and state its engineering significance.

3. What is meant by lubricant starvation?

4. Distinguish central film thickness from minimum film thickness.

5. What is the function of a tribofilm?

Long Answer Questions

1. Explain how pressure–viscosity coupling and elastic deformation produce an EHL film.

2. Discuss the effects of thermal heating and slide-to-roll ratio on traction and film thickness.

3. Explain the Stribeck curve and the physical transitions among boundary, mixed, and full-film lubrication.

4. Compare fully flooded and starved EHL, including causes, consequences, and remedies.

5. Describe asperity-contact models and explain their role in mixed-lubrication analysis.

Analytical / Application Questions

1. A bearing operates at high speed but develops a low lambda ratio after temperature rises. Analyse the likely causes and recommend corrective actions.

2. A gearset shows micropitting only during start-up and low-speed operation. Develop a tribological investigation based on EHL, starvation, traction, and surface chemistry.

3. An engineer proposes replacing a lubricant with a lower-viscosity grade solely to reduce friction. Evaluate the proposal using film thickness, lambda ratio, thermal effects, additive chemistry, and reliability considerations.

22. Multiple Choice Questions with Answers

Question 1

Which coupled effects are primarily responsible for load-carrying capacity in an EHL contact?

A. Pressure-induced viscosity increase and elastic deformation

B. Evaporation and thermal expansion of the lubricant

C. Plastic deformation and lubricant cavitation

D. Surface oxidation and centrifugal pumping

Correct Answer: A. Pressure-induced viscosity increase and elastic deformation

Question 2

If operating temperature increases while all other conditions remain unchanged, the most likely direct effect is

A. An increase in lubricant viscosity and film thickness

B. A reduction in viscosity and possibly film thickness

C. An elimination of elastic deformation

D. A guaranteed transition to boundary lubrication

Correct Answer: B. A reduction in viscosity and possibly film thickness

Question 3

A contact has a minimum film thickness of 0.30 micrometres and composite RMS roughness of 0.10 micrometres. Its lambda ratio is

A. 0.03

B. 0.3

C. 3

D. 30

Correct Answer: C. 3

Question 4

Which condition most directly produces starvation in an EHL contact?

A. Excessive elastic modulus

B. Insufficient lubricant replenishment at the inlet

C. An increase in composite roughness only

D. A reduction in slide-to-roll ratio to zero

Correct Answer: B. Insufficient lubricant replenishment at the inlet

Question 5

In mixed lubrication, the applied load is best understood as being carried by

A. The lubricant film alone

B. Asperities alone

C. A combination of fluid film and asperity contacts

D. The machine housing

Correct Answer: C. A combination of fluid film and asperity contacts

Question 6

A lambda ratio substantially below unity indicates that

A. Full film separation is assured

B. Significant asperity interaction is likely

C. The contact is necessarily starved

D. Traction must be zero

Correct Answer: B. Significant asperity interaction is likely

Question 7

What is the principal role of many anti-wear additives under boundary conditions?

A. To reduce the elastic modulus of the solids

B. To form a protective sacrificial tribofilm

C. To eliminate all lubricant shear

D. To increase surface roughness deliberately

Correct Answer: B. To form a protective sacrificial tribofilm

Question 8

Why can an isothermal, fully flooded film-thickness calculation overpredict performance in a high-speed gear contact?

A. It ignores thermal viscosity loss and possible starvation

B. It assumes that solids are perfectly plastic

C. It excludes Hertzian contact pressure entirely

D. It predicts no pressure–viscosity response

Correct Answer: A. It ignores thermal viscosity loss and possible starvation

Question 9

Increasing slide-to-roll ratio in a gear contact generally tends to increase

A. Sliding heat generation and traction effects

B. The probability of pure rolling

C. The inlet lubricant reservoir

D. The reduced elastic modulus

Correct Answer: A. Sliding heat generation and traction effects

Question 10

Which design response is most technically appropriate when a bearing has a low lambda ratio caused by high operating temperature?

A. Ignore temperature because ambient viscosity is specified

B. Consider thermal control, lubricant selection, surface finish, and supply together

C. Increase roughness to retain lubricant

D. Remove all anti-wear additives

Correct Answer: B. Consider thermal control, lubricant selection, surface finish, and supply together