4. Physics and Models of Friction

Introduction and Context

Friction is the tangential resistance that develops when two bodies move, or tend to move, relative to one another. In engineering, it is neither simply a material constant nor an unavoidable nuisance. It is an emergent response of interacting surfaces, influenced by contact geometry, surface chemistry, deformation, temperature, velocity, loading history, lubrication, wear debris and environmental conditions. The coefficient of friction, μ, is therefore a useful engineering descriptor, but not a universal property of a material pair.

At the macroscopic scale, friction is often represented by the Coulomb relation, F = μN, where F is the friction force and N is the normal load. This relation is valuable for preliminary design, but advanced tribology requires an explanation of why μ changes and when the law fails. Such explanations arise from adhesion, asperity deformation, ploughing, third-body action and the time-dependent behaviour of interfaces. These mechanisms also clarify the differences between static and kinetic friction, the origin of stick-slip, rolling resistance and friction-induced vibration.

The chapter develops physical models that connect microscopic contact events with engineering-scale behaviour. It also considers how friction is measured, modelled and controlled in machine elements such as bearings, gears, brakes, seals, metal forming tools and precision positioning systems.

Learning Objectives

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

1. Distinguish the principal physical mechanisms contributing to friction.

2. Explain static, kinetic, boundary and mixed-contact friction.

3. Formulate and interpret adhesion, deformation, ploughing and third-body models.

4. Describe rate-and-state friction and its relevance to stick-slip.

5. Analyse rolling resistance and distinguish elastic hysteresis from sliding losses.

6. Explain the effects of temperature, velocity, load and surface history.

7. Relate friction laws to friction-induced vibration and dynamic instability.

8. Select suitable models for engineering analysis and identify their limitations.

Friction as an Emergent Contact Phenomenon

Real surfaces are rough at many length scales. When two nominally flat bodies are pressed together, contact occurs only at a collection of asperity summits. The real area of contact, Ar, is much smaller than the apparent area, A. The mean pressure over the real contact area may approach the hardness of the softer material, even when the nominal pressure is modest.

A useful decomposition of friction is

F = Fadh + Fdef + Fpl + F3 + Fvisc,

where Fadh is the adhesive contribution, Fdef is resistance caused by asperity deformation, Fpl is ploughing resistance, F3 is the contribution of third bodies and Fvisc is viscous or fluid-shear resistance. These terms are not always independent. For example, wear particles can alter adhesion, change the real contact area and modify the effective roughness.

The friction coefficient is consequently a system response. It depends on material pair, surface finish, oxide or contaminant films, normal load, sliding speed, temperature and the duration of contact. A value measured in a laboratory tribometer should not be transferred to a machine without considering similarity of contact pressure, speed, temperature, lubrication and surface evolution.

Adhesion and Interfacial Shear

The adhesion model regards friction as the shear required to break microscopic junctions formed where asperities contact. If the average shear strength of these junctions is τ and the total real contact area is Ar, then

Fadh = τAr.

If the asperities deform plastically, Ar is approximately N/H, where H is the hardness of the softer material. The resulting estimate is

μadh = Fadh/N ≈ τ/H.

This relation explains why friction is often associated with the ratio of interfacial shear strength to material hardness. It also shows why hardening a surface may reduce friction by reducing the real area of contact, although hard surfaces do not necessarily have low friction: strong chemical bonding or removal of protective films may increase τ.

Adhesion is particularly important for clean, smooth, conformal contacts under boundary lubrication or vacuum. Oxide films, adsorbed molecules, lubricants and surface treatments commonly reduce direct metallic bonding. In polymers, adhesion may involve molecular attraction and viscoelastic deformation rather than metallic junctions. At very small scales, adhesion can dominate normal contact and cause stiction in microelectromechanical systems.

The adhesive model is limited because τ and Ar are rarely uniform. Junctions have different ages, sizes and temperatures, and their growth may be time dependent. The model is therefore best interpreted as a mechanistic scaling relation rather than an exact predictive law.

Asperity Deformation and Hysteresis

As an asperity passes through a contact, it may deform elastically, elastoplastically or plastically. Energy is dissipated when deformation is irreversible. The tangential force associated with deforming asperities depends on roughness amplitude and wavelength, material yield strength, elastic modulus, Poisson's ratio and sliding direction.

In a purely elastic, perfectly reversible contact, deformation does not necessarily produce net energy loss over a complete cycle. Real materials, however, exhibit plasticity, viscoelasticity and internal damping. A polymer asperity may deform during loading and recover slowly during unloading, producing hysteresis. This is a major source of friction in elastomers sliding on rough surfaces.

Plastic deformation can also generate friction even in the absence of strong adhesion. The energy per unit sliding distance is related to the volume of material plastically displaced and the flow stress. Deformation-based friction becomes important when a hard rough surface slides against a softer material, or when asperities are sufficiently sharp to penetrate the opposing surface.

Ploughing and Cutting Models

Ploughing occurs when a hard asperity, wear particle or engineered texture penetrates a softer surface and pushes material aside. A simplified model relates the ploughing force to the projected cross-sectional area of the groove and the hardness of the softer material:

Fpl ≈ kHAp,

where Ap is the projected ploughing area and k is a factor reflecting geometry and plastic flow. For a conical asperity, the force increases with penetration depth and depends strongly on the included angle.

Ploughing should be distinguished from ordinary asperity deformation. In deformation friction, both surfaces may accommodate contact without a dominant groove. In ploughing, one body acts as a cutting or indentation tool. The mechanism is common in abrasive wear, contaminated lubricants, rough hard coatings and metal forming.

Surface texturing can either increase or decrease ploughing. Shallow, well-designed textures may store debris and reduce uncontrolled contact, whereas sharp or excessively deep features increase penetration and friction. The directionality of grooves can also make friction anisotropic.

Third-Body Friction

A third body is any material located between the two principal surfaces, including wear debris, abrasive particles, transferred films, oxide fragments, lubricant additives and deliberately introduced solid lubricants. Third bodies may roll, slide, become trapped, fracture, compact or form a mechanically mixed layer.

The third-body concept is valuable because it treats wear particles as an active part of the contact rather than merely as a consequence of wear. A compacted debris bed may protect the surfaces and lower friction; a large hard particle may produce severe ploughing; a compliant particle may act as a rolling element. The friction response depends on particle size distribution, hardness, shape, concentration, entrainment and escape paths.

In brakes and clutches, friction films formed from wear products and additives can determine both friction level and stability. Their composition changes with temperature and pressure. Thus, a friction material should be analysed as a dynamic composite contact rather than as a homogeneous solid.

Static and Kinetic Friction

Static friction is the tangential force required to initiate sliding. It adjusts up to a limiting value:

Fs ≤ μsN.

Once sliding begins, the average kinetic friction force is often represented as

Fk = μkN,

with μk commonly, though not universally, smaller than μs. The difference arises from junction growth during rest, stronger adhesive bonds, viscoelastic relaxation and the time needed to establish contact. Static friction is therefore history dependent.

The simple inequality does not mean that static friction is always equal to μsN. Before impending motion, it takes whatever value is needed for equilibrium. Similarly, kinetic friction is not necessarily constant: it may vary with speed, temperature, displacement, wear and contact state.

In lubricated contacts, the transition from boundary to mixed and hydrodynamic lubrication can produce a Stribeck-type relationship in which friction first decreases with increasing speed and then rises because of viscous drag. The controlling parameter is commonly expressed as ηV/P, where η is viscosity, V is entrainment speed and P is a representative load or pressure.

Rate-and-State Friction

Rate-and-state models describe friction as a function of sliding velocity and one or more internal state variables representing contact age, junction population or structural condition. A generic expression is

μ = μ0 + a ln(V/V0) + b ln(θ/θ0),

where V is velocity, θ is a state variable, and a and b are experimentally determined parameters. A state evolution law may take the ageing form

dθ/dt = 1 − Vθ/D,

where D is a characteristic slip distance. During stationary contact, θ increases and junctions strengthen. During sliding, old junctions are replaced and θ decreases.

If a > b, friction tends to increase with velocity and sliding is velocity strengthening. If b > a, friction decreases with velocity and the interface is velocity weakening. Velocity weakening is a key condition for stick-slip when elastic energy is stored in the surrounding system.

Rate-and-state formulations are more realistic than a single coefficient because they represent memory. They are used in seals, earthquake mechanics, brake systems, rock friction and precision drives. Their parameters, however, depend on normal load, temperature, lubrication and the range of velocities used for identification.

Stick-Slip and Friction-Induced Vibration

Stick-slip occurs when a system alternates between periods of sticking and rapid slip. Consider a slider connected to a spring and driven at constant speed. During sticking, the spring force increases while the slider remains stationary. When the force exceeds the available static resistance, the slider slips. If kinetic friction is lower than the force required for breakaway, the slider accelerates, then decelerates and sticks again.

The instability is not determined by friction alone. It depends on system stiffness, mass, damping, driving speed and the slope of the friction–velocity relation. A simplified stability condition for a velocity-weakening interface is that system stiffness must exceed a critical value related to the rate of frictional weakening and the characteristic slip distance. Increasing damping, reducing compliance or modifying the friction law can suppress oscillation.

Friction-induced vibration occurs in brakes, clutches, gears, seals, machine tools and robotic joints. It may appear as squeal, chatter, judder or tonal noise. Mechanisms include negative damping, modal coupling, intermittent contact, geometric variation and fluctuating third-body films. The frequency may be close to a structural natural frequency, but friction can also create self-excited frequencies that shift with temperature and wear.

A practical dynamic analysis should combine a contact model with structural modal analysis. Static coefficient measurements alone cannot predict squeal because the relevant quantities may be friction slope, tangential contact stiffness, delay, damping and spatial variation of friction.

Rolling Resistance

Rolling resistance is the force or torque opposing rolling motion. It is distinct from ideal rolling contact, in which no sliding occurs and no energy is dissipated. Real rolling resistance arises mainly from:

• elastic and viscoelastic hysteresis in wheels, tyres and substrates;
• micro-slip within the contact patch;
• plastic deformation and surface roughness;
• adhesion and contaminants;
• bearing seal and lubricant losses; and
• aerodynamic drag at high speed.

For a wheel of radius R, rolling resistance may be represented by a resisting moment Mr or an equivalent force Fr = Mr/R. A common engineering form is Fr = CrN, where Cr is a rolling-resistance coefficient. Unlike the sliding coefficient, Cr is strongly dependent on radius, load, speed, tyre pressure, material damping and surface condition.

In elastomeric tyres, hysteresis is often dominant: the leading and trailing regions of the contact patch experience different deformation histories, so energy is dissipated within the material. In rigid wheels on hard surfaces, micro-slip, local plasticity and roughness may be more significant. Rolling bearings exhibit additional losses from lubricant churning, cage motion, seal friction and microslip.

Temperature and Thermally Activated Friction

Friction converts mechanical work into heat. The flash temperature at asperity contacts can be much higher than the measured bulk temperature because the contact area is small and heat is generated rapidly. Temperature influences friction by changing hardness, yield strength, viscosity, viscoelastic response, oxidation, adhesion and wear-film chemistry.

A simplified heat partition model divides generated heat between the two bodies according to their thermal properties and contact conditions. In steady sliding, the bulk temperature depends on friction power per unit area, sliding speed, thermal conductivity, heat capacity, geometry and cooling. Transient flashes depend on asperity size and contact duration.

Temperature can produce either positive or negative feedback. If heating lowers viscosity, the lubricant film may thin, increasing asperity contact and friction. If heating lowers polymer modulus, contact pressure may decrease, reducing friction. In brakes, a rise in temperature can cause fade, transfer-film breakdown or, in some materials, friction increase. Thermal expansion can also alter alignment and load distribution.

Thermal effects must therefore be included in high-speed bearings, brakes, metal forming, seals, electrical contacts and space mechanisms. Friction data obtained at room temperature may be misleading when the operating contact experiences substantial flash heating.

Engineering Modelling and Measurement

Model selection should follow the dominant physics and the purpose of the analysis. A Coulomb model is suitable for basic force estimation and multibody simulations where detailed contact evolution is unimportant. An adhesion–hardness model is useful for interpreting clean asperity contacts. Ploughing models are appropriate for abrasive contacts and textured interfaces. Rate-and-state models are preferable when dwell time, velocity history or stick-slip matters. Thermo-mechanical and third-body models are required when temperature and debris evolution control performance.

Tribometers should reproduce the important dimensionless groups and operating conditions: contact pressure, sliding speed, temperature, lubrication regime, surface roughness and duration. A robust test programme includes break-in, steady-state behaviour, transients and repeated runs. Measurements should record friction force, normal load, speed, temperature, wear and, where possible, acoustic or vibration signals.

Common modelling errors include treating μ as a material constant, ignoring run-in, fitting a friction law over too narrow a velocity range, neglecting thermal feedback and applying a sliding model to rolling contact. Numerical simulations should also account for stiffness and damping introduced by fixtures, since these may alter observed stick-slip.

Applied Case Study: Stick-Slip in a Precision Linear Stage

Background

An illustrative precision manufacturing system uses a steel guideway and polymer-coated slider driven by a ballscrew. The stage must position a cutting tool smoothly at low speed. Operators observe periodic positional errors and a low-frequency audible vibration during reversals and very slow motion.

Situation and Problem

The measured breakaway force is substantially higher than the force during motion. At constant commanded speed, the stage alternates between stationary intervals and sudden displacements. Replacing the drive motor does not remove the problem.

Analysis

The contact is operating near the boundary or mixed-lubrication regime. During dwell, polymer relaxation and adhesive junction growth increase static resistance. Once motion begins, junctions are broken and kinetic resistance decreases. The ballscrew and stage provide elastic compliance, storing energy during the sticking phase. The friction–velocity curve is velocity weakening over the relevant low-speed range. The system therefore satisfies the conditions for stick-slip.

Temperature measurements show only a modest bulk rise, but the friction signal changes during long cycles, indicating evolution of the transfer film. A constant-Coulomb simulation predicts smooth motion and fails to reproduce the observed oscillation. A rate-and-state model coupled to the measured stage stiffness reproduces the dwell time and slip amplitude more accurately.

Approach and Solution

The engineering team applies several measures: a low-friction boundary additive is introduced, guideway preload is adjusted to reduce local pressure peaks, structural damping is increased, and the controller includes a low-speed dither signal. The friction law is identified separately for forward and reverse motion and after different dwell times.

Outcome

The modified system exhibits a lower breakaway-to-running friction difference and substantially reduced positional oscillation. The result is not attributed to a single lower coefficient of friction; rather, stability improves because contact ageing, compliance and damping are changed together.

Learning Points

The case demonstrates that friction-induced vibration is a system problem. A friction coefficient measured under steady sliding cannot describe a process dominated by dwell, reversal and low-speed transients. Successful mitigation requires contact physics, structural dynamics, thermal and lubricant considerations, and control-system design.

Limitations, Risks and Professional Considerations

No friction model is universally valid. Simplified laws may be computationally convenient but can conceal dependence on load, speed, temperature and history. Highly detailed models may require parameters that are difficult to measure and may not improve prediction outside the calibration range.

Engineers should avoid extrapolating laboratory results without assessing scale, surface preparation and environment. Contamination, corrosion and lubricant degradation can change the governing mechanism. Safety-critical systems such as brakes, hoists and aerospace actuators require conservative design, traceable testing and monitoring of wear and temperature.

Environmental and professional responsibilities also matter. Lubricant selection should consider toxicity, disposal and leakage. Friction materials must meet applicable safety and emissions requirements. Design decisions should not optimise friction alone while neglecting wear debris, noise, energy consumption or failure consequences.

Current Developments and Best Practices

Contemporary research increasingly combines multiscale contact mechanics, surface chemistry, in situ microscopy, thermal sensing and data-driven modelling. Machine-learning methods can identify friction laws from operating data, but they should be constrained by physical variables and validated under changed conditions. Digital twins of bearings, brakes and robotic joints can integrate friction, wear, temperature and vibration measurements for condition monitoring.

Best practice includes:

• define whether the design requires high friction, low friction or stable friction;
• identify the likely dominant mechanism before selecting a model;
• measure transient as well as steady-state behaviour;
• control surface chemistry, roughness, debris and lubrication;
• include thermal and structural dynamics where power or vibration is significant;
• validate models at representative loads, speeds and temperatures; and
• report uncertainty, repeatability and the evolution of the contact condition.

Chapter Summary

Friction is an emergent response of real contacts, not a fixed material constant. Adhesion arises from shearing microscopic junctions; deformation and hysteresis dissipate energy through irreversible or viscoelastic contact changes; ploughing results from penetration by hard asperities or particles; and third bodies modify contact through rolling, sliding, compaction and film formation. Static friction reflects the maximum resistance before motion, while kinetic friction describes resistance during sliding and may vary with speed and history.

Rate-and-state models represent the memory of an interface and explain why dwell time and velocity can produce stick-slip. Rolling resistance is governed by hysteresis, microslip, plasticity and bearing or lubricant losses rather than by ideal rolling alone. Temperature changes material properties, lubricant films, surface chemistry and wear-film stability. Friction-induced vibration is a coupled contact–structure phenomenon requiring consideration of stiffness, damping and friction–velocity behaviour. Effective engineering practice therefore combines appropriate physical models, representative experiments, thermal and dynamic analysis, and careful attention to uncertainty and contact evolution.

Key Terms

Adhesion: Frictional resistance caused by shear of microscopic junctions formed between contacting surfaces.

Asperity: A microscopic summit or irregularity on a nominally rough surface.

Boundary lubrication: A regime in which surfaces are separated mainly by adsorbed or reacted films rather than a full fluid film.

Coulomb friction: An idealised law in which friction is proportional to normal load and approximately independent of apparent contact area.

Flash temperature: A short-duration local temperature rise at an asperity contact.

Hysteresis: Energy dissipation associated with a material following different loading and unloading paths.

Kinetic friction: Friction acting while two surfaces slide relative to one another.

Mixed lubrication: A regime involving both fluid-film support and direct asperity interaction.

Ploughing: Friction caused by penetration and displacement of material by a hard asperity or particle.

Rate-and-state friction: A friction framework incorporating sliding velocity and internal variables representing contact history.

Real area of contact: The sum of microscopic regions actually touching, usually much smaller than apparent area.

Rolling resistance: Resistance opposing rolling due to hysteresis, microslip, plasticity and related losses.

Stribeck behaviour: Variation of friction with speed, viscosity and load across lubrication regimes.

Static friction: Resistance opposing the initiation of relative motion.

Stick-slip: Alternating sticking and rapid sliding caused by frictional instability and stored elastic energy.

Third body: Material interposed between two primary contacting surfaces, such as debris or a transfer film.

Velocity strengthening: Increase of friction with increasing sliding velocity.

Velocity weakening: Decrease of friction with increasing sliding velocity over a specified range.

Review Questions

Short Answer Questions

1. Why is the coefficient of friction not generally a material constant?

2. State the basic adhesion model and explain the meaning of τ and Ar.

3. Distinguish ploughing friction from adhesion friction.

4. What is meant by a third body in tribology?

5. Define rolling resistance and identify two of its physical sources.

Long Answer Questions

1. Explain the principal mechanisms of friction and discuss how they interact in a real engineering contact.

2. Develop the adhesion–hardness interpretation of friction and critically discuss its assumptions.

3. Explain static and kinetic friction using contact ageing and junction renewal.

4. Describe rate-and-state friction and show how it helps explain stick-slip.

5. Discuss the influence of temperature on friction, lubrication, wear and contact stability.

Analytical / Application Questions

1. A slider is connected to a spring and driven at constant speed. The friction decreases immediately after motion begins and increases again during dwell. Use a rate-and-state interpretation to explain the observed cycle and propose two remedies.

2. A hard textured surface slides against a soft polymer and produces grooves and loose debris. Identify the dominant friction mechanisms and recommend surface or lubrication changes.

3. A brake exhibits increasing noise as temperature rises, although the average normal load remains constant. Construct a coupled explanation involving friction-film evolution, temperature and structural vibration, and state what measurements would be required to verify it.

22. Multiple Choice Questions with Answers

Question 1

Which relation most directly represents the basic adhesion model for friction?

A. Fadh = τAr

B. Fadh = ηV/P

C. Fadh = CrN/R

D. Fadh = kT/V

Correct Answer: A. Fadh = τAr

Question 2

Why can the real area of contact be much smaller than the apparent area while still supporting the load?

A. The contacting asperities carry high local pressures

B. The apparent area is always lubricated completely

C. Normal load acts only on the surrounding material

D. Surface roughness eliminates normal stresses

Correct Answer: A. The contacting asperities carry high local pressures

Question 3

Which mechanism is most characteristic of a hard particle cutting a groove in a softer surface?

A. Adhesion

B. Ploughing

C. Hydrodynamic lubrication

D. Elastic recovery only

Correct Answer: B. Ploughing

Question 4

In a rate-and-state model, an increase in the state variable during rest generally represents:

A. Reduction in normal load

B. Ageing and strengthening of contact junctions

C. Immediate melting of asperities

D. Loss of all surface films

Correct Answer: B. Ageing and strengthening of contact junctions

Question 5

An interface with friction that decreases as velocity increases over the operating range is described as:

A. Velocity strengthening

B. Velocity weakening

C. Purely rolling

D. Thermally neutral

Correct Answer: B. Velocity weakening

Question 6

Which condition most directly promotes stick-slip in a spring-driven slider?

A. Kinetic friction is higher than static friction at all speeds

B. Friction weakens with velocity and elastic energy is stored

C. The system has no compliance

D. The contact has zero normal load

Correct Answer: B. Friction weakens with velocity and elastic energy is stored

Question 7

Which is a major source of rolling resistance in an elastomeric tyre?

A. Perfectly reversible elastic deformation

B. Viscoelastic hysteresis

C. Absence of a contact patch

D. Only aerodynamic drag at all speeds

Correct Answer: B. Viscoelastic hysteresis

Question 8

Why can bulk temperature measurements underestimate thermal damage at a sliding interface?

A. Friction produces no heat at asperities

B. Local flash temperatures can greatly exceed bulk temperature

C. Bulk temperature is always higher than contact temperature

D. Thermal conductivity prevents local heating

Correct Answer: B. Local flash temperatures can greatly exceed bulk temperature

Question 9

Which model is most appropriate when dwell time and velocity history strongly affect friction?

A. Constant Coulomb model only

B. Rate-and-state model

C. Ideal rigid rolling model

D. Geometric area model

Correct Answer: B. Rate-and-state model

Question 10

Why is a constant-friction simulation often inadequate for predicting brake squeal or precision-stage vibration?

A. Structural dynamics and friction–velocity behaviour may control stability

B. Normal load has no influence on vibration

C. Vibration occurs only in lubricated contacts

D. Friction-induced vibration is independent of damping

Correct Answer: A. Structural dynamics and friction–velocity behaviour may control stability