Friction: Difference between revisions

Latest revision as of 09:05, 25 April 2026

The blue plate has more friction on the sloped surface than the green one, and thus slides down more slowly

Simulated blocks with fractal rough surfaces, exhibiting static frictional interactions^[1]

Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding or grinding against each other.^[2]^[3] Types of friction include dry, fluid, lubricated, skin, and internal – an incomplete list. The study of the processes involved is called tribology, and has a history of more than 2,000 years.^[4]

Friction can have dramatic consequences, as illustrated by the use of friction created by rubbing pieces of wood together to start a fire. Another important consequence of many types of friction can be wear, which may lead to performance degradation or damage to components. It is known that frictional energy losses account for about 20% of the total energy expenditure of the world.^[5]^[4]

There are many different contributors to the retarding force in friction, ranging from asperity deformation to the generation of charges and changes in local structure. When two bodies in contact move relative to each other, due to these various contributors some mechanical energy is transformed to heat, the free energy of structural changes, and other types of dissipation. The total dissipated energy per unit distance moved is the retarding frictional force, F=E/d. The complexity of the interactions involved makes the calculation of friction from first principles difficult, and it is often easier to use empirical methods for analysis and the development of theory.^[3]^[2]

Types

There are several types of friction:

Dry friction is a force that opposes the relative lateral motion of two solid surfaces in contact. Dry friction is subdivided into static friction ("stiction") between non-moving surfaces, and kinetic friction between moving surfaces. With the exception of atomic or molecular friction, dry friction generally arises from the interaction of surface features, known as asperities (see Figure).
Fluid friction describes the friction between layers of a viscous fluid that are moving relative to each other.^[6]^[7]
Lubricated friction is a case of fluid friction where a lubricant fluid separates two solid surfaces.^[8]^[9]^[10]
Skin friction is a component of drag, the force resisting the motion of a fluid across the surface of a body.
Internal friction is the force resisting motion between the elements making up a solid material while it undergoes deformation.^[7]^[4]

History

Many ancient authors including Aristotle, Vitruvius, and Pliny the Elder, were interested in the cause and mitigation of friction.^[11] They were aware of differences between static and kinetic friction with Themistius stating in 350 A.D. that "it is easier to further the motion of a moving body than to move a body at rest".^[11]^[12]^[13]^[14]

The classic laws of sliding friction were discovered by Leonardo da Vinci in 1493, a pioneer in tribology, but the laws documented in his notebooks were not published and remained unknown.^[15]^[16]^[17]^[18]^[19]^[20] These laws were rediscovered by Guillaume Amontons in 1699^[21] and became known as Amonton's three laws of dry friction. Amontons presented the nature of friction in terms of surface irregularities and the force required to raise the weight pressing the surfaces together. This view was further elaborated by Bernard Forest de Bélidor^[22] and Leonhard Euler (1750), who derived the angle of repose of a weight on an inclined plane and first distinguished between static and kinetic friction.^[23] John Theophilus Desaguliers (1734) first recognized the role of adhesion in friction.^[24] Microscopic forces cause surfaces to stick together; he proposed that friction was the force necessary to tear the adhering surfaces apart.

The understanding of friction was further developed by Charles-Augustin de Coulomb (1785).^[21] Coulomb investigated the influence of four main factors on friction: the nature of the materials in contact and their surface coatings, the extent of the surface area, the normal pressure (or load), and the length of time that the surfaces remained in contact (time of repose).^[15] Coulomb further considered the influence of sliding velocity, temperature and humidity, in order to decide between the different explanations on the nature of friction that had been proposed. The distinction between static and dynamic friction is made in Coulomb's friction law (see below), although this distinction was already drawn by Johann Andreas von Segner in 1758.^[15] The effect of the time of repose was explained by Pieter van Musschenbroek (1762) by considering the surfaces of fibrous materials, with fibers meshing together, which takes a finite time in which the friction increases.

John Leslie (1766–1832) noted a weakness in the views of Amontons and Coulomb: If friction arises from a weight being drawn up the inclined plane of successive asperities, then why is it not balanced through descending the opposite slope? Leslie was equally skeptical about the role of adhesion proposed by Desaguliers, which should on the whole have the same tendency to accelerate as to retard the motion.^[15] In Leslie's view, friction should be seen as a time-dependent process of flattening, pressing down asperities, which creates new obstacles in what were cavities before.

File:Heat by friction.webm

Heat by friction captured by a thermal camera

In the long course of the development of the law of conservation of energy and of the first law of thermodynamics, friction was recognised as a mode of conversion of mechanical work into heat. In 1798, Benjamin Thompson reported on cannon boring experiments.^[25]

Arthur Jules Morin (1833) developed the concept of sliding versus rolling friction.

In 1842, Julius Robert Mayer frictionally generated heat in paper pulp and measured the temperature rise.^[26] In 1845, Joule published a paper entitled "The Mechanical Equivalent of Heat", in which he specified a numerical value for the amount of mechanical work required to "produce a unit of heat", based on the friction of an electric current passing through a resistor, and on the friction of a paddle wheel rotating in a vat of water.^[27]

Osborne Reynolds (1866) derived the equation of viscous flow. This completed the classic empirical model of friction (static, kinetic, and fluid) commonly used today in engineering.^[16] In 1877, Fleeming Jenkin and J. A. Ewing investigated the continuity between static and kinetic friction.^[28]

In 1907, G.H. Bryan published an investigation of the foundations of thermodynamics, Thermodynamics: an Introductory Treatise dealing mainly with First Principles and their Direct Applications. He noted that for a rough body driven over a rough surface, the mechanical work done by the driver exceeds the mechanical work received by the surface. The lost work is accounted for by heat generated by friction.^[29]

Over the years, for example in his 1879 thesis, but particularly in 1926, Planck advocated regarding the generation of heat by rubbing as the most specific way to define heat, and the prime example of an irreversible thermodynamic process.^[30]

The focus of research during the 20th century has been to understand the physical mechanisms behind friction. Frank Philip Bowden and David Tabor (1950) showed that, at a microscopic level, the actual area of contact between surfaces is a very small fraction of the apparent area.^[17] This actual area of contact, caused by asperities increases with pressure. The development of the atomic force microscope (ca. 1986) enabled scientists to study friction at the atomic scale,^[16] showing that, on that scale, dry friction is the product of the inter-surface shear stress and the contact area. These two discoveries explain Amonton's first law (below); the macroscopic proportionality between normal force and static frictional force between dry surfaces.

Dry friction

Dry friction resists relative lateral motion of two solid surfaces in contact. The two regimes of dry friction are 'static friction' ("stiction") between non-moving surfaces, and kinetic friction (sometimes called sliding friction or dynamic friction) between moving surfaces.

The force of friction is always exerted in a direction that opposes movement (for kinetic friction) or potential movement (for static friction) between the two surfaces. For example, a curling stone sliding along the ice experiences a kinetic force slowing it down. For an example of potential movement, the drive wheels of an accelerating car experience a frictional force pointing forward; if they did not, the wheels would spin, and the rubber would slide backwards along the pavement. Note that it is not the direction of movement of the vehicle they oppose, it is the direction of (potential) sliding between tire and road.

The elementary property of sliding (kinetic) friction were discovered by experiment in the 15th to 18th century and were expressed as three empirical laws:

Amontons' first law: The force of friction is directly proportional to the applied load.
Amontons' second law: The force of friction is independent of the apparent area of contact.
Coulomb's law of friction: Kinetic friction is independent of the sliding velocity.

Coulomb friction, named after Charles-Augustin de Coulomb, is an approximate model used to calculate the force of dry friction. It is governed by the model:^[31] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_\mathrm{f} \leq \mu F_\mathrm{n},} where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_\mathrm{f}} is the force of friction exerted by each surface on the other. It is parallel to the surface, in a direction opposite to the net applied force.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} is the coefficient of friction, which is an empirical property of the contacting materials,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_\mathrm{n}} is the normal force exerted by each surface on the other, directed perpendicular (normal) to the surface.

The Coulomb friction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_\mathrm{f}} may take any value from zero up to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu F_\mathrm{n}} , and the direction of the frictional force against a surface is opposite to the motion that surface would experience in the absence of friction. Thus, in the static case, the frictional force is exactly what it must be in order to prevent motion between the surfaces; it balances the net force tending to cause such motion. In this case, rather than providing an estimate of the actual frictional force, the Coulomb approximation provides a threshold value for this force, above which motion would commence. This maximum force is known as traction.

Static friction

File:Static kinetic friction vs time.png

When the mass is not moving, the object experiences static friction. The friction increases as the applied force increases until the block moves. After the block moves, it experiences kinetic friction, which is less than the maximum static friction.

Static friction is friction between two or more solid objects that are not moving relative to each other. For example, static friction can prevent an object from sliding down a sloped surface. The coefficient of static friction, typically denoted as μ_s, is usually higher than the coefficient of kinetic friction. Static friction is considered to arise as the result of surface roughness features across multiple length scales at solid surfaces. These features, known as asperities are present down to nano-scale dimensions and result in true solid to solid contact existing only at a limited number of points accounting for only a fraction of the apparent or nominal contact area.^[32] The linearity between applied load and true contact area, arising from asperity deformation, gives rise to the linearity between static frictional force and normal force, found for typical Amonton–Coulomb type friction.^[33]

The static friction force must be overcome by an applied force before an object can move. The maximum possible friction force between two surfaces before sliding begins is the product of the coefficient of static friction and the normal force: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_\text{max} = \mu_\mathrm{s} F_\text{n}} . When there is no sliding occurring, the friction force can have any value from zero up to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_\text{max}} . Any force smaller than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_\text{max}} attempting to slide one surface over the other is opposed by a frictional force of equal magnitude and opposite direction. Any force larger than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_\text{max}} overcomes the force of static friction and causes sliding to occur. The instant sliding occurs, static friction is no longer applicable—the friction between the two surfaces is then called kinetic friction. However, an apparent static friction can be observed even in the case when the true static friction is zero.^[34]

An example of static friction is the force that prevents a car wheel from slipping as it rolls on the ground. Even though the wheel is in motion, the patch of the tire in contact with the ground is stationary relative to the ground, so it is static rather than kinetic friction. Upon slipping, the wheel friction changes to kinetic friction. An anti-lock braking system operates on the principle of allowing a locked wheel to resume rotating so that the car maintains static friction.

The maximum value of static friction, when motion is impending, is sometimes referred to as limiting friction,^[35] although this term is not used universally.^[6]

Kinetic friction

Kinetic friction, also known as dynamic friction or sliding friction, occurs when two objects are moving relative to each other and rub together (like a sled on the ground). The coefficient of kinetic friction is typically denoted as μ_k, and is usually less than the coefficient of static friction for the same materials.^[36]^[37] However, Richard Feynman comments that "with dry metals it is very hard to show any difference."^[38] The friction force between two surfaces after sliding begins is the product of the coefficient of kinetic friction and the normal force: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_{k} = \mu_\mathrm{k} F_{n}} . This is responsible for the Coulomb damping of an oscillating or vibrating system.

Role of the normal force

File:Free body diagram2.svg

Free-body diagram for a block on a ramp. Arrows are vectors indicating directions and magnitudes of forces. N is the normal force, mg is the force of gravity, and F_f is the force of friction.

The normal force is defined as the net force compressing two parallel surfaces together, and its direction is perpendicular to the surfaces. In the simple case of a mass resting on a horizontal surface, the only component of the normal force is the force due to gravity, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N=mg\,} . In this case, conditions of equilibrium tell us that the magnitude of the friction force is zero, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_f = 0} . In fact, the friction force always satisfies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_f\le \mu N} , with equality reached only at a critical ramp angle (given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan^{-1}\mu} ) that is steep enough to initiate sliding.

The friction coefficient is an empirical (experimentally measured) structural property that depends only on various aspects of the contacting materials, such as surface roughness. The coefficient of friction is not a function of mass or volume. For instance, a large aluminum block has the same coefficient of friction as a small aluminum block. However, the magnitude of the friction force itself depends on the normal force, and hence on the mass of the block.

Depending on the situation, the calculation of the normal force Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} might include forces other than gravity. If an object is on a level surface and subjected to an external force Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} tending to cause it to slide, then the normal force between the object and the surface is just Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N = mg + P_y} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle mg} is the block's weight and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_y} is the downward component of the external force. Prior to sliding, this friction force is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_f = -P_x} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_x} is the horizontal component of the external force. Thus, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_f \le \mu N} in general. Sliding commences only after this frictional force reaches the value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_f = \mu N} . Until then, friction is whatever it needs to be to provide equilibrium, so it can be treated as simply a reaction.

If the object is on a tilted surface such as an inclined plane, the normal force from gravity is smaller than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle mg} , because less of the force of gravity is perpendicular to the face of the plane. The normal force and the frictional force are ultimately determined using vector analysis, usually via a free body diagram.

In general, process for solving any statics problem with friction is to treat contacting surfaces tentatively as immovable so that the corresponding tangential reaction force between them can be calculated. If this frictional reaction force satisfies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_f \le \mu N} , then the tentative assumption was correct, and it is the actual frictional force. Otherwise, the friction force must be set equal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_f = \mu N} , and then the resulting force imbalance would then determine the acceleration associated with slipping.

File:Free body.svg

Angle of friction, θ, when block just starts to slide

Role of angle

For certain applications, it is more useful to define static friction in terms of the maximum angle before which one of the items will begin sliding. This is called the angle of friction or friction angle. It is defined as: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan{\theta} = \mu_\mathrm{s}} and thus: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta = \arctan{\mu_\mathrm{s}}} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} is the angle from horizontal and μ_s is the static coefficient of friction between the objects.^[39] This formula can also be used to calculate μ_s from empirical measurements of the friction angle.

Coefficient of friction

The coefficient of friction (COF), often symbolized by the Greek letter μ, is a dimensionless scalar value which equals the ratio of the force of friction between two bodies and the force pressing them together, either during or at the onset of slipping. The coefficient of friction depends on the materials used; for example, ice on steel has a low coefficient of friction, while rubber on pavement has a high coefficient of friction. Coefficients of friction range from near zero to greater than one. The coefficient of friction between two surfaces of similar metals is greater than that between two surfaces of different metals; for example, brass has a higher coefficient of friction when moved against brass, but less if moved against steel or aluminum.^[40]

For surfaces at rest relative to each other, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu = \mu_\mathrm{s}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_\mathrm{s}} is the coefficient of static friction. This is usually larger than its kinetic counterpart. The coefficient of static friction exhibited by a pair of contacting surfaces depends upon the combined effects of material deformation characteristics and surface roughness, both of which have their origins in the chemical bonding between atoms in each of the bulk materials and between the material surfaces and any adsorbed material. The fractality of surfaces, a parameter describing the scaling behavior of surface asperities, is known to play an important role in determining the magnitude of the static friction.^[1]

For surfaces in relative motion Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu = \mu_\mathrm{k}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_\mathrm{k}} is the coefficient of kinetic friction. The Coulomb friction is equal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_\mathrm{f}} , and the frictional force on each surface is exerted in the direction opposite to its motion relative to the other surface.

Arthur Morin introduced the term and demonstrated the utility of the coefficient of friction.^[15] The coefficient of friction is an empirical measurement—it has to be measured experimentally, and cannot be found through calculations.^[41] Rougher surfaces tend to have higher effective values. Both static and kinetic coefficients of friction depend on the pair of surfaces in contact; for a given pair of surfaces, the coefficient of static friction is usually larger than that of kinetic friction; in some sets the two coefficients are equal, such as teflon-on-teflon.

Most dry materials in combination have friction coefficient values between 0.3 and 0.6. Values outside this range are rarer, but teflon, for example, can have a coefficient as low as 0.04. A value of zero would mean no friction at all, an elusive property. Rubber in contact with other surfaces can yield friction coefficients from 1 to 2. Occasionally it is maintained that μ is always < 1, but this is not true. While in most relevant applications μ < 1, a value above 1 merely implies that the force required to slide an object along the surface is greater than the normal force of the surface on the object. For example, silicone rubber or acrylic rubber-coated surfaces have a coefficient of friction that can be substantially larger than 1.

While it is often stated that the coefficient of friction is a "material property", it is better categorized as a "system property". Unlike true material properties (such as conductivity, dielectric constant, yield strength), the coefficient of friction for any two materials depends on system variables like temperature, velocity, atmosphere, and also what are now popularly described as aging and deaging timesTemplate:What?, as well as on geometric properties of the interface between the materials, namely surface structure.^[1] For example, a copper pin sliding against a thick copper plate can have a coefficient of friction that varies from 0.6 at low speeds (metal sliding against metal) to below 0.2 at high speeds when the copper surface begins to melt due to frictional heating. The latter speed, of course, does not determine the coefficient of friction uniquely; if the pin diameter is increased so that the frictional heating is removed rapidly, the temperature drops, the pin remains solid, and the coefficient of friction rises to that of a 'low speed' test.^{[citation needed]}

In systems with significant non-uniform stress fields, because local slip occurs before the system slides, the macroscopic coefficient of static friction depends on the applied load, system size, or shape; Amontons' law is not satisfied macroscopically.^[42]

Approximate coefficients of friction

Materials		Static Friction, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_\mathrm{s}}		Kinetic/Sliding Friction, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_\mathrm{k}\,}
Materials		Dry and clean	Lubricated	Dry and clean	Lubricated
Aluminium	Steel	0.61^[43]		0.47^[43]
Aluminium	Aluminium	1.05–1.35^[43]	0.3^[43]	1.4^[43]–1.5^{[citation needed]}
Gold	Gold			2.5^{[citation needed]}
Platinum	Platinum	1.2^[43]	0.25^[43]	3.0^{[citation needed]}
Silver	Silver	1.4^[43]	0.55^[43]	1.5^{[citation needed]}
Alumina ceramic	Silicon nitride ceramic				0.004 (wet)^[44]
BAM (Ceramic alloy AlMgB₁₄)	Titanium boride (TiB₂)	0.04–0.05^[45]	0.02^[46]^[47]
Brass	Steel	0.35–0.51^[43]	0.19^[43]	0.44^[43]
Cast iron	Copper	1.05^[43]		0.29^[43]
Cast iron	Zinc	0.85^[43]		0.21^[43]
Concrete	Rubber	1.0	0.30 (wet)	0.6–0.85^[43]	0.45–0.75 (wet)^[43]
Concrete	Wood	0.62^[43]^[48]
Copper	Glass	0.68^[49]		0.53^[49]
Copper	Steel	0.53^[49]		0.36^[43]^[49]	0.18^[49]
Glass	Glass	0.9–1.0^[43]^[49]	0.005–0.01^[49]	0.4^[43]^[49]	0.09–0.116^[49]
Human synovial fluid	Human cartilage		0.01^[50]		0.003^[50]
Ice	Ice	0.02–0.09^[51]
Polyethene	Steel	0.2^[43]^[51]	0.2^[43]^[51]
PTFE (Teflon)	PTFE (Teflon)	0.04^[43]^[51]	0.04^[43]^[51]		0.04^[43]
Steel	Ice	0.03^[51]
Steel	PTFE (Teflon)	0.04^[43]−0.2^[51]	0.04^[43]		0.04^[43]
Steel	Steel	0.74^[43]−0.80^[51]	0.005–0.23^[49]^[51]	0.42–0.62^[43]^[49]	0.029–0.19^[49]
Wood	Metal	0.2–0.6^[43]^[48]	0.2 (wet)^[43]^[48]	0.49^[49]	0.075^[49]
Wood	Wood	0.25–0.62^[43]^[48]^[49]	0.2 (wet)^[43]^[48]	0.32–0.48^[49]	0.067–0.167^[49]

Sources of friction

File:Asperities.svg

The top image shows asperities under no load. The bottom image depicts the same surface after applying a load.

Friction is due to a range of energy dissipative processes. The base model, first detailed by Bowden and Tabor^[52] is that it involves contact of small protusions on surfaces called asperities. As the load increases the number and the area of asperity contacts increases, leading to the normal force dependence of friction. As these asperities slide past each other both plastic and elastic deformation processes take place, among others. The frictional force is the energy dissipated per unit distance by these, the net result often being the linear Coulomb formula mentioned earlier.^[52]

Experimental studies of dry sliding friction in engineering materials have shown that friction and wear are controlled not only by the nominal contact conditions, but also by near-surface microstructural evolution, debris generation and tribochemical transformations during sliding. Reviews and experimental studies have shown that plastic deformation, fracture, material transfer, mechanical mixing and oxidation can all contribute to the transition between mild and severe wear regimes in dry metal contacts.^[53]^[54]^[55]^[56]

The importance of debris and “third-body” material formed in the contact has also been demonstrated experimentally, since compacted triboparticulates may either accelerate damage or form protective interfacial layers that reduce direct metal-to-metal contact.^[57] In steels, dry sliding tests have shown that hardness level, load and sliding speed can strongly influence friction stability and wear rate, while oxidative wear may become dominant under suitable sliding conditions.^[58]^[59]^[60] Related work on stainless steels and titanium alloys further shows that dry sliding friction is highly sensitive to abrasive particle size, counterface material and the formation or breakdown of protective tribo-oxide layers, which can shift the governing wear mechanism from relatively mild oxidative behaviour to severe delamination or abrasive damage.^[61]^[62]^[63]^[64]^[65] Experimental investigations further indicate that environmental factors such as hydrogen charging, lubrication regime and cyclic loading may significantly influence crack initiation mechanisms and friction stability in structural alloys.^[66]

Breakdown of the Coulomb model

Despite being a simplified model of friction, the Coulomb model is useful in many numerical simulation applications such as multibody systems and granular material. Even its most simple expression encapsulates the fundamental effects of sticking and sliding which are required in many applied cases, although specific algorithms have to be designed in order to efficiently numerically integrate mechanical systems with Coulomb friction and bilateral or unilateral contact.^[67]^[68]^[69]^[70]^[71] Some quite nonlinear effects, such as the so-called Painlevé paradoxes, may be encountered with Coulomb friction.^[72]

The Coulomb approximation follows from the assumptions that: surfaces are in atomically close contact only over a small fraction of their overall area; that this contact area is proportional to the normal force (until saturation, which takes place when all area is in atomic contact); and that the frictional force is proportional to the applied normal force, independently of the contact area; this comes from the asperity work of Bowden and Tabor.^[52] The Coulomb approximation is fundamentally an empirical construct. It is a rule-of-thumb describing the approximate outcome of an extremely complicated physical interaction. The strength of the approximation is its simplicity and versatility. Though the relationship between normal force and frictional force is not exactly linear (and so the frictional force is not entirely independent of the contact area of the surfaces), the Coulomb approximation is an adequate representation of friction for the analysis of many physical systems.

When the surfaces are conjoined, Coulomb friction becomes a very poor approximation (for example, adhesive tape resists sliding even when there is no normal force, or a negative normal force). In this case, the frictional force may depend strongly on the area of contact. Some drag racing tires are adhesive for this reason. However, despite the complexity of the fundamental physics behind friction, the relationships are accurate enough to be useful in many applications.

Dry friction and instabilities

Dry friction can induce several types of instabilities in mechanical systems which display a stable behaviour in the absence of friction.^[73] These instabilities may be caused by the decrease of the friction force with an increasing velocity of sliding, by material expansion due to heat generation during friction (the thermo-elastic instabilities), or by pure dynamic effects of sliding of two elastic materials (the Adams–Martins instabilities). The latter were originally discovered in 1995 by George G. Adams and João Arménio Correia Martins for smooth surfaces^[74]^[75] and were later found in periodic rough surfaces.^[76] In particular, friction-related dynamical instabilities are thought to be responsible for brake squeal and the 'song' of a glass harp,^[77]^[78] phenomena which involve stick and slip, modelled as a drop of friction coefficient with velocity.^[79]

A practically important case is the self-oscillation of the strings of bowed instruments such as the violin, cello, hurdy-gurdy, erhu, etc.

A connection between dry friction and flutter instability in a simple mechanical system has been discovered,^[80] watch the movie Archived 2025-12-11 at the Wayback Machine for more details. Frictional instabilities can lead to the formation of new self-organized patterns (or "secondary structures") at the sliding interface, such as in-situ formed tribofilms which are utilized for the reduction of friction and wear in so-called self-lubricating materials.^[81]

Friction at the nanoscale

In 2008 scientists for the first time were able to move a single atom across a surface, and measure the forces required. Using ultrahigh vacuum and nearly zero temperature (5 K), a modified atomic force microscope was used to drag a cobalt atom, and a carbon monoxide molecule, across surfaces of copper and platinum.^[82]

As of 2012^[update], a single study has demonstrated the potential for an effectively negative coefficient of friction in the low-load regime, meaning that a decrease in normal force leads to an increase in friction. This contradicts everyday experience in which an increase in normal force leads to an increase in friction.^[83] This was reported in the journal Nature in October 2012 and involved the friction encountered by an atomic force microscope stylus when dragged across a graphene sheet in the presence of graphene-adsorbed oxygen.^[83]

The origin of kinetic friction at nanoscale can be rationalized by an energy model.^[84] During sliding, a new surface forms at the back of a sliding true contact, and existing surface disappears at the front of it. Since all surfaces involve the thermodynamic surface energy, work must be spent in creating the new surface, and energy is released as heat in removing the surface. Thus, a force is required to move the back of the contact, and frictional heat is released at the front.

Under certain conditions some materials have very low friction coefficients. An example is (highly ordered pyrolytic) graphite which can have a friction coefficient below 0.01.^[85] This ultralow-friction regime is called superlubricity.^[86]

Fluid friction

Fluid friction occurs between fluid layers that are moving relative to each other. This internal resistance to flow is named viscosity. In everyday terms, the viscosity of a fluid is described as its "thickness". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity. The less viscous the fluid, the greater its ease of deformation or movement.

All real fluids (except superfluids) offer some resistance to shearing and therefore are viscous. For teaching and explanatory purposes it is helpful to use the concept of an inviscid fluid or an ideal fluid which offers no resistance to shearing and so is not viscous.

Lubricated friction

Lubricated friction is a case of fluid friction where a fluid separates two solid surfaces. Lubrication is a technique employed to reduce wear of one or both surfaces in close proximity moving relative to each another by interposing a substance called a lubricant between the surfaces. In most cases the applied load is carried by pressure generated within the fluid due to the frictional viscous resistance to motion of the lubricating fluid between the surfaces. Adequate lubrication allows smooth continuous operation of equipment, with only mild wear, and without excessive stresses or seizures at bearings. When lubrication breaks down, metal or other components can rub destructively over each other, causing heat and possibly damage or failure.^[87]

Skin friction

Skin friction arises from the interaction between the fluid and the skin of an object, and is directly related to the area of the surface of the body that is in contact with the fluid. Skin friction follows the drag equation and rises with the square of the velocity.^[88]

Skin friction is caused by viscous drag in the boundary layer around the object. There are two ways to decrease skin friction: the first is to shape the moving body so that smooth flow is possible, like an airfoil. The second method is to decrease the length and cross-section of the moving object as much as is practicable.^[88]

Internal friction

Internal friction is the force resisting motion between the elements making up a solid material while it undergoes deformation. The best understood case is the drag force on dislocations due to atomic vibrations (phonon drag),^[89] although electron interactions can also play a role.^[90] There are also contributions from dissipative forces on the motion of grain boundaries in metals,^[91] and there are similar phenomena in vitreous materials such as glasses.^[92]

Radiation friction

As a consequence of light pressure, Einstein^[93] in 1909 predicted the existence of "radiation friction" which would oppose the movement of matter. He wrote, "radiation will exert pressure on both sides of the plate. The forces of pressure exerted on the two sides are equal if the plate is at rest. However, if it is in motion, more radiation will be reflected on the surface that is ahead during the motion (front surface) than on the back surface. The backward-acting force of pressure exerted on the front surface is thus larger than the force of pressure acting on the back. Hence, as the resultant of the two forces, there remains a force that counteracts the motion of the plate and that increases with the velocity of the plate. We will call this resultant 'radiation friction' in brief."

Other types of friction

Rolling resistance

Rolling resistance is the force that resists the rolling of a wheel or other circular object along a surface caused by deformations in the object or surface. Generally the force of rolling resistance is less than that associated with kinetic friction.^[94] Typical values for the coefficient of rolling resistance for steel wheels on a track are 0.001.^[95] One of the most common examples of rolling resistance, with a coefficient of around 0.02, is the movement of motor vehicle tires on a road, a process which generates heat and sound as by-products.^[96]

Braking friction

Any wheel equipped with a brake is capable of generating a large retarding force, usually for the purpose of slowing and stopping a vehicle or piece of rotating machinery. Braking friction differs from rolling friction because the coefficient of friction for rolling friction is small whereas the coefficient of friction for braking friction is designed to be large by choice of materials for brake pads.

Triboelectric effect

Rubbing two materials against each other can lead to charge transfer, either electrons or ions. The energy required for this contributes to the friction. In addition, sliding can cause a build-up of electrostatic charge, which can be hazardous if flammable gases or vapours are present. When the static build-up discharges, explosions can be caused by ignition of the flammable mixture. However, as discussed recently^[97] the extent to which triboelectricity contributes to friction is contentious, with some claims that it is large,^[98] others that it is small because charge transfer can occur with superlubricious samples.^[99]

Belt friction

Belt friction is a physical property observed from the forces acting on a belt wrapped around a pulley, when one end is being pulled. The resulting tension, which acts on both ends of the belt, can be modeled by the belt friction equation.

In practice, the theoretical tension acting on the belt or rope calculated by the belt friction equation can be compared to the maximum tension the belt can support. This helps a designer of such a rig to know how many times the belt or rope must be wrapped around the pulley to prevent it from slipping. Mountain climbers and sailing crews demonstrate a standard knowledge of belt friction when accomplishing basic tasks.

Friction reduction

Devices

Devices such as wheels, ball bearings, roller bearings, and air cushion or other types of fluid bearings can change sliding friction into a much smaller type of rolling friction.

Many thermoplastic materials such as nylon, HDPE and PTFE are commonly used in low friction bearings. They are especially useful because the coefficient of friction falls with increasing imposed load.^[100] For improved wear resistance, very high molecular weight grades are usually specified for heavy duty or critical bearings.

Lubricants

A common way to reduce friction is by using a lubricant, such as oil, water, or grease, which is placed between the two surfaces, often dramatically lessening the coefficient of friction. The science of friction and lubrication is called tribology. Lubricant technology is when lubricants are mixed with the application of science, especially to industrial or commercial objectives.

Superlubricity, a recently discovered effect, has been observed in graphite: it is the substantial decrease of friction between two sliding objects, approaching zero levels. A very small amount of frictional energy would still be dissipated.

Lubricants to overcome friction need not always be thin, turbulent fluids or powdery solids such as graphite and talc; acoustic lubrication actually uses sound as a lubricant.

Another way to reduce friction between two parts is to superimpose micro-scale vibration to one of the parts. This can be sinusoidal vibration as used in ultrasound-assisted cutting or vibration noise, known as dither.

Applications

Friction is an important factor in many engineering disciplines.

Transportation

Automobile brakes inherently rely on friction, slowing a vehicle by converting its kinetic energy into heat. Incidentally, dispersing this large amount of heat safely is one technical challenge in designing brake systems. Disk brakes rely on friction between a disc and brake pads that are squeezed transversely against the rotating disc. In drum brakes, brake shoes or pads are pressed outwards against a rotating cylinder (brake drum) to create friction. Since braking discs can be more efficiently cooled than drums, disc brakes have better stopping performance.^[101]
Rail adhesion refers to the grip wheels of a train have on the rails, see Frictional contact mechanics.
Road slipperiness is an important design and safety factor for automobiles^[102]
- Split friction is a particularly dangerous condition arising due to varying friction on either side of a car.
- Road texture affects the interaction of tires and the driving surface.

Measurement

A tribometer is an instrument that measures friction on a surface.
A profilograph is a device used to measure pavement surface roughness.

Household usage

Friction is used to heat and ignite matchsticks (friction between the head of a matchstick and the rubbing surface of the match box).^[103]
Sticky pads are used to prevent object from slipping off smooth surfaces by effectively increasing the friction coefficient between the surface and the object.

References

↑ ^1.0 ^1.1 ^1.2 Hanaor, D.; Gan, Y.; Einav, I. (2016). "Static friction at fractal interfaces". Tribology International. 93: 229–238. arXiv:2106.01473. doi:10.1016/j.triboint.2015.09.016. S2CID 51900923.
↑ ^2.0 ^2.1 Template:MerriamWebsterDictionary
↑ ^3.0 ^3.1 "Friction | Definition, Types, & Formula | Britannica". www.britannica.com. 2024-09-11. Archived from the original on 2024-09-16. Retrieved 2024-10-07.
↑ ^4.0 ^4.1 ^4.2 Ghose, Tia; published, Ailsa Harvey (2022-02-08). "What is Friction?". livescience.com. Archived from the original on 2024-05-20. Retrieved 2024-10-07.
↑ Mitchell, Luke (November 2012). Ward, Jacob (ed.). "The Fiction of Nonfriction". Popular Science. No. 5. 281 (November 2012): 40.
↑ ^6.0 ^6.1 Beer, Ferdinand P.; Johnston, E. Russel Jr. (1996). Vector Mechanics for Engineers (6th ed.). McGraw-Hill. p. 397. ISBN 978-0-07-297688-5.
↑ ^7.0 ^7.1 Meriam, J.L.; Kraige, L.G. (2002). Engineering Mechanics (5th ed.). John Wiley & Sons. p. 328. ISBN 978-0-471-60293-4.
↑ Ruina, Andy; Pratap, Rudra (2002). Introduction to Statics and Dynamics (PDF). Oxford University Press. p. 713. Archived (PDF) from the original on 2019-05-25. Retrieved 2008-12-20.
↑ Hibbeler, R.C. (2007). Engineering Mechanics (11th ed.). Pearson, Prentice Hall. p. 393. ISBN 978-0-13-127146-3.
↑ Soutas-Little, Robert W.; Inman, Balint (2008). Engineering Mechanics. Thomson. p. 329. ISBN 978-0-495-29610-2.
↑ ^11.0 ^11.1 Template:Cite thesis^{[permanent dead link]}
↑ Fishbane, Paul M.; Gasiorowicz, Stephen; Thornton, Stephen T. (1993). Physics for Scientists and Engineers. I (Extended ed.). Englewood Cliffs, New Jersey: Prentice Hall. p. 135. ISBN 978-0-13-663246-7. Themistius first stated around 350 B.C. [sic] that kinetic friction is weaker than the maximum value of static friction.
↑ Hecht, Eugene (2003). Physics: Algebra/Trig (3rd ed.). Cengage Learning. ISBN 978-0-534-37729-8.
↑ Sambursky, Samuel (2014). The Physical World of Late Antiquity. Princeton University Press. pp. 65–66. ISBN 978-1-4008-5898-9. Archived from the original on 2024-10-07. Retrieved 2016-11-01.
↑ ^15.0 ^15.1 ^15.2 ^15.3 ^15.4 Dowson, Duncan (1997). History of Tribology (2nd ed.). Professional Engineering Publishing. ISBN 978-1-86058-070-3.
↑ ^16.0 ^16.1 ^16.2 Armstrong-Hélouvry, Brian (1991). Control of machines with friction. US: Springer. p. 10. ISBN 978-0-7923-9133-3. Archived from the original on 2024-10-07. Retrieved 2020-06-07.
↑ ^17.0 ^17.1 van Beek, Anton. "History of Science Friction". tribology-abc.com. Archived from the original on 2011-08-07. Retrieved 2011-03-24.
↑ Hutchings, Ian M. (2016). "Leonardo da Vinci's studies of friction" (PDF). Wear. 360–361: 51–66. doi:10.1016/j.wear.2016.04.019. Archived (PDF) from the original on 2016-08-03.
↑ Hutchings, Ian M. (2016-08-15). "Leonardo da Vinci's studies of friction". Wear. 360–361: 51–66. doi:10.1016/j.wear.2016.04.019. Archived from the original on 2021-09-18. Retrieved 2019-07-09.
↑ Kirk, Tom (July 22, 2016). "Study reveals Leonardo da Vinci's 'irrelevant' scribbles mark the spot where he first recorded the laws of friction". phys.org. Archived from the original on 2016-07-25. Retrieved 2016-07-26.
↑ ^21.0 ^21.1 Popova, Elena; Popov, Valentin L. (2015-06-01). "The research works of Coulomb and Amontons and generalized laws of friction". Friction. 3 (2): 183–190. doi:10.1007/s40544-015-0074-6.
↑ Forest de Bélidor, Bernard. "Richtige Grund-Sätze der Friction-Berechnung Archived 2021-04-27 at the Wayback Machine" ("Correct Basics of Friction Calculation"), 1737, (in German)
↑ "Leonhard Euler". Friction Module. Nano World. 2002. Archived from the original on 2011-05-07. Retrieved 2011-03-25.
↑ Goedecke, Andreas (2014). Transient Effects in Friction: Fractal Asperity Creep. Springer Science and Business Media. p. 3. ISBN 978-3-7091-1506-0. Archived from the original on 2024-10-07. Retrieved 2020-11-11.
↑ Benjamin Thompson (1798). "An inquiry concerning the source of the heat which is excited by friction," Archived 2024-10-07 at the Wayback Machine Philosophical Transactions of the Royal Society of London, 88 : 80–102. doi:10.1098/rstl.1798.0006
↑ Blundell, S.J., Blundell, K.M. (2006). Concepts in Thermal Physics, Oxford University Press, Oxford UK, ISBN 978-0-19-856769-1, p. 106.
↑ Joule, J.P. (1845)."On the Mechanical Equivalent of Heat". Philosophical Transactions of the Royal Society of London. 140: 61–82. 1850. doi:10.1098/rstl.1850.0004.
↑ Fleeming Jenkin & James Alfred Ewing (1877) "On Friction between Surfaces moving at Low Speeds Archived 2021-09-18 at the Wayback Machine", Philosophical Magazine Series 5, volume 4, pp. 308–10; link from Biodiversity Heritage Library
↑ Bryan, George Hartley (1907). Thermodynamics, an introductory treatise dealing mainly with first principles and their direct applications. Leipzig, Teubner. pp. 48–49. Retrieved 23 June 2023.
↑ Planck, M. (1926). "Über die Begründung des zweiten Hauptsatzes der Thermodynamik", Sitzungsber. Preuss. Akad. Wiss., Phys. Math. Kl., 453—463.
↑ Popov, Valentin L. (2017). "Coulomb's Law of Friction". Contact Mechanics and Friction. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 151–172. doi:10.1007/978-3-662-53081-8_10. ISBN 978-3-662-53080-1. Retrieved 2025-09-06.
↑ multi-scale origins of static friction Archived 2021-09-18 at the Wayback Machine 2016
↑ Greenwood J.A. and JB Williamson (1966). "Contact of nominally flat surfaces". Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. 295 (1442): 300. Bibcode:1966RSPSA.295..300G. doi:10.1098/rspa.1966.0242.
↑ Lua error in package.lua at line 80: module 'Module:Citation/CS1/Suggestions' not found.
↑ Bhavikatti, S.S.; K.G. Rajashekarappa (1994). Engineering Mechanics. New Age International. p. 112. ISBN 978-81-224-0617-7. Archived from the original on 2024-10-07. Retrieved 2007-10-21.
↑ Sheppard, Sheri; Tongue, Benson H.; Anagnos, Thalia (2005). Statics: Analysis and Design of Systems in Equilibrium. Wiley and Sons. p. 618. ISBN 978-0-471-37299-8. In general, for given contacting surfaces, μ_k < μ_s
↑ Meriam, James L.; Kraige, L. Glenn; Palm, William John (2002). Engineering Mechanics: Statics. Wiley and Sons. p. 330. ISBN 978-0-471-40646-4. Kinetic friction force is usually somewhat less than the maximum static friction force.
↑ Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (1964). "The Feynman Lectures on Physics, Vol. I, p. 12–5". Addison-Wesley. Archived from the original on 2021-03-10. Retrieved 2009-10-16.
↑ Nichols, Edward Leamington; Franklin, William Suddards (1898). The Elements of Physics. 1. Macmillan. p. 101. Archived from the original on 2024-10-07. Retrieved 2020-06-07.
↑ Air Brake Association (1921). The Principles and Design of Foundation Brake Rigging. Air brake association. p. 5. Archived from the original on 2024-10-07. Retrieved 2017-07-27.
↑ Lua error in package.lua at line 80: module 'Module:Citation/CS1/Suggestions' not found.
↑ Lua error in package.lua at line 80: module 'Module:Citation/CS1/Suggestions' not found.
↑ ^43.00 ^43.01 ^43.02 ^43.03 ^43.04 ^43.05 ^43.06 ^43.07 ^43.08 ^43.09 ^43.10 ^43.11 ^43.12 ^43.13 ^43.14 ^43.15 ^43.16 ^43.17 ^43.18 ^43.19 ^43.20 ^43.21 ^43.22 ^43.23 ^43.24 ^43.25 ^43.26 ^43.27 ^43.28 ^43.29 ^43.30 ^43.31 ^43.32 ^43.33 ^43.34 ^43.35 "Friction Factors – Coefficients of Friction". Archived from the original on 2019-02-01. Retrieved 2015-04-27.
↑ Ferreira, Vanderlei; Yoshimura, Humberto Naoyuki; Sinatora, Amilton (2012-08-30). "Ultra-low friction coefficient in alumina–silicon nitride pair lubricated with water". Wear. 296 (1–2): 656–659. doi:10.1016/j.wear.2012.07.030.
↑ Tian, Y.; Bastawros, A.F.; Lo, C.C.H.; Constant, A.P.; Russell, A.M.; Cook, B.A. (2003). "Superhard self-lubricating AlMgB[sub 14] films for microelectromechanical devices". Applied Physics Letters. 83 (14): 2781. Bibcode:2003ApPhL..83.2781T. doi:10.1063/1.1615677. Archived from the original on 2024-10-07. Retrieved 2019-01-31.
↑ Kleiner, Kurt (2008-11-21). "Material slicker than Teflon discovered by accident". Archived from the original on 2008-12-20. Retrieved 2008-12-25.
↑ Higdon, C.; Cook, B.; Harringa, J.; Russell, A.; Goldsmith, J.; Qu, J.; Blau, P. (2011). "Friction and wear mechanisms in AlMgB14-TiB2 nanocoatings". Wear. 271 (9–10): 2111–2115. doi:10.1016/j.wear.2010.11.044.
↑ ^48.0 ^48.1 ^48.2 ^48.3 ^48.4 Coefficient of Friction Archived March 8, 2009, at the Wayback Machine. EngineersHandbook.com
↑ ^49.00 ^49.01 ^49.02 ^49.03 ^49.04 ^49.05 ^49.06 ^49.07 ^49.08 ^49.09 ^49.10 ^49.11 ^49.12 ^49.13 ^49.14 ^49.15 ^49.16 Barrett, Richard T. (1 March 1990). "(NASA-RP-1228) Fastener Design Manual". NASA Technical Reports Server. NASA Lewis Research Center: 16. hdl:2060/19900009424. Archived from the original on 7 October 2024. Retrieved 3 August 2020.
↑ ^50.0 ^50.1 "Coefficients of Friction of Human Joints". Archived from the original on 2024-10-07. Retrieved 2015-04-27.
↑ ^51.0 ^51.1 ^51.2 ^51.3 ^51.4 ^51.5 ^51.6 ^51.7 ^51.8 "The Engineering Toolbox: Friction and Coefficients of Friction". Archived from the original on 2013-12-03. Retrieved 2008-11-23.
↑ ^52.0 ^52.1 ^52.2 Bowden, F P; Tabor, D (2001-02-08). The Friction and Lubrication of Solids. Oxford University PressOxford. doi:10.1093/oso/9780198507772.001.0001. ISBN 978-0-19-850777-2.
↑ Rigney, D. A. (1997). "Comments on the sliding wear of metals". Tribology International. 30 (5): 361–367. doi:10.1016/S0301-679X(96)00065-5.
↑ Stott, F. H. (1998). "The role of oxidation in the wear of alloys". Tribology International. 31 (1–3): 61–71. doi:10.1016/S0301-679X(98)00008-5.
↑ Hutchings, I. M. (1992). Tribology: Friction and Wear of Engineering Materials. Butterworth-Heinemann. ISBN 9780750610544.
↑ Zum Gahr, K.-H. (1987). Microstructure and Wear of Materials. Elsevier. ISBN 9780444427540.
↑ Jiang, J.; Stott, F. H.; Stack, M. M. (1998). "The role of triboparticulates in dry sliding wear". Tribology International. 31 (5): 245–256. doi:10.1016/S0301-679X(98)00027-9.
↑ Straffelini, G.; Trabucco, D.; Molinari, A. (2001). "Oxidative wear of heat-treated steels". Wear. 250 (1–12): 485–491. doi:10.1016/S0043-1648(01)00661-5.
↑ Tang, L.; Gao, C.; Huang, J.; Zhang, H.; Chang, W. (2013). "Dry sliding friction and wear behaviour of hardened AISI D2 tool steel with different hardness levels". Tribology International. 66: 165–173. doi:10.1016/j.triboint.2013.05.006.
↑ Zhang, Q. Y.; Chen, K. M.; Wang, L.; Wang, S. Q. (2013). "Characteristics of oxidative wear and oxidative mild wear". Tribology International. 61: 214–223. doi:10.1016/j.triboint.2013.01.003.
↑ Jourani, A.; Bouvier, S. (2015). "Friction and Wear Mechanisms of 316L Stainless Steel in Dry Sliding Contact: Effect of Abrasive Particle Size". Tribology Transactions. 58 (1): 131–139. doi:10.1080/10402004.2014.955229.
↑ Straffelini, G.; Molinari, A. (1999). "Dry sliding wear of Ti–6Al–4V alloy as influenced by the counterface and sliding conditions". Wear. 236 (1–2): 328–338. doi:10.1016/S0043-1648(99)00292-6.
↑ Zhang, Q. Y.; Chen, K. M.; Wang, L.; Wang, S. Q. (2016). "Investigation on tribo-layers and their function of a titanium alloy during dry sliding". Tribology International. 94: 541–549. doi:10.1016/j.triboint.2015.10.032.
↑ Rigney, D. A. (1997). "Comments on the sliding wear of metals". Tribology International. 30 (5): 361–367. doi:10.1016/S0301-679X(96)00065-5.
↑ Stott, F. H. (1998). "The role of oxidation in the wear of alloys". Tribology International. 31 (1–3): 61–71. doi:10.1016/S0301-679X(97)00060-0.
↑ Balitskii, O.I.; Kolesnikov, V.O.; Ivaskevych, L.M.; Havrylyuk, M.R. (2023). "The Influence of Specific Features of Load and Hydrogen Charging on Steel Tribotechnical Properties". Materials Science. 58: 502–512. doi:10.1007/s11003-023-00691-5.
↑ Haslinger, J.; Nedlec, J.C. (1983). "Approximation of the Signorini problem with friction, obeying the Coulomb law" (PDF). Mathematical Methods in the Applied Sciences. 5 (1): 422–437. Bibcode:1983MMAS....5..422H. doi:10.1002/mma.1670050127. hdl:10338.dmlcz/104086. Archived (PDF) from the original on 2024-10-07. Retrieved 2019-09-19.
↑ Alart, P.; Curnier, A. (1991). "A mixed formulation for frictional contact problems prone to Newton like solution method" (PDF). Computer Methods in Applied Mechanics and Engineering. 92 (3): 353–375. Bibcode:1991CMAME..92..353A. doi:10.1016/0045-7825(91)90022-X. Archived (PDF) from the original on 2024-10-07. Retrieved 2024-03-29.
↑ Acary, V.; Cadoux, F.; Lemaréchal, C.; Malick, J. (2011). "A formulation of the linear discrete Coulomb friction problem via convex optimization". Journal of Applied Mathematics and Mechanics. 91 (2): 155–175. Bibcode:2011ZaMM...91..155A. doi:10.1002/zamm.201000073. S2CID 17280625. Archived from the original on 2024-10-07. Retrieved 2018-04-20.
↑ De Saxcé, G.; Feng, Z.-Q. (1998). "The bipotential method: A constructive approach to design the complete contact law with friction and improved numerical algorithms". Mathematical and Computer Modelling. 28 (4): 225–245. doi:10.1016/S0895-7177(98)00119-8.
↑ Simo, J.C.; Laursen, T.A. (1992). "An augmented lagrangian treatment of contact problems involving friction". Computers and Structures. 42 (2): 97–116. doi:10.1016/0045-7949(92)90540-G.
↑ Acary, V.; Brogliato, B. (2008). Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and Electronics. 35. Springer Verlag Heidelberg.
↑ Bigoni, D. (2012). Nonlinear Solid Mechanics: Bifurcation Theory and Material Instability. Cambridge University Press. ISBN 978-1-107-02541-7.
↑ Adams, G.G. (1995). "Self-excited oscillations of two elastic half-spaces sliding with a constant coefficient of friction". Journal of Applied Mechanics. 62 (4): 867–872. Bibcode:1995JAM....62..867A. doi:10.1115/1.2896013.
↑ Martins, J. A. C.; Guimara˜es, J.; Faria, L. O. (1995). "Dynamic surface solutions in linear elasticity and viscoelasticity with frictional boundary conditions". Journal of Vibration and Acoustics. 117 (4): 445–451. doi:10.1115/1.2874477.
↑ Nosonovsky, Mikhail; Adams, George G. (2004). "Vibration and stability of frictional sliding of two elastic bodies with a wavy contact interface". Journal of Applied Mechanics. 71 (2): 154–161. Bibcode:2004JAM....71..154N. doi:10.1115/1.1653684.
↑ Flint, J.; Hultén, Johan (2002). "Lining-deformation-induced modal coupling as squeal generator in a distributed parameter disk brake model". Journal of Sound and Vibration. 254 (1): 1–21. Bibcode:2002JSV...254....1F. doi:10.1006/jsvi.2001.4052.
↑ Kröger, Matthias; Neubauer, Marcus; Popp, Karl (2008). "Experimental investigation on the avoidance of self-excited vibrations". Phil. Trans. R. Soc. A. 366 (1866): 785–810. Bibcode:2008RSPTA.366..785K. doi:10.1098/rsta.2007.2127. PMID 17947204. S2CID 16395796.
↑ Rice, J. R.; Ruina, A. L. (1983). "Stability of Steady Frictional Slipping" (PDF). Journal of Applied Mechanics. 50 (2): 343–349. Bibcode:1983JAM....50..343R. CiteSeerX 10.1.1.161.5207. doi:10.1115/1.3167042. Archived (PDF) from the original on 2010-06-22.
↑ Bigoni, D.; Noselli, G. (2011). "Experimental evidence of flutter and divergence instabilities induced by dry friction". Journal of the Mechanics and Physics of Solids. 59 (10): 2208–2226. Bibcode:2011JMPSo..59.2208B. CiteSeerX 10.1.1.700.5291. doi:10.1016/j.jmps.2011.05.007. Archived from the original on 2020-08-18. Retrieved 2011-11-30.
↑ Nosonovsky, Michael (2013). Friction-Induced Vibrations and Self-Organization: Mechanics and Non-Equilibrium Thermodynamics of Sliding Contact. CRC Press. p. 333. ISBN 978-1-4665-0401-1.
↑ Ternes, Markus; Lutz, Christopher P.; Hirjibehedin, Cyrus F.; Giessibl, Franz J.; Heinrich, Andreas J. (2008-02-22). "The Force Needed to Move an Atom on a Surface" (PDF). Science. 319 (5866): 1066–1069. Bibcode:2008Sci...319.1066T. doi:10.1126/science.1150288. PMID 18292336. S2CID 451375. Archived (PDF) from the original on 2018-07-20.
↑ ^83.0 ^83.1
Deng, Zhao; et al. (October 14, 2012). "Adhesion-dependent negative friction coefficient on chemically modified graphite at the nanoscale". Nature. 11 (12): 1032–7. Bibcode:2012NatMa..11.1032D. doi:10.1038/nmat3452. PMID 23064494.
- "At the nanoscale, graphite can turn friction upside down". R&D Magazine. 2012-10-17. Archived from the original on 2013-07-31.
↑ Makkonen, L (2012). "A thermodynamic model of sliding friction". AIP Advances. 2 (1): 012179. Bibcode:2012AIPA....2a2179M. doi:10.1063/1.3699027.
↑ Lua error in package.lua at line 80: module 'Module:Citation/CS1/Suggestions' not found.
↑ Müser, Martin H. (2015), "Theoretical Studies of Superlubricity", in Gnecco, Enrico; Meyer, Ernst (eds.), Fundamentals of Friction and Wear on the Nanoscale, NanoScience and Technology, Cham: Springer International Publishing, pp. 209–232, Bibcode:2015ffwn.book..209M, doi:10.1007/978-3-319-10560-4_11, ISBN 978-3-319-10559-8, retrieved 2025-04-25
↑ Hamrock, Bernard J. (2004). Fundamentals of fluid film lubrication. Steven R. Schmid, Bo O. Jacobson (2nd ed.). New York: Marcel Dekker. ISBN 0-8247-5120-5. OCLC 55739786.
↑ ^88.0 ^88.1 White, Frank (2011). Fluid Mechanics. New York City, NY: McGraw-Hill. pp. 477–478. ISBN 9780071311212.
↑ Al'shitz, V A; Indenbom, V L (1975-01-31). "Dynamic dragging of dislocations". Soviet Physics Uspekhi. 18 (1): 1–20. doi:10.1070/PU1975v018n01ABEH004689. ISSN 0038-5670.
↑ Kaganov, Moisei I; Kravchenko, Ya V; Natsik, V D (1974-06-30). "Dislocation dragging by electrons in metals". Soviet Physics Uspekhi. 16 (6): 878–891. doi:10.1070/PU1974v016n06ABEH004096. ISSN 0038-5670.
↑ Nowick, A. S. (1953-01-01). "Internal friction in metals". Progress in Metal Physics. 4: 1–70. doi:10.1016/0502-8205(53)90014-8. ISSN 0502-8205.
↑ Zdaniewski, Wieslaw A.; Rindone, Guy E.; Day, Delbert E. (1979-04-01). "The internal friction of glasses". Journal of Materials Science. 14 (4): 763–775. doi:10.1007/BF00550707. ISSN 1573-4803.
↑ Einstein, A. (1909). On the development of our views concerning the nature and constitution of radiation. Translated in: The Collected Papers of Albert Einstein, vol. 2 (Princeton University Press, Princeton, 1989). Princeton, NJ: Princeton University Press. p. 391.
↑ Silliman, Benjamin (1871) Principles of Physics, Or Natural Philosophy, Ivison, Blakeman, Taylor & company publishers
↑ Butt, Hans-Jürgen; Graf, Karlheinz and Kappl, Michael (2006) Physics and Chemistry of Interfaces, Wiley, ISBN 3-527-40413-9
↑ Hogan, C. Michael (1973). "Analysis of highway noise". Water, Air, & Soil Pollution. 2 (3): 387–392. Bibcode:1973WASP....2..387H. doi:10.1007/BF00159677. S2CID 109914430.
↑ Olson, Karl P; Marks, Laurence D (2025-10-01). "Is triboelectricity confusing, confused or complex?". Reports on Progress in Physics. 88 (10): Section 4.1. doi:10.1088/1361-6633/ae08cb. ISSN 0034-4885.
↑ Burgo, Thiago A. L.; Silva, Cristiane A.; Balestrin, Lia B. S.; Galembeck, Fernando (2013-08-12). "Friction coefficient dependence on electrostatic tribocharging". Scientific Reports. 3 (1): 2384. doi:10.1038/srep02384. ISSN 2045-2322. PMC 3740278. PMID 23934227.
↑ Huang, Xuanyu; Xiang, Xiaojian; Nie, Jinhui; Peng, Deli; Yang, Fuwei; Wu, Zhanghui; Jiang, Haiyang; Xu, Zhiping; Zheng, Quanshui (2021-04-15). "Microscale Schottky superlubric generator with high direct-current density and ultralong life". Nature Communications. 12 (1): 2268. doi:10.1038/s41467-021-22371-1. ISSN 2041-1723. PMC 8050059. PMID 33859180.
↑ Valentin L. Popov; Lars Voll; Stephan Kusche; Qiang Li; Svetlana V. Rozhkova (2018). "Generalized master curve procedure for elastomer friction taking into account dependencies on velocity, temperature and normal force". Tribology International. 120: 376–380. arXiv:1604.03407. doi:10.1016/j.triboint.2017.12.047. S2CID 119288819.
↑ "How Do Car Brakes Work?". Wonderopolis. Archived from the original on October 7, 2024. Retrieved November 4, 2018.
↑ Iskander, R.; Stevens, A. "Effectiveness of the Application of High Friction Surfacing-Crash-Reduction.pdf" (PDF). Archived (PDF) from the original on 2017-09-03. Retrieved 2017-09-03.
↑ "How Does Lighting A Match Work?". curiosity.com. Curiosity. November 11, 2015. Archived from the original on November 5, 2018. Retrieved November 4, 2018.

External links

"Friction" . Encyclopædia Britannica. 11 (11th ed.). 1911.
Coefficients of Friction – tables of coefficients, plus many links
Physclips: Mechanics with animations and video clips from the University of New South Wales
Values for Coefficient of Friction – CRC Handbook of Chemistry and Physics
Coefficients of friction of various material pairs in atmosphere and vacuum Archived 2023-05-31 at the Wayback Machine.

Template:Geotechnical engineering

[Hanaor-2016-1] 1.0 ^1.1 ^1.2 Hanaor, D.; Gan, Y.; Einav, I. (2016). "Static friction at fractal interfaces". Tribology International. 93: 229–238. arXiv:2106.01473. doi:10.1016/j.triboint.2015.09.016. S2CID 51900923.

[Merriam-Webster.com_Dictionary-2] 2.0 ^2.1 Template:MerriamWebsterDictionary

[www.britannica.com-2024-3] 3.0 ^3.1 "Friction | Definition, Types, & Formula | Britannica". www.britannica.com. 2024-09-11. Archived from the original on 2024-09-16. Retrieved 2024-10-07.

[Ghose-2022-4] 4.0 ^4.1 ^4.2 Ghose, Tia; published, Ailsa Harvey (2022-02-08). "What is Friction?". livescience.com. Archived from the original on 2024-05-20. Retrieved 2024-10-07.

[Mitchell-2012-5] Mitchell, Luke (November 2012). Ward, Jacob (ed.). "The Fiction of Nonfriction". Popular Science. No. 5. 281 (November 2012): 40.

[Beer-1996-6] 6.0 ^6.1 Beer, Ferdinand P.; Johnston, E. Russel Jr. (1996). Vector Mechanics for Engineers (6th ed.). McGraw-Hill. p. 397. ISBN 978-0-07-297688-5.

[Meriam-2002-7] 7.0 ^7.1 Meriam, J.L.; Kraige, L.G. (2002). Engineering Mechanics (5th ed.). John Wiley & Sons. p. 328. ISBN 978-0-471-60293-4.

[Ruina-2002-8] Ruina, Andy; Pratap, Rudra (2002). Introduction to Statics and Dynamics (PDF). Oxford University Press. p. 713. Archived (PDF) from the original on 2019-05-25. Retrieved 2008-12-20.

[Hibbeler-2007-9] Hibbeler, R.C. (2007). Engineering Mechanics (11th ed.). Pearson, Prentice Hall. p. 393. ISBN 978-0-13-127146-3.

[Soutas-Little-2008-10] Soutas-Little, Robert W.; Inman, Balint (2008). Engineering Mechanics. Thomson. p. 329. ISBN 978-0-495-29610-2.

[Chatterjee-2008-11] 11.0 ^11.1 Template:Cite thesis^{[permanent dead link]}

[12] Fishbane, Paul M.; Gasiorowicz, Stephen; Thornton, Stephen T. (1993). Physics for Scientists and Engineers. I (Extended ed.). Englewood Cliffs, New Jersey: Prentice Hall. p. 135. ISBN 978-0-13-663246-7. Themistius first stated around 350 B.C. [sic] that kinetic friction is weaker than the maximum value of static friction.

[13] Hecht, Eugene (2003). Physics: Algebra/Trig (3rd ed.). Cengage Learning. ISBN 978-0-534-37729-8.

[14] Sambursky, Samuel (2014). The Physical World of Late Antiquity. Princeton University Press. pp. 65–66. ISBN 978-1-4008-5898-9. Archived from the original on 2024-10-07. Retrieved 2016-11-01.

[Dowson-1997-15] 15.0 ^15.1 ^15.2 ^15.3 ^15.4 Dowson, Duncan (1997). History of Tribology (2nd ed.). Professional Engineering Publishing. ISBN 978-1-86058-070-3.

[Armstrong-Hélouvry-1991-16] 16.0 ^16.1 ^16.2 Armstrong-Hélouvry, Brian (1991). Control of machines with friction. US: Springer. p. 10. ISBN 978-0-7923-9133-3. Archived from the original on 2024-10-07. Retrieved 2020-06-07.

[van_Beek-17] 17.0 ^17.1 van Beek, Anton. "History of Science Friction". tribology-abc.com. Archived from the original on 2011-08-07. Retrieved 2011-03-24.

[18] Hutchings, Ian M. (2016). "Leonardo da Vinci's studies of friction" (PDF). Wear. 360–361: 51–66. doi:10.1016/j.wear.2016.04.019. Archived (PDF) from the original on 2016-08-03.

[19] Hutchings, Ian M. (2016-08-15). "Leonardo da Vinci's studies of friction". Wear. 360–361: 51–66. doi:10.1016/j.wear.2016.04.019. Archived from the original on 2021-09-18. Retrieved 2019-07-09.

[20] Kirk, Tom (July 22, 2016). "Study reveals Leonardo da Vinci's 'irrelevant' scribbles mark the spot where he first recorded the laws of friction". phys.org. Archived from the original on 2016-07-25. Retrieved 2016-07-26.

[Popova-2015-21] 21.0 ^21.1 Popova, Elena; Popov, Valentin L. (2015-06-01). "The research works of Coulomb and Amontons and generalized laws of friction". Friction. 3 (2): 183–190. doi:10.1007/s40544-015-0074-6.

[22] Forest de Bélidor, Bernard. "Richtige Grund-Sätze der Friction-Berechnung Archived 2021-04-27 at the Wayback Machine" ("Correct Basics of Friction Calculation"), 1737, (in German)

[23] "Leonhard Euler". Friction Module. Nano World. 2002. Archived from the original on 2011-05-07. Retrieved 2011-03-25.

[Goedecke-2014-24] Goedecke, Andreas (2014). Transient Effects in Friction: Fractal Asperity Creep. Springer Science and Business Media. p. 3. ISBN 978-3-7091-1506-0. Archived from the original on 2024-10-07. Retrieved 2020-11-11.

[25] Benjamin Thompson (1798). "An inquiry concerning the source of the heat which is excited by friction," Archived 2024-10-07 at the Wayback Machine Philosophical Transactions of the Royal Society of London, 88 : 80–102. doi:10.1098/rstl.1798.0006

[26] Blundell, S.J., Blundell, K.M. (2006). Concepts in Thermal Physics, Oxford University Press, Oxford UK, ISBN 978-0-19-856769-1, p. 106.

[27] Joule, J.P. (1845)."On the Mechanical Equivalent of Heat". Philosophical Transactions of the Royal Society of London. 140: 61–82. 1850. doi:10.1098/rstl.1850.0004.

[28] Fleeming Jenkin & James Alfred Ewing (1877) "On Friction between Surfaces moving at Low Speeds Archived 2021-09-18 at the Wayback Machine", Philosophical Magazine Series 5, volume 4, pp. 308–10; link from Biodiversity Heritage Library

[29] Bryan, George Hartley (1907). Thermodynamics, an introductory treatise dealing mainly with first principles and their direct applications. Leipzig, Teubner. pp. 48–49. Retrieved 23 June 2023.

[Planck_1926-30] Planck, M. (1926). "Über die Begründung des zweiten Hauptsatzes der Thermodynamik", Sitzungsber. Preuss. Akad. Wiss., Phys. Math. Kl., 453—463.

[31] Popov, Valentin L. (2017). "Coulomb's Law of Friction". Contact Mechanics and Friction. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 151–172. doi:10.1007/978-3-662-53081-8_10. ISBN 978-3-662-53080-1. Retrieved 2025-09-06.

[32] ulti-scale origins of static friction Archived 2021-09-18 at the Wayback Machine 2016

[33] Greenwood J.A. and JB Williamson (1966). "Contact of nominally flat surfaces". Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. 295 (1442): 300. Bibcode:1966RSPSA.295..300G. doi:10.1098/rspa.1966.0242.

[34] Lua error in package.lua at line 80: module 'Module:Citation/CS1/Suggestions' not found.

[Bhavikatti-1994-35] Bhavikatti, S.S.; K.G. Rajashekarappa (1994). Engineering Mechanics. New Age International. p. 112. ISBN 978-81-224-0617-7. Archived from the original on 2024-10-07. Retrieved 2007-10-21.

[36] Sheppard, Sheri; Tongue, Benson H.; Anagnos, Thalia (2005). Statics: Analysis and Design of Systems in Equilibrium. Wiley and Sons. p. 618. ISBN 978-0-471-37299-8. In general, for given contacting surfaces, μ_k < μ_s

[37] Meriam, James L.; Kraige, L. Glenn; Palm, William John (2002). Engineering Mechanics: Statics. Wiley and Sons. p. 330. ISBN 978-0-471-40646-4. Kinetic friction force is usually somewhat less than the maximum static friction force.

[38] Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (1964). "The Feynman Lectures on Physics, Vol. I, p. 12–5". Addison-Wesley. Archived from the original on 2021-03-10. Retrieved 2009-10-16.

[39] Nichols, Edward Leamington; Franklin, William Suddards (1898). The Elements of Physics. 1. Macmillan. p. 101. Archived from the original on 2024-10-07. Retrieved 2020-06-07.

[Air_Brake_Association-1921-40] Air Brake Association (1921). The Principles and Design of Foundation Brake Rigging. Air brake association. p. 5. Archived from the original on 2024-10-07. Retrieved 2017-07-27.

[Valentin_L._Popov-2014-41] Lua error in package.lua at line 80: module 'Module:Citation/CS1/Suggestions' not found.

[42] Lua error in package.lua at line 80: module 'Module:Citation/CS1/Suggestions' not found.

[Friction_Factors-43] 43.00 ^43.01 ^43.02 ^43.03 ^43.04 ^43.05 ^43.06 ^43.07 ^43.08 ^43.09 ^43.10 ^43.11 ^43.12 ^43.13 ^43.14 ^43.15 ^43.16 ^43.17 ^43.18 ^43.19 ^43.20 ^43.21 ^43.22 ^43.23 ^43.24 ^43.25 ^43.26 ^43.27 ^43.28 ^43.29 ^43.30 ^43.31 ^43.32 ^43.33 ^43.34 ^43.35 "Friction Factors – Coefficients of Friction". Archived from the original on 2019-02-01. Retrieved 2015-04-27.

[Ferreira-2012-44] Ferreira, Vanderlei; Yoshimura, Humberto Naoyuki; Sinatora, Amilton (2012-08-30). "Ultra-low friction coefficient in alumina–silicon nitride pair lubricated with water". Wear. 296 (1–2): 656–659. doi:10.1016/j.wear.2012.07.030.

[Tian-2003-45] Tian, Y.; Bastawros, A.F.; Lo, C.C.H.; Constant, A.P.; Russell, A.M.; Cook, B.A. (2003). "Superhard self-lubricating AlMgB[sub 14] films for microelectromechanical devices". Applied Physics Letters. 83 (14): 2781. Bibcode:2003ApPhL..83.2781T. doi:10.1063/1.1615677. Archived from the original on 2024-10-07. Retrieved 2019-01-31.

[Kleiner,_Kurt-2008-46] Kleiner, Kurt (2008-11-21). "Material slicker than Teflon discovered by accident". Archived from the original on 2008-12-20. Retrieved 2008-12-25.

[Higdon-2011-47] Higdon, C.; Cook, B.; Harringa, J.; Russell, A.; Goldsmith, J.; Qu, J.; Blau, P. (2011). "Friction and wear mechanisms in AlMgB14-TiB2 nanocoatings". Wear. 271 (9–10): 2111–2115. doi:10.1016/j.wear.2010.11.044.

[Coefficient_of_Friction_Archived_March_8-48] 48.0 ^48.1 ^48.2 ^48.3 ^48.4 Coefficient of Friction Archived March 8, 2009, at the Wayback Machine. EngineersHandbook.com

[Barrett-1990-49] 49.00 ^49.01 ^49.02 ^49.03 ^49.04 ^49.05 ^49.06 ^49.07 ^49.08 ^49.09 ^49.10 ^49.11 ^49.12 ^49.13 ^49.14 ^49.15 ^49.16 Barrett, Richard T. (1 March 1990). "(NASA-RP-1228) Fastener Design Manual". NASA Technical Reports Server. NASA Lewis Research Center: 16. hdl:2060/19900009424. Archived from the original on 7 October 2024. Retrieved 3 August 2020.

[Coefficients_of_Friction_of_Human_Joints-50] 50.0 ^50.1 "Coefficients of Friction of Human Joints". Archived from the original on 2024-10-07. Retrieved 2015-04-27.

[The_Engineering_Toolbox-51] 51.0 ^51.1 ^51.2 ^51.3 ^51.4 ^51.5 ^51.6 ^51.7 ^51.8 "The Engineering Toolbox: Friction and Coefficients of Friction". Archived from the original on 2013-12-03. Retrieved 2008-11-23.

[:0-52] 52.0 ^52.1 ^52.2 Bowden, F P; Tabor, D (2001-02-08). The Friction and Lubrication of Solids. Oxford University PressOxford. doi:10.1093/oso/9780198507772.001.0001. ISBN 978-0-19-850777-2.

[53] Rigney, D. A. (1997). "Comments on the sliding wear of metals". Tribology International. 30 (5): 361–367. doi:10.1016/S0301-679X(96)00065-5.

[54] Stott, F. H. (1998). "The role of oxidation in the wear of alloys". Tribology International. 31 (1–3): 61–71. doi:10.1016/S0301-679X(98)00008-5.

[55] Hutchings, I. M. (1992). Tribology: Friction and Wear of Engineering Materials. Butterworth-Heinemann. ISBN 9780750610544.

[56] Zum Gahr, K.-H. (1987). Microstructure and Wear of Materials. Elsevier. ISBN 9780444427540.

[57] Jiang, J.; Stott, F. H.; Stack, M. M. (1998). "The role of triboparticulates in dry sliding wear". Tribology International. 31 (5): 245–256. doi:10.1016/S0301-679X(98)00027-9.

[58] Straffelini, G.; Trabucco, D.; Molinari, A. (2001). "Oxidative wear of heat-treated steels". Wear. 250 (1–12): 485–491. doi:10.1016/S0043-1648(01)00661-5.

[59] Tang, L.; Gao, C.; Huang, J.; Zhang, H.; Chang, W. (2013). "Dry sliding friction and wear behaviour of hardened AISI D2 tool steel with different hardness levels". Tribology International. 66: 165–173. doi:10.1016/j.triboint.2013.05.006.

[60] Zhang, Q. Y.; Chen, K. M.; Wang, L.; Wang, S. Q. (2013). "Characteristics of oxidative wear and oxidative mild wear". Tribology International. 61: 214–223. doi:10.1016/j.triboint.2013.01.003.

[61] Jourani, A.; Bouvier, S. (2015). "Friction and Wear Mechanisms of 316L Stainless Steel in Dry Sliding Contact: Effect of Abrasive Particle Size". Tribology Transactions. 58 (1): 131–139. doi:10.1080/10402004.2014.955229.

[62] Straffelini, G.; Molinari, A. (1999). "Dry sliding wear of Ti–6Al–4V alloy as influenced by the counterface and sliding conditions". Wear. 236 (1–2): 328–338. doi:10.1016/S0043-1648(99)00292-6.

[63] Zhang, Q. Y.; Chen, K. M.; Wang, L.; Wang, S. Q. (2016). "Investigation on tribo-layers and their function of a titanium alloy during dry sliding". Tribology International. 94: 541–549. doi:10.1016/j.triboint.2015.10.032.

[64] Rigney, D. A. (1997). "Comments on the sliding wear of metals". Tribology International. 30 (5): 361–367. doi:10.1016/S0301-679X(96)00065-5.

[65] Stott, F. H. (1998). "The role of oxidation in the wear of alloys". Tribology International. 31 (1–3): 61–71. doi:10.1016/S0301-679X(97)00060-0.

[66] Balitskii, O.I.; Kolesnikov, V.O.; Ivaskevych, L.M.; Havrylyuk, M.R. (2023). "The Influence of Specific Features of Load and Hydrogen Charging on Steel Tribotechnical Properties". Materials Science. 58: 502–512. doi:10.1007/s11003-023-00691-5.

[67] Haslinger, J.; Nedlec, J.C. (1983). "Approximation of the Signorini problem with friction, obeying the Coulomb law" (PDF). Mathematical Methods in the Applied Sciences. 5 (1): 422–437. Bibcode:1983MMAS....5..422H. doi:10.1002/mma.1670050127. hdl:10338.dmlcz/104086. Archived (PDF) from the original on 2024-10-07. Retrieved 2019-09-19.

[68] Alart, P.; Curnier, A. (1991). "A mixed formulation for frictional contact problems prone to Newton like solution method" (PDF). Computer Methods in Applied Mechanics and Engineering. 92 (3): 353–375. Bibcode:1991CMAME..92..353A. doi:10.1016/0045-7825(91)90022-X. Archived (PDF) from the original on 2024-10-07. Retrieved 2024-03-29.

[69] Acary, V.; Cadoux, F.; Lemaréchal, C.; Malick, J. (2011). "A formulation of the linear discrete Coulomb friction problem via convex optimization". Journal of Applied Mathematics and Mechanics. 91 (2): 155–175. Bibcode:2011ZaMM...91..155A. doi:10.1002/zamm.201000073. S2CID 17280625. Archived from the original on 2024-10-07. Retrieved 2018-04-20.

[70] De Saxcé, G.; Feng, Z.-Q. (1998). "The bipotential method: A constructive approach to design the complete contact law with friction and improved numerical algorithms". Mathematical and Computer Modelling. 28 (4): 225–245. doi:10.1016/S0895-7177(98)00119-8.

[71] Simo, J.C.; Laursen, T.A. (1992). "An augmented lagrangian treatment of contact problems involving friction". Computers and Structures. 42 (2): 97–116. doi:10.1016/0045-7949(92)90540-G.

[72] Acary, V.; Brogliato, B. (2008). Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and Electronics. 35. Springer Verlag Heidelberg.

[73] Bigoni, D. (2012). Nonlinear Solid Mechanics: Bifurcation Theory and Material Instability. Cambridge University Press. ISBN 978-1-107-02541-7.

[74] Adams, G.G. (1995). "Self-excited oscillations of two elastic half-spaces sliding with a constant coefficient of friction". Journal of Applied Mechanics. 62 (4): 867–872. Bibcode:1995JAM....62..867A. doi:10.1115/1.2896013.

[75] Martins, J. A. C.; Guimara˜es, J.; Faria, L. O. (1995). "Dynamic surface solutions in linear elasticity and viscoelasticity with frictional boundary conditions". Journal of Vibration and Acoustics. 117 (4): 445–451. doi:10.1115/1.2874477.

[76] Nosonovsky, Mikhail; Adams, George G. (2004). "Vibration and stability of frictional sliding of two elastic bodies with a wavy contact interface". Journal of Applied Mechanics. 71 (2): 154–161. Bibcode:2004JAM....71..154N. doi:10.1115/1.1653684.

[77] Flint, J.; Hultén, Johan (2002). "Lining-deformation-induced modal coupling as squeal generator in a distributed parameter disk brake model". Journal of Sound and Vibration. 254 (1): 1–21. Bibcode:2002JSV...254....1F. doi:10.1006/jsvi.2001.4052.

[78] Kröger, Matthias; Neubauer, Marcus; Popp, Karl (2008). "Experimental investigation on the avoidance of self-excited vibrations". Phil. Trans. R. Soc. A. 366 (1866): 785–810. Bibcode:2008RSPTA.366..785K. doi:10.1098/rsta.2007.2127. PMID 17947204. S2CID 16395796.

[79] Rice, J. R.; Ruina, A. L. (1983). "Stability of Steady Frictional Slipping" (PDF). Journal of Applied Mechanics. 50 (2): 343–349. Bibcode:1983JAM....50..343R. CiteSeerX 10.1.1.161.5207. doi:10.1115/1.3167042. Archived (PDF) from the original on 2010-06-22.

[80] Bigoni, D.; Noselli, G. (2011). "Experimental evidence of flutter and divergence instabilities induced by dry friction". Journal of the Mechanics and Physics of Solids. 59 (10): 2208–2226. Bibcode:2011JMPSo..59.2208B. CiteSeerX 10.1.1.700.5291. doi:10.1016/j.jmps.2011.05.007. Archived from the original on 2020-08-18. Retrieved 2011-11-30.

[81] Nosonovsky, Michael (2013). Friction-Induced Vibrations and Self-Organization: Mechanics and Non-Equilibrium Thermodynamics of Sliding Contact. CRC Press. p. 333. ISBN 978-1-4665-0401-1.

[82] Ternes, Markus; Lutz, Christopher P.; Hirjibehedin, Cyrus F.; Giessibl, Franz J.; Heinrich, Andreas J. (2008-02-22). "The Force Needed to Move an Atom on a Surface" (PDF). Science. 319 (5866): 1066–1069. Bibcode:2008Sci...319.1066T. doi:10.1126/science.1150288. PMID 18292336. S2CID 451375. Archived (PDF) from the original on 2018-07-20.

[Deng-2012-83] 83.0 ^83.1 Deng, Zhao; et al. (October 14, 2012). "Adhesion-dependent negative friction coefficient on chemically modified graphite at the nanoscale". Nature. 11 (12): 1032–7. Bibcode:2012NatMa..11.1032D. doi:10.1038/nmat3452. PMID 23064494.
"At the nanoscale, graphite can turn friction upside down". R&D Magazine. 2012-10-17. Archived from the original on 2013-07-31.

[84] "At the nanoscale, graphite can turn friction upside down". R&D Magazine. 2012-10-17. Archived from the original on 2013-07-31.

[84] Makkonen, L (2012). "A thermodynamic model of sliding friction". AIP Advances. 2 (1): 012179. Bibcode:2012AIPA....2a2179M. doi:10.1063/1.3699027.

[85] Lua error in package.lua at line 80: module 'Module:Citation/CS1/Suggestions' not found.

[86] Müser, Martin H. (2015), "Theoretical Studies of Superlubricity", in Gnecco, Enrico; Meyer, Ernst (eds.), Fundamentals of Friction and Wear on the Nanoscale, NanoScience and Technology, Cham: Springer International Publishing, pp. 209–232, Bibcode:2015ffwn.book..209M, doi:10.1007/978-3-319-10560-4_11, ISBN 978-3-319-10559-8, retrieved 2025-04-25

[87] Hamrock, Bernard J. (2004). Fundamentals of fluid film lubrication. Steven R. Schmid, Bo O. Jacobson (2nd ed.). New York: Marcel Dekker. ISBN 0-8247-5120-5. OCLC 55739786.

[:1-88] 88.0 ^88.1 White, Frank (2011). Fluid Mechanics. New York City, NY: McGraw-Hill. pp. 477–478. ISBN 9780071311212.

[89] Al'shitz, V A; Indenbom, V L (1975-01-31). "Dynamic dragging of dislocations". Soviet Physics Uspekhi. 18 (1): 1–20. doi:10.1070/PU1975v018n01ABEH004689. ISSN 0038-5670.

[90] Kaganov, Moisei I; Kravchenko, Ya V; Natsik, V D (1974-06-30). "Dislocation dragging by electrons in metals". Soviet Physics Uspekhi. 16 (6): 878–891. doi:10.1070/PU1974v016n06ABEH004096. ISSN 0038-5670.

[91] Nowick, A. S. (1953-01-01). "Internal friction in metals". Progress in Metal Physics. 4: 1–70. doi:10.1016/0502-8205(53)90014-8. ISSN 0502-8205.

[92] Zdaniewski, Wieslaw A.; Rindone, Guy E.; Day, Delbert E. (1979-04-01). "The internal friction of glasses". Journal of Materials Science. 14 (4): 763–775. doi:10.1007/BF00550707. ISSN 1573-4803.

[93] Einstein, A. (1909). On the development of our views concerning the nature and constitution of radiation. Translated in: The Collected Papers of Albert Einstein, vol. 2 (Princeton University Press, Princeton, 1989). Princeton, NJ: Princeton University Press. p. 391.

[94] Silliman, Benjamin (1871) Principles of Physics, Or Natural Philosophy, Ivison, Blakeman, Taylor & company publishers

[95] Butt, Hans-Jürgen; Graf, Karlheinz and Kappl, Michael (2006) Physics and Chemistry of Interfaces, Wiley, ISBN 3-527-40413-9

[96] Hogan, C. Michael (1973). "Analysis of highway noise". Water, Air, & Soil Pollution. 2 (3): 387–392. Bibcode:1973WASP....2..387H. doi:10.1007/BF00159677. S2CID 109914430.

[97] Olson, Karl P; Marks, Laurence D (2025-10-01). "Is triboelectricity confusing, confused or complex?". Reports on Progress in Physics. 88 (10): Section 4.1. doi:10.1088/1361-6633/ae08cb. ISSN 0034-4885.

[98] Burgo, Thiago A. L.; Silva, Cristiane A.; Balestrin, Lia B. S.; Galembeck, Fernando (2013-08-12). "Friction coefficient dependence on electrostatic tribocharging". Scientific Reports. 3 (1): 2384. doi:10.1038/srep02384. ISSN 2045-2322. PMC 3740278. PMID 23934227.

[99] Huang, Xuanyu; Xiang, Xiaojian; Nie, Jinhui; Peng, Deli; Yang, Fuwei; Wu, Zhanghui; Jiang, Haiyang; Xu, Zhiping; Zheng, Quanshui (2021-04-15). "Microscale Schottky superlubric generator with high direct-current density and ultralong life". Nature Communications. 12 (1): 2268. doi:10.1038/s41467-021-22371-1. ISSN 2041-1723. PMC 8050059. PMID 33859180.

[100] Valentin L. Popov; Lars Voll; Stephan Kusche; Qiang Li; Svetlana V. Rozhkova (2018). "Generalized master curve procedure for elastomer friction taking into account dependencies on velocity, temperature and normal force". Tribology International. 120: 376–380. arXiv:1604.03407. doi:10.1016/j.triboint.2017.12.047. S2CID 119288819.

[101] "How Do Car Brakes Work?". Wonderopolis. Archived from the original on October 7, 2024. Retrieved November 4, 2018.

[Stevens,_A-102] Iskander, R.; Stevens, A. "Effectiveness of the Application of High Friction Surfacing-Crash-Reduction.pdf" (PDF). Archived (PDF) from the original on 2017-09-03. Retrieved 2017-09-03.

[103] "How Does Lighting A Match Work?". curiosity.com. Curiosity. November 11, 2015. Archived from the original on November 5, 2018. Retrieved November 4, 2018.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

[28]

[29]

[30]

[31]

[32]

[33]

[34]

[35]

[36]

[37]

[38]

[39]

[40]

[41]

[42]

[43]

[44]

[45]

[46]

[47]

[48]

[49]

[50]

[51]

[52]

[53]

[54]

[55]

[56]

[57]

[58]

[59]

[60]

[61]

[62]

[63]

[64]

[65]

[66]

[67]

[68]

[69]

[70]

[71]

[72]

[73]

[74]

[75]

[76]

[77]

[78]

[79]

[80]

[81]

[82]

[83]

[84]

[85]

[86]

[87]

[88]

[89]

[90]

[91]

[92]

[93]

[94]

[95]

[96]

[97]

[98]

[99]

[100]

Friction: Difference between revisions

Latest revision as of 09:05, 25 April 2026

Contents

Types

History

Dry friction

Static friction

Kinetic friction

Role of the normal force

Role of angle

Coefficient of friction

Approximate coefficients of friction

Sources of friction

Breakdown of the Coulomb model

Dry friction and instabilities

Friction at the nanoscale

Fluid friction

Lubricated friction

Skin friction

Internal friction

Radiation friction

Other types of friction

Rolling resistance

Braking friction

Triboelectric effect

Belt friction

Friction reduction

Devices

Lubricants

Applications

Transportation

Measurement

Household usage

See also

References

External links

Navigation menu

@@ Line 2: / Line 2: @@
 {{pp|small=yes}}
 {{Other uses}}
-[[File:Friction Animation 2 Blocks.gif|thumb|360px|Friction between two objects. The blue one has more friction against the sloped surface than the green one.]]
+[[File:Friction Animation 2 Blocks.gif|thumb|360px|The blue plate has more friction on the sloped surface than the green one, and thus slides down more slowly]]
 {{Classical mechanics|cTopic=core}}
 [[File:Friction between surfaces.jpg|300x150px|thumb|Simulated blocks with [[fractal]] rough surfaces, exhibiting static frictional interactions<ref name="Hanaor-2016" />]]
-'''Friction''' is the [[force]] resisting the relative motion of solid surfaces, fluid layers, and material elements [[sliding (motion)|sliding]] against each other.<ref name="Merriam-Webster.com Dictionary">{{MerriamWebsterDictionary|friction}}</ref><ref name="www.britannica.com-2024">{{Cite web |date=2024-09-11 |title=Friction {{!}} Definition, Types, & Formula {{!}} Britannica |url=https://www.britannica.com/science/friction |access-date=2024-10-07 |website=www.britannica.com |language=en |archive-date=2024-09-16 |archive-url=https://web.archive.org/web/20240916091157/https://www.britannica.com/science/friction |url-status=live }}</ref> Types of friction include dry, fluid, lubricated, skin, and internal – an incomplete list. The study of the processes involved is called [[tribology]], and has a history of more than 2000 years.<ref name="Ghose-2022">{{Cite web |last1=Ghose |first1=Tia |last2=published |first2=Ailsa Harvey |date=2022-02-08 |title=What is Friction? |url=https://www.livescience.com/37161-what-is-friction.html |access-date=2024-10-07 |website=livescience.com |language=en |archive-date=2024-05-20 |archive-url=https://web.archive.org/web/20240520084531/https://www.livescience.com/37161-what-is-friction.html |url-status=live }}</ref>
+'''Friction''' is the [[force]] resisting the relative motion of solid surfaces, fluid layers, and material elements [[sliding (motion)|sliding]] or grinding against each other.<ref name="Merriam-Webster.com Dictionary">{{MerriamWebsterDictionary|friction}}</ref><ref name="www.britannica.com-2024">{{Cite web |date=2024-09-11 |title=Friction {{!}} Definition, Types, & Formula {{!}} Britannica |url=https://www.britannica.com/science/friction |access-date=2024-10-07 |website=www.britannica.com |language=en |archive-date=2024-09-16 |archive-url=https://web.archive.org/web/20240916091157/https://www.britannica.com/science/friction |url-status=live }}</ref> Types of friction include dry, fluid, lubricated, skin, and internal – an incomplete list. The study of the processes involved is called [[tribology]], and has a history of more than 2,000 years.<ref name="Ghose-2022">{{Cite web |last1=Ghose |first1=Tia |last2=published |first2=Ailsa Harvey |date=2022-02-08 |title=What is Friction? |url=https://www.livescience.com/37161-what-is-friction.html |access-date=2024-10-07 |website=livescience.com |language=en |archive-date=2024-05-20 |archive-url=https://web.archive.org/web/20240520084531/https://www.livescience.com/37161-what-is-friction.html |url-status=live }}</ref>
-Friction can have dramatic consequences, as illustrated by the use of friction created by [[Fire making#Friction|rubbing pieces of wood together to start a fire]]. Another important consequence of many types of friction can be [[wear]], which may lead to performance degradation or damage to components. It is known that frictional energy losses account for about 20% of the total energy expenditure of the world.<ref name="Mitchell-2012">{{cite journal |author-last=Mitchell |author-first=Luke |date=November 2012 |editor-last=Ward |editor-first=Jacob |title=The Fiction of Nonfriction |journal=[[Popular Science]] |series=No. 5 |volume=281 |issue=November 2012 |page=40 }}</ref><ref>{{Cite web |last1=Ghose |first1=Tia |last2=published |first2=Ailsa Harvey |date=2022-02-08 |title=What is Friction? |url=https://www.livescience.com/37161-what-is-friction.html |access-date=2024-10-07 |website=livescience.com |language=en |archive-date=2024-05-20 |archive-url=https://web.archive.org/web/20240520084531/https://www.livescience.com/37161-what-is-friction.html |url-status=live }}</ref>
+Friction can have dramatic consequences, as illustrated by the use of friction created by [[Fire making#Friction|rubbing pieces of wood together to start a fire]]. Another important consequence of many types of friction can be [[wear]], which may lead to performance degradation or damage to components. It is known that frictional energy losses account for about 20% of the total energy expenditure of the world.<ref name="Mitchell-2012">{{cite journal |author-last=Mitchell |author-first=Luke |date=November 2012 |editor-last=Ward |editor-first=Jacob |title=The Fiction of Nonfriction |journal=[[Popular Science]] |series=No. 5 |volume=281 |issue=November 2012 |page=40 }}</ref><ref name="Ghose-2022"/>
-As briefly discussed later, there are many different contributors to the retarding force in friction, ranging from [[Asperity (materials science)|asperity]] deformation to the generation of [[Triboelectric effect|charges]] and changes in [[Microstructure|local structure]]. When two bodies in contact move relative to each other, due to these various contributors some mechanical energy is transformed to [[heat]], the [[Gibbs free energy|free energy]] of structural changes, and other types of [[dissipation]]. The total dissipated energy per unit distance moved is the retarding frictional force. The complexity of the interactions involved makes the calculation of friction from [[first principle]]s difficult, and it is often easier to use [[empirical method]]s for analysis and the development of theory.<ref name="www.britannica.com-2024" /><ref name="Merriam-Webster.com Dictionary" />
+There are many different contributors to the retarding force in friction, ranging from [[Asperity (materials science)|asperity]] deformation to the generation of [[Triboelectric effect|charges]] and changes in [[Microstructure|local structure]]. When two bodies in contact move relative to each other, due to these various contributors some mechanical energy is transformed to [[heat]], the [[Gibbs free energy|free energy]] of structural changes, and other types of [[dissipation]]. The total dissipated energy per unit distance moved is the '''retarding frictional force''', ''F{{=}}E/d''. The complexity of the interactions involved makes the calculation of friction from [[first principle]]s difficult, and it is often easier to use [[empirical method]]s for analysis and the development of theory.<ref name="www.britannica.com-2024" /><ref name="Merriam-Webster.com Dictionary" />
 ==Types==
@@ Line 47: / Line 47: @@
 | publisher = Oxford University Press
 | page = 713
-| url = http://ruina.tam.cornell.edu/Book/RuinaPratapNoProblems.pdf
+| url = https://ruina.tam.cornell.edu/Book/RuinaPratapNoProblems.pdf
 | access-date = 2008-12-20
 | archive-date = 2019-05-25
@@ Line 74: / Line 74: @@
 Many ancient authors including [[Aristotle]], [[Vitruvius]], and [[Pliny the Elder]], were interested in the cause and mitigation of friction.<ref name="Chatterjee-2008">{{cite thesis |last=Chatterjee |first=Sudipta |year=2008 |title=Tribological Properties of Pseudo-elastic Nickel-titanium |publisher=University of California |isbn=978-0-549-84437-2 |pages=11–12 |url=https://books.google.com/books?id=rX6xfoEaYtQC&pg=PA12 |quote=Classical Greek philosophers like Aristotle, Pliny the Elder and Vitruvius wrote about the existence of friction, the effect of lubricants and the advantages of metal bearings around 350 B.C. }}{{Dead link|date=May 2024 |bot=InternetArchiveBot |fix-attempted=yes }}</ref> They were aware of differences between static and kinetic friction with [[Themistius]] stating in 350 {{Smallcaps2|A.D.}} that "it is easier to further the motion of a moving body than to move a body at rest".<ref name="Chatterjee-2008"/><ref>{{cite book |last1=Fishbane |first1=Paul M. |last2=Gasiorowicz |first2=Stephen |last3=Thornton |first3=Stephen T. |year=1993 |title=Physics for Scientists and Engineers |edition=Extended |volume=I |publisher=Prentice Hall |location=Englewood Cliffs, New Jersey |isbn=978-0-13-663246-7 |page=135 |quote=Themistius first stated around 350 {{Smallcaps2|{{sic|B.C.|expected=A.D.}}}} that kinetic friction is weaker than the maximum value of static friction.}}</ref><ref>{{cite book |last=Hecht |first=Eugene |date=2003 |title=Physics: Algebra/Trig |edition=3rd |publisher=Cengage Learning |isbn=978-0-534-37729-8}}</ref><ref>{{cite book |last=Sambursky |first=Samuel |date=2014 |title=The Physical World of Late Antiquity |publisher=Princeton University Press |isbn=978-1-4008-5898-9 |pages=65–66 |url=https://books.google.com/books?id=Yvz_AwAAQBAJ&pg=PA65 |access-date=2016-11-01 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007091047/https://books.google.com/books?id=Yvz_AwAAQBAJ&pg=PA65#v=onepage&q&f=false |url-status=live }}</ref>
-The classic laws of sliding friction were discovered by [[Leonardo da Vinci]]  in 1493, a pioneer in [[tribology]], but the laws documented in his notebooks were not published and remained unknown.<ref name="Dowson-1997" >{{cite book |last=Dowson |first=Duncan |year=1997 |title=History of Tribology |edition=2nd |publisher=Professional Engineering Publishing |isbn=978-1-86058-070-3}}</ref><ref name="Armstrong-Hélouvry-1991" >{{cite book |last=Armstrong-Hélouvry |first=Brian |year=1991 |title=Control of machines with friction |publisher=Springer |location=US |page=10 |url=https://books.google.com/books?id=0zk_zI3xACgC&pg=PA10 |isbn=978-0-7923-9133-3 |access-date=2020-06-07 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007090932/https://books.google.com/books?id=0zk_zI3xACgC&pg=PA10#v=onepage&q&f=false |url-status=live }}</ref><ref name="van Beek">{{cite web |last=van Beek |first=Anton |title=History of Science Friction |publisher=tribology-abc.com |url=http://www.tribology-abc.com/abc/history.htm |access-date=2011-03-24 |archive-date=2011-08-07 |archive-url=https://web.archive.org/web/20110807185735/http://www.tribology-abc.com/abc/history.htm |url-status=live }}</ref><ref>{{cite journal |last=Hutchings |first=Ian M.  |date=2016 |title=Leonardo da Vinci's studies of friction |journal=Wear  |volume=360–361  |pages=51–66  |doi=10.1016/j.wear.2016.04.019 |url=http://www.ifm.eng.cam.ac.uk/uploads/Hutchings_Leonardo_Friction_2016_v2.pdf |archive-url=https://web.archive.org/web/20160803211351/http://www.ifm.eng.cam.ac.uk/uploads/Hutchings_Leonardo_Friction_2016_v2.pdf |archive-date=2016-08-03 |url-status=live }}</ref><ref>{{cite journal |last=Hutchings |first=Ian M. |date=2016-08-15 |title=Leonardo da Vinci's studies of friction |journal=Wear |doi=10.1016/j.wear.2016.04.019 |volume=360–361 |pages=51–66 |url=https://www.repository.cam.ac.uk/handle/1810/255781 |access-date=2019-07-09 |archive-date=2021-09-18 |archive-url=https://web.archive.org/web/20210918100744/https://www.repository.cam.ac.uk/handle/1810/255781 |url-status=live }}</ref><ref>{{cite web |last=Kirk |first=Tom |date=July 22, 2016 |title=Study reveals Leonardo da Vinci's 'irrelevant' scribbles mark the spot where he first recorded the laws of friction |website=phys.org |url=http://phys.org/news/2016-07-reveals-leonardo-da-vinci-irrelevant.html |access-date=2016-07-26 |archive-date=2016-07-25 |archive-url=https://web.archive.org/web/20160725081116/http://phys.org/news/2016-07-reveals-leonardo-da-vinci-irrelevant.html |url-status=live }}</ref> These laws were rediscovered by [[Guillaume Amontons]] in 1699<ref name="Popova-2015">{{Cite journal|last1=Popova|first1=Elena|last2=Popov|first2=Valentin L.|date=2015-06-01|title=The research works of Coulomb and Amontons and generalized laws of friction|journal=Friction|language=en|volume=3|issue=2|pages=183–190|doi=10.1007/s40544-015-0074-6|doi-access=free}}</ref> and became known as Amonton's three laws of dry friction.   Amontons presented the nature of friction in terms of surface irregularities and the force required to raise the weight pressing the surfaces together. This view was further elaborated by [[Bernard Forest de Bélidor]]<ref>[[Bernard Forest de Bélidor|Forest de Bélidor, Bernard]]. "[http://gallica.bnf.fr/ark:/12148/btv1b53032692q.r=Belidor%2C%20Bernard%20Forest%20de Richtige Grund-Sätze der Friction-Berechnung] {{Webarchive|url=https://web.archive.org/web/20210427035523/http://gallica.bnf.fr/ark:/12148/btv1b53032692q.r=Belidor%2C%20Bernard%20Forest%20de |date=2021-04-27 }}" ("Correct Basics of Friction Calculation"), 1737, (in [[German language|German]])</ref> and [[Leonhard Euler]] (1750), who derived the [[angle of repose]] of a weight on an inclined plane and first distinguished between static and kinetic friction.<ref>{{cite web
+The classic laws of sliding friction were discovered by [[Leonardo da Vinci]] in 1493, a pioneer in [[tribology]], but the laws documented in his notebooks were not published and remained unknown.<ref name="Dowson-1997" >{{cite book |last=Dowson |first=Duncan |year=1997 |title=History of Tribology |edition=2nd |publisher=Professional Engineering Publishing |isbn=978-1-86058-070-3}}</ref><ref name="Armstrong-Hélouvry-1991" >{{cite book |last=Armstrong-Hélouvry |first=Brian |year=1991 |title=Control of machines with friction |publisher=Springer |location=US |page=10 |url=https://books.google.com/books?id=0zk_zI3xACgC&pg=PA10 |isbn=978-0-7923-9133-3 |access-date=2020-06-07 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007090932/https://books.google.com/books?id=0zk_zI3xACgC&pg=PA10#v=onepage&q&f=false |url-status=live }}</ref><ref name="van Beek">{{cite web |last=van Beek |first=Anton |title=History of Science Friction |publisher=tribology-abc.com |url=http://www.tribology-abc.com/abc/history.htm |access-date=2011-03-24 |archive-date=2011-08-07 |archive-url=https://web.archive.org/web/20110807185735/http://www.tribology-abc.com/abc/history.htm |url-status=live }}</ref><ref>{{cite journal |last=Hutchings |first=Ian M.  |date=2016 |title=Leonardo da Vinci's studies of friction |journal=Wear  |volume=360–361  |pages=51–66  |doi=10.1016/j.wear.2016.04.019 |url=http://www.ifm.eng.cam.ac.uk/uploads/Hutchings_Leonardo_Friction_2016_v2.pdf |archive-url=https://web.archive.org/web/20160803211351/http://www.ifm.eng.cam.ac.uk/uploads/Hutchings_Leonardo_Friction_2016_v2.pdf |archive-date=2016-08-03 |url-status=live }}</ref><ref>{{cite journal |last=Hutchings |first=Ian M. |date=2016-08-15 |title=Leonardo da Vinci's studies of friction |journal=Wear |doi=10.1016/j.wear.2016.04.019 |volume=360–361 |pages=51–66 |url=https://www.repository.cam.ac.uk/handle/1810/255781 |access-date=2019-07-09 |archive-date=2021-09-18 |archive-url=https://web.archive.org/web/20210918100744/https://www.repository.cam.ac.uk/handle/1810/255781 |url-status=live }}</ref><ref>{{cite web |last=Kirk |first=Tom |date=July 22, 2016 |title=Study reveals Leonardo da Vinci's 'irrelevant' scribbles mark the spot where he first recorded the laws of friction |website=phys.org |url=http://phys.org/news/2016-07-reveals-leonardo-da-vinci-irrelevant.html |access-date=2016-07-26 |archive-date=2016-07-25 |archive-url=https://web.archive.org/web/20160725081116/http://phys.org/news/2016-07-reveals-leonardo-da-vinci-irrelevant.html |url-status=live }}</ref> These laws were rediscovered by [[Guillaume Amontons]] in 1699<ref name="Popova-2015">{{Cite journal|last1=Popova|first1=Elena|last2=Popov|first2=Valentin L.|date=2015-06-01|title=The research works of Coulomb and Amontons and generalized laws of friction|journal=Friction|language=en|volume=3|issue=2|pages=183–190|doi=10.1007/s40544-015-0074-6|doi-access=free}}</ref> and became known as Amonton's three laws of dry friction. Amontons presented the nature of friction in terms of surface irregularities and the force required to raise the weight pressing the surfaces together. This view was further elaborated by [[Bernard Forest de Bélidor]]<ref>[[Bernard Forest de Bélidor|Forest de Bélidor, Bernard]]. "[http://gallica.bnf.fr/ark:/12148/btv1b53032692q.r=Belidor%2C%20Bernard%20Forest%20de Richtige Grund-Sätze der Friction-Berechnung] {{Webarchive|url=https://web.archive.org/web/20210427035523/http://gallica.bnf.fr/ark:/12148/btv1b53032692q.r=Belidor%2C%20Bernard%20Forest%20de |date=2021-04-27 }}" ("Correct Basics of Friction Calculation"), 1737, (in [[German language|German]])</ref> and [[Leonhard Euler]] (1750), who derived the [[angle of repose]] of a weight on an inclined plane and first distinguished between static and kinetic friction.<ref>{{cite web
    | title = Leonhard Euler
    | work = Friction Module
@@ Line 100: / Line 100: @@
   }}</ref> Microscopic forces cause surfaces to stick together; he proposed that friction was the force necessary to tear the adhering surfaces apart.
-The understanding of friction was further developed by [[Charles-Augustin de Coulomb]] (1785).<ref name="Popova-2015"/> Coulomb investigated the influence of four main factors on friction: the nature of the materials in contact and their surface coatings; the extent of the surface area; the normal pressure (or load); and the length of time that the surfaces remained in contact (time of repose).<ref name="Dowson-1997" /> Coulomb further considered the influence of sliding velocity, temperature and humidity, in order to decide between the different explanations on the nature of friction that had been proposed. The distinction between static and dynamic friction is made in Coulomb's friction law (see below), although this distinction was already drawn by [[Johann Andreas von Segner]] in 1758.<ref name="Dowson-1997" />
+The understanding of friction was further developed by [[Charles-Augustin de Coulomb]] (1785).<ref name="Popova-2015"/> Coulomb investigated the influence of four main factors on friction: the nature of the materials in contact and their surface coatings, the extent of the surface area, the normal pressure (or load), and the length of time that the surfaces remained in contact (time of repose).<ref name="Dowson-1997" /> Coulomb further considered the influence of sliding velocity, temperature and humidity, in order to decide between the different explanations on the nature of friction that had been proposed. The distinction between static and dynamic friction is made in Coulomb's friction law (see below), although this distinction was already drawn by [[Johann Andreas von Segner]] in 1758.<ref name="Dowson-1997" />
 The effect of the time of repose was explained by [[Pieter van Musschenbroek]] (1762) by considering the surfaces of fibrous materials, with fibers meshing together, which takes a finite time in which the friction increases.
@@ Line 110: / Line 110: @@
 [[Arthur Jules Morin]] (1833) developed the concept of sliding versus rolling friction.
-In 1842, [[Julius Robert Mayer]] frictionally generated heat in paper pulp and measured the temperature rise.<ref>Blundell, S.J., Blundell, K.M. (2006). ''Concepts in Thermal Physics'', Oxford University Press, Oxford UK, {{ISBN|978-0-19-856769-1}}, p. 106.</ref> In 1845, Joule published a paper entitled ''The Mechanical Equivalent of Heat'', in which he specified a numerical value for the amount of mechanical work required to "produce a unit of heat", based on the friction of an electric current passing through a resistor, and on the friction of a paddle wheel rotating in a vat of water.<ref>[[James Prescott Joule|Joule, J.P.]] (1845).{{cite journal|last=Joule|first=J.P.|title=On the Mechanical Equivalent of Heat|journal=Philosophical Transactions of the Royal Society of London|date=1850|volume=140|pages=61–82|doi=10.1098/rstl.1850.0004|url=https://archive.org/stream/philtrans00608634/00608634#page/n0/mode/2up|display-authors=0|doi-access=free}}
+In 1842, [[Julius Robert Mayer]] frictionally generated heat in paper pulp and measured the temperature rise.<ref>Blundell, S.J., Blundell, K.M. (2006). ''Concepts in Thermal Physics'', Oxford University Press, Oxford UK, {{ISBN|978-0-19-856769-1}}, p. 106.</ref> In 1845, Joule published a paper entitled "The Mechanical Equivalent of Heat", in which he specified a numerical value for the amount of mechanical work required to "produce a unit of heat", based on the friction of an electric current passing through a resistor, and on the friction of a paddle wheel rotating in a vat of water.<ref>[[James Prescott Joule|Joule, J.P.]] (1845).{{cite journal|last=Joule|first=J.P.|title=On the Mechanical Equivalent of Heat|journal=Philosophical Transactions of the Royal Society of London|date=1850|volume=140|pages=61–82|doi=10.1098/rstl.1850.0004|url=https://archive.org/stream/philtrans00608634/00608634#page/n0/mode/2up|display-authors=0|doi-access=free}}
 </ref>
-[[Osborne Reynolds]] (1866) derived the equation of viscous flow.  This completed the classic empirical model of friction (static, kinetic, and fluid) commonly used today in engineering.<ref name="Armstrong-Hélouvry-1991" /> In 1877,  [[Fleeming Jenkin]] and [[James Alfred Ewing|J. A. Ewing]] investigated the continuity between static and kinetic friction.<ref>[[Fleeming Jenkin]] & [[James Alfred Ewing]] (1877) "[https://www.biodiversitylibrary.org/item/121556#page/322/mode/1up On Friction between Surfaces moving at Low Speeds] {{Webarchive|url=https://web.archive.org/web/20210918100735/https://www.biodiversitylibrary.org/item/121556#page/322/mode/1up |date=2021-09-18 }}", ''[[Philosophical Magazine]]'' Series 5, volume 4, pp. 308–10; link from [[Biodiversity Heritage Library]]</ref>
+[[Osborne Reynolds]] (1866) derived the equation of viscous flow. This completed the classic empirical model of friction (static, kinetic, and fluid) commonly used today in engineering.<ref name="Armstrong-Hélouvry-1991" /> In 1877, [[Fleeming Jenkin]] and [[James Alfred Ewing|J. A. Ewing]] investigated the continuity between static and kinetic friction.<ref>[[Fleeming Jenkin]] & [[James Alfred Ewing]] (1877) "[https://www.biodiversitylibrary.org/item/121556#page/322/mode/1up On Friction between Surfaces moving at Low Speeds] {{Webarchive|url=https://web.archive.org/web/20210918100735/https://www.biodiversitylibrary.org/item/121556#page/322/mode/1up |date=2021-09-18 }}", ''[[Philosophical Magazine]]'' Series 5, volume 4, pp. 308–10; link from [[Biodiversity Heritage Library]]</ref>
 In 1907, [[George H. Bryan|G.H. Bryan]] published an investigation of the foundations of thermodynamics, ''Thermodynamics: an Introductory Treatise dealing mainly with First Principles and their Direct Applications''. He noted that for a rough body driven over a rough surface, the mechanical work done by the driver exceeds the mechanical work received by the surface. The lost work is accounted for by heat generated by friction.<ref>{{cite book |last1=Bryan |first1=George Hartley |title=Thermodynamics, an introductory treatise dealing mainly with first principles and their direct applications |url=https://archive.org/stream/thermodynamicsin00bryauoft/thermodynamicsin00bryauoft_djvu.txt |publisher=Leipzig, Teubner |pages=48–49 |date=1907 |access-date=23 June 2023}}</ref>
@@ Line 119: / Line 119: @@
 Over the years, for example in his 1879 thesis, but particularly in 1926, [[Planck]] advocated regarding the generation of heat by rubbing as the most specific way to define heat, and the prime example of an irreversible thermodynamic process.<ref name="Planck 1926">[[Max Planck|Planck, M.]] (1926). "Über die Begründung des zweiten Hauptsatzes der Thermodynamik", ''Sitzungsber. Preuss. Akad. Wiss., Phys. Math. Kl.'', 453—463.</ref>
-The focus of research during the 20th century has been to understand the physical mechanisms behind friction. [[Frank Philip Bowden]] and [[David Tabor (physicist)|David Tabor]] (1950) showed that, at a [[Microscopic scale|microscopic level]], the actual area of contact between surfaces is a very small fraction of the apparent area.<ref name="van Beek" />  This actual area of contact, caused by asperities increases with pressure. The development of the [[atomic force microscope]] (ca. 1986)  enabled scientists to study friction at the [[Atomic units|atomic scale]],<ref name="Armstrong-Hélouvry-1991" /> showing that, on that scale, dry friction is the product of the inter-surface [[shear stress]] and the contact area. These two discoveries explain Amonton's first law ''(below)''; the macroscopic proportionality between normal force and static frictional force between dry surfaces.
+The focus of research during the 20th century has been to understand the physical mechanisms behind friction. [[Frank Philip Bowden]] and [[David Tabor (physicist)|David Tabor]] (1950) showed that, at a [[Microscopic scale|microscopic level]], the actual area of contact between surfaces is a very small fraction of the apparent area.<ref name="van Beek" /> This actual area of contact, caused by asperities increases with pressure. The development of the [[atomic force microscope]] (ca. 1986) enabled scientists to study friction at the [[Atomic units|atomic scale]],<ref name="Armstrong-Hélouvry-1991" /> showing that, on that scale, dry friction is the product of the inter-surface [[shear stress]] and the contact area. These two discoveries explain Amonton's first law ''(below)''; the macroscopic proportionality between normal force and static frictional force between dry surfaces.
-==Laws of dry friction==
+==Dry friction==
-The elementary property of sliding (kinetic) friction were discovered by experiment in the 15th to 18th centuries and were expressed as three empirical laws:
+Dry friction resists relative lateral motion of two solid surfaces in contact. The two regimes of dry friction are 'static friction' ("[[stiction]]") between non-moving surfaces, and ''kinetic friction'' (sometimes called sliding friction or dynamic friction) between moving surfaces.
-* '''[[Guillaume Amontons#Amontons' Laws of Friction|Amontons']]  First Law''': The force of friction is directly proportional to the applied load.
-* '''Amontons' Second Law''': The force of friction is independent of the apparent area of contact.
-* '''[[Charles-Augustin de Coulomb|Coulomb]]'s Law of Friction''': Kinetic friction is independent of the sliding velocity.
-==Dry friction==
+The force of friction is always exerted in a direction that opposes movement (for kinetic friction) or potential movement (for static friction) between the two surfaces. For example, a [[curling]] stone sliding along the ice experiences a kinetic force slowing it down. For an example of potential movement, the drive wheels of an accelerating car experience a frictional force pointing forward; if they did not, the wheels would spin, and the rubber would slide backwards along the pavement. Note that it is not the direction of movement of the vehicle they oppose, it is the direction of (potential) sliding between tire and road.
-Dry friction resists relative lateral motion of two solid surfaces in contact. The two regimes of dry friction are 'static friction' ("[[stiction]]") between non-moving surfaces, and ''kinetic friction'' (sometimes called sliding friction or dynamic friction) between moving surfaces.
+The elementary property of sliding (kinetic) friction were discovered by experiment in the 15th to 18th century and were expressed as three empirical laws:
+*  [[Guillaume Amontons#Amontons' Laws of Friction|Amontons']] first law: The force of friction is directly proportional to the applied load.
+*  Amontons' second law: The force of friction is independent of the apparent area of contact.
+* [[Charles-Augustin de Coulomb|Coulomb]]'s law of friction: Kinetic friction is independent of the sliding velocity.
-Coulomb friction, named after [[Charles-Augustin de Coulomb]], is an approximate model used to calculate the force of dry friction. It is governed by the model:
+Coulomb friction, named after [[Charles-Augustin de Coulomb]], is an approximate model used to calculate the force of dry friction. It is governed by the model:<ref>{{cite book |last=Popov |first=Valentin L. |chapter=Coulomb's Law of Friction |date=2017 |title=Contact Mechanics and Friction |pages=151–172 |url=http://link.springer.com/10.1007/978-3-662-53081-8_10 |access-date=2025-09-06 |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |language=en |doi=10.1007/978-3-662-53081-8_10 |isbn=978-3-662-53080-1|url-access=subscription }}</ref>
 <math display="block">F_\mathrm{f} \leq \mu F_\mathrm{n},</math>
 where
-* <math>F_\mathrm{f}</math> is the force of friction exerted by each surface on the other.  It is parallel to the surface, in a direction opposite to the net applied force.
+* <math>F_\mathrm{f}</math> is the force of friction exerted by each surface on the other. It is parallel to the surface, in a direction opposite to the net applied force.
 * <math>\mu</math> is the coefficient of friction, which is an empirical property of the contacting materials,
 * <math>F_\mathrm{n}</math> is the [[normal force]] exerted by each surface on the other, directed perpendicular (normal) to the surface.
@@ Line 139: / Line 140: @@
 The Coulomb friction <math>F_\mathrm{f}</math> may take any value from zero up to <math>\mu F_\mathrm{n}</math>, and the direction of the frictional force against a surface is opposite to the motion that surface would experience in the absence of friction. Thus, in the static case, the frictional force is exactly what it must be in order to prevent motion between the surfaces; it balances the net force tending to cause such motion. In this case, rather than providing an estimate of the actual frictional force, the Coulomb approximation provides a threshold value for this force, above which motion would commence. This maximum force is known as [[Traction (engineering)|traction]].
-The force of friction is always exerted in a direction that opposes movement (for kinetic friction) or potential movement (for static friction) between the two surfaces. For example, a [[curling]] stone sliding along the ice experiences a kinetic force slowing it down. For an example of potential movement, the drive wheels of an accelerating car experience a frictional force pointing forward; if they did not, the wheels would spin, and the rubber would slide backwards along the pavement. Note that it is not the direction of movement of the vehicle they oppose, it is the direction of (potential) sliding between tire and road.
+===Static friction===<!-- Traction (engineering) links here -->
+[[File:Static kinetic friction vs time.png|400px|right|thumb|When the mass is not moving, the object experiences static friction. The friction increases as the applied force increases until the block moves. After the block moves, it experiences kinetic friction, which is less than the maximum static friction.]]
+{{Main|Stiction}}
+Static friction is friction between two or more solid objects that are not moving relative to each other. For example, static friction can prevent an object from sliding down a sloped surface. The coefficient of static friction, typically denoted as ''μ''<sub>s</sub>, is usually higher than the coefficient of kinetic friction. Static friction is considered to arise as the result of surface roughness features across multiple length scales at solid surfaces. These features, known as [[asperities]] are present down to nano-scale dimensions and result in true solid to solid contact existing only at a limited number of points accounting for only a fraction of the apparent or nominal contact area.<ref>[https://www.researchgate.net/publication/283675011_Static_friction_at_fractal_interfaces multi-scale origins of static friction] {{Webarchive|url=https://web.archive.org/web/20210918100737/https://www.researchgate.net/publication/283675011_Static_friction_at_fractal_interfaces |date=2021-09-18 }} 2016</ref> The linearity between applied load and true contact area, arising from asperity deformation, gives rise to the linearity between static frictional force and normal force, found for typical Amonton–Coulomb type friction.<ref>{{cite journal | author= Greenwood J.A. and JB Williamson| title= Contact of nominally flat surfaces | journal= Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences | volume=295 | issue=1442 | year=1966 | page= 300 | doi= 10.1098/rspa.1966.0242 | bibcode= 1966RSPSA.295..300G }}</ref>
+The static friction force must be overcome by an applied force before an object can move. The maximum possible friction force between two surfaces before sliding begins is the product of the coefficient of static friction and the normal force: <math>F_\text{max} = \mu_\mathrm{s} F_\text{n}</math>. When there is no sliding occurring, the friction force can have any value from zero up to <math>F_\text{max}</math>. Any force smaller than <math>F_\text{max}</math> attempting to slide one surface over the other is opposed by a frictional force of equal magnitude and opposite direction. Any force larger than <math>F_\text{max}</math> overcomes the force of static friction and causes sliding to occur. The instant sliding occurs, static friction is no longer applicable—the friction between the two surfaces is then called kinetic friction. However, an apparent static friction can be observed even in the case when the true static friction is zero.<ref>{{Cite journal|last1=Nakano|first1=K.|last2=Popov|first2=V. L.|date=2020-12-10|title=Dynamic stiction without static friction: The role of friction vector rotation| url=https://link.aps.org/doi/10.1103/PhysRevE.102.063001| journal=Physical Review E|volume=102|issue=6|article-number=063001| doi=10.1103/PhysRevE.102.063001|pmid=33466084 |bibcode=2020PhRvE.102f3001N |hdl=10131/00013921 |s2cid=230599544 |hdl-access=free|url-access=subscription}}</ref>
+An example of static friction is the force that prevents a car wheel from slipping as it rolls on the ground. Even though the wheel is in motion, the patch of the tire in contact with the ground is stationary relative to the ground, so it is static rather than kinetic friction. Upon slipping, the wheel friction changes to kinetic friction. An [[anti-lock braking system]] operates on the principle of allowing a locked wheel to resume rotating so that the car maintains static friction.
+The maximum value of static friction, when motion is impending, is sometimes referred to as '''limiting friction''',<ref name="Bhavikatti-1994">{{cite book
+| url = https://books.google.com/books?id=4wkLl4NvmWAC&pg=PA112
+| title = Engineering Mechanics
+| last = Bhavikatti
+| first = S.S.
+| author2 = K.G. Rajashekarappa
+| page = 112
+| access-date = 2007-10-21
+| publisher = New Age International
+| isbn = 978-81-224-0617-7
+| year = 1994
+| archive-date = 2024-10-07
+| archive-url = https://web.archive.org/web/20241007091508/https://books.google.com/books?id=4wkLl4NvmWAC&pg=PA112#v=onepage&q&f=false
+| url-status = live
+}}</ref>
+although this term is not used universally.<ref name="Beer-1996" />
+===Kinetic friction===
+'''Kinetic friction''', also known as '''dynamic friction''' or '''sliding friction''', occurs when two objects are moving relative to each other and rub together (like a sled on the ground). The coefficient of kinetic friction is typically denoted as ''μ''<sub>k</sub>, and is usually less than the coefficient of static friction for the same materials.<ref>{{cite book|title=Statics: Analysis and Design of Systems in Equilibrium |publisher=Wiley and Sons|year=2005 |isbn=978-0-471-37299-8|page=618 |quote=In general, for given contacting surfaces, ''μ''<sub>k</sub> < ''μ''<sub>s</sub> |author1=Sheppard, Sheri|author2=Tongue, Benson H.|author3=Anagnos, Thalia|author1-link=Sheri D. Sheppard}}
+</ref><ref>
+{{cite book
+| title = Engineering Mechanics: Statics
+|author1=Meriam, James L. |author2=Kraige, L. Glenn |author3=Palm, William John | publisher = Wiley and Sons
+| year = 2002
+| page= 330
+| quote = Kinetic friction force is usually somewhat less than the maximum static friction force.
+| isbn = 978-0-471-40646-4}}</ref> However, [[Richard Feynman]] comments that "with dry metals it is very hard to show any difference."<ref>{{cite web
+| url = http://hyperphysics.phy-astr.gsu.edu/hbase/frict2.html
+| title = The Feynman Lectures on Physics, Vol. I, p. 12–5
+| publisher = Addison-Wesley
+| year = 1964
+| author1 = Feynman, Richard P.
+| author2 = Leighton, Robert B.
+| author3 = Sands, Matthew
+| access-date = 2009-10-16
+| archive-date = 2021-03-10
+| archive-url = https://web.archive.org/web/20210310234726/http://hyperphysics.phy-astr.gsu.edu/hbase/frict2.html
+| url-status = live
+}}</ref>
+The friction force between two surfaces after sliding begins is the product of the coefficient of kinetic friction and the normal force: <math>F_{k} = \mu_\mathrm{k} F_{n}</math>. This is responsible for the [[Coulomb damping]] of an [[Oscillation#Damped oscillations|oscillating]] or [[Vibration#Types|vibrating]] system.
-===Normal force===
+===Role of the normal force===
 [[Image:Free body diagram2.svg|right|200px|thumb|[[Free-body diagram]] for a block on a ramp. Arrows are [[Euclidean vector|vectors]] indicating directions and magnitudes of forces. ''N'' is the normal force, ''mg'' is the force of [[gravity]], and ''F<sub>f</sub>'' is the force of friction.]]
@@ Line 156: / Line 206: @@
 In general, process for solving any statics problem with friction is to treat contacting surfaces ''tentatively'' as immovable so that the corresponding tangential reaction force between them can be calculated. If this frictional reaction force satisfies <math>F_f \le \mu N</math>, then the tentative assumption was correct, and it is the actual frictional force. Otherwise, the friction force must be set equal to <math>F_f = \mu N</math>, and then the resulting force imbalance would then determine the acceleration associated with slipping.
-===Coefficient of friction===
+[[File:Free body.svg|thumb|Angle of friction, ''θ'', when block just starts to slide]]
+===Role of angle===
+{{For|the maximum angle of static friction between granular materials|Angle of repose}}
+For certain applications, it is more useful to define static friction in terms of the maximum angle before which one of the items will begin sliding. This is called the ''angle of friction'' or ''friction angle''. It is defined as:
+<math display="block">\tan{\theta} = \mu_\mathrm{s}</math>
+and thus:
+<math display="block">\theta = \arctan{\mu_\mathrm{s}}</math>
+where <math>\theta</math> is the angle from horizontal and ''μ<sub>s</sub>'' is the static coefficient of friction between the objects.<ref>{{cite book |last1=Nichols |first1=Edward Leamington |last2=Franklin |first2=William Suddards |title=The Elements of Physics |publisher=Macmillan |page=101 |year=1898 |volume=1 |url=https://books.google.com/books?id=8IlCAAAAIAAJ |access-date=2020-06-07 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007083034/https://books.google.com/books?id=8IlCAAAAIAAJ |url-status=live }}</ref> This formula can also be used to calculate ''μ<sub>s</sub>'' from empirical measurements of the friction angle.
+== Coefficient of friction ==
 {{see also|Coefficient of traction}}
-{{expand section|explanation of why kinetic friction is always lower|date=August 2020}}
 The '''coefficient of friction''' (COF), often symbolized by the Greek letter [[μ]], is a [[dimensionless]] [[scalar (physics)|scalar]] value which equals the ratio of the force of friction between two bodies and the force pressing them together, either during or at the onset of slipping. The coefficient of friction depends on the materials used; for example, ice on steel has a low coefficient of friction, while rubber on pavement has a high coefficient of friction. Coefficients of friction range from near zero to greater than one. The coefficient of friction between two surfaces of similar metals is greater than that between two surfaces of different metals; for example, brass has a higher coefficient of friction when moved against brass, but less if moved against steel or aluminum.<ref name="Air Brake Association-1921">{{cite book|author=Air Brake Association|title=The Principles and Design of Foundation Brake Rigging|url=https://books.google.com/books?id=DoNBAQAAMAAJ&pg=PA5|year=1921|publisher=Air brake association|page=5|access-date=2017-07-27|archive-date=2024-10-07|archive-url=https://web.archive.org/web/20241007090951/https://books.google.com/books?id=DoNBAQAAMAAJ&pg=PA5#v=onepage&q&f=false|url-status=live}}</ref>
@@ Line 169: / Line 229: @@
 Most dry materials in combination have friction coefficient values between 0.3 and 0.6. Values outside this range are rarer, but [[teflon]], for example, can have a coefficient as low as 0.04. A value of zero would mean no friction at all, an elusive property. Rubber in contact with other surfaces can yield friction coefficients from 1 to 2. Occasionally it is maintained that ''μ'' is always < 1, but this is not true. While in most relevant applications ''μ'' < 1, a value above 1 merely implies that the force required to slide an object along the surface is greater than the normal force of the surface on the object. For example, [[silicone rubber]] or [[acrylic rubber]]-coated surfaces have a coefficient of friction that can be substantially larger than 1.
-While it is often stated that the COF is a "material property", it is better categorized as a "system property". Unlike true material properties (such as conductivity, dielectric constant, yield strength), the COF for any two materials depends on system variables like [[temperature]], [[velocity]], [[atmosphere]] and also what are now popularly described as aging and deaging times; as well as on geometric properties of the interface between the materials, namely [[surface roughness|surface structure]].<ref name="Hanaor-2016" /> For example, a [[copper]] pin sliding against a thick copper plate can have a COF that varies from 0.6 at low speeds (metal sliding against metal) to below 0.2 at high speeds when the copper surface begins to melt due to frictional heating. The latter speed, of course, does not determine the COF uniquely; if the pin diameter is increased so that the frictional heating is removed rapidly, the temperature drops, the pin remains solid and the COF rises to that of a 'low speed' test.{{Citation needed|date=December 2008}}
+While it is often stated that the coefficient of friction is a "material property", it is better categorized as a "system property". Unlike true material properties (such as conductivity, dielectric constant, yield strength), the coefficient of friction for any two materials depends on system variables like [[temperature]], [[velocity]], [[atmosphere]], and also what are now popularly described as aging and deaging times{{what?|date=December 2025}}, as well as on geometric properties of the interface between the materials, namely [[surface roughness|surface structure]].<ref name="Hanaor-2016" /> For example, a [[copper]] pin sliding against a thick copper plate can have a coefficient of friction that varies from 0.6 at low speeds (metal sliding against metal) to below 0.2 at high speeds when the copper surface begins to melt due to frictional heating. The latter speed, of course, does not determine the coefficient of friction uniquely; if the pin diameter is increased so that the frictional heating is removed rapidly, the temperature drops, the pin remains solid, and the coefficient of friction rises to that of a 'low speed' test.{{Citation needed|date=December 2008}}
 In systems with significant non-uniform stress fields, because local slip occurs before the system slides, the macroscopic coefficient of static friction depends on the applied load, system size, or shape; [[Friction#Laws of dry friction|Amontons' law]] is not satisfied macroscopically.<ref>{{Cite journal |last1=Otsuki |first1=M. |last2=Matsukawa |first2=H. |date=2013-04-02 |title=Systematic breakdown of Amontons' law of friction for an elastic object locally obeying Amontons' law |journal=Scientific Reports |volume=3 |article-number=1586 |doi=10.1038/srep01586|pmid=23545778 |pmc=3613807 |arxiv=1202.1716 |bibcode=2013NatSR...3.1586O }}</ref>
-====Approximate coefficients of friction====
+=== Approximate coefficients of friction ===
-{{Disputed section|date=November 2021}}
 {| class="wikitable"
 |-
-! colspan="2" rowspan="2" data-sort-type="text"|Materials !! colspan="2"|Static Friction, <math>\mu_\mathrm{s}</math> !! colspan="2"|Kinetic/Sliding Friction, <math>\mu_\mathrm{k}\,</math>
+! colspan="2" rowspan="2" data-sort-type="text" |Materials !! colspan="2" |Static Friction, <math>\mu_\mathrm{s}</math> !! colspan="2" |Kinetic/Sliding Friction, <math>\mu_\mathrm{k}\,</math>
 |-
 |-
-!data-sort-type="number"| Dry and clean !!data-sort-type="number"| Lubricated
+! data-sort-type="number" | Dry and clean !! data-sort-type="number" | Lubricated
-!data-sort-type="number| Dry and clean !!data-sort-type="number"| Lubricated
+! data-sort-type="number" | Dry and clean !! data-sort-type="number" | Lubricated
 |-
 ! Aluminium
@@ Line 194: / Line 253: @@
 }}</ref>
 |
-| 0.47<ref name="Friction Factors"/>
+| 0.47<ref name="Friction Factors" />
 |
 |-
 !Aluminium
 !Aluminium
-|1.05–1.35<ref name="Friction Factors"/>
+|1.05–1.35<ref name="Friction Factors" />
-|0.3<ref name="Friction Factors"/>
+|0.3<ref name="Friction Factors" />
-|1.4<ref name="Friction Factors"/>–1.5{{cn|date=March 2024}}
+|1.4<ref name="Friction Factors" />–1.5{{cn|date=March 2024}}
 |-
 !Gold
@@ Line 212: / Line 271: @@
 !Platinum
 !Platinum
-|1.2<ref name="Friction Factors"/>
+|1.2<ref name="Friction Factors" />
-|0.25<ref name="Friction Factors"/>
+|0.25<ref name="Friction Factors" />
 |3.0{{cn|date=March 2024}}
 |
@@ Line 219: / Line 278: @@
 !Silver
 !Silver
-|1.4<ref name="Friction Factors"/>
+|1.4<ref name="Friction Factors" />
-|0.55<ref name="Friction Factors"/>
+|0.55<ref name="Friction Factors" />
 |1.5{{cn|date=March 2024}}
 |
@@ Line 266: / Line 325: @@
 ! Brass
 ! Steel
-| 0.35–0.51<ref name="Friction Factors"/>
+| 0.35–0.51<ref name="Friction Factors" />
-| 0.19<ref name="Friction Factors"/>
+| 0.19<ref name="Friction Factors" />
-| 0.44<ref name="Friction Factors"/>
+| 0.44<ref name="Friction Factors" />
 |
 |-
 ! Cast iron
 ! Copper
-| 1.05<ref name="Friction Factors"/>
+| 1.05<ref name="Friction Factors" />
 |
-| 0.29<ref name="Friction Factors"/>
+| 0.29<ref name="Friction Factors" />
 |
 |-
 ! Cast iron
 ! Zinc
-| 0.85<ref name="Friction Factors"/>
+| 0.85<ref name="Friction Factors" />
 |
-| 0.21<ref name="Friction Factors"/>
+| 0.21<ref name="Friction Factors" />
 |
 |-
@@ Line 289: / Line 348: @@
 | 1.0
 | 0.30 (wet)
-| 0.6–0.85<ref name="Friction Factors"/>
+| 0.6–0.85<ref name="Friction Factors" />
-| 0.45–0.75 (wet)<ref name="Friction Factors"/>
+| 0.45–0.75 (wet)<ref name="Friction Factors" />
 |-
 ! Concrete
 ! Wood
-| 0.62<ref name="Friction Factors"/><ref name="Coefficient of Friction Archived March 8"/>
+| 0.62<ref name="Friction Factors" /><ref name="Coefficient of Friction Archived March 8" />
 |
 |
@@ Line 310: / Line 369: @@
 | 0.53<ref name="Barrett-1990" />
 |
-| 0.36<ref name="Friction Factors"/><ref name="Barrett-1990" />
+| 0.36<ref name="Friction Factors" /><ref name="Barrett-1990" />
 | 0.18<ref name="Barrett-1990" />
 |-
 ! Glass
 ! Glass
-| 0.9–1.0<ref name="Friction Factors"/><ref name="Barrett-1990" />
+| 0.9–1.0<ref name="Friction Factors" /><ref name="Barrett-1990" />
 | 0.005–0.01<ref name="Barrett-1990" />
-| 0.4<ref name="Friction Factors"/><ref name="Barrett-1990" />
+| 0.4<ref name="Friction Factors" /><ref name="Barrett-1990" />
 | 0.09–0.116<ref name="Barrett-1990" />
 |-
@@ Line 332: / Line 391: @@
 }}</ref>
 |
-| 0.003<ref name="Coefficients of Friction of Human Joints"/>
+| 0.003<ref name="Coefficients of Friction of Human Joints" />
 |-
 ! Ice
 ! Ice
-| 0.02–0.09<ref name="The Engineering Toolbox"/>
+| 0.02–0.09<ref name="The Engineering Toolbox" />
 |
 |
@@ Line 343: / Line 402: @@
 ! [[Polyethene]]
 ! Steel
-| 0.2<ref name="Friction Factors"/><ref name="The Engineering Toolbox"/>
+| 0.2<ref name="Friction Factors" /><ref name="The Engineering Toolbox" />
-| 0.2<ref name="Friction Factors"/><ref name="The Engineering Toolbox"/>
+| 0.2<ref name="Friction Factors" /><ref name="The Engineering Toolbox" />
 |
 |
@@ Line 350: / Line 409: @@
 ! [[PTFE]] (Teflon)
 ! PTFE (Teflon)
-| 0.04<ref name="Friction Factors"/><ref name="The Engineering Toolbox">{{cite web
+| 0.04<ref name="Friction Factors" /><ref name="The Engineering Toolbox">{{cite web
 | url = http://www.engineeringtoolbox.com/friction-coefficients-d_778.html
 | title = The Engineering Toolbox: Friction and Coefficients of Friction
@@ Line 358: / Line 417: @@
 | url-status = live
 }}</ref>
-| 0.04<ref name="Friction Factors"/><ref name="The Engineering Toolbox"/>
+| 0.04<ref name="Friction Factors" /><ref name="The Engineering Toolbox" />
 |
-| 0.04<ref name="Friction Factors"/>
+| 0.04<ref name="Friction Factors" />
 |-
 ! Steel
 ! Ice
-| 0.03<ref name="The Engineering Toolbox"/>
+| 0.03<ref name="The Engineering Toolbox" />
 |
 |
@@ Line 371: / Line 430: @@
 ! Steel
 ! PTFE (Teflon)
-| 0.04<ref name="Friction Factors"/>−0.2<ref name="The Engineering Toolbox"/>
+| 0.04<ref name="Friction Factors" />−0.2<ref name="The Engineering Toolbox" />
-| 0.04<ref name="Friction Factors"/>
+| 0.04<ref name="Friction Factors" />
 |
-| 0.04<ref name="Friction Factors"/>
+| 0.04<ref name="Friction Factors" />
 |-
 ! Steel
 ! Steel
-| 0.74<ref name="Friction Factors"/>−0.80<ref name="The Engineering Toolbox"/>
+| 0.74<ref name="Friction Factors" />−0.80<ref name="The Engineering Toolbox" />
-| 0.005–0.23<ref name="Barrett-1990" /><ref name="The Engineering Toolbox"/>
+| 0.005–0.23<ref name="Barrett-1990" /><ref name="The Engineering Toolbox" />
-| 0.42–0.62<ref name="Friction Factors"/><ref name="Barrett-1990" />
+| 0.42–0.62<ref name="Friction Factors" /><ref name="Barrett-1990" />
 | 0.029–0.19<ref name="Barrett-1990" />
 |-
 ! Wood
 ! Metal
-| 0.2–0.6<ref name="Friction Factors"/><ref name="Coefficient of Friction Archived March 8"/>
+| 0.2–0.6<ref name="Friction Factors" /><ref name="Coefficient of Friction Archived March 8" />
-| 0.2 (wet)<ref name="Friction Factors"/><ref name="Coefficient of Friction Archived March 8"/>
+| 0.2 (wet)<ref name="Friction Factors" /><ref name="Coefficient of Friction Archived March 8" />
 | 0.49<ref name="Barrett-1990" />
 | 0.075<ref name="Barrett-1990" />
@@ Line 392: / Line 451: @@
 ! Wood
 ! Wood
-| 0.25–0.62<ref name="Friction Factors"/><ref name="Coefficient of Friction Archived March 8">[http://www.engineershandbook.com/Tables/frictioncoefficients.htm Coefficient of Friction] {{webarchive |url=https://web.archive.org/web/20090308124246/http://www.engineershandbook.com/Tables/frictioncoefficients.htm |date=March 8, 2009 }}. EngineersHandbook.com</ref><ref name="Barrett-1990">{{cite journal |last1=Barrett |first1=Richard T. |title=(NASA-RP-1228) Fastener Design Manual |url=http://hdl.handle.net/2060/19900009424 |website=NASA Technical Reports Server |publisher=NASA Lewis Research Center |access-date=3 August 2020 |page=16 |date=1 March 1990 |hdl=2060/19900009424 |archive-date=7 October 2024 |archive-url=https://web.archive.org/web/20241007091010/https://ntrs.nasa.gov/citations/19900009424 |url-status=live }}</ref>
+| 0.25–0.62<ref name="Friction Factors" /><ref name="Coefficient of Friction Archived March 8">[http://www.engineershandbook.com/Tables/frictioncoefficients.htm Coefficient of Friction] {{webarchive |url=https://web.archive.org/web/20090308124246/http://www.engineershandbook.com/Tables/frictioncoefficients.htm |date=March 8, 2009 }}. EngineersHandbook.com</ref><ref name="Barrett-1990">{{cite journal |last1=Barrett |first1=Richard T. |title=(NASA-RP-1228) Fastener Design Manual |url=http://hdl.handle.net/2060/19900009424 |website=NASA Technical Reports Server |publisher=NASA Lewis Research Center |access-date=3 August 2020 |page=16 |date=1 March 1990 |hdl=2060/19900009424 |archive-date=7 October 2024 |archive-url=https://web.archive.org/web/20241007091010/https://ntrs.nasa.gov/citations/19900009424 |url-status=live }}</ref>
-| 0.2 (wet)<ref name="Friction Factors"/><ref name="Coefficient of Friction Archived March 8"/>
+| 0.2 (wet)<ref name="Friction Factors" /><ref name="Coefficient of Friction Archived March 8" />
 | 0.32–0.48<ref name="Barrett-1990" />
 | 0.067–0.167<ref name="Barrett-1990" />
 |}
-Under certain conditions some materials have very low friction coefficients. An example is (highly ordered pyrolytic) graphite which can have a friction coefficient below 0.01.<ref>{{cite journal |last=Dienwiebel |first=Martin |title=Superlubricity of Graphite |display-authors=etal |journal=Phys. Rev. Lett. |volume=92 |article-number=126101 |year=2004 |doi=10.1103/PhysRevLett.92.126101 |url=http://www.physics.leidenuniv.nl/sections/cm/ip/group/PDF/Phys.rev.lett/2004/92(2004)12601.pdf |issue=12 |bibcode=2004PhRvL..92l6101D |pmid=15089689 |s2cid=26811802 |access-date=2011-09-01 |archive-date=2011-09-17 |archive-url=https://web.archive.org/web/20110917120623/http://www.physics.leidenuniv.nl/sections/cm/ip/group/PDF/Phys.rev.lett/2004/92(2004)12601.pdf |url-status=live }}</ref>
+== Sources of friction ==
-This ultralow-friction regime is called [[superlubricity]].<ref>{{Citation |last=Müser |first=Martin H. |title=Theoretical Studies of Superlubricity |date=2015 |work=Fundamentals of Friction and Wear on the Nanoscale |series=NanoScience and Technology |pages=209–232 |editor-last=Gnecco |editor-first=Enrico |url=https://link.springer.com/10.1007/978-3-319-10560-4_11 |access-date=2025-04-25 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-10560-4_11 |bibcode=2015ffwn.book..209M |isbn=978-3-319-10559-8 |editor2-last=Meyer |editor2-first=Ernst|url-access=subscription }}</ref>
+[[File:Asperities.svg|thumb|The top image shows asperities under no load. The bottom image depicts the same surface after applying a load.]]
+Friction is due to a range of energy dissipative processes. The base model, first detailed by Bowden and Tabor<ref name=":0">{{Cite book |last=Bowden |first=F P |url=https://academic.oup.com/book/54811 |title=The Friction and Lubrication of Solids |last2=Tabor |first2=D |date=2001-02-08 |publisher=Oxford University PressOxford |isbn=978-0-19-850777-2 |language=en |doi=10.1093/oso/9780198507772.001.0001}}</ref> is that it involves contact of small protusions on surfaces called [[asperities]]. As the load increases the number and the area of asperity contacts increases, leading to the normal force dependence of friction. As these asperities slide past each other both [[Plastic Deformation|plastic]] and [[elastic deformation]] processes take place, among others. The frictional force is the energy dissipated per unit distance by these, the net result often being the linear Coulomb formula mentioned earlier.<ref name=":0" />
-===Static friction===<!-- Traction (engineering) links here -->
-[[File:Static kinetic friction vs time.png|400px|right|thumb|When the mass is not moving, the object experiences static friction.  The friction increases as the applied force increases until the block moves.  After the block moves, it experiences kinetic friction, which is less than the maximum static friction.]]
-{{Main|Stiction}}
-Static friction is friction between two or more solid objects that are not moving relative to each other. For example, static friction can prevent an object from sliding down a sloped surface. The coefficient of static friction, typically denoted as ''μ''<sub>s</sub>, is usually higher than the coefficient of kinetic friction. Static friction is considered to arise as the result of surface roughness features across multiple length scales at solid surfaces. These features, known as [[asperities]] are present down to nano-scale dimensions and result in true solid to solid contact existing only at a limited number of points accounting for only a fraction of the apparent or nominal contact area.<ref>[https://www.researchgate.net/publication/283675011_Static_friction_at_fractal_interfaces multi-scale origins of static friction] {{Webarchive|url=https://web.archive.org/web/20210918100737/https://www.researchgate.net/publication/283675011_Static_friction_at_fractal_interfaces |date=2021-09-18 }} 2016</ref> The linearity between applied load and true contact area, arising from asperity deformation, gives rise to the linearity between static frictional force and normal force, found for typical Amonton–Coulomb type friction.<ref>{{cite journal | author= Greenwood J.A. and JB Williamson| title= Contact of nominally flat surfaces | journal= Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences | volume=295 | issue=1442 | year=1966 | page= 300 | doi= 10.1098/rspa.1966.0242 | bibcode= 1966RSPSA.295..300G }}</ref>
-The static friction force must be overcome by an applied force before an object can move. The maximum possible friction force between two surfaces before sliding begins is the product of the coefficient of static friction and the normal force: <math>F_\text{max} = \mu_\mathrm{s} F_\text{n}</math>. When there is no sliding occurring, the friction force can have any value from zero up to <math>F_\text{max}</math>. Any force smaller than <math>F_\text{max}</math> attempting to slide one surface over the other is opposed by a frictional force of equal magnitude and opposite direction. Any force larger than <math>F_\text{max}</math> overcomes the force of static friction and causes sliding to occur. The instant sliding occurs, static friction is no longer applicable—the friction between the two surfaces is then called kinetic friction. However, an apparent static friction can be observed even in the case when the true static friction is zero.<ref>{{Cite journal|last1=Nakano|first1=K.|last2=Popov|first2=V. L.|date=2020-12-10|title=Dynamic stiction without static friction: The role of friction vector rotation| url=https://link.aps.org/doi/10.1103/PhysRevE.102.063001| journal=Physical Review E|volume=102|issue=6|page=063001| doi=10.1103/PhysRevE.102.063001|pmid=33466084 |bibcode=2020PhRvE.102f3001N |hdl=10131/00013921 |s2cid=230599544 |hdl-access=free}}</ref>
+Experimental studies of dry sliding friction in engineering materials have shown that friction and wear are controlled not only by the nominal contact conditions, but also by near-surface microstructural evolution, debris generation and tribochemical transformations during sliding. Reviews and experimental studies have shown that plastic deformation, fracture, material transfer, mechanical mixing and oxidation can all contribute to the transition between mild and severe wear regimes in dry metal contacts.<ref>{{cite journal
+ |last=Rigney
-An example of static friction is the force that prevents a car wheel from slipping as it rolls on the ground. Even though the wheel is in motion, the patch of the tire in contact with the ground is stationary relative to the ground, so it is static rather than kinetic friction. Upon slipping, the wheel friction changes to kinetic friction.  An [[anti-lock braking system]] operates on the principle of allowing a locked wheel to resume rotating so that the car maintains static friction.
+ |first=D. A.
+ |title=Comments on the sliding wear of metals
-The maximum value of static friction, when motion is impending, is sometimes referred to as '''limiting friction''',<ref name="Bhavikatti-1994">{{cite book
+ |journal=Tribology International
-| url = https://books.google.com/books?id=4wkLl4NvmWAC&pg=PA112
+ |year=1997
-| title = Engineering Mechanics
+ |volume=30
-| last = Bhavikatti
+ |issue=5
-| first = S.S.
+ |pages=361–367
-| author2 = K.G. Rajashekarappa
+ |doi=10.1016/S0301-679X(96)00065-5
-| page = 112
+}}</ref><ref>{{cite journal
-| access-date = 2007-10-21
+ |last=Stott
-| publisher = New Age International
+ |first=F. H.
-| isbn = 978-81-224-0617-7
+ |title=The role of oxidation in the wear of alloys
-| year = 1994
+ |journal=Tribology International
-| archive-date = 2024-10-07
+ |year=1998
-| archive-url = https://web.archive.org/web/20241007091508/https://books.google.com/books?id=4wkLl4NvmWAC&pg=PA112#v=onepage&q&f=false
+ |volume=31
-| url-status = live
+ |issue=1–3
+ |pages=61–71
+ |doi=10.1016/S0301-679X(98)00008-5
+}}</ref><ref>{{cite book
+ |last=Hutchings
+ |first=I. M.
+ |title=Tribology: Friction and Wear of Engineering Materials
+ |publisher=Butterworth-Heinemann
+ |year=1992
+ |isbn=9780750610544
+}}</ref><ref>{{cite book
+ |last=Zum Gahr
+ |first=K.-H.
+ |title=Microstructure and Wear of Materials
+ |publisher=Elsevier
+ |year=1987
+ |isbn=9780444427540
 }}</ref>
-although this term is not used universally.<ref name="Beer-1996"/>
-===Kinetic friction===
+The importance of debris and “third-body” material formed in the contact has also been demonstrated experimentally, since compacted triboparticulates may either accelerate damage or form protective interfacial layers that reduce direct metal-to-metal contact.<ref>{{cite journal
-'''Kinetic friction''', also known as '''dynamic friction''' or '''sliding friction''', occurs when two objects are moving relative to each other and rub together (like a sled on the ground). The coefficient of kinetic friction is typically denoted as ''μ''<sub>k</sub>, and is usually less than the coefficient of static friction for the same materials.<ref>{{cite book|title=Statics: Analysis and Design of Systems in Equilibrium |publisher=Wiley and Sons|year=2005 |isbn=978-0-471-37299-8|page=618 |quote=In general, for given contacting surfaces, ''μ''<sub>k</sub> < ''μ''<sub>s</sub> |author1=Sheppard, Sheri|author2=Tongue, Benson H.|author3=Anagnos, Thalia|author1-link=Sheri D. Sheppard}}
+ |last1=Jiang
-</ref><ref>
+ |first1=J.
-{{cite book
+ |last2=Stott
-| title = Engineering Mechanics: Statics
+ |first2=F. H.
-|author1=Meriam, James L. |author2=Kraige, L. Glenn |author3=Palm, William John | publisher = Wiley and Sons
+ |last3=Stack
-| year = 2002
+ |first3=M. M.
-| page= 330
+ |title=The role of triboparticulates in dry sliding wear
-| quote = Kinetic friction force is usually somewhat less than the maximum static friction force.
+ |journal=Tribology International
-| isbn = 978-0-471-40646-4}}</ref> However, [[Richard Feynman]] comments that "with dry metals it is very hard to show any difference."<ref>{{cite web
+ |year=1998
-| url = http://hyperphysics.phy-astr.gsu.edu/hbase/frict2.html
+ |volume=31
-| title = The Feynman Lectures on Physics, Vol. I, p. 12–5
+ |issue=5
-| publisher = Addison-Wesley
+ |pages=245–256
-| year = 1964
+ |doi=10.1016/S0301-679X(98)00027-9
-| author1 = Feynman, Richard P.
+}}</ref> In steels, dry sliding tests have shown that hardness level, load and sliding speed can strongly influence friction stability and wear rate, while oxidative wear may become dominant under suitable sliding conditions.<ref>{{cite journal
-| author2 = Leighton, Robert B.
+ |last1=Straffelini
-| author3 = Sands, Matthew
+ |first1=G.
-| access-date = 2009-10-16
+ |last2=Trabucco
-| archive-date = 2021-03-10
+ |first2=D.
-| archive-url = https://web.archive.org/web/20210310234726/http://hyperphysics.phy-astr.gsu.edu/hbase/frict2.html
+ |last3=Molinari
-| url-status = live
+ |first3=A.
+ |title=Oxidative wear of heat-treated steels
+ |journal=Wear
+ |year=2001
+ |volume=250
+ |issue=1–12
+ |pages=485–491
+ |doi=10.1016/S0043-1648(01)00661-5
+}}</ref><ref>{{cite journal
+ |last1=Tang
+ |first1=L.
+ |last2=Gao
+ |first2=C.
+ |last3=Huang
+ |first3=J.
+ |last4=Zhang
+ |first4=H.
+ |last5=Chang
+ |first5=W.
+ |title=Dry sliding friction and wear behaviour of hardened AISI D2 tool steel with different hardness levels
+ |journal=Tribology International
+ |year=2013
+ |volume=66
+ |pages=165–173
+ |doi=10.1016/j.triboint.2013.05.006
+}}</ref><ref>{{cite journal
+ |last1=Zhang
+ |first1=Q. Y.
+ |last2=Chen
+ |first2=K. M.
+ |last3=Wang
+ |first3=L.
+ |last4=Wang
+ |first4=S. Q.
+ |title=Characteristics of oxidative wear and oxidative mild wear
+ |journal=Tribology International
+ |year=2013
+ |volume=61
+ |pages=214–223
+ |doi=10.1016/j.triboint.2013.01.003
+}}</ref> Related work on stainless steels and titanium alloys further shows that dry sliding friction is highly sensitive to abrasive particle size, counterface material and the formation or breakdown of protective tribo-oxide layers, which can shift the governing wear mechanism from relatively mild oxidative behaviour to severe delamination or abrasive damage.<ref>{{cite journal
+ |last1=Jourani
+ |first1=A.
+ |last2=Bouvier
+ |first2=S.
+ |title=Friction and Wear Mechanisms of 316L Stainless Steel in Dry Sliding Contact: Effect of Abrasive Particle Size
+ |journal=Tribology Transactions
+ |year=2015
+ |volume=58
+ |issue=1
+ |pages=131–139
+ |doi=10.1080/10402004.2014.955229
+}}</ref><ref>{{cite journal
+ |last1=Straffelini
+ |first1=G.
+ |last2=Molinari
+ |first2=A.
+ |title=Dry sliding wear of Ti–6Al–4V alloy as influenced by the counterface and sliding conditions
+ |journal=Wear
+ |year=1999
+ |volume=236
+ |issue=1–2
+ |pages=328–338
+ |doi=10.1016/S0043-1648(99)00292-6
+}}</ref><ref>{{cite journal
+ |last1=Zhang
+ |first1=Q. Y.
+ |last2=Chen
+ |first2=K. M.
+ |last3=Wang
+ |first3=L.
+ |last4=Wang
+ |first4=S. Q.
+ |title=Investigation on tribo-layers and their function of a titanium alloy during dry sliding
+ |journal=Tribology International
+ |year=2016
+ |volume=94
+ |pages=541–549
+ |doi=10.1016/j.triboint.2015.10.032
+}}</ref><ref>{{cite journal
+ |last=Rigney
+ |first=D. A.
+ |title=Comments on the sliding wear of metals
+ |journal=Tribology International
+ |year=1997
+ |volume=30
+ |issue=5
+ |pages=361–367
+ |doi=10.1016/S0301-679X(96)00065-5
+}}</ref><ref>{{cite journal
+ |last=Stott
+ |first=F. H.
+ |title=The role of oxidation in the wear of alloys
+ |journal=Tribology International
+ |year=1998
+ |volume=31
+ |issue=1–3
+ |pages=61–71
+ |doi=10.1016/S0301-679X(97)00060-0
+}}</ref> Experimental investigations further indicate that environmental factors such as hydrogen charging, lubrication regime and cyclic loading may significantly influence crack initiation mechanisms and friction stability in structural alloys.<ref>{{cite journal
+ |last1=Balitskii
+ |first1=O.I.
+ |last2=Kolesnikov
+ |first2=V.O.
+ |last3=Ivaskevych
+ |first3=L.M.
+ |last4=Havrylyuk
+ |first4=M.R.
+ |title=The Influence of Specific Features of Load and Hydrogen Charging on Steel Tribotechnical Properties
+ |journal=Materials Science
+ |year=2023
+ |volume=58
+ |pages=502–512
+ |doi=10.1007/s11003-023-00691-5
 }}</ref>
-The friction force between two surfaces after sliding begins is the product of the coefficient of kinetic friction and the normal force: <math>F_{k} = \mu_\mathrm{k} F_{n}</math>. This is responsible for the [[Coulomb damping]] of an [[Oscillation#Damped oscillations|oscillating]] or [[Vibration#Types|vibrating]] system.
-New models are beginning to show how kinetic friction can be greater than static friction.<ref name="Volokitin, A. I-2002" >{{cite journal |title=Theory of rubber friction: Nonstationary sliding |journal=Physical Review B |volume=65 |page=134106 |doi=10.1103/PhysRevB.65.134106 |author1=Persson, B.N. |author2=Volokitin, A.I. |year=2002 |bibcode=2002PhRvB..65m4106P |issue=13 |url=http://juser.fz-juelich.de/record/25870/files/17249.pdf |access-date=2019-01-31 |archive-date=2021-09-18 |archive-url=https://web.archive.org/web/20210918100743/https://juser.fz-juelich.de/record/25870/files/17249.pdf |url-status=live }}</ref> In many other cases roughness effects are dominant, for example in rubber to road friction.<ref name="Volokitin, A. I-2002" /> Surface roughness and contact area affect kinetic friction for micro- and nano-scale objects where surface area forces dominate inertial forces.<ref>{{cite book |last=Persson |first=B.N.J. |title=Sliding friction: physical principles and applications |url=https://books.google.com/books?id=1jb-nZMnRGYC&q=kinetic+friction |access-date=2016-01-23 |year=2000 |publisher=Springer |isbn=978-3-540-67192-3 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007091447/https://books.google.com/books?id=1jb-nZMnRGYC&q=kinetic+friction#v=snippet&q=kinetic%20friction&f=false |url-status=live }}</ref>
+== Breakdown of the Coulomb model ==
+Despite being a simplified model of friction, the Coulomb model is useful in many [[numerical simulation]] applications such as [[multibody system]]s and [[granular material]]. Even its most simple expression encapsulates the fundamental effects of sticking and sliding which are required in many applied cases, although specific algorithms have to be designed in order to efficiently [[Numerical integration|numerically integrate]] mechanical systems with Coulomb friction and bilateral or unilateral contact.<ref>{{cite journal |last1=Haslinger |first1=J. |title=Approximation of the Signorini problem with friction, obeying the Coulomb law |journal=Mathematical Methods in the Applied Sciences |volume=5 |issue=1 |pages=422–437 |year=1983 |bibcode=1983MMAS....5..422H |doi=10.1002/mma.1670050127 |last2=Nedlec |first2=J.C. |hdl=10338.dmlcz/104086 |url=http://dml.cz/bitstream/handle/10338.dmlcz/104086/AplMat_29-1984-3_6.pdf |access-date=2019-09-19 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007083034/http://dml.cz/bitstream/handle/10338.dmlcz/104086/AplMat_29-1984-3_6.pdf |url-status=live }}</ref><ref>{{cite journal |last1=Alart |first1=P. |last2=Curnier |first2=A. |title=A mixed formulation for frictional contact problems prone to Newton like solution method |journal=Computer Methods in Applied Mechanics and Engineering |volume=92 |pages=353–375 |year=1991 |bibcode=1991CMAME..92..353A |doi=10.1016/0045-7825(91)90022-X |issue=3 |url=https://hal.science/hal-04264964/file/Alart1991.pdf |access-date=2024-03-29 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007091441/https://hal.science/hal-04264964/file/Alart1991.pdf |url-status=live }}</ref><ref>{{cite journal |last1=Acary |first1=V. |last2=Cadoux |first2=F. |last3=Lemaréchal |first3=C. |last4=Malick |first4=J. |title=A formulation of the linear discrete Coulomb friction problem via convex optimization |journal=Journal of Applied Mathematics and Mechanics |volume=91 |issue=2 |pages=155–175 |year=2011 |doi=10.1002/zamm.201000073 |bibcode=2011ZaMM...91..155A |s2cid=17280625 |url=https://hal.inria.fr/inria-00495734/document |access-date=2018-04-20 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007083036/https://hal.inria.fr/inria-00495734/document |url-status=live }}</ref><ref>{{cite journal |last1=De Saxcé |first1=G. |last2=Feng |first2=Z.-Q. |title=The bipotential method: A constructive approach to design the complete contact law with friction and improved numerical algorithms |journal=Mathematical and Computer Modelling |volume=28 |issue=4 |pages=225–245 |year=1998 |doi=10.1016/S0895-7177(98)00119-8|doi-access=free }}</ref><ref>{{cite journal |last1=Simo |first1=J.C. |last2=Laursen |first2=T.A. |title=An augmented lagrangian treatment of contact problems involving friction |journal=Computers and Structures |volume=42 |issue=2 |pages=97–116 |year=1992 |doi=10.1016/0045-7949(92)90540-G|doi-access=free }}</ref> Some quite [[Nonlinear system#Types of nonlinear dynamic behaviors|nonlinear effects]], such as the so-called [[Painlevé paradox]]es, may be encountered with Coulomb friction.<ref>{{cite book |last1=Acary |first1=V. |last2=Brogliato |first2=B. |title=Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and Electronics |publisher=[[Springer Science+Business Media|Springer Verlag Heidelberg]] |volume=35 |year=2008}}</ref>
-The origin of kinetic friction at nanoscale can be rationalized by an energy model.<ref>{{cite journal |last1=Makkonen |first1=L |year=2012 |title=A thermodynamic model of sliding friction |doi=10.1063/1.3699027 |journal=AIP Advances |volume=2 |issue= 1|page=012179 |bibcode=2012AIPA....2a2179M |doi-access=free }}</ref> During sliding, a new surface forms at the back of a sliding true contact, and existing surface disappears at the front of it. Since all surfaces involve the thermodynamic surface energy, work must be spent in creating the new surface, and energy is released as heat in removing the surface. Thus, a force is required to move the back of the contact, and frictional heat is released at the front.
-[[File:Free body.svg|thumb|Angle of friction, ''θ'', when block just starts to slide]]
+The Coulomb approximation follows from the assumptions that: surfaces are in atomically close contact only over a small fraction of their overall area; that this [[contact area]] is proportional to the normal force (until saturation, which takes place when all area is in atomic contact); and that the frictional force is proportional to the applied normal force, independently of the contact area; this comes from the asperity work of Bowden and Tabor.<ref name=":0" /> The Coulomb approximation is fundamentally an empirical construct. It is a rule-of-thumb describing the approximate outcome of an extremely complicated physical interaction. The strength of the approximation is its simplicity and versatility. Though the relationship between normal force and frictional force is not exactly linear (and so the frictional force is not entirely independent of the contact area of the surfaces), the Coulomb approximation is an adequate representation of friction for the analysis of many physical systems.
-===Angle of friction===
-{{For|the maximum angle of static friction between granular materials|Angle of repose}}
-For certain applications, it is more useful to define static friction in terms of the maximum angle before which one of the items will begin sliding. This is called the ''angle of friction'' or ''friction angle''. It is defined as:
-<math display="block">\tan{\theta} = \mu_\mathrm{s}</math>
-and thus:
-<math display="block">\theta = \arctan{\mu_\mathrm{s}}</math>
-where <math>\theta</math> is the angle from horizontal and ''μ<sub>s</sub>'' is the static coefficient of friction between the objects.<ref>{{cite book |last1=Nichols |first1=Edward Leamington |last2=Franklin |first2=William Suddards |title=The Elements of Physics |publisher=Macmillan |page=101 |year=1898 |volume=1 |url=https://books.google.com/books?id=8IlCAAAAIAAJ |access-date=2020-06-07 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007083034/https://books.google.com/books?id=8IlCAAAAIAAJ |url-status=live }}</ref> This formula can also be used to calculate ''μ<sub>s</sub>'' from empirical measurements of the friction angle.
-===Friction at the atomic level===
-Determining the forces required to move atoms past each other is a challenge in designing [[nanomachines]]. In 2008 scientists for the first time were able to move a single atom across a surface, and measure the forces required. Using ultrahigh vacuum and nearly zero temperature (5 K), a modified atomic force microscope was used to drag a [[cobalt]] atom, and a [[carbon monoxide]] molecule, across surfaces of copper and [[platinum]].<ref>{{Cite journal |last1=Ternes |first1=Markus |last2=Lutz |first2=Christopher P. |last3=Hirjibehedin |first3=Cyrus F. |last4=Giessibl |first4=Franz J. |author-link5= Andreas J. Heinrich |last5=Heinrich |first5=Andreas J. |title=The Force Needed to Move an Atom on a Surface |journal=[[Science (journal)|Science]] |volume=319 |issue=5866 |pages=1066–1069 |date=2008-02-22 |doi=10.1126/science.1150288 |pmid=18292336 |bibcode=2008Sci...319.1066T |s2cid=451375 |url=https://epub.uni-regensburg.de/25284/1/The%20Force%20Needed%20to%20Move%20an%20Atom%20on%20.pdf |archive-url=https://web.archive.org/web/20180720063201/https://epub.uni-regensburg.de/25284/1/The%20Force%20Needed%20to%20Move%20an%20Atom%20on%20.pdf |archive-date=2018-07-20 |url-status=live }}</ref>
-===Limitations of the Coulomb model===
-The Coulomb approximation follows from the assumptions that: surfaces are in atomically close contact only over a small fraction of their overall area; that this [[contact area]] is proportional to the normal force (until saturation, which takes place when all area is in atomic contact); and that the frictional force is proportional to the applied normal force, independently of the contact area. The Coulomb approximation is fundamentally an empirical construct. It is a rule-of-thumb describing the approximate outcome of an extremely complicated physical interaction. The strength of the approximation is its simplicity and versatility. Though the relationship between normal force and frictional force is not exactly linear (and so the frictional force is not entirely independent of the contact area of the surfaces), the Coulomb approximation is an adequate representation of friction for the analysis of many physical systems.
 When the surfaces are conjoined, Coulomb friction becomes a very poor approximation (for example, [[adhesive tape]] resists sliding even when there is no normal force, or a negative normal force). In this case, the frictional force may depend strongly on the area of contact. Some [[drag racing]] tires are adhesive for this reason. However, despite the complexity of the fundamental physics behind friction, the relationships are accurate enough to be useful in many applications.
-===="Negative" coefficient of friction====
+== Dry friction and instabilities ==
-{{As of|2012}}, a single study has demonstrated the potential for an ''effectively negative coefficient of friction in the low-load regime'', meaning that a decrease in normal force leads to an increase in friction. This contradicts everyday experience in which an increase in normal force leads to an increase in friction.<ref name="Deng-2012"/>  This was reported in the journal ''Nature'' in October 2012 and involved the friction encountered by an atomic force microscope stylus when dragged across a graphene sheet in the presence of graphene-adsorbed oxygen.<ref name="Deng-2012">{{cite journal |last1=Deng |first1=Zhao |display-authors=etal |date=October 14, 2012 |title=Adhesion-dependent negative friction coefficient on chemically modified graphite at the nanoscale |journal=[[Nature (journal)|Nature]] |bibcode=2012NatMa..11.1032D |doi=10.1038/nmat3452 |pmid=23064494 |volume=11 |issue=12 |pages=1032–7}}
-* {{cite magazine |date=2012-10-17 |title=At the nanoscale, graphite can turn friction upside down |magazine=R&D Magazine |url=http://www.rdmag.com/news/2012/10/nanoscale-graphite-can-turn-friction-upside-down |archive-url=https://web.archive.org/web/20130731031301/http://www.rdmag.com/news/2012/10/nanoscale-graphite-can-turn-friction-upside-down |archive-date=2013-07-31}}</ref>
-===Numerical simulation of the Coulomb model===
-Despite being a simplified model of friction, the Coulomb model is useful in many [[numerical simulation]] applications such as [[multibody system]]s and [[granular material]]. Even its most simple expression encapsulates the fundamental effects of sticking and sliding which are required in many applied cases, although specific algorithms have to be designed in order to efficiently [[Numerical integration|numerically integrate]] mechanical systems with Coulomb friction and bilateral or unilateral contact.<ref>{{cite journal |last1=Haslinger |first1=J. |title=Approximation of the Signorini problem with friction, obeying the Coulomb law |journal=Mathematical Methods in the Applied Sciences |volume=5 |issue=1 |pages=422–437 |year=1983 |bibcode=1983MMAS....5..422H |doi=10.1002/mma.1670050127 |last2=Nedlec |first2=J.C. |hdl=10338.dmlcz/104086 |url=http://dml.cz/bitstream/handle/10338.dmlcz/104086/AplMat_29-1984-3_6.pdf |access-date=2019-09-19 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007083034/http://dml.cz/bitstream/handle/10338.dmlcz/104086/AplMat_29-1984-3_6.pdf |url-status=live }}</ref><ref>{{cite journal |last1=Alart |first1=P. |last2=Curnier |first2=A. |title=A mixed formulation for frictional contact problems prone to Newton like solution method |journal=Computer Methods in Applied Mechanics and Engineering |volume=92 |pages=353–375 |year=1991 |bibcode=1991CMAME..92..353A |doi=10.1016/0045-7825(91)90022-X |issue=3 |url=https://hal.science/hal-04264964/file/Alart1991.pdf |access-date=2024-03-29 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007091441/https://hal.science/hal-04264964/file/Alart1991.pdf |url-status=live }}</ref><ref>{{cite journal |last1=Acary |first1=V. |last2=Cadoux |first2=F. |last3=Lemaréchal |first3=C. |last4=Malick |first4=J. |title=A formulation of the linear discrete Coulomb friction problem via convex optimization |journal=Journal of Applied Mathematics and Mechanics |volume=91 |issue=2 |pages=155–175 |year=2011 |doi=10.1002/zamm.201000073 |bibcode=2011ZaMM...91..155A |s2cid=17280625 |url=https://hal.inria.fr/inria-00495734/document |access-date=2018-04-20 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007083036/https://hal.inria.fr/inria-00495734/document |url-status=live }}</ref><ref>{{cite journal |last1=De Saxcé |first1=G. |last2=Feng |first2=Z.-Q. |title=The bipotential method: A constructive approach to design the complete contact law with friction and improved numerical algorithms |journal=Mathematical and Computer Modelling |volume=28 |issue=4 |pages=225–245 |year=1998 |doi=10.1016/S0895-7177(98)00119-8|doi-access=free }}</ref><ref>{{cite journal |last1=Simo |first1=J.C. |last2=Laursen |first2=T.A. |title=An augmented lagrangian treatment of contact problems involving friction |journal=Computers and Structures |volume=42 |issue=2 |pages=97–116 |year=1992 |doi=10.1016/0045-7949(92)90540-G|doi-access=free }}</ref> Some quite [[Nonlinear system#Types of nonlinear dynamic behaviors|nonlinear effects]], such as the so-called [[Painlevé paradox]]es, may be encountered with Coulomb friction.<ref>{{cite book |last1=Acary |first1=V. |last2=Brogliato |first2=B. |title=Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and Electronics |publisher=[[Springer Science+Business Media|Springer Verlag Heidelberg]] |volume=35 |year=2008}}</ref>
-===Dry friction and instabilities===
 Dry friction can induce several types of instabilities in mechanical systems which display a stable behaviour in the absence of friction.<ref>{{cite book |last=Bigoni |first=D. |title=Nonlinear Solid Mechanics: Bifurcation Theory and Material Instability |publisher=Cambridge University Press |isbn=978-1-107-02541-7|year=2012}}</ref>
-These instabilities may be caused by the decrease of the friction force with an increasing velocity of sliding, by material expansion due to heat generation during friction (the thermo-elastic instabilities), or by pure dynamic effects of sliding of two elastic materials (the Adams–Martins instabilities). The latter were originally discovered in 1995 by [[George G. Adams (engineer)|George G. Adams]] and [[João Arménio Correia Martins]] for smooth surfaces<ref>{{cite journal |last=Adams |first=G.G. |title=Self-excited oscillations of two elastic half-spaces sliding with a constant coefficient of friction |doi=10.1115/1.2896013 |journal=Journal of Applied Mechanics |year=1995 |volume=62 |issue=4 |pages=867–872 |bibcode=1995JAM....62..867A }}</ref><ref>{{cite journal |last=Martins |first=J.A., Faria, L.O. & Guimarães, J. |title=Dynamic surface solutions in linear elasticity and viscoelasticity with frictional boundary conditions |doi=10.1115/1.2874477 |journal=Journal of Vibration and Acoustics |year=1995 |volume=117 |issue=4 |pages=445–451}}</ref> and were later found in periodic rough surfaces.<ref>{{cite journal |last1=M |first1=Nosonovsky |last2=G. |first2=Adams G. |title=Vibration and stability of frictional sliding of two elastic bodies with a wavy contact interface |doi=10.1115/1.1653684 |journal=Journal of Applied Mechanics |year=2004 |volume=71 |issue=2 |pages=154–161 |bibcode=2004JAM....71..154N }}</ref> In particular, friction-related dynamical instabilities are thought to be responsible for [[Brake#Noise|brake squeal]] and the 'song' of a [[glass harp]],<ref>{{cite journal |last2=J. |first2=Hultén |last1=J. |first1=Flint |title=Lining-deformation-induced modal coupling as squeal generator in a distributed parameter disk brake model |doi=10.1006/jsvi.2001.4052 |bibcode=2002JSV...254....1F |journal= Journal of Sound and Vibration|year=2002 |volume=254 |issue=1 |pages=1–21}}</ref><ref>{{cite journal |last1=M. |first1=Kröger |last2=M. |first2=Neubauer |last3=K. |first3=Popp |s2cid=16395796 |title=Experimental investigation on the avoidance of self-excited vibrations |journal=Phil. Trans. R. Soc. A |doi=10.1098/rsta.2007.2127 |year=2008 |pages=785–810 |volume=366 |issue=1866 |pmid=17947204 |bibcode=2008RSPTA.366..785K }}</ref> phenomena which involve stick and slip, modelled as a drop of friction coefficient with velocity.<ref>{{cite journal |last2=L. |first2=Ruina, A. |last1=R. |first1=Rice, J. |title=Stability of Steady Frictional Slipping |journal=Journal of Applied Mechanics |volume=50 |year=1983 |pages=343–349 |url=http://ruina.tam.cornell.edu/research/topics/friction_and_fracture/stability_steady.pdf |archive-url=https://web.archive.org/web/20100622193459/http://ruina.tam.cornell.edu/research/topics/friction_and_fracture/stability_steady.pdf |archive-date=2010-06-22 |url-status=live |doi=10.1115/1.3167042 |issue=2 |bibcode=1983JAM....50..343R |citeseerx=10.1.1.161.5207 }}</ref>
+These instabilities may be caused by the decrease of the friction force with an increasing velocity of sliding, by material expansion due to heat generation during friction (the thermo-elastic instabilities), or by pure dynamic effects of sliding of two elastic materials (the Adams–Martins instabilities). The latter were originally discovered in 1995 by [[George G. Adams (engineer)|George G. Adams]] and [[João Arménio Correia Martins]] for smooth surfaces<ref>{{cite journal |last=Adams |first=G.G. |title=Self-excited oscillations of two elastic half-spaces sliding with a constant coefficient of friction |doi=10.1115/1.2896013 |journal=Journal of Applied Mechanics |year=1995 |volume=62 |issue=4 |pages=867–872 |bibcode=1995JAM....62..867A }}</ref><ref>{{cite journal | last1=Martins | first1=J. A. C. | last2=Guimara˜es | first2=J. | last3=Faria | first3=L. O. |title=Dynamic surface solutions in linear elasticity and viscoelasticity with frictional boundary conditions |doi=10.1115/1.2874477 |journal=Journal of Vibration and Acoustics |year=1995 |volume=117 |issue=4 |pages=445–451}}</ref> and were later found in periodic rough surfaces.<ref>{{cite journal | last1=Nosonovsky | first1=Mikhail | last2=Adams | first2=George G. |title=Vibration and stability of frictional sliding of two elastic bodies with a wavy contact interface |doi=10.1115/1.1653684 |journal=Journal of Applied Mechanics |year=2004 |volume=71 |issue=2 |pages=154–161 |bibcode=2004JAM....71..154N }}</ref> In particular, friction-related dynamical instabilities are thought to be responsible for [[Brake#Noise|brake squeal]] and the 'song' of a [[glass harp]],<ref>{{cite journal | last1=Flint | first1=J. | last2=Hultén | first2=Johan |title=Lining-deformation-induced modal coupling as squeal generator in a distributed parameter disk brake model |doi=10.1006/jsvi.2001.4052 |bibcode=2002JSV...254....1F |journal= Journal of Sound and Vibration|year=2002 |volume=254 |issue=1 |pages=1–21}}</ref><ref>{{cite journal | last1=Kröger | first1=Matthias | last2=Neubauer | first2=Marcus | last3=Popp | first3=Karl |s2cid=16395796 |title=Experimental investigation on the avoidance of self-excited vibrations |journal=Phil. Trans. R. Soc. A |doi=10.1098/rsta.2007.2127 |year=2008 |pages=785–810 |volume=366 |issue=1866 |pmid=17947204 |bibcode=2008RSPTA.366..785K }}</ref> phenomena which involve stick and slip, modelled as a drop of friction coefficient with velocity.<ref>{{cite journal | last1=Rice | first1=J. R. | last2=Ruina | first2=A. L. |title=Stability of Steady Frictional Slipping |journal=Journal of Applied Mechanics |volume=50 |year=1983 |pages=343–349 |url=https://ruina.tam.cornell.edu/research/topics/friction_and_fracture/stability_steady.pdf |archive-url=https://web.archive.org/web/20100622193459/http://ruina.tam.cornell.edu/research/topics/friction_and_fracture/stability_steady.pdf |archive-date=2010-06-22 |url-status=live |doi=10.1115/1.3167042 |issue=2 |bibcode=1983JAM....50..343R |citeseerx=10.1.1.161.5207 }}</ref>
 A practically important case is the [[self-oscillation]] of the strings of [[bowed instruments]] such as the [[violin]], [[cello]], [[hurdy-gurdy]], [[erhu]], etc.
@@ Line 504: / Line 663: @@
   |archive-url  = https://web.archive.org/web/20200818112658/http://www.ing.unitn.it/~bigoni/
-}}</ref> watch the [http://www.ing.unitn.it/~bigoni/flutter.html movie] {{Webarchive|url=https://web.archive.org/web/20150110065917/http://www.ing.unitn.it/~bigoni/flutter.html |date=2015-01-10 }} for more details.
+}}</ref> watch the [https://bigoni.dicam.unitn.it/flutter.html movie] {{Webarchive|url=https://web.archive.org/web/20251211145054/https://bigoni.dicam.unitn.it/flutter.html|date=2025-12-11}} for more details. Frictional instabilities can lead to the formation of new self-organized patterns (or "secondary structures") at the sliding interface, such as in-situ formed tribofilms which are utilized for the reduction of friction and wear in so-called self-lubricating materials.<ref>{{cite book
-Frictional instabilities can lead to the formation of new self-organized patterns (or "secondary structures") at the sliding interface, such as in-situ formed tribofilms which are utilized for the reduction of friction and wear in so-called self-lubricating materials.<ref>{{cite book
 | title = Friction-Induced Vibrations and Self-Organization: Mechanics and Non-Equilibrium Thermodynamics of Sliding Contact
 | first = Michael
@@ Line 515: / Line 672: @@
 | url = http://www.crcpress.com/product/isbn/9781466504011
 | page = 333}}</ref>
+== Friction at the nanoscale ==
+{{Main|Nanotribology}}
+In 2008 scientists for the first time were able to move a single atom across a surface, and measure the forces required. Using ultrahigh vacuum and nearly zero temperature (5 K), a modified atomic force microscope was used to drag a [[cobalt]] atom, and a [[carbon monoxide]] molecule, across surfaces of copper and [[platinum]].<ref>{{Cite journal |last1=Ternes |first1=Markus |last2=Lutz |first2=Christopher P. |last3=Hirjibehedin |first3=Cyrus F. |last4=Giessibl |first4=Franz J. |author-link5= Andreas J. Heinrich |last5=Heinrich |first5=Andreas J. |title=The Force Needed to Move an Atom on a Surface |journal=[[Science (journal)|Science]] |volume=319 |issue=5866 |pages=1066–1069 |date=2008-02-22 |doi=10.1126/science.1150288 |pmid=18292336 |bibcode=2008Sci...319.1066T |s2cid=451375 |url=https://epub.uni-regensburg.de/25284/1/The%20Force%20Needed%20to%20Move%20an%20Atom%20on%20.pdf |archive-url=https://web.archive.org/web/20180720063201/https://epub.uni-regensburg.de/25284/1/The%20Force%20Needed%20to%20Move%20an%20Atom%20on%20.pdf |archive-date=2018-07-20 |url-status=live }}</ref>
+{{As of|2012}}, a single study has demonstrated the potential for an ''effectively negative coefficient of friction in the low-load regime'', meaning that a decrease in normal force leads to an increase in friction. This contradicts everyday experience in which an increase in normal force leads to an increase in friction.<ref name="Deng-2012">{{cite journal |last1=Deng |first1=Zhao |display-authors=etal |date=October 14, 2012 |title=Adhesion-dependent negative friction coefficient on chemically modified graphite at the nanoscale |journal=[[Nature (journal)|Nature]] |bibcode=2012NatMa..11.1032D |doi=10.1038/nmat3452 |pmid=23064494 |volume=11 |issue=12 |pages=1032–7}}
+*{{cite magazine |date=2012-10-17 |title=At the nanoscale, graphite can turn friction upside down |magazine=R&D Magazine |url=http://www.rdmag.com/news/2012/10/nanoscale-graphite-can-turn-friction-upside-down |archive-url=https://web.archive.org/web/20130731031301/http://www.rdmag.com/news/2012/10/nanoscale-graphite-can-turn-friction-upside-down |archive-date=2013-07-31}}</ref> This was reported in the journal ''Nature'' in October 2012 and involved the friction encountered by an atomic force microscope stylus when dragged across a graphene sheet in the presence of graphene-adsorbed oxygen.<ref name="Deng-2012" />
+The origin of kinetic friction at nanoscale can be rationalized by an energy model.<ref>{{cite journal |last1=Makkonen |first1=L |year=2012 |title=A thermodynamic model of sliding friction |doi=10.1063/1.3699027 |journal=AIP Advances |volume=2 |issue= 1|page=012179 |bibcode=2012AIPA....2a2179M |doi-access=free }}</ref> During sliding, a new surface forms at the back of a sliding true contact, and existing surface disappears at the front of it. Since all surfaces involve the thermodynamic surface energy, work must be spent in creating the new surface, and energy is released as heat in removing the surface. Thus, a force is required to move the back of the contact, and frictional heat is released at the front.
+Under certain conditions some materials have very low friction coefficients. An example is (highly ordered pyrolytic) graphite which can have a friction coefficient below 0.01.<ref>{{cite journal |last=Dienwiebel |first=Martin |title=Superlubricity of Graphite |display-authors=etal |journal=Phys. Rev. Lett. |volume=92 |article-number=126101 |year=2004 |doi=10.1103/PhysRevLett.92.126101 |url=http://www.physics.leidenuniv.nl/sections/cm/ip/group/PDF/Phys.rev.lett/2004/92(2004)12601.pdf |issue=12 |bibcode=2004PhRvL..92l6101D |pmid=15089689 |s2cid=26811802 |access-date=2011-09-01 |archive-date=2011-09-17 |archive-url=https://web.archive.org/web/20110917120623/http://www.physics.leidenuniv.nl/sections/cm/ip/group/PDF/Phys.rev.lett/2004/92(2004)12601.pdf |url-status=live }}</ref>
+This ultralow-friction regime is called [[superlubricity]].<ref>{{Citation |last=Müser |first=Martin H. |chapter=Theoretical Studies of Superlubricity |date=2015 |title=Fundamentals of Friction and Wear on the Nanoscale |series=NanoScience and Technology |pages=209–232 |editor-last=Gnecco |editor-first=Enrico |url=https://link.springer.com/10.1007/978-3-319-10560-4_11 |access-date=2025-04-25 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-10560-4_11 |bibcode=2015ffwn.book..209M |isbn=978-3-319-10559-8 |editor2-last=Meyer |editor2-first=Ernst|url-access=subscription }}</ref>
 ==Fluid friction{{anchor|Fluid}}==
@@ Line 526: / Line 695: @@
 {{Main|Lubrication}}
-Lubricated friction is a case of fluid friction where a fluid separates two solid surfaces. Lubrication is a technique employed to reduce wear of one or both surfaces in close proximity moving relative to each another by interposing a substance called a lubricant between the surfaces.
+Lubricated friction is a case of fluid friction where a fluid separates two solid surfaces. Lubrication is a technique employed to reduce wear of one or both surfaces in close proximity moving relative to each another by interposing a substance called a lubricant between the surfaces. In most cases the applied load is carried by pressure generated within the fluid due to the frictional viscous resistance to motion of the lubricating fluid between the surfaces. Adequate lubrication allows smooth continuous operation of equipment, with only mild wear, and without excessive stresses or seizures at bearings. When lubrication breaks down, metal or other components can rub destructively over each other, causing heat and possibly damage or failure.<ref>{{Cite book |last=Hamrock |first=Bernard J. |title=Fundamentals of fluid film lubrication |date=2004 |publisher=Marcel Dekker |others=Steven R. Schmid, Bo O. Jacobson |isbn=0-8247-5120-5 |edition=2nd |location=New York |oclc=55739786}}</ref>
-In most cases the applied load is carried by pressure generated within the fluid due to the frictional viscous resistance to motion of the lubricating fluid between the surfaces. Adequate lubrication allows smooth continuous operation of equipment, with only mild wear, and without excessive stresses or seizures at bearings. When lubrication breaks down, metal or other components can rub destructively over each other, causing heat and possibly damage or failure.
 ==Skin friction==
 {{Main|Parasitic drag}}
-Skin friction arises from the interaction between the fluid and the skin of the body, and is directly related to the area of the surface of the body that is in contact with the fluid. Skin friction follows the [[drag equation]] and rises with the square of the velocity.
+Skin friction arises from the interaction between the fluid and the skin of an object, and is directly related to the area of the surface of the body that is in contact with the fluid. Skin friction follows the [[drag equation]] and rises with the square of the velocity.<ref name=":1">{{Cite book |last=White |first=Frank |title=Fluid Mechanics |publisher=McGraw-Hill |year=2011 |isbn=9780071311212 |location=New York City, NY |pages=477–478}}</ref>
-Skin friction is caused by viscous drag in the [[boundary layer]] around the object. There are two ways to decrease skin friction: the first is to shape the moving body so that smooth flow is possible, like an airfoil. The second method is to decrease the length and cross-section of the moving object as much as is practicable.
+Skin friction is caused by viscous drag in the [[boundary layer]] around the object. There are two ways to decrease skin friction: the first is to shape the moving body so that smooth flow is possible, like an airfoil. The second method is to decrease the length and cross-section of the moving object as much as is practicable.<ref name=":1" />
 ==Internal friction==
 {{Main|Plastic deformation of solids}}
 {{See also|Deformation (engineering)}}
-Internal friction is the force resisting motion between the elements making up a solid material while it undergoes [[deformation (engineering)|deformation]].
+Internal friction is the force resisting motion between the elements making up a solid material while it undergoes [[deformation (engineering)|deformation]]. The best understood case is the drag force on [[Dislocation|dislocations]] due to atomic vibrations ([[phonon]] drag),<ref>{{Cite journal |last=Al'shitz |first=V A |last2=Indenbom |first2=V L |date=1975-01-31 |title=Dynamic dragging of dislocations |url=https://ufn.ru/en/articles/1975/1/a/ |journal=Soviet Physics Uspekhi |volume=18 |issue=1 |pages=1–20 |doi=10.1070/PU1975v018n01ABEH004689 |issn=0038-5670}}</ref> although electron interactions can also play a role.<ref>{{Cite journal |last=Kaganov |first=Moisei I |last2=Kravchenko |first2=Ya V |last3=Natsik |first3=V D |date=1974-06-30 |title=Dislocation dragging by electrons in metals |url=https://ufn.ru/en/articles/1974/6/i/ |journal=Soviet Physics Uspekhi |volume=16 |issue=6 |pages=878–891 |doi=10.1070/PU1974v016n06ABEH004096 |issn=0038-5670}}</ref> There are also contributions from dissipative forces on the motion of [[grain boundaries]] in metals,<ref>{{Cite journal |last=Nowick |first=A. S. |date=1953-01-01 |title=Internal friction in metals |url=https://www.sciencedirect.com/science/article/pii/0502820553900148 |journal=Progress in Metal Physics |volume=4 |pages=1–70 |doi=10.1016/0502-8205(53)90014-8 |issn=0502-8205}}</ref> and there are similar phenomena in vitreous materials such as [[Glass|glasses]].<ref>{{Cite journal |last=Zdaniewski |first=Wieslaw A. |last2=Rindone |first2=Guy E. |last3=Day |first3=Delbert E. |date=1979-04-01 |title=The internal friction of glasses |url=https://doi.org/10.1007/BF00550707 |journal=Journal of Materials Science |language=en |volume=14 |issue=4 |pages=763–775 |doi=10.1007/BF00550707 |issn=1573-4803}}</ref>
-[[Plastic deformation]] in solids is an irreversible change in the internal molecular structure of an object. This change may be due to either (or both) an applied force or a change in temperature. The change of an object's shape is called strain. The force causing it is called [[stress (mechanics)|stress]].
-[[Elastic deformation]] in solids is reversible change in the internal molecular structure of an object. Stress does not necessarily cause permanent change. As deformation occurs, internal forces oppose the applied force. If the applied stress is not too large these opposing forces may completely resist the applied force, allowing the object to assume a new equilibrium state and to return to its original shape when the force is removed. This is known as elastic deformation or elasticity.
 ==Radiation friction==
@@ Line 554: / Line 718: @@
 {{Main|Rolling resistance}}
-Rolling resistance is the force that resists the rolling of a wheel or other circular object along a surface caused by deformations in the object or surface. Generally the force of rolling resistance is less than that associated with kinetic friction.<ref>Silliman, Benjamin (1871) ''Principles of Physics, Or Natural Philosophy'', Ivison, Blakeman, Taylor & company publishers</ref> Typical values for the coefficient of rolling resistance are 0.001.<ref>Butt, Hans-Jürgen; Graf, Karlheinz and Kappl, Michael (2006) ''Physics and Chemistry of Interfaces'', Wiley, {{ISBN|3-527-40413-9}}</ref>
+Rolling resistance is the force that resists the rolling of a wheel or other circular object along a surface caused by deformations in the object or surface. Generally the force of rolling resistance is less than that associated with kinetic friction.<ref>Silliman, Benjamin (1871) ''Principles of Physics, Or Natural Philosophy'', Ivison, Blakeman, Taylor & company publishers</ref> Typical values for the coefficient of rolling resistance for steel wheels on a track are 0.001.<ref>Butt, Hans-Jürgen; Graf, Karlheinz and Kappl, Michael (2006) ''Physics and Chemistry of Interfaces'', Wiley, {{ISBN|3-527-40413-9}}</ref>
-One of the most common examples of rolling resistance is the movement of [[motor vehicle]] tires on a [[road]], a process which generates heat and [[roadway noise|sound]] as by-products.<ref>{{cite journal |last1=Hogan |first1=C. Michael |s2cid=109914430 |title=Analysis of highway noise |journal=Water, Air, & Soil Pollution |volume=2 |pages=387–392 |year=1973 |doi=10.1007/BF00159677 |issue=3|bibcode=1973WASP....2..387H }}</ref>
+One of the most common examples of rolling resistance, with a coefficient of around 0.02, is the movement of [[motor vehicle]] tires on a [[road]], a process which generates heat and [[roadway noise|sound]] as by-products.<ref>{{cite journal |last1=Hogan |first1=C. Michael |s2cid=109914430 |title=Analysis of highway noise |journal=Water, Air, & Soil Pollution |volume=2 |pages=387–392 |year=1973 |doi=10.1007/BF00159677 |issue=3|bibcode=1973WASP....2..387H }}</ref>
 ===Braking friction===
@@ Line 563: / Line 727: @@
 {{Main|Triboelectric effect}}
-Rubbing two materials against each other can lead to charge transfer, either electrons or ions. The energy required for this contributes to the friction. In addition, sliding can cause a build-up of [[electrostatic charge]], which can be hazardous if flammable gases or vapours are present. When the static build-up discharges, [[explosion]]s can be caused by ignition of the flammable mixture.
+Rubbing two materials against each other can lead to charge transfer, either electrons or ions. The energy required for this contributes to the friction. In addition, sliding can cause a build-up of [[electrostatic charge]], which can be hazardous if flammable gases or vapours are present. When the static build-up discharges, [[explosion]]s can be caused by ignition of the flammable mixture. However, as discussed recently<ref>{{Cite journal |last=Olson |first=Karl P |last2=Marks |first2=Laurence D |date=2025-10-01 |title=Is triboelectricity confusing, confused or complex? |url=https://iopscience.iop.org/article/10.1088/1361-6633/ae08cb |journal=Reports on Progress in Physics |volume=88 |issue=10 |pages=Section 4.1 |doi=10.1088/1361-6633/ae08cb |issn=0034-4885}}</ref> the extent to which triboelectricity contributes to friction is contentious, with some claims that it is large,<ref>{{Cite journal |last=Burgo |first=Thiago A. L. |last2=Silva |first2=Cristiane A. |last3=Balestrin |first3=Lia B. S. |last4=Galembeck |first4=Fernando |date=2013-08-12 |title=Friction coefficient dependence on electrostatic tribocharging |url=https://www.nature.com/articles/srep02384 |journal=Scientific Reports |language=en |volume=3 |issue=1 |pages=2384 |doi=10.1038/srep02384 |issn=2045-2322 |pmc=3740278 |pmid=23934227}}</ref> others that it is small because charge transfer can occur with superlubricious samples.<ref>{{Cite journal |last=Huang |first=Xuanyu |last2=Xiang |first2=Xiaojian |last3=Nie |first3=Jinhui |last4=Peng |first4=Deli |last5=Yang |first5=Fuwei |last6=Wu |first6=Zhanghui |last7=Jiang |first7=Haiyang |last8=Xu |first8=Zhiping |last9=Zheng |first9=Quanshui |date=2021-04-15 |title=Microscale Schottky superlubric generator with high direct-current density and ultralong life |url=https://www.nature.com/articles/s41467-021-22371-1 |journal=Nature Communications |language=en |volume=12 |issue=1 |pages=2268 |doi=10.1038/s41467-021-22371-1 |issn=2041-1723 |pmc=8050059 |pmid=33859180}}</ref>
 ===Belt friction===
@@ Line 573: / Line 737: @@
 In practice, the theoretical tension acting on the belt or rope calculated by the belt friction equation can be compared to the maximum tension the belt can support. This helps a designer of such a rig to know how many times the belt or rope must be wrapped around the pulley to prevent it from slipping. Mountain climbers and sailing crews demonstrate a standard knowledge of belt friction when accomplishing basic tasks.
-==Reduction==
+==Friction reduction==
 ===Devices===
@@ Line 588: / Line 752: @@
 Another way to reduce friction between two parts is to superimpose micro-scale vibration to one of the parts. This can be sinusoidal vibration as used in ultrasound-assisted cutting or vibration noise, known as [[dither]].
 ==Applications==
 Friction is an important factor in many [[engineering]] disciplines.
@@ Line 632: / Line 797: @@
 * [https://web.archive.org/web/20070601020244/http://www.physclips.unsw.edu.au/ Physclips: Mechanics with animations and video clips] from the University of New South Wales
 * [https://books.google.com/books?id=WDll8hA006AC&pg=PT2503&lpg=PT2503 Values for Coefficient of Friction] – ''[[CRC Handbook of Chemistry and Physics]]''
-* [http://www.tribonet.org/wiki/friction-coefficients-in-atmosphere-and-vacuum/ Coefficients of friction of various material pairs in atmosphere and vacuum].
+* [http://www.tribonet.org/wiki/friction-coefficients-in-atmosphere-and-vacuum/ Coefficients of friction of various material pairs in atmosphere and vacuum] {{Webarchive|url=https://web.archive.org/web/20230531225136/https://www.tribonet.org/wiki/friction-coefficients-in-atmosphere-and-vacuum/ |date=2023-05-31 }}.
 {{Geotechnical engineering}}

Friction: Difference between revisions

Latest revision as of 09:05, 25 April 2026

Types

History

Dry friction

Static friction

Kinetic friction

Role of the normal force

Role of angle

Coefficient of friction

Approximate coefficients of friction

Sources of friction

Breakdown of the Coulomb model

Dry friction and instabilities

Friction at the nanoscale

Fluid friction

Lubricated friction

Skin friction

Internal friction

Radiation friction

Other types of friction

Rolling resistance

Braking friction

Triboelectric effect

Belt friction

Friction reduction

Devices

Lubricants

Applications

Transportation

Measurement

Household usage

See also

References

External links

Navigation menu

Search