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90-481: [REDACTED] Look up constitutive in Wiktionary, the free dictionary. Constitutive may refer to: In physics, a constitutive equation is a relation between two physical quantities In ecology , a constitutive defense is one that is always active, as opposed to an inducible defense Constitutive theory of statehood In biochemistry and pharmacology,

180-419: A constitutively active receptor produces a biological response in the absence of a bound ligand In genetics, a constitutive gene is always expressed – see constitutive expression Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Constitutive . If an internal link led you here, you may wish to change the link to point directly to

270-506: A consequence of the definition, the speed of light in matter is for special case of vacuum; ε = ε 0 and μ = μ 0 , The piezooptic effect relates the stresses in solids σ to the dielectric impermeability a , which are coupled by a fourth-rank tensor called the piezooptic coefficient Π (units K ): There are several laws which describe the transport of matter, or properties of it, in an almost identical way. In every case, in words they read: Friction Friction

360-401: 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. As briefly discussed later, there are many different contributors to the retarding force in friction, ranging from asperity deformation to

450-645: A graphene sheet in the presence of graphene-adsorbed oxygen. 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. Some quite nonlinear effects , such as

540-428: A homogeneous effective medium (valid for excitations with wavelengths much larger than the scale of the inhomogeneity). The theoretical modeling of the continuum-approximation properties of many real materials often rely upon experimental measurement as well. For example, ε of an insulator at low frequencies can be measured by making it into a parallel-plate capacitor , and ε at optical-light frequencies

630-410: A material is transparent; and metals with finite conductivity often are approximated at microwave or longer wavelengths as perfect metals with infinite conductivity (forming hard barriers with zero skin depth of field penetration). Some man-made materials such as metamaterials and photonic crystals are designed to have customized permittivity and permeability. The theoretical calculation of

720-481: A material's constitutive equations is a common, important, and sometimes difficult task in theoretical condensed-matter physics and materials science . In general, the constitutive equations are theoretically determined by calculating how a molecule responds to the local fields through the Lorentz force . Other forces may need to be modeled as well such as lattice vibrations in crystals or bond forces. Including all of

810-411: 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. For certain applications, it

900-400: A parameter describing the scaling behavior of surface asperities, is known to play an important role in determining the magnitude of the static friction. For surfaces in relative motion μ = μ k {\displaystyle \mu =\mu _{\mathrm {k} }} , where μ k {\displaystyle \mu _{\mathrm {k} }} is

990-472: 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. 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. The focus of research during

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1080-515: 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. This was reported in the journal Nature in October 2012 and involved the friction encountered by an atomic force microscope stylus when dragged across

1170-506: A suitable volume to form a continuum approximation. These continuum approximations often require some type of quantum mechanical analysis such as quantum field theory as applied to condensed matter physics . See, for example, density functional theory , Green–Kubo relations and Green's function . A different set of homogenization methods (evolving from a tradition in treating materials such as conglomerates and laminates ) are based upon approximation of an inhomogeneous material by

1260-419: A threshold value for this force, above which motion would commence. This maximum force is known as 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,

1350-420: A uniform shear flow : with u ( y ) the variation of the flow velocity u in the cross-flow (transverse) direction y . In general, for a Newtonian fluid, the relationship between the elements τ ij of the shear stress tensor and the deformation of the fluid is given by where v i are the components of the flow velocity vector in the corresponding x i coordinate directions, e ij are

1440-434: 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. As of 2012 ,

1530-409: Is (highly ordered pyrolytic) graphite which can have a friction coefficient below 0.01. This ultralow-friction regime is called superlubricity . 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 ,

1620-498: 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

1710-408: Is an empirical measurement   —   it has to be measured experimentally , and cannot be found through calculations. 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

1800-495: Is an approximate model used to calculate the force of dry friction. It is governed by the model: F f ≤ μ F n , {\displaystyle F_{\mathrm {f} }\leq \mu F_{\mathrm {n} },} where The Coulomb friction F f {\displaystyle F_{\mathrm {f} }} may take any value from zero up to μ F n {\displaystyle \mu F_{\mathrm {n} }} , and

1890-425: Is easier to further the motion of a moving body than to move a body at rest". 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. These laws were rediscovered by Guillaume Amontons in 1699 and became known as Amonton's three laws of dry friction. Amontons presented

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1980-429: Is expressed as a simple proportionality using a parameter taken to be a property of the material, such as electrical conductivity or a spring constant . However, it is often necessary to account for the directional dependence of the material, and the scalar parameter is generalized to a tensor . Constitutive relations are also modified to account for the rate of response of materials and their non-linear behavior. See

2070-437: 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. For surfaces at rest relative to each other, μ = μ s {\displaystyle \mu =\mu _{\mathrm {s} }} , where μ s {\displaystyle \mu _{\mathrm {s} }}

2160-424: Is impending, is sometimes referred to as limiting friction , although this term is not used universally. 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

2250-415: 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

2340-562: 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: tan ⁡ θ = μ s {\displaystyle \tan {\theta }=\mu _{\mathrm {s} }} and thus: θ = arctan ⁡ μ s {\displaystyle \theta =\arctan {\mu _{\mathrm {s} }}} where θ {\displaystyle \theta }

2430-479: Is no sliding occurring, the friction force can have any value from zero up to F max {\displaystyle F_{\text{max}}} . Any force smaller than F max {\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 F max {\displaystyle F_{\text{max}}} overcomes

2520-412: Is often measured by ellipsometry . These constitutive equations are often used in crystallography , a field of solid-state physics . The (absolute) refractive index of a medium n (dimensionless) is an inherently important property of geometric and physical optics defined as the ratio of the luminal speed in vacuum c 0 to that in the medium c : where ε is the permittivity and ε r

2610-456: 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

2700-434: Is on a level surface and subjected to an external force P {\displaystyle P} tending to cause it to slide, then the normal force between the object and the surface is just N = m g + P y {\displaystyle N=mg+P_{y}} , where m g {\displaystyle mg} is the block's weight and P y {\displaystyle P_{y}}

2790-423: 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 N = m g {\displaystyle N=mg\,} . In this case, conditions of equilibrium tell us that the magnitude of the friction force is zero , F f = 0 {\displaystyle F_{f}=0} . In fact,

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2880-409: 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

2970-573: Is responsible for the Coulomb damping of an oscillating or vibrating system. New models are beginning to show how kinetic friction can be greater than static friction. In many other cases roughness effects are dominant, for example in rubber to road friction. Surface roughness and contact area affect kinetic friction for micro- and nano-scale objects where surface area forces dominate inertial forces. The origin of kinetic friction at nanoscale can be rationalized by an energy model. During sliding,

3060-466: 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,

3150-460: Is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. 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. Friction can have dramatic consequences, as illustrated by the use of friction created by rubbing pieces of wood together to start

3240-412: Is the angle from horizontal and μ s is the static coefficient of friction between the objects. This formula can also be used to calculate μ s from empirical measurements of the friction angle. 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

3330-497: Is the downward component of the external force. Prior to sliding, this friction force is F f = − P x {\displaystyle F_{f}=-P_{x}} , where P x {\displaystyle P_{x}} is the horizontal component of the external force. Thus, F f ≤ μ N {\displaystyle F_{f}\leq \mu N} in general. Sliding commences only after this frictional force reaches

3420-462: Is useful to examine the following special cases. In the absence of magnetic or dielectric materials, the constitutive relations are simple: where ε 0 and μ 0 are two universal constants, called the permittivity of free space and permeability of free space, respectively. In an ( isotropic ) linear material, where P is proportional to E , and M is proportional to B , the constitutive relations are also straightforward. In terms of

3510-537: 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. The linearity between applied load and true contact area, arising from asperity deformation, gives rise to

3600-860: The atomic scale , 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 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. Coulomb friction, named after Charles-Augustin de Coulomb ,

3690-400: The coefficient of kinetic friction . The Coulomb friction is equal to F f {\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. The coefficient of friction

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3780-451: The coefficient of restitution , defined by Newton's experimental impact law : which depends on the materials A and B are made from, since the collision involves interactions at the surfaces of A and B. Usually 0 ≤ e ≤ 1 , in which e = 1 for completely elastic collisions, and e = 0 for completely inelastic collisions . It is possible for e ≥ 1 to occur – for superelastic (or explosive) collisions. The drag equation gives

3870-477: The drag force D on an object of cross-section area A moving through a fluid of density ρ at velocity v (relative to the fluid) where the drag coefficient (dimensionless) c d depends on the geometry of the object and the drag forces at the interface between the fluid and object. For a Newtonian fluid of viscosity μ , the shear stress τ is linearly related to the strain rate (transverse flow velocity gradient ) ∂ u /∂ y (units s ). In

3960-404: The 'song' of a glass harp , phenomena which involve stick and slip, modelled as a drop of friction coefficient with velocity. 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, watch

4050-435: 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. 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

4140-408: The additional coupling constants ξ and ζ : In practice, some materials properties have a negligible impact in particular circumstances, permitting neglect of small effects. For example: optical nonlinearities can be neglected for low field strengths; material dispersion is unimportant when frequency is limited to a narrow bandwidth ; material absorption can be neglected for wavelengths for which

4230-406: 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

4320-428: The article Linear response function . The first constitutive equation (constitutive law) was developed by Robert Hooke and is known as Hooke's law . It deals with the case of linear elastic materials . Following this discovery, this type of equation, often called a "stress-strain relation" in this example, but also called a "constitutive assumption" or an "equation of state" was commonly used. Walter Noll advanced

4410-435: The classic empirical model of friction (static, kinetic, and fluid) commonly used today in engineering. In 1877, Fleeming Jenkin and J. A. Ewing investigated the continuity between static and kinetic friction. 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

4500-455: The components of the strain rate tensor, Δ is the volumetric strain rate (or dilatation rate) and δ ij is the Kronecker delta . The ideal gas law is a constitutive relation in the sense the pressure p and volume V are related to the temperature T , via the number of moles n of gas: where R is the gas constant (J⋅K ⋅mol ). In both classical and quantum physics ,

4590-424: The constitutive equation plays a major role. See Linear constitutive equations and Nonlinear correlation functions . Friction is a complicated phenomenon. Macroscopically, the friction force F between the interface of two materials can be modelled as proportional to the reaction force R at a point of contact between two interfaces through a dimensionless coefficient of friction μ f , which depends on

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4680-462: The constitutive relations are not linear, except approximately. Calculating the constitutive relations from first principles involves determining how P and M are created from a given E and B . These relations may be empirical (based directly upon measurements), or theoretical (based upon statistical mechanics , transport theory or other tools of condensed matter physics ). The detail employed may be macroscopic or microscopic , depending upon

4770-402: The constitutive relations describing the electrical response of various materials, such as permittivities , permeabilities , conductivities and so forth. It is necessary to specify the relations between displacement field D and E , and the magnetic H-field H and B , before doing calculations in electromagnetism (i.e. applying Maxwell's macroscopic equations). These equations specify

4860-855: The constitutive relations). As a result, various approximation schemes are typically used. For example, in real materials, complex transport equations must be solved to determine the time and spatial response of charges, for example, the Boltzmann equation or the Fokker–Planck equation or the Navier–Stokes equations . For example, see magnetohydrodynamics , fluid dynamics , electrohydrodynamics , superconductivity , plasma modeling . An entire physical apparatus for dealing with these matters has developed. See for example, linear response theory , Green–Kubo relations and Green's function (many-body theory) . These complex theories provide detailed formulas for

4950-501: 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 N {\displaystyle N} might include forces other than gravity. If an object

5040-433: 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

5130-433: 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 normal force is defined as the net force compressing two parallel surfaces together, and its direction

5220-459: 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). 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

5310-556: 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 F f ≤ μ N {\displaystyle F_{f}\leq \mu N} , then

5400-406: 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. An example of static friction is the force that prevents a car wheel from slipping as it rolls on

5490-451: The forces leads to changes in the molecule which are used to calculate P and M as a function of the local fields. The local fields differ from the applied fields due to the fields produced by the polarization and magnetization of nearby material; an effect which also needs to be modeled. Further, real materials are not continuous media ; the local fields of real materials vary wildly on the atomic scale. The fields need to be averaged over

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5580-402: 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 . 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

5670-489: The friction force always satisfies F f ≤ μ N {\displaystyle F_{f}\leq \mu N} , with equality reached only at a critical ramp angle (given by tan − 1 ⁡ μ {\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

5760-486: The frictional heating is removed rapidly, the temperature drops, the pin remains solid and the COF rises to that of a 'low speed' test. 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. Under certain conditions some materials have very low friction coefficients. An example

5850-413: The generation of charges and changes in local structure . Friction is not itself a fundamental force , it is a non-conservative force – work done against friction is path dependent. In the presence of friction, some mechanical energy is transformed to heat as well as the free energy of the structural changes and other types of dissipation , so mechanical energy is not conserved. The complexity of

5940-441: 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

6030-457: 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. 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. In

6120-502: The intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Constitutive&oldid=1208170337 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Constitutive equation Some constitutive equations are simply phenomenological ; others are derived from first principles . A common approximate constitutive equation frequently

6210-537: 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. There are several types of friction: Many ancient authors including Aristotle , Vitruvius , and Pliny the Elder , were interested in the cause and mitigation of friction. They were aware of differences between static and kinetic friction with Themistius stating in 350 A.D. that "it

6300-411: The interface between the materials, namely surface structure . 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

6390-408: The law defines the spring constant (or elasticity constant) k in a scalar equation, stating the tensile/compressive force is proportional to the extended (or contracted) displacement x : meaning the material responds linearly. Equivalently, in terms of the stress σ , Young's modulus E , and strain ε (dimensionless): In general, forces which deform solids can be normal to a surface of

6480-401: The level necessary to the problem under scrutiny. In general, the constitutive relations can usually still be written: but ε and μ are not, in general, simple constants, but rather functions of E , B , position and time, and tensorial in nature. Examples are: As a variation of these examples, in general materials are bianisotropic where D and B depend on both E and H , through

6570-530: The linearity between static frictional force and normal force, found for typical Amonton–Coulomb type friction. 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: F max = μ s F n {\displaystyle F_{\text{max}}=\mu _{\mathrm {s} }F_{\text{n}}} . When there

6660-498: 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. 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

6750-447: The material (normal forces), or tangential (shear forces), this can be described mathematically using the stress tensor : where C is the elasticity tensor and S is the compliance tensor . Several classes of deformations in elastic materials are the following: The relative speed of separation v separation of an object A after a collision with another object B is related to the relative speed of approach v approach by

6840-409: 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 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. John Theophilus Desaguliers (1734) first recognized

6930-443: The pair of materials: This can be applied to static friction (friction preventing two stationary objects from slipping on their own), kinetic friction (friction between two objects scraping/sliding past each other), or rolling (frictional force which prevents slipping but causes a torque to exert on a round object). The stress-strain constitutive relation for linear materials is commonly known as Hooke's law . In its simplest form,

7020-444: The polarization P and the magnetization M they are: where χ e and χ m are the electric and magnetic susceptibilities of a given material respectively. In terms of D and H the constitutive relations are: where ε and μ are constants (which depend on the material), called the permittivity and permeability , respectively, of the material. These are related to the susceptibilities by: For real-world materials,

7110-437: The precise dynamics of a system form a set of coupled differential equations , which are almost always too complicated to be solved exactly, even at the level of statistical mechanics . In the context of electromagnetism, this remark applies to not only the dynamics of free charges and currents (which enter Maxwell's equations directly), but also the dynamics of bound charges and currents (which enter Maxwell's equations through

7200-436: The relative permittivity of the medium, likewise μ is the permeability and μ r are the relative permeability of the medium. The vacuum permittivity is ε 0 and vacuum permeability is μ 0 . In general, n (also ε r ) are complex numbers . The relative refractive index is defined as the ratio of the two refractive indices. Absolute is for one material, relative applies to every possible pair of interfaces; As

7290-491: The response of bound charge and current to the applied fields and are called constitutive relations. Determining the constitutive relationship between the auxiliary fields D and H and the E and B fields starts with the definition of the auxiliary fields themselves: where P is the polarization field and M is the magnetization field which are defined in terms of microscopic bound charges and bound current respectively. Before getting to how to calculate M and P it

7380-408: The role of adhesion in friction. 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). Coulomb investigated the influence of four main factors on friction: the nature of the materials in contact and their surface coatings;

7470-405: The same materials. However, Richard Feynman comments that "with dry metals it is very hard to show any difference." The friction force between two surfaces after sliding begins is the product of the coefficient of kinetic friction and the normal force: F k = μ k F n {\displaystyle F_{k}=\mu _{\mathrm {k} }F_{n}} . This

7560-865: The so-called Painlevé paradoxes , may be encountered with Coulomb friction. Dry friction can induce several types of instabilities in mechanical systems which display a stable behaviour in the absence of friction. 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 and were later found in periodic rough surfaces. In particular, friction-related dynamical instabilities are thought to be responsible for brake squeal and

7650-435: The temperature rise. 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. Osborne Reynolds (1866) derived the equation of viscous flow. This completed

7740-456: The tentative assumption was correct, and it is the actual frictional force. Otherwise, the friction force must be set equal to F f = μ N {\displaystyle F_{f}=\mu N} , and then the resulting force imbalance would then determine the acceleration associated with slipping. The coefficient of friction (COF), often symbolized by the Greek letter μ ,

7830-417: 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

7920-427: The use of constitutive equations, clarifying their classification and the role of invariance requirements, constraints, and definitions of terms like "material", "isotropic", "aeolotropic", etc. The class of "constitutive relations" of the form stress rate = f (velocity gradient, stress, density) was the subject of Walter Noll 's dissertation in 1954 under Clifford Truesdell . In modern condensed matter physics ,

8010-402: The value F f = μ N {\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 m g {\displaystyle mg} , because less of

8100-412: Was already drawn by Johann Andreas von Segner in 1758. 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

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