# Friction

Friction can be defined as the force between surfaces in contact that resists their relative tangential motion (slipping). Friction is a passive resistance that tends to hinder the relative motion of two bodies in contact. Passive resistance, which produces the loss of dynamic work in contact between bodies in relative motion, can be distinguished in various respects:

• geometric aspect (contact extension): point-like, or extended to a line or surface;
• kinematic aspect (form of relative motion): sliding or rolling; in particular, in extended contact, the relative motion occurs at any time around an axis to which all the contact normals are incident, provided that the collision is excluded and that the bodies are non-deformable;
• nature of bodies: contact between rigid or deformable solids, between solid and fluid, between fluids;
• state of the surfaces: smooth or rough;
• friction form: dry, lubricated.

One of the simpler characteristics of friction is that it is parallel to the contact surface between systems and always in a direction that opposes motion or attempted motion of the systems relative to each other.

## Friction coefficient

The friction coefficient is a dimensionless quantity and its value depends both on the typology of the materials in contact and on the treatment of the surfaces (which determines the surface roughness and therefore the interaction forces between the atoms of the materials in contact).

Friction is a macroscopic phenomenon of resistance, due to the roughness of the material surfaces. It is a phenomenon that occurs everywhere in the contact between two material bodies, even if only minimally, both in static (static friction) and dynamic (dynamic friction in relative motion) conditions, giving rise to a resisting force called friction force.

The effects of friction manifest a dissipation of mechanical energy (or kinetic energy) that is transformed into heat, which reduces the efficiency of the movement but in some cases is of fundamental utility, for example, if it is not looking for a free movement but an adhesion or relative motion control.

The friction coefficient is equal to the ratio between the friction force developed between the two bodies and the force that keeps them in contact.

The static friction coefficient is always greater, or at most, equal to the dynamic friction coefficient for the same surfaces in contact. From the microscopic point of view, it is due to the interaction forces between the atoms of the materials in contact. This implies that the first detachment force required (i.e. to cause the bodies to begin to slide in relative motion) is greater than that required to keep them in sliding.

The static friction coefficient is equal to the tangent of the maximum angle that can be reached between the two forces before one of the two bodies begins to slide along the other (friction angle).

## Types of friction

In mechanics there are basically three types of friction:

• dry friction: occurs between two bodies in contact on non-lubricated surfaces.
• fluid (viscous) friction: occurs on the surfaces of two bodies in relative motion between which a liquid or gaseous lubricant is interposed, whereby viscous forces of interaction develop between the material body and the molecules of the fluid (liquid or gas) with which it is in relative motion. This viscous friction force is linked to a dimensionless number called Reynolds number.
• internal friction of materials: manifests itself as a non-perfect elasticity of real bodies when they are deformed.

Depending on the presence or absence of relative motion and its type, the following are also distinguished:

• static friction (typical of quiet bodies);
• kinetic or dynamic friction (typical of bodies in relative motion);
• sliding friction;
• rolling friction.

## Static friction

The static friction force represents the force that must be overcome in order to set bodies in motion, and is defined by the formula:

$F_s=\mu_s\cdot F_{\perp}$

wherewith $$F_{\perp}$$ indicates the force that acts perpendicularly against the contact surface of the two bodies and with $$\mu_s$$ the static friction coefficient, which depends on the materials and surfaces involved and which has no unit of measurement (it is a pure number).

Very often the perpendicular force $$F_{\perp}$$ coincides with the weight force, but it is not always so. More generally, the component perpendicular to the surface of the resultant of the forces applied to the body must be considered. For example, imagine pressing down a block on a table with one hand downwards; in this case the component perpendicular to the surface is given by the sum of the weight force and the force exerted by the hand.

## Kinetic friction

Kinetic friction, also known as dynamic friction, occurs when two objects are already in relative motion to each other and rub together. The coefficient of kinetic friction is typically denoted as $$\mu_k$$, and is usually less than the coefficient of static friction for the same materials. The formula of dynamic friction force is structurally identical to that of static friction force.

$F_k=\mu_k\cdot F_{\perp}$

With the same materials and surfaces in contact, the dynamic friction coefficient has lower values than the static one. In general, therefore:

$\mu_k<\mu_s$

If the external force exerted on the moving body is less than the kinetic friction, the body slows down in its motion; on the contrary, it accelerates. If, however, the two forces are perfectly equal in value, the resultant force will be null and therefore the body will move with uniform rectilinear motion.

The origin of kinetic friction at the nanoscale can be explained by thermodynamics. Upon sliding, new surface forms at the back of sliding true contact, and the 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.

## The dynamic manifestation of friction

Between the surfaces of two bodies in contact develops a tangential force able to oppose their relative motion. Its dynamic manifestation is, therefore, the deviation (of a certain friction angle) of the straight line of action of the mutual contact force from the perpendicular, on the side of which the tangential component opposes the relative motion of the body to which it is applied (with respect to the other).

Considering the contact between two solid bodies, and assuming a punctiform contact between the respective surfaces (for example between two very small spheres), the contact force exchanged between the two bodies passes through the common contact point and its straight line of action can assume an inclination of various sizes with respect to the contact normal (of the tangent in the common point). This inclination is not determined only by the geometry of the bond.

The mutual force can be distinguished from the perpendicular component, whose straight line of action is the contact perpendicular, and the tangential component, lying in the tangent contact plane. It is shown that for a given perpendicular component, there is a maximum for the tangential component of the contact force, i.e. a maximum of the inclination of the force itself on the perpendicular; this maximum value depends on the conditions that characterize the contact locally.

## Sliding friction

Sliding friction is defined as the phenomenon of resistance to motion due to the sliding between the surfaces of two material bodies in contact which have a certain surface roughness.

In the case of immobility related to contact, the inclination (angle of friction) of the force exchanged between the common points of two bodies can take any of the values between zero and a maximum, which depends on the conditions that characterize the contact locally.

For this kinematic condition, the inclination of the contact force is determined exclusively by the condition of equilibrium with the other forces urging the body, arbitrarily chosen, and is independent of the characteristics of the contact, as long as the action useful to obtain the equilibrium is developed in contact.

In other words, a tangential action is defined as sliding friction in the case of relative motion; while adhesion is defined as that between elements in contact in the absence of sliding.

### Forms sliding friction

The extent of friction and the friction coefficient depends on the geometric, kinematic and dynamic conditions that arise in contact between the surfaces of the bodies, but in particular, it is extremely sensitive to the chemical and physical state of the parts in contact.

Imperceptible changes in this state can cause as many variations in the friction coefficient of the order of hundreds of times. For this reason, the possibility of encountering significant variations in the friction coefficient for minimal and often uncontrollable causes must always be kept in mind when machines and related mechanisms operating, especially for safety issues. There are at least three forms of sliding friction:

• dry friction between bodies: between solids with a clean surface (in the chemical meaning, ie at the single monomolecular layer level). For dry bodies, Coulomb experimentally deduced that the frictional resistance is proportional to the perpendicular load and does not depend on the sliding speed or the extension of the contact surface, while it depends only on the nature of the materials. The characteristic manifestation of the sliding friction is the deviation of the perpendicular of the line of action of the contact force so that the force component in the tangent plane is directly opposite to the relative speed of the body to which it is applied (with respect to the body to be where it comes from); the amplitude of the deviation depends only on the nature of the materials. In fact, due to the geometric characteristics, the state of surface finish also influences, that is, the greater or lesser relief of the small and extensive irregularities of the surface;
• friction between perfectly lubricated bodies: it is characterized by the presence between the two solids of a layer of lubricating fluid which, while being of small thickness, admits sliding of thin fluid layers on the inside, making sure that between the two bodies there is no direct contact. The dynamic connection is established exclusively through the lubricant layer, and therefore the force applied to each of the two bodies follows laws expressing exclusive properties of the fluid. It is for this reason that the friction magnitude of this form, is greatly influenced by the sliding velocity and the temperature of the lubricant, this, in turn, being dependent on the work dissipated by friction within the fluid layer. It can be observed that even the air can creep between the facing surfaces and act as a lubricant, developing significant forces on them due to the rapid relative motion;
• friction between bodies coated with a layer whose smallness is the extreme conceivable: the thickness of a single molecule, or even of only a few molecules. The coating usually consists of lubricant molecules, for example oils or greases. These molecules have an active end and stick to it orthogonally to the surface with great greed and strength; they can also join together in the form of chain links, together forming a layer similar to that of the threads of a fabric and having a very strong resistance to tearing, neither more nor less than a solid layer.

### Influence of speed

The independence of the friction coefficient from speed is approximate and can be considered admissible only for modest speed variations. In fact, the range of variation of the sliding speed for which there is a fundamental interest in knowing the behavior of the friction coefficient is extremely extensive: from values below one millimeter per second, up to values in the order of a hundred meters per second, as for example happens for wheels and brakes.

Different behaviors of friction can take place, it can be said that starting from the lowest speeds, the friction coefficient between dry bodies first decreases significantly, and then grows already starting from speed values of the order of one centimeter per second. When the speed becomes of the order of one meter per second and more, the friction coefficient value decreases again.

### Influence of pressure

Even pressure can contribute to characterizing the value of the friction coefficient: in fact, it produces an alteration of the shape of the surfaces in contact. Starting from the lowest values to end at values corresponding to the limit of permanent crushing deformation, as in the case of extremely localized (point-like) contacts. For example, enormous pressures occur in the contact between wheel and rail of railway vehicles and in that between the rolling elements of roller and ball bearings.

However, the variation of the friction coefficient as the pressure changes is relatively modest. Often with the increase in pressure starting from very small values, the friction coefficient first slightly decreases while towards maximum pressures there is a significant increase in the friction coefficient.

### Fluidostatic effect of pressure

If on the contact surface between two bodies there is the presence of fluid, among all the forces involved that develop, it is also necessary to consider that of the action of the fluid, in particular the fluidostatic force.

If no trace of the fluid is present at the mating surfaces of two bodies, then the static action of the fluid will be absent. It is observed that during the relative motion, and more easily the faster it is, the instantaneous expulsion of the fluid from limited portions of the surface can occur. This is one of the reasons why there are anomalies and irregularities in the extent of friction.

### Effect of molecular actions

The local load on the contact surface elements expressed as a unitary pressure can also be very large. The extreme proximity of the material elements allows the manifestation to an appreciable extent of molecular actions, of the type of adhesion and cohesion.

## Rolling friction

Rolling friction is defined as the phenomenon of resistance to motion due to the rolling between the surfaces of two material bodies in contact. The resistance produced by rolling friction is, in general, much lower than that generated by sliding friction.

### Laws of the rolling friction of Charles-Augustin de Coulomb

1. law: the rolling friction is proportional to the normal component on the contact surface (for example in the case of horizontal surfaces we have the weight force);
2. law: rolling friction depends on the nature and state of the bodies in contact. This occurs similarly to sliding friction; an example of how this affects the friction force is that it is much easier to drive a car on asphalt than on dirt;
3. law: the rolling friction is inversely proportional to the radius of the rolling body (therefore its width): this is because the greater width of the contact surface causes a lower sinking of the body, as the unit pressure of the weight is lower; besides, the increase in the radius of the body results in an increase in the lever arm and the moment of rotation.

### Rolling friction coefficient

Since the geometric manifestation of friction is the displacement of the force application point between the two members, this expression is chosen as the ratio of the displacement $$u$$ to the radius of curvature of the roller $$r$$:

$\mu_r=\dfrac{u}{r}$

which takes the name of rolling friction coefficient.

### Causes of rolling friction

The characters of the contact on which the displacement of the point of application of the mutual force and therefore of the rolling friction depends, are various. The following considerations apply:

• perfect elasticity: rolling friction is null; if the materials are perfectly elastic and the shape of the bodies is perfectly regular and free of roughness, the distribution of pressures in the contact meniscus is symmetrical with respect to the normal plane of contact $$\pi_n$$. Under this assumption, the line of action of the resultant of the pressures is on the symmetry plane $$\pi_n$$ and $$u$$ is zero;
• perfect non-elasticity: on the contrary, the material may be perfectly plastic, and then the ground level is lowered and does not rise again after passage. Most of the contact occurs at the roller’s advancing face and the $$U$$ point naturally moves away from $$\pi_n$$;
• elastic hysteresis: Elastic hysteresis occurs when the material is imperfectly elastic. For it, the unit tension in the material is not simply a function of the deformation, but also of the sign of its variation: for the same deformation, the tension is greater if the deformation is increasing. The distribution of pressures at contact is therefore not symmetrical, but in fact, in the front there are higher pressures than in the back;
• crushing and impact: the surface irregularities of the walls in contact give rise to loss of work:
• by crushing of the walls that yield plastically due to the excessive concentration of the load on them;
• by impact due to the kinematically incorrect contacts that occur in this case.
• rigid body creep: a cause of loss in imperfect rolling comes from contact creep occurring in the relative motion of the two bodies considered rigid. In this regard it should be noted that, if we consider with all rigor the motion of the individual parts taking into account their deformation, we must first define what is meant by relative motion between the two bodies, in order to have the possibility of establishing the corresponding axis of rotation;
• creep due to local deformation: Due to the deformation around the contact region, the shape and size of the surfaces change. For the two bodies the deformations of the contact areas are not the same, in fact there can be two cases:
• the tangential action that is manifested at the contact is sufficient to prevent sliding corresponding to the different shape that the surface elements are gradually acquiring, as the stress varies according to their position along the meniscus of contact between the two bodies. In this case the work of friction for sliding is null; but it is clear that internal sliding is produced in the mass of the two bodies, the more sensitive the closer the volume element is to the contact surface. To this deformation of the volume elements corresponds work lost by elastic hysteresis;
• the tangential action of contact is insufficient to prevent sliding at the contact; in any case it hinders it. Therefore, the phenomenon considered above is repeated to a reduced extent, with consequent loss of dynamic energy; in addition, in this case there is the loss of work corresponding to the sliding.

The resistant force $$R=F_v$$ opposed by rolling friction, is the lesser the greater the radius of curvature $$r$$ of the rolling body, and is calculated by the following equation:

$F_v=\dfrac{\mu_vF_{\perp}}{r}$

where $$\mu_v$$ is the coefficient of rolling friction.

## History of friction

The invention of the wheel and its use for the transport of loads and people date back to the fourth millennium BC. It must undoubtedly have been preceded by a long period of experience of protohistoric human communities in the field of transporting large stones and other heavy loads necessary for the buildings (which we now call megalithic) that characterize the latter part of the Neolithic and the beginning of the metal age in the regions of western Europe and those around the eastern Mediterranean.

In these periods, alongside or in place of direct dragging on the ground and towing on sleds, it can be assumed that a technique based on the use, as rollers, of cylindrical sections of tree trunks was used (an idea probably related to practices of successive overturns used to move stones of modest size). The tree trunks used as rollers were perhaps the ancestors of the wheels that came when the problem of connecting the wheels by means of axles to a structure capable of bearing the load was overcome. It is very likely that these inventions go to Mesopotamian proto-history since the first archaeological finds and the first representations of wheeled chariots (dated to the first half of the third millennium BC) come from the city of Ur.

The extraordinary singularity of the rolling of a wheel lies in the role, paradoxical at first sight, of the static friction. In fact, due to static friction, non-cylindrical or non-spherical solid systems (which therefore can only translate) are “kept still” on their support base. In fact, without static friction, a great number of things that “must stay still” could not be: chairs, tables and all other furniture would move with the slightest push in a horizontal direction; screws could not be tightened; one could not lean a ladder against the wall, much less climb on it, and so on.

All this creates, on the level of conceptual representations, a strong correlation between static friction and immobility of things on their supports and makes the effect of static friction on rolling quite paradoxical. But the idea of paradox falls away if the shape of the rolling objects is taken into account. Even a cylinder or a sphere, in fact, are held still with respect to the support by the forces of static friction; but on these two geometric shapes, these forces can act only on a portion of the contact surface which ideally, if the support is perfectly flat, is constituted by the points of a straight line segment (for the cylinder) and even by a single point for the sphere and which in reality is however very limited with respect to the entire surface of the solid.

This implies that, while keeping the support area still, the static friction cannot hinder the rotational movements of the entire solid around this restricted area. This occurs not only for cylinder and sphere but also for any solid with a convex shape, for example an ellipsoid, or a prismatic solid that is overturning around an edge. The peculiarity of cylinder and sphere lies in the fact that, given their particular symmetry, they also enjoy the possibility of being in indifferent equilibrium (if they are homogeneous).

Another feature of rolling that is decisive for the success of wheel (or roller) based transport systems is the extraordinary energy advantage over sliding based systems In rolling, the static frictional forces that hold the contact area on the base stationary, for this reason, do not perform work and therefore have no role in the energy balances. This means that if the wheels and base were perfectly rigid, the only dissipation of mechanical energy for a wagon would be due to the dynamic friction active in the wheel axles (abstracting, of course, from the air resistance).

In reality, perfectly rigid materials do not exist: this entails the need to take into account the possible deformations during rolling and their consequences on motion, both geometric and energetic. In many significant cases, it is possible to ignore the effect of the deformability of the materials on the geometric aspects of the rolling and on the value of the frictional force necessary to avoid sliding. On the other hand, it is never possible to neglect the energetic consequences of the deformability of materials, under penalty of affirming the possibility of creating perpetual motions.

The parameter that, from a phenomenological point of view, interprets the dissipation of mechanical energy due to the deformability of the materials is the rolling friction coefficient. Although it is never zero, it is nevertheless related to energy losses that are much lower than those due to sliding or air resistance.