
Contact mechanics is the study of the
deformation of
solids
Solid is a state of matter where molecules are closely packed and can not slide past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree to which they are resisted dependent upon the ...
that touch each other at one or more points.
A central distinction in contact mechanics is between
stresses acting
perpendicular
In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...
to the contacting bodies' surfaces (known as
normal stress
In continuum mechanics, stress is a physical quantity that describes forces present during deformation. For example, an object being pulled apart, such as a stretched elastic band, is subject to ''tensile'' stress and may undergo elongati ...
) and
friction
Friction 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 t ...
al stresses acting
tangentially between the surfaces (
shear stress
Shear stress (often denoted by , Greek alphabet, Greek: tau) is the component of stress (physics), stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross secti ...
). Normal contact mechanics or frictionless contact mechanics focuses on normal stresses caused by applied
normal force
In mechanics, the normal force F_n is the component of a contact force that is perpendicular to the surface that an object contacts. In this instance '' normal'' is used in the geometric sense and means perpendicular, as opposed to the meanin ...
s and by the
adhesion
Adhesion is the tendency of dissimilar particles or interface (matter), surfaces to cling to one another. (Cohesion (chemistry), Cohesion refers to the tendency of similar or identical particles and surfaces to cling to one another.)
The ...
present on surfaces in close contact, even if they are clean and dry.
''
Frictional contact mechanics'' emphasizes the effect of friction forces.
Contact mechanics is part of mechanical
engineering
Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...
. The physical and mathematical formulation of the subject is built upon the
mechanics of materials and
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a ''continuous medium'' (also called a ''continuum'') rather than as discrete particles.
Continuum mec ...
and focuses on computations involving
elastic
Elastic is a word often used to describe or identify certain types of elastomer, Elastic (notion), elastic used in garments or stretch fabric, stretchable fabrics.
Elastic may also refer to:
Alternative name
* Rubber band, ring-shaped band of rub ...
,
viscoelastic
In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both Viscosity, viscous and Elasticity (physics), elastic characteristics when undergoing deformation (engineering), deformation. Viscous mate ...
, and
plastic
Plastics are a wide range of synthetic polymers, synthetic or Semisynthesis, semisynthetic materials composed primarily of Polymer, polymers. Their defining characteristic, Plasticity (physics), plasticity, allows them to be Injection moulding ...
bodies in
static or
dynamic contact. Contact mechanics provides necessary information for the safe and energy efficient design of technical systems and for the study of
tribology
Tribology is the science and engineering of understanding friction, lubrication and wear phenomena for interacting surfaces in relative Motion (physics), motion. It is highly interdisciplinary, drawing on many academic fields, including physics, c ...
,
contact stiffness,
electrical contact resistance and
indentation hardness. Principles of contacts mechanics are implemented towards applications such as locomotive wheel-rail contact,
coupling devices,
braking systems,
tire
A tire (North American English) or tyre (Commonwealth English) is a ring-shaped component that surrounds a Rim (wheel), wheel's rim to transfer a vehicle's load from the axle through the wheel to the ground and to provide Traction (engineeri ...
s,
bearings,
combustion engines,
mechanical linkage
A mechanical linkage is an assembly of systems connected so as to manage forces and movement. The movement of a body, or link, is studied using geometry so the link is considered to be rigid. The connections between links are modeled as pro ...
s,
gasket
Some seals and gaskets
A gasket is a mechanical seal which fills the space between two or more mating surfaces, generally to prevent leakage from or into the joined objects while under compression. It is a deformable material that is used to c ...
seals,
metalworking
Metalworking is the process of shaping and reshaping metals in order to create useful objects, parts, assemblies, and large scale structures. As a term, it covers a wide and diverse range of processes, skills, and tools for producing objects on e ...
, metal forming,
ultrasonic welding,
electrical contacts
An electrical contact is an electrical circuit component found in electrical switches, relays, connectors and circuit breakers. Each contact is a piece of electrically conductive material, typically metal. When a pair of contacts touch, they ...
, and many others. Current challenges faced in the field may include
stress analysis
Stress may refer to:
Science and medicine
* Stress (biology)
Stress, whether physiological, biological or psychological, is an organism's response to a stressor, such as an environmental condition or change in life circumstances. When s ...
of contact and coupling members and the influence of
lubrication
Lubrication is the process or technique of using a lubricant to reduce friction and wear and tear in a contact between two surfaces. The study of lubrication is a discipline in the field of tribology.
Lubrication mechanisms such as fluid-lubr ...
and material
design
A design is the concept or proposal for an object, process, or system. The word ''design'' refers to something that is or has been intentionally created by a thinking agent, and is sometimes used to refer to the inherent nature of something ...
on
friction
Friction 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 t ...
and
wear
Wear is the damaging, gradual removal or deformation of material at solid surfaces. Causes of wear can be mechanical (e.g., erosion) or chemical (e.g., corrosion). The study of wear and related processes is referred to as tribology.
Wear in ...
. Applications of contact mechanics further extend into the
micro
Micro may refer to:
Measurement
* micro- (μ), a metric prefix denoting a factor of 10−6
Places
* Micro, North Carolina, town in U.S.
People
* DJ Micro, (born Michael Marsicano) an American trance DJ and producer
* Chii Tomiya (都宮 � ...
- and
nanotechnological realm.
The original work in contact mechanics dates back to 1881 with the publication of the paper "On the contact of elastic solids"
[H. Hertz, 1881, Über die berührung fester elastischer Körper, ''Journal für die reine und angewandte Mathematik'' 92, pp.156-171. (For English version, see: Hertz, H., 1896. On the contact of elastic solids, In: ]
Miscellaneous Papers, Chapter V, pp.146-162
'. by Hertz, H. and Lenard P., translated by Jones, D. E. and Schott G.A., London: Macmillan. "Über die Berührung fester elastischer Körper" by
Heinrich Hertz
Heinrich Rudolf Hertz (; ; 22 February 1857 – 1 January 1894) was a German physicist who first conclusively proved the existence of the electromagnetic waves predicted by James Clerk Maxwell's equations of electromagnetism.
Biography
Heinri ...
. Hertz attempted to understand how the optical properties of multiple, stacked
lenses
A lens is a transmissive optical device that focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements''), ...
might change with the
force
In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
holding them together. Hertzian contact stress refers to the localized stresses that develop as two curved surfaces come in contact and deform slightly under the imposed loads. This amount of deformation is dependent on the
modulus of elasticity of the material in contact. It gives the contact stress as a function of the normal contact force, the radii of curvature of both bodies and the modulus of elasticity of both bodies. Hertzian contact stress forms the foundation for the equations for load bearing capabilities and
fatigue
Fatigue is a state of tiredness (which is not sleepiness), exhaustion or loss of energy. It is a signs and symptoms, symptom of any of various diseases; it is not a disease in itself.
Fatigue (in the medical sense) is sometimes associated wit ...
life in bearings, gears, and any other bodies where two surfaces are in contact.
History

Classical contact mechanics is most notably associated with Heinrich Hertz.
[Hertz, H. R., 1882, Über die Berührung fester elastischer Körper und Über die Härte, ''Verhandlungen des Vereins zur Beförderung des Gewerbefleisscs'', Berlin: Verein zur Beförderung des Gewerbefleisses, pp.449-463 (For English version, see: Hertz, H., 1896. On the contact of rigid elastic solids and on hardness, In: ]
Miscellaneous Papers, Chapter VI, pp.163-183
'. by Hertz, H. and Lenard P., translated by Jones, D. E. and Schott G.A., London: Macmillan. In 1882, Hertz solved the contact problem of two elastic bodies with curved surfaces. This still-relevant classical solution provides a foundation for modern problems in contact mechanics. For example, in
mechanical engineering
Mechanical engineering is the study of physical machines and mechanism (engineering), mechanisms that may involve force and movement. It is an engineering branch that combines engineering physics and engineering mathematics, mathematics principl ...
and
tribology
Tribology is the science and engineering of understanding friction, lubrication and wear phenomena for interacting surfaces in relative Motion (physics), motion. It is highly interdisciplinary, drawing on many academic fields, including physics, c ...
, ''Hertzian contact stress'' is a description of the stress within mating parts. The Hertzian contact stress usually refers to the stress close to the area of contact between two spheres of different radii.
It was not until nearly one hundred years later that
Kenneth L. Johnson,
Kevin Kendall, and
Alan D. Roberts found a similar solution for the case of
adhesive
Adhesive, also known as glue, cement, mucilage, or paste, is any non-metallic substance applied to one or both surfaces of two separate items that binds them together and resists their separation.
The use of adhesives offers certain advantage ...
contact.
This theory was rejected by
Boris Derjaguin and co-workers
who proposed a different theory of adhesion
in the 1970s. The Derjaguin model came to be known as the Derjaguin–Muller–Toporov (DMT) model (after Derjaguin, M. V. Muller and Yu. P. Toporov),
[ and the Johnson et al. model came to be known as the Johnson–Kendall–Roberts (JKR) model for adhesive elastic contact. This rejection proved to be instrumental in the development of the David Tabor] and later Daniel Maugis parameters that quantify which contact model (of the JKR and DMT models) represent adhesive contact better for specific materials.
Further advancement in the field of contact mechanics in the mid-twentieth century may be attributed to names such as Frank Philip Bowden and Tabor. Bowden and Tabor were the first to emphasize the importance of surface roughness for bodies in contact. Through investigation of the surface roughness, the true contact area between friction partners is found to be less than the apparent contact area. Such understanding also drastically changed the direction of undertakings in tribology. The works of Bowden and Tabor yielded several theories in contact mechanics of rough surfaces.
The contributions of J. F. Archard (1957) must also be mentioned in discussion of pioneering works in this field. Archard concluded that, even for rough elastic surfaces, the contact area is approximately proportional to the normal force
In mechanics, the normal force F_n is the component of a contact force that is perpendicular to the surface that an object contacts. In this instance '' normal'' is used in the geometric sense and means perpendicular, as opposed to the meanin ...
. Further important insights along these lines were provided by James A. Greenwood and J. B. P. Williamson (1966), A. W. Bush (1975), and Bo N. J. Persson (2002). The main findings of these works were that the true contact surface in rough materials is generally proportional to the normal force, while the parameters of individual micro-contacts (pressure and size of the micro-contact) are only weakly dependent upon the load.
Classical solutions for non-adhesive elastic contact
The theory of contact between elastic bodies can be used to find contact areas and indentation depths for simple geometries. Some commonly used solutions are listed below. The theory used to compute these solutions is discussed later in the article. Solutions for multitude of other technically relevant shapes, e.g. the truncated cone, the worn sphere, rough profiles, hollow cylinders, etc. can be found in
Contact between a sphere and a half-space
An elastic sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
of radius
In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...
indents an elastic half-space where total deformation is , causing a contact area of radius
:
The applied force is related to the displacement by
:
where
:
and , are the elastic moduli and , the Poisson's ratio
In materials science and solid mechanics, Poisson's ratio (symbol: ( nu)) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value ...
s associated with each body.
The distribution of normal pressure in the contact area as a function of distance from the center of the circle is
:
where is the maximum contact pressure given by
:
The radius of the circle is related to the applied load by the equation
:
The total deformation is related to the maximum contact pressure by
:
The maximum shear stress occurs in the interior at for .
Contact between two spheres
For contact between two spheres of radii and , the area of contact is a circle of radius . The equations are the same as for a sphere in contact with a half plane except that the effective radius is defined as
:
Contact between two crossed cylinders of equal radius
This is equivalent to contact between a sphere of radius and a plane.
Contact between a rigid cylinder with flat end and an elastic half-space
If a rigid cylinder
A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infinite ...
is pressed into an elastic half-space, it creates a pressure distribution described by
:
where is the radius of the cylinder and
:
The relationship between the indentation depth and the normal force is given by
:
Contact between a rigid conical indenter and an elastic half-space
In the case of indentation
__FORCETOC__
In the written form of many languages, indentation describes empty space ( white space) used before or around text to signify an important aspect of the text such as:
* Beginning of a paragraph
* Hierarchy subordinate concept
* Qu ...
of an elastic half-space of Young's modulus using a rigid conical indenter, the depth of the contact region and contact radius are related by[
:
with defined as the angle between the plane and the side surface of the cone. The total indentation depth is given by:
:
The total force is
:
The pressure distribution is given by
:
The stress has a ]logarithm
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
ic singularity at the tip of the cone.
Contact between two cylinders with parallel axes
In contact between two cylinders with parallel axes, the force is linearly proportional to the length of cylinders ''L'' and to the indentation depth ''d'':
:
The radii of curvature are entirely absent from this relationship. The contact radius is described through the usual relationship
:
with
:
as in contact between two spheres. The maximum pressure is equal to
:
Bearing contact
The contact in the case of bearings is often a contact between a convex surface (male cylinder or sphere) and a concave surface (female cylinder or sphere: bore or hemispherical cup).
Method of dimensionality reduction
Some contact problems can be solved with the method of dimensionality reduction (MDR). In this method, the initial three-dimensional system is replaced with a contact of a body with a linear elastic or viscoelastic foundation (see fig.). The properties of one-dimensional systems coincide exactly with those of the original three-dimensional system, if the form of the bodies is modified and the elements of the foundation are defined according to the rules of the MDR. MDR is based on the solution to axisymmetric contact problems first obtained by Ludwig Föppl (1941) and Gerhard Schubert (1942)
However, for exact analytical results, it is required that the contact problem is axisymmetric and the contacts are compact.
Hertzian theory of non-adhesive elastic contact
The classical theory of contact focused primarily on non-adhesive contact where no tension force is allowed to occur within the contact area, i.e., contacting bodies can be separated without adhesion forces. Several analytical and numerical approaches have been used to solve contact problems that satisfy the no-adhesion condition. Complex forces and moments are transmitted between the bodies where they touch, so problems in contact mechanics can become quite sophisticated. In addition, the contact stresses are usually a nonlinear function of the deformation. To simplify the solution procedure, a frame of reference
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system, whose origin (mathematics), origin, orientation (geometry), orientation, and scale (geometry), scale have been specified in physical space. It ...
is usually defined in which the objects (possibly in motion relative to one another) are static. They interact through surface tractions (or pressures/stresses) at their interface.
As an example, consider two objects which meet at some surface in the (,)-plane with the -axis assumed normal to the surface. One of the bodies will experience a normally-directed pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...
distribution and in-plane surface traction
Traction, traction force or tractive force is a force used to generate motion between a body and a tangential surface, through the use of either dry friction or shear force.
It has important applications in vehicles, as in ''tractive effort''.
...
distributions and over the region . In terms of a Newtonian force balance, the forces:
:
must be equal and opposite to the forces established in the other body. The moments corresponding to these forces:
:
are also required to cancel between bodies so that they are kinematically immobile.
Assumptions in Hertzian theory
The following assumptions are made in determining the solutions of Hertzian contact problems:
* The strains are small and within the elastic limit.
* The surfaces are continuous and non-conforming (implying that the area of contact is much smaller than the characteristic dimensions of the contacting bodies).
* Each body can be considered an elastic half-space.
* The surfaces are frictionless.
Additional complications arise when some or all these assumptions are violated and such contact problems are usually called non-Hertzian.
Analytical solution techniques
Analytical solution methods for non-adhesive contact problem can be classified into two types based on the geometry of the area of contact. A conforming contact is one in which the two bodies touch at multiple points before any deformation takes place (i.e., they just "fit together"). A non-conforming contact is one in which the shapes of the bodies are dissimilar enough that, under zero load, they only touch at a point (or possibly along a line). In the non-conforming case, the contact area is small compared to the sizes of the objects and the stresses are highly concentrated in this area. Such a contact is called ''concentrated'', otherwise it is called ''diversified''.
A common approach in linear elasticity
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed by prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechani ...
is to superpose a number of solutions each of which corresponds to a point load acting over the area of contact. For example, in the case of loading of a half-plane
In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example ...
, the Flamant solution is often used as a starting point and then generalized to various shapes of the area of contact. The force and moment balances between the two bodies in contact act as additional constraints to the solution.
Point contact on a (2D) half-plane
A starting point for solving contact problems is to understand the effect of a "point-load" applied to an isotropic, homogeneous, and linear elastic half-plane, shown in the figure to the right. The problem may be either plane stress or plane strain. This is a boundary value problem
In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satis ...
of linear elasticity subject to the traction boundary condition
In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satis ...
s:
:
where is the Dirac delta function
In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
. The boundary conditions state that there are no shear stresses on the surface and a singular normal force P is applied at (0, 0). Applying these conditions to the governing equations of elasticity produces the result
:
for some point, , in the half-plane. The circle shown in the figure indicates a surface on which the maximum shear stress is constant. From this stress field, the strain components and thus the displacements of all material points may be determined.
Line contact on a (2D) half-plane
= Normal loading over a region
=
Suppose, rather than a point load , a distributed load is applied to the surface instead, over the range