Contact mechanics is the study of the
deformation of
solids that touch each other at one or more points.
[Johnson, K. L, 1985, Contact mechanics, Cambridge University Press.][Popov, Valentin L., 2010, ''Contact Mechanics and Friction. Physical Principles and Applications'', Springer-Verlag, 362 p., .] A central distinction in contact mechanics is between
stresses acting
perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
to the contacting bodies' surfaces (known as
normal stress) and
friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of ...
al stresses acting
tangentially between the surfaces (
shear stress
Shear stress, often denoted by ( Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. '' Normal stress'', on ...
). Normal contact mechanics or frictionless contact mechanics focuses on normal stresses caused by applied
normal forces and by the
adhesion
Adhesion is the tendency of dissimilar particles or surfaces to cling to one another ( cohesion refers to the tendency of similar or identical particles/surfaces to cling to one another).
The forces that cause adhesion and cohesion can ...
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 use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
. The physical and mathematical formulation of the subject is built upon the
mechanics of materials
The field of strength of materials, also called mechanics of materials, typically refers to various methods of calculating the stresses and strains in structural members, such as beams, columns, and shafts. The methods employed to predict the re ...
and
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such mo ...
and focuses on computations involving
elastic
Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics.
Elastic may also refer to:
Alternative name
* Rubber band, ring-shaped band of rubber used to hold objects togethe ...
,
viscoelastic, and
plastic
Plastics are a wide range of synthetic or semi-synthetic materials that use polymers as a main ingredient. Their plasticity makes it possible for plastics to be moulded, extruded or pressed into solid objects of various shapes. This adapta ...
bodies in
static or
dynamic
Dynamics (from Greek δυναμικός ''dynamikos'' "powerful", from δύναμις ''dynamis'' "power") or dynamic may refer to:
Physics and engineering
* Dynamics (mechanics)
** Aerodynamics, the study of the motion of air
** Analytical dyn ...
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 interacting surfaces in relative motion. It includes the study and application of the principles of friction, lubrication and wear. Tribology is highly interdisciplinary, drawing on many academic fi ...
,
contact stiffness,
electrical contact resistance and
indentation hardness. Principles of contacts mechanics are implemented towards applications such as locomotive wheel-rail contact,
coupling
A coupling is a device used to connect two shafts together at their ends for the purpose of transmitting power. The primary purpose of couplings is to join two pieces of rotating equipment while permitting some degree of misalignment or end mov ...
devices,
braking systems,
tire
A tire (American English) or tyre (British 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 (engineering), t ...
s,
bearings,
combustion engines
An internal combustion engine (ICE or IC engine) is a heat engine in which the combustion of a fuel occurs with an oxidizer (usually air) in a combustion chamber that is an integral part of the working fluid flow circuit. In an internal combu ...
,
mechanical linkage
A mechanical linkage is an assembly of systems connected 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 providing i ...
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 ...
seals,
metalworking
Metalworking is the process of shaping and reshaping metals 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 every scale ...
, 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 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-lubric ...
and material
design
A design is a plan or specification for the construction of an object or system or for the implementation of an activity or process or the result of that plan or specification in the form of a prototype, product, or process. The verb ''to design' ...
on
friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of ...
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 ち ...
- 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."Ueber 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. The uni ...
. Hertz was attempting to understand how the optical properties of multiple, stacked
lenses might change with the
force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
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
An elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is ...
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 describes a state of tiredness that does not resolve with rest or sleep. In general usage, fatigue is synonymous with extreme tiredness or exhaustion that normally follows prolonged physical or mental activity. When it does not resolve ...
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 that may involve force and movement. It is an engineering branch that combines engineering physics and mathematics principles with materials science, to design, analyze, manufacture, ...
and
tribology
Tribology is the science and engineering of interacting surfaces in relative motion. It includes the study and application of the principles of friction, lubrication and wear. Tribology is highly interdisciplinary, drawing on many academic fi ...
, ''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
Johnson
Johnson is a surname of Anglo-Norman origin meaning "Son of John". It is the second most common in the United States and 154th most common in the world. As a common family name in Scotland, Johnson is occasionally a variation of ''Johnston'', a ...
, Kendall, and 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 advant ...
contact.
[K. L. Johnson and K. Kendall and A. D. Roberts, Surface energy and the contact of elastic solids, Proc. R. Soc. Lond. A 324 (1971) 301-313] This theory was rejected by
Boris Derjaguin
Boris Vladimirovich Derjaguin (or Deryagin; russian: Бори́с Влади́мирович Деря́гин) (9 August 1902 in Moscow – 16 May 1994) was a Soviet and Russian chemist. As a member of the Russian Academy of Sciences, he laid the ...
and co-workers
[D. Maugis, Contact, Adhesion and Rupture of Elastic Solids, Springer-Verlag, Solid-State Sciences, Berlin 2000, ] who proposed a different theory of adhesion
[Derjaguin, BV and Muller, VM and Toporov, Y.P., 1975, ''Effect of contact deformations on the adhesion of particles'', Journal of Colloid and Interface Science, 53(2), pp. 314-326] in the 1970s. The Derjaguin model came to be known as the DMT (after Derjaguin, Muller and Toporov) model,
[ and the Johnson et al. model came to be known as the JKR (after Johnson, Kendall and Roberts) model for adhesive elastic contact. This rejection proved to be instrumental in the development of the Tabor][D. Tabor, The hardness of solids, J. Colloid Interface Sci. 58 (1977) 145-179] and later Maugis[D. Maugis, Adhesion of spheres: The JKR-DMT transition using a Dugdale model, J. Colloid Interface Sci. 150 (1992) 243--269] 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 Bowden and Tabor
Tabor may refer to:
Places
Czech Republic
* Tábor, a town in the South Bohemian Region
** Tábor District, the surrounding district
* Tábor, a village and part of Velké Heraltice in the Moravian-Silesian Region
Israel
* Mount Tabor, Galilee ...
. Bowden and Tabor were the first to emphasize the importance of surface roughness for bodies in contact.[Bowden, FP and Tabor, D., 1939, ''The area of contact between stationary and between moving surfaces'', Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 169(938), pp. 391--413.][Bowden, F.P. and Tabor, D., 2001, The friction and lubrication of solids, Oxford University Press.] 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 Archard (1957)[Archard, JF, 1957, ''Elastic deformation and the laws of friction'', Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 243(1233), pp.190--205.] 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. Further important insights along these lines were provided by Greenwood and Williamson (1966), Bush (1975), and 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 (i.e., pressure, 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 () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
of radius
In classical geometry, a radius (plural, : radii) 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 name comes from the latin ''radius'', ...
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 ratios 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 (from ) 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 ...
is pressed into an elastic half-space, it creates a pressure distribution described by[Sneddon, I. N., 1965, ''The Relation between Load and Penetration in the Axisymmetric Boussinesq Problem for a Punch of Arbitrary Profile.'' Int. J. Eng. Sci. v. 3, pp. 47–57.]
:
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 of an elastic half-space of Young's modulus using a rigid conical
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines c ...
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 is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
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'':[Popov, V.L., "Contact Mechanics and Friction: Physical Principles and Applications"]
:
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).
The 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.[Popov, V.L., ''Method of reduction of dimensionality in contact and friction mechanics: A linkage between micro and macro scales,'' Friction, 2013, v.1, N. 1, pp.41–62.][Popov, V.L. and Heß, M., Methode der Dimensionsreduktion in Kontaktmechanik und Reibung, Springer, 2013.] 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, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both math ...
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 a ...
distribution and in-plane surface traction 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.[Shigley, J.E., Mischke, C.R., 1989, Mechanical Engineering Design, Fifth Edition, Chapter 2, McGraw-Hill, Inc, 1989, .] 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 due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mec ...
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, 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 mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to t ...
of linear elasticity subject to the traction boundary conditions:
:
where is the Dirac delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...
. 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