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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 ...
R indents an elastic half-space where total deformation is d, causing a contact area of radius : a = \sqrt The applied force F is related to the displacement d by : F = \frac E^*R^\fracd^\frac where : \frac = \frac + \frac and E_1,E_2 are the elastic moduli and \nu_1,\nu_2 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 : p(r) = p_0\left(1 - \frac\right)^\frac where p_0 is the maximum contact pressure given by : p_0 = \frac = \frac\left(\frac\right)^\frac The radius of the circle is related to the applied load F by the equation : a^3 = \cfrac The total deformation d is related to the maximum contact pressure by : d = \frac = \left(\frac\right)^\frac The maximum shear stress occurs in the interior at z \approx 0.49a for \nu = 0.33.


Contact between two spheres

For contact between two spheres of radii R_1 and R_2, the area of contact is a circle of radius a. The equations are the same as for a sphere in contact with a half plane except that the effective radius R is defined as : \frac = \frac + \frac


Contact between two crossed cylinders of equal radius

This is equivalent to contact between a sphere of radius R 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 : p(r) = p_0\left(1 - \frac\right)^ where R is the radius of the cylinder and : p_0 = \fracE^*\frac The relationship between the indentation depth and the normal force is given by : F = 2RE^*d


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 E using a rigid conical indenter, the depth of the contact region \epsilon and contact radius a are related by : \epsilon = a\tan(\theta) with \theta defined as the angle between the plane and the side surface of the cone. The total indentation depth d is given by: : d = \frac\epsilon The total force is : F = \frac a^2 \tan(\theta) = \frac\frac The pressure distribution is given by : p\left(r\right) = \frac \ln\left(\frac + \sqrt\right) = \frac \cosh^\left(\frac\right) 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'': :F \approx \fracE^*Ld The radii of curvature are entirely absent from this relationship. The contact radius is described through the usual relationship :a = \sqrt with :\frac = \frac + \frac as in contact between two spheres. The maximum pressure is equal to :p_0 = \left(\frac\right)^\frac


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 S in the (x,y)-plane with the z-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 p_z=p(x,y)=q_z(x,y) 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 q_x=q_x(x,y) and q_y=q_y(x,y) over the region S. In terms of a Newtonian force balance, the forces: : P_z = \int_S p(x,y)~ \mathrmA ~;~~ Q_x = \int_S q_x(x,y)~ \mathrmA ~;~~ Q_y = \int_S q_y(x,y)~ \mathrmA must be equal and opposite to the forces established in the other body. The moments corresponding to these forces: : M_x = \int_S y~q_z(x,y)~ \mathrmA ~;~~ M_y = \int_S -x~q_z(x,y)~ \mathrmA ~;~~ M_z = \int_S ~q_y(x,y) - y~q_x(x,y) \mathrmA 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: :\sigma_(x, 0) = 0 ~;~~ \sigma_z(x, z) = -P\delta(x, z) where \delta(x, z) 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 :\begin \sigma_ & = -\frac\frac \\ \sigma_ & = -\frac\frac \\ \sigma_ & = -\frac\frac \end for some point, (x, y), 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 P, a distributed load p(x) is applied to the surface instead, over the range a. The principle of linear superposition can be applied to determine the resulting stress field as the solution to the
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
equations: :\begin \sigma_ &= -\frac\int_a^b\frac ~;~~ \sigma_ = -\frac\int_a^b\frac \\ pt \sigma_ &= -\frac\int_a^b\frac \end


= Shear loading over a region

= The same principle applies for loading on the surface in the plane of the surface. These kinds of tractions would tend to arise as a result of friction. The solution is similar the above (for both singular loads Q and distributed loads q(x)) but altered slightly: :\begin \sigma_ &= -\frac\int_a^b\frac ~;~~ \sigma_ = -\frac\int_a^b\frac \\ pt \sigma_ &= -\frac\int_a^b\frac \end These results may themselves be superposed onto those given above for normal loading to deal with more complex loads.


Point contact on a (3D) half-space

Analogously to the Flamant solution for the 2D half-plane, fundamental solutions are known for the linearly elastic 3D half-space as well. These were found by Boussinesq for a concentrated normal load and by Cerruti for a tangential load. See the section on this 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 ...
.


Numerical solution techniques

Distinctions between conforming and non-conforming contact do not have to be made when numerical solution schemes are employed to solve contact problems. These methods do not rely on further assumptions within the solution process since they base solely on the general formulation of the underlying equations. Besides the standard equations describing the deformation and motion of bodies two additional inequalities can be formulated. The first simply restricts the motion and deformation of the bodies by the assumption that no penetration can occur. Hence the gap h between two bodies can only be positive or zero :h \ge 0 where h = 0 denotes contact. The second assumption in contact mechanics is related to the fact, that no tension force is allowed to occur within the contact area (contacting bodies can be lifted up without adhesion forces). This leads to an inequality which the stresses have to obey at the contact interface. It is formulated for the normal stress \sigma_n = \mathbf \cdot \mathbf. At locations where there is contact between the surfaces the gap is zero, i.e. h = 0, and there the normal stress is different than zero, indeed, \sigma_n < 0. At locations where the surfaces are not in contact the normal stress is identical to zero; \sigma_n = 0, while the gap is positive; i.e., h > 0. This type of complementarity formulation can be expressed in the so-called Kuhn–Tucker form, viz. :h \ge 0\,, \quad \sigma_n \le 0\,, \quad \sigma_n\,h = 0\,. These conditions are valid in a general way. The mathematical formulation of the gap depends upon the kinematics of the underlying theory of the solid (e.g., linear or nonlinear solid in two- or three dimensions, beam or
shell Shell may refer to: Architecture and design * Shell (structure), a thin structure ** Concrete shell, a thin shell of concrete, usually with no interior columns or exterior buttresses Science Biology * Seashell, a hard outer layer of a marine ani ...
model). By restating the normal stress \sigma_n in terms of the contact pressure, p; i.e., p = -\sigma_n the Kuhn-Tucker problem can be restated as in standard complementarity form i.e. h \ge 0\,,\quad p \ge 0\,,\quad p\,h = 0\,. In the linear elastic case the gap can be formulated as = h_0 + + u, where h_0 is the rigid body separation, g is the geometry/topography of the contact (cylinder and roughness) and u is the elastic deformation/deflection. If the contacting bodies are approximated as linear elastic half spaces, the Boussinesq-Cerruti integral equation solution can be applied to express the deformation (u) as a function of the contact pressure (p); i.e.,u = \int_\infty^\infty K(x - s)p(s)ds, where K(x - s) = \frac\ln, x - s, for line loading of an elastic half space and K(x - s) = \frac\frac for point loading of an elastic half-space. After discretization the linear elastic contact mechanics problem can be stated in standard Linear Complementarity Problem (LCP) form. : \begin \mathbf &= \mathbf_0 + \mathbf + \mathbf, \\ \mathbf \cdot \mathbf &= 0,\,\,\,\mathbf \geq 0,\,\,\, \mathbf \geq 0,\\ \end where \mathbf is a matrix, whose elements are so called influence coefficients relating the contact pressure and the deformation. The strict LCP formulation of the CM problem presented above, allows for direct application of well-established numerical solution techniques such as Lemke's pivoting algorithm. The Lemke algorithm has the advantage that it finds the numerically exact solution within a finite number of iterations. The MATLAB implementation presented b
Almqvist et al.
is one example that can be employed to solve the problem numerically. In addition, an example code for an LCP solution of a 2D linear elastic contact mechanics problem has also been made public at MATLAB file exchange b
Almqvist et al.


Contact between rough surfaces

When two bodies with rough surfaces are pressed against each other, the true contact area formed between the two bodies, A, is much smaller than the apparent or nominal contact area A_0. The mechanics of contacting rough surfaces are discussed in terms of normal contact mechanics and static frictional interactions. Natural and engineering surfaces typically exhibit roughness features, known as asperities, across a broad range of length scales down to the molecular level, with surface structures exhibiting self affinity, also known as surface fractality. It is recognized that the self affine structure of surfaces is the origin of the linear scaling of true contact area with applied pressure. Assuming a model of shearing welded contacts in
tribological Tribology is the science and engineering of understanding friction, lubrication and wear phenomena for interacting surfaces in relative motion. It is highly interdisciplinary, drawing on many academic fields, including physics, chemistry, material ...
interactions, this ubiquitously observed linearity between contact area and pressure can also be considered the origin of the linearity of the relationship between static friction and applied normal force. In contact between a "random rough" surface and an elastic half-space, the true contact area is related to the normal force F by : A=\fracF with h' equal to the root mean square (also known as the quadratic mean) of the surface slope and \kappa \approx2 . The median pressure in the true contact surface : p_ =\frac\approx\fracE^*h' can be reasonably estimated as half of the effective elastic modulus E^* multiplied with the root mean square of the surface slope h' .


An overview of the GW model

Greenwood and Williamson in 1966 (GW) proposed a theory of elastic contact mechanics of rough surfaces which is today the foundation of many theories in tribology (friction, adhesion, thermal and electrical conductance, wear, etc.). They considered the contact between a smooth rigid plane and a nominally flat deformable rough surface covered with round tip asperities of the same radius R. Their theory assumes that the deformation of each asperity is independent of that of its neighbours and is described by the Hertz model. The heights of asperities have a random distribution. The probability that asperity height is between z and z + dz is \phi(z)dz. The authors calculated the number of contact spots n, the total contact area A_r and the total load P in general case. They gave those formulas in two forms: in the basic and using standardized variables. If one assumes that N asperities covers a rough surface, then the expected number of contacts is : n = N\int_d^\infty \phi(z) dz The expected total area of contact can be calculated from the formula : A_a = N\pi R \int_d^\infty (z - d) \phi(z) dz and the expected total force is given by : P = \fracN E_r \sqrt \int_d^\infty (z - d)^\frac \phi(z) dz where: : R, radius of curvature of the microasperity, : z, height of the microasperity measured from the profile line, : d, close the surface, : E_r = \left(\frac + \frac\right)^, composite Young's modulus of elasticity, : E_i, modulus of elasticity of the surface, : \nu_i, Poisson's surface coefficients. Greenwood and Williamson introduced standardized separation h = d/\sigma and standardized height distribution \phi^*(s) whose standard deviation is equal to one. Below are presented the formulas in the standardized form. : \begin F_n(h) &= \int_h^\infty (s - h)^n \phi^*(s) ds \\ n &= \eta A_n F_0(h) \\ A_a &= \pi \eta A R \sigma F_1(h) \\ P &= \frac \eta A E_r \sqrt \sigma^\frac F_\frac(h) \end where: : d is the separation, : A is the nominal contact area, : \eta is the surface density of asperities, : E^* is the effective Young modulus. ''A'' and P can be determined when the F_n(h) terms are calculated for the given surfaces using the convolution of the surface roughness \phi^*(s). Several studies have followed the suggested curve fits for F_n(h) assuming a Gaussian surface high distribution with curve fits presented by Arcoumanis et al. and Jedynak among others. It has been repeatedly observed that engineering surfaces do not demonstrate Gaussian surface height distributions e.g. Peklenik. Leighton et al. presented fits for crosshatched IC engine cylinder liner surfaces together with a process for determining the F_n(h) terms for any measured surfaces. Leighton et al. demonstrated that Gaussian fit data is not accurate for modelling any engineered surfaces and went on to demonstrate that early running of the surfaces results in a gradual transition which significantly changes the surface topography, load carrying capacity and friction. Recently the exact approximants to A_r and P were published by Jedynak. They are given by the following rational formulas, which are approximants to the integrals F_n(h). They are calculated for the Gaussian distribution of asperities, which have been shown to be unrealistic for engineering surface but can be assumed where friction, load carrying capacity or real contact area results are not critical to the analysis. : F_(h) = \frac\exp\left(-\frac\right) For F_1(h) the coefficients are :\begin[] [a_0, a_1, a_2, a_3] &= [0.398942280401, 0.159773702775, 0.0389687688311, 0.00364356495452] \\[] [b_1, b_2, b_3, b_4, b_5, b_6] &= \left[1.653807476138, 1.170419428529, 0.448892964428, 0.0951971709160, 0.00931642803836, -6.383774657279 \times 10^\right] \end The maximum relative error is 9.93 \times 10^%. For F_\frac(h) the coefficients are : \begin[] [a_0, a_1, a_2, a_3] &= [0.430019993662, 0.101979509447, 0.0229040629580, 0.000688602924] \\[] [b_1, b_2, b_3, b_4, b_5,b_6] &= [1.671117125984, 1.199586555505, 0.46936532151, 0.102632881122, 0.010686348714, 0.0000517200271] \end The maximum relative error is 1.91 \times 10^%. The paper also contains the exact expressions for F_n(h) : \begin F_1(h) &= \frac \exp\left(-\frach^2\right) - \frac h\, \operatorname\left(\frac\right) \\ F_\frac(h) &= \frac\exp\left(-\frac\right) \sqrt \left(\left(h^2 + 1\right) K_\left(\frac\right) - h^2 K_\left(\frac\right)\right) \end where erfc(z) means the complementary error function and K_\nu(z) is the modified Bessel function of the second kind. For the situation where the asperities on the two surfaces have a Gaussian height distribution and the peaks can be assumed to be spherical, the average contact pressure is sufficient to cause yield when p_\text = 1.1\sigma_y \approx 0.39 \sigma_0 where \sigma_y is the uniaxial
yield stress In materials science and engineering, the yield point is the point on a stress–strain curve that indicates the limit of elasticity (physics), elastic behavior and the beginning of plasticity (physics), plastic behavior. Below the yield point ...
and \sigma_0 is the indentation hardness. Greenwood and Williamson defined a dimensionless parameter \Psi called the plasticity index that could be used to determine whether contact would be elastic or plastic. The Greenwood-Williamson model requires knowledge of two statistically dependent quantities; the standard deviation of the surface roughness and the curvature of the asperity peaks. An alternative definition of the plasticity index has been given by Mikic. Yield occurs when the pressure is greater than the uniaxial yield stress. Since the yield stress is proportional to the indentation hardness \sigma _0, Mikic defined the plasticity index for elastic-plastic contact to be : \Psi = \frac > \frac~. In this definition \Psi represents the micro-roughness in a state of complete plasticity and only one statistical quantity, the rms slope, is needed which can be calculated from surface measurements. For \Psi < \frac, the surface behaves elastically during contact. In both the Greenwood-Williamson and Mikic models the load is assumed to be proportional to the deformed area. Hence, whether the system behaves plastically or elastically is independent of the applied normal force.


An overview of the GT model

The model proposed by John A. Greenwood and John H. Tripp (GT), extended the GW model to contact between two rough surfaces. The GT model is widely used in the field of elastohydrodynamic analysis. The most frequently cited equations given by the GT model are for the asperity contact area : A_a = \pi^2 (\eta\beta\sigma)^2 AF_2(\lambda), and load carried by asperities :P =\frac\pi (\eta\beta\sigma)^2 \sqrt E' AF_\frac(\lambda), where: : \eta\beta\sigma, roughness parameter, : A, nominal contact area, : \lambda, Stribeck oil film parameter, first defined by Stribeck \cite as \lambda = h/\sigma, : E', effective elastic modulus, : F_2, F_\frac(\lambda), statistical functions introduced to match the assumed Gaussian distribution of asperities. Matthew Leighton et al. presented fits for crosshatched IC engine cylinder liner surfaces together with a process for determining the F_n(h) terms for any measured surfaces. Leighton et al. demonstrated that Gaussian fit data is not accurate for modelling any engineered surfaces and went on to demonstrate that early running of the surfaces results in a gradual transition which significantly changes the surface topography, load carrying capacity and friction. The exact solutions for A_a and P are firstly presented by Jedynak. They are expressed by F_n as follows. They are calculated for the Gaussian distribution of asperities, which have been shown to be unrealistic for engineering surface but can be assumed where friction, load carrying capacity or real contact area results are not critical to the analysis. : \begin F_2 &= \frac \left(h^2 + 1\right)\operatorname \left(\frac\right) - \frac\exp\left(-\frac\right) \\ F_\frac &= \frac\exp\left(-\frac\right) h^\frac \left(\left(2h^2 + 3\right) K_\frac \left(\frac\right) - \left(2h^2 + 5\right) K_\frac\left(\frac\right)\right) \end where erfc(z) means the complementary error function and K_\nu(z) is the modified Bessel function of the second kind. In paper one can find comprehensive review of existing approximants to F_\frac. New proposals give the most accurate approximants to F_\frac and F_2, which are reported in the literature. They are given by the following rational formulas, which are very exact approximants to integrals F_n(h). They are calculated for the Gaussian distribution of asperities : F_n(h) = \frac\exp\left(-\frac\right) For F_2(h) the coefficients are : \begin[] [a_0, a_1, a_2, a_3] &= [0.5, 0.182536384941, 0.039812283118, 0.003684879001] \\[] [b_1, b_2, b_3, b_4, b_5, b_6] &= [1.960841785003, 1.708677456715, 0.856592986083, 0.264996791567, 0.049257843893, 0.004640740133] \end The maximum relative error is 1.68 \times 10^%. For F_\frac(h) the coefficients are :\begin[] [a_0, a_1, a_2, a_3] &= [0.616634218997, 0.108855827811, 0.023453835635, 0.000449332509] \\[] [b_1, b_2, b_3, b_4, b_5, b_6] &= [1.919948267476, 1.635304362591, 0.799392556572, 0.240278859212, 0.043178653945, 0.003863334276] \end The maximum relative error is 4.98 \times 10^%.


Adhesive contact between elastic bodies

When two solid surfaces are brought into close proximity, they experience attractive
van der Waals force In molecular physics and chemistry, the van der Waals force (sometimes van der Waals' force) is a distance-dependent interaction between atoms or molecules. Unlike ionic or covalent bonds, these attractions do not result from a chemical elec ...
s. R. S. Bradley's van der Waals model provides a means of calculating the tensile force between two rigid spheres with perfectly smooth surfaces. The Hertzian model of contact does not consider adhesion possible. However, in the late 1960s, several contradictions were observed when the Hertz theory was compared with experiments involving contact between rubber and glass spheres. It was observed that, though Hertz theory applied at large loads, at low loads * the area of contact was larger than that predicted by Hertz theory, * the area of contact had a non-zero value even when the load was removed, and * there was even strong adhesion if the contacting surfaces were clean and dry. This indicated that adhesive forces were at work. The Johnson-Kendall-Roberts (JKR) model and the Derjaguin-Muller-Toporov (DMT) models were the first to incorporate adhesion into Hertzian contact.


Bradley model of rigid contact

It is commonly assumed that the surface force between two atomic planes at a distance z from each other can be derived from the
Lennard-Jones potential In computational chemistry, molecular physics, and physical chemistry, the Lennard-Jones potential (also termed the LJ potential or 12-6 potential; named for John Lennard-Jones) is an intermolecular pair potential. Out of all the intermolecul ...
. With this assumption : F(z) = \cfrac\left left(\cfrac\right)^ - \left(\cfrac\right)^\right where F is the force (positive in compression), 2\gamma is the total surface energy of ''both'' surfaces per unit area, and z_0 is the equilibrium separation of the two atomic planes. The Bradley model applied the Lennard-Jones potential to find the force of adhesion between two rigid spheres. The total force between the spheres is found to be : F_a(z) = \cfrac\left cfrac\left(\cfrac\right)^ - \left(\cfrac\right)^\right~;~~ \frac = \frac + \frac where R_1,R_2 are the radii of the two spheres. The two spheres separate completely when the ''pull-off force'' is achieved at z = z_0 at which point : F_a = F_c = -4\gamma\pi R .


JKR model of elastic contact

To incorporate the effect of adhesion in Hertzian contact, Johnson, Kendall, and Roberts formulated the JKR theory of adhesive contact using a balance between the stored
elastic energy Elastic energy is the mechanical potential energy stored in the configuration of a material or physical system as it is subjected to elastic deformation by work performed upon it. Elastic energy occurs when objects are impermanently compressed ...
and the loss in
surface energy In surface science, surface energy (also interfacial free energy or surface free energy) quantifies the disruption of intermolecular bonds that occurs when a surface is created. In solid-state physics, surfaces must be intrinsically less energe ...
. The JKR model considers the effect of contact pressure and adhesion only inside the area of contact. The general solution for the pressure distribution in the contact area in the JKR model is : p(r) = p_0\left(1 - \frac\right)^\frac + p_0'\left(1 - \frac\right)^ Note that in the original Hertz theory, the term containing p_0' was neglected on the ground that tension could not be sustained in the contact zone. For contact between two spheres : p_0 = \frac ;\quad p_0' = -\left(\frac\right)^\frac where a\, is the radius of the area of contact, F is the applied force, 2\gamma is the total surface energy of both surfaces per unit contact area, R_i,\, E_i,\, \nu_i,~~i = 1, 2 are the radii, Young's moduli, and Poisson's ratios of the two spheres, and : \frac = \frac + \frac ;\quad \frac = \frac + \frac The approach distance between the two spheres is given by : d = \frac\left(p_0 + 2p_0'\right) = \frac The Hertz equation for the area of contact between two spheres, modified to take into account the surface energy, has the form : a^3 = \frac\left(F + 6\gamma\pi R + \sqrt\right) When the surface energy is zero, \gamma = 0, the Hertz equation for contact between two spheres is recovered. When the applied load is zero, the contact radius is : a^3 = \frac The tensile load at which the spheres are separated (i.e., a = 0) is predicted to be : F_\text = -3\gamma\pi R\, This force is also called the pull-off force. Note that this force is independent of the moduli of the two spheres. However, there is another possible solution for the value of a at this load. This is the critical contact area a_\text, given by : a_\text^3 = \frac If we define the work of adhesion as : \Delta\gamma = \gamma_1 + \gamma_2 - \gamma_ where \gamma_1, \gamma_2 are the adhesive energies of the two surfaces and \gamma_ is an interaction term, we can write the JKR contact radius as : a^3 = \frac\left(F + 3\Delta\gamma\pi R + \sqrt\right) The tensile load at separation is : F = -\frac\Delta\gamma\pi R\, and the critical contact radius is given by : a_\text^3 = \frac The critical depth of penetration is : d_\text = \frac = \left(R^\frac \frac\right)^\frac


DMT model of elastic contact

The Derjaguin–Muller–Toporov (DMT) model is an alternative model for adhesive contact which assumes that the contact profile remains the same as in Hertzian contact but with additional attractive interactions outside the area of contact. The radius of contact between two spheres from DMT theory is : a^3 = \cfrac\left(F + 4\gamma\pi R\right) and the pull-off force is : F_c = -4\gamma\pi R\, When the pull-off force is achieved the contact area becomes zero and there is no singularity in the contact stresses at the edge of the contact area. In terms of the work of adhesion \Delta\gamma : a^3 = \cfrac\left(F + 2\Delta\gamma\pi R\right) and : F_c = -2\Delta\gamma\pi R\,


Tabor parameter

In 1977, Tabor showed that the apparent contradiction between the JKR and DMT theories could be resolved by noting that the two theories were the extreme limits of a single theory parametrized by the Tabor parameter (\mu) defined as : \mu := \frac \approx \left frac\right\frac where z_0 is the equilibrium separation between the two surfaces in contact. The JKR theory applies to large, compliant spheres for which \mu is large. The DMT theory applies for small, stiff spheres with small values of \mu. Subsequently, Derjaguin and his collaborators by applying Bradley's surface force law to an elastic half space, confirmed that as the Tabor parameter increases, the pull-off force falls from the Bradley value 2\pi R\Delta\gamma to the JKR value (3/2)\pi R\Delta\gamma. More detailed calculations were later done by Greenwood revealing the S-shaped load/approach curve which explains the jumping-on effect. A more efficient method of doing the calculations and additional results were given by Feng


Maugis–Dugdale model of elastic contact

Further improvement to the Tabor idea was provided by Maugis who represented the surface force in terms of a Dugdale cohesive zone approximation such that the work of adhesion is given by : \Delta\gamma = \sigma_0~h_0 where \sigma_0 is the maximum force predicted by the Lennard-Jones potential and h_0 is the maximum separation obtained by matching the areas under the Dugdale and Lennard-Jones curves (see adjacent figure). This means that the attractive force is constant for z_0 \le z \le z_0 + h_0. There is not further penetration in compression. Perfect contact occurs in an area of radius a and adhesive forces of magnitude \sigma_0 extend to an area of radius c > a. In the region a < r < c, the two surfaces are separated by a distance h(r) with h(a) = 0 and h(c) = h_0. The ratio m is defined as : m := \frac. In the Maugis–Dugdale theory, the surface traction distribution is divided into two parts - one due to the Hertz contact pressure and the other from the Dugdale adhesive stress. Hertz contact is assumed in the region -a < r < a. The contribution to the surface traction from the Hertz pressure is given by : p^H(r) = \left(\frac\right)\left(1 - \frac\right)^\frac where the Hertz contact force F^H is given by : F^H = \frac The penetration due to elastic compression is : d^H = \frac The vertical displacement at r = c is : u^H(c) = \cfrac \left ^2\left(2 - m^2\right)\sin^\left(\frac\right) + a^2\sqrt\right and the separation between the two surfaces at r = c is : h^H(c) = \frac - d^H + u^H(c) The surface traction distribution due to the adhesive Dugdale stress is : p^D(r) = \begin -\frac\cos^\left frac\right& \quad \text \quad r \le a \\ -\sigma_0 & \quad \text \quad a \le r \le c \end The total adhesive force is then given by : F^D = -2\sigma_0 m^2 a^2\left cos^\left(\frac\right) + \frac\sqrt\right The compression due to Dugdale adhesion is : d^D = -\left(\frac\right)\sqrt and the gap at r = c is : h^D(c) = \left(\frac\right)\left sqrt\cos^\left(\frac\right) + 1 - m\right The net traction on the contact area is then given by p(r) = p^H(r) + p^D(r) and the net contact force is F = F^H + F^D. When h(c) = h^H(c) + h^D(c) = h_0 the adhesive traction drops to zero. Non-dimensionalized values of a, c, F, d are introduced at this stage that are defied as : \bar = \alpha a ~;~~ \bar := \alpha c ~;~~ \bar := \alpha^2 Rd ~;~~ \alpha := \left(\frac\right)^\frac ~;~~ \bar := \pi c^2 ~;~~ \bar = \frac In addition, Maugis proposed a parameter \lambda which is equivalent to the Tabor parameter \mu . This parameter is defined as : \lambda := \sigma_0\left(\frac\right)^\frac\approx 1.16\mu where the step cohesive stress \sigma_0 equals to the theoretical stress of the Lennard-Jones potential : \sigma_\text = \frac Zheng and Yu suggested another value for the step cohesive stress : \sigma_0 = \exp\left(-\frac\right) \cdot \frac \approx 0.588 \frac to match the Lennard-Jones potential, which leads to : \lambda \approx 0.663\mu Then the net contact force may be expressed as : \bar = \bar^3 - \lambda \bar^2\left sqrt + m^2 \sec^ m\right and the elastic compression as : \bar = \bar^2 - \frac~\lambda \bar\sqrt The equation for the cohesive gap between the two bodies takes the form : \frac\left left(m^2 - 2\right)\sec^ m + \sqrt\right+ \frac\left sqrt\sec^ m - m + 1\right= 1 This equation can be solved to obtain values of c for various values of a and \lambda. For large values of \lambda, m \rightarrow 1 and the JKR model is obtained. For small values of \lambda the DMT model is retrieved.


Carpick–Ogletree-Salmeron (COS) model

The Maugis–Dugdale model can only be solved iteratively if the value of \lambda is not known a-priori. The Carpick–Ogletree–Salmeron (COS) approximate solution (after Robert Carpick, D. Frank Ogletree and Miquel Salmeron)simplifies the process by using the following relation to determine the contact radius a: : a = a_0(\beta) \left(\frac\right)^\frac where a_0 is the contact area at zero load, and \beta is a transition parameter that is related to \lambda by : \lambda \approx -0.924 \ln(1 - 1.02\beta) The case \beta = 1 corresponds exactly to JKR theory while \beta = 0 corresponds to DMT theory. For intermediate cases 0 < \beta < 1 the COS model corresponds closely to the Maugis–Dugdale solution for 0.1 < \lambda < 5.


Influence of contact shape

Even in the presence of perfectly smooth surfaces, geometry can come into play in form of the macroscopic shape of the contacting region. When a rigid punch with flat but oddly shaped face is carefully pulled off its soft counterpart, its detachment occurs not instantaneously but detachment fronts start at pointed corners and travel inwards, until the final configuration is reached which for macroscopically isotropic shapes is almost circular. The main parameter determining the adhesive strength of flat contacts occurs to be the maximum linear size of the contact. The process of detachment can as observed experimentally can be seen in the film.


See also

* * * * * * * * (ECR) * * * * * * * * * * * * * * *


References


External links



A MATLAB routine to solve the linear elastic contact mechanics problem entitled; "An LCP solution of the linear elastic contact mechanics problem" is provided at the file exchange at MATLAB Central.

Contact mechanics calculator.

detailed calculations and formulae of JKR theory for two spheres.
[5
/nowiki>">">[5
/nowiki> A Matlab code for Hertz contact analysis (includes line, point and elliptical cases).
[6
/nowiki>]: JKR, MD, and DMT models of adhesion (Matlab routines). {{Authority control Bearings (mechanical) Mechanical engineering Solid mechanics