Contact Mechanics

Stresses in a contact area loaded simultaneously

with a normal and a tangential force. Stresses

were made visible using photoelasticity.

Contact mechanicsFrom Wikipedia, the free encyclopedia

Contact mechanics is the study of the deformation of solids that

touch each other at one or more points.[1][2] The physical andmathematical formulation of the subject is built upon the mechanicsof materials and continuum mechanics and focuses oncomputations involving elastic, viscoelastic, and plastic bodies instatic or dynamic contact. Central aspects in contact mechanics arethe pressures and adhesion acting perpendicular to the contactingbodies' surfaces, the normal direction, and the frictional stressesacting tangentially between the surfaces. This page focuses mainlyon the normal direction, i.e. on frictionless contact mechanics.Frictional contact mechanics is discussed separately.

Contact mechanics is foundational to the field of mechanicalengineering; it provides necessary information for the safe andenergy efficient design of technical systems and for the study oftribology and indentation hardness. Principles of contactsmechanics can be applied in areas such as locomotive wheel-railcontact, coupling devices, braking systems, tires, bearings, combustion engines, mechanical linkages, gasket seals,metalworking, metal forming, ultrasonic welding, electrical contacts, and many others. Current challenges faced in thefield may include stress analysis of contact and coupling members and the influence of lubrication and material design onfriction and wear. Applications of contact mechanics further extend into the micro- and nanotechnological realm.

The original work in contact mechanics dates back to 1882 with the publication of the paper "On the contact of elastic

solids"[3] ("Ueber die Berührung fester elastischer Körper" (http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=251917)) by Heinrich Hertz. Hertz was attempting to understand how the optical properties of multiple,stacked lenses might change with the force holding them together. Hertzian contact stress refers to the localized stressesthat develop as two curved surfaces come in contact and deform slightly under the imposed loads. This amount ofdeformation is dependent on the modulus of elasticity of the material in contact. It gives the contact stress as a function ofthe normal contact force, the radii of curvature of both bodies and the modulus of elasticity of both bodies. Hertziancontact stress forms the foundation for the equations for load bearing capabilities and fatigue life in bearings, gears, andany other bodies where two surfaces are in contact.

Contents

1 History2 Classical solutions for non-adhesive elastic contact

2.1 Contact between a sphere and an elastic half-space2.2 Contact between two spheres2.3 Contact between two crossed cylinders of equal radius2.4 Contact between a rigid cylinder and an elastic half-space2.5 Contact between a rigid conical indenter and an elastic half-space2.6 Contact between two cylinders with parallel axes2.7 The Method of Dimensionality Reduction

3 Hertzian theory of non-adhesive elastic contact

When a sphere is pressed against an elastic

material, the contact area increases.

3.1 Assumptions in Hertzian theory3.2 Analytical solution techniques

3.2.1 Point contact on a (2D) half-plane3.2.2 Line contact on a (2D) half-plane

3.2.2.1 Normal loading over a region3.2.2.2 Shear loading over a region

3.2.3 Point contact on a (3D) half-space3.3 Numerical solution techniques

4 Non-adhesive contact between rough surfaces5 Adhesive contact between elastic bodies

5.1 Bradley model of rigid contact5.2 Johnson-Kendall-Roberts (JKR) model of elastic contact5.3 Derjaguin-Muller-Toporov (DMT) model of elastic contact5.4 Tabor coefficient5.5 Maugis-Dugdale model of elastic contact5.6 Carpick-Ogletree-Salmeron (COS) model

6 See also7 References8 External links

History

Classical contact mechanics is most notably associated with

Heinrich Hertz.[4] In 1882, Hertz solved the contact problem oftwo elastic bodies with curved surfaces. This still-relevant classicalsolution provides a foundation for modern problems in contactmechanics. For example, in mechanical engineering and tribology,Hertzian contact stress is a description of the stress within matingparts. The Hertzian contact stress usually refers to the stress closeto the area of contact between two spheres of different radii.

It was not until nearly one hundred years later that Johnson,Kendall, and Roberts found a similar solution for the case of

adhesive contact.[5] This theory was rejected by Boris Derjaguin

and co-workers[6] who proposed a different theory of adhesion[7]

in the 1970s. The Derjaguin model came to be known as the DMT

(after Derjaguin, Muller and Toporov) model,[7] and the Johnsonet al. model came to be known as the JKR (after Johnson, Kendalland Roberts) model for adhesive elastic contact. This rejection

proved to be instrumental in the development of the Tabor[8] and

later Maugis[6][9] parameters that quantify which contact model (ofthe JKR and DMT models) represent adhesive contact better forspecific materials.

Further advancement in the field of contact mechanics in the mid-twentieth century may be attributed to names such asBowden and Tabor. Bowden and Tabor were the first to emphasize the importance of surface roughness for bodies in

contact.[10][11] Through investigation of the surface roughness, the true contact area between friction partners is found to

Contact between a sphere and an

elastic half-space

be less than the apparent contact area. Such understanding also drastically changed the direction of undertakings intribology. The works of Bowden and Tabor yielded several theories in contact mechanics of rough surfaces.

The contributions of Archard (1957)[12] must also be mentioned in discussion of pioneering works in this field. Archardconcluded 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),[13] Bush (1975),[14]

and Persson (2002).[15] The main findings of these works were that the true contact surface in rough materials isgenerally proportional to the normal force, while the parameters of individual micro-contacts (i.e., pressure, size of themicro-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 simplegeometries. Some commonly used solutions are listed below. The theory used to compute these solutions is discussedlater in the article.

Contact between a sphere and an elastic half-space

An elastic sphere of radius indents an elastic half-space to depth , andthus creates 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.

Contact between two spheres

For contact between two spheres of radii and , the area of contact is a circle of radius . The distribution of

normal traction in the contact area as a function of distance from the center of the circle is[1]

where is the maximum contact pressure given by

Contact between

two spheres

Contact between two crossed

cylinders of equal radius

Contact between a rigid cylindrical

indenter and an elastic half-space

where the effective radius is defined as

The radius of the circle is related to the applied load by the equation

The depth of indentation is related to the maximum contact pressure by

The maximum shear stress occurs in the interior at for .

Contact between two crossed cylinders of equal radius

This is equivalent to contact between a sphere of radius and a plane (seeabove).

Contact between a rigid cylinder and an elastic half-space

If a rigid cylinder is pressed into an elastic half-space, it creates a pressure

distribution described by[16]

where is the radius of the cylinder and

The relationship between the indentation depth and the normal force is givenby

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 indenter, the depth of the

contact region and contact radius are related by[16]

Contact between a rigid conical

indenter and an elastic half-space

Contact between two cylinders with

parallel axes

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 logarithmic 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 linearlyproportional to the indentation depth:

The radii of curvature are entirely absent from this relationship. The contactradius is described through the usual relationship

with

as in contact between two spheres. The maximum pressure is equal to

The Method of Dimensionality Reduction

Contact between a sphere and an

elastic half-space and one-dimensional

replaced model

Many contact problems can be solved easily with the Method ofDimensionality Reduction. In this method, the initial three-dimensional systemis replaced with a contact of a body with a linear elastic or viscoelasticfoundation (see Fig). The properties of one-dimensional systems coincidehereby exactly with those of the original three-dimensional system, if the formof the bodies is modified and the elements of the foundation are defined

according to the rules of the RMD. [17] [18]

Hertzian theory of non-adhesive elasticcontact

The classical theory of contact focused primarily on non-adhesive contact where no tension force is allowed to occurwithin the contact area, i.e., contacting bodies can be separated without adhesion forces. Several analytical andnumerical approaches have been used to solve contact problems that satisfy the no-adhesion condition. Complex forcesand moments are transmitted between the bodies where they touch, so problems in contact mechanics can become quitesophisticated. In addition, the contact stresses are usually a nonlinear function of the deformation. To simplify the solutionprocedure, a frame of reference is usually defined in which the objects (possibly in motion relative to one another) arestatic. 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 normalto the surface. One of the bodies will experience a normally-directed pressure 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,

each body can be considered an elastic half-space, i.e., the area of contact is much smaller than the characteristic

radius of the body,the surfaces are continuous and non-conforming, and

the surfaces are frictionless.

Additional complications arise when some or all these assumptions are violated and such contact problems are usuallycalled non-Hertzian.

Analytical solution techniques

Contact between two spheres.

Schematic of the loading on a plane by force P at

a point (0,0).

Analytical solution methods for non-adhesive contact problem canbe classified into two types based on the geometry of the area of

contact.[19] A conforming contact is one in which the two bodiestouch at multiple points before any deformation takes place (i.e.,they just "fit together"). A non-conforming contact is one inwhich the shapes of the bodies are dissimilar enough that, underzero load, they only touch at a point (or possibly along a line). Inthe non-conforming case, the contact area is small compared to thesizes of the objects and the stresses are highly concentrated in thisarea. Such a contact is called concentrated, otherwise it is calleddiversified.

A common approach in linear elasticity is to superpose a numberof solutions each of which corresponds to a point load acting overthe area of contact. For example, in the case of loading of a half-plane, the Flamant solution is often used as a starting point andthen generalized to various shapes of the area of contact. The forceand moment balances between the two bodies in contact act asadditional constraints to the solution.

Point contact on a (2D) half-plane

Main article: Flamant solution

A starting point for solving contact problems is to understand theeffect of a "point-load" applied to an isotropic, homogeneous, andlinear elastic half-plane, shown in the figure to the right. Theproblem may be either be plane stress or plane strain. This is aboundary value problem of linear elasticity subject to the tractionboundary conditions:

where is the Dirac delta function. The boundary

conditions state that there are no shear stresses on the surface anda singular normal force P is applied at (0,0). Applying theseconditions to the governing equations of elasticity produces theresult

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 bedetermined.

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

. The principle of linear superposition can be applied to determine the resulting stress field as the solution tothe integral equations:

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 toarise as a result of friction. The solution is similar the above (for both singular loads and distributed loads ) but

altered slightly:

These results may themselves be superposed onto those given above for normal loading to deal with more complexloads.

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 3Dhalf-space as well. These were found by Boussinesq for a concentrated normal load and by Cerutti for a tangential load.See the section on this in Linear elasticity.

Numerical solution techniques

Distinctions between conforming and non-conforming contact do not have to be made when numerical solution schemesare 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 [20] [21] [22] [23] .[24] Besides the standardequations describing the deformation and motion of bodies two additional inequalities can be formulated. The first simplyrestricts the motion and deformation of the bodies by the assumption that no penetration can occur. Hence the gap between two bodies can only be positive or zero

where 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 leadsto an inequality which the stresses have to obey at the contact interface. It is formulated for the contact pressure

Since for contact, , the contact pressure is always negative, , and further for non contact the gap is

open, , and the contact pressure is zero, , the so-called Kuhn–Tucker form of the contact constraints

can be written as

These conditions are valid in a general way. The mathematical formulation of the gap depends upon the kinematics of theunderlying theory of the solid (e.g., linear or nonlinear solid in two- or three dimensions, beam or shell model).

Non-adhesive contact between rough surfaces

When two bodies with rough surfaces are pressed into each other, the true contact area is much smaller than theapparent contact area . In contact between a "random rough" surface and an elastic half-space, the true contact area

is related to the normal force by[1][25][26][27]

with equal to the root mean square (also known as the quadratic mean) of the surface slope and . The medianpressure in the true contact surface

can be reasonably estimated as half of the effective elastic modulus multiplied with the root mean square of the

surface slope .

For the situation where the asperities on the two surfaces have a Gaussian height distribution and the peaks can be

assumed to be spherical,[25] the average contact pressure is sufficient to cause yield when

where is the uniaxial yield stress and is the indentation hardness.[1] Greenwood and Williamson[25] defined adimensionless parameter 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 deviationof the surface roughness and the curvature of the asperity peaks. An alternative definition of the plasticity index has been

given by Mikic.[26] Yield occurs when the pressure is greater than the uniaxial yield stress. Since the yield stress isproportional to the indentation hardness , Micic defined the plasticity index for elastic-plastic contact to be

In this definition 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 , 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.[1]

Adhesive contact between elastic bodies

When two solid surfaces are brought into close proximity, they experience attractive van der Waals forces. Bradley's van

der Waals model[28] provides a means of calculating the tensile force between two rigid spheres with perfectly smoothsurfaces. The Hertzian model of contact does not consider adhesion possible. However, in the late 1960s, severalcontradictions were observed when the Hertz theory was compared with experiments involving contact between rubberand glass spheres.

It was observed[5] 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 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 from each other can bederived from the Lennard-Jones potential. With this assumption

where is the force (positive in compression), is the total surface energy of both surfaces per unit area, and 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. Thetotal force between the spheres is found to be

where are the radii of the two spheres.

The two spheres separate completely when the pull-off force is achieved at at which point

Schematic of contact area for the JKR model.

JKR test with a rigid bead on a

deformable planar material: complete

cycle

Johnson-Kendall-Roberts (JKR) model of elastic contact

To incorporate the effect of adhesion in Hertzian

contact, Johnson, Kendall, and Roberts[5]

formulated the JKR theory of adhesive contactusing a balance between the stored elasticenergy and the loss in surface energy. The JKRmodel considers the effect of contact pressureand adhesion only inside the area of contact.The general solution for the pressure distributionin the contact area in the JKR model is

Note that in the original Hertz theory, the term containing was neglected on

the ground that tension could not be sustained in the contact zone. For contactbetween two spheres

where is the radius of the area of contact, is the applied force, is the

total surface energy of both surfaces per unit contact area, are the radii, Young's moduli, and Poisson's ratios

of the two spheres, and

The approach distance between the two spheres is given by

The Hertz equation for the area of contact between two spheres, modified to take into account the surface energy, hasthe form

When the surface energy is zero, , the Hertz equation for contact between two spheres is recovered. When the

applied load is zero, the contact radius is

The tensile load at which the spheres are separated, i.e., , is predicted to be

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 at this load. This is the critical contact area , given by

If we define the work of adhesion as

where are the adhesive energies of the two surfaces and is an interaction term, we can write the JKRcontact radius as

The tensile load at separation is

and the critical contact radius is given by

The critical depth of penetration is

Derjaguin-Muller-Toporov (DMT) model of elastic contact

The Derjaguin-Muller-Toporov (DMT) model[29][30] is an alternative model for adhesive contact which assumes that thecontact profile remains the same as in Hertzian contact but with additional attractive interactions outside the area ofcontact.

The area of contact between two spheres from DMT theory is

and the pull-off force is

When the pull-off force is achieved the contact area becomes zero and there is no singularity in the contact stresses at theedge of the contact area.

In terms of the work of adhesion

and

Tabor coefficient

In 1977, Tabor[31] showed that the apparent contradiction between the JKR and DMT theories could be resolved bynoting that the two theories were the extreme limits of a single theory parametrized by the Tabor coefficient ( ) definedas

where is the equilibrium separation between the two surfaces in contact. The JKR theory applies to large, compliantspheres for which is large. The DMT theory applies for small, stiff spheres with small values of .

Maugis-Dugdale model of elastic contact

Further improvement to the Tabor idea was provided by Maugis[9] who represented the surface force in terms of aDugdale cohesive zone approximation such that the work of adhesion is given by

where is the maximum force predicted by the Lennard-Jones potential and is the maximum separation obtained bymatching the areas under the Dugdale and Lennard-Jones curves (see adjacent figure). This means that the attractiveforce is constant for . There is not further penetration in compression. Perfect contact occurs in anarea of radius and adhesive forces of magnitude extend to an area of radius . In the region ,the two surfaces are separated by a distance with and . The ratio is defined as

.

In the Maugis-Dugdale theory,[32] the surface traction distribution is divided into two parts - one due to the Hertz contactpressure and the other from the Dugdale adhesive stress. Hertz contact is assumed in the region . Thecontribution to the surface traction from the Hertz pressure is given by

Schematic of contact area for the Maugis-Dugdale model.

where the Hertz contact force is given by

The penetration due to elastic compression is

The vertical displacement at is

and the separation between the two surfaces at is

The surface traction distribution due to the adhesive Dugdale stress is

The total adhesive force is then given by

The compression due to Dugdale adhesion is

and the gap at is

The net traction on the contact area is then given by and the net contact force is

. When the adhesive traction drops to zero.

Non-dimensionalized values of are introduced at this stage that are defied as

In addition, Maugis proposed a parameter which is equivalent to the Tabor coefficient. This parameter is defined as

where the step cohesive stress equals to the theoretical stress of the Lennard-Jones potential

Zheng and Yu [33] suggested another value for the step cohesive stress

to match the Lennard-Jones potential, which leads to

Then the net contact force may be expressed as

and the elastic compression as

The equation for the cohesive gap between the two bodies takes the form

This equation can be solved to obtain values of for various values of and . For large values of , and theJKR model is obtained. For small values of the DMT model is retrieved.

Carpick-Ogletree-Salmeron (COS) model

The Maugis-Dugdale model can only be solved iteratively if the value of is not known a-priori. The Carpick-Ogletree-

Salmeron approximate solution [34] simplifies the process by using the following relation to determine the contact radius :

where is the contact area at zero load, and is a transition parameter that is related to by

The case corresponds exactly to JKR theory while corresponds to DMT theory. For intermediate

cases the COS model corresponds closely to the Maugis-Dugdale solution for .

See also

AdhesiveAdhesive bondingAdhesive dermatitis

Adhesive surface forces

Bearing capacity

BioadhesivesContact dynamicsDispersive adhesion

Electrostatic generatorEnergetically modified cement

Frictional contact mechanicsGalling

GoniometerNon-smooth mechanics

Plastic wrapShock (mechanics)Signorini problem

Surface tensionSynthetic setae

Unilateral contactWetting

References

1. ̂a b c d e Johnson, K. L, 1985, Contact mechanics, Cambridge University Press.

2. ^ Popov, Valentin L., 2010, Contact Mechanics and Friction. Physical Principles and Applications, Springer-Verlag,362 p., ISBN 978-3-642-10802-0.

3. ^ H. Hertz, Über die berührung fester elastischer Körper (On the contact of rigid elastic solids). In: MiscellaneousPapers (http://www.archive.org/details/cu31924012500306). Jones and Schott, Editors, J. reine und angewandteMathematik 92, Macmillan, London (1896), p. 156 English translation: Hertz, H.

4. ^ Hertz, H. R., 1882, Ueber die Beruehrung elastischer Koerper (On Contact Between Elastic Bodies), in Gesammelte

Werke (Collected Works), Vol. 1, Leipzig, Germany, 1895.

5. ̂a b c K. L. Johnson and K. Kendall and A. D. Roberts, Surface energy and the contact of elastic solids, Proc. R. Soc.London A 324 (1971) 301-313

6. ̂a b D. Maugis, Contact, Adhesion and Rupture of Elastic Solids, Springer-Verlag, Solid-State Sciences, Berlin 2000,ISBN 3-540-66113-1

7. ̂a b B. V. Derjaguin and V. M. Muller and Y. P. Toporov, Effect of contact deformations on the adhesion of particles,J. Colloid Interface Sci. 53 (1975) 314--325

8. ^ D. Tabor, The hardness of solids, J. Colloid Interface Sci. 58 (1977) 145-179

9. ̂a b D. Maugis, Adhesion of spheres: The JKR-DMT transition using a Dugdale model, J. Colloid Interface Sci. 150(1992) 243--269

10. ^ , Bowden, FP and Tabor, D., 1939, The area of contact between stationary and between moving surfaces, Proceedingsof the Royal Society of London. Series A, Mathematical and Physical Sciences, 169(938), pp. 391--413.

11. ^ Bowden, F.P. and Tabor, D., 2001, The friction and lubrication of solids, Oxford University Press.

12. ^ Archard, JF, 1957, Elastic deformation and the laws of friction, Proceedings of the Royal Society of London. SeriesA, Mathematical and Physical Sciences, 243(1233), pp.190--205.

13. ^ Greenwood, JA and Williamson, JBP., 1966, Contact of nominally flat surfaces, Proceedings of the Royal Society ofLondon. Series A, Mathematical and Physical Sciences, pp. 300-319.

14. ^ Bush, AW and Gibson, RD and Thomas, TR., 1975, The elastic contact of a rough surface, Wear, 35(1), pp. 87-111.

15. ^ Persson, BNJ and Bucher, F. and Chiaia, B., 2002, Elastic contact between randomly rough surfaces: Comparison oftheory with numerical results, Physical Review B, 65(18), p. 184106.

16. ̂a b Sneddon, I. N., 1965, The Relation between Load and Penetration in the Axisymmetric Boussinesq Problem for aPunch of Arbitrary Profile. Int. J. Eng. Sci. v. 3, pp. 47–57.

17. ^ Popov, V.L., Method of reduction of dimensionality in contact and friction mechanics: A linkage between micro andmacro scales, Friction, 2013, v.1, N. 1, pp.41–62.

18. ^ Popov, V.L. and Heß, M., Methode der Dimensionsreduktion in Kontaktmechanik und Reibung, Springer,2013.

19. ^ Shigley, J.E., Mischke, C.R., 1989, Mechanical Engineering Design, Fifth Edition, Chapter 2, McGraw-Hill, Inc,1989, ISBN 0-07-056899-5.

20. ^ Kalker, J.J. 1990, Three-Dimensional Elastic Bodies in Rolling Contact. (Kluwer Academic Publishers: Dordrecht).

21. ^ Wriggers, P. 2006, Computational Contact Mechanics. 2nd ed. (Springer Verlag: Heidelberg).

22. ^ Laursen, T. A., 2002, Computational Contact and Impact Mechanics: Fundamentals of Modeling InterfacialPhenomena in Nonlinear Finite Element Analysis, (Springer Verlag: New York).

23. ^ Acary V. and Brogliato B., 2008,Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanicsand Electronics. Springer Verlag, LNACM 35, Heidelberg.

24. ^ Popov, Valentin L., 2009, Kontaktmechanik und Reibung. Ein Lehr- und Anwendungsbuch von der Nanotribologie biszur numerischen Simulation, Springer-Verlag, 328 S., ISBN 978-3-540-88836-9.

25. ̂a b c Greenwood, J. A. and Williamson, J. B. P., (1966), Contact of nominally flat surfaces, Proceedings of the RoyalSociety of London. Series A, Mathematical and Physical Sciences, vol. 295, pp. 300--319.

26. ̂a b Mikic, B. B., (1974), Thermal contact conductance; theoretical considerations, International Journal of Heat andMass Transfer, 17(2), pp. 205-214.

27. ^ Hyun, S., and M.O. Robbins, 2007, Elastic contact between rough surfaces: Effect of roughness at large and smallwavelengths. Tribology International, v.40, pp. 1413-1422.

28. ^ Bradley, RS., 1932, The cohesive force between solid surfaces and the surface energy of solids, PhilosophicalMagazine Series 7, 13(86), pp. 853--862.

29. ^ 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.

30. ^ Muller, VM and Derjaguin, BV and Toporov, Y.P., 1983, On two methods of calculation of the force of sticking of anelastic sphere to a rigid plane, Colloids and Surfaces, 7(3), pp. 251-259.

31. ^ Tabor, D., 1977, Surface forces and surface interactions, Journal of Colloid and Interface Science, 58(1), pp. 2-13.

32. ^ Johnson, KL and Greenwood, JA, 1997, An adhesion map for the contact of elastic spheres, Journal of Colloid andInterface Science, 192(2), pp. 326-333.

33. ^ Zheng, Z.J. and Yu, J.L., 2007, Using the Dugdale approximation to match a specific interaction in the adhesivecontact of elastic objects, Journal of Colloid and Interface Science, 310(1), pp. 27-34.

34. ^ Carpick, R.W. and Ogletree, D.F. and Salmeron, M., 1999, A general equation for fitting contact area and friction vsload measurements, Journal of colloid and interface science, 211(2), pp. 395-400.

External links

[1] (http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19970025228_1997043322.pdf): More about contact

stresses and the evolution of bearing stress equations can be found in this publication by NASA Glenn ResearchCenter head the NASA Bearing, Gearing and Transmission Section, Erwin Zaretsky.

Online engineering calculator to calculate contact stress for spherical and cylindrical contacts.(http://www.amesweb.info/HertzianContact/HertzianContact.aspx)

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