Contact mechanics
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Contact mechanicsis the study of thedeformationofsolidsthat touch each other at one or more points.[1][2]A central distinction in contact mechanics is betweenstressesactingperpendicularto the contacting bodies' surfaces (known asnormal stress) andfrictionalstresses actingtangentiallybetween the surfaces (shear stress).Normal contact mechanicsorfrictionless contact mechanicsfocuses on normal stresses caused by appliednormal forcesand by theadhesionpresent on surfaces in close contact, even if they are clean and dry. Frictional contact mechanicsemphasizes the effect of friction forces.
Contact mechanics is part of mechanicalengineering.The physical and mathematical formulation of the subject is built upon themechanics of materialsandcontinuum mechanicsand focuses on computations involvingelastic,viscoelastic,andplasticbodies instaticordynamiccontact. Contact mechanics provides necessary information for the safe and energy efficient design of technical systems and for the study oftribology,contact stiffness,electrical contact resistanceandindentation hardness.Principles of contacts mechanics are implemented towards applications such as locomotive wheel-rail contact,couplingdevices,brakingsystems,tires,bearings,combustion engines,mechanical linkages,gasketseals,metalworking,metal forming,ultrasonic welding,electrical contacts,and many others. Current challenges faced in the field may includestress analysisof contact and coupling members and the influence oflubricationand materialdesignonfrictionandwear.Applications of contact mechanics further extend into themicro- andnanotechnologicalrealm.
The original work in contact mechanics dates back to 1881 with the publication of the paper "On the contact of elastic solids"[3]("Über die Berührung fester elastischer Körper") byHeinrich Hertz.Hertz was attempting to understand how the optical properties of multiple, stackedlensesmight change with theforceholding 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 themodulus of elasticityof 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 andfatiguelife in bearings, gears, and any other bodies where two surfaces are in contact.
History
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Hertz_contact_animated.gif/290px-Hertz_contact_animated.gif)
Classical contact mechanics is most notably associated with Heinrich Hertz.[3][4]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, inmechanical engineeringandtribology,Hertzian contact stressis 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 thatJohnson,Kendall, and Roberts found a similar solution for the case ofadhesivecontact.[5]This theory was rejected byBoris Derjaguinand 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 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[8]and later Maugis[6][9]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 asBowdenandTabor.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 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)[12]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 thenormal 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 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
[edit]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[16]
Contact between a sphere and a half-space
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/8/82/Contact_sphere-plane.jpg/220px-Contact_sphere-plane.jpg)
An elasticsphereofradiusindents an elastichalf-spacewhere total deformation is,causing a contact area of radius
The applied forceis related to the displacementby[4]
where
and,are theelastic moduliand,thePoisson's ratiosassociated with each body.
The distribution of normal pressure in the contact area as a function of distance from the center of the circle is[1]
whereis the maximum contact pressure given by
The radius of the circle is related to the applied loadby the equation
The total deformationis related to the maximum contact pressure by
The maximum shear stress occurs in the interior atfor.
Contact between two spheres
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Kontakt_Kugel_Kugel.jpg/120px-Kontakt_Kugel_Kugel.jpg)
![](https://upload.wikimedia.org/wikipedia/commons/thumb/4/49/Kontakt_gekreuzter_Zylinder.jpg/220px-Kontakt_gekreuzter_Zylinder.jpg)
For contact between two spheres of radiiand,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 radiusis defined as[4]
Contact between two crossed cylinders of equal radius
[edit]This is equivalent to contact between a sphere of radiusand aplane.
Contact between a rigid cylinder with flat end and an elastic half-space
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Kontakt_Zylindrischer_Indenter_Ebene.jpg/220px-Kontakt_Zylindrischer_Indenter_Ebene.jpg)
If a rigidcylinderis pressed into an elastic half-space, it creates a pressure distribution described by[17]
whereis 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
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Kontakt_Kegel_Ebene.jpg/220px-Kontakt_Kegel_Ebene.jpg)
In the case ofindentationof an elastic half-space of Young's modulususing a rigidconicalindenter, the depth of the contact regionand contact radiusare related by[17]
withdefined as the angle between the plane and the side surface of the cone. The total indentation depthis given by:
The total force is
The pressure distribution is given by
The stress has alogarithmicsingularityat the tip of the cone.
Contact between two cylinders with parallel axes
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Kontakt_paralleler_Zylinder.jpg/220px-Kontakt_paralleler_Zylinder.jpg)
In contact between two cylinders with parallel axes, the force is linearly proportional to the length of cylindersLand to the indentation depthd:[18]
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
[edit]The contact in the case ofbearingsis often a contact between a convex surface (male cylinder or sphere) and a concave surface (female cylinder or sphere:boreorhemispherical cup).
The Method of Dimensionality Reduction
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/4/40/WikipediabilderKap_4.jpg/220px-WikipediabilderKap_4.jpg)
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.[19][20]MDR is based on the solution to axisymmetric contact problems first obtained by Ludwig Föppl (1941) and Gerhard Schubert (1942)[21]
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
[edit]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 andmomentsare 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, aframe of referenceis 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 surfacein the (,)-plane with the-axis assumed normal to the surface. One of the bodies will experience a normally-directedpressuredistributionand in-planesurface tractiondistributionsandover the region.In terms of aNewtonianforce 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 arekinematicallyimmobile.
Assumptions in Hertzian theory
[edit]The following assumptions are made in determining the solutions ofHertziancontact 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 callednon-Hertzian.
Analytical solution techniques
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Hertz.svg/290px-Hertz.svg.png)
Analytical solution methods for non-adhesive contact problem can be classified into two types based on the geometry of the area of contact.[22]Aconforming contactis one in which the two bodies touch at multiple points before any deformation takes place (i.e., they just "fit together" ). Anon-conforming contactis 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 thestressesare highly concentrated in this area. Such a contact is calledconcentrated,otherwise it is calleddiversified.
A common approach inlinear elasticityis tosuperposea 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 ahalf-plane,theFlamant solutionis 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
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Point_contact_fig.svg/290px-Point_contact_fig.svg.png)
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 eitherplane stressorplane strain.This is aboundary value problemof linear elasticity subject to the tractionboundary conditions:
whereis theDirac delta function.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, thestraincomponents and thus thedisplacementsof all material points may be determined.
Line contact on a (2D) half-plane
[edit]Normal loading over a region
[edit]Suppose, rather than a point load,a distributed loadis 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 to theintegralequations:
Shear loading over a region
[edit]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 loadsand distributed loads) but altered slightly:
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
[edit]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 byBoussinesqfor a concentrated normal load and by Cerruti for a tangential load. See the section on this inLinear elasticity.
Numerical solution techniques
[edit]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.[23][24][25][26][27]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 gapbetween two bodies can only be positive or zero
wheredenotes 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.
At locations where there is contact between the surfaces the gap is zero, i.e.,and there the normal stress is different than zero, indeed,.At locations where the surfaces are not in contact the normal stress is identical to zero;,while the gap is positive; i.e.,.This type of complementarity formulation can be expressed in the so-calledKuhn–Tuckerform, viz.
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,beamorshellmodel). By restating the normal stressin terms of the contact pressure,;i.e.,the Kuhn-Tucker problem can be restated as in standard complementarity form i.e.In the linear elastic case the gap can be formulated aswhereis the rigid body separation,is the geometry/topography of the contact (cylinder and roughness) andis 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 () as a function of the contact pressure (); i.e.,wherefor line loading of an elastic half space andfor point loading of an elastic half-space.[1]
After discretization the linear elastic contact mechanics problem can be stated in standard Linear Complementarity Problem (LCP) form.[28]
whereis 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 asLemke'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 byAlmqvist 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 byAlmqvist et al.
Contact between rough surfaces
[edit]When two bodies with rough surfaces are pressed against each other, the true contact area formed between the two bodies,,is much smaller than the apparent or nominal contact area.The mechanics of contacting rough surfaces are discussed in terms of normal contact mechanics and static frictional interactions.[29]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 assurface 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.[30][31]Assuming a model of shearing welded contacts intribologicalinteractions, 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.[29]
In contact between a "random rough" surface and an elastic half-space, the true contact area is related to the normal forceby[1][31][32][33]
withequal to the root mean square (also known as the quadratic mean) of the surface slope and.The median pressure in the true contact surface
can be reasonably estimated as half of the effective elastic modulusmultiplied with the root mean square of the surface slope.
An overview of the GW model
[edit]Greenwood and Williamson in 1966 (GW)[31]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 betweenandis.The authors calculated the number of contact spots n, the total contact areaand 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
The expected total area of contact can be calculated from the formula
and the expected total force is given by
where:
- R, radius of curvature of the microasperity,
- z, height of the microasperity measured from the profile line,
- d, close the surface,
- ,composite Young's modulus of elasticity,
- ,modulus of elasticity of the surface,
- ,Poisson's surface coefficients.
Greenwood and Williamson introduced standardized separationand standardized height distributionwhose standard deviation is equal to one. Below are presented the formulas in the standardized form.
where:
- d is the separation,
- is the nominal contact area,
- is the surface density of asperities,
- is the effective Young modulus.
andcan be determined when theterms are calculated for the given surfaces using the convolution of the surface roughness.[34]Several studies have followed the suggested curve fits forassuming a Gaussian surface high distribution with curve fits presented by Arcoumanis et al.[35]and Jedynak[36]among others. It has been repeatedly observed that engineering surfaces do not demonstrate Gaussian surface height distributions e.g. Peklenik.[37]Leighton et al.[38]presented fits for crosshatched IC engine cylinder liner surfaces together with a process for determining theterms for any measured surfaces. Leighton et al.[38]demonstrated that Gaussian fit data is not accurate for modelling any engineered surfaces and went on to demonstrate[39]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 toandwere published by Jedynak.[36]They are given by the following rational formulas, which are approximants to the integrals.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.[38]
Forthe coefficients are
The maximum relative error is.
Forthe coefficients are
The maximum relative error is.The paper[36]also contains the exact expressions for
where erfc(z) means the complementary error function andis 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,[31]the average contact pressure is sufficient to cause yield whenwhereis the uniaxialyield stressandis the indentation hardness.[1]Greenwood and Williamson[31]defined a dimensionless parametercalled theplasticity indexthat 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.[32]Yield occurs when the pressure is greater than the uniaxial yield stress. Since the yield stress is proportional to the indentation hardness,Mikic defined the plasticity index for elastic-plastic contact to be
In this definitionrepresents 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]
An overview of the GT model
[edit]The model proposed by Greenwood and Tripp (GT),[40]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
and load carried by asperities
where:
- ,roughness parameter,
- ,nominal contact area,
- ,Stribeck oil film parameter, first defined by Stribeck \cite{gt} as,
- ,effective elastic modulus,
- ,statistical functions introduced to match the assumed Gaussian distribution of asperities.
Leighton et al.[38]presented fits for crosshatched IC engine cylinder liner surfaces together with a process for determining theterms for any measured surfaces. Leighton et al.[38]demonstrated that Gaussian fit data is not accurate for modelling any engineered surfaces and went on to demonstrate[39]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 forandare firstly presented by Jedynak.[36]They are expressed byas 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.[38]
where erfc(z) means the complementary error function andis the modified Bessel function of the second kind.
In paper[36]one can find comprehensive review of existing approximants to.New proposals give the most accurate approximants toand,which are reported in the literature. They are given by the following rational formulas, which are very exact approximants to integrals.They are calculated for the Gaussian distribution of asperities
Forthe coefficients are
The maximum relative error is.
Forthe coefficients are
The maximum relative error is.
Adhesive contact between elastic bodies
[edit]When two solid surfaces are brought into close proximity, they experience attractivevan der Waals forces.Bradley's van der Waals model[41]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[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 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
[edit]It is commonly assumed that the surface force between two atomic planes at a distancefrom each other can be derived from theLennard-Jones potential.With this assumption
whereis the force (positive in compression),is the total surface energy ofbothsurfaces per unit area, andis 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
whereare the radii of the two spheres.
The two spheres separate completely when thepull-off forceis achieved atat which point
Johnson-Kendall-Roberts (JKR) model of elastic contact
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/9/98/JKRModel.svg/400px-JKRModel.svg.png)
![](https://upload.wikimedia.org/wikipedia/commons/thumb/f/f4/JKR-full-cycle.gif/220px-JKR-full-cycle.gif)
To incorporate the effect of adhesion in Hertzian contact, Johnson, Kendall, andRoberts[5]formulated the JKR theory of adhesive contact using a balance between the storedelastic energyand the loss insurface energy.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
Note that in the original Hertz theory, the term containingwas neglected on the ground that tension could not be sustained in the contact zone. For contact between two spheres
whereis the radius of the area of contact,is the applied force,is the total surface energy ofbothsurfaces 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, has the 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 thepull-off force.Note that this force is independent of the moduli of the two spheres. However, there is another possible solution for the value ofat this load. This is the critical contact area,given by
If we define thework of adhesionas
whereare the adhesive energies of the two surfaces andis an interaction term, we can write the JKR contact 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
[edit]The Derjaguin-Muller-Toporov (DMT) model[7][42]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
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 the edge of the contact area.
In terms of the work of adhesion
and
Tabor parameter
[edit]In 1977, Tabor[43]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 theTabor parameter() defined as
whereis the equilibrium separation between the two surfaces in contact. The JKR theory applies to large, compliant spheres for whichis large. The DMT theory applies for small, stiff spheres with small values of.
Subsequently, Derjaguin and his collaborators[44]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 valueto the JKR value.More detailed calculations were later done by Greenwood[45]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[46]
Maugis-Dugdale model of elastic contact
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/8/8b/MaugisDugdale.svg/400px-MaugisDugdale.svg.png)
Further improvement to the Tabor idea was provided by Maugis[9]who represented the surface force in terms of a Dugdalecohesive zoneapproximation such that the work of adhesion is given by
whereis the maximum force predicted by the Lennard-Jones potential andis 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.There is not further penetration in compression. Perfect contact occurs in an area of radiusand adhesive forces of magnitudeextend to an area of radius.In the region,the two surfaces are separated by a distancewithand.The ratiois defined as
- .
In the Maugis-Dugdale theory,[47]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.The contribution to the surface traction from the Hertz pressure is given by
where the Hertz contact forceis given by
The penetration due to elastic compression is
The vertical displacement atis
and the separation between the two surfaces atis
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 atis
The net traction on the contact area is then given byand the net contact force is.Whenthe adhesive traction drops to zero.
Non-dimensionalized values ofare introduced at this stage that are defied as
In addition, Maugis proposed a parameterwhich is equivalent to the Tabor parameter.This parameter is defined as
where the step cohesive stressequals to the theoretical stress of the Lennard-Jones potential
Zheng and Yu[48]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 offor various values ofand.For large values of,and the JKR model is obtained. For small values ofthe DMT model is retrieved.
Carpick-Ogletree-Salmeron (COS) model
[edit]The Maugis-Dugdale model can only be solved iteratively if the value ofis not known a-priori. The Carpick-Ogletree-Salmeron approximate solution[49]simplifies the process by using the following relation to determine the contact radius:
whereis the contact area at zero load, andis a transition parameter that is related toby
The casecorresponds exactly to JKR theory whilecorresponds to DMT theory. For intermediate casesthe COS model corresponds closely to the Maugis-Dugdale solution for.
Influence of contact shape
[edit]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.[50]The process of detachment can as observed experimentally can be seen in the film.[51]
See also
[edit]- Adhesive– Non-metallic material used to bond various materials together
- Adhesive bonding– Joining technique used in manufacture and repair
- Adhesion railway– Railway relying on adhesion to move trains
- Adhesive surface forces– Molecular property
- Bearing capacity– Capacity of soil to support loads
- Collision– Instance of two or more bodies physically contacting each other within a short period of time
- Contact dynamics– Motion of multibody systems
- Contact resistance– Electrical resistance attributed to contacting interfaces (ECR)
- Dispersive adhesion– Adhesion between materials due to intermolecular interactions
- Electrostatic generator– Device that generates electrical charge on a high voltage electrode
- Energetically modified cement– Class of cements, mechanically processed to transform reactivity
- Frictional contact mechanics– Study of the deformation of bodies in the presence of frictional effects
- Friction drive– Mechanical power transmission by friction between components
- Galling– Form of wear caused by adhesion between sliding surfaces
- Goniometer– Angle measuring instrument
- Non-smooth mechanics– Modeling approach in mechanics
- Plastic wrap– Thin plastic film used for sealing food
- Rolling (metalworking)– Metal forming process
- Shock (mechanics)– Sudden transient acceleration
- Signorini problem– Elastostatics problem in linear elasticity
- Surface tension– Tendency of a liquid surface to shrink to reduce surface area
- Tribology– Science and engineering of interacting surfaces in relative motion
- Unilateral contact– Mechanical constraint which prevents penetration between two bodies
- Wetting– Ability of a liquid to maintain contact with a solid surface
References
[edit]- ^abcdefJohnson, K.L. (1985).Contact Mechanics.Cambridge University Press.ISBN978-0-521-25576-9.
- ^Popov, V.L. (2010).Contact Mechanics and Friction: Physical Principles and Applications.Springer Berlin Heidelberg. p. 362.ISBN978-3-642-10803-7.
- ^abH. Hertz, 1881, Über die berührung fester elastischer Körper,Journal für die reine und angewandte Mathematik92, 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.
- ^abcHertz, 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.
- ^abcJohnson, K. L.; Kendall, K.; Roberts, A. D. (1971-09-08)."Surface energy and the contact of elastic solids".Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.324(1558). The Royal Society: 301–313.Bibcode:1971RSPSA.324..301J.doi:10.1098/rspa.1971.0141.ISSN0080-4630.S2CID137730057.
- ^abMaugis, D. (2000-02-03).Contact, Adhesion and Rupture of Elastic Solids.Springer Science & Business Media.ISBN3-540-66113-1.
- ^abcDerjaguin, B.V; Muller, V.M; Toporov, Yu.P (1975). "Effect of contact deformations on the adhesion of particles".Journal of Colloid and Interface Science.53(2). Elsevier BV: 314–326.Bibcode:1975JCIS...53..314D.doi:10.1016/0021-9797(75)90018-1.ISSN0021-9797.
- ^Tabor, D (1970-01-01). "The hardness of solids".Reviews of Physics in Technology.1(3). IOP Publishing: 145–179.doi:10.1088/0034-6683/1/3/i01.ISSN0034-6683.
- ^abMaugis, Daniel (1992). "Adhesion of spheres: The JKR-DMT transition using a dugdale model".Journal of Colloid and Interface Science.150(1). Elsevier BV: 243–269.Bibcode:1992JCIS..150..243M.doi:10.1016/0021-9797(92)90285-t.ISSN0021-9797.
- ^"The area of contact between stationary and moving surfaces".Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.169(938). The Royal Society: 391–413. 1939-02-07.doi:10.1098/rspa.1939.0005.ISSN0080-4630.
- ^Bowden, F.P.; Tabor, D. (2001).The Friction and Lubrication of Solids.International series of monographs on physics. Vol. 1. Clarendon Press.ISBN978-0-19-850777-2.
- ^Archard, J. F. (1957-12-24). "Elastic deformation and the laws of friction".Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.243(1233). The Royal Society: 190–205.Bibcode:1957RSPSA.243..190A.doi:10.1098/rspa.1957.0214.ISSN0080-4630.S2CID138707812.
- ^Greenwood, J. A.; Williamson, J. B. P. (1966-12-06). "Contact of nominally flat surfaces".Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.295(1442). The Royal Society: 300–319.Bibcode:1966RSPSA.295..300G.doi:10.1098/rspa.1966.0242.ISSN0080-4630.S2CID137430238.
- ^Bush, A.W.; Gibson, R.D.; Thomas, T.R. (1975). "The elastic contact of a rough surface".Wear.35(1). Elsevier BV: 87–111.doi:10.1016/0043-1648(75)90145-3.ISSN0043-1648.
- ^Persson, B. N. J.; Bucher, F.; Chiaia, B. (2002-04-29). "Elastic contact between randomly rough surfaces: Comparison of theory with numerical results".Physical Review B.65(18). American Physical Society (APS): 184106.Bibcode:2002PhRvB..65r4106P.doi:10.1103/physrevb.65.184106.ISSN0163-1829.
- ^Popov, Valentin L.; Heß, Markus; Willert, Emanuel (2019).Handbook of Contact Mechanics: Exact Solutions of Axisymmetric Contact Problems.Berlin Heidelberg: Springer-Verlag.ISBN9783662587089.
- ^abSneddon, Ian N. (1965). "The relation between load and penetration in the axisymmetric boussinesq problem for a punch of arbitrary profile".International Journal of Engineering Science.3(1). Elsevier BV: 47–57.doi:10.1016/0020-7225(65)90019-4.ISSN0020-7225.
- ^Popov, Valentin L. (2010-03-10).Contact Mechanics and Friction.Springer Science & Business Media.ISBN978-3-642-10803-7.
- ^Popov, Valentin L. (2013)."Method of reduction of dimensionality in contact and friction mechanics: A linkage between micro and macro scales".Friction.1(1). Springer Science and Business Media LLC: 41–62.doi:10.1007/s40544-013-0005-3.ISSN2223-7690.S2CID256405038.
- ^Popov, Valentin L.; Heß, Markus (2013-10-01).Methode der Dimensionsreduktion in Kontaktmechanik und Reibung(in German). Springer-Verlag.ISBN978-3-642-32673-8.
- ^Popova, Elena; Popov, Valentin L. (2020)."Ludwig Föppl and Gerhard Schubert: Unknown classics of contact mechanics".ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik.100(9): e202000203.Bibcode:2020ZaMM..100E0203P.doi:10.1002/zamm.202000203.
- ^Shigley, Joseph Edward; Mischke, Charles R. (1989). "Ch. 2".Mechanical Engineering Design(5th ed.). McGraw-Hill Science, Engineering & Mathematics.ISBN0-07-056899-5.
- ^Kalker, J.J.(1990).Three-Dimensional Elastic Bodies in Rolling Contact.Solid Mechanics and Its Applications. Springer Netherlands.ISBN978-0-7923-0712-9.
- ^Wriggers, P. (2006).Computational Contact Mechanics(2nd ed.). Springer Berlin Heidelberg.ISBN978-3-540-32609-0.
- ^Laursen, T.A. (2003).Computational Contact and Impact Mechanics: Fundamentals of Modeling Interfacial Phenomena in Nonlinear Finite Element Analysis.Engineering online library. Springer Berlin Heidelberg.ISBN978-3-540-42906-7.
- ^ Acary, V.; Brogliato, B. (2008).Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics.Lecture Notes in Applied and Computational Mechanics. Springer Berlin Heidelberg.ISBN978-3-540-75392-6.
- ^Popov, Valentin L. (2009).Kontaktmechanik und Reibung(in German). Springer Science & Business Media.ISBN978-3-540-88836-9.
- ^Cottle, R.; Pang, J.; Stone, R. (2009-01-01).The Linear Complementarity Problem.Classics in Applied Mathematics. Society for Industrial and Applied Mathematics.doi:10.1137/1.9780898719000.ISBN9780898716863.
- ^abHanaor, D.; Gan, Y.; Einav, I. (2016)."Static friction at fractal interfaces".Tribology International.93:229–238.arXiv:2106.01473.doi:10.1016/j.triboint.2015.09.016.S2CID51900923.
- ^Zhai, Chongpu; Hanaor, Dorian; Gan, Yixiang (2017). "Contact stiffness of multiscale surfaces by truncation analysis".International Journal of Mechanical Sciences.131–132. Elsevier BV: 305–316.doi:10.1016/j.ijmecsci.2017.07.018.ISSN0020-7403.
- ^abcdeGreenwood, J. A.; Williamson, J. B. P. (1966-12-06). "Contact of nominally flat surfaces".Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.295(1442). The Royal Society: 300–319.Bibcode:1966RSPSA.295..300G.doi:10.1098/rspa.1966.0242.ISSN0080-4630.S2CID137430238.
- ^abMikić, B.B. (1974). "Thermal contact conductance; theoretical considerations".International Journal of Heat and Mass Transfer.17(2). Elsevier BV: 205–214.doi:10.1016/0017-9310(74)90082-9.ISSN0017-9310.
- ^Hyun, Sangil; Robbins, Mark O. (2007). "Elastic contact between rough surfaces: Effect of roughness at large and small wavelengths".Tribology International.40(10–12). Elsevier BV: 1413–1422.doi:10.1016/j.triboint.2007.02.003.ISSN0301-679X.
- ^Leighton; et al. (2016)."Surface-specific flow factors for prediction of friction of cross-hatched surfaces"(PDF).Surface Topography: Metrology and Properties.4(2): 025002.Bibcode:2016SuTMP...4b5002L.doi:10.1088/2051-672x/4/2/025002.S2CID111631084.
- ^Arcoumanis, C.; Ostovar, P.; Mortier, R. (1997-10-01)."Mixed Lubrication Modelling of Newtonian and Shear Thinning Liquids in a Piston-Ring Configuration".SAE Technical Paper Series.Vol. 1. Warrendale, PA.doi:10.4271/972924.
- ^abcdeJedynak, Radosław (2019). "Exact and approximate solutions of the infinite integrals of the asperity height distribution for the Greenwood-Williamson and the Greenwood-Tripp asperity contact models".Tribology International.130.Elsevier BV: 206–215.doi:10.1016/j.triboint.2018.09.009.ISSN0301-679X.S2CID139894096.
- ^Peklenik, J. (September 1967)."Paper 24: New Developments in Surface Characterization and Measurements by Means of Random Process Analysis".Proceedings of the Institution of Mechanical Engineers, Conference Proceedings.182(11): 108–126.doi:10.1243/pime_conf_1967_182_309_02.ISSN0367-8849.
- ^abcdefLeighton, M.; Morris, N.; Rahmani, R.; Rahnejat, H. (2017-01-01)."Surface specific asperity model for prediction of friction in boundary and mixed regimes of lubrication"(PDF).Meccanica.52(1): 21–33.doi:10.1007/s11012-016-0397-z.ISSN1572-9648.S2CID54710212.
- ^abLeighton, M; Morris, N; Gore, M; Rahmani, R; Rahnejat, H; King, PD (2016-08-05)."Boundary interactions of rough non-Gaussian surfaces"(PDF).Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology.230(11): 1359–1370.doi:10.1177/1350650116656967.ISSN1350-6501.S2CID53347629.
- ^Greenwood, J. A.; Tripp, J. H. (1970). "The Contact of Two Nominally Flat Rough Surfaces".Proceedings of the Institution of Mechanical Engineers.185(1). SAGE Publications: 625–633.doi:10.1243/pime_proc_1970_185_069_02.ISSN0020-3483.
- ^Bradley, R.S. (1932). "LXXIX.The cohesive force between solid surfaces and the surface energy of solids".The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science.13(86). Informa UK Limited: 853–862.doi:10.1080/14786449209461990.ISSN1941-5982.
- ^Muller, V.M.; Derjaguin, B.V.; Toporov, Yu.P. (1983). "On two methods of calculation of the force of sticking of an elastic sphere to a rigid plane".Colloids and Surfaces.7(3). Elsevier BV: 251–259.doi:10.1016/0166-6622(83)80051-1.ISSN0166-6622.
- ^TABOR, D. (1977). "Surface Forces and Surface Interactions".Plenary and Invited Lectures.Elsevier. p. 3–14.doi:10.1016/b978-0-12-404501-9.50009-2.ISBN9780124045019.
- ^Muller, V.M; Yushchenko, V.S; Derjaguin, B.V (1980). "On the influence of molecular forces on the deformation of an elastic sphere and its sticking to a rigid plane".Journal of Colloid and Interface Science.77(1). Elsevier BV: 91–101.Bibcode:1980JCIS...77...91M.doi:10.1016/0021-9797(80)90419-1.ISSN0021-9797.
- ^Greenwood, J. A. (1997-06-08). "Adhesion of elastic spheres".Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.453(1961). The Royal Society: 1277–1297.Bibcode:1997RSPSA.453.1277G.doi:10.1098/rspa.1997.0070.ISSN1364-5021.S2CID124217562.
- ^Feng, James Q. (2000). "Contact behavior of spherical elastic particles: a computational study of particle adhesion and deformations".Colloids and Surfaces A: Physicochemical and Engineering Aspects.172(1–3). Elsevier BV: 175–198.doi:10.1016/s0927-7757(00)00580-x.ISSN0927-7757.
- ^Johnson, K.L.; Greenwood, J.A. (1997). "An Adhesion Map for the Contact of Elastic Spheres".Journal of Colloid and Interface Science.192(2). Elsevier BV: 326–333.Bibcode:1997JCIS..192..326J.doi:10.1006/jcis.1997.4984.ISSN0021-9797.PMID9367554.
- ^Zheng, Zhijun; Yu, Jilin (2007). "Using the Dugdale approximation to match a specific interaction in the adhesive contact of elastic objects".Journal of Colloid and Interface Science.310(1). Elsevier BV: 27–34.Bibcode:2007JCIS..310...27Z.doi:10.1016/j.jcis.2007.01.042.ISSN0021-9797.PMID17335843.
- ^Carpick, Robert W.; Ogletree, D.Frank; Salmeron, Miquel (1999). "A General Equation for Fitting Contact Area and Friction vs Load Measurements".Journal of Colloid and Interface Science.211(2). Elsevier BV: 395–400.Bibcode:1999JCIS..211..395C.doi:10.1006/jcis.1998.6027.ISSN0021-9797.PMID10049556.
- ^Popov, Valentin L.; Pohrt, Roman; Li, Qiang (2017-09-01)."Strength of adhesive contacts: Influence of contact geometry and material gradients".Friction.5(3): 308–325.doi:10.1007/s40544-017-0177-3.ISSN2223-7690.
- ^Friction Physics (2017-12-06).Science friction: Adhesion of complex shapes.Archivedfrom the original on 2021-12-15.Retrieved2018-01-02– via YouTube.
External links
[edit]- [1]: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.
- [2]:Contact mechanics calculator.
- [3]:detailed calculations and formulae of JKR theory for two spheres.
- [5]:A Matlab code for Hertz contact analysis (includes line, point and elliptical cases).
- [6]:JKR, MD, and DMT models of adhesion (Matlab routines).