{"history":[],"article":{"id":"860cb0f6-a62d-4853-8e5d-bd8526330cbd","title":"Contact mechanics","slug":"contact-mechanics","content":":::summary\n:::\n\n# \n**Contact mechanics** is the study of the [deformation](https://bliptext.com/articles/deformation-mechanics) of [solids](https://bliptext.com/articles/solids) that touch each other at one or more points.\nA central distinction in contact mechanics is between [stresses](https://bliptext.com/articles/stress-mechanics) acting [perpendicular](https://bliptext.com/articles/perpendicular) to the contacting bodies' surfaces (known as [normal stress](https://bliptext.com/articles/normal-stress)) and [frictional](https://bliptext.com/articles/friction) [stresses](https://bliptext.com/articles/stress-mechanics) acting [tangentially](https://bliptext.com/articles/tangential-and-normal-components) between the surfaces ([shear stress](https://bliptext.com/articles/shear-stress)).\n**Normal contact mechanics** or **frictionless contact mechanics** focuses on normal stresses caused by applied [normal forces](https://bliptext.com/articles/normal-force) and by the [adhesion](https://bliptext.com/articles/adhesion) present on surfaces in close contact, even if they are clean and dry.\n[Frictional contact mechanics](https://bliptext.com/articles/frictional-contact-mechanics) emphasizes the effect of friction forces.\nContact mechanics is part of mechanical [engineering](https://bliptext.com/articles/engineering).\nThe physical and mathematical formulation of the subject is built upon the [mechanics of materials](https://bliptext.com/articles/mechanics-of-materials) and [continuum mechanics](https://bliptext.com/articles/continuum-mechanics) and focuses on computations involving [elastic](https://bliptext.com/articles/elasticity-physics), visco[elastic](https://bliptext.com/articles/elasticity-physics), and [plastic](https://bliptext.com/articles/plastic-deformation) bodies in [static](https://bliptext.com/articles/statics) or [dynamic](https://bliptext.com/articles/dynamics-physics) contact.\nContact mechanics provides necessary information for the safe and energy efficient design of technical systems and for the study of [tribology](https://bliptext.com/articles/tribology), [contact stiffness](https://bliptext.com/articles/normal-contact-stiffness), [electrical contact resistance](https://bliptext.com/articles/contact-resistance) and [indentation hardness](https://bliptext.com/articles/indentation-hardness).\nPrinciples of contacts mechanics are implemented towards applications such as locomotive wheel-rail contact, [coupling](https://bliptext.com/articles/coupling) devices, [braking](https://bliptext.com/articles/brake) systems, [tires](https://bliptext.com/articles/tire), [bearings](https://bliptext.com/articles/bearing-mechanical), [combustion engines](https://bliptext.com/articles/internal-combustion-engine), [mechanical linkages](https://bliptext.com/articles/linkage-mechanical), [gasket](https://bliptext.com/articles/gasket) seals, [metalworking](https://bliptext.com/articles/metalworking), metal forming, [ultrasonic welding](https://bliptext.com/articles/ultrasonic-welding), [electrical contacts](https://bliptext.com/articles/electrical-connector), and many others.\nCurrent challenges faced in the field may include [stress analysis](https://bliptext.com/articles/stress-analysis) of contact and coupling members and the influence of [lubrication](https://bliptext.com/articles/lubrication) and material [design](https://bliptext.com/articles/design) on [friction](https://bliptext.com/articles/friction) and [wear](https://bliptext.com/articles/wear).\nApplications of contact mechanics further extend into the [micro](https://bliptext.com/articles/microtechnology)- and [nanotechnological](https://bliptext.com/articles/nanotechnology) realm.\nThe original work in contact mechanics dates back to 1881 with the publication of the paper \"On the contact of elastic solids\" \"Über die Berührung fester elastischer Körper\" by [Heinrich Hertz](https://bliptext.com/articles/heinrich-hertz).\nHertz attempted to understand how the optical properties of multiple, stacked [lenses](https://bliptext.com/articles/lens-optics) might change with the [force](https://bliptext.com/articles/force) holding them together.\nHertzian contact stress refers to the localized stresses that develop as two curved surfaces come in contact and deform slightly under the imposed loads.\nThis amount of deformation is dependent on the [modulus of elasticity](https://bliptext.com/articles/modulus-of-elasticity) of the material in contact.\nIt 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.\nHertzian contact stress forms the foundation for the equations for load bearing capabilities and [fatigue](https://bliptext.com/articles/fatigue-material) life in bearings, gears, and any other bodies where two surfaces are in contact.\n# [Stresses in a contact area loaded simultaneously with a normal and a tangential force. Stresses were made visible using photoelasticity. | upload.wikimedia.org/wikipedia/commons/1/18/Kontakt///Spannungsoptik.JPG]\n\n# History\nClassical contact mechanics is most notably associated with Heinrich Hertz.\nIn 1882, Hertz solved the contact problem of two elastic bodies with curved surfaces.\nThis still-relevant classical solution provides a foundation for modern problems in contact mechanics.\nFor example, in [mechanical engineering](https://bliptext.com/articles/mechanical-engineering) and [tribology](https://bliptext.com/articles/tribology), Hertzian contact stress is a description of the stress within mating parts.\nThe Hertzian contact stress usually refers to the stress close to the area of contact between two spheres of different radii.\nIt was not until nearly one hundred years later that [Kenneth L. Johnson](https://bliptext.com/articles/kenneth-l-johnson), [Kevin Kendall](https://bliptext.com/articles/kevin-kendall), and [Alan D. Roberts](https://bliptext.com/articles/alan-d-roberts) found a similar solution for the case of [adhesive](https://bliptext.com/articles/adhesion) contact.\nThis theory was rejected by [Boris Derjaguin](https://bliptext.com/articles/boris-derjaguin) and co-workers who proposed a different theory of adhesion in the 1970s.\nThe 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.\nThis rejection proved to be instrumental in the development of the [David Tabor](https://bliptext.com/articles/david-tabor-physicist) and later Daniel Maugis parameters that quantify which contact model (of the JKR and DMT models) represent adhesive contact better for specific materials.\nFurther advancement in the field of contact mechanics in the mid-twentieth century may be attributed to names such as [Frank Philip Bowden](https://bliptext.com/articles/frank-philip-bowden) and Tabor.\nBowden and Tabor were the first to emphasize the importance of surface roughness for bodies in contact.\nThrough investigation of the surface roughness, the true contact area between friction partners is found to be less than the apparent contact area.\nSuch understanding also drastically changed the direction of undertakings in tribology.\nThe works of Bowden and Tabor yielded several theories in contact mechanics of rough surfaces.\nThe contributions of J. F. Archard (1957) must also be mentioned in discussion of pioneering works in this field.\nArchard concluded that, even for rough elastic surfaces, the contact area is approximately proportional to the [normal force](https://bliptext.com/articles/normal-force).\nFurther important insights along these lines were provided by Jonh A. Greenwood and J. B. P. Williamson (1966), A. W. Bush (1975), and Bo N. J. Persson (2002).\nThe 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.\n\n# Classical solutions for non-adhesive elastic contact\nThe theory of contact between elastic bodies can be used to find contact areas and indentation depths for simple geometries.\nSome commonly used solutions are listed below.\nThe theory used to compute these solutions is discussed later in the article.\nSolutions for multitude of other technically relevant shapes, e.g. the truncated cone, the worn sphere, rough profiles, hollow cylinders, etc. can be found in\n\n## Contact between a sphere and a half-space\nAn elastic [sphere](https://bliptext.com/articles/sphere) of [radius](https://bliptext.com/articles/radius) R indents an elastic [half-space](https://bliptext.com/articles/half-space-geometry) where total deformation is d, causing a contact area of [radius](https://bliptext.com/articles/radius)\nThe applied force F is related to the displacement d by\nwhere\nand E_1,E_2 are the [elastic moduli](https://bliptext.com/articles/elastic-modulus) and \\nu_1,\\nu_2 the [Poisson's ratios](https://bliptext.com/articles/poisson-s-ratio) associated with each body.\nThe distribution of normal pressure in the contact area as a function of distance from the center of the circle is\nwhere p_0 is the maximum contact pressure given by\nThe radius of the circle is related to the applied load F by the equation\nThe total deformation d is related to the maximum contact pressure by\nThe maximum shear stress occurs in the interior at for \\nu = 0.33.\n# [Contact of an elastic sphere with an elastic half-space | upload.wikimedia.org/wikipedia/commons/8/82/Contact///sphere-plane.jpg]\n\n## Contact between two spheres\nFor 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\n# [Contact between two spheres | upload.wikimedia.org/wikipedia/commons/4/43/Kontakt///Kugel///Kugel.jpg]\n# [Contact between two crossed cylinders of equal radius | upload.wikimedia.org/wikipedia/commons/4/49/Kontakt///gekreuzter///Zylinder.jpg]\n\n## Contact between two crossed cylinders of equal radius\nThis is equivalent to contact between a sphere of radius R and a [plane](https://bliptext.com/articles/plane-geometry).\n\n## Contact between a rigid cylinder with flat end and an elastic half-space\nIf a rigid [cylinder](https://bliptext.com/articles/cylinder-geometry) is pressed into an elastic half-space, it creates a pressure distribution described by\nwhere R is the radius of the cylinder and\nThe relationship between the indentation depth and the normal force is given by\n# [Contact between a rigid cylindrical indenter and an elastic half-space | upload.wikimedia.org/wikipedia/commons/1/1b/Kontakt///Zylindrischer///Indenter///Ebene.jpg]\n\n## Contact between a rigid conical indenter and an elastic half-space\nIn the case of [indentation](https://bliptext.com/articles/indentation-hardness) of an elastic half-space of Young's modulus E using a rigid [conical](https://bliptext.com/articles/cone-geometry) indenter, the depth of the contact region \\epsilon and contact radius a are related by\nwith \\theta defined as the angle between the plane and the side surface of the cone.\nThe total indentation depth d is given by:\nThe total force is\nThe pressure distribution is given by\nThe stress has a [logarithmic](https://bliptext.com/articles/logarithm) [singularity](https://bliptext.com/articles/mathematical-singularity) at the tip of the cone.\n# [Contact between a rigid conical indenter and an elastic half-space | upload.wikimedia.org/wikipedia/commons/c/c3/Kontakt///Kegel///Ebene.jpg]\n\n## Contact between two cylinders with parallel axes\nIn contact between two cylinders with parallel axes, the force is linearly proportional to the length of cylinders L and to the indentation depth d:\nThe radii of curvature are entirely absent from this relationship.\nThe contact radius is described through the usual relationship\nwith\nas in contact between two spheres.\nThe maximum pressure is equal to\n# [Contact between two cylinders with parallel axes | upload.wikimedia.org/wikipedia/commons/e/ef/Kontakt///paralleler///Zylinder.jpg]\n\n## Bearing contact\nThe contact in the case of [bearings](https://bliptext.com/articles/bearing-mechanical) is often a contact between a convex surface (male cylinder or sphere) and a concave surface (female cylinder or sphere: [bore](https://bliptext.com/articles/boring-manufacturing) or [hemispherical cup](https://bliptext.com/articles/spherical-bearing)).\n\n## Method of dimensionality reduction\nSome contact problems can be solved with the method of dimensionality reduction (MDR).\nIn this method, the initial three-dimensional system is replaced with a contact of a body with a linear elastic or viscoelastic foundation (see fig.).\nThe 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.\nMDR is based on the solution to axisymmetric contact problems first obtained by [Ludwig Föppl](https://bliptext.com/articles/ludwig-f-ppl) (1941) and Gerhard Schubert (1942)\nHowever, for exact analytical results, it is required that the contact problem is axisymmetric and the contacts are compact.\n# [Contact between a sphere and an elastic half-space and one-dimensional replaced model | upload.wikimedia.org/wikipedia/commons/4/40/WikipediabilderKap///4.jpg]\n\n# Hertzian theory of non-adhesive elastic contact\nThe 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.\nSeveral analytical and numerical approaches have been used to solve contact problems that satisfy the no-adhesion condition.\nComplex forces and [moments](https://bliptext.com/articles/moment-physics) are transmitted between the bodies where they touch, so problems in contact mechanics can become quite sophisticated.\nIn addition, the contact stresses are usually a nonlinear function of the deformation.\nTo simplify the solution procedure, a [frame of reference](https://bliptext.com/articles/frame-of-reference) is usually defined in which the objects (possibly in motion relative to one another) are static.\nThey interact through surface tractions (or pressures/stresses) at their interface.\nAs 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.\nOne of the bodies will experience a normally-directed [pressure](https://bliptext.com/articles/pressure) distribution and in-plane [surface traction](https://bliptext.com/articles/surface-traction) distributions and over the region S. In terms of a [Newtonian](https://bliptext.com/articles/newtonian-mechanics) force balance, the forces:\nmust be equal and opposite to the forces established in the other body.\nThe moments corresponding to these forces:\nare also required to cancel between bodies so that they are [kinematically](https://bliptext.com/articles/kinematics) immobile.\n\n## Assumptions in Hertzian theory\nThe following assumptions are made in determining the solutions of **Hertzian** contact problems:\nAdditional complications arise when some or all these assumptions are violated and such contact problems are usually called **non-Hertzian**.\n\n## Analytical solution techniques\nAnalytical solution methods for non-adhesive contact problem can be classified into two types based on the geometry of the area of contact.\nA **conforming contact** is one in which the two bodies touch at multiple points before any deformation takes place (i.e., they just \"fit together\").\nA **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).\nIn the non-conforming case, the contact area is small compared to the sizes of the objects and the [stresses](https://bliptext.com/articles/stress-physics) are highly concentrated in this area.\nSuch a contact is called concentrated, otherwise it is called diversified.\nA common approach in [linear elasticity](https://bliptext.com/articles/linear-elasticity) is to [superpose](https://bliptext.com/articles/superposition-principle) a number of solutions each of which corresponds to a point load acting over the area of contact.\nFor example, in the case of loading of a [half-plane](https://bliptext.com/articles/half-plane), the [Flamant solution](https://bliptext.com/articles/flamant-solution) is often used as a starting point and then generalized to various shapes of the area of contact.\nThe force and moment balances between the two bodies in contact act as additional constraints to the solution.\n# [Contact between two spheres | upload.wikimedia.org/wikipedia/commons/7/7a/Hertz.svg]\n\n### Point contact on a (2D) half-plane\nA 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.\nThe problem may be either [plane stress](https://bliptext.com/articles/plane-stress) or [plane strain](https://bliptext.com/articles/plane-strain).\nThis is a [boundary value problem](https://bliptext.com/articles/boundary-value-problem) of linear elasticity subject to the traction [boundary conditions](https://bliptext.com/articles/boundary-condition):\nwhere is the [Dirac delta function](https://bliptext.com/articles/dirac-delta-function).\nThe boundary conditions state that there are no shear stresses on the surface and a singular normal force P is applied at (0, 0).\nApplying these conditions to the governing equations of elasticity produces the result\nfor some point, (x, y), in the half-plane.\nThe circle shown in the figure indicates a surface on which the maximum shear stress is constant.\nFrom this stress field, the [strain](https://bliptext.com/articles/strain-materials-science) components and thus the [displacements](https://bliptext.com/articles/displacement-field-mechanics) of all material points may be determined.\n# [Schematic of the loading on a plane by force P at a point (0, 0) | upload.wikimedia.org/wikipedia/commons/5/5a/Point///contact///fig.svg]\n\n### Line contact on a (2D) half-plane\n\n#### Normal loading over a region\nSuppose, rather than a point load P, a distributed load p(x) is applied to the surface instead, over the range a 0.\nThis type of complementarity formulation can be expressed in the so-called [Kuhn–Tucker](https://bliptext.com/articles/karush-kuhn-tucker-conditions) form, viz.\nThese conditions are valid in a general way.\nThe 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](https://bliptext.com/articles/beam-theory) or [shell](https://bliptext.com/articles/plate-theory) model).\nBy restating the normal stress \\sigma_n in terms of the contact pressure, p; 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 as 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.\nIf 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., where for line loading of an elastic half space and for point loading of an elastic half-space.\nAfter discretization the linear elastic contact mechanics problem can be stated in standard Linear Complementarity Problem (LCP) form.\nwhere \\mathbf{C} is a matrix, whose elements are so called influence coefficients relating the contact pressure and the deformation.\nThe 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](https://bliptext.com/articles/lemke-s-algorithm).\nThe Lemke algorithm has the advantage that it finds the numerically exact solution within a finite number of iterations.\nThe MATLAB implementation presented by [Almqvist et al.](https://bliptext.com/articles/undefined) is one example that can be employed to solve the problem numerically.\nIn 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 by [Almqvist et al.](https://bliptext.com/articles/undefined)\n\n# Contact between rough surfaces\nWhen 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.\nThe mechanics of contacting rough surfaces are discussed in terms of normal contact mechanics and static frictional interactions.\nNatural 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](https://bliptext.com/articles/fractal).\nIt is recognized that the self affine structure of surfaces is the origin of the linear scaling of true contact area with applied pressure.\nAssuming a model of shearing welded contacts in [tribological](https://bliptext.com/articles/tribology) 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.\nIn contact between a \"random rough\" surface and an elastic half-space, the true contact area is related to the normal force F by with h' equal to the root mean square (also known as the quadratic mean) of the surface slope and.\nThe median pressure in the true contact surface\ncan be reasonably estimated as half of the effective elastic modulus E^* multiplied with the root mean square of the surface slope h'.\n\n## An overview of the GW model\nGreenwood 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.).\nThey 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.\nThe heights of asperities have a random distribution.\nThe probability that asperity height is between z and z + dz is \\phi(z)dz.\nThe authors calculated the number of contact spots n, the total contact area A_r and the total load P in general case.\nThey gave those formulas in two forms: in the basic and using standardized variables.\nIf one assumes that N asperities covers a rough surface, then the expected number of contacts is\nThe expected total area of contact can be calculated from the formula\nand the expected total force is given by\nwhere:\nGreenwood and Williamson introduced standardized separation and standardized height distribution \\phi^*(s) whose standard deviation is equal to one.\nBelow are presented the formulas in the standardized form.\nwhere:\nA 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).\nSeveral 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.\nIt has been repeatedly observed that engineering surfaces do not demonstrate Gaussian surface height distributions e.g. Peklenik.\nLeighton 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.\nLeighton 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.\nRecently the exact approximants to A_r and P were published by Jedynak.\nThey are given by the following rational formulas, which are approximants to the integrals F_n(h).\nThey 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.\nFor F_1(h) the coefficients are\nThe maximum relative error is.\nFor the coefficients are\nThe maximum relative error is.\nThe paper also contains the exact expressions for F_n(h)\nwhere erfc(z) means the complementary error function and K_\\nu(z) is the modified Bessel function of the second kind.\nFor 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 where \\sigma_y is the uniaxial [yield stress](https://bliptext.com/articles/yield-stress) and \\sigma_0 is the indentation hardness.\nGreenwood and Williamson defined a dimensionless parameter \\Psi called the **plasticity index** that could be used to determine whether contact would be elastic or plastic.\nThe Greenwood-Williamson model requires knowledge of two statistically dependent quantities; the standard deviation of the surface roughness and the curvature of the asperity peaks.\nAn alternative definition of the plasticity index has been given by Mikic.\nYield occurs when the pressure is greater than the uniaxial yield stress.\nSince the yield stress is proportional to the indentation hardness \\sigma _0, Mikic defined the plasticity index for elastic-plastic contact to be\nIn 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.\nFor, the surface behaves elastically during contact.\nIn both the Greenwood-Williamson and Mikic models the load is assumed to be proportional to the deformed area.\nHence, whether the system behaves plastically or elastically is independent of the applied normal force.\n\n## An overview of the GT model\nThe model proposed by John A. Greenwood and John H. Tripp (GT), extended the GW model to contact between two rough surfaces.\nThe GT model is widely used in the field of elastohydrodynamic analysis.\nThe most frequently cited equations given by the GT model are for the asperity contact area\nand load carried by asperities\nwhere:\nMatthew 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.\nLeighton 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.\nThe exact solutions for A_a and P are firstly presented by Jedynak.\nThey are expressed by F_n as follows.\nThey 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.\nwhere erfc(z) means the complementary error function and K_\\nu(z) is the modified Bessel function of the second kind.\nIn paper one can find comprehensive review of existing approximants to.\nNew proposals give the most accurate approximants to and F_2, which are reported in the literature.\nThey are given by the following rational formulas, which are very exact approximants to integrals F_n(h).\nThey are calculated for the Gaussian distribution of asperities\nFor F_2(h) the coefficients are\nThe maximum relative error is.\nFor the coefficients are\nThe maximum relative error is.\n\n# Adhesive contact between elastic bodies\nWhen two solid surfaces are brought into close proximity, they experience attractive [van der Waals forces](https://bliptext.com/articles/van-der-waals-force).\nR. S. Bradley's van der Waals model provides a means of calculating the tensile force between two rigid spheres with perfectly smooth surfaces.\nThe Hertzian model of contact does not consider adhesion possible.\nHowever, in the late 1960s, several contradictions were observed when the Hertz theory was compared with experiments involving contact between rubber and glass spheres.\nIt was observed that, though Hertz theory applied at large loads, at low loads\nThis indicated that adhesive forces were at work.\nThe Johnson-Kendall-Roberts (JKR) model and the Derjaguin-Muller-Toporov (DMT) models were the first to incorporate adhesion into Hertzian contact.\n\n## Bradley model of rigid contact\nIt 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](https://bliptext.com/articles/lennard-jones-potential).\nWith this assumption\nwhere 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.\nThe Bradley model applied the Lennard-Jones potential to find the force of adhesion between two rigid spheres.\nThe total force between the spheres is found to be where R_1,R_2 are the radii of the two spheres.\nThe two spheres separate completely when the pull-off force is achieved at z = z_0 at which point\n\n## JKR model of elastic contact\nTo 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](https://bliptext.com/articles/elastic-energy) and the loss in [surface energy](https://bliptext.com/articles/surface-energy).\nThe JKR model considers the effect of contact pressure and adhesion only inside the area of contact.\nThe general solution for the pressure distribution in the contact area in the JKR model is\nNote 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.\nFor contact between two spheres\nwhere 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,\nare the radii, Young's moduli, and Poisson's ratios of the two spheres, and\nThe approach distance between the two spheres is given by\nThe Hertz equation for the area of contact between two spheres, modified to take into account the surface energy, has the form\nWhen the surface energy is zero, \\gamma = 0, the Hertz equation for contact between two spheres is recovered.\nWhen the applied load is zero, the contact radius is\nThe tensile load at which the spheres are separated (i.e., a = 0) is predicted to be\nThis force is also called the **pull-off force**.\nNote that this force is independent of the moduli of the two spheres.\nHowever, there is another possible solution for the value of a at this load.\nThis is the critical contact area a_\\text{c}, given by\nIf we define the **work of adhesion** as\nwhere are the adhesive energies of the two surfaces and \\gamma_{12} is an interaction term, we can write the JKR contact radius as\nThe tensile load at separation is\nand the critical contact radius is given by\nThe critical depth of penetration is\n# [Schematic of contact area for the JKR model | upload.wikimedia.org/wikipedia/commons/9/98/JKRModel.svg]\n\n## DMT model of elastic contact\nThe 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.\nThe 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.\nIn terms of the work of adhesion\nand\n\n## Tabor parameter\nIn 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\nwhere z_0 is the equilibrium separation between the two surfaces in contact.\nThe JKR theory applies to large, compliant spheres for which \\mu is large.\nThe DMT theory applies for small, stiff spheres with small values of \\mu.\nSubsequently, 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 to the JKR value.\nMore detailed calculations were later done by Greenwood revealing the S-shaped load/approach curve which explains the jumping-on effect.\nA more efficient method of doing the calculations and additional results were given by Feng\n\n## Maugis–Dugdale model of elastic contact\nFurther improvement to the Tabor idea was provided by Maugis who represented the surface force in terms of a Dugdale [cohesive zone](https://bliptext.com/articles/cohesive-zone) approximation such that the work of adhesion is given by\nwhere \\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).\nThis means that the attractive force is constant for.\nThere is not further penetration in compression.\nPerfect 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.\nThe ratio m is defined as\nIn 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.\nHertz contact is assumed in the region -a < r < a. The contribution to the surface traction from the Hertz pressure is given by\nwhere the Hertz contact force F^H is given by\nThe penetration due to elastic compression is\nThe vertical displacement at r = c is\nand the separation between the two surfaces at r = c is\nThe surface traction distribution due to the adhesive Dugdale stress is\nThe total adhesive force is then given by\nThe compression due to Dugdale adhesion is\nand the gap at r = c is\nThe net traction on the contact area is then given by and the net contact force is.\nWhen the adhesive traction drops to zero.\nNon-dimensionalized values of a, c, F, d are introduced at this stage that are defied as\nIn addition, Maugis proposed a parameter \\lambda which is equivalent to the Tabor parameter \\mu.\nThis parameter is defined as\nwhere the step cohesive stress \\sigma_0 equals to the theoretical stress of the Lennard-Jones potential\nZheng and Yu suggested another value for the step cohesive stress\nto match the Lennard-Jones potential, which leads to\nThen the net contact force may be expressed as\nand the elastic compression as\nThe equation for the cohesive gap between the two bodies takes the form\nThis equation can be solved to obtain values of c for various values of a and \\lambda.\nFor large values of \\lambda, and the JKR model is obtained.\nFor small values of \\lambda the DMT model is retrieved.\n# [Schematic of contact area for the Maugis-Dugdale model | upload.wikimedia.org/wikipedia/commons/8/8b/MaugisDugdale.svg]\n\n## Carpick–Ogletree-Salmeron (COS) model\nThe Maugis–Dugdale model can only be solved iteratively if the value of \\lambda is not known a-priori.\nThe Carpick–Ogletree–Salmeron (COS) approximate solution (after [Robert Carpick](https://bliptext.com/articles/robert-carpick), D. Frank Ogletree and Miquel Salmeron)simplifies the process by using the following relation to determine the contact radius a:\nwhere a_0 is the contact area at zero load, and \\beta is a transition parameter that is related to \\lambda by\nThe case \\beta = 1 corresponds exactly to JKR theory while \\beta = 0 corresponds to DMT theory.\nFor intermediate cases the COS model corresponds closely to the Maugis–Dugdale solution for.\n\n## Influence of contact shape\nEven in the presence of perfectly smooth surfaces, geometry can come into play in form of the macroscopic shape of the contacting region.\nWhen 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.\nThe main parameter determining the adhesive strength of flat contacts occurs to be the maximum linear size of the contact.\nThe process of detachment can as observed experimentally can be seen in the film.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T09:24:48.462Z","updated_at":"2025-01-26T09:24:48.462Z","search_vector":"'contact':1 'mechan':2","search_content":" \n \n\n# \n**Contact mechanics** is the study of the [deformation](https://bliptext.com/articles/deformation-mechanics) of [solids](https://bliptext.com/articles/solids) that touch each other at one or more points.\nA central distinction in contact mechanics is between [stresses](https://bliptext.com/articles/stress-mechanics) acting [perpendicular](https://bliptext.com/articles/perpendicular) to the contacting bodies' surfaces (known as [normal stress](https://bliptext.com/articles/normal-stress)) and [frictional](https://bliptext.com/articles/friction) [stresses](https://bliptext.com/articles/stress-mechanics) acting [tangentially](https://bliptext.com/articles/tangential-and-normal-components) between the surfaces ([shear stress](https://bliptext.com/articles/shear-stress)).\n**Normal contact mechanics** or **frictionless contact mechanics** focuses on normal stresses caused by applied [normal forces](https://bliptext.com/articles/normal-force) and by the [adhesion](https://bliptext.com/articles/adhesion) present on surfaces in close contact, even if they are clean and dry.\n[Frictional contact mechanics](https://bliptext.com/articles/frictional-contact-mechanics) emphasizes the effect of friction forces.\nContact mechanics is part of mechanical [engineering](https://bliptext.com/articles/engineering).\nThe physical and mathematical formulation of the subject is built upon the [mechanics of materials](https://bliptext.com/articles/mechanics-of-materials) and [continuum mechanics](https://bliptext.com/articles/continuum-mechanics) and focuses on computations involving [elastic](https://bliptext.com/articles/elasticity-physics), visco[elastic](https://bliptext.com/articles/elasticity-physics), and [plastic](https://bliptext.com/articles/plastic-deformation) bodies in [static](https://bliptext.com/articles/statics) or [dynamic](https://bliptext.com/articles/dynamics-physics) contact.\nContact mechanics provides necessary information for the safe and energy efficient design of technical systems and for the study of [tribology](https://bliptext.com/articles/tribology), [contact stiffness](https://bliptext.com/articles/normal-contact-stiffness), [electrical contact resistance](https://bliptext.com/articles/contact-resistance) and [indentation hardness](https://bliptext.com/articles/indentation-hardness).\nPrinciples of contacts mechanics are implemented towards applications such as locomotive wheel-rail contact, [coupling](https://bliptext.com/articles/coupling) devices, [braking](https://bliptext.com/articles/brake) systems, [tires](https://bliptext.com/articles/tire), [bearings](https://bliptext.com/articles/bearing-mechanical), [combustion engines](https://bliptext.com/articles/internal-combustion-engine), [mechanical linkages](https://bliptext.com/articles/linkage-mechanical), [gasket](https://bliptext.com/articles/gasket) seals, [metalworking](https://bliptext.com/articles/metalworking), metal forming, [ultrasonic welding](https://bliptext.com/articles/ultrasonic-welding), [electrical contacts](https://bliptext.com/articles/electrical-connector), and many others.\nCurrent challenges faced in the field may include [stress analysis](https://bliptext.com/articles/stress-analysis) of contact and coupling members and the influence of [lubrication](https://bliptext.com/articles/lubrication) and material [design](https://bliptext.com/articles/design) on [friction](https://bliptext.com/articles/friction) and [wear](https://bliptext.com/articles/wear).\nApplications of contact mechanics further extend into the [micro](https://bliptext.com/articles/microtechnology)- and [nanotechnological](https://bliptext.com/articles/nanotechnology) realm.\nThe original work in contact mechanics dates back to 1881 with the publication of the paper \"On the contact of elastic solids\" \"Über die Berührung fester elastischer Körper\" by [Heinrich Hertz](https://bliptext.com/articles/heinrich-hertz).\nHertz attempted to understand how the optical properties of multiple, stacked [lenses](https://bliptext.com/articles/lens-optics) might change with the [force](https://bliptext.com/articles/force) holding them together.\nHertzian contact stress refers to the localized stresses that develop as two curved surfaces come in contact and deform slightly under the imposed loads.\nThis amount of deformation is dependent on the [modulus of elasticity](https://bliptext.com/articles/modulus-of-elasticity) of the material in contact.\nIt 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.\nHertzian contact stress forms the foundation for the equations for load bearing capabilities and [fatigue](https://bliptext.com/articles/fatigue-material) life in bearings, gears, and any other bodies where two surfaces are in contact.\n# [Stresses in a contact area loaded simultaneously with a normal and a tangential force. Stresses were made visible using photoelasticity. | upload.wikimedia.org/wikipedia/commons/1/18/Kontakt///Spannungsoptik.JPG]\n\n# History\nClassical contact mechanics is most notably associated with Heinrich Hertz.\nIn 1882, Hertz solved the contact problem of two elastic bodies with curved surfaces.\nThis still-relevant classical solution provides a foundation for modern problems in contact mechanics.\nFor example, in [mechanical engineering](https://bliptext.com/articles/mechanical-engineering) and [tribology](https://bliptext.com/articles/tribology), Hertzian contact stress is a description of the stress within mating parts.\nThe Hertzian contact stress usually refers to the stress close to the area of contact between two spheres of different radii.\nIt was not until nearly one hundred years later that [Kenneth L. Johnson](https://bliptext.com/articles/kenneth-l-johnson), [Kevin Kendall](https://bliptext.com/articles/kevin-kendall), and [Alan D. Roberts](https://bliptext.com/articles/alan-d-roberts) found a similar solution for the case of [adhesive](https://bliptext.com/articles/adhesion) contact.\nThis theory was rejected by [Boris Derjaguin](https://bliptext.com/articles/boris-derjaguin) and co-workers who proposed a different theory of adhesion in the 1970s.\nThe 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.\nThis rejection proved to be instrumental in the development of the [David Tabor](https://bliptext.com/articles/david-tabor-physicist) and later Daniel Maugis parameters that quantify which contact model (of the JKR and DMT models) represent adhesive contact better for specific materials.\nFurther advancement in the field of contact mechanics in the mid-twentieth century may be attributed to names such as [Frank Philip Bowden](https://bliptext.com/articles/frank-philip-bowden) and Tabor.\nBowden and Tabor were the first to emphasize the importance of surface roughness for bodies in contact.\nThrough investigation of the surface roughness, the true contact area between friction partners is found to be less than the apparent contact area.\nSuch understanding also drastically changed the direction of undertakings in tribology.\nThe works of Bowden and Tabor yielded several theories in contact mechanics of rough surfaces.\nThe contributions of J. F. Archard (1957) must also be mentioned in discussion of pioneering works in this field.\nArchard concluded that, even for rough elastic surfaces, the contact area is approximately proportional to the [normal force](https://bliptext.com/articles/normal-force).\nFurther important insights along these lines were provided by Jonh A. Greenwood and J. B. P. Williamson (1966), A. W. Bush (1975), and Bo N. J. Persson (2002).\nThe 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.\n\n# Classical solutions for non-adhesive elastic contact\nThe theory of contact between elastic bodies can be used to find contact areas and indentation depths for simple geometries.\nSome commonly used solutions are listed below.\nThe theory used to compute these solutions is discussed later in the article.\nSolutions for multitude of other technically relevant shapes, e.g. the truncated cone, the worn sphere, rough profiles, hollow cylinders, etc. can be found in\n\n## Contact between a sphere and a half-space\nAn elastic [sphere](https://bliptext.com/articles/sphere) of [radius](https://bliptext.com/articles/radius) R indents an elastic [half-space](https://bliptext.com/articles/half-space-geometry) where total deformation is d, causing a contact area of [radius](https://bliptext.com/articles/radius)\nThe applied force F is related to the displacement d by\nwhere\nand E_1,E_2 are the [elastic moduli](https://bliptext.com/articles/elastic-modulus) and \\nu_1,\\nu_2 the [Poisson's ratios](https://bliptext.com/articles/poisson-s-ratio) associated with each body.\nThe distribution of normal pressure in the contact area as a function of distance from the center of the circle is\nwhere p_0 is the maximum contact pressure given by\nThe radius of the circle is related to the applied load F by the equation\nThe total deformation d is related to the maximum contact pressure by\nThe maximum shear stress occurs in the interior at for \\nu = 0.33.\n# [Contact of an elastic sphere with an elastic half-space | upload.wikimedia.org/wikipedia/commons/8/82/Contact///sphere-plane.jpg]\n\n## Contact between two spheres\nFor 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\n# [Contact between two spheres | upload.wikimedia.org/wikipedia/commons/4/43/Kontakt///Kugel///Kugel.jpg]\n# [Contact between two crossed cylinders of equal radius | upload.wikimedia.org/wikipedia/commons/4/49/Kontakt///gekreuzter///Zylinder.jpg]\n\n## Contact between two crossed cylinders of equal radius\nThis is equivalent to contact between a sphere of radius R and a [plane](https://bliptext.com/articles/plane-geometry).\n\n## Contact between a rigid cylinder with flat end and an elastic half-space\nIf a rigid [cylinder](https://bliptext.com/articles/cylinder-geometry) is pressed into an elastic half-space, it creates a pressure distribution described by\nwhere R is the radius of the cylinder and\nThe relationship between the indentation depth and the normal force is given by\n# [Contact between a rigid cylindrical indenter and an elastic half-space | upload.wikimedia.org/wikipedia/commons/1/1b/Kontakt///Zylindrischer///Indenter///Ebene.jpg]\n\n## Contact between a rigid conical indenter and an elastic half-space\nIn the case of [indentation](https://bliptext.com/articles/indentation-hardness) of an elastic half-space of Young's modulus E using a rigid [conical](https://bliptext.com/articles/cone-geometry) indenter, the depth of the contact region \\epsilon and contact radius a are related by\nwith \\theta defined as the angle between the plane and the side surface of the cone.\nThe total indentation depth d is given by:\nThe total force is\nThe pressure distribution is given by\nThe stress has a [logarithmic](https://bliptext.com/articles/logarithm) [singularity](https://bliptext.com/articles/mathematical-singularity) at the tip of the cone.\n# [Contact between a rigid conical indenter and an elastic half-space | upload.wikimedia.org/wikipedia/commons/c/c3/Kontakt///Kegel///Ebene.jpg]\n\n## Contact between two cylinders with parallel axes\nIn contact between two cylinders with parallel axes, the force is linearly proportional to the length of cylinders L and to the indentation depth d:\nThe radii of curvature are entirely absent from this relationship.\nThe contact radius is described through the usual relationship\nwith\nas in contact between two spheres.\nThe maximum pressure is equal to\n# [Contact between two cylinders with parallel axes | upload.wikimedia.org/wikipedia/commons/e/ef/Kontakt///paralleler///Zylinder.jpg]\n\n## Bearing contact\nThe contact in the case of [bearings](https://bliptext.com/articles/bearing-mechanical) is often a contact between a convex surface (male cylinder or sphere) and a concave surface (female cylinder or sphere: [bore](https://bliptext.com/articles/boring-manufacturing) or [hemispherical cup](https://bliptext.com/articles/spherical-bearing)).\n\n## Method of dimensionality reduction\nSome contact problems can be solved with the method of dimensionality reduction (MDR).\nIn this method, the initial three-dimensional system is replaced with a contact of a body with a linear elastic or viscoelastic foundation (see fig.).\nThe 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.\nMDR is based on the solution to axisymmetric contact problems first obtained by [Ludwig Föppl](https://bliptext.com/articles/ludwig-f-ppl) (1941) and Gerhard Schubert (1942)\nHowever, for exact analytical results, it is required that the contact problem is axisymmetric and the contacts are compact.\n# [Contact between a sphere and an elastic half-space and one-dimensional replaced model | upload.wikimedia.org/wikipedia/commons/4/40/WikipediabilderKap///4.jpg]\n\n# Hertzian theory of non-adhesive elastic contact\nThe 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.\nSeveral analytical and numerical approaches have been used to solve contact problems that satisfy the no-adhesion condition.\nComplex forces and [moments](https://bliptext.com/articles/moment-physics) are transmitted between the bodies where they touch, so problems in contact mechanics can become quite sophisticated.\nIn addition, the contact stresses are usually a nonlinear function of the deformation.\nTo simplify the solution procedure, a [frame of reference](https://bliptext.com/articles/frame-of-reference) is usually defined in which the objects (possibly in motion relative to one another) are static.\nThey interact through surface tractions (or pressures/stresses) at their interface.\nAs 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.\nOne of the bodies will experience a normally-directed [pressure](https://bliptext.com/articles/pressure) distribution and in-plane [surface traction](https://bliptext.com/articles/surface-traction) distributions and over the region S. In terms of a [Newtonian](https://bliptext.com/articles/newtonian-mechanics) force balance, the forces:\nmust be equal and opposite to the forces established in the other body.\nThe moments corresponding to these forces:\nare also required to cancel between bodies so that they are [kinematically](https://bliptext.com/articles/kinematics) immobile.\n\n## Assumptions in Hertzian theory\nThe following assumptions are made in determining the solutions of **Hertzian** contact problems:\nAdditional complications arise when some or all these assumptions are violated and such contact problems are usually called **non-Hertzian**.\n\n## Analytical solution techniques\nAnalytical solution methods for non-adhesive contact problem can be classified into two types based on the geometry of the area of contact.\nA **conforming contact** is one in which the two bodies touch at multiple points before any deformation takes place (i.e., they just \"fit together\").\nA **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).\nIn the non-conforming case, the contact area is small compared to the sizes of the objects and the [stresses](https://bliptext.com/articles/stress-physics) are highly concentrated in this area.\nSuch a contact is called concentrated, otherwise it is called diversified.\nA common approach in [linear elasticity](https://bliptext.com/articles/linear-elasticity) is to [superpose](https://bliptext.com/articles/superposition-principle) a number of solutions each of which corresponds to a point load acting over the area of contact.\nFor example, in the case of loading of a [half-plane](https://bliptext.com/articles/half-plane), the [Flamant solution](https://bliptext.com/articles/flamant-solution) is often used as a starting point and then generalized to various shapes of the area of contact.\nThe force and moment balances between the two bodies in contact act as additional constraints to the solution.\n# [Contact between two spheres | upload.wikimedia.org/wikipedia/commons/7/7a/Hertz.svg]\n\n### Point contact on a (2D) half-plane\nA 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.\nThe problem may be either [plane stress](https://bliptext.com/articles/plane-stress) or [plane strain](https://bliptext.com/articles/plane-strain).\nThis is a [boundary value problem](https://bliptext.com/articles/boundary-value-problem) of linear elasticity subject to the traction [boundary conditions](https://bliptext.com/articles/boundary-condition):\nwhere is the [Dirac delta function](https://bliptext.com/articles/dirac-delta-function).\nThe boundary conditions state that there are no shear stresses on the surface and a singular normal force P is applied at (0, 0).\nApplying these conditions to the governing equations of elasticity produces the result\nfor some point, (x, y), in the half-plane.\nThe circle shown in the figure indicates a surface on which the maximum shear stress is constant.\nFrom this stress field, the [strain](https://bliptext.com/articles/strain-materials-science) components and thus the [displacements](https://bliptext.com/articles/displacement-field-mechanics) of all material points may be determined.\n# [Schematic of the loading on a plane by force P at a point (0, 0) | upload.wikimedia.org/wikipedia/commons/5/5a/Point///contact///fig.svg]\n\n### Line contact on a (2D) half-plane\n\n#### Normal loading over a region\nSuppose, rather than a point load P, a distributed load p(x) is applied to the surface instead, over the range a 0.\nThis type of complementarity formulation can be expressed in the so-called [Kuhn–Tucker](https://bliptext.com/articles/karush-kuhn-tucker-conditions) form, viz.\nThese conditions are valid in a general way.\nThe 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](https://bliptext.com/articles/beam-theory) or [shell](https://bliptext.com/articles/plate-theory) model).\nBy restating the normal stress \\sigma_n in terms of the contact pressure, p; 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 as 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.\nIf 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., where for line loading of an elastic half space and for point loading of an elastic half-space.\nAfter discretization the linear elastic contact mechanics problem can be stated in standard Linear Complementarity Problem (LCP) form.\nwhere \\mathbf{C} is a matrix, whose elements are so called influence coefficients relating the contact pressure and the deformation.\nThe 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](https://bliptext.com/articles/lemke-s-algorithm).\nThe Lemke algorithm has the advantage that it finds the numerically exact solution within a finite number of iterations.\nThe MATLAB implementation presented by [Almqvist et al.](https://bliptext.com/articles/undefined) is one example that can be employed to solve the problem numerically.\nIn 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 by [Almqvist et al.](https://bliptext.com/articles/undefined)\n\n# Contact between rough surfaces\nWhen 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.\nThe mechanics of contacting rough surfaces are discussed in terms of normal contact mechanics and static frictional interactions.\nNatural 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](https://bliptext.com/articles/fractal).\nIt is recognized that the self affine structure of surfaces is the origin of the linear scaling of true contact area with applied pressure.\nAssuming a model of shearing welded contacts in [tribological](https://bliptext.com/articles/tribology) 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.\nIn contact between a \"random rough\" surface and an elastic half-space, the true contact area is related to the normal force F by with h' equal to the root mean square (also known as the quadratic mean) of the surface slope and.\nThe median pressure in the true contact surface\ncan be reasonably estimated as half of the effective elastic modulus E^* multiplied with the root mean square of the surface slope h'.\n\n## An overview of the GW model\nGreenwood 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.).\nThey 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.\nThe heights of asperities have a random distribution.\nThe probability that asperity height is between z and z + dz is \\phi(z)dz.\nThe authors calculated the number of contact spots n, the total contact area A_r and the total load P in general case.\nThey gave those formulas in two forms: in the basic and using standardized variables.\nIf one assumes that N asperities covers a rough surface, then the expected number of contacts is\nThe expected total area of contact can be calculated from the formula\nand the expected total force is given by\nwhere:\nGreenwood and Williamson introduced standardized separation and standardized height distribution \\phi^*(s) whose standard deviation is equal to one.\nBelow are presented the formulas in the standardized form.\nwhere:\nA 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).\nSeveral 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.\nIt has been repeatedly observed that engineering surfaces do not demonstrate Gaussian surface height distributions e.g. Peklenik.\nLeighton 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.\nLeighton 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.\nRecently the exact approximants to A_r and P were published by Jedynak.\nThey are given by the following rational formulas, which are approximants to the integrals F_n(h).\nThey 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.\nFor F_1(h) the coefficients are\nThe maximum relative error is.\nFor the coefficients are\nThe maximum relative error is.\nThe paper also contains the exact expressions for F_n(h)\nwhere erfc(z) means the complementary error function and K_\\nu(z) is the modified Bessel function of the second kind.\nFor 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 where \\sigma_y is the uniaxial [yield stress](https://bliptext.com/articles/yield-stress) and \\sigma_0 is the indentation hardness.\nGreenwood and Williamson defined a dimensionless parameter \\Psi called the **plasticity index** that could be used to determine whether contact would be elastic or plastic.\nThe Greenwood-Williamson model requires knowledge of two statistically dependent quantities; the standard deviation of the surface roughness and the curvature of the asperity peaks.\nAn alternative definition of the plasticity index has been given by Mikic.\nYield occurs when the pressure is greater than the uniaxial yield stress.\nSince the yield stress is proportional to the indentation hardness \\sigma _0, Mikic defined the plasticity index for elastic-plastic contact to be\nIn 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.\nFor, the surface behaves elastically during contact.\nIn both the Greenwood-Williamson and Mikic models the load is assumed to be proportional to the deformed area.\nHence, whether the system behaves plastically or elastically is independent of the applied normal force.\n\n## An overview of the GT model\nThe model proposed by John A. Greenwood and John H. Tripp (GT), extended the GW model to contact between two rough surfaces.\nThe GT model is widely used in the field of elastohydrodynamic analysis.\nThe most frequently cited equations given by the GT model are for the asperity contact area\nand load carried by asperities\nwhere:\nMatthew 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.\nLeighton 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.\nThe exact solutions for A_a and P are firstly presented by Jedynak.\nThey are expressed by F_n as follows.\nThey 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.\nwhere erfc(z) means the complementary error function and K_\\nu(z) is the modified Bessel function of the second kind.\nIn paper one can find comprehensive review of existing approximants to.\nNew proposals give the most accurate approximants to and F_2, which are reported in the literature.\nThey are given by the following rational formulas, which are very exact approximants to integrals F_n(h).\nThey are calculated for the Gaussian distribution of asperities\nFor F_2(h) the coefficients are\nThe maximum relative error is.\nFor the coefficients are\nThe maximum relative error is.\n\n# Adhesive contact between elastic bodies\nWhen two solid surfaces are brought into close proximity, they experience attractive [van der Waals forces](https://bliptext.com/articles/van-der-waals-force).\nR. S. Bradley's van der Waals model provides a means of calculating the tensile force between two rigid spheres with perfectly smooth surfaces.\nThe Hertzian model of contact does not consider adhesion possible.\nHowever, in the late 1960s, several contradictions were observed when the Hertz theory was compared with experiments involving contact between rubber and glass spheres.\nIt was observed that, though Hertz theory applied at large loads, at low loads\nThis indicated that adhesive forces were at work.\nThe Johnson-Kendall-Roberts (JKR) model and the Derjaguin-Muller-Toporov (DMT) models were the first to incorporate adhesion into Hertzian contact.\n\n## Bradley model of rigid contact\nIt 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](https://bliptext.com/articles/lennard-jones-potential).\nWith this assumption\nwhere 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.\nThe Bradley model applied the Lennard-Jones potential to find the force of adhesion between two rigid spheres.\nThe total force between the spheres is found to be where R_1,R_2 are the radii of the two spheres.\nThe two spheres separate completely when the pull-off force is achieved at z = z_0 at which point\n\n## JKR model of elastic contact\nTo 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](https://bliptext.com/articles/elastic-energy) and the loss in [surface energy](https://bliptext.com/articles/surface-energy).\nThe JKR model considers the effect of contact pressure and adhesion only inside the area of contact.\nThe general solution for the pressure distribution in the contact area in the JKR model is\nNote 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.\nFor contact between two spheres\nwhere 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,\nare the radii, Young's moduli, and Poisson's ratios of the two spheres, and\nThe approach distance between the two spheres is given by\nThe Hertz equation for the area of contact between two spheres, modified to take into account the surface energy, has the form\nWhen the surface energy is zero, \\gamma = 0, the Hertz equation for contact between two spheres is recovered.\nWhen the applied load is zero, the contact radius is\nThe tensile load at which the spheres are separated (i.e., a = 0) is predicted to be\nThis force is also called the **pull-off force**.\nNote that this force is independent of the moduli of the two spheres.\nHowever, there is another possible solution for the value of a at this load.\nThis is the critical contact area a_\\text{c}, given by\nIf we define the **work of adhesion** as\nwhere are the adhesive energies of the two surfaces and \\gamma_{12} is an interaction term, we can write the JKR contact radius as\nThe tensile load at separation is\nand the critical contact radius is given by\nThe critical depth of penetration is\n# [Schematic of contact area for the JKR model | upload.wikimedia.org/wikipedia/commons/9/98/JKRModel.svg]\n\n## DMT model of elastic contact\nThe 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.\nThe 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.\nIn terms of the work of adhesion\nand\n\n## Tabor parameter\nIn 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\nwhere z_0 is the equilibrium separation between the two surfaces in contact.\nThe JKR theory applies to large, compliant spheres for which \\mu is large.\nThe DMT theory applies for small, stiff spheres with small values of \\mu.\nSubsequently, 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 to the JKR value.\nMore detailed calculations were later done by Greenwood revealing the S-shaped load/approach curve which explains the jumping-on effect.\nA more efficient method of doing the calculations and additional results were given by Feng\n\n## Maugis–Dugdale model of elastic contact\nFurther improvement to the Tabor idea was provided by Maugis who represented the surface force in terms of a Dugdale [cohesive zone](https://bliptext.com/articles/cohesive-zone) approximation such that the work of adhesion is given by\nwhere \\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).\nThis means that the attractive force is constant for.\nThere is not further penetration in compression.\nPerfect 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.\nThe ratio m is defined as\nIn 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.\nHertz contact is assumed in the region -a < r < a. The contribution to the surface traction from the Hertz pressure is given by\nwhere the Hertz contact force F^H is given by\nThe penetration due to elastic compression is\nThe vertical displacement at r = c is\nand the separation between the two surfaces at r = c is\nThe surface traction distribution due to the adhesive Dugdale stress is\nThe total adhesive force is then given by\nThe compression due to Dugdale adhesion is\nand the gap at r = c is\nThe net traction on the contact area is then given by and the net contact force is.\nWhen the adhesive traction drops to zero.\nNon-dimensionalized values of a, c, F, d are introduced at this stage that are defied as\nIn addition, Maugis proposed a parameter \\lambda which is equivalent to the Tabor parameter \\mu.\nThis parameter is defined as\nwhere the step cohesive stress \\sigma_0 equals to the theoretical stress of the Lennard-Jones potential\nZheng and Yu suggested another value for the step cohesive stress\nto match the Lennard-Jones potential, which leads to\nThen the net contact force may be expressed as\nand the elastic compression as\nThe equation for the cohesive gap between the two bodies takes the form\nThis equation can be solved to obtain values of c for various values of a and \\lambda.\nFor large values of \\lambda, and the JKR model is obtained.\nFor small values of \\lambda the DMT model is retrieved.\n# [Schematic of contact area for the Maugis-Dugdale model | upload.wikimedia.org/wikipedia/commons/8/8b/MaugisDugdale.svg]\n\n## Carpick–Ogletree-Salmeron (COS) model\nThe Maugis–Dugdale model can only be solved iteratively if the value of \\lambda is not known a-priori.\nThe Carpick–Ogletree–Salmeron (COS) approximate solution (after [Robert Carpick](https://bliptext.com/articles/robert-carpick), D. Frank Ogletree and Miquel Salmeron)simplifies the process by using the following relation to determine the contact radius a:\nwhere a_0 is the contact area at zero load, and \\beta is a transition parameter that is related to \\lambda by\nThe case \\beta = 1 corresponds exactly to JKR theory while \\beta = 0 corresponds to DMT theory.\nFor intermediate cases the COS model corresponds closely to the Maugis–Dugdale solution for.\n\n## Influence of contact shape\nEven in the presence of perfectly smooth surfaces, geometry can come into play in form of the macroscopic shape of the contacting region.\nWhen 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.\nThe main parameter determining the adhesive strength of flat contacts occurs to be the maximum linear size of the contact.\nThe process of detachment can as observed experimentally can be seen in the film.","revisionCount":0},"hasMore":false,"nextCursor":null}