Contact mechanics

Contact of an elastic sphere with an elastic half-space

An elastic sphere of radius ${\displaystyle R}$ indents an elastic half-space where total deformation is ${\displaystyle d}$, causing a contact area of radius

${\displaystyle a={\sqrt {Rd}}}$

The applied force ${\displaystyle F}$ is related to the displacement ${\displaystyle d}$ by [4]

${\displaystyle F={\frac {4}{3}}E^{*}R^{\frac {1}{2}}d^{\frac {3}{2}}}$

where

${\displaystyle {\frac {1}{E^{*}}}={\frac {1-\nu _{1}^{2}}{E_{1}}}+{\frac {1-\nu _{2}^{2}}{E_{2}}}}$

and ${\displaystyle E_{1}}$,${\displaystyle E_{2}}$ are the elastic moduli and ${\displaystyle \nu _{1}}$,${\displaystyle \nu _{2}}$ the Poisson's ratios associated with each body.

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

${\displaystyle p(r)=p_{0}\left(1-{\frac {r^{2}}{a^{2}}}\right)^{\frac {1}{2}}}$

where ${\displaystyle p_{0}}$ is the maximum contact pressure given by

${\displaystyle p_{0}={\frac {3F}{2\pi a^{2}}}={\frac {1}{\pi }}\left({\frac {6F{E^{*}}^{2}}{R^{2}}}\right)^{\frac {1}{3}}}$

The radius of the circle is related to the applied load ${\displaystyle F}$ by the equation

${\displaystyle a^{3}={\cfrac {3FR}{4E^{*}}}}$

The total deformation ${\displaystyle d}$ is related to the maximum contact pressure by

${\displaystyle d={\frac {a^{2}}{R}}=\left({\frac {9F^{2}}{16{E^{*}}^{2}R}}\right)^{\frac {1}{3}}}$

The maximum shear stress occurs in the interior at ${\displaystyle z\approx 0.49a}$ for ${\displaystyle \nu =0.33}$.

Contact between two spheres.

Contact between two crossed cylinders of equal radius.

For contact between two spheres of radii ${\displaystyle R_{1}}$ and ${\displaystyle R_{2}}$, the area of contact is a circle of radius ${\displaystyle a}$. The equations are the same as for a sphere in contact with a half plane except that the effective radius ${\displaystyle R}$ is defined as [4]

${\displaystyle {\frac {1}{R}}={\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}}$

This is equivalent to contact between a sphere of radius ${\displaystyle R}$ and a plane.

Contact between a rigid cylindrical indenter and an elastic half-space.

If a rigid cylinder is pressed into an elastic half-space, it creates a pressure distribution described by[17]

${\displaystyle p(r)=p_{0}\left(1-{\frac {r^{2}}{R^{2}}}\right)^{-{\frac {1}{2}}}}$

where ${\displaystyle R}$ is the radius of the cylinder and

${\displaystyle p_{0}={\frac {1}{\pi }}E^{*}{\frac {d}{R}}}$

The relationship between the indentation depth and the normal force is given by

${\displaystyle F=2RE^{*}d}$

Contact between a rigid conical indenter and an elastic half-space.

In the case of indentation of an elastic half-space of Young's modulus ${\displaystyle E}$ using a rigid conical indenter, the depth of the contact region ${\displaystyle \epsilon }$ and contact radius ${\displaystyle a}$ are related by[17]

${\displaystyle \epsilon =a\tan(\theta )}$

with ${\displaystyle \theta }$ defined as the angle between the plane and the side surface of the cone. The total indentation depth ${\displaystyle d}$ is given by:

${\displaystyle d={\frac {\pi }{2}}\epsilon }$

The total force is

${\displaystyle F={\frac {\pi E}{2\left(1-\nu ^{2}\right)}}a^{2}\tan(\theta )={\frac {2E}{\pi \left(1-\nu ^{2}\right)}}{\frac {d^{2}}{\tan(\theta )}}}$

The pressure distribution is given by

${\displaystyle p\left(r\right)={\frac {Ed}{\pi a\left(1-\nu ^{2}\right)}}\ln \left({\frac {a}{r}}+{\sqrt {\left({\frac {a}{r}}\right)^{2}-1}}\right)={\frac {Ed}{\pi a\left(1-\nu ^{2}\right)}}\cosh ^{-1}\left({\frac {a}{r}}\right)}$

The stress has a logarithmic singularity at the tip of the cone.

Contact between two cylinders with parallel axes

In contact between two cylinders with parallel axes, the force is linearly proportional to the length of cylinders L and to the indentation depth d:[18]

${\displaystyle F\approx {\frac {\pi }{4}}E^{*}Ld}$

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

${\displaystyle a={\sqrt {Rd}}}$

with

${\displaystyle {\frac {1}{R}}={\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}}$

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

${\displaystyle p_{0}=\left({\frac {E^{*}F}{LR}}\right)^{\frac {1}{2}}}$

The contact in the case of bearings is often a contact between a convex surface (male cylinder or sphere) and a concave surface (female cylinder or sphere: bore or hemispherical cup).

Contact between a sphere and an elastic half-space and one-dimensional replaced model.

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.

The following assumptions are made in determining the solutions of Hertzian contact problems:

• The strains are small and within the elastic limit.
• The surfaces are continuous and non-conforming (implying that the area of contact is much smaller than the characteristic dimensions of the contacting bodies).
• Each body can be considered an elastic half-space.
• The surfaces are frictionless.

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

Contact between two spheres.

Analytical solution methods for non-adhesive contact problem can be classified into two types based on the geometry of the area of contact.[22] A conforming contact is one in which the two bodies touch at multiple points before any deformation takes place (i.e., they just "fit together"). A non-conforming contact is one in which the shapes of the bodies are dissimilar enough that, under zero load, they only touch at a point (or possibly along a line). In the non-conforming case, the contact area is small compared to the sizes of the objects and the stresses are highly concentrated in this area. Such a contact is called concentrated, otherwise it is called diversified.

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

Schematic of the loading on a plane by force P at a point (0, 0).

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

${\displaystyle \sigma _{xz}(x,0)=0~;~~\sigma _{z}(x,z)=-P\delta (x,z)}$

where ${\displaystyle \delta (x,z)}$ is the Dirac 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

${\displaystyle {\begin{aligned}\sigma _{xx}&=-{\frac {2P}{\pi }}{\frac {x^{2}z}{\left(x^{2}+z^{2}\right)^{2}}}\\\sigma _{zz}&=-{\frac {2P}{\pi }}{\frac {z^{3}}{\left(x^{2}+z^{2}\right)^{2}}}\\\sigma _{xz}&=-{\frac {2P}{\pi }}{\frac {xz^{2}}{\left(x^{2}+z^{2}\right)^{2}}}\end{aligned}}}$

for some point, ${\displaystyle (x,y)}$, in the half-plane. The circle shown in the figure indicates a surface on which the maximum shear stress is constant. From this stress field, the strain components and thus the displacements of all material points may be determined.

Suppose, rather than a point load ${\displaystyle P}$, a distributed load ${\displaystyle p(x)}$ is applied to the surface instead, over the range ${\displaystyle a<x<b}$. The principle of linear superposition can be applied to determine the resulting stress field as the solution to the integral equations:

${\displaystyle {\begin{aligned}\sigma _{xx}&=-{\frac {2z}{\pi }}\int _{a}^{b}{\frac {p\left(x'\right)\left(x-x'\right)^{2}\,dx'}{\left[\left(x-x'\right)^{2}+z^{2}\right]^{2}}}~;~~\sigma _{zz}=-{\frac {2z^{3}}{\pi }}\int _{a}^{b}{\frac {p\left(x'\right)\,dx'}{\left[\left(x-x'\right)^{2}+z^{2}\right]^{2}}}\\[3pt]\sigma _{xz}&=-{\frac {2z^{2}}{\pi }}\int _{a}^{b}{\frac {p\left(x'\right)\left(x-x'\right)\,dx'}{\left[\left(x-x'\right)^{2}+z^{2}\right]^{2}}}\end{aligned}}}$

The same principle applies for loading on the surface in the plane of the surface. These kinds of tractions would tend to arise as a result of friction. The solution is similar the above (for both singular loads ${\displaystyle Q}$ and distributed loads ${\displaystyle q(x)}$) but altered slightly:

${\displaystyle {\begin{aligned}\sigma _{xx}&=-{\frac {2}{\pi }}\int _{a}^{b}{\frac {q\left(x'\right)\left(x-x'\right)^{3}\,dx'}{\left[\left(x-x'\right)^{2}+z^{2}\right]^{2}}}~;~~\sigma _{zz}=-{\frac {2z^{2}}{\pi }}\int _{a}^{b}{\frac {q\left(x'\right)\left(x-x'\right)\,dx'}{\left[\left(x-x'\right)^{2}+z^{2}\right]^{2}}}\\[3pt]\sigma _{xz}&=-{\frac {2z}{\pi }}\int _{a}^{b}{\frac {q\left(x'\right)\left(x-x'\right)^{2}\,dx'}{\left[\left(x-x'\right)^{2}+z^{2}\right]^{2}}}\end{aligned}}}$

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

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

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 gap ${\displaystyle h}$ between two bodies can only be positive or zero

${\displaystyle h\geq 0}$

where ${\displaystyle h=0}$ denotes contact. The second assumption in contact mechanics is related to the fact, that no tension force is allowed to occur within the contact area (contacting bodies can be lifted up without adhesion forces). This leads to an inequality which the stresses have to obey at the contact interface. It is formulated for the normal stress ${\displaystyle \sigma _{n}=\mathbf {t} \cdot \mathbf {n} }$.

At locations where there is contact between the surfaces the gap is zero, i.e. ${\displaystyle h=0}$, and there the normal stress is different than zero, indeed, ${\displaystyle \sigma _{n}<0}$. At locations where the surfaces are not in contact the normal stress is identical to zero; ${\displaystyle \sigma _{n}=0}$, while the gap is positive; i.e., ${\displaystyle h>0}$. This type of complementarity formulation can be expressed in the so-called Kuhn–Tucker form, viz.

${\displaystyle h\geq 0\,,\quad \sigma _{n}\leq 0\,,\quad \sigma _{n}\,h=0\,.}$

These conditions are valid in a general way. The mathematical formulation of the gap depends upon the kinematics of the underlying theory of the solid (e.g., linear or nonlinear solid in two- or three dimensions, beam or shell model). By restating the normal stress ${\displaystyle \sigma _{n}}$ in terms of the contact pressure, ${\displaystyle p}$; i.e., ${\displaystyle p=-\sigma _{n}}$ the Kuhn-Tucker problem can be restated as in standard complementarity form i.e.

$${\displaystyle h\geq 0\,,\quad p\geq 0\,,\quad p\,h=0\,.}$$

In the linear elastic case the gap can be formulated as

$${\displaystyle {h}=h_{0}+{g}+u,}$$

where ${\displaystyle h_{0}}$is the rigid body separation, ${\displaystyle g}$ is the geometry/topography of the contact (cylinder and roughness) and ${\displaystyle u}$ is the elastic deformation/deflection. If the contacting bodies are approximated as linear elastic half spaces, the Boussinesq-Cerruti integral equation solution can be applied to express the deformation (${\displaystyle u}$) as a function of the contact pressure (${\displaystyle p}$); i.e.,

$${\displaystyle u=\int _{\infty }^{\infty }K(x-s)p(s)ds,}$$

where

$${\displaystyle K(x-s)={\frac {2}{\pi E^{*}}}\ln |x-s|}$$

for line loading of an elastic half space and

$${\displaystyle K(x-s)={\frac {1}{\pi E^{*}}}{\frac {1}{\sqrt {\left(x_{1}-s_{1}\right)^{2}+\left(x_{2}-s_{2}\right)^{2}}}}}$$

for 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]

${\displaystyle {\begin{aligned}\mathbf {h} &=\mathbf {h} _{0}+\mathbf {g} +\mathbf {Cp} ,\\\mathbf {h} \cdot \mathbf {p} &=0,\,\,\,\mathbf {p} \geq 0,\,\,\,\mathbf {h} \geq 0,\\\end{aligned}}}$

where ${\displaystyle \mathbf {C} }$ is a matrix, whose elements are so called influence coefficients relating the contact pressure and the deformation. The strict LCP formulation of the CM problem presented above, allows for direct application of well-established numerical solution techniques such as Lemke's pivoting algorithm. The Lemke algorithm has the advantage that it finds the numerically exact solution within a finite number of iterations. The MATLAB implementation presented by Almqvist et al. is one example that can be employed to solve the problem numerically. In addition, an example code for an LCP solution of a 2D linear elastic contact mechanics problem has also been made public at MATLAB file exchange by Almqvist et al.

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 between ${\displaystyle z}$ and ${\displaystyle z+dz}$ is ${\displaystyle \phi (z)dz}$. The authors calculated the number of contact spots n, the total contact area ${\displaystyle A_{r}}$ and the total load P in general case. They gave those formulas in two forms: in the basic and using standardized variables. If one assumes that N asperities covers a rough surface, then the expected number of contacts is

${\displaystyle n=N\int _{d}^{\infty }\phi (z)dz}$

The expected total area of contact can be calculated from the formula

${\displaystyle A_{a}=N\pi R\int _{d}^{\infty }(z-d)\phi (z)dz}$

and the expected total force is given by

${\displaystyle P={\frac {4}{3}}NE_{r}{\sqrt {R}}\int _{d}^{\infty }(z-d)^{\frac {3}{2}}\phi (z)dz}$

where:

R, radius of curvature of the microasperity,

z, height of the microasperity measured from the profile line,

d, close the surface,

${\displaystyle E_{r}=\left({\frac {1-\nu _{1}^{2}}{E_{1}}}+{\frac {1-\nu _{2}^{2}}{E_{2}}}\right)^{-1}}$, composite Young's modulus of elasticity,

${\displaystyle E_{i}}$, modulus of elasticity of the surface,

${\displaystyle \nu _{i}}$, Poisson's surface coefficients.

Greenwood and Williamson introduced standardized separation ${\displaystyle h=d/\sigma }$ and standardized height distribution ${\displaystyle \phi ^{*}(s)}$ whose standard deviation is equal to one. Below are presented the formulas in the standardized form.

${\displaystyle {\begin{aligned}F_{n}(h)&=\int _{h}^{\infty }(s-h)^{n}\phi ^{*}(s)ds\\n&=\eta A_{n}F_{0}(h)\\A_{a}&=\pi \eta AR\sigma F_{1}(h)\\P&={\frac {4}{3}}\eta AE_{r}{\sqrt {R}}\sigma ^{\frac {3}{2}}F_{\frac {3}{2}}(h)\end{aligned}}}$

where:

d is the separation,

${\displaystyle A}$ is the nominal contact area,

${\displaystyle \eta }$ is the surface density of asperities,

${\displaystyle E^{*}}$ is the effective Young modulus.

${\displaystyle A}$ and ${\displaystyle P}$ can be determined when the ${\displaystyle F_{n}(h)}$ terms are calculated for the given surfaces using the convolution of the surface roughness ${\displaystyle \phi ^{*}(s)}$.[34] Several studies have followed the suggested curve fits for ${\displaystyle F_{n}(h)}$ assuming 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 the ${\displaystyle F_{n}(h)}$ terms 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 to ${\displaystyle A_{r}}$ and ${\displaystyle P}$ were published by Jedynak.[36] They are given by the following rational formulas, which are approximants to the integrals ${\displaystyle F_{n}(h)}$. They are calculated for the Gaussian distribution of asperities, which have been shown to be unrealistic for engineering surface but can be assumed where friction, load carrying capacity or real contact area results are not critical to the analysis.[38]

${\displaystyle F_{n}(h)={\frac {a_{0}+a_{1}h+a_{2}h^{2}+a_{3}h^{3}}{1+b_{1}h+b_{2}h^{2}+b_{3}h^{3}+b_{4}h^{4}+b_{5}h^{5}+b_{6}h^{6}}}\exp \left(-{\frac {h^{2}}{2}}\right)}$

For ${\displaystyle F_{1}(h)}$ the coefficients are

${\displaystyle {\begin{aligned}[][a_{0},a_{1},a_{2},a_{3}]&=[0.398942280401,0.159773702775,0.0389687688311,0.00364356495452]\\[][b_{1},b_{2},b_{3},b_{4},b_{5},b_{6}]&=\left[1.653807476138,1.170419428529,0.448892964428,0.0951971709160,0.00931642803836,-6.383774657279\times 10^{-6}\right]\end{aligned}}}$

The maximum relative error is ${\displaystyle 9.93\times 10^{-8}\%}$.

For ${\displaystyle F_{\frac {3}{2}}(h)}$ the coefficients are

${\displaystyle {\begin{aligned}[][a_{0},a_{1},a_{2},a_{3}]&=[0.430019993662,0.101979509447,0.0229040629580,0.000688602924]\\[][b_{1},b_{2},b_{3},b_{4},b_{5},b_{6}]&=[1.671117125984,1.199586555505,0.46936532151,0.102632881122,0.010686348714,0.0000517200271]\end{aligned}}}$

The maximum relative error is ${\displaystyle 1.91\times 10^{-7}\%}$. The paper[36] also contains the exact expressions for ${\displaystyle F_{n}(h)}$

${\displaystyle {\begin{aligned}F_{1}(h)&={\frac {1}{\sqrt {2\pi }}}\exp \left(-{\frac {1}{2}}h^{2}\right)-{\frac {1}{2}}h\,\operatorname {erfc} \left({\frac {h}{\sqrt {2}}}\right)\\F_{\frac {3}{2}}(h)&={\frac {1}{4{\sqrt {\pi }}}}\exp \left(-{\frac {h^{2}}{4}}\right){\sqrt {h}}\left(\left(h^{2}+1\right)K_{\frac {1}{4}}\left({\frac {h^{2}}{4}}\right)-h^{2}K_{\frac {3}{4}}\left({\frac {h^{2}}{4}}\right)\right)\end{aligned}}}$

where erfc(z) means the complementary error function and ${\displaystyle K_{\nu }(z)}$ is the modified Bessel function of the second kind.

For the situation where the asperities on the two surfaces have a Gaussian height distribution and the peaks can be assumed to be spherical,[31] the average contact pressure is sufficient to cause yield when ${\displaystyle p_{\text{av}}=1.1\sigma _{y}\approx 0.39\sigma _{0}}$ where ${\displaystyle \sigma _{y}}$ is the uniaxial yield stress and ${\displaystyle \sigma _{0}}$ is the indentation hardness.[1] Greenwood and Williamson[31] defined a dimensionless parameter ${\displaystyle \Psi }$ called the plasticity index that could be used to determine whether contact would be elastic or plastic.

The Greenwood-Williamson model requires knowledge of two statistically dependent quantities; the standard deviation of the surface roughness and the curvature of the asperity peaks. An alternative definition of the plasticity index has been given by Mikic.[32] Yield occurs when the pressure is greater than the uniaxial yield stress. Since the yield stress is proportional to the indentation hardness ${\displaystyle \sigma _{0}}$, Mikic defined the plasticity index for elastic-plastic contact to be

${\displaystyle \Psi ={\frac {E^{*}h'}{\sigma _{0}}}>{\frac {2}{3}}~.}$

In this definition ${\displaystyle \Psi }$ represents the micro-roughness in a state of complete plasticity and only one statistical quantity, the rms slope, is needed which can be calculated from surface measurements. For ${\displaystyle \Psi <{\frac {2}{3}}}$, 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]

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

${\displaystyle A_{a}=\pi ^{2}(\eta \beta \sigma )^{2}AF_{2}(\lambda ),}$

and load carried by asperities

${\displaystyle P={\frac {8{\sqrt {2}}}{15}}\pi (\eta \beta \sigma )^{2}{\sqrt {\frac {\sigma }{\beta }}}E'AF_{\frac {5}{2}}(\lambda ),}$

where:

${\displaystyle \eta \beta \sigma }$, roughness parameter,

${\displaystyle A}$, nominal contact area,

${\displaystyle \lambda }$, Stribeck oil film parameter, first defined by Stribeck \cite{gt} as ${\displaystyle \lambda =h/\sigma }$,

${\displaystyle E'}$, effective elastic modulus,

${\displaystyle F_{2},F_{\frac {5}{2}}(\lambda )}$, 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 the ${\displaystyle F_{n}(h)}$ terms 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 for ${\displaystyle A_{a}}$ and ${\displaystyle P}$ are firstly presented by Jedynak.[36] They are expressed by ${\displaystyle F_{n}}$ as follows. They are calculated for the Gaussian distribution of asperities, which have been shown to be unrealistic for engineering surface but can be assumed where friction, load carrying capacity or real contact area results are not critical to the analysis.[38]

${\displaystyle {\begin{aligned}F_{2}&={\frac {1}{2}}\left(h^{2}+1\right)\operatorname {erfc} \left({\frac {h}{\sqrt {2}}}\right)-{\frac {h}{\sqrt {2\pi }}}\exp \left(-{\frac {h^{2}}{2}}\right)\\F_{\frac {5}{2}}&={\frac {1}{8{\sqrt {\pi }}}}\exp \left(-{\frac {h^{2}}{4}}\right)h^{\frac {3}{2}}\left(\left(2h^{2}+3\right)K_{\frac {3}{4}}\left({\frac {h^{2}}{4}}\right)-\left(2h^{2}+5\right)K_{\frac {1}{4}}\left({\frac {h^{2}}{4}}\right)\right)\end{aligned}}}$

where erfc(z) means the complementary error function and ${\displaystyle K_{\nu }(z)}$ is the modified Bessel function of the second kind.

In paper [36] one can find comprehensive review of existing approximants to ${\displaystyle F_{\frac {5}{2}}}$. New proposals give the most accurate approximants to ${\displaystyle F_{\frac {5}{2}}}$ and ${\displaystyle F_{2}}$, which are reported in the literature. They are given by the following rational formulas, which are very exact approximants to integrals ${\displaystyle F_{n}(h)}$. They are calculated for the Gaussian distribution of asperities

${\displaystyle F_{n}(h)={\frac {a_{0}+a_{1}h+a_{2}h^{2}+a_{3}h^{3}}{1+b_{1}h+b_{2}h^{2}+b_{3}h^{3}+b_{4}h^{4}+b_{5}h^{5}+b_{6}h^{6}}}\exp \left(-{\frac {h^{2}}{2}}\right)}$

For ${\displaystyle F_{2}(h)}$ the coefficients are

${\displaystyle {\begin{aligned}[][a_{0},a_{1},a_{2},a_{3}]&=[0.5,0.182536384941,0.039812283118,0.003684879001]\\[][b_{1},b_{2},b_{3},b_{4},b_{5},b_{6}]&=[1.960841785003,1.708677456715,0.856592986083,0.264996791567,0.049257843893,0.004640740133]\end{aligned}}}$

The maximum relative error is ${\displaystyle 1.68\times 10^{-7}\%}$.

For ${\displaystyle F_{\frac {5}{2}}(h)}$ the coefficients are

${\displaystyle {\begin{aligned}[][a_{0},a_{1},a_{2},a_{3}]&=[0.616634218997,0.108855827811,0.023453835635,0.000449332509]\\[][b_{1},b_{2},b_{3},b_{4},b_{5},b_{6}]&=[1.919948267476,1.635304362591,0.799392556572,0.240278859212,0.043178653945,0.003863334276]\end{aligned}}}$

The maximum relative error is ${\displaystyle 4.98\times 10^{-8}\%}$.

It is commonly assumed that the surface force between two atomic planes at a distance ${\displaystyle z}$ from each other can be derived from the Lennard-Jones potential. With this assumption

${\displaystyle F(z)={\cfrac {16\gamma }{3z_{0}}}\left[\left({\cfrac {z}{z_{0}}}\right)^{-9}-\left({\cfrac {z}{z_{0}}}\right)^{-3}\right]}$

where ${\displaystyle F}$ is the force (positive in compression), ${\displaystyle 2\gamma }$ is the total surface energy of both surfaces per unit area, and ${\displaystyle z_{0}}$ is the equilibrium separation of the two atomic planes.

The Bradley model applied the Lennard-Jones potential to find the force of adhesion between two rigid spheres. The total force between the spheres is found to be

${\displaystyle F_{a}(z)={\cfrac {16\gamma \pi R}{3}}\left[{\cfrac {1}{4}}\left({\cfrac {z}{z_{0}}}\right)^{-8}-\left({\cfrac {z}{z_{0}}}\right)^{-2}\right]~;~~{\frac {1}{R}}={\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}}$

where ${\displaystyle R_{1},R_{2}}$ are the radii of the two spheres.

The two spheres separate completely when the pull-off force is achieved at ${\displaystyle z=z_{0}}$ at which point

${\displaystyle F_{a}=F_{c}=-4\gamma \pi R.}$

Schematic of contact area for the JKR model.

JKR test with a rigid bead on a deformable planar material: complete cycle

To incorporate the effect of adhesion in Hertzian contact, Johnson, Kendall, and Roberts[5] formulated the JKR theory of adhesive contact using a balance between the stored elastic energy and the loss in surface 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

${\displaystyle p(r)=p_{0}\left(1-{\frac {r^{2}}{a^{2}}}\right)^{\frac {1}{2}}+p_{0}'\left(1-{\frac {r^{2}}{a^{2}}}\right)^{-{\frac {1}{2}}}}$

Note that in the original Hertz theory, the term containing ${\displaystyle p_{0}'}$ was neglected on the ground that tension could not be sustained in the contact zone. For contact between two spheres

${\displaystyle p_{0}={\frac {2aE^{*}}{\pi R}};\quad p_{0}'=-\left({\frac {4\gamma E^{*}}{\pi a}}\right)^{\frac {1}{2}}}$

where ${\displaystyle a\,}$ is the radius of the area of contact, ${\displaystyle F}$ is the applied force, ${\displaystyle 2\gamma }$ is the total surface energy of both surfaces per unit contact area, ${\displaystyle R_{i},\,E_{i},\,\nu _{i},~~i=1,2}$ are the radii, Young's moduli, and Poisson's ratios of the two spheres, and

${\displaystyle {\frac {1}{R}}={\frac {1}{R_{1}}}+{\frac {1}{R_{2}}};\quad {\frac {1}{E^{*}}}={\frac {1-\nu _{1}^{2}}{E_{1}}}+{\frac {1-\nu _{2}^{2}}{E_{2}}}}$

The approach distance between the two spheres is given by

${\displaystyle d={\frac {\pi a}{2E^{*}}}\left(p_{0}+2p_{0}'\right)={\frac {a^{2}}{R}}}$

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

${\displaystyle a^{3}={\frac {3R}{4E^{*}}}\left(F+6\gamma \pi R+{\sqrt {12\gamma \pi RF+(6\gamma \pi R)^{2}}}\right)}$

When the surface energy is zero, ${\displaystyle \gamma =0}$, the Hertz equation for contact between two spheres is recovered. When the applied load is zero, the contact radius is

${\displaystyle a^{3}={\frac {9R^{2}\gamma \pi }{E^{*}}}}$

The tensile load at which the spheres are separated (i.e., ${\displaystyle a=0}$) is predicted to be

${\displaystyle F_{\text{c}}=-3\gamma \pi R\,}$

This force is also called the pull-off force. Note that this force is independent of the moduli of the two spheres. However, there is another possible solution for the value of ${\displaystyle a}$ at this load. This is the critical contact area ${\displaystyle a_{\text{c}}}$, given by

${\displaystyle a_{\text{c}}^{3}={\frac {9R^{2}\gamma \pi }{4E^{*}}}}$

If we define the work of adhesion as

${\displaystyle \Delta \gamma =\gamma _{1}+\gamma _{2}-\gamma _{12}}$

where ${\displaystyle \gamma _{1},\gamma _{2}}$ are the adhesive energies of the two surfaces and ${\displaystyle \gamma _{12}}$ is an interaction term, we can write the JKR contact radius as

${\displaystyle a^{3}={\frac {3R}{4E^{*}}}\left(F+3\Delta \gamma \pi R+{\sqrt {6\Delta \gamma \pi RF+(3\Delta \gamma \pi R)^{2}}}\right)}$

The tensile load at separation is

${\displaystyle F=-{\frac {3}{2}}\Delta \gamma \pi R\,}$

and the critical contact radius is given by

${\displaystyle a_{\text{c}}^{3}={\frac {9R^{2}\Delta \gamma \pi }{8E^{*}}}}$

The critical depth of penetration is

${\displaystyle d_{\text{c}}={\frac {a_{c}^{2}}{R}}=\left(R^{\frac {1}{2}}{\frac {9\Delta \gamma \pi }{4E^{*}}}\right)^{\frac {2}{3}}}$

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

${\displaystyle a^{3}={\cfrac {3R}{4E^{*}}}\left(F+4\gamma \pi R\right)}$

and the pull-off force is

${\displaystyle F_{c}=-4\gamma \pi R\,}$

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

In terms of the work of adhesion ${\displaystyle \Delta \gamma }$

${\displaystyle a^{3}={\cfrac {3R}{4E^{*}}}\left(F+2\Delta \gamma \pi R\right)}$

and

${\displaystyle F_{c}=-2\Delta \gamma \pi R\,}$

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 the Tabor parameter (${\displaystyle \mu }$) defined as

${\displaystyle \mu$ :={\frac {d_{c}}{z_{0}}}\approx \left[{\frac {R(\Delta \gamma )^{2}}{{E^{*}}^{2}z_{0}^{3}}}\right]^{\frac {1}{3}}}

where ${\displaystyle z_{0}}$ is the equilibrium separation between the two surfaces in contact. The JKR theory applies to large, compliant spheres for which ${\displaystyle \mu }$ is large. The DMT theory applies for small, stiff spheres with small values of ${\displaystyle \mu }$.

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 value ${\displaystyle 2\pi R\Delta \gamma }$ to the JKR value ${\displaystyle (3/2)\pi R\Delta \gamma }$. 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]

Schematic of contact area for the Maugis-Dugdale model.

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

${\displaystyle \Delta \gamma =\sigma _{0}~h_{0}}$

where ${\displaystyle \sigma _{0}}$ is the maximum force predicted by the Lennard-Jones potential and ${\displaystyle h_{0}}$ is the maximum separation obtained by matching the areas under the Dugdale and Lennard-Jones curves (see adjacent figure). This means that the attractive force is constant for ${\displaystyle z_{0}\leq z\leq z_{0}+h_{0}}$. There is not further penetration in compression. Perfect contact occurs in an area of radius ${\displaystyle a}$ and adhesive forces of magnitude ${\displaystyle \sigma _{0}}$ extend to an area of radius ${\displaystyle c>a}$. In the region ${\displaystyle a<r<c}$, the two surfaces are separated by a distance ${\displaystyle h(r)}$ with ${\displaystyle h(a)=0}$ and ${\displaystyle h(c)=h_{0}}$. The ratio ${\displaystyle m}$ is defined as

${\displaystyle m:={\frac {c}{a}}}$.

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 ${\displaystyle -a<r<a}$. The contribution to the surface traction from the Hertz pressure is given by

${\displaystyle p^{H}(r)=\left({\frac {3F^{H}}{2\pi a^{2}}}\right)\left(1-{\frac {r^{2}}{a^{2}}}\right)^{\frac {1}{2}}}$

where the Hertz contact force ${\displaystyle F^{H}}$ is given by

${\displaystyle F^{H}={\frac {4E^{*}a^{3}}{3R}}}$

The penetration due to elastic compression is

${\displaystyle d^{H}={\frac {a^{2}}{R}}}$

The vertical displacement at ${\displaystyle r=c}$ is

${\displaystyle u^{H}(c)={\cfrac {1}{\pi R}}\left[a^{2}\left(2-m^{2}\right)\sin ^{-1}\left({\frac {1}{m}}\right)+a^{2}{\sqrt {m^{2}-1}}\right]}$

and the separation between the two surfaces at ${\displaystyle r=c}$ is

${\displaystyle h^{H}(c)={\frac {c^{2}}{2R}}-d^{H}+u^{H}(c)}$

The surface traction distribution due to the adhesive Dugdale stress is

${\displaystyle p^{D}(r)={\begin{cases}-{\frac {\sigma _{0}}{\pi }}\cos ^{-1}\left[{\frac {2-m^{2}-{\frac {r^{2}}{a^{2}}}}{m^{2}\left(1-{\frac {r^{2}}{m^{2}a^{2}}}\right)}}\right]&\quad {\text{for}}\quad r\leq a\\-\sigma _{0}&\quad {\text{for}}\quad a\leq r\leq c\end{cases}}}$

The total adhesive force is then given by

${\displaystyle F^{D}=-2\sigma _{0}m^{2}a^{2}\left[\cos ^{-1}\left({\frac {1}{m}}\right)+{\frac {1}{m^{2}}}{\sqrt {m^{2}-1}}\right]}$

The compression due to Dugdale adhesion is

${\displaystyle d^{D}=-\left({\frac {2\sigma _{0}a}{E^{*}}}\right){\sqrt {m^{2}-1}}}$

and the gap at ${\displaystyle r=c}$ is

${\displaystyle h^{D}(c)=\left({\frac {4\sigma _{0}a}{\pi E^{*}}}\right)\left[{\sqrt {m^{2}-1}}\cos ^{-1}\left({\frac {1}{m}}\right)+1-m\right]}$

The net traction on the contact area is then given by ${\displaystyle p(r)=p^{H}(r)+p^{D}(r)}$ and the net contact force is ${\displaystyle F=F^{H}+F^{D}}$. When ${\displaystyle h(c)=h^{H}(c)+h^{D}(c)=h_{0}}$ the adhesive traction drops to zero.

Non-dimensionalized values of ${\displaystyle a,c,F,d}$ are introduced at this stage that are defied as

${\displaystyle {\bar {a}}=\alpha a~;~~{\bar {c}}:=\alpha c~;~~{\bar {d}}:=\alpha ^{2}Rd~;~~\alpha$ :=\left({\frac {4E^{*}}{3\pi \Delta \gamma R^{2}}}\right)^{\frac {1}{3}}~;~~{\bar {A}}:=\pi c^{2}~;~~{\bar {F}}={\frac {F}{\pi \Delta \gamma R}}}

In addition, Maugis proposed a parameter ${\displaystyle \lambda }$ which is equivalent to the Tabor parameter ${\displaystyle \mu }$ . This parameter is defined as

${\displaystyle \lambda$ :=\sigma _{0}\left({\frac {9R}{2\pi \Delta \gamma {E^{*}}^{2}}}\right)^{\frac {1}{3}}\approx 1.16\mu }

where the step cohesive stress ${\displaystyle \sigma _{0}}$ equals to the theoretical stress of the Lennard-Jones potential

${\displaystyle \sigma _{\text{th}}={\frac {16\Delta \gamma }{9{\sqrt {3}}z_{0}}}}$

Zheng and Yu [48] suggested another value for the step cohesive stress

${\displaystyle \sigma _{0}=\exp \left(-{\frac {223}{420}}\right)\cdot {\frac {\Delta \gamma }{z_{0}}}\approx 0.588{\frac {\Delta \gamma }{z_{0}}}}$

to match the Lennard-Jones potential, which leads to

${\displaystyle \lambda \approx 0.663\mu }$

Then the net contact force may be expressed as

${\displaystyle {\bar {F}}={\bar {a}}^{3}-\lambda {\bar {a}}^{2}\left[{\sqrt {m^{2}-1}}+m^{2}\sec ^{-1}m\right]}$

and the elastic compression as

${\displaystyle {\bar {d}}={\bar {a}}^{2}-{\frac {4}{3}}~\lambda {\bar {a}}{\sqrt {m^{2}-1}}}$