Research Article  Open Access
Ali Sepehri, Kambiz Farhang, "ClosedForm Equations for Contact Force and Moment in Elastic Contact of Rough Surfaces", Modelling and Simulation in Engineering, vol. 2011, Article ID 739562, 14 pages, 2011. https://doi.org/10.1155/2011/739562
ClosedForm Equations for Contact Force and Moment in Elastic Contact of Rough Surfaces
Abstract
It is reasonable to expect that, when two nominally flat rough surfaces are brought into contact by an applied resultant force, they must support, in addition to the compressive load, an induced moment. The existence of a net applied moment would imply noneven distribution of contact force so that there are more asperities in contact over one region of the nominal area. In this paper, we consider the contact between two rectangular rough surfaces that provide normal and tangential contact force as well as contact moment to counteract the net moment imposed by the applied forces. The surfaces are permitted to develop slight angular misalignment, and thereby contact moment is derived. Through this scheme, it is possible to also define elastic contribution to friction since the halfplane tangential contact force on one side of an asperity is no longer balanced by the halfplane tangential force component on the opposite side. The elastic friction force, however, is shown to be of a much smaller order than the contact normal force. Approximate closedform equations are found for contact force and moment for the contact of rough surfaces.
1. Introduction
The seminal work of Greenwood and Tripp [1] and Greenwood and Williamson [2] established the early models for elastic contact of nominally flat rough surfaces. These models have been used extensively in the realm of tribology and contact mechanics of rough surfaces. Many published works have appeared [3–35] since the publication of GW and GT models that, in one way or another, are inspired by [1, 2]. Almost an entire body of work [3–26, 28–35] ignores altogether the effect of contact moment. Hess and Soom [27] are perhaps the first and only ones to address contact moment. But their treatment of contact moment was limited to the application of the GW model in which only normal contact force is considered. Sepehri and Farhang [28] developed a threedimensional model of elastic interaction between two rough surfaces. In this model asperity shouldershoulder contact is considered leading to the formulation of both halfplane tangential and normal components of contact force.
This paper develops the governing equations for the elastic contact of two nominally flat rough surfaces subject simultaneously to applied force and moment. To treat contact moment, the surfaces are allowed to sustain a slight angular misalignment so as to counteract the effect of applied moment. It is shown that the existence of a net moment induces a net tangential force as the halfplane tangential forces imposed on the surfaces no longer cancel. The net tangential force due to elastic interaction is found to be several orders of magnitude smaller than the normal contact force. Approximate equations are found, using a multistep optimization procedure that provides accuracy within seven percent.
2. Elastic Contact
When two rough surfaces are brought into contact by an applied resultant force, the surfaces must support, in addition to the compressive load, an induced moment. The existence of a net applied moment would imply nonuniform distribution of contact force so that there are more asperities in contact over one region of the nominal area. It is assumed that the moment would result in a slight angular misalignment of the mean planes of the two surfaces.
We consider the contact between two rectangular rough surfaces that provide normal and tangential contact force as well as contact moment to counteract the net moment imposed. Thus, the mean plane separation is no longer constant and is described below using a linear function of the following form: where denotes the separation when the normal force is applied symmetrically so that no net moment is supported by the surfaces. When a net moment is additionally imposed by either the application of an applied tangential force or asymmetric application of the normal force or both, the surfaces would experience a slight relative rotation so that the normal pressure redistributes to accommodate the net applied moment. Let x_{0}, measured from the center of the plate nominal area of contact along, correspond to the location along the mean plane, at which the mean plane separation is in the relative tilted configuration (Figure 1). Let be the mean plane slope of surface with respect to after applying the external moment.
We begin by examining the contact between two asperities as shown in Figure 2. An asperity on surface interacts with an asperity on surface . Such an interaction occurs, in general, obliquely as shown in the figure so that the Hertz contact force along the common normal to the two asperities is not orthogonal to the mean planes of the two surfaces. Therefore, the Hertz contact force for interaction of two asperities can be expressed as where is the combined Young’s modulus of elasticity for the two surfaces is the combined radius of curvature at the contact and is the interference (approach) along the normal to the asperity contact patch. is the tangential offset of the mating asperities so that when the asperities interfere along the normal to the mean planes. For , asperities interfere along an oblique line, as shown in an enlarged view of the interference in Figure 3.
For the asperity interaction shown in Figures 2 and 3, the interference as well as equivalent radius of curvature must be found corresponding to the midpoint of contact, located approximately midway between the intersections of undeformed asperities. Denoting by and , respectively, the tangential offset of the intersection midpoint from the peak of asperity on surfaces and , it follows that where is the equivalent summit radius of the asperities given by Alternatively, the offset may be written as follows: A note of caution in the above is the difference between and ; the first is the equivalent radius of curvature of the asperity summits, whereas the second represents the equivalent radius of curvature at a contact of the two asperities. Using a quadratic approximation for asperity shape near its summit, we find The approach, , is (Figures 2 and 3) where is the interference defined by Greenwood and Williamson [2]. It is found in terms of the local mean plane separation, , and asperity heights and offset as follows: and in (9) are the asperity shape functions of surfaces and , respectively, and the local mean plane separation: where is the local orientation of an asperity on surface relative to the asperity on surface , as shown in Figure 4. Note that in (9) the change of due to mean plane tilt (order of 10^{−6}) is negligible compared to that of the local mean plane separation (order of 10^{−3}). That is, the change in separation due to relative angular rotation has a most profound effect through its influence on changing the mean plane separation. Its effect through relative rotation of two asperities is negligible. Using a quadratic approximation near the summit of each asperity, (9) reduces to Therefore, the asperity interference can be found by combining (8), (10), (11), and (12) that yield Finally, substitution from (7) and (13) in (2) yields
The asperity contact force in (14) is directed along the normal to the contact patch. It yields two components as shown in Figure 3. Denoting by and the components of the asperity contact force along the normal and tangential directions, respectively, we find, with the help of (10) and (14), where z = z_{1}+ z_{2} the height sum of two asperities.
3. Normal Force
All the normal components of various contact forces are parallel (Figure 4) and can be algebraically summed by statistical means to obtain the total normal force of one surface on another.
Let (Figure 4), be an infinitesimal area on surface located at from an asperity on an infinitesimal area Figure 5 on surface located at from the middle of the two rectangular surfaces: The total normal force exerted on an asperity of the differential area at height by all asperities on surface is obtained by the following sum: where is the density function associated with the asperity height distribution on surface . The above triple integral could be simplified if an averaging method is used for the integration over the angular coordinate in the range of to . We present the average representation of the integrand as follows: The approximation provides relative error within 1.16 percent. So (17) yields, the normal force on all asperities of the differential area at height by all asperities on surface , can be found by The total normal force on all asperities of the differential area due to all asperities on surface is obtained as follows where is the density function for summit height sum distribution of asperity summits on and . The total normal force on all asperities of surface due to all asperities on surface can be found by integration along Every length parameter in (22) is normalized with respect to the standard deviation of asperity height sum. Define , and consider hereafter , , and as normalized values using as the normalization parameter. The limits of are determined by the roots of the following equation (setting interference, , equal to zero) We find where is the normalized value of and and are now used to denote normalized values using as the normalization parameter. For a Gaussian distribution of the asperity height sum we find the following normalized form for the equation describing total normal contact force between surfaces and where In the absence of a moment, the total normal force is [28] where So we have The results of numerical solution confirm the above equation. For a plate of known size and prescribed normal force and moment, one can solve the above equation along with the moment equation to find and . is the location at which the mean plane separation equals that with no applied moment. We refer to this as the position of initial separation. It is essential that the relation between and other parameters is established since it greatly facilitates the solution of problems involving contact moment. only depends on , , and and it is not a function of . We find, using a multistep optimization procedure, the following approximate equation for over , to , and Recall that the large values assigned to a are due to their normalization with respect to the standard deviation of asperity height sum, . values are given in the Table 1. Figures 6, 7, and 8, respectively, illustrate for , for , and for . The approximate function in (31) provides estimates of to within 0.6 percent accuracy over the entire domain of , and .

4. Tangential Force
The tangential components due to various interactions cannot be algebraically added as they are projections of contact force onto the mean plane and depend on circumferential position of asperities on surface (Figures 9 and 10). We are interested in formulating the cumulative effect of xcomponent of tangential force after applying the moment. Hereafter as we generate result for the xcomponent of the tangential force, we will refer to this as the “tangential force” and denote the force component . The goal here is to account for the tangential force that an asperity would experience on each side. Accumulation or summation of such forces would establish the total tangential load on an asperity from both sides, that is, due to all contacts at positive slopes and negative slopes.
It can be shown that the total tangential force (along ) on an asperity of the differential area at height by all asperities on surface is found by The integral (32) can be reduced through the following approximation of the integration over the range of to : The approximation provides a relative error less than 0.83 percent. We find, the tangential force on all asperities in the differential area at height by all asperities on surface , can be found by The total tangential force on all asperities of the differential area due to all asperities on surface is obtained as follows: The total tangential force on all asperities of surface due to all asperities on surface can be found by integration along : For a Gaussian distribution of asperity height sum the total tangential force between surfaces and in normalized form is where and and , are defined in (27) and (24), respectively.
5. Contact Moment
Contact moment about the yaxis consists of two terms: (1) sum of moments about each asperity and (2) moment of normal force about the yaxis due to movement of the point of exertion of the resultant normal force , as shown in Figure 1. First we consider to find total moment about all asperities. Moment about the yaxis on an asperity in the differential area at height due to all asperities at height confined in and located at is The total moment on an asperity of the differential area at height by all asperities on surface is found by where we have employed the approximation in (33). , the moment on all asperities of the differential area at height by all asperities on surface , can be found by The total moment on all asperities in the differential area due to all asperities on surface is obtained as follows: The total moment on all asperities of surface due to all asperities on surface can be found by integration along For a Gaussian distribution of asperity height sum the total moment between surfaces and is where is given by (39) and and , are defined in (27) and (24), respectively. The second term of moment due to shifting of the location of the resultant normal load is where Therefore, the total moment between surfaces and can be written as where Numerical results show that the first moment term (order of 10^{5}) is negligible in comparison to the second term (order of 10^{12}). The force center, , is obtained by Friction coefficient due to elastic deformation is equal to, , and in the above two equations are given by (29), (39), and (49), respectively.
6. Results
Figures 11 and 12, respectively, depict the dimensionless normal and tangential contact force components, and , versus dimension length, , for , , and plate dimensionless length . Since each slope corresponds to an applied moment, the figures show the pressure and shear distribution for various applied moments. Figures 13, 14, and 15 illustrate the dependence of , in each case holding two of the parameters fixed. Figure 13 shows for and , Figure 14 depicts for and , and Figure 15 represents for and . In these figures corresponds to an applied normal force such that a small value of represents a large normal force and large corresponds to a small applied normal force. Therefore, for large applied normal force it is shown in Figures 13, 14, and 15 that the sensitivity of dimensionless contact moment increases with asperity radius of curvature sum , slope , and nominal dimension . Similar observation can be made about the dimensionless contact force . Figures 16, 17 and 18, respectively, show , and for the same cases.
Figures 19, 20, and 21, respectively, demonstrate the relation between the location of the resultant normal contact force, , while, as before, holding two of the parameters constant. These are shown as , , and . Figures 22, 23, and 24, respectively, represent , , and . Figures 19 and 22 show that the force center and elastic friction coefficient are independent of .
7. Approximate Equations
In this section we introduce approximate equations for the integral functions of contact moment and tangential force. We find, using a multistep optimization procedure, the following general approximate equations for and : where as well as where To assess the accuracy of the approximation in (52), we define the following error between the dimensionless critical contact moment and its approximation in percent error form: The error is plotted over the ranges , and for and in Figure 25, demonstrating that the accuracy of approximation is within seven percent (7%). Indeed the approximate function in (52) yields accuracy to within 7 percent over the entire domain of , and .
A similar study for error associated with the approximation of yields accuracy within 7 percent over the entire range of , and . Figure 26 depicts over and for and .
Approximate functions for the force center, , and elastic friction coefficient, , are where I_{Na} is given by [28] The approximate functions in (57) provide estimates of and to within 5%.
8. Concluding Remarks
This paper has addressed a methodology for treating contact moment in the interaction of two nominally flat rough surfaces. The method is based on an extension of GT model [28] in which the asperity elastic shouldershoulder contact is considered, yielding resultant asperity contact force in a slanted orientation with respect to the mean planes of the surfaces.
In consideration of contact moment, the method allows a slight relative angular rotation (tilting) of the mean planes. Hence the applied moment is balanced by a nonuniform distribution of normal contact pressure. In addition to the nonuniform distribution of normal force, relative tilting of the mean planes is shown to result in imbalance of the and halfplane tangential forces, leading to a definition of elastic friction force. The results have shown that friction force due to elastic contact is very small so that the coefficient of friction is found to be less than .
Approximate equations were forwarded for the integral functions of contact moment and tangential force. These equations were shown to provide accuracy within seven percent over ranges of mean plane separation, asperity summit radius of curvature sum, and slope and plate dimension. The approximate equations greatly simplify solution of problems involving elastic contact of rough surfaces.
Nomenclature
:  Hertz contact force 
:  Combined Young’s modulus 
:  Dimensionless average summit radius of asperities on the surfaces 1 and 2 
:  Combined asperity summit radius of curvature 
:  Equivalent asperity radius of curvature at a contact 
:  Interference 
:  Interference defined by Greenwood and Tripp 
:  Contact angle between two asperities 
:  Asperities shape functions 
:  Number of asperities per unit nominal area on 
:  Number of asperities per unit nominal area on 
:  Asperity contact force along normal direction 
:  Asperity contact force along tangential direction 
:  Standard deviation of asperity height summation 
:  Young's moduli 
:  Poisson ratios 
:  Initial mean plane separation 
:  Local mean plane separation 
:  Local mean plane separation around an asperity 
:  Dimensionless initial mean plane separation 
:  Dimensionless local mean plane separation 
:  Heights of asperities measured from the mean asperity heights 
:  Height sum of two asperities 
:  Dimensionless asperity heights sum 
:  Dimensionless mean separation 
:  Tangential offset of the mating asperities 
:  Dimensionless asperity summit radius of curvature sum 
:  Mean plane slope of surface due to moment 
:  Position of initial separation 
:  Total normal contact force 
:  Total tangential force 
:  Contact moment (first term) 
:  Dimensionless contact moment (first term) 
:  Contact moment (second term) 
:  Dimensionless contact moment (second term) 
:  Dimensionless normal force 
:  Dimensionless tangential force 
:  Dimensionless contact moment 
:  Location of force center 
:  Elastic friction coefficient 
:  Length and width of rectangular surfaces 
:  Approximate function for 
:  Approximate function for 
:  Approximate function for 
:  Approximate function for 
:  Approximate function for 
:  Percent error between and 
:  Percent error between and . 
References
 J. A. Greenwood and J. H. Tripp, “The contact of two nominally flat rough surfaces,” Proceedings of the Institution of Mechanical Engineers, vol. 185, pp. 625–633, 1970. View at: Google Scholar
 J. A. Greenwood and J. B. P. Williamson, “Contact of nominally flat surfaces,” Proceedings of the Royal Society of London, vol. 295, pp. 300–319, 1966. View at: Google Scholar
 J. I. McCool, “Extending the capability of the Greenwood Williamson microcontact model,” Journal of Tribology, vol. 122, no. 3, pp. 496–502, 2000. View at: Publisher Site  Google Scholar
 J. I. McCool, “NonGaussian effects in microcontact,” International Journal of Machine Tools and Manufacture, vol. 32, no. 12, pp. 115–123, 1992. View at: Google Scholar
 F. M. Borodich and A. B. Mosolov, “Fractal roughness in contact problems,” Journal of Applied Mathematics and Mechanics, vol. 56, no. 5, pp. 681–690, 1992. View at: Google Scholar
 A. A. Polycarpou and I. Etsion, “Analytical approximations in modeling contacting rough surfaces,” Journal of Tribology, vol. 121, no. 2, pp. 234–239, 1999. View at: Google Scholar
 N. Yu and A. A. Polycarpou, “Contact of rough surfaces with asymmetric distribution of asperity heights,” Journal of Tribology, vol. 124, no. 2, pp. 367–376, 2002. View at: Publisher Site  Google Scholar
 N. Yu and A. A. Polycarpou, “Combining and contacting of two rough surfaces with asymmetric distribution of asperity heights,” Journal of Tribology, vol. 126, no. 2, pp. 225–232, 2004. View at: Publisher Site  Google Scholar
 A. Hariri, J. W. Zu, and R. B. Mrad, “npoint asperity model for contact between nominally flat surfaces,” Journal of Tribology, vol. 128, no. 3, pp. 505–514, 2006. View at: Publisher Site  Google Scholar
 J. F. Antoine, C. Visa, C. Sauvey, and G. Abba, “Approximate analytical model for Hertzian elliptical contact problems,” Journal of Tribology, vol. 128, no. 3, pp. 660–664, 2006. View at: Publisher Site  Google Scholar
 M. Ciavarella and F. Leoci, “An assessment of the GreenwoodWilliamson and other asperities models with, special reference to electrical conductance,” Journal of Tribology, vol. 128, no. 1, pp. 10–17, 2006. View at: Publisher Site  Google Scholar
 M. Bahrami, M. M. Yovanovich, and J. R. Culham, “A compact model for spherical rough contacts,” Journal of Tribology, vol. 127, no. 4, pp. 884–889, 2005. View at: Publisher Site  Google Scholar
 J. J. Wu, “The properties of asperities of real surfaces,” Journal of Tribology, vol. 123, no. 4, pp. 872–883, 2001. View at: Publisher Site  Google Scholar
 Y. Lin and T. C. Ovaert, “A rough surface contact model for general anisotropic materials,” Journal of Tribology, vol. 126, no. 1, pp. 41–49, 2004. View at: Google Scholar
 Y. A. Karpenko and A. Akay, “A numerical method for analysis of extended rough wavy surfaces in contact,” Journal of Tribology, vol. 124, no. 4, pp. 668–679, 2002. View at: Publisher Site  Google Scholar
 W. R. Chang, I. Etsion, and D. B. Bogy, “An elasticplastic model for the contact of rough surfaces,” Journal of Tribology, vol. 109, no. 2, pp. 257–263, 1987. View at: Google Scholar
 J. Halling, R. D. Arnell, and K. A. Nuri, “The elasticplastic contact of rough surfaces and its relevance in the study of wear,” Proceedings of the Institution of Mechanical Engineers Part H, vol. 202, no. C4, pp. 269–274, 1988. View at: Google Scholar
 H. So and D. C. Liu, “An elasticplastic model for the contact of anisotropic rough surfaces,” Wear, vol. 146, no. 2, pp. 201–218, 1991. View at: Google Scholar
 A. Majumdar and B. Bhushan, “Fractal model of elasticplastic contact between rough surfaces,” Journal of Tribology, vol. 113, no. 1, pp. 1–11, 1991. View at: Google Scholar
 A. Majumdar and B. Bhushan, “Elasticplastic contact model for bifractal surfaces,” Wear, vol. 153, no. 1, pp. 53–64, 1992. View at: Google Scholar
 W. Yan and K. Komvopoulos, “Contact analysis of elasticplastic fractal surfaces,” Journal of Applied Physics, vol. 84, no. 7, pp. 3617–3624, 1998. View at: Google Scholar
 K. Komvopoulos and W. Yan, “Threedimensional elasticplastic fractal analysis of surface adhesion in microelectromechanical systems,” Journal of Tribology, vol. 120, no. 4, pp. 808–813, 1998. View at: Google Scholar
 J. Abdo and K. Farhang, “Elasticplastic contact model for rough surfaces based on plastic asperity concept,” International Journal of NonLinear Mechanics, vol. 40, no. 4, pp. 495–506, 2005. View at: Publisher Site  Google Scholar
 K. Komvopoulos and W. Yan, “A fractal analysis of stiction in microelectromechanical systems,” Journal of Tribology, vol. 119, no. 3, pp. 391–400, 1997. View at: Google Scholar
 H. Gao, G. C. Barber, and M. Shillor, “Numerical simulation of engagement of a wet clutch with skewed surface roughness,” Journal of Tribology, vol. 124, no. 2, pp. 305–312, 2002. View at: Publisher Site  Google Scholar
 H. Gao and G. C. Barber, “Microcontact model for paperbased wet friction materials,” Journal of Tribology, vol. 124, no. 2, pp. 414–419, 2002. View at: Publisher Site  Google Scholar
 D. P. Hess and A. Soom, “Normal and angular motions at rough planar contacts during sliding with friction,” Journal of Tribology, vol. 114, no. 3, pp. 567–578, 1992. View at: Google Scholar
 A. Sepehri and K. Farhang, “On elastic interaction of nominally flat rough surfaces,” Journal of Tribology, vol. 130, no. 1, 2008. View at: Google Scholar
 W. R. Chang, I. Etsion, and D. B. Bogy, “Static friction coefficient model for metallic rough surfaces,” Journal of Tribology, vol. 110, no. 1, pp. 57–63, 1988. View at: Google Scholar
 L. Kogut and I. Etsion, “A static friction model for elasticplastic contacting rough surfaces,” Journal of Tribology, vol. 126, no. 1, pp. 34–40, 2004. View at: Google Scholar
 X. Shi and A. A. Polycarpou, “A dynamic friction model for unlubricated rough planar surfaces,” Journal of Tribology, vol. 125, no. 4, pp. 788–796, 2003. View at: Publisher Site  Google Scholar
 S. Wang and K. Komvopoulos, “Static friction and initiation of slip at magnetic headdisk interfaces,” Journal of Tribology, vol. 122, no. 1, pp. 246–256, 2000. View at: Publisher Site  Google Scholar
 S. Wang, “Real contact area of fractalregular surfaces and its implications in the law of friction,” Journal of Tribology, vol. 126, no. 1, pp. 1–8, 2004. View at: Google Scholar
 N. Yu, S. R. Pergande, and A. A. Polycarpou, “Static friction model for rough surfaces with asymmetric distribution of asperity heights,” Journal of Tribology, vol. 126, no. 3, pp. 626–629, 2004. View at: Publisher Site  Google Scholar
 J. Yang and K. Komvopoulos, “A mechanics approach to static friction of elasticplastic fractal surfaces,” Journal of Tribology, vol. 127, no. 2, pp. 315–324, 2005. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2011 Ali Sepehri and Kambiz Farhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.