# Closed-Form Equations for Contact Force and Moment in Elastic Contact of Rough Surfaces

Authors: Ali Sepehri; Kambiz Farhang
Journal: Modelling and Simulation in Engineering
 (2011)

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## 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 half-plane tangential contact force on one side of an asperity is no longer balanced by the half-plane 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 closed-form equations are found for contact force and moment for the contact of rough surfaces.


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## Body

## 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 three-dimensional model of elastic interaction between two rough surfaces. In this model asperity shoulder-shoulder contact is considered leading to the formulation of both half-plane 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 half-plane 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(a×b) 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:(1)d=d0-m(x-x0),
where d0 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 x0, measured from the center of the plate nominal area of contact alongx, correspond to the location along the mean plane, at which the mean plane separation is d0 in the relative tilted configuration (Figure 1). Let m be the mean plane slope of surface S1 with respect to S2 after applying the external moment.Figure 1
Contacting two rough surfaces with a slight rotation of the mean plane.We begin by examining the contact between two asperities as shown in Figure2. An asperity on surface S1 interacts with an asperity on surface S2. 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(2)fa=43E′β(r)1/2w3/2,
where E′ is the combined Young’s modulus of elasticity for the two surfaces(3)1E′=1-ν12E1+1-ν22E2,β(r) is the combined radius of curvature at the contact and w is the interference (approach) along the normal to the asperity contact patch. r is the tangential offset of the mating asperities so that when r=0 the asperities interfere along the normal to the mean planes. For r≠0, asperities interfere along an oblique line, as shown in an enlarged view of the interference in Figure 3.Figure 2
Asperity contact.Figure 3
Overlap region showing normal and oblique interference.For the asperity interaction shown in Figures2 and 3, the interference as well as equivalent radius of curvature must be found corresponding to the mid-point of contact, located approximately midway between the intersections of undeformed asperities. Denoting by r1 and r2, respectively, the tangential offset of the intersection mid-point from the peak of asperity on surfaces S1 and S2, it follows that(4)r1=ββ2r,r2=ββ1r,
where β is the equivalent summit radius of the asperities given by(5)1β=1β1+1β2.
Alternatively, the offset may be written as follows:(6)r1=β1βsr,r2=β2βsr,βs=β1+β2,
A note of caution in the above is the difference between β and β(r); 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(7)β(r)=β(1+r2βs2)3/2.
The approach, w, is (Figures 2 and 3)(8)w=w1cos⁡α,
where w1 is the interference defined by Greenwood and Williamson [2]. It is found in terms of the local mean plane separation, d’, and asperity heights and offset as follows:(9)w1=z1+z2-d′-f1(r1)-f2(r2),(10)cos⁡α=(1+r2βs2)-1/2,sin⁡α=(1+r2βs2)-1/2rβs,f1 and f2 in (9) are the asperity shape functions of surfaces S1 and S2, respectively, and d’the local mean plane separation:(11)d′=d-mrcos⁡θ,
where θ is the local orientation of an asperity on surface S2 relative to the asperity on surface S1, as shown in Figure 4. Note that in (9) the change of z1 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(12)w1=z1+z2-d′-r22βs.
Therefore, the asperity interference can be found by combining (8), (10), (11), and (12) that yield(13)w=(z1+z2-d+mrcos⁡θ-r22βs)(1+r2βs2)-1/2.
Finally, substitution from (7) and (13) in (2) yields(14)fa=43E′β1/2(z1+z2-d+mrcos⁡θ-r22βs)3/2.Figure 4
Differential area(da=rdrdθ) on surface S2 and the normal component of contact force (normal to the reference mean plane).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 fN and ft the components of the asperity contact force along the normal and tangential directions, respectively, we find, with the help of (10) and (14), (15)fN=43E′β1/2(z-d+mrcos⁡θ-r22βs)3/2(1+r2βs2)-1/2,ft=43E′β1/2(z-d+mrcos⁡θ-r22βs)3/2(1+r2βs2)-1/2rβs,
where z = z1+ z2 the height sum of two asperities.


## 3. Normal Force

All the normal components of various contact forces(dFN′s) are parallel (Figure 4) and can be algebraically summed by statistical means to obtain the total normal force of one surface on another.LetdA (Figure 4), be an infinitesimal area on surface S2 located at r from an asperity on an infinitesimal area dA′ Figure 5 on surface S1 located at x from the middle of the two rectangular surfaces: (16)dA=rdrdθ,dA′=bdx.
The total normal force exerted on an asperity of the differential area dA′ at height z1 by all asperities on surface S2 is obtained by the following sum:(17)(FN)z1=43E′η2β1/2∫z2∫r∫θ(z-d+mrcos⁡θ-r22βs)3/2×(1+r2βs2)-1/2ϕ2(z2)rdrdθdz2,
where ϕ2(z2) is the density function associated with the asperity height distribution on surface S2. The above triple integral could be simplified if an averaging method is used for the integration over the angular coordinate θ in the range of -π/2 to π/2. We present the average representation of the integrand as follows: (18)∫-π/2π/2(z-d+mrcos⁡θ-r22βs)3/2dθ≈(z-d+23mr-r22βs)3/2π.
The approximation provides relative error within 1.16 percent. So (17) yields(19)(FN)z1=43πE′η2β1/2∫z2∫r(z-d+23mr-r22βs)3/2×(1+r2βs2)-1/2ϕ2(z2)rdrdz2.dFN, the normal force on all asperities of the differential area dA′ at height z1 by all asperities on surface S2, can be found by(20)dFN=43πE′η1η2bβ1/2×∫z2∫r(z-d+23mr-r22βs)3/2(1+r2βs2)-1/2×ϕ1(z1)ϕ2(z2)rdrdz1dz2dx.
The total normal force on all asperities of the differential area dA′ due to all asperities on surface S2 is obtained as follows(21)(FN)dA′=43πE′η1η2bβ1/2∫z∫r(z-d+23mr-r22βs)3/2×(1+r2βs2)-1/2ϕ(z)rdrdzdx,
where ϕ(z) is the density function for summit height sum distribution of asperity summits on S1 and S2. The total normal force on all asperities of surface S1 due to all asperities on surface S2 can be found by integration along x(22)FN=43πE′η1η2bβ1/2∫x∫z∫r(z-d+23mr-r22βs)3/2×(1+r2βs2)-1/2ϕ(z)rdrdzdx.
Every length parameter in (22) is normalized with respect to the standard deviation of asperity height sum. Define s=z/σ, h=d/σ and consider hereafter r, β, and βs as normalized values using σ as the normalization parameter.  The limits of r are determined by the roots of the following equation (setting interference, w, equal to zero)(23)s-h+23mr-r22βs=0.
We find(24)r1=23βsm+(23βsm)2+2βs(s-h),r2=23βsm-(23βsm)2+2βs(s-h),
where(25)h=h0-m(x-x0).h0 is the normalized value of d0 and x and x0 are now used to denote normalized values using σ 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 S1 and S2(26)FN(h0,βs,m,a)=Cπ2IN(h0,βs,m,a),
where(27)C=832πE′η1η2bβ1/2σ4,IN(h0,βs,m,a)=∫-(a/2)a/2(∫h∞∫0r1(s-h+23mr-r22βs)3/2×(1+r2βs2)-1/2e-s2/2rdrds+∫h∞∫0r2(s-h+23mr-r22βs)3/2×(1+r2βs2)-1/2e-s2/2rdrds)dx.
In the absence of a moment, the total normal force is [28](28)FN(h0,βs)=CπaIN(h0,βs),
where(29)IN(h0,βs)=∫h0∞∫02βs(s-h0)(s-h0-r22βs)3/2.×(1+r2βs2)-1/2e-s2/2rdrds.
So we have(30)IN(h0,βs,m,a)=2aIN(h0,βs).
The results of numerical solution confirm the above equation. For a plate of known size a×b and prescribed normal force and moment, one can solve the above equation along with the moment equation to find x0 and m. x0 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 x0 and other parameters is established since it greatly facilitates the solution of problems involving contact moment. x0 only depends on h0, m, and a and it is not a function of βs. We find, using a multistep optimization procedure, the following approximate equation for x0 over h0=1to4, m=3×10-7 to 3×10-5, and a=50000to250000(31)x0(h0,m,a)=∑i=14∑j=14∑k=15ci,j,k(m×105)i-1(h0)j-1(a×10-4)k-1.
Recall that the large values assigned to a are due to their normalization with respect to the standard deviation of asperity height sum, σ. ci,j,k values are given in the Table 1. Figures 6, 7, and 8, respectively, illustrate x0(h0,m) for a=150000, x0(h0,a) for m=2×10-5, and x0(m,a) for h0=1. The approximate function in (31) provides estimates of x0 to within 0.6 percent accuracy over the entire domain of h0,m, and a.Table 1
The values ofci,j,k.
kc1,1,kc1,2,kc1,3,kc1,4,k16.79643 × 103−1.32902 × 1046.43709 × 103−8.31116 × 1022−2.71370 × 1035.30015 × 103−2.53736 × 1033.24476 × 10233.53998 × 102−6.87494 × 1023.23027 × 102−4.07732 × 1014−1.82659 × 1013.48897 × 101−1.60638 × 1012.00326 × 10053.1917 × 10−1−6.00360 × 10−12.72020 × 10−1−3.35900 × 10−2kc2,1,kc2,2,kc2,3,kc2,4,k1−2.74514 × 1045.28941 × 104−2.46289 × 1043.05682 × 10321.09554 × 104−2.10216 × 1049.65395 × 103−1.18604 × 1033−1.36820 × 1032.73353 × 103−1.21540 × 1031.47538 × 10247.35955 × 101−1.36314 × 1026.01116 × 101−7.20771 × 1005−1.28057 × 1002.33444 × 100−1.01436 × 1001.20460 × 10−1kc3,1,kc3,2,kc3,3,kc3,4,k11.80741 × 104−3.37842 × 1041.49086 × 104−1.75153 × 1032−7.20575 × 1031.33487 × 104−5.80095 × 1036.74219 × 10239.37846 × 102−1.70451 × 1037.25202 × 102−8.31789 × 1014−4.83447 × 1018.48724 × 101−3.56787 × 1014.02650 × 10058.30100 × 10−1−1.45519 × 1005.99160 × 10−1−6.71400 × 10−2kc4,1,kc4,2,kc4,3,kc4,4,k1−2.80312 × 1035.05179 × 103−2.10227 × 1032.30629 × 10221.11487 × 103−1.98646 × 1038.13468 × 102−8.81519 × 1013−1.44454 × 1022.52229 × 102−1.01106 × 1021.07859 × 10147.34839 × 100−1.25728 × 1014.94501 × 100−5.19150 × 10−15−1.26410 × 10−12.16950 × 10−1−8.31500 × 10−28.81712 × 10−3Figure 5
Differential area(dA′=bdx) on surface S1.Figure 6
x0(h0,m) for a=150000.Figure 7
x0(h0,a) for m=2×10-5.Figure 8
x0(m,a) for h0=1.


## 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 surfaceS2 (Figures 9 and 10). We are interested in formulating the cumulative effect of x-component of tangential force after applying the moment. Hereafter as we generate result for the x-component of the tangential force, we will refer to this as the “tangential force” and denote the force component Fx. 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.Figure 9
Schematic showing the tangential components of contact force exerted by two asperities ofS2 on an asperity on S1.Figure 10
Differential area(dA=rdrdθ) and  x-component of tangential force.It can be shown that the total tangential force (alongx) on an asperity of the differential area dA′ at height z1 by all asperities on surface S2 is found by(32)(Fx)z1=43E′η2β1/2βs∫z2∫r∫θ(z-d+mrcos⁡θ-r22βs)3/2×(1+r2βs2)-1/2cos⁡θϕ2(z2)r2drdθdz2.
The integral (32) can be reduced through the following approximation of the θ-integration over the range of -π/2 to π/2:(33)∫-π/2π/2(z-d+mrcos⁡θ-r22βs)3/2cos⁡θdθ≈2(z-d+45mr-r22βs)3/2.
The approximation provides a relative error less than 0.83 percent. We find(34)(Fx)z1=83E′η2β1/2βs∫z2∫r(z-d+45mr-r22βs)3/2×(1+r2βs2)-1/2ϕ2(z2)r2drdz2.dFx, the tangential force on all asperities in the differential area dA′ at height z1 by all asperities on surface S2, can be found by(35)dFx=83E′η1η2bβ1/2βs×∫z2∫r(z-d+45mr-r22βs)3/2(1+r2βs2)-1/2×ϕ1(z1)ϕ2(z2)r2drdz1dz2dx.
The total tangential force on all asperities of the differential area dA′ due to all asperities on surface S2 is obtained as follows:(36)(Fx)dA′=83E′η1η2bβ1/2βs×∫z∫r(z-d+45mr-r22βs)3/2×(1+r2βs2)-1/2ϕ(z)r2drdzdx.
The total tangential force on all asperities of surface S1 due to all asperities on surface S2 can be found by integration along x:(37)Fx=83E′η1η2bβ1/2βs×∫x∫z∫r(z-d+45mr-r22βs)3/2×(1+r2βs2)-1/2ϕ(z)r2drdzdx.
For a Gaussian distribution of asperity height sum the total tangential force between surfaces S1 and S2 in normalized form is(38)Fx(h0,βs,m,a)=CβsIx(h0,βs,m,a),
where (39)Ix(h0,βs,m,a)=∫-(a/2)a/2(∫h∞∫0r1(s-h+45mr-r22βs)3/2×(1+r2βs2)-1/2e-s2/2r2drds+∫h∞∫0r2(s-h+45mr-r22βs)3/2×(1+r2βs2)-1/2e-s2/2r2drds)dx
and C and r1, r2 are defined in (27) and (24), respectively.


## 5. Contact Moment

Contact moment about they-axis consists of two terms: (1) sum of moments about each asperity (M1) and (2) moment of normal force about the y-axis (M2) due to movement of the point of exertion of the resultant normal force (xN), as shown in Figure 1. First we consider M1 to find total moment about all asperities. Moment about the y-axis on an asperity in the differential area dA′ at height z1 due to all asperities at height z2 confined in (rdr,dθ) and located at r is(40)(dM1)z1=43E′η2β1/2(z-d+mrcos⁡θ-r22βs)3/2×(1+r2βs2)-1/2cos⁡θϕ2(z2)r2drdθdz2.
The total moment on an asperity of the differential area dA′ at height z1 by all asperities on surface S2 is found by(41)(M1)z1=83E′η2β1/2∫z2∫r(z-d+45mr-r22βs)3/2×(1+r2βs2)-1/2ϕ2(z2)r2drdz2,
where we have employed the approximation in (33).  dM, the moment on all asperities of the differential area dA′ at height z1 by all asperities on surface S2, can be found by(42)dM1=83E′η1η2bβ1/2×∫z2∫r(z-d+45mr-r22βs)3/2(1+r2βs2)-1/2×ϕ1(z1)ϕ2(z2)r2drdz1dz2dx.
The total moment on all asperities in the differential area dA′due to all asperities on surface S2 is obtained as follows:(43)(M1)dA′=83E′η1η2bβ1/2×∫z∫r(z-d+45mr-r22βs)3/2×(1+r2βs2)-1/2ϕ(z)r2drdzdx.
The total moment on all asperities of surface S1 due to all asperities on surface S2 can be found by integration along x(44)M1=83E′η1η2bβ1/2×∫x∫z∫r(z-d+45mr-r22βs)3/2×(1+r2βs2)-1/2ϕ(z)r2drdzdx.
For a Gaussian distribution of asperity height sum the total moment between surfaces S1 and S2 is
(45)M1(h0,βs,m,a)=CσIM1(h0,βs,m,a),
where IM1(h0,βs,m,a)=Ix(h0,βs,m,a) is given by (39) and C and r1, r2 are defined in (27) and (24), respectively. The second term of moment (M2) due to shifting of the location of the resultant normal load is(46)M2(h0,βs,m,a)=Cπσ2IM2(h0,βs,m,a),
where (47)IM2(h0,βs,m,a)=∫-(a/2)a/2(∫h∞∫0r1(s-h+23mr-r22βs)3/2×(1+r2βs2)-1/2e-s2/2rdrds+∫h∞∫0r2(s-h+23mr-r22βs)3/2×(1+r2βs2)-1/2e-s2/2rdrds)xdx.
Therefore, the total moment between surfaces S1 and S2 can be written as(48)M(h0,βs,m,a)=CσIM(h0,βs,m,a),
where (49)IM(h0,βs,m,a)=IM1(h0,βs,m,a)+π2IM2(h0,βs,m,a).
Numerical results show that the first moment term IM1 (order of 105) is negligible in comparison to the second term IM2 (order of 1012). The force center, xN, is obtained by(50)xN(h0,βs,m,a)=IM(h0,βs,m,a)2aIN(h0,βs).
Friction coefficient due to elastic deformation is equal to(51)μ(h0,βs,m,a)=Ix(h0,βs,m,a)πaβsIN(h0,βs).IN(h0,βs), Ix(h0,βs,m,a), and IM(h0,βs,m,a) in the above two equations are given by (29), (39), and (49), respectively.


## 6. Results

Figures11 and 12, respectively, depict the dimensionless normal and tangential contact force components, IN and Ix, versus dimension length, x, for h0=1, βs=500, and plate dimensionless length a=250000. 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 IM(h0,βs,m,a), in each case holding two of the parameters fixed.  Figure 13 shows IM(h0,βs) for m=2×10-5  and a=150000, Figure 14 depicts IM(h0,m) for βs=500 and a=150000, and Figure 15 represents IM(h0,a) for βs=500 and m=2×10-5. In these figures h0 corresponds to an applied normal force such that a small value of h0 represents a large normal force and large h0 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 βs, slope m, and nominal dimension a.  Similar observation can be made about the dimensionless contact force Ix. Figures 16, 17 and 18, respectively, show Ix(h0,βs), Ix(h0,m) and Ix(h0,a) for the same cases.Figure 11
Dimensionless normal force,IN, versus x for  h0=1, βs=500, and a=250000.Figure 12
Dimensionless tangential force,Ix, versus x for h0=1, βs=500, and a=250000.Figure 13
IM(h0,βs) for m=2×10-5  and a=150000.Figure 14
IM(h0,m) for βs=500 and a=150000.Figure 15
IM(h0,a) for βs=500  and m=2×10-5.Figure 16
Ix(h0,βs) for m=2×10-5  and a=150000.Figure 17
Ix(h0,m) for βs=500  and a= 150000.Figure 18
Ix(h0,a) for βs=500  and m=2×10-5.Figures19, 20, and 21, respectively, demonstrate the relation between the location of the resultant normal contact force, xN(h0,βs,m,a),while, as before, holding two of the parameters constant. These are shown as xN(h0,βs),xN(h0,m), and xN(h0,a). Figures 22, 23, and 24, respectively, represent μ(h0,βs),μ(h0,m), and μ(h0,a). Figures 19 and 22 show that the force center and elastic friction coefficient are independent of βs.Figure 19
xN(h0,βs) for m=2×10-5 and a=150000.Figure 20
xN(h0,m) for βs=500 and a=150000.Figure 21
xN(h0,a) for βs=500 and m=2×10-5.Figure 22
μ(h0,βs) for m=2×10-5 and a=150000.Figure 23
μ(h0,m) for βs=500 and a=150000.Figure 24
μ(h0,a) for βs=500 and m=2×10-5.


## 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 forIM and Ix:(52)IMa(h0,βs,m,a)=α1(βs,m,a)eα2(m,a)h0α3(m,a),α1(βs,m,a)=f1(m)(a×10-4)f2(m)βs-f3(m)(a×10-4)f4(m),α2(m,a)=-(f5(m)(a×10-4)+f6(m)),α3(m,a)=f7(m)(a×10-4)+f8(m),
where(53)f1(m)=6.30326×1012m1.33929,f2(m)=-2.03970×104m+3.05009,f3(m)=5.96853×1011m1.20289,f4(m)=-2.94562×104m+3.22846,f5(m)=6.52400×102m-1.31000×10-3,f6(m)=9.49280×10-1m-1.38200×10-2,f7(m)=-6.08955×106m1.93643,f8(m)=1.65800×103m+1.61790,
as well as(54)Ixa(h0,βs,m,a)=α1(βs,m,a)eα2(m,a)h0α3(m,a),α1(βs,m,a)=g1(m)(a×10-4)g2(m)(βs)g3(m),α2(m,a)=-(g4(m)(a×10-4-g5(m))2+g6(m)),α3(m,a)=g7(m)(a×10-4-g8(m))2+g9(m),
where(55)g1(m)=9.39400×10-2(m×105)8.73060×10-1,g2(m)=9.92810×10-1(m×105)7.14000×10-2,g3(m)=-3.88000×10-4(m×105)+2.00278,g4(m)=-5.35496×10-4(m×105-1.79100)2+4.26462,g5(m)=2.98406(m×105-2.67800)2+8.02206,g6(m)=6.78500×10-2(m×105-1.49100)2+1.33360,g7(m)=6.36775×10-5(m×105-3.36200)2-3.94300×10-4,g8(m)=4.61410(m×105-2.30800)2-1.03711×101,g9(m)=-3.40500×10-2(m×105-1.57000)2-1.56729.
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:(56)EM(h0,βs,m,a)=IM(h0,βs,m,a)-IMa(h0,βs,m,a)IM(h0,βs,m,a)×100.
The error is plotted over the ranges h0=1to4, and βs=100to2000 for m=2×10-5 and a=150000 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 h0,βs,m, and a.Figure 25
EM(h0,βs) for m=2×10-5  and a=150000,(52).A similar study for error associated with the approximation ofEx(h0,βs,m,a) yields accuracy within 7 percent over the entire range of h0,βs,m, and a. Figure 26 depicts Ex(h0,βs) over h0=1to4 and βs=100to2000 for m=2×10-5 and a=150000.Figure 26
Ex(h0,βs) for m=2×10-5  and a=150000,(54).Approximate functions for the force center,xN, and elastic friction coefficient, μ, are(57)xNa(h0,m,a)=IMa(h0,βs,m,a)2aINa(h0,βs),μa(h0,m,a)=Ixa(h0,βs,m,a)πaβsINa(h0,βs),
where INa is given by [28](58)INa(h0,βs)=(2.95810×10-1βs-1.11760×10-1)eα1h0α2,α1=-1.35102,α2=1.55700.
The approximate functions in (57) provide estimates of xN and μ 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 shoulder-shoulder 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+x and -x half-plane 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 3×10-5.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.



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*Source: 739562-2011-10-18.xml*