Closed-Form 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 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.

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:𝑑=𝑑₀−𝑚𝑥−𝑥₀, (1) 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 Figure 2. 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 as𝑓_𝑎=43𝐸^′𝛽(𝑟)^1/2𝑤^3/2, (2) where 𝐸^′ is the combined Young’s modulus of elasticity for the two surfaces1=𝐸′1−𝜈²₁𝐸₁+1−𝜈²₂𝐸₂, (3)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 Figures 2 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 that𝑟₁=𝛽𝛽₂𝑟,𝑟₂=𝛽𝛽₁𝑟, (4) where 𝛽 is the equivalent summit radius of the asperities given by1𝛽=1𝛽₁+1𝛽₂. (5) Alternatively, the offset may be written as follows:𝑟₁=𝛽₁𝛽_𝑠𝑟,𝑟₂=𝛽₂𝛽_𝑠𝑟,𝛽_𝑠=𝛽₁+𝛽₂, (6) 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 find𝛽(𝑟)𝑟=𝛽1+²𝛽²_𝑠^3/2. (7) The approach, 𝑤, is (Figures 2 and 3)𝑤=𝑤₁cos𝛼, (8) 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:𝑤₁=𝑧₁+𝑧₂−𝑑^′−𝑓₁𝑟₁−𝑓₂𝑟₂, (9)𝑟cos𝛼=1+²𝛽²_𝑠^−1/2𝑟,sin𝛼=1+²𝛽²_𝑠^−1/2𝑟𝛽_𝑠, (10)𝑓₁ and 𝑓₂ in (9) are the asperity shape functions of surfaces 𝑆₁ and 𝑆₂, respectively, and 𝑑^′the local mean plane separation:𝑑^′=𝑑−𝑚𝑟cos𝜃, (11) 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 to𝑤₁=𝑧₁+𝑧₂−𝑑^′−𝑟²2𝛽_𝑠. (12) Therefore, the asperity interference can be found by combining (8), (10), (11), and (12) that yield𝑤=𝑧₁+𝑧₂𝑟−𝑑+𝑚𝑟cos𝜃−²2𝛽_𝑠𝑟1+²𝛽²_𝑠^−1/2. (13) Finally, substitution from (7) and (13) in (2) yields𝑓_𝑎=43𝐸^′𝛽^1/2𝑧₁+𝑧₂𝑟−𝑑+𝑚𝑟cos𝜃−²2𝛽_𝑠^3/2. (14)

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 𝑓_𝑁 and 𝑓_𝑡 the components of the asperity contact force along the normal and tangential directions, respectively, we find, with the help of (10) and (14), 𝑓_𝑁=43𝐸^′𝛽^1/2𝑟𝑧−𝑑+𝑚𝑟cos𝜃−²2𝛽_𝑠^3/2𝑟1+²𝛽²_𝑠^−1/2,𝑓_𝑡=43𝐸^′𝛽^1/2𝑟𝑧−𝑑+𝑚𝑟cos𝜃−²2𝛽_𝑠^3/2𝑟1+²𝛽²_𝑠^−1/2𝑟𝛽_𝑠, (15) where z = z₁+ z₂ the height sum of two asperities.

3. Normal Force

All the normal components of various contact forces (𝑑𝐹^′_𝑁𝑠) are parallel (Figure 4) and can be algebraically summed by statistical means to obtain the total normal force of one surface on another.

Let 𝑑𝐴 (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: 𝑑𝐴=𝑟𝑑𝑟𝑑𝜃,𝑑𝐴^′=𝑏𝑑𝑥. (16) 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:𝐹_𝑁𝑧1=43𝐸^′𝜂₂𝛽^1/2_{𝑧2𝑟𝜃}𝑟𝑧−𝑑+𝑚𝑟cos𝜃−²2𝛽_𝑠^3/2×𝑟1+²𝛽²_𝑠^−1/2𝜙₂𝑧₂𝑟𝑑𝑟𝑑𝜃𝑑𝑧₂, (17) 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: ^𝜋/2_−𝜋/2𝑟𝑧−𝑑+𝑚𝑟cos𝜃−²2𝛽_𝑠^3/2≈2𝑑𝜃𝑧−𝑑+3𝑟𝑚𝑟−²2𝛽_𝑠^3/2𝜋. (18) The approximation provides relative error within 1.16 percent. So (17) yields𝐹_𝑁𝑧1=43𝜋𝐸′𝜂₂𝛽^1/2_𝑧2𝑟2𝑧−𝑑+3𝑟𝑚𝑟−²2𝛽_𝑠^3/2×𝑟1+²𝛽²_𝑠^−1/2𝜙₂𝑧₂𝑟𝑑𝑟𝑑𝑧₂. (19)𝑑𝐹_𝑁, the normal force on all asperities of the differential area 𝑑𝐴^′ at height 𝑧₁ by all asperities on surface 𝑆₂, can be found by𝑑𝐹_𝑁=43𝜋𝐸′𝜂₁𝜂₂𝑏𝛽^1/2×_𝑧2𝑟2𝑧−𝑑+3𝑟𝑚𝑟−²2𝛽_𝑠^3/2𝑟1+²𝛽²_𝑠^−1/2×𝜙₁𝑧₁𝜙₂𝑧₂𝑟𝑑𝑟𝑑𝑧₁𝑑𝑧₂𝑑𝑥. (20) The total normal force on all asperities of the differential area 𝑑𝐴^′ due to all asperities on surface 𝑆₂ is obtained as follows𝐹_{𝑁𝑑𝐴′}=43𝜋𝐸^′𝜂₁𝜂₂𝑏𝛽^1/2_𝑧𝑟2𝑧−𝑑+3𝑟𝑚𝑟−²2𝛽_𝑠^3/2×𝑟1+²𝛽²_𝑠^−1/2𝜙(𝑧)𝑟𝑑𝑟𝑑𝑧𝑑𝑥, (21) 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 𝑥𝐹_𝑁=43𝜋𝐸^′𝜂₁𝜂₂𝑏𝛽^1/2_𝑥𝑧𝑟2𝑧−𝑑+3𝑟𝑚𝑟−²2𝛽_𝑠^3/2×𝑟1+²𝛽²_𝑠^−1/2𝜙(𝑧)𝑟𝑑𝑟𝑑𝑧𝑑𝑥. (22) 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 𝛽_𝑠 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)2𝑠−ℎ+3𝑟𝑚𝑟−²2𝛽_𝑠=0. (23) We find𝑟₁=23𝛽_𝑠𝑚+23𝛽_𝑠𝑚²+2𝛽_𝑠(),𝑟𝑠−ℎ₂=23𝛽_𝑠𝑚−23𝛽_𝑠𝑚²+2𝛽_𝑠(𝑠−ℎ), (24) whereℎ=ℎ₀−𝑚𝑥−𝑥₀. (25)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 𝑆₂𝐹_𝑁ℎ₀,𝛽_𝑠,𝑚,𝑎=𝐶𝜋2𝐼_𝑁ℎ₀,𝛽_𝑠,𝑚,𝑎, (26) where8𝐶=3√𝐸2𝜋^′𝜂₁𝜂₂𝑏𝛽^1/2𝜎⁴,𝐼_𝑁ℎ₀,𝛽_𝑠,𝑚,𝑎=^𝑎/2_−(𝑎/2)^∞_ℎ^𝑟1₀2𝑠−ℎ+3𝑟𝑚𝑟−²2𝛽_𝑠^3/2×𝑟1+²𝛽²_𝑠^−1/2𝑒^−𝑠2/2+𝑟𝑑𝑟𝑑𝑠_∞ℎ𝑟202𝑠−ℎ+3𝑟𝑚𝑟−²2𝛽_𝑠^3/2×𝑟1+²𝛽²_𝑠^−1/2𝑒^−𝑠2/2𝑟𝑑𝑟𝑑𝑠𝑑𝑥. (27) In the absence of a moment, the total normal force is [28]𝐹_𝑁ℎ₀,𝛽_𝑠=𝐶𝜋𝑎𝐼_𝑁ℎ₀,𝛽_𝑠, (28) where𝐼_𝑁ℎ₀,𝛽_𝑠=^∞_ℎ0^{2𝛽𝑠(𝑠−ℎ0)}₀𝑠−ℎ₀−𝑟²2𝛽_𝑠^3/2.×𝑟1+²𝛽²_𝑠^−1/2𝑒^−𝑠2/2𝑟𝑑𝑟𝑑𝑠. (29) So we have𝐼_𝑁ℎ₀,𝛽_𝑠,𝑚,𝑎=2𝑎𝐼_𝑁ℎ₀,𝛽_𝑠. (30) 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 𝛽_𝑠. 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=50000to250000𝑥₀ℎ₀=,𝑚,𝑎_{44𝑖=15𝑗=1𝑘=1}𝑐_{𝑖,𝑗,𝑘}𝑚×10^5𝑖−1ℎ₀^𝑗−1𝑎×10^−4𝑘−1. (31) Recall that the large values assigned to a are due to their normalization with respect to the standard deviation of asperity height sum, 𝜎. 𝑐_{𝑖,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 of 𝑐_{𝑖,𝑗,𝑘}.

Figure 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 surface 𝑆₂ (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 𝐹_𝑥. 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 of 𝑆2 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 (along 𝑥) on an asperity of the differential area 𝑑𝐴^′ at height 𝑧₁ by all asperities on surface 𝑆₂ is found by𝐹_𝑥𝑧1=43𝐸^′𝜂₂𝛽^1/2𝛽_{𝑠𝑧2𝑟𝜃}𝑟𝑧−𝑑+𝑚𝑟cos𝜃−²2𝛽_𝑠^3/2×𝑟1+²𝛽²_𝑠^−1/2cos𝜃𝜙₂𝑧₂𝑟²𝑑𝑟𝑑𝜃𝑑𝑧₂. (32) The integral (32) can be reduced through the following approximation of the 𝜃-integration over the range of −𝜋/2 to 𝜋/2:^𝜋/2_−𝜋/2𝑟𝑧−𝑑+𝑚𝑟cos𝜃−²2𝛽_𝑠^3/24cos𝜃𝑑𝜃≈2𝑧−𝑑+5𝑟𝑚𝑟−²2𝛽_𝑠^3/2. (33) The approximation provides a relative error less than 0.83 percent. We find𝐹_𝑥𝑧1=83𝐸^′𝜂₂𝛽^1/2𝛽_{𝑠𝑧2𝑟}4𝑧−𝑑+5𝑟𝑚𝑟−²2𝛽_𝑠^3/2×𝑟1+²𝛽²_𝑠^−1/2𝜙₂𝑧₂𝑟²𝑑𝑟𝑑𝑧₂. (34)𝑑𝐹_𝑥, the tangential force on all asperities in the differential area 𝑑𝐴^′ at height 𝑧₁ by all asperities on surface 𝑆₂, can be found by𝑑𝐹_𝑥=83𝐸^′𝜂₁𝜂₂𝑏𝛽^1/2𝛽_𝑠×_𝑧2𝑟4𝑧−𝑑+5𝑟𝑚𝑟−²2𝛽_𝑠^3/2𝑟1+²𝛽²_𝑠^−1/2×𝜙₁𝑧₁𝜙₂𝑧₂𝑟²𝑑𝑟𝑑𝑧₁𝑑𝑧₂𝑑𝑥. (35) The total tangential force on all asperities of the differential area 𝑑𝐴^′ due to all asperities on surface 𝑆₂ is obtained as follows:𝐹_{𝑥𝑑𝐴′}=83𝐸′𝜂₁𝜂₂𝑏𝛽^1/2𝛽_𝑠×_𝑧𝑟4𝑧−𝑑+5𝑟𝑚𝑟−²2𝛽_𝑠^3/2×𝑟1+²𝛽²_𝑠^−1/2𝜙(𝑧)𝑟²𝑑𝑟𝑑𝑧𝑑𝑥. (36) The total tangential force on all asperities of surface 𝑆₁ due to all asperities on surface 𝑆₂ can be found by integration along 𝑥:𝐹_𝑥=83𝐸′𝜂₁𝜂₂𝑏𝛽^1/2𝛽_𝑠×_𝑥𝑧𝑟4𝑧−𝑑+5𝑟𝑚𝑟−²2𝛽_𝑠^3/2×𝑟1+²𝛽²_𝑠^−1/2𝜙(𝑧)𝑟²𝑑𝑟𝑑𝑧𝑑𝑥. (37) For a Gaussian distribution of asperity height sum the total tangential force between surfaces 𝑆₁ and 𝑆₂ in normalized form is𝐹_𝑥ℎ₀,𝛽_𝑠=𝐶,𝑚,𝑎𝛽_𝑠𝐼_𝑥ℎ₀,𝛽_𝑠,𝑚,𝑎, (38) where 𝐼_𝑥ℎ₀,𝛽_𝑠,𝑚,𝑎=^𝑎/2_−(𝑎/2)^∞_ℎ^𝑟1₀4𝑠−ℎ+5𝑟𝑚𝑟−²2𝛽_𝑠^3/2×𝑟1+²𝛽²_𝑠^−1/2𝑒^−𝑠2/2𝑟²+𝑑𝑟𝑑𝑠_∞ℎ𝑟204𝑠−ℎ+5𝑟𝑚𝑟−²2𝛽_𝑠^3/2×𝑟1+²𝛽²_𝑠^−1/2𝑒^−𝑠2/2𝑟²𝑑𝑟𝑑𝑠𝑑𝑥 (39) and 𝐶 and 𝑟₁, 𝑟₂ are defined in (27) and (24), respectively.

5. Contact Moment

Contact moment about the y-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 (𝑥_𝑁), 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 is𝑑𝑀_1𝑧1=43𝐸^′𝜂₂𝛽^1/2𝑟𝑧−𝑑+𝑚𝑟cos𝜃−²2𝛽_𝑠^3/2×𝑟1+²𝛽²_𝑠^−1/2cos𝜃𝜙₂𝑧₂𝑟²𝑑𝑟𝑑𝜃𝑑𝑧₂. (40) The total moment on an asperity of the differential area 𝑑𝐴^′ at height 𝑧₁ by all asperities on surface 𝑆₂ is found by𝑀_1𝑧1=83𝐸^′𝜂₂𝛽^1/2_𝑧2𝑟4𝑧−𝑑+5𝑟𝑚𝑟−²2𝛽_𝑠^3/2×𝑟1+²𝛽²_𝑠^−1/2𝜙₂𝑧₂𝑟²𝑑𝑟𝑑𝑧₂, (41) 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 by𝑑𝑀₁=83𝐸′𝜂₁𝜂₂𝑏𝛽^1/2×_𝑧2𝑟4𝑧−𝑑+5𝑟𝑚𝑟−²2𝛽_𝑠^3/2𝑟1+²𝛽²_𝑠^−1/2×𝜙₁𝑧₁𝜙₂𝑧₂𝑟²𝑑𝑟𝑑𝑧₁𝑑𝑧₂𝑑𝑥. (42) The total moment on all asperities in the differential area 𝑑𝐴^′due to all asperities on surface 𝑆₂ is obtained as follows:𝑀_1𝑑𝐴′=83𝐸^′𝜂₁𝜂₂𝑏𝛽^1/2×_𝑧𝑟4𝑧−𝑑+5𝑟𝑚𝑟−²2𝛽_𝑠^3/2×𝑟1+²𝛽²_𝑠^−1/2𝜙(𝑧)𝑟²𝑑𝑟𝑑𝑧𝑑𝑥. (43) The total moment on all asperities of surface 𝑆₁ due to all asperities on surface 𝑆₂ can be found by integration along 𝑥𝑀₁=83𝐸^′𝜂₁𝜂₂𝑏𝛽^1/2×_𝑥𝑧𝑟4𝑧−𝑑+5𝑟𝑚𝑟−²2𝛽_𝑠^3/2×𝑟1+²𝛽²_𝑠^−1/2𝜙(𝑧)𝑟²𝑑𝑟𝑑𝑧𝑑𝑥. (44) For a Gaussian distribution of asperity height sum the total moment between surfaces 𝑆₁ and 𝑆₂ is 𝑀₁ℎ₀,𝛽_𝑠,𝑚,𝑎=𝐶𝜎𝐼_𝑀1ℎ₀,𝛽_𝑠,𝑚,𝑎, (45) where 𝐼_𝑀1(ℎ₀,𝛽_𝑠,𝑚,𝑎)=𝐼_𝑥(ℎ₀,𝛽_𝑠,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 is𝑀₂ℎ₀,𝛽_𝑠,𝑚,𝑎=𝐶𝜋𝜎2𝐼_𝑀2ℎ₀,𝛽_𝑠,𝑚,𝑎, (46) where 𝐼_𝑀2ℎ₀,𝛽_𝑠,𝑚,𝑎=^𝑎/2_−(𝑎/2)^∞_ℎ^𝑟1₀2𝑠−ℎ+3𝑟𝑚𝑟−²2𝛽_𝑠^3/2×𝑟1+²𝛽²_𝑠^−1/2𝑒^−𝑠2/2+𝑟𝑑𝑟𝑑𝑠_∞ℎ𝑟202𝑠−ℎ+3𝑟𝑚𝑟−²2𝛽_𝑠^3/2×𝑟1+²𝛽²_𝑠^−1/2𝑒^−𝑠2/2𝑟𝑑𝑟𝑑𝑠𝑥𝑑𝑥. (47) Therefore, the total moment between surfaces 𝑆₁ and 𝑆₂ can be written as𝑀ℎ₀,𝛽_𝑠,𝑚,𝑎=𝐶𝜎𝐼_𝑀ℎ₀,𝛽_𝑠,𝑚,𝑎, (48) where 𝐼_𝑀ℎ₀,𝛽_𝑠,𝑚,𝑎=𝐼_𝑀1ℎ₀,𝛽_𝑠+𝜋,𝑚,𝑎2𝐼_𝑀2ℎ₀,𝛽_𝑠,𝑚,𝑎. (49) Numerical results show that the first moment term 𝐼_𝑀1 (order of 10⁵) is negligible in comparison to the second term 𝐼_𝑀2 (order of 10¹²). The force center, 𝑥_𝑁, is obtained by𝑥_𝑁ℎ₀,𝛽_𝑠=𝐼,𝑚,𝑎^𝑀ℎ₀,𝛽_𝑠,𝑚,𝑎2𝑎𝐼_𝑁ℎ₀,𝛽_𝑠. (50) Friction coefficient due to elastic deformation is equal to𝜇ℎ₀,𝛽_𝑠=𝐼,𝑚,𝑎^𝑥ℎ₀,𝛽_𝑠,𝑚,𝑎𝜋𝑎𝛽_𝑠𝐼_𝑁ℎ₀,𝛽_𝑠. (51)𝐼_𝑁(ℎ₀,𝛽_𝑠), 𝐼_𝑥(ℎ₀,𝛽_𝑠,𝑚,𝑎), and 𝐼_𝑀(ℎ₀,𝛽_𝑠,𝑚,𝑎) 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, 𝐼_𝑁 and 𝐼_𝑥, versus dimension length, 𝑥, for ℎ₀=1, 𝛽_𝑠=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 𝐼_𝑀(ℎ₀,𝛽_𝑠,𝑚,𝑎), in each case holding two of the parameters fixed. Figure 13 shows 𝐼_𝑀(ℎ₀,𝛽_𝑠) for 𝑚=2×10⁻⁵ and 𝑎=150000, Figure 14 depicts 𝐼_𝑀(ℎ₀,𝑚) for 𝛽_𝑠=500 and 𝑎=150000, and Figure 15 represents 𝐼_𝑀(ℎ₀,𝑎) for 𝛽_𝑠=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 𝛽_𝑠, slope 𝑚, and nominal dimension 𝑎. Similar observation can be made about the dimensionless contact force 𝐼_𝑥. Figures 16, 17 and 18, respectively, show 𝐼_𝑥(ℎ₀,𝛽_𝑠), 𝐼_𝑥(ℎ₀,𝑚) and 𝐼_𝑥(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.

Figures 19, 20, and 21, respectively, demonstrate the relation between the location of the resultant normal contact force, 𝑥_𝑁(ℎ₀,𝛽_𝑠,𝑚,𝑎), while, as before, holding two of the parameters constant. These are shown as 𝑥_𝑁(ℎ₀,𝛽_𝑠), 𝑥_𝑁(ℎ₀,𝑚), and 𝑥_𝑁(ℎ₀,𝑎). Figures 22, 23, and 24, respectively, represent 𝜇(ℎ₀,𝛽_𝑠), 𝜇(ℎ₀,𝑚), and 𝜇(ℎ₀,𝑎). Figures 19 and 22 show that the force center and elastic friction coefficient are independent of 𝛽_𝑠.

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 for 𝐼_𝑀 and 𝐼_𝑥:𝐼_𝑀𝑎ℎ₀,𝛽_𝑠,𝑚,𝑎=𝛼₁𝛽_𝑠,𝑚,𝑎𝑒^{𝛼2(𝑚,𝑎)ℎ𝛼30(𝑚,𝑎)},𝛼₁𝛽_𝑠,𝑚,𝑎=𝑓₁(𝑚)𝑎×10^{−4𝑓2(𝑚)}𝛽_𝑠−𝑓₃(𝑚)𝑎×10^{−4𝑓4(𝑚)},𝛼₂(𝑚,𝑎)=−𝑓₅(𝑚)𝑎×10⁻⁴+𝑓₆(𝑚,𝛼₃)(𝑚,𝑎)=𝑓₇(𝑚)𝑎×10⁻⁴+𝑓₈(𝑚), (52) where𝑓₁(𝑚)=6.30326×10¹²𝑚^1.33929,𝑓₂(𝑚)=−2.03970×10⁴𝑓𝑚+3.05009,₃(𝑚)=5.96853×10¹¹𝑚^1.20289,𝑓₄(𝑚)=−2.94562×10⁴𝑓𝑚+3.22846,₅(𝑚)=6.52400×10²𝑚−1.31000×10⁻³,𝑓₆(𝑚)=9.49280×10⁻¹𝑚^{−1.38200×10−2},𝑓₇(𝑚)=−6.08955×10⁶𝑚^1.93643,𝑓₈(𝑚)=1.65800×10³𝑚+1.61790, (53) as well as𝐼_𝑥𝑎ℎ₀,𝛽_𝑠,𝑚,𝑎=𝛼₁𝛽_𝑠,𝑚,𝑎𝑒^{𝛼2(𝑚,𝑎)ℎ𝛼30(𝑚,𝑎)},𝛼₁𝛽_𝑠,𝑚,𝑎=𝑔₁(𝑚)𝑎×10^{−4𝑔2(𝑚)}𝛽_𝑠^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), (54) where𝑔₁(𝑚)=9.39400×10⁻²𝑚×10^{58.73060×10−1},𝑔₂(𝑚)=9.92810×10⁻¹𝑚×10^{57.14000×10−2},𝑔₃(𝑚)=−3.88000×10⁻⁴𝑚×10⁵𝑔+2.00278,₄(𝑚)=−5.35496×10⁻⁴𝑚×10⁵−1.79100²𝑔+4.26462,₅(𝑚)=2.98406𝑚×10⁵−2.67800²𝑔+8.02206,₆(𝑚)=6.78500×10⁻²𝑚×10⁵−1.49100²𝑔+1.33360,₇(𝑚)=6.36775×10⁻⁵𝑚×10⁵−3.36200²−3.94300×10⁻⁴,𝑔₈(𝑚)=4.61410𝑚×10⁵−2.30800²−1.03711×10¹,𝑔₉(𝑚)=−3.40500×10⁻²𝑚×10⁵−1.57000²−1.56729. (55) 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:𝐸_𝑀ℎ₀,𝛽_𝑠=𝐼,𝑚,𝑎^𝑀ℎ₀,𝛽_𝑠,𝑚,𝑎−𝐼_𝑀𝑎ℎ₀,𝛽_𝑠,𝑚,𝑎𝐼_𝑀ℎ₀,𝛽_𝑠,𝑚,𝑎×100. (56) The error is plotted over the ranges ℎ₀=1to4, and 𝛽_𝑠=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,𝛽_𝑠,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 of 𝐸_𝑥(ℎ₀,𝛽_𝑠,𝑚,𝑎) yields accuracy within 7 percent over the entire range of ℎ₀,𝛽_𝑠,𝑚, and 𝑎. Figure 26 depicts 𝐸_𝑥(ℎ₀,𝛽_𝑠) over ℎ₀=1to4 and 𝛽_𝑠=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, 𝑥_𝑁, and elastic friction coefficient, 𝜇, are𝑥_𝑁𝑎ℎ₀=𝐼,𝑚,𝑎^𝑀𝑎ℎ₀,𝛽_𝑠,𝑚,𝑎2𝑎𝐼_𝑁𝑎ℎ₀,𝛽_𝑠,𝜇_𝑎ℎ₀=𝐼,𝑚,𝑎^𝑥𝑎ℎ₀,𝛽_𝑠,𝑚,𝑎𝜋𝑎𝛽_𝑠𝐼_𝑁𝑎ℎ₀,𝛽_𝑠, (57) where I_Na is given by [28]𝐼_𝑁𝑎ℎ₀,𝛽_𝑠=2.95810×10⁻¹𝛽_𝑠−1.11760×10⁻¹𝑒^{𝛼1ℎ𝛼20},𝛼₁=−1.35102,𝛼₂=1.55700. (58) The approximate functions in (57) provide estimates of 𝑥_𝑁 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.

Nomenclature

𝑓𝑎:	Hertz contact force
𝐸′:	Combined Young’s modulus
𝛽1,2:	Dimensionless average summit radius of asperities on the surfaces 1 and 2
𝛽:	Combined asperity summit radius of curvature
𝛽(𝑟):	Equivalent asperity radius of curvature at a contact
𝑤:	Interference
𝑤1:	Interference defined by Greenwood and Tripp
𝛼:	Contact angle between two asperities
𝑓1,2:	Asperities shape functions
𝜂1:	Number of asperities per unit nominal area on 𝑆1
𝜂2:	Number of asperities per unit nominal area on 𝑆2
𝑓𝑁:	Asperity contact force along normal direction
𝑓𝑡:	Asperity contact force along tangential direction
𝜎:	Standard deviation of asperity height summation
𝐸1,2:	Young's moduli
𝜈1,2:	Poisson ratios
𝑑0:	Initial mean plane separation
𝑑:	Local mean plane separation
𝑑′:	Local mean plane separation around an asperity
ℎ0:	Dimensionless initial mean plane separation
ℎ:	Dimensionless local mean plane separation
𝑧1,2:	Heights of asperities measured from the mean asperity heights
𝑧:	Height sum of two asperities
𝑠:	Dimensionless asperity heights sum
ℎ:	Dimensionless mean separation
𝑟:	Tangential offset of the mating asperities
𝛽𝑠:	Dimensionless asperity summit radius of curvature sum
𝑚:	Mean plane slope of surface 𝑆1 due to moment
𝑥0:	Position of initial separation
𝐹𝑁:	Total normal contact force
𝐹𝑥:	Total tangential force
𝑀1:	Contact moment (first term)
𝐼𝑀1:	Dimensionless contact moment (first term)
𝑀2:	Contact moment (second term)
𝐼𝑀2:	Dimensionless contact moment (second term)
𝐼𝑁:	Dimensionless normal force
𝐼𝑥:	Dimensionless tangential force
𝐼𝑀:	Dimensionless contact moment
𝑥𝑁:	Location of force center
𝜇:	Elastic friction coefficient
𝑎,𝑏:	Length and width of rectangular surfaces
𝐼𝑥𝑎:	Approximate function for 𝐼𝑥
𝐼𝑀𝑎:	Approximate function for 𝐼𝑀
𝐼𝑁𝑎:	Approximate function for 𝐼𝑁
𝑥𝑁𝑎:	Approximate function for 𝑥𝑁
𝜇𝑎:	Approximate function for 𝜇
𝐸𝑥:	Percent error between 𝐼𝑥 and 𝐼𝑥𝑎
𝐸𝑀:	Percent error between 𝐼𝑀 and 𝐼𝑀𝑎.