Abstract

Three-dimensional elastic-plastic contact of two nominally flat rough surfaces is considered. Equations governing the shoulder-shoulder contact of asperities are derived based on the asperity constitutive relations from a finite element model of the elastic-plastic interaction proposed by Kogut and Etsion (2002), in which asperity scale constitutive relations are derived using piecewise approximate functions. An analytical fusion technique is developed to combine the piecewise asperity level constitutive relations. Shoulder-shoulder asperity contact yields a slanted contact force consisting of two components, one in the normal direction and a half-plane tangential component. Statistical summation of the asperity level contact force components and asperity level contact area results in the total contact force and total contact area formulae between two rough surfaces. Approximate equations are developed in closed form for contact force components and contact area.

1. Introduction

The GW theory [1] of contact between nominally flat rough surfaces has been preferred by numerous researchers as it benefits from relatively simple representation of a rough surface. It is based on a statistical account of a rough surface in which three parameters are identified. These include (1) standard deviation of asperity height distribution, σ; (2) average asperity summit radius of curvature, β; (3) area asperity density, η. The GW model treats both elastic and plastic contacts and it presumes that asperity contacts occur independent of each other, that is, no influence from adjacent local contacts on a given asperity contact. In the treatment of elastic interaction, GW model relies on the presumption of the Hertz contact. The GW model has been followed by numerous other studies, as summarized in the review paper by Adams and Nosonovsky [2], which take into account various aspects of surface topography such as contact between two rough surfaces, nonuniform radii of the asperities, non-Gaussian distributions of the asperity summit heights, anisotropy, and plasticity. The work proposed by Greenwood and Tripp [3] extended the GW model to contact between two rough surfaces. Greenwood and Tripp (GT) demonstrated that the contact between two rough surfaces could be treated as that between a flat and a rough surface if the composite statistics of the two surfaces are employed. Namely, Gaussian distribution of heights is in terms of the height sum distribution of the surfaces and the standard deviation of asperity height sum distribution is employed in the formulation of contact. This simplification required a modified function related to the interference of asperities involving the integration of interference function over the range of asperity tangential offset. McCool [4] extended GW microcontact model to include skewness in the distribution of surface summit heights and the presence of a surface coating of prescribed thickness and compliance. Recently, Sepehri and Farhang [5] developed an elastic model for two nominally flat rough surfaces in which asperity shoulder-shoulder contact was permitted to derive formulae for elastic contact of two rough surfaces.

A major contribution to the modeling of nominally flat rough surfaces is the work in 1987 by Chang et al. [6], who proposed a method for treating elastic-plastic contact of rough surfaces. This model, widely known as the CEB model, is based on volume conservation of an asperity during its plastic flow. The CEB model enjoys the simplicity of the GW model while providing a predictive tool for contact problems not amenable to an elastic contact assumption. Many publications have appeared since the CEB model that are based on the CEB or are inspired by the method employed by the CEB model [7–35]. Many researchers have employed statistical models for the elastic-plastic contact of rough surfaces [7–17]. Others have advocated the use of deterministic methods based on fractal characterization of roughness [18–35].

Another approach is to use the finite element method (FEM) to study the elastic-plastic contact of a single asperity contact. Kogut and Etsion (KE) [36] performed such an FEM analysis of an elastic-perfectly plastic spherical asperity in contact with a rigid flat. The KE model was then used to give empirical expressions for the contact area, the contact force and the average contact pressure as functions of the interference. Jackson and Green [37] also studied an elastic-perfectly plastic hemisphere in frictionless contact with a rigid flat using the FEM and with material yielding based on the Von Mises criterion. This model went farther into the elastic-plastic regime and also examined a wider range of conditions. The finer meshes provided more accurate results over the entire range of deformation. Etsion et al. [38] and then Jackson et al. [39] analyzed different aspects of single unloading of an elastic-plastically loaded sphere in contact with a rigid flat for a wide range of sphere material properties and radii. Jackson et al. [40] used a semianalytical model and finite element model to generate empirical equations describing the tangential and normal contact forces between sliding elastic-plastic spheres.

The FEM based models can be used as building blocks to study multi-asperity contacts with mixed elastic-plastic deformation. Kogut and Etsion [41] and Jackson and Green [42] used the FE models in [36, 37] in conjunction with the GW methodology [1] to present an elastic-plastic model for the contact of rough surfaces. Similarly, Kucharski et al. [43] investigated elastic-plastic contact between a hemisphere and a rigid plane using the FEM and combined the resulting relations with a statistical description of rough surfaces.

In this paper, we consider elastic-plastic contact of nominally flat rough surfaces. Equations governing the shoulder-shoulder contact of asperities are derived based on the asperity-asperity constitutive relations from a finite element model of the elastic-plastic interaction proposed by Kogut and Etsion [36]. Shoulder-shoulder asperity contact yields a slanted contact force consisting of both tangential (parallel to mean plane) and normal components. An analytical fusion technique is developed to combine the piecewise asperity level constitutive relations for contact force and real contact area. Statistical summation of tangential contact force component along an arbitrary tangential direction yields the half-plane tangential contact force. Similarly, statistical summation of contact force along the normal direction obtains the elastic-plastic normal contact force formulae for two rough surfaces. Approximate equations are developed in closed form for contact force components and contact area as a function of mean plane separation, sum of curvature radii of asperity summits, and plasticity index.

2. Elastic-Plastic Contact

Consider the elastic-plastic contact of two nominally flat rough surfaces. As shown in Figure 1, let and be the critical interferences of the Surface 1 and the Surface 2 , respectively. Note that the critical interference on a surface defines the plastic asperities on that surface as illustrated by the dashed curves in Figure 1.

Figure 1 

Elastic-plastic contact of two rough surfaces-for , elastic-plastic behavior would be dominated by the Surface 2.

Let , then elastic-plastic behavior would be primarily by the asperities on . Hence, we consider as the critical interference for the inception of plastic deformation for the contact of the two rough surfaces. For simplicity we denote the lower critical interference by . It should be noted that prior to interference of with plastic asperities of there is only elastic contribution. Any elastic-plastic contribution would be due to the interference of the asperities on and the plastic asperities of .

Since in general asperities meet in a shoulder-to-shoulder contact, a contact force between two asperities would be slanted, giving rise to both normal and tangential force. This is illustrated in Figure 2 wherein the interference between shoulders of two asperities and the resulting contact force are depicted. It can be shown by considering the geometry of interference between surface asperities (Figure 2) that the interference is [5] where is the sum of curvature radii of asperity summits and the tangential offset of the mating asperities so that when the asperities interfere along the normal to the mean planes. In (1) and (2) the parameters have been normalized with respect to the standard deviation of asperity height sum σ, so that s is , is and and are the normalized values using as the normalization parameter.

Figure 2 

Asperity contact: Overlap region showing normal and oblique interferences-Elastic-plastic force and its components.

Kogut and Etsion [36], using an FEA model, obtained the following piecewise fits for contact load and area of contact between a deformable sphere and a rigid flat:

Elastic Range:

Elastic-Plastic Range (1):

Elastic-Plastic Range (2):
where is the ratio of interference to the critical interference

and the critical interference is that corresponding to the onset of plastic flow proposed by Greenwood and Williamson [1] where is the equivalent radius of curvature of asperity summit and is hardness of softer material, that is, Surface 2. The hardness coefficient, , is related to the Poisson ratio by and the hardness is assumed . Alternatively, from the Jackson and Green model [37]where is related to the Poisson ratio by . in (7a) and (7b) is the combined Young’s modulus for the two surfaces where and are Young’s Moduli and Poisson ratios of two contacting materials, respectively. in (3)–(5) is the ratio of contact load to the load at critical interference, . Likewise, is the ratio of contact area to the contact area at critical interference, ; where and are, respectively, Here we propose, the following continuous form of the (3)–(5). Equations (11) and (12) were obtained by fusing the piecewise equations (3)) to (5) for asperity scale contact force and area, using appropriate sets of analytical filters, and by optimizing the cutoff points. Figure 3 depicts the percent error between the continuous functions in (11) and (12) and the piecewise equations (3)) to (5). As shown in the figure, the accuracy is within 3 percent of the piecewise functions for the entire domain of .

Figure 3 

Comparison of the continuous functions for asperity contact force and area obtained through fusion ((11) and (12)) with the piecewise ((3), (4), and (5)).

The asperity contact force in (11) is directed along the normal to the contact patch. It yields two components as shown in Figure 2. Denoting and the components of the asperity contact force along the normal and tangential (parallel to the mean plane) direction, respectively, we find, with the help of (2) and (9), The asperity contact area with the help of (10) can be found as

3. Normal Force

The normal components of various contact forces are parallel and can be algebraically summed by statistical means to obtain the total normal force of one surface on another. Statistical summation of asperity normal force components yields the total normal contact force between the two rough surfaces as follows: where and are the number of asperity per unit nominal area on and , respectively, and is the nominal area. is the statistical integral. For a Gaussian distribution of asperity height sum it is It is noteworthy to mention that in (15) and (16) the parameters are normalized with respect to the standard deviation of asperity height sum, , so that is , is , and , , , and are the normalized values using as the normalization parameter.

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 (Figure 4). In considering the tangential component of contact force, we seek the components of the tangential contact force along an axis of interest, for instance tangential force component along the -axis, depicted in Figure 4. We are interested in formulating the cumulative effect of -component of tangential force along the positive direction (as shown in Figure 4). Hereafter, as we generate result for the -component of the tangential force due to positive contact slope, we will refer to this as the “tangential force” and denote by the force component . The goal here is to account for the tangential force of an asperity that would be experienced on each side, and therefore accumulation or summation of such forces would establish the tangential load on a surface from each side, that is, due to all contacts at positive slope.

Figure 4 

Schematic showing the tangential components of contact force exerted by asperities of on an asperity on .

Tangential force due to all asperities at height confined in area and at radial distance can be found as where is the density function associated with asperity heights on the surface and , the number of asperities per unit nominal area on . The component of this force along +x is By considering Figure 4, the force due to all asperities in the +x half plane at height and distance would be obtained by integration of (18) over to , resulting in

Using (19), accounting for the contribution of all asperities and considering a Gaussian distribution of asperity height sum, it can be shown that the component of the tangential force between surfaces and , along +x, may be found using the following equation: where

5. Contact Area

All the asperity contact areas can be algebraically summed by statistical means to obtain the total contact area of one surface on another. Statistical summation of asperity contact area yields the total contact area between the two rough surfaces as follows: where, and are the number of asperity per unit nominal area on and , respectively, and is the nominal area. For a Gaussian distribution of asperity height sum so that

6. Approximate Equations

In this section we introduce approximate equations for the integral functions of normal and tangential forces as well as contact area. Based on the dominant physical interaction, we define three ranges for critical interference or corresponding plasticity index to be able to find the most accurate fitting functions.

Elastic Range:
, or ,

Elastic-Plastic Range (1):
, or ,

Elastic-Plastic Range (2):
, or .

The approximate function for each integral is denoted using an additional letter “a” in the subscript to signify approximation. For instance, the approximations to dimensionless normal contact force component, , is denoted and is given as follows:

Elastic Range:
or

Elastic-Plastic Range (1):
or

Elastic-Plastic Range (2):
or Figures 5, 6, and 7 illustrate over to 4 and to 2000 for the values of plasticity index of 0.1 in the elastic range, 1 in the elastic-plastic range (1), and 5 in the elastic-plastic range (2), respectively. To assess the accuracy of the approximation in (24), we define the following error between the dimensionless normal contact force component and its approximation in percent error form

Figure 5 

for   , Elastic Range.

Figure 6 

for   , Elastic-Plastic Range (1).

Figure 7 

for   , Elastic-Plastic Range (2).

The approximate function in (24) yields accuracy to within 7 percent (7%) over the entire domain of , and .

The approximate equation for the dimensionless tangential contact force component, , is where for the elastic range for the elastic-plastic range (1) and for the elastic-plastic range (2) Figure 8 illustrates over to 4 and to 2000 for the values of plasticity index of 0.1. A similar observation applies to the results relevant to the elastic-plastic ranges. Assess the accuracy of the approximation in (29) by defining the following error between the dimensionless load component and its approximation: Similar accuracy (7%) is obtained by (29) for the half-plane tangential force component.

Figure 8 

for   , Elastic Range.

In the same way, we find the approximate equation for the dimensionless contact area, , as follows where for the elastic range for the elastic-plastic range (1) and for the elastic-plastic range (2) Figure 9 depicts over to 4 and to 2000 for the values of plasticity index of 0.1. Define the percent error as follows: Using the above, we find that the approximate function in (34) yields accuracy to within 8 percent (8%) over the entire domain of , and .

Figure 9 

for   , Elastic Range.

7. Comparison with CEB-Based Model

The model based on CEB [17] extended the CEB model to handle the oblique contact of asperities on two rough surfaces in contact. From extension to CEB model [17] we have where where where We define As shown in Figures 10, 11, and 12, for elastic contact ( or ), both the present and the extension to CEB models yield identical results as would be expected. However, large differences (of up to 45% in the contact load and contact area for a given separation) are found for or . It is interesting to note from Figures 9 to 11 that error between the two models does not depend on asperity summit radius of curvature sum, .

Figure 10 

for .

Figure 11 

for .

Figure 12 

for .

8. Concluding Remarks

The asperity level constitutive equations were presented based on the work by Kogut and Etsion [36]. Kogut and Etsion [36] developed a finite element model of an elastic plastic sphere in contact with a rigid flat. Based on the FEA results, they established the relation between contact force and interference and contact area and interference for different ranges of interference ratio.

This paper developed continuous constitutive asperity equations relating (1) the asperity contact force to interference and (2) asperity area of contact to the interference. This was accomplished by devising an analytical fusion technique to combine the piecewise equations of Kogut and Etsion. The resulting continuous function was accurate to within 3 percent of the piecewise functions. Therefore, the analytical fusion technique successfully removed the discontinuity presented in [36] and thereby facilitated the ensuing development that included the derivation of the normal and tangential contact force components and contact area between two rough surfaces in a three-dimensional account of elastic-plastic contact. It should be noted that asperities experiencing interference larger than 110 times the critical interference would introduce error due to the limitation of the KE model.

Consideration of shoulder-shoulder asperity contact yielded contact force in a slanted orientation due to contact slope. Thereby, giving rise to both normal and tangential contact force components. Statistical summation of +x half-plane tangential contact force component resulted in the formulation of the tangential force impinged upon one surface by the other due to the cumulative effect of interactions in a half plane. In the absence of an applied tangential force the net tangential force transferred between the two surfaces is zero due to symmetry of interactions about an asperity. Similarly, statistical summation of the asperity contact force along the normal direction and asperity contact area, respectively, yielded the total normal contact force and contact area formula for two rough surfaces.

Approximate equations were forwarded for the integral functions of contact force components and contact area. These equations were shown to provide accuracy within seven and eight percent, respectively, for contact force components and contact area over ranges of mean plane separation, asperity summit radius of curvature sum, and plasticity index. The approximate equations greatly simplify solution of problems involving elastic-plastic contact of rough surfaces.

A comparison with the approximate elastic-plastic CEB model showed identical results for elastic contacts having plasticity index values below 0.6 but substantial differences for elastic-plastic contacts with plasticity index values above 1.

Nomenclature