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Modelling and Simulation in Engineering
Volume 2011 (2011), Article ID 561828, 11 pages
http://dx.doi.org/10.1155/2011/561828
Research Article

A Finite Element-Based Elastic-Plastic Model for the Contact of Rough Surfaces

Department of Mechanical Engineering and Energy Processes, Southern Illinois University Carbondale, Carbondale, Il 62901, USA

Received 24 March 2011; Revised 18 June 2011; Accepted 20 June 2011

Academic Editor: Xiaosheng Gao

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.

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 [735]. Many researchers have employed statistical models for the elastic-plastic contact of rough surfaces [717]. Others have advocated the use of deterministic methods based on fractal characterization of roughness [1835].

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 𝑤𝑐1 and 𝑤𝑐2 be the critical interferences of the Surface 1 (𝑆1) and the Surface 2 (𝑆2), 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.

561828.fig.001
Figure 1: Elastic-plastic contact of two rough surfaces-for 𝑤𝑐2<𝑤𝑐1, elastic-plastic behavior would be dominated by the Surface 2.

Let 𝑤𝑐2<𝑤𝑐1, then elastic-plastic behavior would be primarily by the asperities on 𝑆2. Hence, we consider 𝑤𝑐2 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 𝑤𝑐. It should be noted that prior to interference of 𝑆1 with plastic asperities of 𝑆2 there is only elastic contribution. Any elastic-plastic contribution would be due to the interference of the asperities on 𝑆1 and the plastic asperities of 𝑆2.

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]𝑟𝑤=𝑠22𝛽𝑠𝑟1+2𝛽2𝑠1/2,𝑟(1)cos𝛼=1+2𝛽2𝑠1/2𝑟;sin𝛼=1+2𝛽2𝑠1/2𝑟𝛽𝑠,(2) where 𝛽𝑠 is the sum of curvature radii of asperity summits and 𝑟 the tangential offset of the mating asperities so that when 𝑟=0 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 (𝑧1+𝑧2)/𝜎, is 𝑑/𝜎 and 𝑟 and 𝛽𝑠 are the normalized values using 𝜎 as the normalization parameter.

561828.fig.002
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:𝑤cr=0-1
𝑃cr𝑤cr=𝑤cr3/2,𝐴cr𝑤cr=𝑤cr,(3)

Elastic-Plastic Range (1): 𝑤cr=1-6
𝑃cr𝑤cr𝑤=1.03cr1.425,𝐴cr𝑤cr𝑤=0.93cr1.136,(4)

Elastic-Plastic Range (2): 𝑤cr=6-110
𝑃cr𝑤cr𝑤=1.40cr1.263,𝐴cr𝑤cr𝑤=0.94cr1.146,(5) where 𝑤cr is the ratio of interference to the critical interference

𝑤cr=𝑤𝑤𝑐,(6) and the critical interference is that corresponding to the onset of plastic flow proposed by Greenwood and Williamson [1] 𝑤𝑐=𝛽𝜋𝐾𝐻2𝐸2,(7a) where 𝛽=𝛽1𝛽2/𝛽𝑠 is the equivalent radius of curvature of asperity summit and 𝐻 is hardness of softer material, that is, Surface 2. The hardness coefficient, 𝐾, is related to the Poisson ratio by 𝐾=0.454+0.41𝜈 and the hardness is assumed 𝐻=2.8𝑆𝑦. Alternatively, from the Jackson and Green model [37]𝑤𝑐=𝛽𝜋𝐶𝑆𝑦2𝐸2,(7b)where 𝐶 is related to the Poisson ratio by 𝐶=1.295exp(0.736𝜈). 𝐸 in (7a) and (7b) is the combined Young’s modulus for the two surfaces1=𝐸1𝜈21𝐸1+1𝜈22𝐸2,(8) where 𝐸1,𝐸2 and 𝜈1,𝜈2 are Young’s Moduli and Poisson ratios of two contacting materials, respectively. 𝑃cr in (3)–(5) is the ratio of contact load to the load at critical interference, 𝑃/𝑃𝑐. Likewise, 𝐴cr is the ratio of contact area to the contact area at critical interference, 𝐴/𝐴𝑐; where 𝑃𝑐 and 𝐴𝑐 are, respectively,𝑃𝑐=43𝐸𝛽1/2𝑤𝑐3/2,𝐴(9)𝑐=𝜋𝛽𝑤𝑐.(10) Here we propose, the following continuous form of the (3)–(5). 𝑃cr𝑤cr=5𝑤1+cr2𝑤8+5cr2𝑤3cr0.21+2𝑤cr/134×𝑤1.03cr1.425+1.402𝑤cr134𝑤cr1.263,𝐴(11)cr𝑤cr=21+5𝑤cr/6221+5𝑤cr/62𝑒2(𝑤cr)0.71+2𝑤cr/134×𝑤0.93cr1.136+0.942𝑤cr134𝑤cr1.146.(12) 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 𝑤cr.

561828.fig.003
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 𝑓𝑁 and 𝑓𝑡 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), 𝑓𝑁=43𝐸𝛽1/2𝑤𝑐3/2𝑃cr𝑤cr𝑟1+2𝛽2𝑠1/2,𝑓𝑡=43𝐸𝛽1/2𝑤𝑐3/2𝑃cr𝑤cr𝑟1+2𝛽2𝑠1/2𝑟𝛽𝑠.(13) The asperity contact area with the help of (10) can be found as𝐴𝑠=𝜋𝛽𝑤𝑐𝐴cr𝑤cr.(14)

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:𝐹𝑁,𝛽𝑠,𝑤𝑐=832𝜋𝜋𝐸𝜂1𝜂2𝐴𝑛𝛽1/2𝑤𝑐3/2𝜎4𝐼𝑁,𝛽𝑠,𝑤𝑐,(15) where 𝜂1 and 𝜂2 are the number of asperity per unit nominal area on 𝑆1 and 𝑆2, respectively, and 𝐴𝑛 is the nominal area. 𝐼𝑁 is the statistical integral. For a Gaussian distribution of asperity height sum it is𝐼𝑁,𝛽𝑠,𝑤𝑐=2𝛽𝑠0(𝑠)𝑃cr𝑤cr𝑟1+2𝛽2𝑠1/2𝑒𝑠2/2𝑟𝑑𝑟𝑑𝑠.(16) It is noteworthy to mention that in (15) and (16) the parameters are normalized with respect to the standard deviation of asperity height sum, 𝜎, so that 𝑠 is 𝑧/𝜎, is 𝑑/𝜎, and 𝑟, 𝛽, 𝛽𝑠, and 𝑤𝑐 are the normalized values using 𝜎 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 𝑆2 (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 𝑥-axis, depicted in Figure 4. We are interested in formulating the cumulative effect of 𝑥-component of tangential force along the positive 𝑥 direction (as shown in Figure 4). Hereafter, as we generate result for the 𝑥-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 𝐹𝑥. 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.

561828.fig.004
Figure 4: Schematic showing the tangential components of contact force exerted by asperities of 𝑆2 on an asperity on 𝑆1.

Tangential force due to all asperities at height 𝑧2 confined in area 𝑑𝐴 and at radial distance 𝑟 can be found as𝑑𝐹𝑡𝑧1=43𝐸𝜂2𝛽1/2𝛽𝑠𝑤𝑐3/2𝑃cr𝑤cr𝑟1+2𝛽2𝑠1/2𝜙2𝑧2𝑟2𝑑𝑟𝑑𝑧2𝑑𝜃,(17) where 𝜙2(𝑧2) is the density function associated with asperity heights on the surface 𝑆2 and 𝜂2, the number of asperities per unit nominal area on 𝑆2. The component of this force along +x is𝑑𝐹𝑥𝑧1=43𝐸𝜂2𝛽1/2𝛽𝑠𝑤𝑐3/2𝑃cr𝑤cr𝑟1+2𝛽2𝑠1/2𝜙2𝑧2×𝑟2𝑑𝑟𝑑𝑧2cos𝜃𝑑𝜃.(18) By considering Figure 4, the force due to all asperities in the +x half plane at height 𝑧2 and distance 𝑟 would be obtained by integration of (18) over 𝜃=𝜋/2 to 𝜋/2, resulting in𝑑𝐹𝑥𝑧1=83𝐸𝜂2𝛽1/2𝛽𝑠𝑤𝑐3/2𝑃cr𝑤cr𝑟1+2𝛽2𝑠1/2×𝜙2𝑧2𝑟2𝑑𝑟𝑑𝑧2.(19)

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 𝑆1 and 𝑆2, along +x, may be found using the following equation:𝐹𝑥,𝛽𝑠,𝑤𝑐=83𝐸2𝜋𝜂1𝜂2𝐴𝑛𝛽1/2𝛽𝑠𝑤𝑐3/2𝜎4𝐼𝑥,𝛽𝑠,𝑤𝑐,(20) where𝐼𝑥,𝛽𝑠,𝑤𝑐=2𝛽𝑠0(𝑠)𝑃cr𝑤cr𝑟1+2𝛽2𝑠1/2𝑒𝑠2/2𝑟2𝑑𝑟𝑑𝑠.(21)

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:𝐴,𝛽𝑠,𝑤𝑐=2𝜋2𝜂2𝜋1𝜂2𝐴𝑛𝛽𝑤𝑐𝜎4𝐼𝐴,𝛽𝑠,𝑤𝑐,(22) where, 𝜂1 and 𝜂2 are the number of asperity per unit nominal area on 𝑆1 and 𝑆2, respectively, and 𝐴𝑛 is the nominal area. For a Gaussian distribution of asperity height sum so that𝐼𝐴,𝛽𝑠,𝑤𝑐=2𝛽𝑠0(𝑠)𝐴cr𝑤cr𝑒𝑠2/2𝑟𝑑𝑟𝑑𝑠.(23)

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 (𝜓=1/𝑤𝑐) to be able to find the most accurate fitting functions.

Elastic Range:
𝑤𝑐=2.75-200, or 𝜓=0.07-0.6,

Elastic-Plastic Range (1):
𝑤𝑐=0.16-2.75, or 𝜓=0.6-2.5,

Elastic-Plastic Range (2):
𝑤𝑐=0.0156-0.16, or 𝜓=2.5-8.

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, 𝐼𝑁, is denoted 𝐼𝑁𝑎 and is given as follows: 𝐼𝑁𝑎,𝛽𝑠,𝑤𝑐=𝛼1𝛽𝑠,𝑤𝑐𝑒𝛼2(𝑤𝑐)𝛼3𝑐)(𝑤.(24)

Elastic Range:
𝑤𝑐=2.75-200 or 𝜓=0.07-0.6𝛼1𝛽𝑠,𝑤𝑐=2.82550×101𝑤𝑐1.48175𝛽𝑠1.00460,𝛼2𝑤𝑐𝛼=1.35000,3𝑤𝑐=1.55700.(25)

Elastic-Plastic Range (1):
𝑤𝑐=0.16-2.75 or 𝜓=0.6-2.5𝛼1𝛽𝑠,𝑤𝑐=2.90330×101𝑤𝑐1.45649𝛽𝑠1.00295,𝛼2𝑤𝑐𝑤=1.34779𝑐7.81366×103,𝛼3𝑤𝑐𝑤=1.55821𝑐2.23723×103.(26)

Elastic-Plastic Range (2):
𝑤𝑐=0.0156-0.16 or 𝜓=2.5-8𝛼1𝛽𝑠,𝑤𝑐=4.02510×101𝑤𝑐1.26820𝛽𝑠1.00230,𝛼2𝑤𝑐=5.53530×101𝑤𝑐1.66900𝛼1.30000,3𝑤𝑐=3.87770×101𝑤𝑐3.20000×1022+1.57160.(27) Figures 5, 6, and 7 illustrate 𝐼𝑁 over =1 to 4 and 𝛽𝑠=100 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

561828.fig.005
Figure 5: 𝐼𝑁(,𝛽𝑠) for 𝑤𝑐=100  (𝜓=0.1), Elastic Range.
561828.fig.006
Figure 6: 𝐼𝑁(,𝛽𝑠) for 𝑤𝑐=1  (𝜓=1), Elastic-Plastic Range (1).
561828.fig.007
Figure 7: 𝐼𝑁(,𝛽𝑠) for 𝑤𝑐=0.04  (𝜓=5), Elastic-Plastic Range (2).

𝐸𝑁,𝛽𝑠,𝑤𝑐=𝐼𝑁,𝛽𝑠,𝑤𝑐𝐼𝑁𝑎,𝛽𝑠,𝑤𝑐𝐼𝑁,𝛽𝑠,𝑤𝑐×100.(28) The approximate function in (24) yields accuracy to within 7 percent (7%) over the entire domain of ,𝛽𝑠, and 𝑤𝑐.

The approximate equation for the dimensionless tangential contact force component, 𝐼𝑥, is𝐼𝑥𝑎,𝛽𝑠,𝑤𝑐=𝛼1𝛽𝑠,𝑤𝑐𝑒𝛼2(𝑤𝑐)𝛼3𝑐)(𝑤,(29) where for the elastic range𝛼1𝛽𝑠,𝑤𝑐=2.42720×101𝑤𝑐1.48021𝛽𝑠1.50570,𝛼2𝑤𝑐𝛼=1.45000,3𝑤𝑐=1.53100,(30) for the elastic-plastic range (1)𝛼1𝛽𝑠,𝑤𝑐=2.50300×101𝑤𝑐1.45916𝛽𝑠1.50435,𝛼2𝑤𝑐𝑤=1.45046𝑐6.76706×103,𝛼3𝑤𝑐𝑤=1.53092𝑐1.92673×103,(31) and for the elastic-plastic range (2)𝛼1𝛽𝑠,𝑤𝑐=3.50250×101𝑤𝑐1.26946𝛽𝑠1.50320,𝛼2𝑤𝑐=5.75910×101𝑤𝑐1.65700𝛼1.40369,3𝑤𝑐=3.83420×101𝑤𝑐3×1022+1.54278.(32) Figure 8 illustrates 𝐼𝑥 over =1 to 4 and 𝛽𝑠=100 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: 𝐸𝑥,𝛽𝑠,𝑤𝑐=𝐼𝑥,𝛽𝑠,𝑤𝑐𝐼𝑥𝑎,𝛽𝑠,𝑤𝑐𝐼𝑥,𝛽𝑠,𝑤𝑐×100.(33) Similar accuracy (7%) is obtained by (29) for the half-plane tangential force component.

561828.fig.008
Figure 8: 𝐼𝑥(,𝛽𝑠) for 𝑤𝑐=100  (𝜓=0.1), Elastic Range.

In the same way, we find the approximate equation for the dimensionless contact area, 𝐼𝐴, as follows𝐼𝐴𝑎,𝛽𝑠,𝑤𝑐=𝛼1𝛽𝑠,𝑤𝑐𝑒𝛼2(𝑤𝑐)𝛼3𝑐)(𝑤,(34) where for the elastic range𝛼1𝛽𝑠,𝑤𝑐=3.29230×101𝑤𝑐1.03048𝛽𝑠9.99650×101,𝛼2𝑤𝑐𝑤=1.23087𝑐6.26624×103,𝛼3𝑤𝑐𝑤=1.59327𝑐1.56574×103,(35) for the elastic-plastic range (1)𝛼1𝛽𝑠,𝑤𝑐=3.10750×101𝑤𝑐1.06230𝛽𝑠1.00350,𝛼2𝑤𝑐𝑤=1.24888𝑐1.13200×102,𝛼3𝑤𝑐𝑤=1.58657𝑐3.07496×103,(36) and for the elastic-plastic range (2)𝛼1𝛽𝑠,𝑤𝑐=2.79650×101𝑤𝑐1.14807𝛽𝑠1.00210,𝛼2𝑤𝑐𝛼=1.277003𝑤𝑐=1.57850.(37) Figure 9 depicts 𝐼𝐴 over =1 to 4 and 𝛽𝑠=100 to 2000 for the values of plasticity index of 0.1. Define the percent error as follows:𝐸𝐴,𝛽𝑠,𝑤𝑐=𝐼𝐴,𝛽𝑠,𝑤𝑐𝐼𝐴𝑎,𝛽𝑠,𝑤𝑐𝐼𝐴,𝛽𝑠,𝑤𝑐×100.(38) Using the above, we find that the approximate function in (34) yields accuracy to within 8 percent (8%) over the entire domain of ,𝛽𝑠, and 𝑤𝑐.

561828.fig.009
Figure 9: 𝐼𝐴(,𝛽𝑠) for 𝑤𝑐=100  (𝜓=0.1), 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 have𝐹𝑁-CEB,𝛽𝑠,𝑤𝑐=𝐶𝑁-CEB𝐼𝑁-CEB,𝛽𝑠,𝑤𝑐,(39) where 𝐶𝑁-CEB=832𝜋𝜋𝐸𝜂1𝜂2𝐴𝑛𝛽1/2𝜎4,𝐼𝑁-CEB,𝛽𝑠,𝑤𝑐=𝐼𝑁𝑒,𝛽𝑠𝐼𝑁𝑒𝑐,𝛽𝑠,𝑤𝑐𝑤+3𝑐1/2𝐼𝑁𝑒𝑝1,𝛽𝑠,𝑤𝑐32𝑤𝑐3/2𝐼𝑁𝑒𝑝2,𝛽𝑠,𝑤𝑐,𝐼𝑁𝑒,𝛽𝑠=2𝛽𝑠0(𝑠)𝑟𝑠22𝛽𝑠3/2𝑟1+2𝛽2𝑠1/2𝑒𝑠2/2𝐼𝑟𝑑𝑟𝑑𝑠,𝑁𝑒𝑐,𝛽𝑠,𝑤𝑐=+𝑤𝑐2𝛽𝑠0(𝑠)𝑟𝑠22𝛽𝑠3/2𝑟1+2𝛽2𝑠1/2𝑒𝑠2/2𝐼𝑟𝑑𝑟𝑑𝑠,𝑁𝑒𝑝1,𝛽𝑠,𝑤𝑐=+𝑤𝑐2𝛽𝑠0(𝑠)𝑟𝑠22𝛽𝑠𝑟1+2𝛽2𝑠1/2𝑒𝑠2/2𝐼𝑟𝑑𝑟𝑑𝑠,𝑁𝑒𝑝2,𝛽𝑠,𝑤𝑐=+𝑤𝑐2𝛽𝑠0(𝑠)𝑟1+2𝛽2𝑠𝑒𝑠2/2𝐹𝑟𝑑𝑟𝑑𝑠,(40)𝑥-CEB,𝛽𝑠,𝑤𝑐=𝐶𝑥-CEB𝐼𝑥-CEB,𝛽𝑠,𝑤𝑐,(41) where 𝐶𝑥-CEB=83𝐸2𝜋𝜂1𝜂2𝐴𝑛𝛽1/2𝛽𝑠𝜎4,𝐼𝑥-CEB,𝛽𝑠,𝑤𝑐=𝐼𝑥𝑒,𝛽𝑠𝐼𝑥𝑒𝑐,𝛽𝑠,𝑤𝑐𝑤+3𝑐1/2𝐼𝑥𝑒𝑝1,𝛽𝑠,𝑤𝑐32𝑤𝑐3/2𝐼𝑥𝑒𝑝2,𝛽𝑠,𝑤𝑐,𝐼𝑥𝑒,𝛽𝑠=2𝛽𝑠0(𝑠)𝑟𝑠22𝛽𝑠3/2𝑟1+2𝛽2𝑠1/2𝑒𝑠2/2𝑟2𝐼𝑑𝑟𝑑𝑠,𝑥𝑒𝑐,𝛽𝑠,𝑤𝑐=+𝑤𝑐2𝛽𝑠0(𝑠)𝑟𝑠22𝛽𝑠3/2𝑟1+2𝛽2𝑠1/2𝑒𝑠2/2𝑟2𝐼𝑑𝑟𝑑𝑠,𝑥𝑒𝑝1,𝛽𝑠,𝑤𝑐=+𝑤𝑐2𝛽𝑠0(𝑠)𝑟𝑠22𝛽𝑠𝑟1+2𝛽2𝑠1/2𝑒𝑠2/2𝑟2𝐼𝑑𝑟𝑑𝑠,𝑥𝑒𝑝2,𝛽𝑠,𝑤𝑐=+𝑤𝑐2𝛽𝑠0(𝑠)𝑟1+2𝛽2𝑠𝑒𝑠2/2𝑟2𝐴𝑑𝑟𝑑𝑠,(42)CEB,𝛽𝑠,𝑤𝑐=𝐶𝐴-CEB𝐼𝐴-CEB,𝛽𝑠,𝑤𝑐,(43) where 𝐶𝐴-CEB=2𝜋2𝜂2𝜋1𝜂2𝐴𝑛𝛽𝜎4,𝐼𝐴-CEB,𝛽𝑠,𝑤𝑐=𝐼𝐴𝑒,𝛽𝑠+𝐼𝐴𝑒𝑝1,𝛽𝑠,𝑤𝑐𝑤𝑐𝐼𝐴𝑒𝑝2,𝛽𝑠,𝑤𝑐,𝐼𝐴𝑒,𝛽𝑠=2𝛽𝑠0(𝑠)𝑟𝑠22𝛽𝑠𝑟1+2𝛽2𝑠𝑒𝑠2/2𝐼𝑟𝑑𝑟𝑑𝑠,𝐴𝑒𝑝1,𝛽𝑠,𝑤𝑐=+𝑤𝑐2𝛽𝑠0(𝑠)𝑟𝑠22𝛽𝑠𝑟1+2𝛽2𝑠𝑒𝑠2/2𝐼𝑟𝑑𝑟𝑑𝑠,𝐴𝑒𝑝2,𝛽𝑠,𝑤𝑐=+𝑤𝑐2𝛽𝑠0(𝑠)𝑟1+2𝛽2𝑠3/2𝑒𝑠2/2𝑟𝑑𝑟𝑑𝑠.(44) We define 𝐸𝑁-CEB,𝑤𝑐=𝐼𝑁,𝛽𝑠,𝑤𝑐𝑤𝑐3/2𝐼𝑁-CEB,𝛽𝑠,𝑤𝑐𝐼𝑁,𝛽𝑠,𝑤𝑐𝑤𝑐3/2𝐸100,𝑥-CEB,𝑤𝑐=𝐼𝑥,𝛽𝑠,𝑤𝑐𝑤𝑐3/2𝐼𝑥-CEB,𝛽𝑠,𝑤𝑐𝐼𝑥,𝛽𝑠,𝑤𝑐𝑤𝑐3/2𝐸100,𝐴-CEB,𝑤𝑐=𝐼𝐴,𝛽𝑠,𝑤𝑐𝑤𝑐𝐼𝐴-CEB,𝛽𝑠,𝑤𝑐𝐼𝐴,𝛽𝑠,𝑤𝑐𝑤𝑐100.(45) As shown in Figures 10, 11, and 12, for elastic contact (𝑤𝑐>3 or 𝜓<0.6), 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 𝑤𝑐<1 or 𝜓>1. 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, 𝛽𝑠.

561828.fig.0010
Figure 10: 𝐸𝑁-CEB(,𝑤𝑐) for 𝛽𝑠=500.
561828.fig.0011
Figure 11: 𝐸𝑥-CEB(,𝑤𝑐) for 𝛽𝑠=500.
561828.fig.0012
Figure 12: 𝐸𝐴-CEB(,𝑤𝑐) for 𝛽𝑠=500.

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

𝑤:Dimensionless interference
𝑤1:Dimensionless interference defined in GT [3]
𝛼:Contact angle between two asperities
𝑟:Dimensionless asperity tangential offset
𝐴𝑛:Nominal area
𝐸:Combined Young’s modulus
𝐻:Hardness of the softer material (= H2)
𝐾:Maximum contact pressure factor
𝜎:Standard deviation of asperity height sum
:Dimensionless mean separation
𝑧1,𝑧2:Heights of asperities on the surfaces 1 and 2 measured from the mean asperity heights
𝑠:Dimensionless asperity heights sum
𝛽1,𝛽2:Dimensionless average summit radius of asperities on the surfaces 1 and 2
𝛽:Combined asperity summit radius of curvature
𝛽𝑠:Dimensionless asperity summit radius of curvature sum
𝑑:Mean separation
𝑤𝑐:Smaller dimensionless critical interference
𝜓:Plasticity index
𝑤cr:Ratio of interference to the interference for onset of plastic flow
𝑆𝑦:Yield strength
𝜂1,𝜂2:Asperity areal density for the surfaces 1 and 2
𝑃cr:Dimensionless contact load in KE [36]
𝐴cr:Dimensionless contact area in KE [36]
𝐸1,2:Young's moduli
𝜈1,2:Poisson ratios
𝑃𝑐:Contact load at critical interference
𝐴𝑐:Contact area at critical interference
𝑓𝑁:Component of the asperity contact force along the normal direction
𝑓𝑡:Component of the asperity contact force along the tangential direction
𝐴𝑠:Asperity contact area
𝐹𝑁:Total normal contact force
𝐼𝑁:Dimensionless total normal contact force
𝐹𝑥:Total half-plane tangential contact force
𝐼𝑥:Dimensionless total half-plane tangential contact force
𝐴:Total contact area
𝐼𝐴:Dimensionless total contact area
𝐼𝑁𝑎:Approximate function for 𝐼𝑁
𝐼𝑥𝑎:Approximate function for 𝐼𝑥
𝐼𝐴𝑎:Approximate function for 𝐼𝐴
𝐸():Percent error between 𝐼() and 𝐼()𝑎.

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