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# Evolution of the Contact Area with Normal Load for Rough Surfaces: from Atomic to Macroscopic Scales

- Shiping Huang
^{1, 2, 3, 4}Email authorView ORCID ID profile

**Received:**22 August 2017**Accepted:**3 November 2017**Published:**13 November 2017

## Abstract

The evolution of the contact area with normal load for rough surfaces has great fundamental and practical importance, ranging from earthquake dynamics to machine wear. This work bridges the gap between the atomic scale and the macroscopic scale for normal contact behavior. The real contact area, which is formed by a large ensemble of discrete contacts (clusters), is proven to be much smaller than the apparent surface area. The distribution of the discrete contact clusters and the interaction between them are key to revealing the mechanism of the contacting solids. To this end, Green’s function molecular dynamics (GFMD) is used to study both how the contact cluster evolves from the atomic scale to the macroscopic scale and the interaction between clusters. It is found that the interaction between clusters has a strong effect on their formation. The formation and distribution of the contact clusters is far more complicated than that predicted by the asperity model. Ignorance of the interaction between them leads to overestimating the contacting force. In real contact, contacting clusters are smaller and more discrete due to the interaction between the asperities. Understanding the exact nature of the contact area with the normal load is essential to the following research on friction.

## Keywords

- Fractal surface
- Asperity model
- Green’s function molecular dynamics
- Contact area

## Background

Most macroscopic surfaces are considered to be rough and fractal [1, 2]. The contact behavior between rough surfaces is much more complicated than that of perfectly smooth surfaces [3, 4]. The real contact area is formed by a large ensemble of discrete contact regions (clusters), which is much smaller than the apparent surface area. The normal force and the size, shape, and distribution of the contact clusters are key to revealing the contact behavior, which is essential for the following studies on friction [5–7].

To obtain the relationship between the contact area and the load, numerous models have been proposed since the 1960s [1, 8–14]. Among them, the asperity model is the simplest and most popular one. In one of the early applications of the asperity model, Greenwood and Williamson [8] describe the roughness of the contact interface by assuming that asperities have the same radii but different heights. Since then, the asperity model has prevailed and a vast amount of literature has appeared in this field. Whitehouse and Archard [15] developed the Greenwood and Williamson (G-W) model by accounting for the random radii of curvature of the asperity tips. Nayak [16–18] introduced the techniques of random process theory [19, 20] into the analysis of Gaussian roughness, which was subsequently used by Bush et al. [9] in rough surface contact.

One of the basic assumptions in the asperity model is that the interaction between the asperities can be neglected, which indicates that the potential contact asperities can be determined by the surface geometry in advance. However, this assumption may lead to inaccurate estimations of the contact force and contact area. To obtain the evolution of the contacting clusters and the interaction between them, we utilize Green’s function molecular dynamics (GFMD) [21–23] to study the fractal rough surface.

This work is to bridge the gap between the atomic scale and the macroscopic scale for normal contact behavior. The evolution of the contact area from atomic to macroscopic scales is demonstrated through numerical examples with the consideration of the asperity interactions. In the subsequent discussion, we first briefly present our approaches for the fractal surface generation, the GFMD model, the contacting cluster detection algorithm, and the numerical experimental design. We then focus on the forming and development of the contacting cluster and the influence of these processes on the interface’s behaviors.

## Methods

### Rough Fractal Surface Generation

*w*

_{ H }), the minimum frequency (

*w*

_{ L }), the Hurst exponent (

*H*), and the standard deviation of the amplitude (

*P*). The surface’s basic statistical parameters, such as RMS (root mean square) height \( \left(\sqrt{M_0}\right) \), RMS slope \( \left(\sqrt{M_2}\right) \)

_{,}and RMS curvature \( \left(\sqrt{M_4}\right) \), are the key parameters for the interface’s behaviors, where

*M*

_{ i }is the

*i*th radial moment of the surface spectrum [19, 20]. It is worth noting that the surface statistical parameter

*M*

_{ i }is related to the profile statistical parameters

*m*

_{ i }by the following equation: \( {M}_0={m}_0,{M}_2=2{m}_2,{M}_4=\frac{4}{3}{m}_4 \). It is well known that the asperity density

*n*(surface summits or valleys) can be determined by the following equation:

*N*is expressed by

*A*

_{0}is the apparent surface area. For the self-affine fractal surface, the surface statistical parameters are related to the input parameters (

*w*,

*H*,

*P*) by the following equation:

*B*is the surface roughness constant, which is related to

*P*. Equations (3) and (2) indicate that the fractal surface summit/valley number is dependent on the wavelength and the Hurst exponent. Detailed discussions of the fractal surface statistical properties can be found in the literature [25, 26].

In the Fourier transform algorithm, as a typical example, we set the Hurst component to be *H* = 0.5, the maximum frequency to be *w*
_{
L
} = 1/(24*σ*), the minimum frequency to be *w*
_{
H
} = 1/(256*σ*), the standard deviation of the frequency amplitude to be *P* = 0.69, and the system size to be 512 × 512 atoms (with initial spacing equal to 1.12*σ*). These input parameters subsequently generate the surface with the following statistical parameters: surface RMS slope \( \sqrt{M_2}=0.077 \) and RMS curvature \( \sqrt{M_4}= 0.0077 \)
*.* The total number of surface summits/valleys is 150 based on Eq. (2), while by counting the number numerically, the surface summit number is 158 and the valley number is 159. The error is within 5%, which suggests that the system size is acceptable in a statistical sense. In fact, when we increase the system size up to 2048 × 2048 atoms (with initial spacing equal to 1.12*σ*), the results for the statistical parameters are consistent with those of the smaller system.

### GFMD Model

*ε*is the depth of the potential well,

*σ*is the finite distance at which the inter-particle potential is zero, and

*r*is the distance between the particles. We take

*ε*,

*σ*, and

*ε*/

*σ*as the energy, distance unit, and force unit, respectively. According to the LJ potential, we know that when

*r*= 2

^{1/6}

*σ*≈ 1.12

*σ*, the inter-particle force is zero. When

*r*> 1.12

*σ*, the inter-particle force is attractive; when

*r*< 1.12

*σ*, the inter-particle force is repulsive. Since we do not consider adhesion in this work, the cutoff distance is set to be 1.12

*σ*. The crystal structure used for the atomic layer is face-centered cubic (FCC). Due to symmetry, we only take the interface’s layer to form the surface geometry as shown in Fig. 1, and the elastic block below the flat surface is simulated by Green’s function.

### Contacting Cluster Recognition Method

Step 1. Start the row search and get the new atom data, that is, search the atoms from row to row.

Step 2. Determine if the atom is in contact. If it is not in contact, go back to step 1. If it is in contact, move to the next step.

Step 3. Compare the current atom with the previous atom in the same row. If the previous atom is also in contact, merge the atom into the cluster to which the previous atom belongs, then label the atom with the same number as the previous atom. If the previous atom is not in contact, label the atom with a new number that is the previous number plus one.

Step 4. Determine whether it is the last atom; if not, go back to step 1, or go to the column search process.

Step 5. Start the column search and get the new atom data, that is, search the atoms from column to column.

Step 6. Determine if the atom is in contact. If it is not in contact, go back to step 5. If it is in contact, move to the next step.

Step 7. Compare the current atom with the previous atom in the same column. If the previous atom is also in contact and belongs to a different cluster, merge the current cluster into the cluster to which the previous atom belongs, then label the atoms with the same number and store them. If the previous atom is not in contact or belongs to the same cluster, move to the next step.

Step 8. Determine whether the current atom is the last atom; if not, go back to step 5, or the search process is done.

### Numerical Experimental Design

*E**, which is written as

*E*

_{1}and

*E*

_{2}are the elastic modulus of the upper surface and the lower surface, respectively. For simplicity, we consider a rigid rough surface contacting with an elastic smooth surface and then study the formation and development of the contacting cluster and its force-area behavior. In the following discussion, we will use the surface generated above (the upper surface is rigid and rough (

*E*

_{1}= ∞), and the lower surface is smooth and elastic (

*E*

_{2}= 3

*ε*/

*σ*

^{3})) to study the contact behavior, where both of

*v*

_{1}and

*v*

_{2}are set to be 0.5. Our system size is 512 × 512 atoms (with initial spacing equal to 1.12

*σ*), and periodic boundary conditions are used in the

*x*-

*y*plane. The elastic block depth is set to be 1024 atomic layers (with initial spacing equal to 1.12

*σ*). In a regular molecular dynamics simulation, the system will be comprised of 268,697,600 atoms; the GFMD model reduces the number to 524,288 (two layers of atoms), as seen in Fig. 1. We gradually push the rough surface (on the top) into the flat elastic surface. The loading of the rigid surface is controlled by the displacement. Each displacement loading step is set to 0.01

*σ*, and the GFMD algorithm will update each atom’s position until the atomic force meets the convergence criteria

*L*

^{1}-norm = 0.01

*ε/σ.*The maximum iteration number is set to be 50,000 to avoid an endless loop.

## Results and Discussion

### Contacting Clusters’ Distribution and Development

### Contact Area-Load Relationship

*F*can be expressed as

*Z*

_{ i }is the asperity height,

*d*is the displacement applied at the rigid surface, and

*f*is the asperity contact force based on the Hertz contact theory.

*F*in the asperity model is 1.80 times than that predicted by GFMD, and

*F*in the G-W model is 1.54 times than that predicted by GFMD. This can be explained by the RMS slope of the contact clusters. It is known that the normal load is proportional to the RMS slope, that is, \( L\propto \sqrt{M_2} \). In the GFMD model, the contact area is composed of a greater number of clusters, whose penetrations are shallower than that of the asperity model. Since the asperity tip’s slope is smaller, the RMS slope for the contact cluster in the GFMD model is also smaller. Figure 10 shows the contact cluster’s RMS slopes for the three models. It can be seen that the contact clusters’ RMS slope in GFMD is less than that of surface RMS slope of 0.077, while both of the other two models’ contact cluster RMS slopes are larger than that of the surface RMS slope.

## Conclusions

To find the evolution of the contact area from atomic scale to macroscopic scale, the rough fractal surface contact problem has been studied using the GFMD model. We defined the atomic contact by the existence of a force larger than zero and studied three different length scales in the same system. It is found that the inter-asperity interaction is essential to the forming of a contact cluster. Some clusters are close enough that they can merge into a big one. The real contact region is far more complicated than that predicted by surface geometry due to elastic deformation in the elastic smooth surface. Most of the contact cluster locations are coincident with the surface summits/valleys. However, the cluster’s size is smaller, and its formation is not determined by the surface asperity heights. As the contact area increases, both summits and valleys can form as contact clusters. In the GFMD model, the force is much smaller than that of the asperity model, while the contact cluster number in the GFMD model is much larger. The RMS slope of the contact clusters in the GFMD model is smaller than that of the asperity model, which explains why the asperity model leads to higher pressure. Our findings suggest that the real contact area cannot be predicted simply by the surface geometry. The actual contact area with the normal load is of importance for the following research on friction.

## Declarations

### Acknowledgements

Shiping Huang thanks Mark O. Robbins (Johns Hopkins University), Tristan A. Sharp (University of Pennsylvania), and Lars Pastewka (Albert-Ludwigs-Universität Freiburg) for providing the GFMD codes and many useful discussions during his postdoctoral research at the Department of Physics and Astronomy, Johns Hopkins University.

### Funding

This material is based on the work supported by the National Natural Science Foundation of China under Grant Nos. 11202080 and11672108 and supported by “the Fundamental Research Funds for the Central Universities”. Part of this work is funded by the Research Fund of The State Key Laboratory of Coal Resources and Safe Mining, CUMT (12KF07).

### Competing Interests

The author declares no competing interests.

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## Authors’ Affiliations

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