Elsevier

Wear

Volume 259, Issues 7–12, July–August 2005, Pages 1408-1423
Wear

A numerical three-dimensional contact model for rough, multilayered elastic/plastic solid surfaces

https://doi.org/10.1016/j.wear.2005.02.014Get rights and content

Abstract

A numerical three-dimensional contact model is presented to investigate the contact behavior of rough, multilayered elastic-perfectly plastic solid surfaces. The model is based on a variational principle in which the contact pressure distributions are those which minimize the total complementary potential energy. The quasi-Newton method is used to find the minimum. The influence coefficients of the displacements and stresses for a multilayered contact model are determined using the Papkovich–Neuber potentials with a fast fourier transform (FFT) based scheme. Contact analyses are performed to study the effects of layer-to-substrate ratios of stiffness and surface roughness on the contact statistics of rough, two-layered elastic/plastic solids. Local contact pressure profiles, von Mises stresses, tensile and shear stress contours, and the maximum displacement as a function of material properties and applied load are generated. The results yield insight into the effects of stiffness of layers and substrates, surface roughness and applied load on the contact performance. Optimum layer parameters are identified to provide low friction/stiction and wear.

Introduction

The deposition of thin layers, ranging from a few nanometers to a few microns, is an effective way to improve the tribological properties of contacting surfaces because it affects the maximum contact pressure, the real area of contact and surface and subsurface stresses [1], [2], [3]. The maximum contact pressure, real area of contact and surface and subsurface stresses contribute to friction and wear of two contacting rough surfaces, which are functions of surface roughness, material properties and interfacial loading conditions. Repeated surface interactions and surface and subsurface stresses, developed at the interface, result in the formation of wear particles and eventual failure [4]. So a lower contact pressure, smaller real area of contact and lower surface and subsurface stresses are preferred to minimize friction/stiction and wear. Appropriately chosen layers can reduce coefficient of friction and wear rate without having to change the bulk material. The use of these layers often requires the deposition of supporting interlayers to provide the necessary adhesion and toughness, which makes the design of a multilayered solid necessary.

Several numerical models have been developed to find the pressure distribution for a layered solid subjected to prescribed loading and boundary displacement. For analyses of a single asperity contact, stress and deformation equations are written in terms of harmonic functions, e.g., Green function in two dimensions and Papkovich–Neuber potentials in three-dimensions (3D). Fast fourier transformation (FFT) is commonly used to speed up the computation. Numerical stress and deformation analyses for rigid or elastic cylindrical and spherical indenters contacting an elastic half space bonded to a elastic layer in frictionless and frictional contacts have been performed by several researchers. Kral and Komvopoulos [5] developed a 3D model to analyze the elastic–plastic contact of a rigid smooth sphere sliding on a layered, smooth and flat solid; Plumet and Dubourg [6] studied the contact of a smooth rigid ellipsoid with a layered elastic half space using FFT scheme.

Multiple asperity contact analysis of two elastic, single-layered rough surfaces has been carried out by various investigators. Merriman and Kannel [7] developed a 2D model to analyze the effect of surface roughness on contact stresses for the case of soft layered, rough elastic cylinder in contact with a rough flat surface. In their study, the deflection was first assumed and the pressure distribution was calculated using Green function approach. Cole and Sayles [8] and Mao et al. [9] developed 2D models to analyze the sliding line contact under both normal and tangential loads of two elastic rough flat surfaces, one of which had a layer. They used Green function to obtain the so-called influence coefficients matrix for their 2D models. Nogi and Kato [10] used Papkovich–Neuber potentials to formulate the 3D problem for a rough surface in contact with a layered rough surface. A conventional matrix inversion technique and iterative process were used to obtain the contact pressure distribution and real area of contact, and the conjugate gradient method [11], [12] was applied to solve the system of linear equations which relate pressure to displacement, for unknown pressures during the iterative process. These techniques are good for rough surfaces with moderate number of contact points. With the increase of contact points and layers, the influence matrices can become very large and possibly ill-conditioned due to round-off errors. This may result in non-convergence in some cases. In order to avoid this problem, a variational principle was introduced by Tian and Bhushan [13] for a homogeneous solid contact problem. According to the variational principle, the actual contact pressure distributions are those which minimize the total complementary potential energy. They used the Newton method to find the minimum complementary potential energy. The variational approach was extended by Peng and Bhushan [14] who used the Papkovich–Neuber potentials to derive the influence matrix to compute the contact statistics of a 3D single-layered rough surfaces contact model. The quasi-Newton method, a bounded constrained indefinite quadratic programming method for solving optimization problem, was used to find the minimum complementary potential energy. This method has been proven to be a feasible and efficient way to guarantee the uniqueness of the solution even in the case when the matrix inversion method does not result in iterative convergence.

Many applications have a multilayered structure. Examples are the multilayered construction of magnetic heads and disks in magnetic rigid disk drives, head tape devices, and various components in microelectromechanical systems (MEMS) devices. Layers applied in these devices are as thin as 1 nm. The tribological properties of these layered solids directly depend on the layer properties, e.g., the layer thickness, surface roughness, coefficient of friction, stiffness and hardness ratios of the layers to the substrate. In order to obtain optimal layers, e.g., a top layer possessing a longer life as well as lower friction, it is necessary to investigate friction and wear mechanisms of contact of multilayered solid with rough surfaces through theoretical or empirical analyses. The current model extends a single-layered contact model for the contact of rough, multilayered solid surfaces. The Papcovich–Neuber potentials were used to derive the influence coefficient matrices for multilayered solid, which leads to a set of linear equations, and Gaussian elimination is used to solve these linear equations. Contact analyses are performed to identify optimum layer parameters in order to provide low friction/stiction and wear.

Section snippets

Multiple asperity contact of multiple layered rough surfaces

Schematic of multiple asperity contact of a rough, multilayered rough surface in contact with another surface is shown in Fig. 1. Based on the variational principle, an algorithm is developed to find the minimum value of the total complementary potential energy of the rough multilayered elastic and elastic-perfectly plastic layered solid surfaces in contact. The influence coefficients matrix which describes the applied pressure–displacement and applied pressure–stress relations in a layered

Results and discussion

The analysis carried out here is for a rigid surface in a frictionless contact with a rough elastic-perfectly plastic two-layered solid surface. The upper surface is assumed to be rigid to simplify the problem. Young's modulus of the substrate E3 is taken as 100 GPa. The pressure is normalized by E3. Poisson's ratios are taken as 0.3 for all the cases. The hardness of the substrate is taken as 0.05E3. Two sets of simulations are performed to study the stiffness, normal contact pressure, surface

Conclusions

A numerical 3D model for the contact analysis of rough, two-layered elastic and elastic-perfectly plastic surfaces has been developed. The model is based on the variational principle according to which the actual contact pressure distributions are those with minimum total complementary potential energy. The quasi-Newton method is used to find the minimum. The influence coefficients for rough, multilayered solid are determined using the Papkovich–Neuber potentials with an FFT scheme.

The results

Acknowledgements

Financial support for this research is provided by the industrial membership of the Nanotribology Laboratory for Information Storage and MEMS/NEMS (NLIM). Thanks are also given to Dr. Wei Peng for the useful discussions during the course of the work.

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