Proceedings of WTC2005 World Tribology Congress III

September 12-16, 2005, Washington, D.C., USA

WTC2005-63815

A TWO-DIMENSIONAL ADAPTIVE ELASTO-PLASTIC CONTACT MODEL OF ROUGH SURFACES

Tianxiang Liu, Geng Liu, Qin Xie

School of Mechatronic Engineering, Northwestern Polytechnical University, Xi’an, 710072, P.R. China

Proceedings of WTC2005 World Tribology Congress III

September 12-16, 2005, Washington, D.C., USA

WTC2005-63815

ABSTRACT

When contact problems are solved by numerical approaches, the surface profile is usually described by a series of discrete nodes with the same intervals along the coordinate axis. An adaptive-surface-based elasto-plastic asperity contact model is presented in this paper. Such a model is developed in order to reduce the computing time by removing the surface nodes that have little influence on the contact behavior of rough surfaces. The removed nodes are determined by setting a threshold. Thus, the contact problems can be described by fewer surface nodes but have similar results to the ones of the original surface. The adaptive asperity contact model is solved by using the element-free Galerkin-finite element (EFG-FE) coupling method because of its flexibility in domain descritization and versatility in node arrangements. The effects of different thresholds on the contact pressure distributions, real contact area, and the elasto-plastic stress fields in the contacting bodies are investigated and discussed. The results show that the computational time will dramatically reduce to about 50% when the relative error is about 5%.

INTRODUCTION

Real surfaces are not perfectly smooth, where micro/nano irregular protuberances exist. The description of rough surfaces usually includes experimental measurement and mathematical description, such as the GW statistical contact model [1] and the fractal model [2], etc. However, a surface profile is often described by a series of discrete nodes with the same intervals along coordinate axes no matter the surface is generated by mathematical methods or digitized with a measuring instrument.

In computer graphics, many researches have noticed the adaptive description of surface topography that reduces the number of control points on surface while maintaining interpolation precise [3]. Based on these ideas, the rough surfaces in contact problems can also be described as new adaptive ones in which the nodes on the rough parts of the surfaces are fine while those on the smooth parts are coarse. However, if the adaptive technique is used, it is somewhat

difficult to construct the FEM meshes to adapt the surface description. The flexibility in node arrangement of element-free Galerkin-finite element (EFG-FE) coupling method [4] can make an adaptive-surface-based elasto-plastic asperity contact model possible. Details for such a model are presented in this paper to show how the requirement of irregular surface description after adaptive process is satisfied. A few factors influencing the elasto-plastic contact of adaptive rough surface are studied. MODEL DESCRIPTION

The adaptive-surface-based asperity contact model developed in this work utilizes an threshold, ε , to decide which part of the curve should be scattered into coarse or fine nodes. For a curve consisting of three nodes, if the vertical distance of the middle node, p2, to the section of the line connected by the two nodes, p1 and p3, adjacent to the middle node, is less than a given threshold,ε , the middle node can be removed. Otherwise, the middle node should remain.

After generating the adaptive surface, the linear programming technique combined with the initial-stiffness method and EFG-FE coupling method can be utilized to establish an incremental iteration algorithm for the elasto-plastic contact problems of real rough surfaces [4, 5]. RESULTS AND DISCUSSION

Considering a rough surface described originally by 129 nodes, five values of threshold, 3100.5 −×=ε , 4100.5 −× ,

4100.2 −× , 4103.1 −× , and 5100.5 −× corresponding to NPA=65, 74, 85, 96 and 109 are used to investigate the influence of thresholds on the distributions of contact pressures, where NPA is the number of nodes which remained on the surface after adaptive process. The length of the calculation region is L=0.128mm. The nominal applied load is P=50N, which corresponds to a nominal pressure of 390.625MPa. Fig. 1 gives the comparison of the contact pressure when the threshold, 5100.5 −×=ε , with that obtained with non-adaptive method. The two curves are very close. Fig. 2 gives

1 Copyright © 2005 by ASME

the relationship between the calculation time and the relative errors of the maximum contact pressures of the four contact regions, shown in Fig. 1 (Peak 1 to Peak 4), under different thresholds gave above. It shows that when the relative error less than 5%, the computing time is about 50% of that used in the non-adaptive calculation.

Figure 3 shows the comparison of the von Mises stress contours in the meshless region between the adaptive calculation when threshold 5100.5 −×=ε and those obtained with the non-adaptive method. The two stress distributions are very similar.

Figure 4 gives the relationship between the non-dimensional average gap ( rmshh /T= ) and relative errors of the contact pressures, p , to that from the non-adaptive model with different thresholds, where Th is the average gap of the surface after deformation, and rms the roots mean square roughness. It shows that the relative errors of contact pressure for NPA=109 with different average gap are smaller than 7.04%, and the errors become larger when the surface nodes were further reduced.

CONCLUSIONS

An adaptive-surface-based contact model is presented. Such a model is used to remove the nodes that have little influence on the description of a nodal-expressed numerical rough surface. The influences of different thresholds on the contact pressure distribution, the stress distribution in the elasto-plastic contact are studied by using the EFG-FE coupling method. The results indicate that the adaptive model can save the calculation time to about 50% as compared with non-adaptive method when the relative errors are about 5%.

ACKNOWLEDGMENTS The research was supported by the National Natural

Science Foundation of China (50475146), the Specialized Research Fund for the Doctoral Program of Higher Education of MOE (20030699035), the Natural Science Foundation of Shaanxi Province (2004E225) and the Doctorate Creation Foundation of Northwestern Polytechnical University (CX200312).

REFERENCES [1] Greenwood, J.A., Williamson, J.B.P., 1966, “The Contact

of Nominally Flat Surfaces,” Proc. R. Soc. London, Ser. A, A295, pp. 300-319

[2] Borri-Brunetto, M., Carpinteri, A., and Chiaia, B., 1998, “Lacunarity of the Contact Domain Between Elastic Bodies with Rough Boundaries,” Probamat-21st Century: Probabilities and Materials, G. Frantziskonis, ed., Kluwer, Dordrecht, pp. 45-64

[3] Piegl, Les. A., Tiller, W., 2000, “Reducing Control Points in Surface Interpolation,” IEEE Comput. Graphics Appl., 20(5), pp. 70-74

[4] Liu, T., Liu, G., Wang, Q., 2005, “An Element-Free Galerkin-Finite Element Coupling Method for Elasto-Plastic Contact Problems,” ASME J. Tribol., to appear

[5] Liu, G., Zhu, J., Yu, L., Wang, Q., 2001, “Elasto-Plastic Contact of Rough Surfaces,” STLE Tribol. Trans., 44, pp. 437-443

FIGURES

0.10 0.20 0.30 0.40 0.50 0.60Time/Time(Non-adaptive)

-60.00

-20.00

20.00

-80.00

-40.00

0.00

40.00

Rel

ativ

e er

ror (

%)

Peak 1Peak 2Peak 3Peak 4

Figure 1. The comparison of the contact pressures

Figure 2. Relative errors of the maximum contact pressures

(a) 5100.5 −×=ε , 109A =NP

(b) Non-adaptive, 129A =NP Figure 3. The von Mises stress contours

Figure 4. Non-dimensional average gap vs contact pressures

0.00 0.20 0.40 0.60 0.80 1.00X/L

0.00

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4.00

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/Cyd

Non-adaptiveAdaptive

Peak 4

Peak 3

Peak 2

Peak 1 , NPA=129, NPA=109

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5

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0.0 0.5 1.0 1.5 2.0 2.5Non-dimensional average gap

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)

NP =65NP =74NP =85NP =96NP =109A

AAAA

2 Copyright © 2005 by ASME