Tribology Letters

, 66:146

# Theoretical and Finite Element Analysis of Static Friction Between Multi-Scale Rough Surfaces

• Xianzhang Wang
• Yang Xu
• Robert L. Jackson
Original Paper

## Abstract

The current work considers the multi-scale nature of roughness in a new model that predicts the static friction coefficient. This work is based upon a previous rough surface contact model, which used stacked elastic–plastic 3-D sinusoids to model the asperities at multiple scales of roughness. A deterministic model of a three-dimensional deformable rough surface pressed against a rigid flat surface is also carried out using the finite element method (FEM). The accuracy of the deterministic FEM model is also considered. At the beginning of contact, which is surface-point contact, the asperities or peaks are isolated, sharp, and the contact areas consist of an inadequate number of elements and sources of error. In this range of contact, the results are not presented as real or accurate. As the normal load increases, the number of the contact elements become larger, and thus, the results become more accurate. That is, the deterministic FEM results are most accurate at high loads. Spectral interpolation is used to smooth the geometry in between the original measured nodes. The effects of normal load and plasticity index on static friction are then analyzed. The results predicted by the theoretical model are also compared to other existing rough surface friction contact models and the FEM results. They are in a good qualitative agreement, especially for higher loads and higher plasticity indices. The FEM model also has significant error, but it is more accurate at higher loads where the proposed multi-scale static friction model and FEM model are in better agreement.

## Keywords

Contact mechanics Static friction Surface roughness analysis and models

## Nomenclature

A

Area of contact

A0

Contact area under normal preload only

An

Nominal contact area of the surface

Ar

Real contact area

As

Contact area at sliding inception

C

Critical yield stress coefficient

E

Elastic modulus

E′

$${E \mathord{\left/ {\vphantom {E {\left( {1 - {v^2}} \right)}}} \right. \kern-0pt} {\left( {1 - {v^2}} \right)}}$$

f

Spatial frequency (reciprocal of wavelength)

F

Contact force for single asperity

Ff

Friction force

Fn

$$F_{n}^{*}$$

Ft

L

Scan length

N

Number of nodes

$${N_e}$$

Number of elements

$${p^{\text{*}}}$$

Average pressure to cause complete contact (Elastic)

$$p_{{ep}}^{*}$$

Average pressure to cause complete contact (Elastic–plastic)

$$\bar {p}$$

Average pressure over the entire surface

$${S_y}$$

Yield strength

$${u_x}$$

Displacement in the x direction

## Greek symbols

$$\lambda$$

Asperity wavelength

$$\Delta$$

Asperity amplitude

$${\Delta _c}$$

Critical asperity amplitude

$$\psi$$

Sinusoidal asperity parameter

$$\Psi$$

Plasticity index

$$\sigma$$

Standard derivation on the surface heights

$${\sigma _s}$$

Standard derivation on the asperity heights

$$\nu$$

Poisson’s ratio

$${\tau _c}$$

Critical interfacial shear strength

$${\omega _0}$$

$${\omega _c}$$

Critical interference (full stick condition)

$${\omega _{cs}}$$

Critical interference (perfect slip condition)

$${\mu _s}$$

Static friction coefficient

$${\mu _s}$$

Static friction coefficient

## Subscripts

c

Critical value at the onset of plastic deformation (full stick condition)

ave

Average value

max

Maximum value

ep

Elastic–plastic

JGH

From model by Johnson, Greenwood, and Higgson [1]

x

In the x direction

## References

1. 1.
Kogut, L., Etsion, I.: A static friction model for elastic-plastic contacting rough surfaces. J. Tribol. 126(1), 34–40 (2004).
2. 2.
Bowden, F.P., Bowden, T.D.: Friction and lubrication of solids. Clarendon, Oxford, (1954)Google Scholar
3. 3.
Kogut, L., Etsion, I.: A semi-analytical solution for the sliding inception of a spherical contact. J. Tribol. 125(3), 499–506 (2003).
4. 4.
Brizmer, V., Kligerman, Y., Etsion, I.: Elastic–plastic spherical contact under combined normal and tangential loading in full stick. Tribol. Lett. 25(1), 61–70 (2007)
5. 5.
Eriten, M., Polycarpou, A.A., Bergman, L.A.: Physics-based modeling for partial slip behavior of spherical contacts. Int. J. Solids Struct. 47(18–19), 2554–2567 (2010).
6. 6.
Wu, A.Z., Shi, X., Polycarpou, A.A.: An Elastic–plastic spherical contact model under combined normal and tangential loading. J. Appl. Mech. 79(5) (2012).
7. 7.
Wang, X., Xu, Y., Jackson, R.L.: Elastic–Plastic sinusoidal waviness contact under combined normal and tangential loading. Tribol Lett 65(2), 45 (2017)
8. 8.
Chang, W.R., Etsion, I., Bogy, D.B.: Static friction coefficient model for metallic rough surfaces. J. Tribol. 110(1), 57–63 (1988)
9. 9.
Kogut, L., Etsion, I.: A finite element based elastic-plastic model for the contact of rough surfaces. Tribol. T. 46(3), 383–390 (2003). doi:
10. 10.
Cohen, D., Kligerman, Y., Etsion, I.: A model for contact and static friction of nominally flat rough surfaces under full stick contact condition. J. Tribol. 130(3) (2008).
11. 11.
Li, L., Etsion, I., Talke, F.E.: Contact area and static friction of rough surfaces with high plasticity index. J. Tribol. 132(3) (2010).
12. 12.
Greenwood, J., Williamson, J.P.: Contact of nominally flat surfaces. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 1966, pp. 300–319. The Royal SocietyGoogle Scholar
13. 13.
Etsion, I., Amit, M.: The effect of small normal loads on the static friction coefficient for very smooth surfaces. J. Tribol. 115(3), 406–410 (1993). doi:
14. 14.
Cohen, D., Kligerman, Y., Etsion, I.: The effect of surface roughness on static friction and junction growth of an elastic-plastic spherical contact. J. Tribol. 131(2) (2009).
15. 15.
Mccool, J.I.: Relating profile instrument measurements to the functional performance of rough surfaces. J. Tribol. 109(2), 264–270 (1987)
16. 16.
Jackson, R.L., Green, I.: A finite element study of elasto-plastic hemispherical contact against a rigid flat. J. Tribol. 127(2), 343–354 (2005).
17. 17.
Kogut, L., Jackson, R.L.: A comparison of contact modeling utilizing statistical and fractal approaches. J. Tribol. 128(1), 213–217 (2006). doi:
18. 18.
Jackson, R.L., Green, I.: On the modeling of elastic contact between rough surfaces. Trib. Trans. 54(2), 300–314 (2011)
19. 19.
Tayebi, N., Polycarpou, A.A.: Modeling the effect of skewness and kurtosis on the static friction coefficient of rough surfaces. Tribol. Int. 37(6), 491–505 (2004).
20. 20.
Lee, C.H., Polycarpou, A.A.: Static friction experiments and verification of an improved elastic-plastic model including roughness effects. J. Tribol. 129(4), 754–760 (2007).
21. 21.
Lee, C.H., Eriten, M., Polycarpou, A.A.: Application of elastic-plastic static friction models to rough surfaces with asymmetric asperity distribution. J. Tribol. 132(3), 031602 (2010).
22. 22.
Dickey, R.D.I., Jackson, R.L., Flowers, G.T.: Measurements of the static friction coefficient between tin surfaces and comparison to a theoretical model. J. Tribol. 133(3), 031408 (2011).
23. 23.
Archard, J.: Elastic deformation and the laws of friction. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 1957, pp. 190–205. The Royal Society (1957)Google Scholar
24. 24.
Ciavarella, M., Demelio, G., Barber, J.R., Jang, Y.H.: Linear elastic contact of the Weierstrass profile. Proc. Royal Soc. A 456, 387–405 (2000).
25. 25.
Westergaard, H.M.: Bearing prssures and cracks. Journal of Applied Mechanics-Transactions of the Asme 6, 49–53 (1939)Google Scholar
26. 26.
Jackson, R.L., Streator, J.L.: A multi-scale model for contact between rough surfaces. Wear. 261(11–12), 1337–1347 (2006).
27. 27.
Gao, Y.F., Bower, A.F.: Elastic-plastic contact of a rough surface with Weierstrass profile. Proc. Royal Soc. A 462(2065), 319–348 (2006).
28. 28.
Wilson, W.E., Angadi, S.V., Jackson, R.L.: Surface separation and contact resistance considering sinusoidal elastic–plastic multi-scale rough surface contact. Wear 268(1), 190–201 (2010)
29. 29.
Pastewka, L., Robbins, M.O.: Contact between rough surfaces and a criterion for macroscopic adhesion. Proc. Natl. Acad. Sci. USA. 111(9), 3298–3303 (2014).
30. 30.
Polonsky, I.A., Keer, L.M.: Fast methods for solving rough contact problems: a comparative study. J. Tribol. 122(1), 36–41 (2000). doi:
31. 31.
Yang, C., Persson, B.N.J., Israelachvili, J., Rosenberg, K.: Contact mechanics with adhesion: interfacial separation and contact area. Europhys. Lett. 84(4) (2008).
32. 32.
Persson, B.N.J., Scaraggi, M.: Theory of adhesion: role of surface roughness. J. Chem. Phys. 141(12) (2014).
33. 33.
Wu, J.J.: Numerical analyses on elliptical adhesive contact. J. Phys. D 39(9), 1899–1907 (2006).
34. 34.
Ilincic, S., Vernes, A., Vorlaufer, G., Hunger, H., Dorr, N., Franek, F.: Numerical estimation of wear in reciprocating tribological experiments. Proc. Inst. Mech. Eng. J 227(J5), 510–519 (2013).
35. 35.
Solhjoo, S., Vakis, A.I.: Continuum mechanics at the atomic scale: Insights into non-adhesive contacts using molecular dynamics simulations. J. Appl. Phys. 120(21) (2016).
36. 36.
Rostami, A., Jackson, R.L.: Predictions of the average surface separation and stiffness between contacting elastic and elastic-plastic sinusoidal surfaces. Proc. Inst. Mech. Eng. J 227(12), 1376–1385 (2013).
37. 37.
Rostami, A., Streator, J.L.: Study of liquid-mediated adhesion between 3D rough surfaces: a spectral approach. Tribol. Int. 84, 36–47 (2015)
38. 38.
Medina, S., Dini, D.: A numerical model for the deterministic analysis of adhesive rough contacts down to the nano-scale. Int. J. Solids Struct. 51(14), 2620–2632 (2014).
39. 39.
Putignano, C., Afferrante, L., Carbone, G., Demelio, G.: A new efficient numerical method for contact mechanics of rough surfaces. Int. J. Solids Struct. 49(2), 338–343 (2012).
40. 40.
Afferrante, L., Carbone, G., Demelio, G.: Interacting and coalescing Hertzian asperities: a new multiasperity contact model. Wear. 278, 28–33 (2012).
41. 41.
Muser, M.H., Dapp, W.B., Bugnicourt, R., Sainsot, P., Lesaffre, N., Lubrecht, T.A., Persson, B.N.J., Harris, K., Bennett, A., Schulze, K., Rohde, S., Ifju, P., Sawyer, W.G., Angelini, T., Esfahani, H.A., Kadkhodaei, M., Akbarzadeh, S., Wu, J.J., Vorlaufer, G., Vernes, A., Solhjoo, S., Vakis, A.I., Jackson, R.L., Xu, Y., Streator, J., Rostami, A., Dini, D., Medina, S., Carbone, G., Bottiglione, F., Afferrante, L., Monti, J., Pastewka, L., Robbins, M.O., Greenwood, J.A.: Meeting the contact-mechanics challenge. Tribol. Lett. 65(4) (2017).
42. 42.
Ning, Y., Polycarpou, A.A.: Extracting summit roughness parameters from random Gaussian surfaces accounting for asymmetry of the summit heights. J. Tribol. 126(4), 761–766 (2004).
43. 43.
Yu, N., Polycarpou, A.A.: Combining and contacting of two rough surfaces with asymmetric distribution of asperity heights. J. Tribol. 126(2), 225–232 (2004).
44. 44.
Xu, Y.: An analysis of elastic rough contact models. (2012)Google Scholar
45. 45.
Jackson, R.L., Bhavnani, S.H., Ferguson, T.P.: A multi-scale model of thermal contact resistance between rough surfaces. ASME J. Heat Transfer 130, 081301 (2008)
46. 46.
Jackson, R.L., Ghaednia, H., Elkady, Y.A., Bhavnani, S.H., Knight, R.W.: A closed-form multiscale thermal contact resistance model. IEEE Trans. Compon. Packag. Manuf. Technol. 2(7), 1158–1171 (2012)
47. 47.
Almeida, L., Ramadoss, R., Jackson, R., Ishikawa, K., Yu, Q.: Laterally actuated multicontact MEMS relay fabricated using MetalMUMPS process: experimental characterization and multiscale contact modeling. J. Micro/Nanolith. MEMS MOEMS 6(2), 023009 (2007)
48. 48.
Johnson, K.L., Greenwood, J.A., Higginson, J.G.: The contact of elastic regular wavy surfaces. Int. J. Mech. Sci. 27(6), 383 (1985). doi:
49. 49.
Jackson, R.L., Krithivasan, V., Wilson, W.E.: The pressure to cause complete contact between elastic-plastic sinusoidal surfaces. Proc. Inst. Mech. Eng. J 222(J7), 857–863 (2008).
50. 50.
Krithivasan, V., Jackson, R.L.: An analysis of three-dimensional elasto-plastic sinusoidal contact. Tribol. Lett. 27(1), 31–43 (2007).
51. 51.
Ghaednia, H., Wang, X., Saha, S., Jackson, R.L., Xu, Y., Sharma, A.: A Review of elastic-plastic contact mechanics. Appl. Mech. Rev. 69(6), 060804 (2017)
52. 52.
Jackson, R.L.: An analytical solution to an archard-type fractal rough surface contact model. Tribol. T. 53(4), 543–553 (2010).
53. 53.
Sahli, R., Pallares, G., Ducottet, C., Ali, B., Al Akhrass, I.E., Guibert, S., Scheibert, M.: J.: Evolution of real contact area under shear and the value of static friction of soft materials. Proc. Natl. Acad. Sci. USA. 115(3), 471–476 (2018).
54. 54.
Ovcharenko, A., Halperin, G., Etsion, I., Varenberg, M.: A novel test rig for in situ and real time optical measurement of the contact area evolution during pre-sliding of a spherical contact. Tribol. Lett. 23(1), 55–63 (2006).
55. 55.
Brizmer, V., Kligerman, Y., Etsion, I.: A model for junction growth of a spherical contact under full stick condition. J. Tribol. 129(4), 783–790 (2007).
56. 56.
Ovcharenko, A., Halperin, G., Etsion, I.: In situ and real-time optical investigation of junction growth in spherical elastic-plastic contact. Wear. 264(11–12), 1043–1050 (2008).
57. 57.
Zolotarevskiy, V., Kligerman, Y., Etsion, I.: The evolution of static friction for elastic–plastic spherical contact in pre-sliding. J. Tribol. 133(3) (2011).
58. 58.
Jackson, R.L., Green, I.: A statistical model of elasto-plastic asperity contact between rough surfaces. Tribol. Int. 39(9), 906–914 (2006).

## Authors and Affiliations

1. 1.Mechanical Engineering DepartmentAuburn UniversityAuburnUSA