Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

The influence of fractal dimension in the microcontact of three-dimensional elastic-plastic fractal surfaces

  • 328 Accesses

  • 2 Citations


Microcontact affects the fabrication and assembly of MEMS/NEMS system, the contact between thin film microcavity and microcantilever beam, and the motion of microstructure. In this work, in the microcontact of three-dimensional elastic-plastic Weierstrass-Mandelbrot (W-M) fractal surfaces, influence of fractal dimension was studied based on a comprehensive contact model. With increasing fractal dimension, maximum microcontact force in the plastic deformation zone shows parabolic change; comparably, the intermediate force in elastic zone parabolically varies. For both the forces, the minimum values are obtained when the fractal dimension is 2.5. Besides, in the plastic deformation zone, the real contact areas increase with the fractal dimension. Experiments were completed to compare with the numerical analysis. The results show that the simulated contact force curve is in line with the experimental load curve when Young’s modulus E and hardness H are equal to the actual measured values. Nevertheless, it will greatly deviate from the experimental load curve when E and H differ from the measured values.

This is a preview of subscription content, log in to check access.


  1. 1.

    Jazairy A (2013) From MEMS to NEMS: smart chips with senses and muscles. Symp Des Test Integration Packing MEMS/MOEMS (DTIP) 20:1–3

  2. 2.

    Zhang WM, Yan H, Peng ZK, Meng G (2014) Electrostatic pull-in instability in MEMS/NEMS: a review. Sensor Actuat A-Phys 214:187–218. https://doi.org/10.1016/j.sna.2014.04.025

  3. 3.

    Du H, Chau FS, Zhou GY (2016) Mechanically-tunable photonic devices with on-chip integrated MEMS/NEMS actuators. Micromachines 7(4):69. https://doi.org/10.3390/mi7040069

  4. 4.

    Jiang ZD, Zhou XY, Zhu Q, Zhao ZX, Wang HR, Prewett PD, Jiang K (2009) Recent study on the related problems in evaluation of mechanical properties for MEMS materials. Int J Appl Mech 1(04):765–779. https://doi.org/10.1142/S1758825109000332

  5. 5.

    Hoang TV, Wu L, Paquay S, Golinval JC, Arnst M, Noels L (2017) A stochastic multi-scale model for predicting MEMS stiction failure. Micro Nanomech 5:1–8

  6. 6.

    Broer W, Palasantzas G, Knoester J, Svetovoy VB (2013) Significance of the Casimir force and surface roughness for actuation dynamics of MEMS. Phys Rev B 87(12):125413. https://doi.org/10.1103/PhysRevB.87.125413

  7. 7.

    Qiu H, Wang H, Ke F (2013) Instability of contact resistance in MEMS and NEMS DC switches under low force: the role of alien films on the contact surface. Sensors 13:16360–16371

  8. 8.

    Morrow CA (2003) Adhesive rough surface contact, Pittsburgh University PH.D. dissertation

  9. 9.

    Basu A, Adams G, McGruer NE (2016) A review of micro-contact physics, materials, and failure mechanisms in direct-contact RF MEMS switches. J Micromech Microeng 26: 104004

  10. 10.

    Toler BF, Coutu RAJ, McBride JW (2013) A review of micro-contact physics for microelectromechanical systems (MEMS) metal contact switches. J Micromech Microeng 23(10):103001. https://doi.org/10.1088/0960-1317/23/10/103001

  11. 11.

    Wu Y, Peroulis D (2015) Contact behavior evolution induced by damage growth in radio-frequency microelectromechanical system switches. J Appl Phys 117(6):064505. https://doi.org/10.1063/1.4907803

  12. 12.

    Lin QJ, Yang SM, Wang CY, Ding JJ, Jiang ZD (2014) Multifractal analysis for Cu/Ti bilayer thin films. Surf Interface Anal 45:1223–1227

  13. 13.

    Sayles RS, Thomas TR (1978) Surface-topography as a nonstationary random process. Nature 271(5644):431–434. https://doi.org/10.1038/271431a0

  14. 14.

    Borodich FM, Pepelyshev A, Savencu Q (2016) Statistical approaches to description of rough engineering surfaces at nano and microscales. Tribol Int 103:197–207. https://doi.org/10.1016/j.triboint.2016.06.043

  15. 15.

    Le Goic G, Bigerelle M, Samper S, Favreliere H, Pillet M (2015) Multiscale roughness analysis of engineering surfaces: a comparison of methods for the investigation of functional correlations. Mech Syst Signal Process 66-67:437–457

  16. 16.

    Jiang CX, Lu ZX, Jin Z, Memon MS (2017) Evaluation of fractal dimension of soft terrain surface. J Terramech 70:27–34. https://doi.org/10.1016/j.jterra.2017.01.003

  17. 17.

    Lin, QJ, Yang SM, Wang CY, Yuan GY (2012) Surface characterization of Cu/Ti thin films by fractal analysis. 12th IEEE International Conference on Nanotechnology (IEEE-NANO 2012) 1–4

  18. 18.

    Shi JP, Cao XS, Zhu H (2014) Tangential contact stiffness of rough cylindrical faying surfaces based on the fractal theory. J Tribol-T Asme 136(4):041401. https://doi.org/10.1115/1.4028042

  19. 19.

    Wu DG, Gang TQ, Fan XG, Wu JL (2014) Surface roughness measuring based on the theory of fractal geometry. Design Manuf and Mech 551:434–438

  20. 20.

    Yehoda JE, Messier R (1985) Are thin-film physical structures fractals. Appl Surf Sci s22–3:590–595

  21. 21.

    Majumdar A, Bhushan B (1991) Fractal model of elastic-plastic contact between rough surfaces. J Tribol 113(1):11

  22. 22.

    Majumdar A, Bhushan B (1990) Role of fractal geometry in roughness characterization and contact mechanics of surfaces. J Tribol 112(2):205–216. https://doi.org/10.1115/1.2920243

  23. 23.

    Yan W, Komvopoulos K (1998) Contact analysis of elastic-plastic fractal surfaces. J Appl Phys 84(7):3617–3624. https://doi.org/10.1063/1.368536

  24. 24.

    Buzio R, Boragno C, Valbusa U (2003) Contact mechanics and friction of fractal surfaces probed by atomic force microscopy. Wear 254:917–923

  25. 25.

    Ji CC, Jiang W (2016) Revising elastic-plastic contact models of fractal surfaces. Adv Intell Syst Res 138:123–127

  26. 26.

    Chen JX, Yang F, Luo KY, Wu Y, Niu CP, Rong MZ (2016) Study on contact spots of fractal rough surfaces based on three-dimensional Weierstrass-Mandelbrot function. Proceeding of 62nd IEEE Holm Conference on Electrical Contacts: 198–204

  27. 27.

    Jain R, Pitchumani R (2017) Fractal model for wettability of rough surfaces. Langmuir 33(28):7181–7190. https://doi.org/10.1021/acs.langmuir.7b01524

  28. 28.

    Berry MV, Lewis ZV (1980) On the Weierstrass-Mandelbrot fractal function. Proc R Soc Lond Ser A Math Phys Sci 370(1743):459–484. https://doi.org/10.1098/rspa.1980.0044

  29. 29.

    Greenwood JA, Williamson JBP (1966) Contact of nominally flat surfaces. Proc R Soc London Ser A 295(1442):300–319. https://doi.org/10.1098/rspa.1966.0242

  30. 30.

    Mandelbrot BB (1982) The fractal geometry of nature. Freeman, San Francisco

  31. 31.

    Yuan Y, Gan L, Liu K, Yang XH (2017) Elastoplastic contact mechanics model of rough surface based on fractal theory. Chin J Mech Eeg 30(1):207–215. https://doi.org/10.3901/CJME.2016.0624.079

Download references


The authors would like to thank the financial supports by National Natural Science Foundation of China (No. 51505368, No.91748207), 973 Program (No. 2015CB057402), China Postdoctoral Science Foundation (No. 2017M613114), Shaanxi Postdoctoral Science Foundation (No. 2017BSHEDZZ69), the Fundamental Research Funds for the Central Universities (No. xjj2016011, No. xjj2017077), the fund of the State Key Lab of Digital Manufacturing Equipment & Technology (No. DMETKF2016006), Open Foundation of the State Key Laboratory of Fluid Power and Mechatronic Systems (GZKF-201617), Research Project of State Key Laboratory of Mechanical System and Vibration (MSV201813), and 111 Program (No. B12016). We also appreciate the support from the International Joint Laboratory for Micro/Nano Manufacturing and Measurement Technologies.

Author information

Correspondence to Chenying Wang.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lin, Q., Meng, Q., Wang, C. et al. The influence of fractal dimension in the microcontact of three-dimensional elastic-plastic fractal surfaces. Int J Adv Manuf Technol 104, 17–25 (2019). https://doi.org/10.1007/s00170-018-1660-3

Download citation


  • Microcontact
  • Fractal surface
  • Fractal dimension
  • Elastic-plastic
  • Microcontact force