A computational method for equivalent area distribution characterized by the normal contact behavior of two rough surfaces

Abstract

The contact state of surfaces significantly influences the dynamic properties of assemblies, where the contact area is the key parameter to describe the contact state of assemblies. To address the computational challenges associated with contact area distribution, an improved nominal contact area distribution model is proposed for accurately characterizing the contact state of surfaces. Based on the distribution pattern of contact area, a cumulative quantity function for nominal contact area in logarithmic form is established. And its superiority is proved by comparing with two traditional functions. Based on this, a fractal dimension conversion formula was derived to equate the contact of two rough surfaces to the contact of single rough surface, thereby the optimized expression for the nominal contact area distribution function is obtained. The proposed model is verified through actual contact analysis of the anchorage system (anchor plate and anchor base plate). The results show that the proposed cumulative quantity function for nominal contact area and equivalent fractal dimension function exhibit excellent fitting accuracy and feasibility. In the contact between two rough surfaces, the surface with the smaller fractal dimension dominates the contact behavior. The effectiveness and applicability of the proposed model in complex contact problems are verified through case studies. This research provides new theoretical tools for multiscale mechanical analysis of complex contact surfaces, which has important application value in these fields such as precision manufacturing and tribological optimization.

Graphical abstract

Keywords

fractal theory / rough surface / pressure grade / equivalent fractal dimension / normal contact area distribution function

References

[1] Cheng S , Meng X , Li R , Liu R , Zhang R , Sun K , Ye W , Zhao F . Rough surface damping contact model and its space mechanism application. International Journal of Mechanical Sciences, 2022, 214: 106899

[2] Sun Y , Xiao H , Xu J . Investigation into the interfacial stiffness ratio of stationary contacts between rough surfaces using an equivalent thin layer. International Journal of Mechanical Sciences, 2019, 163: 105147

[3] Chen J , Liu D , Wang C , Zhang W , Zhu L . A fractal contact model of rough surfaces considering detailed multi-scale effects. Tribology International, 2022, 176: 107920

[4] Majumdar A , Bhushan B . Role of fractal geometry in roughness characterization and contact mechanics of surfaces. Journal of Tribology, 1990, 112(2): 205–216

[5] Yan W , Komvopoulos K . Contact analysis of elastic–plastic fractal surfaces. Journal of Applied Physics, 1998, 84(7): 3617–3624

[6] Miao X , Huang X . A complete contact model of a fractal rough surface. Wear, 2014, 309(1-2): 146–151

[7] Liou J , Tsai C , Lin J . A microcontact model developed for sphere- and cylinder-based fractal bodies in contact with a rigid flat surface. Wear, 2010, 268(3-4): 431–442

[8] Liou J L , Lin J F . A modified fractal microcontact model developed for asperity heights with variable morphology parameters. Wear, 2010, 268(1): 133–1449

[9] Xu K , Yuan Y , Chen J . The effects of size distribution functions on contact between fractal rough surfaces. AIP Advances, 2018, 8: 075317

[10] Yuan Y , Cheng Y , Liu K , Gan L . A revised Majumdar and Bushan model of elastoplastic contact between rough surfaces. Applied Surface Science, 2017, 425: 1138–1157

[11] Zhang X , Shen H , Zhang F , Zhang G , Xia H , Liu J , Ao X , Luo J , Zhu X , Zhang J . et al. A multi-scale contact model for rough surfaces under loading and unloading conditions. Tribology Letters, 2025, 73(2): 70

[12] Wang Y H , Zhang X L , Wen S H , Chen Y . Fractal loading model of the joint interface considering strain hardening of materials. Advances in Materials Science and Engineering, 2019, 2019(1442): 1–14

[13] Tikhomirov V P , Izmerov M A . Fractal model of contact interaction of nominally flat surfaces. AIP Conference Proceedings, 2021, 2340: 060004

[14] Wang R , Zhu L , Zhu C . Research on fractal model of normal contact stiffness for mechanical joint considering asperity interaction. International Journal of Mechanical Sciences, 2017, 134: 357–369

[15] Zhao Y S , Song X L , Cai L G , Liu Z , Cheng Q . Surface fractal topography-based contact stiffness determination of spindle–toolholder joint. Proceedings of the Institution of Mechanical Engineers. Part C: Journal of Mechanical Engineering Science, 2016, 230(4): 602–610

[16] Pan W , Li X , Wang L , Guo N , Yang Z . Influence of contact stiffness of joint surfaces on oscillation system based on the fractal theory. Archive of Applied Mechanics, 2017, 88(4): 525–541

[17] Pan W , Sun Y , Li X , Song H , Guo J . Contact mechanics modeling of fractal surface with complex multi-stage actual loading deformation. Applied Mathematical Modelling, 2024, 128: 58–81

[18] Yuan Y , Xu K , Zhao K . The loading–unloading model of contact between fractal rough surfaces. International Journal of Precision Engineering and Manufacturing, 2020, 21(6): 1047–1063

[19] Yuan Y , Chen J , Zhang L . Loading–unloading contact model between three-dimensional fractal rough surfaces. AIP Advances, 2018, 8(7): 075017

[20] Yuan Y , Xu K , Zhao K . A fractal model of contact between rough surfaces for a complete loading–unloading process. Proceedings of the Institution of Mechanical Engineers. Part C: Journal of Mechanical Engineering Science, 2020, 234(14): 2923–2935

[21] Yu X , Sun Y , Zhao D , Wu S . A revised contact stiffness model of rough curved surfaces based on the length scale. Tribology International, 2021, 164: 107206

[22] Liu Y , Wang Y , Chen X , Yu H . A spherical conformal contact model considering frictional and microscopic factors based on fractal theory. Chaos, Solitons, and Fractals, 2018, 111: 96–107

[23] Shen J , Xu S , Liu W , Yang J . Fractal model of normal contact stiffness between two spheres of joint interfaces with simulation. Mechanics, 2017, 23(5): 703–713

[24] Li X , Yue B , Wang D , Liang Y , Sun D . Dynamic characteristics of cylinders’ joint surfaces considering friction and elastic–plastic deformation based on fractal theory. Australian Journal of Mechanical Engineering, 2017, 15(1): 11–18

[25] Yuan Y , Gan L , Liu K , Yang X . Elastoplastic contact mechanics model of rough surface based on fractal theory. Chinese Journal of Mechanical Engineering, 2016, 30(1): 207–215

[26] Sun J , Ji Z , Zhang Y , Yu Q , Ma C . A contact mechanics model for rough surfaces based on a new fractal characterization method. International Journal of Applied Mechanics, 2018, 10(06): 1850069

[27] Tian H , Chen B , He K , Dong Y , Zhong X , Wang X , Xi N . A new fractal model of elastic, elastoplastic and plastic normal contact stiffness for slow sliding interface considering dynamic friction and strain hardening. Journal of Shanghai Jiaotong University (Science), 2017, 22(5): 589–601

[28] Lan G , Sun W , Zhang X , Chen Y , Tan W , Li X . A three-dimensional fractal model of the normal contact characteristics of two contacting rough surfaces. AIP Advances, 2021, 11: 055023

[29] Zhou Y , Xiao Y , He Y , Zhang Z . A detailed finite element analysis of composite bolted joint dynamics with multiscale modeling of contacts between rough surfaces. Composite Structures, 2020, 236: 111874

[30] Zhao Y , Yang C , Cai L , Shi W , Liu Z . Surface contact stress-based nonlinear virtual material method for dynamic analysis of bolted joint of machine tool. Precision Engineering, 2016, 43: 230–240

[31] Zhang Z , Xiao Y , Xie Y , Su Z . Effects of contact between rough surfaces on the dynamic responses of bolted composite joints: Multiscale modeling and numerical simulation. Composite Structures, 2019, 211: 13–23

[32] Zhou H , Long X , Meng G , Liu X . A stiffness model for bolted joints considering asperity interactions of rough surface contact. Journal of Tribology, 2022, 144(1): 011501

[33] Zhao Y , Xu J , Cai L , Shi W , Liu Z . Stiffness and damping model of bolted joint based on the modified three-dimensional fractal topography. Proceedings of the Institution of Mechanical Engineers. Part C: Journal of Mechanical Engineering Science, 2017, 231(2): 279–293

[34] Ausloos M , Berman D H . A multivariate Weierstrass-Mandelbrot function. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 1819, 1985(400): 331–350

[35] Mandelbrot B. The Fractal Geometry of Nature. New York, NY: W.H. Freeman and Company, 1977, 116–121

[36] Li X , Wang W , Zhao M . Fractal prediction model for tangential contact damping of jointsurface considering friction factors and its simulation. Journal of Mechanical Engineering, 2012, 48(23): 46–50

[37] Xie Y , Chen S , Wan X , Tse P . A preliminary numerical study on the interactions between nonlinear ultrasonic guided waves and a single crack in bone materials with motivation to the evaluation of micro cracks in long bones. IEEE Access, 2020, 8: 169169–169182

[38] Xu T , Moore L D , Gallant J C . Fractals, fractal dimensions and landscapes-a review. Geomorphology, 1993, 8(4): 245–262

[39] Kulatilake P H S W , Du S G , Ankah M L Y , Yong R , Sunkpal D T , Zhao X , Liu G , Wu R . Non-stationarity, heterogeneity, scale effects, and anisotropy investigations on natural rock joint roughness using the variogram method. Bulletin of Engineering Geology and the Environment, 2021, 80(8): 6121–6143

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