The present paper deals with a finite-element-based static loading–unloading analysis of a functionally graded rough surface contact with fractal characteristics. Two different gradation models, namely elastic and plastic gradations, are adopted. In these models, one out of yield strength and Young's modulus is varied spatially according to exponential functions, while the other is kept constant. In both these material models, separate inhomogeneity parameters control the variation of material properties. The gradation is such that throughout the top of the rough surface properties remain constant with variations in the depth direction being controlled by the above-mentioned parameters. Different fractal surfaces with different levels of roughness (governed by the values of fractal dimension and fractal roughness) have been analysed. The influence of the gradation parameters on the contact properties, viz. contact force, contact area, contact stress, etc., are investigated for both loading and unloading phases. It was found that for most of the loading phase, higher elastic, as well as plastic gradation parameter, causes higher contact force and contact area. However, in the case of the unloading of elastically graded surfaces, this trend is not maintained throughout. For the cases, where a substantial amount of yielding takes place during loading near the contact surface, the resulting contact area is found to be higher for the unloading phase in comparison with the same during the loading phase. The trend of plastic yielding at the vicinity of the contact surface is studied for varying gradation parameters. It is observed that the higher volume of yielded material is obtained for the higher value of elastic gradation parameter. On the other hand, the higher value of plastic gradation parameter causes more yielding to take place at the vicinity of the contact surface. Additionally, the effect of gradation on the energy dissipation due to plasticity after complete unloading is explored in detail.
ZhuravlevVA. On the question of theoretical justification of the Amontons–Coulomb law for friction of unlubricated surfaces. J Tech Phys1940; 10: 1447–1452.
2.
GreenwoodJAWilliamsonJBP. Contact of nominally flat surfaces. Proc R Soc London Ser A Math Phys Sci1966; 295: 300–319.
3.
HertzH. Ueber die berührung fester elastischer körper. J für die reine und Angew Math1882; 92: 156–171.
4.
BushAWGibsonRDThomasTR. The elastic contact of a rough surface. Wear1975; 35: 87–111.
WhitehouseDJArchardJF. The properties of random surfaces of significance in their contact. Proc R Soc London A Math Phys Sci1970; 316: 97–121.
7.
HisakadoT. Effect of surface roughness on contact between solid surfaces. Wear1974; 28: 217–234.
8.
OnionsRAArchardJF. The contact of surfaces having a random structure. J Phys D Appl Phys1973; 6: 302.
9.
SaylesRSThomasTR. Surface topography as a nonstationary random process. Nature1978; 271: 431–434.
10.
MandelbrotBB. How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science1967; 156: 636–638.
11.
MandelbrotBBBlumenA. Fractal geometry: what is it, and what does it do? [and discussion]. Proc R Soc A Math Phys Eng Sci1989; 423: 3–16.
12.
AusloosMBermanDH. A multivariate Weierstrass-Mandelbrot function. Proc R Soc A Math Phys Eng Sci1985; 400: 331–350.
13.
BerryMVLewisZV. On the Weierstrass-Mandelbrot fractal function. Proc R Soc A Math Phys Eng Sci1980; 370: 459–484.
14.
MajumdarABhushanB. Role of fractal geometry in roughness characterization and contact mechanics of surfaces. J Tribol1990; 112: 205–216.
15.
MajumdarABhushanB. Fractal model of elastic-plastic contact between rough surfaces. J Tribol1991; 113: 1–11.
16.
MajumdarATienCL. Fractal characterization and simulation of rough surfaces. Wear1990; 136: 313–327.
17.
LingFF. Fractals, engineering surfaces and tribology. Wear1990; 136: 141–156.
18.
LiouJLLinJF. A new method developed for fractal dimension and topothesy varying with the mean separation of two contact surfaces. J Tribol2006; 128: 515–524.
19.
LiouJLLinJF. A new microcontact model developed for variable fractal dimension, topothesy, density of asperity, and probability density function of asperity heights. J Appl Mech Trans ASME2007; 74: 603–613.
CiavarellaMMuroloGDemelioG, et al.Elastic contact stiffness and contact resistance for the Weierstrass profile. J Mech Phys Solids2004; 52: 1247–1265.
22.
SahooPRoy ChowdhurySK. A fractal analysis of adhesive wear at the contact between rough solids. Wear2002; 253: 924–934.
23.
SahooPRoy ChowdhurySK. A fractal analysis of adhesive friction between rough solids in gentle sliding. Proc Inst Mech Eng Part J J Eng Tribol2000; 214: 583–595.
24.
MoragYEtsionI. Resolving the contradiction of asperities plastic to elastic mode transition in current contact models of fractal rough surfaces. Wear2007; 262: 624–629.
25.
YuanYChengYLiuK, et al.A revised Majumdar and Bushan model of elastoplastic contact between rough surfaces. Appl Surf Sci2017; 425: 1138–1157.
26.
PohrtRPopovVL. Normal contact stiffness of elastic solids with fractal rough surfaces. Phys Rev Lett2012; 108: 104301.
27.
ZhangDXiaYScarpaF, et al.Interfacial contact stiffness of fractal rough surfaces. Sci Rep2017; 7: 12874.
28.
ZhaiCHanaorDGanY. Contact stiffness of multiscale surfaces by truncation analysis. Int J Mech Sci2017; 131–132: 305–316.
29.
XiaoHSunYChenZ. Fractal modeling of normal contact stiffness for rough surface contact considering the elastic–plastic deformation. J Brazilian Soc Mech Sci Eng2019; 41: 11.
30.
HyunSPeiLMolinariJ-F, et al.Finite-element analysis of contact between elastic self-affine surfaces. Phys Rev E2004; 70: 026117.
31.
PeiLHyunSMolinariJF, et al.Finite element modeling of elasto-plastic contact between rough surfaces. J Mech Phys Solids2005; 53: 2385–2409.
32.
SahooPGhoshN. Finite element contact analysis of fractal surfaces. J Phys D Appl Phys2007; 40: 4245–4252.
33.
XiaoHShaoYBrennanMJ. On the contact stiffness and nonlinear vibration of an elastic body with a rough surface in contact with a rigid flat surface. Eur J Mech - A/Solids2015; 49: 321–328.
SureshS. Graded materials for resistance to contact deformation and damage. Science2001; 292: 2447–2451.
36.
MossakovskiiVI. Pressure of a circular die [punch] on an elastic half-space, whose modulus of elasticity is an exponential function of depth. J Appl Math Mech1958; 22: 168–171.
37.
RostovtsevNAKhranevskaiaIE. The solution of the boussinesq problem for a half-space whose modulus of elasticity is a power function of the depth. J Appl Math Mech1971; 35: 1000–1009.
38.
GiannakopoulosAESureshS. Indentation of solids with gradients in elastic properties: part I. Point force. Int J Solids Struct1997; 34: 2357–2392.
39.
GiannakopoulosAESureshS. Indentation of solids with gradients in elastic properties: part II. Axisymmetric indentors. Int J Solids Struct1997; 34: 2393–2428.
40.
SureshSGiannakopoulosAEAlcaláJ. Spherical indentation of compositionally graded materials: theory and experiments. Acta Mater1997; 45: 1307–1321.
41.
AizikovichSMAlexandrovVMKalkerJJ, et al.Analytical solution of the spherical indentation problem for a half-space with gradients with the depth elastic properties. Int J Solids Struct2002; 39: 2745–2772.
LiuTJZhangCWangYS, et al.The axisymmetric stress analysis of double contact problem for functionally graded materials layer with arbitrary graded materials properties. Int J Solids Struct2016; 96: 229–239.
44.
GulerMAErdoganF. Contact mechanics of graded coatings. Int J Solids Struct2004; 41: 3865–3889.
45.
GulerMAErdoganF. Contact mechanics of two deformable elastic solids with graded coatings. Mech Mater2006; 38: 633–647.
46.
NakamuraTWangTSampathS. Determination of properties of graded materials by inverse analysis and instrumented indentation. Acta Mater2000; 48: 4293–4306.
47.
GiannakopoulosAE. Indentation of plastically graded substrates by sharp indentors. Int J Solids Struct2002; 39: 2495–2515.
48.
KleckaMASubhashGArakereNK. Experimental and numerical modeling of surface indentation response of plastically graded materials. J Eng Mater Technol Trans ASME 2013; 135: 041004.
49.
ChoiISDetorAJSchwaigerR, et al.Mechanics of indentation of plastically graded materials—II: experiments on nanocrystalline alloys with grain size gradients. J Mech Phys Solids2008; 56: 172–183.
50.
PaggiMZavariseG. Contact mechanics of microscopically rough surfaces with graded elasticity. Eur J Mech - A/Solids2011; 30: 696–704.
51.
JinFGuoX. Mechanics of axisymmetric adhesive contact of rough surfaces involving power-law graded materials. Int J Solids Struct2013; 50: 3375–3386.
52.
ScaraggiMComingioD. Rough contact mechanics for viscoelastic graded materials: the role of small-scale wavelengths on rubber friction. Int J Solids Struct2017; 125: 276–296.
53.
JanaTMitraASahooP. Contact behavior of functionally graded fractal surface. Proc Inst Mech Eng Part L J Mater Des Appl2020; 234: 1355–1374.
54.
PerssonBNJ. On the fractal dimension of rough surfaces. Tribol Lett2014; 54: 99–106.
55.
JingJFengPWeiS, et al.Investigation on the surface morphology of Si3N4 ceramics by a new fractal dimension calculation method. Appl Surf Sci2016; 387: 812–821.
56.
ChouCCLinHH. Fractal dimension and surface topography on the diamond deposition of seeded WC–Co substrates. J Appl Phys2010; 107: 073510.
57.
JhaSJainVK. Nanofinishing Techniques. In: Micromanufacturing and Nanotechnology, Berlin/Heidelberg: Springer-Verlag, 2006, pp. 171–195.
58.
HosokawaHShimojimaKMabuchiM. Surface roughness of a co-less WC micro-die fabricated by focused-ion-beam machining. Philos Mag Lett2009; 84: 149–155.
59.
ZhangDLiuJChenY, et al.Investigation on S-136 steel surface planarization by chemical mechanical polishing. Microelectron Eng2015; 134: 47–53.
60.
BuzioRBoragnoCValbusaU. Contact mechanics and friction of fractal surfaces probed by atomic force microscopy. Wear2003; 254: 917–923.
61.
JanaTMitraASahooP. Unloading analysis of elastically and plastically graded hemispherical contact with rigid flat. Tribol Int2020; 142: 105973.
62.
KadinYKligermanYEtsionI. Multiple loading–unloading of an elastic–plastic spherical contact. Int J Solids Struct2006; 43: 7119–7127.
63.
ZaitYKligermanYEtsionI. Unloading of an elastic–plastic spherical contact under stick contact condition. Int J Solids Struct2010; 47: 990–997.
64.
EtsionIKligermanYKadinY. Unloading of an elastic–plastic loaded spherical contact. Int J Solids Struct2005; 42: 3716–3729.
