A Contact and Friction Model of Galvanized Sheet Based on Fractal Rough Surfaces
Leigang Wang1, a,Yao Huang1, bYatingXu1,c
1School of Materials Science and Technology, JiangsuUniversity, Zhenjiang212013, China
,,
Abstract:Hot-dip galvannealed sheet and hot-dip galvanized sheet as examples, the surface topography was measured in application of profile meter and the fractal parameters were calculated based on fractal theory.Moreover, the contact model between sheet and mold was studied, and the relationship between real contact area and pressure was attained. According to the contact model, the friction model concerning surface topography of galvanized sheet and pressure was established.Compared with the results from experiments, the friction coefficients of theoretical values are in agreement with experimental values. As a result, the friction model has been proved feasible within the error less than 10%.
Keywords:galvanized sheet,real contact area,fractal theory,contact model,friction model
0 Introduction
Every friction pair surface is rough at the micro-scale.This contact characterization of rough surface has a significant impact on the phenomena of micro contact, friction, abrasion,thermal conductivity and lubrication [1-2]. As a result, research onparameters of rough surfacesin tribology is always of immense importance[1-2].Traditional model of surface topography used to be representedin two-dimensional parameters,nevertheless, 2D parameters are too simple to be of uniqueness and accuracy. With fractal theory put forward, fractal parameters have been widely used in characterization of rough surface topography and establishment of contact model in consideration of self-similarity and scale invariance. Miao [3]hasbuilt a complete contact model of fractal rough surfaces which overcame the shortcomings of the M-B model. The results show that the relationship between contact load and real contact area is linear; as the contact area increases, the share of the plastic contact area decreases and the contact stiffness increases. Morag [4] has presented a modified elastic-plastic contact model of a single fractal asperity. The revised model shows that as the load and contact area increase, a transition from elastic to plastic contact mode takes place in this order.Ding [5] has established the fractal contact model based on base length considering asperity deformation properties, fractal theory and friction effect.The relationship of fractal dimension, contact force and real contact area was studied by using numerical simulation analysis.
The surface topography of galvanized sheet was measured in this paper in application of optical profile meter.On basis of fractal theory, fractal parameters were first calculated through MATLAB. Real contact state between galvanized sheet and mold was then simplified to establish a initial fractal contact model. Eventually, the friction model which conforms to actual situation was built and experimentally verified.
1 Contact model offractal rough surface
1.1 Deformation properties of contact surface
In Majumdar- Bhushan[6] model, two rough contact surfaces are replaced by an equivalent rough surface and an ideal rigid flat surface. It is shown in Fig.1, wherel isthe length scale of a fractal asperity, δ is the asperity height.In M-B model, the asperity height is defined as:
δ = GD−1a (2−D)/2(1)
where D is fractal dimension, G is the fractal roughness parameter and a iscontact area.
Fig.1 The MB model[6]: (a) contact between a rough surface and a flat surface, and (b) the geometry of a contact point of length scale l
The critical deformation (δc) is used to examine the type of deformation by comparison with magnitude of asperity deformation (its height δ). Whenδ is larger than δc, the type of deformation is plastic; otherwise, the type of deformation is elastic. δc, the critical deformation, is in the form as:
(2)
K is defined as a factor related to the hardness H as well as yield strengthσy, namelyK=H/σy, andфis a factor concerned with the yield strengthσy and the elastic modulus E asф=σy/E. In addition, σy is the yield strength of softer material.
From Eq. (1) and (2), the critical contact area is defined as:
(3)
1.2 Elastic-plastic contact load of rough surface
At the level al (largest contact area)>ac (critical contact area), the contact load P is [7]:
(4)
In Eq. (4), fronthalfof the right side means elastic load, and the latter means plastic load [7]. n (a) is size distribution of contact points, and as is the smallest contact area. The contact load P in the al ac range is [7]:
(5)
1.3 Real contact area of rough surface
According to M-B model, the size distribution of contact spots n(a)[7] is:
(6)
It can be seen from Eq.(6), the time contact area a→0, the number of contact points is infinite. Because of fractal feature of rough surfaces in scale of nanometer or smaller, thus in hypothesis of minimum contact area of single asperity in rough surfaces approaches zero[7], in that case, the total real contact areaAr is related to n(a) as follows:
(1<D<2)(7)
1.4 Relation between real contact area and pressure
Eq. (7) substituted into Eq. (4), the total contact load for the level ofal ac can be gotten:
(a) IfD≠1.5:
(8)
(b) IfD=1.5:
(9)
In Eq.(8) and (9), the parameters are non-dimensional:
; ;;
Where Aa is nominal contact area, and E is the equivalent Hertzian elastic modulus.
Similarly, at the level al ac, the contact load is:
(10)
2 Surface topographies of galvanized sheets
2.1 Materials and methods
The hot-dip galvannealed(GA)sheet in number of SP781BQ and galvanized(GI) sheet of HC420/780DPD+Z used in the study are from Shanghai Baosteel Group Corporation. Rectangular sheet samples in dimension of 30 mm×100 mm are used to measure the surface topographies in application of WykoNT1100 optical profile meter. All samples are treated with anhydrous ethanol to keep surface finish. In the sample, three micro areas are selected randomly as a measurement area forsurface profile curve and 3D surface topography. The surface roughness values (Ra) of GA sheet SP781BQ and GI sheet HC420/780DPD+Z are measured 0.8893 and 0.525μm respectively.
2.2 Surface topographiesof galvanized sheets
The surface profile curves of SP781BQ and HC420/780DPD+Z are shown in Fig.2, and Fig.3 presentsthe 3D surface topographies of galvanized sheets. As showedin Fig.2 and Fig.3, the surface profile of SP781BQ hasa larger range of fluctuation than HC420/780DPD+Z, and the uneven degree of surface is larger than HC420/780DPD+Z as well.
(a) SP781BQ (b)HC420/780DPD+Z
Fig. 2 Surface profile curves of galvanized sheets
(a) SP781BQ (b)HC420/780DPD+Z
Fig. 3 3Dsurface topographies of galvanized sheets
2.3 Fractal characterization of galvanized sheets
The dimension of surface profile curveis calculated to judge the fractal characteristics of galvanized sheets. At the level 1<D<2, the surfaces have fractal characteristics. In this paper, the collection points of surface profile curve are computedin root-mean-square method, and the specific calculation process is finishedin the MATLAB platform. The double logarithm chart of root-mean-square measurement σ (τ) and correlation length τ isdescribed in Fig.4. In Fig.4, the logarithm of root-mean-square measurement is near linear relation with the logarithm of correlation length.
According to calculation, fractal dimension and fractal roughness parameter of SP781BQ hot–dip galvannealed sheet are 1.52 and 0.23μm correspondingly,comparativelythe dimension and parameter of HC420/780DPD+Z hot-dip galvanized sheet are 1.6 and 0.11μm. Both of the fractal dimensions meet the range 1<D<2. Hence,the surfaces of galvanized sheets have fractal characteristics.
(a) SP781BQ (b) HC420/780DPD+Z
Fig.4 The double logarithm chart of galvanized sheets
3 Fractal contact models of galvanized sheets
3.1 Contact model of SP781BQ hot–dip galvannealed sheet
The mechanical properties and surface morphology parameters of SP781BQ are shown below:
Mechanical properties:
, whereE1 andν1 are the Young’s modules and Poisson’s ratios of galvanized surface, the values are 0.27 and 130GPa; andrelatively E2 and ν2 are the Young’s modules and Poisson’s ratios of mold surface, the values are 0.3 and 200GPa. The parameters of material performance:
Surface morphology parameters:
The fractal dimension D is 1.52, the fractal roughness parameter G is 0.23μm, and the nominal contact area Aa=2×30mm2.
Based on the mechanical properties and surface morphology parameters of SP781BQ, the parameters can be figured out:
;;
;
Above parameterssubstitutedinto Eq.(8), the relation between real contact areaand pressure P can be gotten in the following form:
(11)
The relation between real contact area and pressure is described in Fig.5 which can be concluded that the real contact area rises as the pressure increases.
Fig.5 Relation of real contact area and pressure between galvanized sheet and mold (SP781BQ)
3.2 Contact model of HC420/780DPD+Z hot-dip galvanized sheet
The mechanical properties and surface morphology parameters of HC420/780DPD+Z are shown below:
Mechanical properties:
The Equivalent Hertzian elastic modulus:
The parameters of material performance:
Surface morphology parameters:
The fractal dimension D is 1.60, the fractal roughness parameter G is 0.11μm, and the nominal contact area Aa=2×30mm2.
Based on the mechanical properties and surface morphology parameters of HC420/780DPD+Z, the parameters can be figured out:
; ;
;
Above parameters substituted into Eq.(8), the relation between real contact areaand pressure P can be gotten in the following form:
(12)
4 Friction models of galvanized sheets
According to theory of adhesive friction, the calculation formula of friction can be found:
(13)
Where F is the friction resistance, Fa is the adhesion force, and Fe is the plowing force. Ar is the real contact area, and τb is the shear strength of softer material. In general, theproportion ofplowing force in friction is too less to calculate. In consequence, the simplified formula of friction coefficient is calculated in the form:
(14)
Where μ means the friction coefficient and P means the pressure.
4.1 Friction model of SP781BQ hot–dip galvannealed sheet
The shear strength of SP781BQ zinc coating is 113MPa, the friction model is:
;(15)
4.2 Friction model of HC420/780DPD+Z hot-dip galvanized sheet
The shear strength of HC420/780DPD+Z zinc coating is 90MPa, the friction model is:
;(16)
The relation between friction coefficient and positive pressure is shown in Fig.6,which indicates that the friction coefficient decreases with the increasing pressure. From the figure, the friction coefficient of hot–dip galvannealed sheet shows a massive decline at first, thenas the pressure reaches 3000N, the tendency of decline becomes slowly. By contrast, the friction coefficient of hot-dip galvanized sheet is in a slightdowntrend.
Fig.6 Friction coefficient calculated by friction model under different pressures
4.3 Experimental verification of friction model
Flat plate experiment is conducted to testfriction coefficientsof galvanized sheet under different pressures. The friction coefficients arecalculated from friction model and details of the experiment are shown in Table 2 and Table 3.
Table 2 Friction coefficients of theoretical value and experimental value (SP781BQ)
Pressure/N / 1000 / 1500 / 2000Theoretical value / 0.1898 / 0.1627 / 0.1463
Experimental value / 0.2055 / 0.1793 / 0.1584
Error / 7.6% / 9.2% / 7.6%
Table 3 Friction coefficients of theoretical value and experimental value (HC420/780DPD+Z)
Pressure/N / 1000 / 1500 / 2000Theoretical value / 0.135 / 0.1332 / 0.1296
Experimental value / 0.1501 / 0.1439 / 0.1375
Error / 10% / 7.4% / 5.7%
Itcan be seen from Table 2 and Table 3, friction coefficients of theoretical valuesare close to experimental values, and the error between them is less than 10%. Due to the ignored plowing force, theoretical values are slightly smaller than experimental values.
5 Conclusion
(1) It has been showed that galvanized sheet has fractal characteristics. Fractal dimension and fractal roughness parameter of SP781BQ hot–dip galvannealed sheet are 1.52 and 0.23μm respectively. Likewise, fractal dimension and fractal roughness parameter of HC420/780DPD+Z hot-dip galvanized sheet are 1.6 and 0.11μm accordingly.
(2) On the basis of fractal theory, the elastic-plastic contact model has been established and the relation between real contact area and pressure has beengotten as well. The model demonstrates that real contact area rises with the increase of pressure.
(3)Friction model is established on the foundation of the fractal contact model. Experimental verification is carried out through flat plate experiment. The results indicate that the error of friction coefficients between theoretical values and experimental valuesis within 10%. Thus, the friction model is relatively accurate.
Acknowledgement
This work is supported by the National Natural Science Foundation of China (51275216).
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