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Coupled Effects of Fractal Roughness and Self-Lubricating Composite Porosity on Lubrication andWearYan Lu a & Zuomin Liu aa Institute of Tribology Wuhan University of Technology Luoshi Road 205 , Wuhan , 430070 ,ChinaAccepted author version posted online: 25 Feb 2013.Published online: 30 Apr 2013.

To cite this article: Yan Lu & Zuomin Liu (2013) Coupled Effects of Fractal Roughness and Self-Lubricating Composite Porosityon Lubrication and Wear, Tribology Transactions, 56:4, 581-591, DOI: 10.1080/10402004.2012.711437

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Tribology Transactions, 56: 581-591, 2013Copyright C© Society of Tribologists and Lubrication EngineersISSN: 1040-2004 print / 1547-397X onlineDOI: 10.1080/10402004.2012.711437

Coupled Effects of Fractal Roughness and Self-LubricatingComposite Porosity on Lubrication and Wear

YAN LU and ZUOMIN LIUInstitute of Tribology

Wuhan University of TechnologyLuoshi Road 205, Wuhan 430070, China

The lubricant characteristics of porous self-lubricating com-

posites with a realistic rough surface are incorporated into

an improved elastohydrodynamic model. The evolved model

demonstrates that the wear rate can be measured by exam-

ining the lubricant distribution at various fractal dimensions

and porosities. The results show that the physical nature of the

rough surface topography and the composite’s physical prop-

erties must be understood, because the relative contact area is

enlarged and friction forces are increased by the increase in

the fractal dimension and the porosity. It is obvious that the

method can significantly improve the lubricant properties to

avoid wear by controlling these two coupled effects. The re-

search also indicates that optimization of the design the mi-

crostructure of the porous self-lubricating composite should fo-

cus on the porosity based on the wear rather than the amount

of lubricant.

KEY WORDS

Self-Lubricating Composites; Partial EHL; Roughness Ef-fects; Film Geometry in Hydrodynamics; Unlubricated Wear

INTRODUCTION

For economical, ecological, and technical reasons there hasbeen a tendency for some years to introduce self-lubricating ma-terials for mechanical applications under severe sliding condi-tions. High-temperature porous self-lubricating composites area new material with an ordered microporous substrate and in-filtrated with soft metal lubricant through the microporous (Liu(1); Wang and Liu (2)). During the sliding process, the soft metallubricant in the composite is diffused through its micropores andspread on the surface, which generates a thin hydrodynamic fluidfilm. Porous self-lubrication composites, as powder metallurgi-cal materials with additives of solid lubricants, have been widelyused in many fields when oil or greases does not meet the ad-vanced requirements of modern technology. Unfortunately, thelubricant and wear properties of such materials are very poorin many cases. Therefore, the composite’s porosity has two side

Manuscript received January 16, 2012Manuscript accepted July 7, 2012

Review led by Dong Zhu

effects on the lubricant characteristics and wear resistance prop-erties. One is that the amount of lubricant is determined by thesoft metal lubricant phases. The other is that the continuity ofsubstrate hard phases was destroyed by the solid lubricant phaseand thus reduced wear resistance of the composites. The rest ofthis article investigates a new approach to designing the porosityof composites based on the lubricant properties.

The conventional hydrodynamic lubrication theory is basedon the assumption of a perfectly smooth surface (Naduvinamaniand Siddangouda (3)). However, it has been shown that suchan assumption is unrealistic, especially to study this activity withsmall fluid film thicknesses. The existence of surface roughnessmay result in direct solid contact when the lubricant film thicknessis below a certain limit (Larsson (4); Tonder (5); Jagadeesha, et al.(6); Hartl, et al. (7)). The direct solid contact is the major cause ofsurface damage such as sliding wear, friction, scuffing, and pittingdue to contact fatigue. Therefore, original surface roughness andtopography are important factors in component failure analysis(Yuan, et al. (8); Ramalho and Miranda (9); Litwin (10)).

The elastic deformation of the surface has a dramatic influ-ence on the elastic hydrodynamic lubrication (EHL) problem(Sofuoglu and Ozer (11); Hu and Zhu (12)). Therefore, it is nec-essary to pay careful attention to the material’s physical param-eters. It is well known that the composite’s microstructure con-tributes to its physical properties (Voigt (13); Reuss (14); Hashin(15); Kalaprasad and Joseph (16)). Therefore, the microstructureof the composite is another important factor in the componentfailure analysis.

The present work will focus on investigating the coupled ef-fects of the fractal rough surface and the composite’s porosity onthe hydrodynamic fluid film to action optimal design of porousself-lubricating composites. Due to the limitation of our experi-mental apparatus to manufacture samples of the porous compos-ites with different fractal dimensions and measure the thin lubri-cant film, controlled experiments to confirm the feasibility of themodel will be a subject of further research.

STATEMENT OF THE PROBLEM

Self-Lubrication Mechanisms for Hydrodynamic Model

As described in the previous section, self-lubricating com-posites are a novel tribological functional material designed formechanical components under elevated temperature conditions.

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582 Y. LU AND Z. LIU

NOMENCLATURE

a = Semi-width of Hertzian contact, (3RW/2E)1/3

C = Specific wear rate coefficient (m2/N)D = Fractal dimensionE = Elastic modulusE = Dimensionless elastic modulus, βEEh = Hirsh elastic modulus, Eh = αEv + (1 − α)Er

Ei = Elastic modulus of multiphasesEl = Metal lubricant phases elastic modulusEr = Reuss elastic modulus, E−1

r = (�f i/Ei)−1

Ev = Voigt elastic modulus, Ev = �f iEi

Fp = Normal forceFs = Friction forceFs = Dimensionless friction force, Fs/Fp

f i = Volume fractionf 0 = PorosityG = Amplitude parameter of fractal roughnessh = Film thicknessL = Sample sizeLs = Fractal sample lengthM = Superimposed ridgesn = Super limitp = Hydrodynamic pressurepl = Lubricant contact pressureps = Solid contact pressureP0 = Maximum Hertzian pressure, 3W/2πa2

R = Effective radiusS = Shear-thinning effects factoru = Surface deformation

V = Velocity, V1 + V2

V = Dimensionless velocity, η0V/ERV1 = Velocity of rough surface in the x directionV2 = Velocity of smooth surface in the x directionW = LoadW = Dimensionless load parameter, W/ER2

Ws = Wear ratex, y = Coordinatesx, y = Dimensionless coordinates, x/a, y/az = Surface point heightZ0 = Roelands viscosity-pressure, where, Z0 = 22 × 109/

[(1/1.96 × 108)(ln η0 − ln 6.31 × 10−5)]α = Hirsh semi-empirical constantβ = Barus viscosity–pressure index, 22 GPa−1

γ = Density of frequenciesη = Viscosity of lubricantη = Relative viscosity, η/η0

η0 = Ambient viscosity of lubricantλ = Ratio of film thickness to surface roughness, h/σ

ρ = Density of lubricantρ = Dimensionless density of lubricantρ0 = Ambient density of lubricantσ = Root-mean-square roughness, σ = (σ2

1 + σ22)1/2

σ1, σ2 = Root mean square roughness of two contact surfacesτs = Shear stressφm,n = Random phase in the interval [0, 2π] = Calculated area of hydrodynamic lubrication l = Relative lubricant area s = Relative contact area

The following points should be noted with regard to the self-lubrication mechanisms:

1. The situation discussed in this article is one in which the tem-perature has increased sufficiently for surface layers of thesolid lubricant to melt (Liu (1)).

2. Figure 1, which shows the liquid state and the fluid flowingtrack of the molten metallic after sliding under high tempera-ture, illustrates that the molten metallic phase in the compos-ite is squeezed out under the driving force of the contact stressto provide liquid lubrication during the sliding process.

3. The flow behavior of the molten metallic fluid analogues tothat of the nonmetallic liquid lubricant has been observed byexperimental and theoretical research on the liquid properties(Stewart and Weinberg (17), (18); Hattori, et al. (19); Wonsooand Tong-Seek (20)). So the liquid metallic can act as normalfluids during the elastohydrodynamic lubrication. Its lubricantproperties can also refer from it.

In addition, the surface roughness is defined as the arithmeticaverage of the distance between the high and low point of a sur-face, and porosity determines its value change during the slid-ing contact process, so it plays an important role in satisfactoryoperation of elastohydrodynamic lubrication of sliding contacts.In order to address the issue above, we propose an improvedconventional elastohydrodynamic model that is combined to theelastohydrodynamic lubrication model as in Fig. 2, which showsthe elastohydrodynamic lubrication of a porous self-lubricating

composite with a rough surface sliding. It helps us to understandthe significant effects of reduced friction and wear and providessmooth running as well as a satisfactory life span for the machineelements. There will be a thin film between the two moving sur-faces, but if the pressures on the elastohydrodynamically lubri-cated machine elements are too high or the running speeds aretoo low, the lubricant film will be penetrated. Some contact willtake place between the asperities and mixed lubrication will oc-cur. The lubrication mechanism is governed by the physical prop-erties of the composite and its surface topography. Therefore, thedesign of the microstructure is optimized and the roughness ofthe composite is reduced, so that the machine elements will not

Fig. 1—Solid lubricants diffuse out as liquid molten metal lubricationunder elevated temperature: (a) liquid molten metallic lubricantdiffusing out (SEM 500×) and (b) the flowing state of liquid moltenmetallic lubricant (EDX 600×).

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Coupled Fractal Effects 583

Fig. 2—Hydrodynamic lubrication generated between the rough surfaceof a porous self-lubricating composite and a smooth surface.

encounter the solid contact problem. The present work demon-strates the self-lubrication mechanism for a hydrodynamic model(Section 2.1), and then discribes an evolution of the conventionalEHL model by incorporating the rough porous composite’s phys-ical properties (Section 2.2). The problem requires the simultane-ous solution of the equations mentioned below, using a numeri-cal method similar to those described in detail by Hamrock andDowson (21) and Lubrecht (22). Assume that the two surfacesshown in Fig. 2 are moving in the x-direction with the relativespeed of the surface, which means the V is vector sum of the sur-face velocities V = V1 + V2 (here V1 = 0). The lubricant distribu-tion is governed by the Reynolds equation (Verner and Lubrecht(29)):

div(

h3ρ(p)η(p)

S∇p)

= 6V∂(ρ(p)h)

∂x[1]

Note that a boundary condition of pressure p = 0 at the edges ofthe solution domain as well as a normal cavitational condition onthe outlet side must be satisfied when solving this equation.

The shear-thinning effects factor S, which was modeled by theYang and Wen (23) is denoted as S = τ

τ′ Sinh( τ′

τ). Note that in the

previous section, we assumed that the temperature was too highto neglect the small incremental sensitivity to shear rate. So S =1, which means that the shear thinning effect can be neglectedreasonably.

Since the assumption that the temperature is sufficient to meltthe surface layers, the viscosity of the liquid molten metallic per-formance depends on pressure from the literature (Wonsoo andTong-Seek (20)), and due to the limitation of our experimen-tal apparatus to measure the realistic viscosity of the moltenmetal lubricant, we assume that it follows the Roelands equation(Roelands, et al. (24)):

η(p) = η0 exp {(ln η0 − ln (6.31 × 10−5))

× [−1 + (1 + p/(1.96 × 108))Z0 ]}ρ(p) = ρ0(1 + (0.6 × 10−9p)/(1 + 1.7 × 10−9p)).

h is the distance between the surfaces. The formulas for the ge-ometry of the composite rough surface are as follows:

h(x, y) = h0 + x2 + y2

2R+ z(x, y) + u(x, y)

u(x, y) = 2πE

∫ ∫

p√(x − x′)2 + (y − y′)2

dx′dy′

Note that z is a three-dimensional roughness profile for fractalsurfaces. R is the effective radius of the contact area. In order todetermine the film thickness distribution h0, the integrating equa-tion W = ∫ ∫

pdxdy should be used.

Properties Involved in Improving the ConventionalEHL Model

The above-mentioned equations form the conventional EHLmodel. It only contains the elastic modulus property, which isfixed for the certain material. So it will do nothing to and in opti-mum design of rough porous composites. Therefore, the porosityand fractal dimension incorporated into the governing equations,which helps to make the optimization research unique.

Fractal Model of Rough Surface

Because the surface roughness plays an important role inthe contact performance and lubricant film formation, a realis-tic representation of a rough surface is necessary to obtain accu-rate information regarding mixed lubrication. Stochastic modelswere developed by Tonder (5) and others using selected statisticparameters to represent the characteristics of rough surface lu-brication that exhibit dependencies on instrument resolution andsample length. They deal only with the global effect of surfacetopography and provide no detailed information about local pres-sure peaks, local film thickness fluctuations, or asperity deforma-tions, which are usually critical for the study of lubrication break-down and surface failure mechanisms.

The fractal model uses scale-independent parameters tomathematically describe the concepts of self-similarity and self-affinity. It is essential to use it to characterize rough surfaces.Fractals are geometric structures that have a similar appearanceregardless of the scale at which they are being observed. In or-der to investigate the effects of different values of fractal dimen-sions on the performance of the mixed lubrication, a computerprogram was developed to generate rough surfaces with fractaldimensions and fractal roughness based on a technique proposedby Komvopoulos (25).

z(x, y) = L(

GL

)(D−2) (ln γ

M

)1/2

×M∑

m=1

nmax∑n=1

γ(D−3)n{

cos φm,n − cos[

2πγn(x2 + y2)1/2

L

× cos(

tan−1(y

x

)− πm

M

)+ φm,n

]}[2]

Here, x and y are the Cartesian coordinates of a surface pointof height z; D is the fractal dimension (2 < D < 3), G is the am-plitude parameter, which is referred to as the fractal roughness;γ(γ > 1) is a parameter that controls the density of frequencies

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584 Y. LU AND Z. LIU

Fig. 3—Two-dimensional surface topography of a 3D fractal surface fromEq. [2] for G = 2.35 × 10−3 μm and D = 2.3. The RMS roughnessof this surface is σ = 6.5 μm (color figure available online).

z(x, y) (usually γ = 1.5), M is the number of superimposed ridges,φm,n is the random phase in the interval [0, 2π]; L is the samplesize; the upper limit n is given by nmax = int[log(L/Ls/ log γ)],it must equal to the numerical size of 2D Reynolds equation,where int[. . .] denotes the integer part of the number in thebrackets; and Ls is the fractal sample length. Figure 3 shows aplane view of a 3D fractal surface constructed from Eq. [2] us-ing G = 2.35×10−3μm and D = 2.3. The root mean square (rms)roughness of this surface is σ = 6.5 μm.

Physical Parameters of the Self-Lubricating Composite Basedon the Hirsch Model

The porous self-lubricating composite was composed of a cer-met matrix (TiC-Fe-Cr-W-Mo-V) with three-dimensional inter-penetrated micropores and a soft metal lubricant (Pb-Sn-Cu-Ag)infiltrating the channels of the matrix; its microstructure is shownin Fig. 4. Each component of cermet matrix’s and metal lubri-cant’s volume ratio (1:2) and the elastic modulus (American So-ciety for Metals (30)) is shown in (Tables 1 and 2). Accordingto the microstructure of the composite’s three-dimensional inter-penetrated micropores the Hirsh model (Fig. 5) is the equivalentmodel for its physical parameters. The expression of the modulusis a semi-empirical formula, where the constant value is deter-

TABLE 1—COMPONENT OF CERMET MATRIX’S VOLUME FRACTION

AND THE ELASTIC MODULUS

Components Fe Cr W Mo V TiC

Volume fraction (%) 6.9 13.3 7.0 17.2 35.6 20Elastic modulus (GPa) 211 279 411 329 128 350

TABLE 2—COMPONENTS OF METAL LUBRICANT’S VOLUME

FRACTION AND THE ELASTIC MODULUS

Components Pb Sn Cu Ag

Volume fraction (%) 2.377 24.583 47.343 25.696Elastic modulus (GPa) 16 50 130 83

mined by the test. In this article, the value α = 50% was chosen.Therefore, the Hirsch model can be simplified as Eq. [3]:

Eh = αEv + (1 − α)Er, [3]

where Ev = �f iEi is the Voigt modulus (Voigt (13)), Er−1 =

(�f i/Ei)−1 is the Reuss modulus (Reuss (14)), f i is the volumefraction, Ei is the elastic modulus, and Eh = (Ev + Er)/2.

According to Eq. [3], the cermet matrix phase and themetal lubricant phase moduli can be determined as Es =229.4 GPa, El = 85.8 GPa; then,

E(f 0) = 12

[Es(1 − f 0) + Elf 0] + EsEl

2[Esf 0 + El(1 − f 0)], [4]

where f 0 is the volume fraction of the metal lubricant phase,which is also called porosity.

The model presented here takes the composite’s characteris-tics into consideration. It uses the properties of the composite asthe new variables to improve the conventional EHL model. Thisapproach means that the model has distinctive features. Differentlubrication statuses will be determined by the various propertiesinput, so the optimum parameters for design of the rough porouscomposite can be easily chosen.

Design Optimization of Composite

It is obvious that the composite’s physical parameters and itssurface topography are two important factors that determine thevalue of the film thickness motivated by the question of how thetwo parameters of porous self-lubricating composite affect the lu-brication situation. The following part will discuss the mathemat-ical model for calculating these two parameters.

Fig. 4—Typical topographies of micropore before and after infiltration with solid lubricant: (a) micropores in cross section; (b) micropores in verticalsection; and (c) composite infiltrated with metal lubricant.

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Coupled Fractal Effects 585

Fig. 5—Hirsch model of the composite microstructure.

Criterion. Engineers use the λ ratio of film thickness to surfaceroughness to estimate the lubrication situations of the contact-ing bodies. It is a dimensionless film parameter defined to distin-guish the lubrication regimes possible for rough surfaces lubrica-tion (Hamrock, et al. (27)):

λ = h(x, y)/σ, [5]

where σ = (σ21 + σ2

2)2, here σ1 and σ2 are the root mean squareof two contacting surfaces. λ is defined as the ratio of the averagefilm thickness from the smooth surface solution to the compositeRMS roughness. It is generally believed that full-film thicknessis considered to exist when its value is larger than 3 and if λ hasa value less than 3. The surface asperities will force direct solid-to-solid contact, resulting in wear as a major source of failure.According to this theory, it is used as a criterion for determiningthe relative contact area.

s : λ ≤ 3 is the relative contact area l : λ > 3 is the relative lubricant area

Wear Rate. Due to the presence of a solid contact in a thinlubrication area, the effect of running-in wear on asperities ofa rough surface is important. From the wear theory, the totalwear rate Ws = CVFs, where C is the specific wear rate coefficient(m2/N). Fs is the total shear stress applied in the contact area,which is also called friction force. So the friction force Fs at therough surface (h = 0) in the x, z-plane is given by (Komvopoulos,et al. (27)):

Fs =∫ ∫

s

τsdxdy =∫ ∫

s

(−1

2∂p∂x

h + η pVh

)dxdy [6]

The dimensionless form given as Fs = Fs/Fp , here Fp =∫ ∫ s

pdxdy. The life span of the mechanical components will beadversely affected by this high friction force resulting from directsolid contact.

The algorithm so far described is the method for solving thecoupled effects of the fractal dimension of the rough surface andthe porosity of the porous self-lubricating composite on the wearproblem. Based on the lubricant distribution, the relationship be-tween the effects and the friction force can be obtained. The re-sults show that the model represents a significant optimizationtechnique for the porous self-lubricating composite’s microstruc-tural design. The optimization is useful for the composite userto derive a maximum benefit from the available resources. Fig-ure 6 shows a flowchart of the logic in optimal design of thecomposite.

Fig. 6—Flowchart for computing design optimization parameters of thecomposite based on coupled effects.

However, the following points should be noted during the nu-merical solution in practical applications:

1. The model presented here is an evolved full thin-film mixedEHL model. This means that the film will not be separated,although the film thickness h decreases within the asperitiesdomain. The consideration is based on the objective of this ar-ticle: to find optimized values of all parameters. Here the con-cept of relative (relative contact area) has been introduced tobe the standard for determining the area where asperity con-tact occurs. The optimum parameters can be chosen by com-paring the size of these areas under this stand. The real has thesame function as the relative because the relative finally be-comes real under severe working conditions and their changetendencies are equal. Briefly, the result will not change whenthe standard vary from relative to real (when h = 0); there-fore, the requests of any high accuracy calculated when thefilm thickness h = 0 does not make any sense to my problem.

2. This article assumes sufficient lubricant as the condition of hy-drodynamic lubrication independent of the porosity becauseall pores with adequate length have be fully filled with lubri-cant (see Fig. 4).

3. It should be noted that the Moes parameter L is fixed at avalue twice of the width of Hertzian contact and M is fixed at3. In order to make sure the investigated conditions remain

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586 Y. LU AND Z. LIU

Fig. 7—Contour figures of the film distribution with different fractal dimensions: (a) D = 2.1; (b) D = 2.4; and (c) D = 2.7 and E = 200 (color figure availableonline).

the same in all analysis cases, the Moes parameters should befixed.

RESULTS AND DISCUSSIONS

The primary objectives of this section are to predict the cou-pled effects of fractal dimension and physical parameters of aporous self-lubricate composite on the relative contact area bythe numerical method described above and to investigate the

wear rate behavior at different λ ratios. A series of cases wasinvestigating using the same fixed operating conditions: U =1.0×10−12, W = 2.0×10−5, R = 10 mm, η0 = 0.078 Pa · s, ρ0 =870 kg/m3, and the Moes parameters was fixed at L = 4×a, M =3. Different surface fractal dimensions and physical parameterswere used.

Note that the physical properties of the composite were re-inforced by some novel additives such as nano-composite (Tjong

(a) (b)

Fig. 8—Curves showing (a) the effect of various fractal dimensions on both dimensionless friction force and the relative contact area and (b) correlationbetween the friction force and relative contact area when E = 200 (color figure available online).

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Coupled Fractal Effects 587

Fig. 9—Contour figures of the film distribution with different elastic moduli: (a) E = 200; (b) D = 400; (c) D = 600; (d) D = 800, and D = 2.2 (color figureavailable online).

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588 Y. LU AND Z. LIU

(a) (b)

Fig. 10—Curves showing (a) the effect of various elastic moduli on both dimensionless friction force and the relative contact area and (b) correlationbetween the friction force and relative contact area when D = 2.2 (color figure available online).

(26)), so choose E modulus in the range 100–1000 Mpa and poros-ity values from 10 to 40% (details are provided in the followingsections).

Fractal Dimension Surface Influence

The applied modulus was fixed at E = 200 GPa. The corre-sponding dimensionless parameter was E = 4,400. The results arepresented for some types of fractal surfaces with different frac-tal dimensions but the same root mean square σ = 0.4 μm. Thelubrication states are shown by contour figures of the film distri-bution of λ (as defined in the previous subsection), as followingFig. 7. Solutions were successfully obtained for a wide range offractal dimensions. In the top row, the white areas surrounded bythe thick black line are the relative contact areas with λ ≤ 3. Thebottom row shows are the partial enlargement of the relative con-tact areas, which shows details of the fluid film distribution withinthe contact area. It can be seen that the values of the minimumfluid film shown above the color bar are 1.3319, 1.2007, and 0.7416corresponding to results of the fractal dimension of 2.1, 2.4, and2.7. Its decline along with the relative contact area increase showsthe results of the fractal dimension increase. This indicates thatthe breakdown of lubrication layers of fluid is caused by surfaceroughness. A significant increase in the relative contact area dueto the increase in the fractal dimension is observed.

As mentioned above, a linear growth in relative contact areaand the friction force was observed (shown Fig. 8a). The relativecontact area and friction force on the high fractal dimensionsurface was high. Figure 8b shows the correlation between thefriction force and relative contact area. The upward trendingcurve between them indicates that the tendencies of contact areaand the friction force were the same. The contact area may be thebest for interpretation of the value of friction force. The resultsshow that this phenomenon must be dependent on the rough sur-face topography. Concerning the effect of rough surface topogra-

phy on friction, the relative contact area and friction force on thelow fractal dimension was smaller than that on the high fractaldimension.

Physical Parameter Influence

The applied modulus was fixed at fractal dimension D = 2.2.Assume that the elastic modulus was changed in a very widerange from 100 to 1,000 GPa, because the physical–mechanicalproperties of the composite were reinforced by some noveladditives (Duckworth (28)). The corresponding dimensionlessparameter E = 2,200–22,000. In order to deduce the fluid filmsdistribution that is affected only by the composite physicalparameter (elastic modulus), the modulus value of E need to beadjusted by setting the load, velocity and maximum pressuresvalue. The lubrication states are shown by contour figures of thefilm distribution of λ, as following Fig. 9. In the upper figures, thewhite areas surrounded by the thick black line are the relativecontact areas with λ ≤ 3. The lower figures show the partialenlargement of the relative contact areas, which shows the detailof the fluid film distribution within the contact area. It is obviousthat the minimum fluid film increased from 1.3191, 1.7899, 2.3607,to 2.7879 as E increased from 200, 400, 600, to 800 and caused therelative contact area and friction force to decrease accordingly.It can be deduced that when its value is large enough, twosurfaces that are completely separated by a thick fluid film willhave a small relative contact or have no point of the contact atall. The resulting friction force will then produce only a smallwear.

On the whole, a linear decline in relative contact area andfriction force is demonstrated in Fig. 10a. The relative contactarea and frictional force on the high dimensionless modulus wassmall. It should be pointed out that the physical properties of thecomposite have a strong influence on the fluid film distribution.The high modulus resulted in decreased contact area and friction

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Coupled Fractal Effects 589

Fig. 11—Coupled effects of various fractal dimensions (from 2.1 to 2.8)and various elastic modului (from 100 to 1,000) (color figureavailable online).

force, providing a lower wear rate. Figure 10b shows the corre-lation between the friction force and relative contact area, whichemphasizes that the small contact area obviously contributed todecreasing the friction force.

Fig. 12—Coupled effects of various fractal dimensions (from 2.1 to 2.8)and porosities (from 10 to 90) on dimensionless friction force(color figure available online).

Couple effect. The coupled effect is an influence of both fractalrough surface and the composite’s porosity on lubricant and wear.Both factors of the fractal dimension and the physical parametersaffect hydrodynamic lubrication. The coupled effect will be seen

Fig. 13—Overview of optimized design for the microstructure of porous self-lubricating composites (color figure available online).

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590 Y. LU AND Z. LIU

more clearly in the curve shown Fig. 11, which shows comparisonof profiles of friction under the coupled effect. It can be seenthat when E decreased, the friction force increased. Therefore, itmust be considered as a major factor when analyzing the relativecontact area, and when it has a small value the influence of thefrictional force will be more dramatic. It also shows that whenthe fractal dimension is high enough, its influence can not beignored, because the frictional force increases rapidly and itsvalue becomes much larger than the lower fractal dimensionincrease dramatically. It can also be seen that, compared to thelow fractal dimension, D, a higher value could make the frictionforce. In addition to the effect of the elastic modulus, the fractaldimension of the rough surface in the film thickness equationplays a significant role. The fractal dimension enhances theroughness effect because it is easier for the lubricant to leak outfrom the asperities contact when the fractal dimension increases.

Porous Self-Lubricating Composite

The physical properties of porous selflubricant composite de-scribed in the previous section will be determined by its poros-ity. It can be seen in Fig. 12 that according to a porosity value inthe range 10–40% (the porosity of the metal composite will notbe greater than 40%; (Hamrock, et al. (27))) the friction forcewill increase with porosity. Therefore, once the lubrication is atthe desired level, choose the porosity as small as possible. Atthis point, the high fractal dimension also causes a large frictionalforce. Combining these two effects will be the best way to designan optimum composite that, different precision requirements andsevere working conditions, will avoid high friction force.

The method outlined above is shown as an overview in Fig. 13.It illustrates that the developed model can be effectively usedin the absence of porous self-lubricated composite design proce-dures and can be attributed to the fact that they represent a stableself-lubricating fluid and life span. This is attributed to two majorsets of variables: the fractal dimension and porosity. A strong re-lationship between wear rate and coupled effects is quite obviousand an optimal design algorithm can be used to establish it. Then,the wear life span of the porous self-lubricating mechanical el-ements the actual upper limit of the porosity with the differentfractal dimensions can be predicted.

Controlled experiments to confirm the feasibility of the modelwill be the main aim of our further research. The present workpresents a theoretical method for designing optimum porous self-lubricant composites with a rough surface. Due to the limitationof our experimental apparatus for manufacturing samples of theporous composites with different fractal dimensions and measur-ing the thin lubricant film, we still need to determine how to over-come the difficulties mentioned above.

CONCLUSIONS

The following conclusions can be drawn based on the numer-ical model, the results, and the discussions presented above:

1. The conventional EHL model has been improved for themixed lubricating problem based on computer-simulated 3Dfractal rough surfaces. In this model, the generated rough sur-face topographies were used to investigate the effect of the

fractal dimension on the relative contact area ratio. The re-sults showed that the surface texture is important in ensuringproper lubrication and a higher friction force will be generatedwith an increased fractal dimension due to the lubricating thinfluid film.

2. The porosity of the self-lubricating composite can determinethe results of the relative contact area due to its influence onthe elastic modulus. The calculated results based on the vari-ous values of the dimensionless elastic modulus analyzed forfriction force showed that the frictional force increased withan increase in porosity under the same fractal dimensions.

3. It was observed that the higher the porosity and the fractaldimension, the greater the frictional force was. This indicatesthat the optimization of the porosity of the composite basedon the friction force will significantly improved the lubricatingproperties.

The model developed can be effectively used to study the cou-pled effects of the porous self-lubrication composite and its sur-face fractal dimensions on the frictional force. This is the majorfactor causing wear problems. The results can provide the basictheory to aid in design optimized microstructure of the porouscomposite under severe conditions to meet different precisionrequests and improve the composite life span. Thus, any futurestudies attempting to connect these effects to the design opti-mization of porous self-lubricating composites should focus moreon the effect of porosity on the lubricating fluid film distributionrather than the amount of lubricant.

ACKNOWLEDGEMENT

The authors thank the National Natural Science Foundationof P.R. China for financial support (ID 51075311). Professor J. R.Barber of the University of Michigan is acknowledged for fruitfuldiscussions during this study.

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