Research ArticleFractal Loading Model of the Joint Interface Considering StrainHardening of Materials

Yanhui Wang 12 Xueliang Zhang 1 Shuhua Wen1 and Yonghui Chen1

1School of Mechanical Engineering Taiyuan University of Science and Technology Taiyuan 030024 China2Department of Mechanical and Electrical Engineering Shanxi Institute of Energy Jinzhong 030600 China

Correspondence should be addressed to Xueliang Zhang zhang_xue_lsinacom

Received 2 August 2018 Revised 7 December 2018 Accepted 6 February 2019 Published 3 March 2019

Academic Editor Fabio Minghini

Copyright copy 2019 YanhuiWang et alis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Based on fractal geometry theory the deformation state of the four stages of the asperity elastic first elastoplastic secondelastoplastic and fully plastic deformation and comprehensively considering the hardness of the asperity changes with the amountof deformation in elastoplastic deformation stage due to strain hardening are considered thereby establishing a single-loadingmodel of the joint interface By introducing the pushing coefficient and the asperity frequency exponent each critical frequencyexponents of asperity is obtained and the relationship between the normal contact load and the contact area of the firstelastoplastic deformation phase and the second elastoplastic deformation phase of the single asperity in the case of taking intoaccount the change in hardness is inferred eventually deducing the relationship between the total contact load of the jointinterface and the contact area e analysis results show that in the elastoplastic deformation stage when the deformation isconstant the asperity load considering the hardness change is smaller than the unconsidered load and the difference increaseswith the increase of the deformation amount e establishment of the model provides a theoretical basis for further research onthe elastoplastic contact of joint interfaces

1 Introduction

e joint formed by assembly of mechanical parts is calledthe joint interface which plays an important role in thetransmission of motion load and energy in the normaloperation of mechanical system e joint interface presentsa series of different curvature radius and the height asperitiesat the microscopic scale e contact between the jointinterface is discontinuous and only occurs at higher asperitylead to the real contact area accounts for only a small part ofthe nominal contact area resulting in the situation of largeload on a small contact area As a result researches on theproperties of interfacial contact and stress analysis are verycomplex [1] erefore to study the deformation behaviorand the accurate modeling of joint interface is an importantissue for in-depth understanding of the mechanism offriction wear lubrication heat conduction etc

Statistical and fractal contact models for solving contactproblems on joint interface have been used widely in this

field Statistical contact model was originally put forward byGreenwood and Williamson (GW model) and improved bymany subsequent researchers [2ndash4] Zhao et al deduced anew elastoplastic contact model of joint interface whichdescribing a long transition period from elastic deformationto fully plastic deformation of joint interface It is shown thatthe elastoplastic contact of the asperity plays an importantrole in the microscopic contact behavior of the joint in-terface [5] Kogut and Etsion established the contact modelbetween a single asperity and a rigid flat by means of finiteelement analysis and obtained the relationship betweencontact area and contact load of a single asperity duringloading and unloading [6 7] Kadin et al applied theconclusion of Etsion to the whole joint interface and got astatistical model of single loading and unloading of jointinterface According to his conclusion plastic deformationand residual stress may occur in the process of loading andunloading e actual contact area of the asperity duringunloading is larger than the actual contact area of the loading

HindawiAdvances in Materials Science and EngineeringVolume 2019 Article ID 2108162 14 pageshttpsdoiorg10115520192108162

process [8 9] However the value of statistical parametersdepends largely on the filter or resolution of the roughnessmeasuring instrument so it is not unique for a jointinterface

e fractal model was first proposed by Majumdar andBhushan (MB model) in 1990 e model holds that thedeformation of microconvex body changes from plasticdeformation to elastic deformation with the increase ofload contrary to the traditional contact study [10ndash12]Many scholars put forward many kinds of fractal modelsbased on MB fractal model and obtained more accuratecontact mechanical properties of joint interface Wang et almodified the area distribution density function of asperityin MB model and obtained the modified model of MBelastic and plastic contact [13 14] Morag and Etsionestablished the elastoplastic contact fractal model of asingle asperity and explained the contradiction between thedeformation sequence of asperity from plastic deformationto elastic deformation in MB model and the classical Hertzcontact theory [15] Tian et al modified the model furthertaking into account the change of material hardness withthe change of surface depth in elastoplastic stage andestablished a new single-loading model of joint interfaceHowever the model only takes into account the transitionfrom elastic to elastoplastic and elastoplastic to fully plasticdeformation stage of the asperity e description of theelastoplastic deformation stage is seldom involved in themodel [16] Yuan et al proposed an improved model of thefractal elastoplastic contact model of rough surface basedon the MB model so as to deduce a model of the totalcontact load and the total actual contact area [17] How-ever the model does not take into account the strainhardening phenomenon of the joint interface material thatis the hardness of the material is no longer a constantvalue but will change with the increase of the amount ofdeformation Hardness is an important index to charac-terize the mechanical properties of materials such aselasticity plasticity strength and toughness e change ofhardness value is directly related to the accuracy of cal-culation According to the strain hardening criterion theaverage hardness increases with the increase of de-formation e degree of plastic deformation increases thedegree of work hardening and dislocation strengtheningincreases and the hardness of the material increases Basedon the above research results and fractal theory a newhardness change function is expected to be constructed inthis paper considering that the hardness of the materialchanges with the deformation amount of the asperity in theelastoplastic deformation stage In this paper the criticalconditions of elastic elastoplastic and plastic deformationof asperity are studied and the four deformation ranges arerevised and a fractal theoretical model describing the singleloading of the joint interface is proposed It is expected thatthe microscopic and macroscopic contact state of thesurface of the interface can be more scientifically andreasonably described in order to provide some theoreticalbasis for the research of contact friction wear and lu-brication on the surface of mechanical parts

2 Fractal Model of a Single Asperity

Majumdar et al show that the contours of joint interfacetopography in practical engineering have fractal charac-teristics mathematical characteristics are continuity non-differentiability and self-affinity [10 11] e joint interfaceprofile can be described by theWeierstrassndashMandelbrot (W-M) function which is expressed as

Z(x) G(Dminus1)

1113944

infin

nnmin

cos 2πcnx( 1113857

c(2minusD)n (1ltDlt 2 cgt 1) (1)

where x is the horizontal coordinate of the profile functionof the joint interface and the corresponding function valueis the height of the profile D is the fractal dimension of thesurface profile (for a physically continuous surface1ltDlt 2) G is the length scale parameter of the surfacewhich reflects the amplitude of Z(x) and is the measure-ment constant nmin is the lowest frequency exponent cor-responding to the profile and cn determines the spectrum ofsurface roughness which is the frequency density controlparameter cgt 1 e actual surface profile has an unstablerandomness [18] and its lowest frequency is related to thesample length which is given by cnmin 1L In order tosatisfy the requirements for high spectral density and forphase randomization c 15

21 Elastic Deformation of a Single Asperity On the mi-croscopic scale the contact between the two joint interfacesis essentially a contact between the asperity and the asperitywhich can be simplified as a contact between a series ofequivalent asperities on the joint interface and a rigid flatsurface Assuming that the joint interface is isotropic thereis no interaction between the asperity and the asperityduring the contact process and no large deformation willoccur e equation before deformation of the asperity withfrequency exponent n is obtained as follows

zn(x) GDminus1cos πcnx( 1113857

c(2minusD)nminus

12cnltxlt

12cn

1113888 1113889 (2)

Figure 1 shows an asperity in equivalent joint interfacecontacts with a rigid flat surface e height of the asperity ishn the interference of the asperity is ωn during the loadingprocess and the size of the substrate of the asperity is lnAccording to equation (2) the curvature radius of an as-perity with frequency exponent n at any point x is obtainedas follows

ρn(x) 1 + dzn(x)dx1113858 1113859

21113966 1113967

32

d2zn(x)dx2

11138681113868111386811138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868111386811138681113868

1 + π2G2(Dminus1)c2(1minusD)n( 1113857sin2 πcnx( 11138571113960 1113961

32

minus π2GDminus1cminusDn( 1113857cos πcnx( 11138571113868111386811138681113868

1113868111386811138681113868

(3)

When x 0 the curvature radius of asperity isminimum

2 Advances in Materials Science and Engineering

ρnmin(x) Rn cminusn D

π2GDminus1(4)

e height before deformation of the asperity is

hn zn(0) GDminus1

c(2minusD)n (5)

In the loading process of asperity the deformation willincrease with the increase of normal contact load Accordinglyasperity will change from elastic deformation to elastoplasticdeformation and then to fully plastic deformation

e elastic critical interference of asperity at initial yieldis [16]

ωnec 3πKH

4Eprime1113874 1113875

2R (6)

where K 0454 + 041υ υ is the Poisson ratio of the softermaterial H is the hardness of the softer material H 28Yand Eprime is the equivalent elastic modulus (1Eprime)

(1minus υ1primeE1) + (1minus υ2primeE2) E1 and E2 are respectively elasticmodulus of two objects in contact with each other υ1prime and υ2primeare respectively the corresponding Poisson ratios

When ωn ltωnec an lt anec the asperity is in a state ofelastic deformation According to Hertz theory the contactarea of the asperity is

an πRωn (7)

Substituting equation (6) in equation (7) the criticalcontact area of the elastic asperity is

anec πRωnec π3πKH

4Eprime1113874 1113875

2R2

1π

3KHcminusn D

4G(Dminus1)Eprime1113888 1113889

2

(8)

According to the Hertz contact theory the normal loadon a single asperity is

fne 43EprimeR12

n ω32n (9)

Substituting equations (4) and (7) in equation (9) we canobtain

fne 4Eprimea32

n π12GDminus1

3cminusDn (10)

According to equations (6) (8) and (9) we can get thecritical contact load of the elastic asperity

fnec K middot H middot anec (11)

22 Elastic-Plastic Deformation of a Single AsperityLiterature [6] through the finite element analysis of a singleasperity it is concluded that when the asperity actual de-formation is greater than the elastic critical interference(ωn gtωnec) the yield phenomenon begins to appear and theelastoplastic deformation of the asperity occurs Accordingto the results of [6] the elastoplastic deformation of asperitycan be divided into two different stages according to theratio ωnωnec namely the first elastoplastic deformationstage when 1ltωnωnec le 6 and the second elastoplasticdeformation stage when 6ltωnωnec le 110 Define ωnepc

6ωnec as the first elastoplastic critical interference where theactual contact area is anepc e actual deformation of as-perity ωnpc 110ωnec is defined as the second elastoplasticcritical interference and the actual contact area is anpc erelationship between contact area-deformation and contactload-deformation in the elastoplastic deformation stage ofasperity is [6]

an

anec 093

ωn

ωnec1113888 1113889

1136

fnep1

fnec 103

ωn

ωnec1113888 1113889

1425

ωnec ltωn le 6ωnec( 1113857

an

anec 094

ωn

ωnec1113888 1113889

1146

fnep2

fnec 140

ωn

ωnec1113888 1113889

1263

6ωnec ltωn le 110ωnec( 1113857

(12)

From the above equations we can get

X

ln

h n

w n

O

Z

Figure 1 Diagram of single asperity loading

Advances in Materials Science and Engineering 3

anepc 71197anec

fnep1 KH times 11282aminus02544nec a

12544n anec lt an lt anepc1113872 1113873

(13)

anpc 2053827anec

fnep2 KH times 14988aminus01021nec a

11021n anepc lt an lt anpc1113872 1113873

(14)

where fnec is contact load for ω ωnec and fnep1 and fnep2are contact loads in the first elastoplastic stage and thesecond elastoplastic stage respectively Both fnep1 and fnep2obtained above are related to the hardness (H) of thematerial However according to the plastic strengtheningprinciple the hardness is not a constant when the materialyields but a function related to the deformation that is itchanges with the deformation erefore it is not accurateto describe elastoplastic deformation by the above formulaIn order to express the characteristics of elastoplastic de-formation more accurately the concept of limit meangeometric hardness is introduced

According to equations (13) and (14) HG(a) is fittedinto the following segmented relations

e first elastoplastic deformation stage is

HG1 an( 1113857 c1Yan

anec1113888 1113889

c2

anec lt an le anepc1113872 1113873 (15)

e second elastoplastic deformation stage is

HG2 an( 1113857 c3Yan

anec1113888 1113889

c4

anepc lt an le anpc1113872 1113873 (16)

where c1 c2 c3 and c4 are the coefficients to be solved

(1) Equation (15) should satisfy two limiting conditions

HG1 anec( 1113857 pea anec( 1113857 (17)

HG1 anepc1113872 1113873 pepa1 anepc1113872 1113873 (18)

where pea(a) is the average contact pressure of theasperity in elastic stage which is given bypea(a) (fnea) pepa1(a) is the average contactpressure of the asperity in the first elastoplastic de-formation stage and is given by pepa1(a) (fnep1a)Substituting equations (11) and (15) in equation (17)we can obtain

c1Y KH (19)

c1 28K (20)

Substituting equations (13) and (15) in equation (18)we can obtain

KH times 11282aminus02544nec times 71197anec( 1113857

02544 28KY times 71197c2

(21)

Derived from equation (21)

c2 ln 11282 times 7119702544( 1113857

ln 71197 (22)

Considering the change of hardness the normal con-tact load of a single asperity in the first elastoplasticstage is

fnep1prime HG1(a) middot an (23)

Substituting equations (15) (20) and (22) in equation(23) new equations are yielded

fnep1prime 28KYaminusc2neca

c2+1n (24)

(2) Equation (16) should satisfy two limiting conditions

HG2 anepc1113872 1113873 pepa1 anepc1113872 1113873 (25)

HG2 anpc1113872 1113873 pepa2 anpc1113872 1113873 (26)

where pepa2(a) (fnep2a) is the average contactpressure of the asperity in the second elastoplastic stageSubstituting equations (13) and (16) in equation (25)we can obtain

c3(71197)c4 K times 28 times 11282 times(71197)

02544 (27)

Substituting equations (14) and (16) in equation (26)we can obtain

c3(2053827)c4 K times 28 times 14988 times(2053827)

01021

(28)

Simultaneous equations (27) and (28) obtained

c4 ln 11282 + 02544 ln 71197minus ln 14988minus 01021 ln 2053827

ln 71197minus ln 2053827

(29)

Substituting equation (29) in equation (27) we canobtain

c3 K times 315896 times(71197)02544minusc4 (30)

Considering the change of hardness the normal con-tact load of a single asperity in the second elastoplasticstage is

fnep2prime HG2(a) middot an K times 315896 times(71197)02544minusc4Y

middot aminusc4nec middot a

c4+1n

(31)

23 Full Plastic Deformation of a Single Asperity As thedeformation continues to increase when ωn gt 110ωnec thecontact area an gt anpc and the asperity enters the stage of fullplastic deformation At this stage the hardness of the ma-terial is no longer affected by the deformation and can beregarded as a constant When the hardness of the material is

4 Advances in Materials Science and Engineering

given according to literature [7] the contact load andcontact area of the asperity at this stage can be expressed as

fnp Han

an 2πRnωn(32)

In conclusion with the increase of load and deformationthe contact area of the same asperity increases graduallyie anec lt anepc lt anpc With the increase of the load andcontact area the asperity underwent elastic deformationfirst elastoplastic deformation second elastoplastic de-formation and full plastic deformation successively Underconstant load and deformation the actual contact area of theasperity is related to the radius of curvature at the vertex ofthe asperity

24 Asperityrsquos Frequency Exponent n When using W-Mfunction to describe the surface profile of an asperity theprofile function is related to the asperityrsquos frequency ex-ponent In other words the radius of curvature at the vertexof the asperity and the height of the asperity vary with thefrequency exponent when the load is constant According tothe equations (5)ndash(7) it was found that the value of hn Rnand ωnec correlated with the frequency exponent When thefrequency exponent is constant the deformation of theasperity is not greater than the height of the asperityunder the action of the load In order to obtain the criticalvalue of the frequency exponent we take hn ωnec ie(GDminus1c(2minusD)nec) (3KH4Eprime)2 middot (cminusnecDGDminus1)

e elastic critical frequency exponent can be obtainedas follows

nec intln 3KH4Eprime( 1113857

2middot G2(1minusD)1113960 1113961

2(Dminus 1)ln c

⎧⎨

⎩

⎫⎬

⎭ (33)

where int is the integer part of the value in the parenthesisSimilarly the first elastoplastic critical frequency expo-

nent can be obtained

nepc intln 6 3KH4Eprime( 1113857

2middot G2(1minusD)1113960 1113961

2(Dminus 1)ln c

⎧⎨

⎩

⎫⎬

⎭ (34)

e second elastoplastic critical frequency exponent canbe obtained

npc intln 110 3KH4Eprime( 1113857

2middot G2(1minusD)1113960 1113961

2(Dminus 1)ln c

⎧⎨

⎩

⎫⎬

⎭ (35)

From the above when the asperity frequency exponent isnmin lt nle nec elastic deformation only takes place in theseasperities under contact load When nec lt nle nepc elasticdeformation or the first elastoplastic deformation can takeplace in these asperities When nepc lt nle npc elastic de-formation the first elastoplastic deformation or the secondelastoplastic deformation can take place in these asperities and

full plastic deformation never occur When npc lt nle nmaxelastic deformation elastoplastic deformation or full plasticdeformation can take place in these asperities

3 Actual Contact Area and Normal ContactLoad of Joint Interface

According to reference [10] when the asperity frequencyexponent is n the area distribution density function of theasperity on the joint interface is defined as

nn(a) 12

D middotaD2nl

a(D+2)2 0lt ale anl 1ltDlt 2( 1113857 (36)

where anl represents the largest contact area when theasperityrsquos frequency exponent is n

In order to simplify equation (36) we define the areadistribution function of the asperity of any frequency ex-ponent as nn(a) Mn(a) According to reference [17] theactual contact area of joint interface is

Ar 1113944

nmax

nnmin

1113946anl

0nn(a)a da M 1113944

nmax

nnmin

1113946anl

0n(a)a da (37)

where M (al1113936nmaxnnmin

anl)(nmin le nle nmax al max anl1113864 1113865)

31 When the Frequency Exponent Belongs to nmin lt nle necWhen the frequency exponent belongs to nmin lt nle nec evenif these asperities are completely deformed only elasticdeformation will occur and anl lt anec In this case the actualcontact area of the joint interface is defined as Ar1

Ar1 1113944

nec

nnmin

1113946anl

0Mn(a)ada

MD

2minusD1113944

nec

nnmin

anl (38)

In this case the contact load of the joint interface is asfollows

Fr1 1113944

nec

nnmin

1113946anl

0fneMn(a)da (39)

Substituting equation (11) in equation (39) we canobtain

Fr1 MD

3minusD1113944

nec

nnmin

4Eπ12G(Dminus1)

3cminusDna32nl (40)

32 When the Frequency Exponent Belongs to nec lt nle nepcWhen the frequency exponent belongs to nec lt nle nepc forthe case anec lt anl le anepc elastic deformation or the firstelastoplastic deformation may take place in these asperitiesAt this point the actual contact area of the joint interfaceconsists of two parts the elastic deformation stage and thefirst elastoplastic deformation stage

Advances in Materials Science and Engineering 5

Ar2 Are + Arep1 (41)

Are 1113944

nepc

nnec+11113946

anec

0Mn(a)ada

MD

2minusD1113944

nepc

nnec+1a

(2minusD)2nec a

D2nl

(42)

For the determined frequency exponent the maximumactual contact area of the asperity appears at the maximumdeformation amount ωn where the maximum value of theelastic deformation phase ωn appears at ωnec whereuponformula (42) is simplified to

Are MD

2minusD1113944

nepc

nnec+1anec

MD

(2minusD)π1113944

nepc

nnec+1

3KHcminusDn

4G(Dminus 1)Eprime1113888 1113889

2

Arep1 1113944

nepc

nnec+11113946

anl

anec

Mn(a)ada MD

2minusD

middot 1113944

nepc

nnec+1a

(2minusD)2nl minus a

(2minusD)2nec1113960 1113961a

D2nl

(43)

e contact load is given by

Fr2 Fre + Frep1 (44)

Fre 1113944

nepc

nnec+11113946

anec

0fneMn(a)da

9MD(KH)3

16(3minusD) EprimeπGDminus1( 11138572

middot 1113944

nepc

nnec+1cminus2 Dn

(45)

Frep1 1113944

nepc

nnec+11113946

anl

anec

fnep1prime Mn(a)da (46)

Substituting equations (24) and (36) in equation (46) wecan obtain

Frep1 28KYMD

2c2 minusD + 21113944

nepc

nnec+1aminusc2neca

c2+1nl minus a

(2minusD)2nec a

D2nl1113960 1113961 (47)

33 When the Frequency Exponent Belongs to nepc lt nle npcWhen the frequency exponent belongs to nepc lt nle npc forthe case anepc lt anl le anpc elastic deformation the firstelastoplastic deformation or the second elastoplastic de-formation may take place in these asperities At this point

the actual contact area of the joint interface consists ofthree parts the elastic deformation stage the first elas-toplastic deformation stage and the second elastoplasticdeformation stage

Ar3 Are + Arep1 + Arep2

Are 1113944

npc

nnepc+11113946

anec

0Mn(a)ada

MD

(2minusD)π

middot 1113944

npc

nnepc+1

3KHcminusDn

4GDminus1Eprime1113888 1113889

2

Arep1 1113944

npc

nnepc+11113946

anepc

anec

Mn(a)ada

MD

π(2minusD)71197minus 71197D2

1113872 1113873 1113944

npc

nnepc+1

3KHcminusDn

4GDminus1Eprime1113888 1113889

2

Arep2 1113944

npc

nnepc+11113946

anl

anepc

Mn(a)ada

MD

(2minusD)1113944

npc

nnepc+1a

(2minusD)2nl minus 71197anec( 1113857

(2minusD)21113960 1113961a

D2nl

(48)

In this case the contact load of the joint interface is asfollows

Fr3 Fre + Frep1 + Frep2

Fre 1113944

npc

nnepc+11113946

anec

0fneMn(a)da

9MD(KH)3

16(3minusD) EprimeπGDminus1( 11138572 1113944

npc

nnepc+1cminus2 Dn

Frep1 28KYMD

2c2 minusD + 271197c2+1 minus 71197D2

1113872 1113873

middot 1113944

npc

nnepc+1

1π

3KHcminusDn

4GDminus 1Eprime1113888 1113889

2

(49)

When the second elastoplastic deformation occurs thenormal contact load of the joint interface is as follows

6 Advances in Materials Science and Engineering

Frep2 1113944

npc

nnepc+11113946

anl

anepc

fnep2prime Mn(a)da (50)

Substituting equations (31) and (38) in equation (46) wecan obtain

Frep2 2106 times 7119702544minusc4KYMD

2c4 minusD + 2

middot 1113944

npc

nnepc+1aminusc4neca

c4+1nl minus 71197c4minus(D2)+1

a1minus(D2)nec a

(D2)nl1113872 1113873

(51)

34 When the Frequency Exponent Belongs to npc lt nWhen the frequency exponent belongs to npc lt n elasticdeformation elastoplastic deformation or full plastic de-formation may take place in these asperities e actualcontact area of the joint interface can be evaluated as

Ar4 Are + Arep1 + Arep2 + Arp

Are 1113944

nmax

nnpc+11113946

anec

0Mn(a)ada

MD

2minusD1113944

nmax

nnpc+1anec

Arep1 1113944

nmax

nnpc+11113946

anepc

anec

Mn(a)ada

MD

2minusD71197minus 71197D2

1113872 1113873 1113944

nmax

nnpc+1anec

Arep2 1113944

nmax

nnpc+11113946

anpc

anepc

Mn(a)ada

MD

2minusD2053827minus 711971minus(D2)

middot 2053827D21113872 1113873 1113944

nmax

nnpc+1anec

Arp 1113944

nmax

nnpc+11113946

anl

anpc

Mn(a)ada

MD

2minusD1113944

nmax

nnpc+1a

(2minusD)2nl minus 2053827anec( 1113857

(2minusD)21113960 1113961a

D2nl

(52)

In this case the contact load of the joint interface is asfollows

Fr4 Fre + Frep1 + Frep2 + Frp

Fre 1113944

nmax

nnpc+11113946

anec

0fneMn(a)da

MDKH

(3minusD)π1113944

nmax

nnpc+1anec

Frep1 1113944

nmax

nnpc+11113946

anepc

anec

fnep1prime Mn(a)da

28KYMD

2c2 minusD + 271197c2+1 minus 71197D2

1113872 1113873 1113944

nmax

nnpc+1anec

Frep2 2106 times 7119702544minusc4KYMD

2c4 minusD + 2

middot 1113944

nmax

nnpc+1aminusc4neca

c4+1nl minus 71197c4minus(D2)+1

a1minus(D2)nec a

D2nl1113872 1113873

2106 times 7119702544minusc4KYMD

2c4 minusD + 2

middot 2053827c4+1 minus 2053827D2middot 71197c4minus(D2)+1

1113872 1113873

middot 1113944

nmax

nnpc+1anec

Frp 1113944

nmax

nnpc+11113946

anl

anpc

fnpMn(a)da

MHD

2minusD1113944

nmax

nnpc+1a

D2nl a

1minus(D2)nl minus 2053827anec( 1113857

1minus(D2)1113960 1113961

(53)

For all frequency exponents the total actual contact areaof the joint interface is

Ar Ar1 + Ar2 + Ar3 + Ar4 (54)

e total contact load of the joint interface is

Fr Fr1 + Fr2 + Fr3 + Fr4 (55)

e total real contact area and the total contact load in anondimensional form can be written as follows

Alowastr

Ar

Aa

Flowastr

Fr

AaE

(56)

where Aa is the nominal contact area and is given byAa L2L 1cnmin

Advances in Materials Science and Engineering 7

4 Results Analysis

In order to further analyze the above calculation results theparameters of equivalent joint interface are taken as shownin Table 1 [16]

Figure 2 shows the relation between all critical contactareas and frequency exponents of single asperity whenD 15 It can be seen from the figure that as for one definiteasperity when frequency exponent n is certain elastic criticalcontact area is minimum followed by the first elastoplasticcritical contact area and the second elastoplastic criticalcontact area is maximum With gradual increase of contactload the contact area increases e single asperity is firstlysubject to elastic deformation followed by the first elasto-plastic deformation the second elastoplastic deformation andfully plastic deformation successively which is consistent withtypical contact mechanics theory As for different asperitieswith increase of frequency exponent all critical contact areasdecrease correspondingly which shows that elastic criticalcontact area the first elastoplastic critical contact area and thesecond elastoplastic critical contact area are all related tofrequency exponent n

Figure 3 shows the relation curve between fractal di-mension D and critical frequency exponent n of asperityWhen fractal dimension is definite elastic critical frequencyexponent nec the first elastoplastic critical frequency ex-ponent nepc and the second elastoplastic critical frequencyexponent npc increase gradually As shown in Figure 3 whenDlt 106 nec nepc and npc are all negative As for asperitieswith minimum value of frequency exponent being 0 elasticdeformation elastoplastic deformation and fully plasticdeformation will all occur When D 113 nec and nepc arenegative and npc is positive At this time as for asperitieswith minimum value of frequency exponent being 0 elasticdeformation elastoplastic deformation will occur exceptfully plastic deformation

For D 15 G 25 times 10minus9 m H 55 times 109 Nm2 wecan obtain the elastic critical frequency exponent nec 32the first elastoplastic critical frequency exponent nepc 36and the second elastoplastic critical frequency exponentnpc 43 ese asperities whose frequency exponents rangefrom 20 to 32 are only under elastic deformation Elasticdeformation and the first elastoplastic deformation canoccur in these asperities whose frequency exponents rangefrom 33 to 36 Elastic deformation the first elastoplasticdeformation and the second elastoplastic deformation canoccur in these asperities whose frequency exponents rangefrom 37 to 43 When frequency exponents range from 43 to50 all deformations types can occur in these asperities

Figure 4 shows relation comparison diagram betweencontact load and contact area of single asperity with andwithout hardness change at the first elastoplastic stage ecomparison diagram is simulation result when n 33 It canbe seen from the figure that with gradual increase of contactarea with contact area of single asperity over 32 times 10minus13 m2contact load of the same asperity with hardness change willbe less than that without hardness change In addition as theamount of deformation increases the difference betweenthem tends to increase

Figure 5 shows relation comparison diagram betweencontact load and contact area of single asperity with andwithout hardness change at the second elastoplastic stagee comparison diagram is simulation result when n 37 Itcan be seen from the figure that when the deformation is

Table 1 e parameters of equivalent joint interface

Parameters ValuesEquivalent elastic modulus Eprime 72 times 1010 Nm2

Poissonrsquos ratio υ 017Initial hardness H 55 times 109 Nm2

Profile scale parameter G 25 times 10minus9 mFractal dimension D 1ltDlt 2Frequency exponent n 20sim50

20 25 30 35 40 45 5010ndash1510ndash1410ndash1310ndash1210ndash1110ndash1010ndash910ndash810ndash710ndash610ndash510ndash410ndash3

Criti

cal c

onta

ct ar

eas

of a

singl

e asp

erity

(am

2 )

Asperity levels (n)

Second elastoplastic critical contact areaFirst elastoplastic critical contact areaElastic critical contact area

Figure 2 e relationship between critical contact area and fre-quency exponent of a single asperity

10 11 12 13 14 15 16 17 18 19 20ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

0

20

40

Asp

erity

freq

uenc

y ex

pone

nts (n)

e fractal dimension (D)

Elastic critical frequency exponentsFirst elastoplastic critical frequency exponentsSecond elastoplastic critical frequency exponents

Figure 3 e relationship between fractal dimension D andcritical frequency exponent n of a single asperity

8 Advances in Materials Science and Engineering

definite contact load of the same asperity with hardnesschange will be less than that without hardness change Inaddition with increase of deformation amount the differ-ence between them tends to increase which is consistentwith the change trend at the first elastoplastic stage

Figure 6 shows the relation between limit mean geo-metric hardness and contact area (logarithm) of single as-perity at the first elastoplastic deformation stage Figure 6(a)shows relation curve that D 11 13 15 17 when n 34Figure 6(b) shows relation curve that n 32 33 34 35 whenD 15 It can be seen from Figure 6 that the limit meangeometric hardness of single asperity is related to contactarea fractal dimension and frequency exponent in the firstelastoplastic deformation stage e limit mean geometrichardness increases with increase of contact area When n is

definite the relation between limit mean geometric hardnessand contact area of asperity is related to fractal dimension De larger the D is the more obvious the relation curvebetween them changes when D is definite the relationbetween limit mean geometric hardness and contact area ofasperity is related to frequency exponent n e smaller n isthe more obvious the relation curve between them changes

Figure 7 shows the relation between limit mean geo-metric hardness and contact area (logarithm) of single as-perity at the second elastoplastic deformation stageFigure 7(a) shows the relation curve n 40 that D 11 1315 17 when n 40 Figure 7(b) shows the relation curvethat n 36 38 40 42 when D 15

During loading the contact area increases with the in-crease of deformation of a single asperity e ratio of de-formation to the natural height of the asperity is defined asthe pushing coefficient namely the pushing coefficientk ωnhn 0le kle 09 When fractal dimension is 15 we willresearch the relation between contact load and contact areaof single asperity with frequency exponent n being 30 35and 40 respectively during loading

When n 30 the asperity will only be subject to elasticdeformation During loading even the pushing coefficient k

is maximum no plastic deformation will occur e relationbetween contact area and contact load is fsima15 approxi-mately as shown in Figure 8(a)

As is shown in Figure 8(b) when n 35 elastic de-formation and the first elastoplastic deformation may takeplace in the asperity during loading When the pushingcoefficient k is less than 0247 the asperity will under elasticdeformation At this time the relation between contact areaand contact load is fsima15 approximately when the pushingcoefficient is over 0247 the first elastoplastic deformationoccurs At this time the relation between contact area andcontact load is fsima11093 approximately As is shown inFigure 8(c) when n 40 elastic deformation the firstelastoplastic deformation and the second elastoplastic

0000 0001 0002 0003 0004 0005Contact load of a single asperity in the

first elastoplastic deformation regime (fN)

00

12 times 10ndash12

10 times 10ndash12

80 times 10ndash13

60 times 10ndash13

40 times 10ndash13

20 times 10ndash13

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

Change of hardness is consideredChange of hardness is not considered

Figure 4 e relationship between contact load and contact area of single asperity in the first elastoplastic deformation stage

0000 0001 0002 0003 0004 00050

2 times 10ndash13

4 times 10ndash13

6 times 10ndash13

8 times 10ndash13

1 times 10ndash12

1 times 10ndash12

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

Contact load of a single asperity in the second elastoplastic deformation regime (f2N)

Change of hardness is not consideredChange of hardness is considered

Figure 5e relationship between contact load and contact area ofsingle asperity in the second elastoplastic deformation stage

Advances in Materials Science and Engineering 9

deformation may take place in the asperity during loadingWhen the pushing coefficient is greater than 01954 theasperity begins to enter the second elastoplastic de-formation the relation between contact area and contactload is fsima10977 approximately When n 45 and thepushing coefficient is greater than 0472 the asperity beginsto enter fully plastic deformation the relation betweencontact area and contact load is fsima approximately

Figure 9 shows that when the minimum frequency ex-ponent is 20 and the maximum value is 32 the actual contactarea of the joint interface increases with the increase of thetotal contact load and the relation between them isFlowastr simAlowast15

r approximately During the whole deformationprocess the joint interface appears to be of elastic property

Figure 10 shows the relation between dimensionless totalreal contact area and the dimensionless total contact loading

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash9

10ndash15

10ndash14

10ndash13

10ndash12

10ndash11

10ndash10

D = 11 n = 34D = 13 n = 34

D = 15 n = 34D = 17 n = 34

25 times 109 30 times 109 35 times 109 40 times 109 45 times 109 50 times 109 55 times 109

Limit mean geometric hardness in the firstelastoplastic deformation regime HG1(Nm2)

(a)C

onta

ct ar

eas o

f a si

ngle

aspe

rity

(am

2 )

10ndash10

10ndash15

10ndash14

10ndash13

10ndash12

10ndash11

n = 32 D = 15n = 33 D = 15

n = 34 D = 15n = 35 D = 15

25 times 109 30 times 109 35 times 109 40 times 109 45 times 109 50 times 109 55 times 109

Limit mean geometric hardness in the firstelastoplastic deformation regime HG1(Nm2)

(b)

Figure 6 e relationship between limit mean geometric hardness and contact for single asperity during the first elastoplastic deformationstage (a) n 34 11leDle 17 (b) D 15 32le nle 35

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash12

10ndash18

10ndash17

10ndash16

10ndash15

10ndash14

10ndash13

D = 11 n = 40D = 13 n = 40

D = 15 n = 40D = 17 n = 40

50 times 109 55 times 109 60 times 109 65 times 109 70 times 109 75 times 109

Limit mean geometric hardness in the secondelastoplastic deformation regime HG2(Nm2)

(a)

50 times 109 55 times 109 60 times 109 65 times 109 70 times 109 75 times 109

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash10

10ndash11

10ndash12

10ndash13

10ndash18

10ndash17

10ndash16

10ndash15

10ndash14

n = 36 D = 15n = 38 D = 15

n = 40 D = 15n = 42 D = 15

Limit mean geometric hardness in the secondelastoplastic deformation regime HG2(Nm2)

(b)

Figure 7 e relationship between limit mean geometric hardness and contact for single asperity during the second elastoplastic de-formation stage (a) n 40 11leDle 17 (b) D 15 36le nle 42

10 Advances in Materials Science and Engineering

of the joint interface when minimum value and maximumvalue of frequency exponent of asperities are respectively 20and 36When the pushing coefficient k is 01648 the asperitybegins to be subject to the first elastoplastic deformation andthe relation between dimensionless total real contact areaand the dimensionless total contact loading is FlowastrsimAlowast15

r

approximately e joint interface appears to be of elasticproperty As the load continues increasing when thepushing coefficient k is 05564 at this pointFlowastr gt 69023 times 10minus3 the relation between above load and areais approximately FlowastrsimAlowast11093

r e joint interface presentselastoplastic properties At this time the first elastoplasticdeformation takes place in these asperities whose frequencyexponents range 33sim36

Figure 11 shows the relation between dimensionless totalreal contact area and the dimensionless total contact loadingof the joint interface when minimum value and maximumvalue of frequency exponent of asperities are respectively 44and 50When the pushing coefficient k is 00621 the asperity

begins to be subject to the second elastoplastic deformationat this point 00016ltFlowastr lt 0023 and the relation betweendimensionless total real contact area and the dimensionlesstotal contact loading is FlowastrsimAlowast10977

r approximately Whenthe pushing coefficient k is 07076 the asperity begins to besubject to the fully plastic deformation at this pointFlowastr gt 0023 and the relation between dimensionless total realcontact area and the dimensionless total contact loading isapproximately FlowastrsimAlowastr

5 Conclusions

(1) e contact mechanical properties of a single as-perity on a joint interface are related to the frequencyexponent of the asperity while the frequency ex-ponent of an asperity is related to the fractal di-mension and the profile scale parameter In thispaper the critical frequency exponents of each de-formation stage of a single asperity are obtained and

000 005 010 015 020 025 030 03500

10 times 10ndash11

20 times 10ndash11

30 times 10ndash11

40 times 10ndash11

50 times 10ndash11

60 times 10ndash11

Contact load of a single asperity (fN)

n = 30

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

(a)

Contact load of a single asperity (fN)

n = 35

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

50

times10

ndash5

10

times10

ndash4

104 times 10ndash4

10 times 10ndash13

80 times 10ndash14

60 times 10ndash14

40 times 10ndash14

20 times 10ndash14

363 times 10ndash14

15

times10

ndash4

20

times10

ndash4

25

times10

ndash4

30

times10

ndash4

35

times10

ndash400

00

(b)

Contact load of a single asperity (fN)

n = 40

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

4 times 10ndash17

3 times 10ndash17

2 times 10ndash17

1 times 10ndash17

0

20

times10

ndash8

40

times10

ndash8

60

times10

ndash8

80

times10

ndash8

10

times10

ndash700

(c)

Figure 8 e relationship between contact load and contact area of a single asperity during loading

Advances in Materials Science and Engineering 11

the deformation characteristics of the asperity underdifferent frequency exponents are obtained

(2) e normal contact load of a single asperity in theelastoplastic stage is related to the hardness of thematerial When the material yields the hardness H isnot a constant but a function related to the amountof deformation In this paper the limit mean geo-metric hardness is introduced to express the normalcontact load of a single asperity in the first andsecond elastoplastic deformation stages consideringthe change of hardness

(3) e relationship between the contact load andcontact area of a single asperity in the first andsecond elastoplastic deformation stages considering

the change of material hardness and not consideringthe change of hardness is compared respectivelyWhen the contact area of the asperity is the same thecontact load with the change of hardness is smallerthan that without the change of hardness and thedifference between them increases with the increaseof deformation

(4) e limit mean geometric hardness of a single as-perity is related to the contact area fractal di-mension and frequency exponent during theelastoplastic deformation stage and the limit meangeometric hardness increases with the increase of thecontact area When the frequency exponent isconstant the relationship between the limit meangeometric hardness and the contact area of the as-perity is related to the fractal dimension e largerthe fractal dimension is the more obvious the re-lationship curve between them is When the fractaldimension is constant the relationship between thelimit mean geometric hardness of the asperity andthe contact area is related to the frequency exponente smaller the frequency exponent is the moreobvious the change of the curve is

(5) e relationship between dimensionless total realcontact area and the dimensionless total contactloading of joint interface at each deformation stageis obtained by introducing the pushing coefficientand the critical pushing coefficient exists Whenthe pushing coefficient exceeds this value someasperities begin to deform in the next stage andthe inflection point appears on the relationshipcurve

(6) e current model still has some limitations on theapplicable scope of materials and further researchand improvement are needed to make it more ap-plicable in the future

09

08

07

06

05

04

03

02

01

000000 0002 0004 0006 0008 0010 0012 0014

Nondimensional total contact load (Flowastr)

Non

dim

ensio

nal t

otal

cont

act a

rea (

Alowastr)

nmin = 20 nmax = 36

Figure 10 e relationship between dimensionless total realcontact area and the dimensionless total contact loading for20le nle 36

10

09

08

07

06

05

04

03

02

01Non

dim

ensio

nal t

otal

cont

act a

rea (

Alowastr)

000000 0005 0010 0015 0020 0025 0030 0035

Nondimensional total contact load (Flowastr)

nmin = 44 nmax = 50

Figure 11 e relationship between dimensionless total realcontact area and the dimensionless total contact loading for44le nle 50

10

09

08

07

06

05

04

03

02

01

Nondimensional total contact load (Flowastr1)

Non

dim

ensio

nal t

otal

real

cont

act a

rea (

Alowastr1

)

0000 10 times 10ndash3 20 times 10ndash3 30 times 10ndash3 40 times 10ndash3 50 times 10ndash3

nmin = 20 nmax = 32

Figure 9e relationship between dimensionless total real contactarea and the dimensionless total contact loading for 20le nle 32

12 Advances in Materials Science and Engineering

Nomenclature

ωn Interference of the asperityωnec Elastic critical interference of the asperityωnepc First elastoplastic critical interference of the

asperityωnpc Second elastoplastic critical interference of the

asperityan Contact area of the asperityanec Elastic critical contact area of the asperityanepc First elastoplastic critical contact area of the

asperityanpc Second elastoplastic critical contact area of the

asperityfne Normal load in the elastic deformation of a single

asperityfnec Normal contact load for ω ωnecfnep1 Normal contact load of a single asperity in the

first elastoplastic stagefnep2 Normal contact load of a single asperity in the

second elastoplastic stagefnp Normal contact load of a single asperity in the full

plastic deformation stagefnep1prime Normal contact load of a single asperity in the

first elastoplastic stage considering the change ofhardness

fnep2prime Normal contact load of a single asperity in thesecond elastoplastic stage considering the changeof hardness

HG1(an) Limit mean geometric hardness in the firstelastoplastic deformation stage

HG2(an) Limit mean geometric hardness in the secondelastoplastic deformation stage

nec Elastic critical frequency exponentnepc First elastoplastic critical frequency exponentnpc Second elastoplastic critical frequency exponentAr Actual contact area of the joint interfaceAr1 Actual contact area of the joint interface for

nmin lt nle necAr2 Actual contact area of the joint interface for

nec lt nle nepcAr3 Actual contact area of the joint interface for

nepc lt nle npcAr4 Actual contact area of the joint interface for

npc lt n

Fr1 Actual contact load of the joint interface fornmin lt nle nec

Fr2 Actual contact load of the joint interface fornec lt nle nepc

Fr3 Actual contact load of the joint interface fornepc lt nle npc

Fr4 Actual contact load of the joint interface fornpc lt n

pea(a) Average contact pressure of the asperity in elasticstage

pepa1(a) Average contact pressure of the asperity in thefirst elastoplastic deformation stage

pepa2(a) Average contact pressure of the asperity in thesecond elastoplastic stage

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (grant no 51275328) and the NaturalScience Foundation Project of Shanxi Province (grant no201601D011062)

References

[1] B Zhao S Zhang P Wang and Y Hai ldquoLoading-unloadingnormal stiffness model for power-law hardening surfacesconsidering actual surface topographyrdquo Tribology In-ternational vol 90 pp 332ndash342 2015

[2] J Greenwood and J Williamson ldquoContact of nominally flatsurfacesrdquo Mathematical Physical and Engineering Sciencesvol 295 no 1442 pp 299ndash319 1966

[3] J A Greenwood and J H Tripp ldquoe contact of twonominally flat rough surfacesrdquo Proceedings of the Institution ofMechanical Engineers vol 185 no 1 pp 625ndash633 1970

[4] W R Chang I Etsion and D B Bogy ldquoAn elastic-plasticmodel for the contact of rough surfacesrdquo Journal of Tribologyvol 109 no 2 pp 257ndash263 1987

[5] Y Zhao D M Maietta and L Chang ldquoAn asperity micro-contact model incorporating the transition from elastic de-formation to fully plastic flowrdquo Journal of Tribology vol 122no 1 pp 86ndash93 2000

[6] L Kogut and I Etsion ldquoElastic-plastic contact analysis ofasphere and a rigid flatrdquo Journal of Applied Mechanics vol 69no 5 pp 657ndash662 2002

[7] I Etsion Y Kligerman and Y Kadin ldquoUnloading of anelastic-plastic loaded spherical contactrdquo International Journalof Solids and Structures vol 42 no 13 pp 3716ndash3729 2005

[8] Y Kadin Y Kligerman and I Etsion ldquoUnloading an elastic-plastic contact of rough surfacesrdquo Journal of the Mechanicsand Physics of Solids vol 54 no 12 pp 2652ndash2674 2006

[9] Y Kadin Y Kligerman and I Etsion ldquoMultiple loading-unloading of an elastic-plastic spherical contactrdquo In-ternational Journal of Solids and Structures vol 43 no 22-23pp 7119ndash7127 2006

[10] A Majumdar and B Bhushan ldquoRole of fractal geometry inroughness characterization and contact mechanics of sur-facesrdquo Journal of Tribology vol 112 no 2 pp 205ndash216 1990

[11] A Majumdar and C L Tien ldquoFractal characterization andsimulation of rough surfacesrdquo Wear vol 136 no 2pp 313ndash327 1990

[12] AMajumdar and B Bhushan ldquoFractal model of elastic-plasticcontact between rough surfacesrdquo Journal of Tribology vol 113no 1 pp 1ndash11 1991

[13] S Wang and K Komvopoulos ldquoA fractal theory of the in-terfacial temperature distribution in the slow sliding regimePart I-elastic contact and heat transfer analysisrdquo Journal ofTribology vol 116 no 4 pp 812ndash822 1994

[14] S Wang and K Komvopoulos ldquoA fractal theory of the in-terfacial temperature distribution in the slow sliding regime

Advances in Materials Science and Engineering 13

Part II-multiple domains elastoplastic contacts and appli-cationsrdquo Journal of Tribology vol 116 no 4 pp 824ndash8321994

[15] Y Morag and I Etsion ldquoResolving the contradiction of as-perities plastic to elastic mode transition in current contactmodels of fractal rough surfacesrdquo Wear vol 262 no 5-6pp 624ndash629 2007

[16] H Tian X Zhong and C Zhao ldquoOne loading model of jointinterface considering elastoplastic and variation of hardnesswith surface depthrdquo Journal of Mechanical Engineeringvol 51 no 5 pp 90ndash104 2015

[17] Y Yuan Y Cheng K Liu et al ldquoA revised Majumdar andBushan model of elastoplastic contact between rough sur-facesrdquo Applied Surface Science vol 425 pp 1138ndash1157 2017

[18] R S Sayles and T R omas ldquoSurface topography as anonstationary random processrdquo Nature vol 271 no 5644pp 431ndash434 1978

14 Advances in Materials Science and Engineering

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Submit your manuscripts atwwwhindawicom

ρnmin(x) Rn cminusn D

π2GDminus1(4)

e height before deformation of the asperity is

hn zn(0) GDminus1

c(2minusD)n (5)

In the loading process of asperity the deformation willincrease with the increase of normal contact load Accordinglyasperity will change from elastic deformation to elastoplasticdeformation and then to fully plastic deformation

e elastic critical interference of asperity at initial yieldis [16]

ωnec 3πKH

4Eprime1113874 1113875

2R (6)

where K 0454 + 041υ υ is the Poisson ratio of the softermaterial H is the hardness of the softer material H 28Yand Eprime is the equivalent elastic modulus (1Eprime)

(1minus υ1primeE1) + (1minus υ2primeE2) E1 and E2 are respectively elasticmodulus of two objects in contact with each other υ1prime and υ2primeare respectively the corresponding Poisson ratios

When ωn ltωnec an lt anec the asperity is in a state ofelastic deformation According to Hertz theory the contactarea of the asperity is

an πRωn (7)

Substituting equation (6) in equation (7) the criticalcontact area of the elastic asperity is

anec πRωnec π3πKH

4Eprime1113874 1113875

2R2

1π

3KHcminusn D

4G(Dminus1)Eprime1113888 1113889

2

(8)

According to the Hertz contact theory the normal loadon a single asperity is

fne 43EprimeR12

n ω32n (9)

Substituting equations (4) and (7) in equation (9) we canobtain

fne 4Eprimea32

n π12GDminus1

3cminusDn (10)

According to equations (6) (8) and (9) we can get thecritical contact load of the elastic asperity

fnec K middot H middot anec (11)

22 Elastic-Plastic Deformation of a Single AsperityLiterature [6] through the finite element analysis of a singleasperity it is concluded that when the asperity actual de-formation is greater than the elastic critical interference(ωn gtωnec) the yield phenomenon begins to appear and theelastoplastic deformation of the asperity occurs Accordingto the results of [6] the elastoplastic deformation of asperitycan be divided into two different stages according to theratio ωnωnec namely the first elastoplastic deformationstage when 1ltωnωnec le 6 and the second elastoplasticdeformation stage when 6ltωnωnec le 110 Define ωnepc

6ωnec as the first elastoplastic critical interference where theactual contact area is anepc e actual deformation of as-perity ωnpc 110ωnec is defined as the second elastoplasticcritical interference and the actual contact area is anpc erelationship between contact area-deformation and contactload-deformation in the elastoplastic deformation stage ofasperity is [6]

an

anec 093

ωn

ωnec1113888 1113889

1136

fnep1

fnec 103

ωn

ωnec1113888 1113889

1425

ωnec ltωn le 6ωnec( 1113857

an

anec 094

ωn

ωnec1113888 1113889

1146

fnep2

fnec 140

ωn

ωnec1113888 1113889

1263

6ωnec ltωn le 110ωnec( 1113857

(12)

From the above equations we can get

X

ln

h n

w n

O

Z

Figure 1 Diagram of single asperity loading

Advances in Materials Science and Engineering 3

anepc 71197anec

fnep1 KH times 11282aminus02544nec a

12544n anec lt an lt anepc1113872 1113873

(13)

anpc 2053827anec

fnep2 KH times 14988aminus01021nec a

11021n anepc lt an lt anpc1113872 1113873

(14)

where fnec is contact load for ω ωnec and fnep1 and fnep2are contact loads in the first elastoplastic stage and thesecond elastoplastic stage respectively Both fnep1 and fnep2obtained above are related to the hardness (H) of thematerial However according to the plastic strengtheningprinciple the hardness is not a constant when the materialyields but a function related to the deformation that is itchanges with the deformation erefore it is not accurateto describe elastoplastic deformation by the above formulaIn order to express the characteristics of elastoplastic de-formation more accurately the concept of limit meangeometric hardness is introduced

According to equations (13) and (14) HG(a) is fittedinto the following segmented relations

e first elastoplastic deformation stage is

HG1 an( 1113857 c1Yan

anec1113888 1113889

c2

anec lt an le anepc1113872 1113873 (15)

e second elastoplastic deformation stage is

HG2 an( 1113857 c3Yan

anec1113888 1113889

c4

anepc lt an le anpc1113872 1113873 (16)

where c1 c2 c3 and c4 are the coefficients to be solved

(1) Equation (15) should satisfy two limiting conditions

HG1 anec( 1113857 pea anec( 1113857 (17)

HG1 anepc1113872 1113873 pepa1 anepc1113872 1113873 (18)

where pea(a) is the average contact pressure of theasperity in elastic stage which is given bypea(a) (fnea) pepa1(a) is the average contactpressure of the asperity in the first elastoplastic de-formation stage and is given by pepa1(a) (fnep1a)Substituting equations (11) and (15) in equation (17)we can obtain

c1Y KH (19)

c1 28K (20)

Substituting equations (13) and (15) in equation (18)we can obtain

KH times 11282aminus02544nec times 71197anec( 1113857

02544 28KY times 71197c2

(21)

Derived from equation (21)

c2 ln 11282 times 7119702544( 1113857

ln 71197 (22)

Considering the change of hardness the normal con-tact load of a single asperity in the first elastoplasticstage is

fnep1prime HG1(a) middot an (23)

Substituting equations (15) (20) and (22) in equation(23) new equations are yielded

fnep1prime 28KYaminusc2neca

c2+1n (24)

(2) Equation (16) should satisfy two limiting conditions

HG2 anepc1113872 1113873 pepa1 anepc1113872 1113873 (25)

HG2 anpc1113872 1113873 pepa2 anpc1113872 1113873 (26)

where pepa2(a) (fnep2a) is the average contactpressure of the asperity in the second elastoplastic stageSubstituting equations (13) and (16) in equation (25)we can obtain

c3(71197)c4 K times 28 times 11282 times(71197)

02544 (27)

Substituting equations (14) and (16) in equation (26)we can obtain

c3(2053827)c4 K times 28 times 14988 times(2053827)

01021

(28)

Simultaneous equations (27) and (28) obtained

c4 ln 11282 + 02544 ln 71197minus ln 14988minus 01021 ln 2053827

ln 71197minus ln 2053827

(29)

Substituting equation (29) in equation (27) we canobtain

c3 K times 315896 times(71197)02544minusc4 (30)

Considering the change of hardness the normal con-tact load of a single asperity in the second elastoplasticstage is

fnep2prime HG2(a) middot an K times 315896 times(71197)02544minusc4Y

middot aminusc4nec middot a

c4+1n

(31)

23 Full Plastic Deformation of a Single Asperity As thedeformation continues to increase when ωn gt 110ωnec thecontact area an gt anpc and the asperity enters the stage of fullplastic deformation At this stage the hardness of the ma-terial is no longer affected by the deformation and can beregarded as a constant When the hardness of the material is

4 Advances in Materials Science and Engineering

given according to literature [7] the contact load andcontact area of the asperity at this stage can be expressed as

fnp Han

an 2πRnωn(32)

In conclusion with the increase of load and deformationthe contact area of the same asperity increases graduallyie anec lt anepc lt anpc With the increase of the load andcontact area the asperity underwent elastic deformationfirst elastoplastic deformation second elastoplastic de-formation and full plastic deformation successively Underconstant load and deformation the actual contact area of theasperity is related to the radius of curvature at the vertex ofthe asperity

24 Asperityrsquos Frequency Exponent n When using W-Mfunction to describe the surface profile of an asperity theprofile function is related to the asperityrsquos frequency ex-ponent In other words the radius of curvature at the vertexof the asperity and the height of the asperity vary with thefrequency exponent when the load is constant According tothe equations (5)ndash(7) it was found that the value of hn Rnand ωnec correlated with the frequency exponent When thefrequency exponent is constant the deformation of theasperity is not greater than the height of the asperityunder the action of the load In order to obtain the criticalvalue of the frequency exponent we take hn ωnec ie(GDminus1c(2minusD)nec) (3KH4Eprime)2 middot (cminusnecDGDminus1)

e elastic critical frequency exponent can be obtainedas follows

nec intln 3KH4Eprime( 1113857

2middot G2(1minusD)1113960 1113961

2(Dminus 1)ln c

⎧⎨

⎩

⎫⎬

⎭ (33)

where int is the integer part of the value in the parenthesisSimilarly the first elastoplastic critical frequency expo-

nent can be obtained

nepc intln 6 3KH4Eprime( 1113857

2middot G2(1minusD)1113960 1113961

2(Dminus 1)ln c

⎧⎨

⎩

⎫⎬

⎭ (34)

e second elastoplastic critical frequency exponent canbe obtained

npc intln 110 3KH4Eprime( 1113857

2middot G2(1minusD)1113960 1113961

2(Dminus 1)ln c

⎧⎨

⎩

⎫⎬

⎭ (35)

From the above when the asperity frequency exponent isnmin lt nle nec elastic deformation only takes place in theseasperities under contact load When nec lt nle nepc elasticdeformation or the first elastoplastic deformation can takeplace in these asperities When nepc lt nle npc elastic de-formation the first elastoplastic deformation or the secondelastoplastic deformation can take place in these asperities and

full plastic deformation never occur When npc lt nle nmaxelastic deformation elastoplastic deformation or full plasticdeformation can take place in these asperities

3 Actual Contact Area and Normal ContactLoad of Joint Interface

According to reference [10] when the asperity frequencyexponent is n the area distribution density function of theasperity on the joint interface is defined as

nn(a) 12

D middotaD2nl

a(D+2)2 0lt ale anl 1ltDlt 2( 1113857 (36)

where anl represents the largest contact area when theasperityrsquos frequency exponent is n

In order to simplify equation (36) we define the areadistribution function of the asperity of any frequency ex-ponent as nn(a) Mn(a) According to reference [17] theactual contact area of joint interface is

Ar 1113944

nmax

nnmin

1113946anl

0nn(a)a da M 1113944

nmax

nnmin

1113946anl

0n(a)a da (37)

where M (al1113936nmaxnnmin

anl)(nmin le nle nmax al max anl1113864 1113865)

31 When the Frequency Exponent Belongs to nmin lt nle necWhen the frequency exponent belongs to nmin lt nle nec evenif these asperities are completely deformed only elasticdeformation will occur and anl lt anec In this case the actualcontact area of the joint interface is defined as Ar1

Ar1 1113944

nec

nnmin

1113946anl

0Mn(a)ada

MD

2minusD1113944

nec

nnmin

anl (38)

In this case the contact load of the joint interface is asfollows

Fr1 1113944

nec

nnmin

1113946anl

0fneMn(a)da (39)

Substituting equation (11) in equation (39) we canobtain

Fr1 MD

3minusD1113944

nec

nnmin

4Eπ12G(Dminus1)

3cminusDna32nl (40)

32 When the Frequency Exponent Belongs to nec lt nle nepcWhen the frequency exponent belongs to nec lt nle nepc forthe case anec lt anl le anepc elastic deformation or the firstelastoplastic deformation may take place in these asperitiesAt this point the actual contact area of the joint interfaceconsists of two parts the elastic deformation stage and thefirst elastoplastic deformation stage

Advances in Materials Science and Engineering 5

Ar2 Are + Arep1 (41)

Are 1113944

nepc

nnec+11113946

anec

0Mn(a)ada

MD

2minusD1113944

nepc

nnec+1a

(2minusD)2nec a

D2nl

(42)

For the determined frequency exponent the maximumactual contact area of the asperity appears at the maximumdeformation amount ωn where the maximum value of theelastic deformation phase ωn appears at ωnec whereuponformula (42) is simplified to

Are MD

2minusD1113944

nepc

nnec+1anec

MD

(2minusD)π1113944

nepc

nnec+1

3KHcminusDn

4G(Dminus 1)Eprime1113888 1113889

2

Arep1 1113944

nepc

nnec+11113946

anl

anec

Mn(a)ada MD

2minusD

middot 1113944

nepc

nnec+1a

(2minusD)2nl minus a

(2minusD)2nec1113960 1113961a

D2nl

(43)

e contact load is given by

Fr2 Fre + Frep1 (44)

Fre 1113944

nepc

nnec+11113946

anec

0fneMn(a)da

9MD(KH)3

16(3minusD) EprimeπGDminus1( 11138572

middot 1113944

nepc

nnec+1cminus2 Dn

(45)

Frep1 1113944

nepc

nnec+11113946

anl

anec

fnep1prime Mn(a)da (46)

Substituting equations (24) and (36) in equation (46) wecan obtain

Frep1 28KYMD

2c2 minusD + 21113944

nepc

nnec+1aminusc2neca

c2+1nl minus a

(2minusD)2nec a

D2nl1113960 1113961 (47)

33 When the Frequency Exponent Belongs to nepc lt nle npcWhen the frequency exponent belongs to nepc lt nle npc forthe case anepc lt anl le anpc elastic deformation the firstelastoplastic deformation or the second elastoplastic de-formation may take place in these asperities At this point

the actual contact area of the joint interface consists ofthree parts the elastic deformation stage the first elas-toplastic deformation stage and the second elastoplasticdeformation stage

Ar3 Are + Arep1 + Arep2

Are 1113944

npc

nnepc+11113946

anec

0Mn(a)ada

MD

(2minusD)π

middot 1113944

npc

nnepc+1

3KHcminusDn

4GDminus1Eprime1113888 1113889

2

Arep1 1113944

npc

nnepc+11113946

anepc

anec

Mn(a)ada

MD

π(2minusD)71197minus 71197D2

1113872 1113873 1113944

npc

nnepc+1

3KHcminusDn

4GDminus1Eprime1113888 1113889

2

Arep2 1113944

npc

nnepc+11113946

anl

anepc

Mn(a)ada

MD

(2minusD)1113944

npc

nnepc+1a

(2minusD)2nl minus 71197anec( 1113857

(2minusD)21113960 1113961a

D2nl

(48)

In this case the contact load of the joint interface is asfollows

Fr3 Fre + Frep1 + Frep2

Fre 1113944

npc

nnepc+11113946

anec

0fneMn(a)da

9MD(KH)3

16(3minusD) EprimeπGDminus1( 11138572 1113944

npc

nnepc+1cminus2 Dn

Frep1 28KYMD

2c2 minusD + 271197c2+1 minus 71197D2

1113872 1113873

middot 1113944

npc

nnepc+1

1π

3KHcminusDn

4GDminus 1Eprime1113888 1113889

2

(49)

When the second elastoplastic deformation occurs thenormal contact load of the joint interface is as follows

6 Advances in Materials Science and Engineering

Frep2 1113944

npc

nnepc+11113946

anl

anepc

fnep2prime Mn(a)da (50)

Substituting equations (31) and (38) in equation (46) wecan obtain

Frep2 2106 times 7119702544minusc4KYMD

2c4 minusD + 2

middot 1113944

npc

nnepc+1aminusc4neca

c4+1nl minus 71197c4minus(D2)+1

a1minus(D2)nec a

(D2)nl1113872 1113873

(51)

34 When the Frequency Exponent Belongs to npc lt nWhen the frequency exponent belongs to npc lt n elasticdeformation elastoplastic deformation or full plastic de-formation may take place in these asperities e actualcontact area of the joint interface can be evaluated as

Ar4 Are + Arep1 + Arep2 + Arp

Are 1113944

nmax

nnpc+11113946

anec

0Mn(a)ada

MD

2minusD1113944

nmax

nnpc+1anec

Arep1 1113944

nmax

nnpc+11113946

anepc

anec

Mn(a)ada

MD

2minusD71197minus 71197D2

1113872 1113873 1113944

nmax

nnpc+1anec

Arep2 1113944

nmax

nnpc+11113946

anpc

anepc

Mn(a)ada

MD

2minusD2053827minus 711971minus(D2)

middot 2053827D21113872 1113873 1113944

nmax

nnpc+1anec

Arp 1113944

nmax

nnpc+11113946

anl

anpc

Mn(a)ada

MD

2minusD1113944

nmax

nnpc+1a

(2minusD)2nl minus 2053827anec( 1113857

(2minusD)21113960 1113961a

D2nl

(52)

In this case the contact load of the joint interface is asfollows

Fr4 Fre + Frep1 + Frep2 + Frp

Fre 1113944

nmax

nnpc+11113946

anec

0fneMn(a)da

MDKH

(3minusD)π1113944

nmax

nnpc+1anec

Frep1 1113944

nmax

nnpc+11113946

anepc

anec

fnep1prime Mn(a)da

28KYMD

2c2 minusD + 271197c2+1 minus 71197D2

1113872 1113873 1113944

nmax

nnpc+1anec

Frep2 2106 times 7119702544minusc4KYMD

2c4 minusD + 2

middot 1113944

nmax

nnpc+1aminusc4neca

c4+1nl minus 71197c4minus(D2)+1

a1minus(D2)nec a

D2nl1113872 1113873

2106 times 7119702544minusc4KYMD

2c4 minusD + 2

middot 2053827c4+1 minus 2053827D2middot 71197c4minus(D2)+1

1113872 1113873

middot 1113944

nmax

nnpc+1anec

Frp 1113944

nmax

nnpc+11113946

anl

anpc

fnpMn(a)da

MHD

2minusD1113944

nmax

nnpc+1a

D2nl a

1minus(D2)nl minus 2053827anec( 1113857

1minus(D2)1113960 1113961

(53)

For all frequency exponents the total actual contact areaof the joint interface is

Ar Ar1 + Ar2 + Ar3 + Ar4 (54)

e total contact load of the joint interface is

Fr Fr1 + Fr2 + Fr3 + Fr4 (55)

e total real contact area and the total contact load in anondimensional form can be written as follows

Alowastr

Ar

Aa

Flowastr

Fr

AaE

(56)

where Aa is the nominal contact area and is given byAa L2L 1cnmin

Advances in Materials Science and Engineering 7

4 Results Analysis

In order to further analyze the above calculation results theparameters of equivalent joint interface are taken as shownin Table 1 [16]

Figure 2 shows the relation between all critical contactareas and frequency exponents of single asperity whenD 15 It can be seen from the figure that as for one definiteasperity when frequency exponent n is certain elastic criticalcontact area is minimum followed by the first elastoplasticcritical contact area and the second elastoplastic criticalcontact area is maximum With gradual increase of contactload the contact area increases e single asperity is firstlysubject to elastic deformation followed by the first elasto-plastic deformation the second elastoplastic deformation andfully plastic deformation successively which is consistent withtypical contact mechanics theory As for different asperitieswith increase of frequency exponent all critical contact areasdecrease correspondingly which shows that elastic criticalcontact area the first elastoplastic critical contact area and thesecond elastoplastic critical contact area are all related tofrequency exponent n

Figure 3 shows the relation curve between fractal di-mension D and critical frequency exponent n of asperityWhen fractal dimension is definite elastic critical frequencyexponent nec the first elastoplastic critical frequency ex-ponent nepc and the second elastoplastic critical frequencyexponent npc increase gradually As shown in Figure 3 whenDlt 106 nec nepc and npc are all negative As for asperitieswith minimum value of frequency exponent being 0 elasticdeformation elastoplastic deformation and fully plasticdeformation will all occur When D 113 nec and nepc arenegative and npc is positive At this time as for asperitieswith minimum value of frequency exponent being 0 elasticdeformation elastoplastic deformation will occur exceptfully plastic deformation

For D 15 G 25 times 10minus9 m H 55 times 109 Nm2 wecan obtain the elastic critical frequency exponent nec 32the first elastoplastic critical frequency exponent nepc 36and the second elastoplastic critical frequency exponentnpc 43 ese asperities whose frequency exponents rangefrom 20 to 32 are only under elastic deformation Elasticdeformation and the first elastoplastic deformation canoccur in these asperities whose frequency exponents rangefrom 33 to 36 Elastic deformation the first elastoplasticdeformation and the second elastoplastic deformation canoccur in these asperities whose frequency exponents rangefrom 37 to 43 When frequency exponents range from 43 to50 all deformations types can occur in these asperities

Figure 4 shows relation comparison diagram betweencontact load and contact area of single asperity with andwithout hardness change at the first elastoplastic stage ecomparison diagram is simulation result when n 33 It canbe seen from the figure that with gradual increase of contactarea with contact area of single asperity over 32 times 10minus13 m2contact load of the same asperity with hardness change willbe less than that without hardness change In addition as theamount of deformation increases the difference betweenthem tends to increase

Figure 5 shows relation comparison diagram betweencontact load and contact area of single asperity with andwithout hardness change at the second elastoplastic stagee comparison diagram is simulation result when n 37 Itcan be seen from the figure that when the deformation is

Table 1 e parameters of equivalent joint interface

Parameters ValuesEquivalent elastic modulus Eprime 72 times 1010 Nm2

Poissonrsquos ratio υ 017Initial hardness H 55 times 109 Nm2

Profile scale parameter G 25 times 10minus9 mFractal dimension D 1ltDlt 2Frequency exponent n 20sim50

20 25 30 35 40 45 5010ndash1510ndash1410ndash1310ndash1210ndash1110ndash1010ndash910ndash810ndash710ndash610ndash510ndash410ndash3

Criti

cal c

onta

ct ar

eas

of a

singl

e asp

erity

(am

2 )

Asperity levels (n)

Second elastoplastic critical contact areaFirst elastoplastic critical contact areaElastic critical contact area

Figure 2 e relationship between critical contact area and fre-quency exponent of a single asperity

10 11 12 13 14 15 16 17 18 19 20ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

0

20

40

Asp

erity

freq

uenc

y ex

pone

nts (n)

e fractal dimension (D)

Elastic critical frequency exponentsFirst elastoplastic critical frequency exponentsSecond elastoplastic critical frequency exponents

Figure 3 e relationship between fractal dimension D andcritical frequency exponent n of a single asperity

8 Advances in Materials Science and Engineering

definite contact load of the same asperity with hardnesschange will be less than that without hardness change Inaddition with increase of deformation amount the differ-ence between them tends to increase which is consistentwith the change trend at the first elastoplastic stage

Figure 6 shows the relation between limit mean geo-metric hardness and contact area (logarithm) of single as-perity at the first elastoplastic deformation stage Figure 6(a)shows relation curve that D 11 13 15 17 when n 34Figure 6(b) shows relation curve that n 32 33 34 35 whenD 15 It can be seen from Figure 6 that the limit meangeometric hardness of single asperity is related to contactarea fractal dimension and frequency exponent in the firstelastoplastic deformation stage e limit mean geometrichardness increases with increase of contact area When n is

definite the relation between limit mean geometric hardnessand contact area of asperity is related to fractal dimension De larger the D is the more obvious the relation curvebetween them changes when D is definite the relationbetween limit mean geometric hardness and contact area ofasperity is related to frequency exponent n e smaller n isthe more obvious the relation curve between them changes

Figure 7 shows the relation between limit mean geo-metric hardness and contact area (logarithm) of single as-perity at the second elastoplastic deformation stageFigure 7(a) shows the relation curve n 40 that D 11 1315 17 when n 40 Figure 7(b) shows the relation curvethat n 36 38 40 42 when D 15

During loading the contact area increases with the in-crease of deformation of a single asperity e ratio of de-formation to the natural height of the asperity is defined asthe pushing coefficient namely the pushing coefficientk ωnhn 0le kle 09 When fractal dimension is 15 we willresearch the relation between contact load and contact areaof single asperity with frequency exponent n being 30 35and 40 respectively during loading

When n 30 the asperity will only be subject to elasticdeformation During loading even the pushing coefficient k

is maximum no plastic deformation will occur e relationbetween contact area and contact load is fsima15 approxi-mately as shown in Figure 8(a)

As is shown in Figure 8(b) when n 35 elastic de-formation and the first elastoplastic deformation may takeplace in the asperity during loading When the pushingcoefficient k is less than 0247 the asperity will under elasticdeformation At this time the relation between contact areaand contact load is fsima15 approximately when the pushingcoefficient is over 0247 the first elastoplastic deformationoccurs At this time the relation between contact area andcontact load is fsima11093 approximately As is shown inFigure 8(c) when n 40 elastic deformation the firstelastoplastic deformation and the second elastoplastic

0000 0001 0002 0003 0004 0005Contact load of a single asperity in the

first elastoplastic deformation regime (fN)

00

12 times 10ndash12

10 times 10ndash12

80 times 10ndash13

60 times 10ndash13

40 times 10ndash13

20 times 10ndash13

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

Change of hardness is consideredChange of hardness is not considered

Figure 4 e relationship between contact load and contact area of single asperity in the first elastoplastic deformation stage

0000 0001 0002 0003 0004 00050

2 times 10ndash13

4 times 10ndash13

6 times 10ndash13

8 times 10ndash13

1 times 10ndash12

1 times 10ndash12

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

Contact load of a single asperity in the second elastoplastic deformation regime (f2N)

Change of hardness is not consideredChange of hardness is considered

Figure 5e relationship between contact load and contact area ofsingle asperity in the second elastoplastic deformation stage

Advances in Materials Science and Engineering 9

deformation may take place in the asperity during loadingWhen the pushing coefficient is greater than 01954 theasperity begins to enter the second elastoplastic de-formation the relation between contact area and contactload is fsima10977 approximately When n 45 and thepushing coefficient is greater than 0472 the asperity beginsto enter fully plastic deformation the relation betweencontact area and contact load is fsima approximately

Figure 9 shows that when the minimum frequency ex-ponent is 20 and the maximum value is 32 the actual contactarea of the joint interface increases with the increase of thetotal contact load and the relation between them isFlowastr simAlowast15

r approximately During the whole deformationprocess the joint interface appears to be of elastic property

Figure 10 shows the relation between dimensionless totalreal contact area and the dimensionless total contact loading

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash9

10ndash15

10ndash14

10ndash13

10ndash12

10ndash11

10ndash10

D = 11 n = 34D = 13 n = 34

D = 15 n = 34D = 17 n = 34

25 times 109 30 times 109 35 times 109 40 times 109 45 times 109 50 times 109 55 times 109

Limit mean geometric hardness in the firstelastoplastic deformation regime HG1(Nm2)

(a)C

onta

ct ar

eas o

f a si

ngle

aspe

rity

(am

2 )

10ndash10

10ndash15

10ndash14

10ndash13

10ndash12

10ndash11

n = 32 D = 15n = 33 D = 15

n = 34 D = 15n = 35 D = 15

25 times 109 30 times 109 35 times 109 40 times 109 45 times 109 50 times 109 55 times 109

Limit mean geometric hardness in the firstelastoplastic deformation regime HG1(Nm2)

(b)

Figure 6 e relationship between limit mean geometric hardness and contact for single asperity during the first elastoplastic deformationstage (a) n 34 11leDle 17 (b) D 15 32le nle 35

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash12

10ndash18

10ndash17

10ndash16

10ndash15

10ndash14

10ndash13

D = 11 n = 40D = 13 n = 40

D = 15 n = 40D = 17 n = 40

50 times 109 55 times 109 60 times 109 65 times 109 70 times 109 75 times 109

Limit mean geometric hardness in the secondelastoplastic deformation regime HG2(Nm2)

(a)

50 times 109 55 times 109 60 times 109 65 times 109 70 times 109 75 times 109

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash10

10ndash11

10ndash12

10ndash13

10ndash18

10ndash17

10ndash16

10ndash15

10ndash14

n = 36 D = 15n = 38 D = 15

n = 40 D = 15n = 42 D = 15

Limit mean geometric hardness in the secondelastoplastic deformation regime HG2(Nm2)

(b)

Figure 7 e relationship between limit mean geometric hardness and contact for single asperity during the second elastoplastic de-formation stage (a) n 40 11leDle 17 (b) D 15 36le nle 42

10 Advances in Materials Science and Engineering

of the joint interface when minimum value and maximumvalue of frequency exponent of asperities are respectively 20and 36When the pushing coefficient k is 01648 the asperitybegins to be subject to the first elastoplastic deformation andthe relation between dimensionless total real contact areaand the dimensionless total contact loading is FlowastrsimAlowast15

r

approximately e joint interface appears to be of elasticproperty As the load continues increasing when thepushing coefficient k is 05564 at this pointFlowastr gt 69023 times 10minus3 the relation between above load and areais approximately FlowastrsimAlowast11093

r e joint interface presentselastoplastic properties At this time the first elastoplasticdeformation takes place in these asperities whose frequencyexponents range 33sim36

Figure 11 shows the relation between dimensionless totalreal contact area and the dimensionless total contact loadingof the joint interface when minimum value and maximumvalue of frequency exponent of asperities are respectively 44and 50When the pushing coefficient k is 00621 the asperity

begins to be subject to the second elastoplastic deformationat this point 00016ltFlowastr lt 0023 and the relation betweendimensionless total real contact area and the dimensionlesstotal contact loading is FlowastrsimAlowast10977

r approximately Whenthe pushing coefficient k is 07076 the asperity begins to besubject to the fully plastic deformation at this pointFlowastr gt 0023 and the relation between dimensionless total realcontact area and the dimensionless total contact loading isapproximately FlowastrsimAlowastr

5 Conclusions

(1) e contact mechanical properties of a single as-perity on a joint interface are related to the frequencyexponent of the asperity while the frequency ex-ponent of an asperity is related to the fractal di-mension and the profile scale parameter In thispaper the critical frequency exponents of each de-formation stage of a single asperity are obtained and

000 005 010 015 020 025 030 03500

10 times 10ndash11

20 times 10ndash11

30 times 10ndash11

40 times 10ndash11

50 times 10ndash11

60 times 10ndash11

Contact load of a single asperity (fN)

n = 30

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

(a)

Contact load of a single asperity (fN)

n = 35

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

50

times10

ndash5

10

times10

ndash4

104 times 10ndash4

10 times 10ndash13

80 times 10ndash14

60 times 10ndash14

40 times 10ndash14

20 times 10ndash14

363 times 10ndash14

15

times10

ndash4

20

times10

ndash4

25

times10

ndash4

30

times10

ndash4

35

times10

ndash400

00

(b)

Contact load of a single asperity (fN)

n = 40

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

4 times 10ndash17

3 times 10ndash17

2 times 10ndash17

1 times 10ndash17

0

20

times10

ndash8

40

times10

ndash8

60

times10

ndash8

80

times10

ndash8

10

times10

ndash700

(c)

Figure 8 e relationship between contact load and contact area of a single asperity during loading

Advances in Materials Science and Engineering 11

the deformation characteristics of the asperity underdifferent frequency exponents are obtained

(2) e normal contact load of a single asperity in theelastoplastic stage is related to the hardness of thematerial When the material yields the hardness H isnot a constant but a function related to the amountof deformation In this paper the limit mean geo-metric hardness is introduced to express the normalcontact load of a single asperity in the first andsecond elastoplastic deformation stages consideringthe change of hardness

(3) e relationship between the contact load andcontact area of a single asperity in the first andsecond elastoplastic deformation stages considering

the change of material hardness and not consideringthe change of hardness is compared respectivelyWhen the contact area of the asperity is the same thecontact load with the change of hardness is smallerthan that without the change of hardness and thedifference between them increases with the increaseof deformation

(4) e limit mean geometric hardness of a single as-perity is related to the contact area fractal di-mension and frequency exponent during theelastoplastic deformation stage and the limit meangeometric hardness increases with the increase of thecontact area When the frequency exponent isconstant the relationship between the limit meangeometric hardness and the contact area of the as-perity is related to the fractal dimension e largerthe fractal dimension is the more obvious the re-lationship curve between them is When the fractaldimension is constant the relationship between thelimit mean geometric hardness of the asperity andthe contact area is related to the frequency exponente smaller the frequency exponent is the moreobvious the change of the curve is

(5) e relationship between dimensionless total realcontact area and the dimensionless total contactloading of joint interface at each deformation stageis obtained by introducing the pushing coefficientand the critical pushing coefficient exists Whenthe pushing coefficient exceeds this value someasperities begin to deform in the next stage andthe inflection point appears on the relationshipcurve

(6) e current model still has some limitations on theapplicable scope of materials and further researchand improvement are needed to make it more ap-plicable in the future

09

08

07

06

05

04

03

02

01

000000 0002 0004 0006 0008 0010 0012 0014

Nondimensional total contact load (Flowastr)

Non

dim

ensio

nal t

otal

cont

act a

rea (

Alowastr)

nmin = 20 nmax = 36

Figure 10 e relationship between dimensionless total realcontact area and the dimensionless total contact loading for20le nle 36

10

09

08

07

06

05

04

03

02

01Non

dim

ensio

nal t

otal

cont

act a

rea (

Alowastr)

000000 0005 0010 0015 0020 0025 0030 0035

Nondimensional total contact load (Flowastr)

nmin = 44 nmax = 50

Figure 11 e relationship between dimensionless total realcontact area and the dimensionless total contact loading for44le nle 50

10

09

08

07

06

05

04

03

02

01

Nondimensional total contact load (Flowastr1)

Non

dim

ensio

nal t

otal

real

cont

act a

rea (

Alowastr1

)

0000 10 times 10ndash3 20 times 10ndash3 30 times 10ndash3 40 times 10ndash3 50 times 10ndash3

nmin = 20 nmax = 32

Figure 9e relationship between dimensionless total real contactarea and the dimensionless total contact loading for 20le nle 32

12 Advances in Materials Science and Engineering

Nomenclature

ωn Interference of the asperityωnec Elastic critical interference of the asperityωnepc First elastoplastic critical interference of the

asperityωnpc Second elastoplastic critical interference of the

asperityan Contact area of the asperityanec Elastic critical contact area of the asperityanepc First elastoplastic critical contact area of the

asperityanpc Second elastoplastic critical contact area of the

asperityfne Normal load in the elastic deformation of a single

asperityfnec Normal contact load for ω ωnecfnep1 Normal contact load of a single asperity in the

first elastoplastic stagefnep2 Normal contact load of a single asperity in the

second elastoplastic stagefnp Normal contact load of a single asperity in the full

plastic deformation stagefnep1prime Normal contact load of a single asperity in the

first elastoplastic stage considering the change ofhardness

fnep2prime Normal contact load of a single asperity in thesecond elastoplastic stage considering the changeof hardness

HG1(an) Limit mean geometric hardness in the firstelastoplastic deformation stage

HG2(an) Limit mean geometric hardness in the secondelastoplastic deformation stage

nec Elastic critical frequency exponentnepc First elastoplastic critical frequency exponentnpc Second elastoplastic critical frequency exponentAr Actual contact area of the joint interfaceAr1 Actual contact area of the joint interface for

nmin lt nle necAr2 Actual contact area of the joint interface for

nec lt nle nepcAr3 Actual contact area of the joint interface for

nepc lt nle npcAr4 Actual contact area of the joint interface for

npc lt n

Fr1 Actual contact load of the joint interface fornmin lt nle nec

Fr2 Actual contact load of the joint interface fornec lt nle nepc

Fr3 Actual contact load of the joint interface fornepc lt nle npc

Fr4 Actual contact load of the joint interface fornpc lt n

pea(a) Average contact pressure of the asperity in elasticstage

pepa1(a) Average contact pressure of the asperity in thefirst elastoplastic deformation stage

pepa2(a) Average contact pressure of the asperity in thesecond elastoplastic stage

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (grant no 51275328) and the NaturalScience Foundation Project of Shanxi Province (grant no201601D011062)

References

[1] B Zhao S Zhang P Wang and Y Hai ldquoLoading-unloadingnormal stiffness model for power-law hardening surfacesconsidering actual surface topographyrdquo Tribology In-ternational vol 90 pp 332ndash342 2015

[2] J Greenwood and J Williamson ldquoContact of nominally flatsurfacesrdquo Mathematical Physical and Engineering Sciencesvol 295 no 1442 pp 299ndash319 1966

[3] J A Greenwood and J H Tripp ldquoe contact of twonominally flat rough surfacesrdquo Proceedings of the Institution ofMechanical Engineers vol 185 no 1 pp 625ndash633 1970

[4] W R Chang I Etsion and D B Bogy ldquoAn elastic-plasticmodel for the contact of rough surfacesrdquo Journal of Tribologyvol 109 no 2 pp 257ndash263 1987

[5] Y Zhao D M Maietta and L Chang ldquoAn asperity micro-contact model incorporating the transition from elastic de-formation to fully plastic flowrdquo Journal of Tribology vol 122no 1 pp 86ndash93 2000

[6] L Kogut and I Etsion ldquoElastic-plastic contact analysis ofasphere and a rigid flatrdquo Journal of Applied Mechanics vol 69no 5 pp 657ndash662 2002

[7] I Etsion Y Kligerman and Y Kadin ldquoUnloading of anelastic-plastic loaded spherical contactrdquo International Journalof Solids and Structures vol 42 no 13 pp 3716ndash3729 2005

[8] Y Kadin Y Kligerman and I Etsion ldquoUnloading an elastic-plastic contact of rough surfacesrdquo Journal of the Mechanicsand Physics of Solids vol 54 no 12 pp 2652ndash2674 2006

[9] Y Kadin Y Kligerman and I Etsion ldquoMultiple loading-unloading of an elastic-plastic spherical contactrdquo In-ternational Journal of Solids and Structures vol 43 no 22-23pp 7119ndash7127 2006

[10] A Majumdar and B Bhushan ldquoRole of fractal geometry inroughness characterization and contact mechanics of sur-facesrdquo Journal of Tribology vol 112 no 2 pp 205ndash216 1990

[11] A Majumdar and C L Tien ldquoFractal characterization andsimulation of rough surfacesrdquo Wear vol 136 no 2pp 313ndash327 1990

[12] AMajumdar and B Bhushan ldquoFractal model of elastic-plasticcontact between rough surfacesrdquo Journal of Tribology vol 113no 1 pp 1ndash11 1991

[13] S Wang and K Komvopoulos ldquoA fractal theory of the in-terfacial temperature distribution in the slow sliding regimePart I-elastic contact and heat transfer analysisrdquo Journal ofTribology vol 116 no 4 pp 812ndash822 1994

[14] S Wang and K Komvopoulos ldquoA fractal theory of the in-terfacial temperature distribution in the slow sliding regime

Advances in Materials Science and Engineering 13

Part II-multiple domains elastoplastic contacts and appli-cationsrdquo Journal of Tribology vol 116 no 4 pp 824ndash8321994

[15] Y Morag and I Etsion ldquoResolving the contradiction of as-perities plastic to elastic mode transition in current contactmodels of fractal rough surfacesrdquo Wear vol 262 no 5-6pp 624ndash629 2007

[16] H Tian X Zhong and C Zhao ldquoOne loading model of jointinterface considering elastoplastic and variation of hardnesswith surface depthrdquo Journal of Mechanical Engineeringvol 51 no 5 pp 90ndash104 2015

[17] Y Yuan Y Cheng K Liu et al ldquoA revised Majumdar andBushan model of elastoplastic contact between rough sur-facesrdquo Applied Surface Science vol 425 pp 1138ndash1157 2017

[18] R S Sayles and T R omas ldquoSurface topography as anonstationary random processrdquo Nature vol 271 no 5644pp 431ndash434 1978

14 Advances in Materials Science and Engineering

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Submit your manuscripts atwwwhindawicom

anepc 71197anec

fnep1 KH times 11282aminus02544nec a

12544n anec lt an lt anepc1113872 1113873

(13)

anpc 2053827anec

fnep2 KH times 14988aminus01021nec a

11021n anepc lt an lt anpc1113872 1113873

(14)

where fnec is contact load for ω ωnec and fnep1 and fnep2are contact loads in the first elastoplastic stage and thesecond elastoplastic stage respectively Both fnep1 and fnep2obtained above are related to the hardness (H) of thematerial However according to the plastic strengtheningprinciple the hardness is not a constant when the materialyields but a function related to the deformation that is itchanges with the deformation erefore it is not accurateto describe elastoplastic deformation by the above formulaIn order to express the characteristics of elastoplastic de-formation more accurately the concept of limit meangeometric hardness is introduced

According to equations (13) and (14) HG(a) is fittedinto the following segmented relations

e first elastoplastic deformation stage is

HG1 an( 1113857 c1Yan

anec1113888 1113889

c2

anec lt an le anepc1113872 1113873 (15)

e second elastoplastic deformation stage is

HG2 an( 1113857 c3Yan

anec1113888 1113889

c4

anepc lt an le anpc1113872 1113873 (16)

where c1 c2 c3 and c4 are the coefficients to be solved

(1) Equation (15) should satisfy two limiting conditions

HG1 anec( 1113857 pea anec( 1113857 (17)

HG1 anepc1113872 1113873 pepa1 anepc1113872 1113873 (18)

where pea(a) is the average contact pressure of theasperity in elastic stage which is given bypea(a) (fnea) pepa1(a) is the average contactpressure of the asperity in the first elastoplastic de-formation stage and is given by pepa1(a) (fnep1a)Substituting equations (11) and (15) in equation (17)we can obtain

c1Y KH (19)

c1 28K (20)

Substituting equations (13) and (15) in equation (18)we can obtain

KH times 11282aminus02544nec times 71197anec( 1113857

02544 28KY times 71197c2

(21)

Derived from equation (21)

c2 ln 11282 times 7119702544( 1113857

ln 71197 (22)

Considering the change of hardness the normal con-tact load of a single asperity in the first elastoplasticstage is

fnep1prime HG1(a) middot an (23)

Substituting equations (15) (20) and (22) in equation(23) new equations are yielded

fnep1prime 28KYaminusc2neca

c2+1n (24)

(2) Equation (16) should satisfy two limiting conditions

HG2 anepc1113872 1113873 pepa1 anepc1113872 1113873 (25)

HG2 anpc1113872 1113873 pepa2 anpc1113872 1113873 (26)

where pepa2(a) (fnep2a) is the average contactpressure of the asperity in the second elastoplastic stageSubstituting equations (13) and (16) in equation (25)we can obtain

c3(71197)c4 K times 28 times 11282 times(71197)

02544 (27)

Substituting equations (14) and (16) in equation (26)we can obtain

c3(2053827)c4 K times 28 times 14988 times(2053827)

01021

(28)

Simultaneous equations (27) and (28) obtained

c4 ln 11282 + 02544 ln 71197minus ln 14988minus 01021 ln 2053827

ln 71197minus ln 2053827

(29)

Substituting equation (29) in equation (27) we canobtain

c3 K times 315896 times(71197)02544minusc4 (30)

Considering the change of hardness the normal con-tact load of a single asperity in the second elastoplasticstage is

fnep2prime HG2(a) middot an K times 315896 times(71197)02544minusc4Y

middot aminusc4nec middot a

c4+1n

(31)

23 Full Plastic Deformation of a Single Asperity As thedeformation continues to increase when ωn gt 110ωnec thecontact area an gt anpc and the asperity enters the stage of fullplastic deformation At this stage the hardness of the ma-terial is no longer affected by the deformation and can beregarded as a constant When the hardness of the material is

4 Advances in Materials Science and Engineering

given according to literature [7] the contact load andcontact area of the asperity at this stage can be expressed as

fnp Han

an 2πRnωn(32)

In conclusion with the increase of load and deformationthe contact area of the same asperity increases graduallyie anec lt anepc lt anpc With the increase of the load andcontact area the asperity underwent elastic deformationfirst elastoplastic deformation second elastoplastic de-formation and full plastic deformation successively Underconstant load and deformation the actual contact area of theasperity is related to the radius of curvature at the vertex ofthe asperity

24 Asperityrsquos Frequency Exponent n When using W-Mfunction to describe the surface profile of an asperity theprofile function is related to the asperityrsquos frequency ex-ponent In other words the radius of curvature at the vertexof the asperity and the height of the asperity vary with thefrequency exponent when the load is constant According tothe equations (5)ndash(7) it was found that the value of hn Rnand ωnec correlated with the frequency exponent When thefrequency exponent is constant the deformation of theasperity is not greater than the height of the asperityunder the action of the load In order to obtain the criticalvalue of the frequency exponent we take hn ωnec ie(GDminus1c(2minusD)nec) (3KH4Eprime)2 middot (cminusnecDGDminus1)

e elastic critical frequency exponent can be obtainedas follows

nec intln 3KH4Eprime( 1113857

2middot G2(1minusD)1113960 1113961

2(Dminus 1)ln c

⎧⎨

⎩

⎫⎬

⎭ (33)

where int is the integer part of the value in the parenthesisSimilarly the first elastoplastic critical frequency expo-

nent can be obtained

nepc intln 6 3KH4Eprime( 1113857

2middot G2(1minusD)1113960 1113961

2(Dminus 1)ln c

⎧⎨

⎩

⎫⎬

⎭ (34)

e second elastoplastic critical frequency exponent canbe obtained

npc intln 110 3KH4Eprime( 1113857

2middot G2(1minusD)1113960 1113961

2(Dminus 1)ln c

⎧⎨

⎩

⎫⎬

⎭ (35)

From the above when the asperity frequency exponent isnmin lt nle nec elastic deformation only takes place in theseasperities under contact load When nec lt nle nepc elasticdeformation or the first elastoplastic deformation can takeplace in these asperities When nepc lt nle npc elastic de-formation the first elastoplastic deformation or the secondelastoplastic deformation can take place in these asperities and

full plastic deformation never occur When npc lt nle nmaxelastic deformation elastoplastic deformation or full plasticdeformation can take place in these asperities

3 Actual Contact Area and Normal ContactLoad of Joint Interface

According to reference [10] when the asperity frequencyexponent is n the area distribution density function of theasperity on the joint interface is defined as

nn(a) 12

D middotaD2nl

a(D+2)2 0lt ale anl 1ltDlt 2( 1113857 (36)

where anl represents the largest contact area when theasperityrsquos frequency exponent is n

In order to simplify equation (36) we define the areadistribution function of the asperity of any frequency ex-ponent as nn(a) Mn(a) According to reference [17] theactual contact area of joint interface is

Ar 1113944

nmax

nnmin

1113946anl

0nn(a)a da M 1113944

nmax

nnmin

1113946anl

0n(a)a da (37)

where M (al1113936nmaxnnmin

anl)(nmin le nle nmax al max anl1113864 1113865)

31 When the Frequency Exponent Belongs to nmin lt nle necWhen the frequency exponent belongs to nmin lt nle nec evenif these asperities are completely deformed only elasticdeformation will occur and anl lt anec In this case the actualcontact area of the joint interface is defined as Ar1

Ar1 1113944

nec

nnmin

1113946anl

0Mn(a)ada

MD

2minusD1113944

nec

nnmin

anl (38)

In this case the contact load of the joint interface is asfollows

Fr1 1113944

nec

nnmin

1113946anl

0fneMn(a)da (39)

Substituting equation (11) in equation (39) we canobtain

Fr1 MD

3minusD1113944

nec

nnmin

4Eπ12G(Dminus1)

3cminusDna32nl (40)

32 When the Frequency Exponent Belongs to nec lt nle nepcWhen the frequency exponent belongs to nec lt nle nepc forthe case anec lt anl le anepc elastic deformation or the firstelastoplastic deformation may take place in these asperitiesAt this point the actual contact area of the joint interfaceconsists of two parts the elastic deformation stage and thefirst elastoplastic deformation stage

Advances in Materials Science and Engineering 5

Ar2 Are + Arep1 (41)

Are 1113944

nepc

nnec+11113946

anec

0Mn(a)ada

MD

2minusD1113944

nepc

nnec+1a

(2minusD)2nec a

D2nl

(42)

For the determined frequency exponent the maximumactual contact area of the asperity appears at the maximumdeformation amount ωn where the maximum value of theelastic deformation phase ωn appears at ωnec whereuponformula (42) is simplified to

Are MD

2minusD1113944

nepc

nnec+1anec

MD

(2minusD)π1113944

nepc

nnec+1

3KHcminusDn

4G(Dminus 1)Eprime1113888 1113889

2

Arep1 1113944

nepc

nnec+11113946

anl

anec

Mn(a)ada MD

2minusD

middot 1113944

nepc

nnec+1a

(2minusD)2nl minus a

(2minusD)2nec1113960 1113961a

D2nl

(43)

e contact load is given by

Fr2 Fre + Frep1 (44)

Fre 1113944

nepc

nnec+11113946

anec

0fneMn(a)da

9MD(KH)3

16(3minusD) EprimeπGDminus1( 11138572

middot 1113944

nepc

nnec+1cminus2 Dn

(45)

Frep1 1113944

nepc

nnec+11113946

anl

anec

fnep1prime Mn(a)da (46)

Substituting equations (24) and (36) in equation (46) wecan obtain

Frep1 28KYMD

2c2 minusD + 21113944

nepc

nnec+1aminusc2neca

c2+1nl minus a

(2minusD)2nec a

D2nl1113960 1113961 (47)

33 When the Frequency Exponent Belongs to nepc lt nle npcWhen the frequency exponent belongs to nepc lt nle npc forthe case anepc lt anl le anpc elastic deformation the firstelastoplastic deformation or the second elastoplastic de-formation may take place in these asperities At this point

the actual contact area of the joint interface consists ofthree parts the elastic deformation stage the first elas-toplastic deformation stage and the second elastoplasticdeformation stage

Ar3 Are + Arep1 + Arep2

Are 1113944

npc

nnepc+11113946

anec

0Mn(a)ada

MD

(2minusD)π

middot 1113944

npc

nnepc+1

3KHcminusDn

4GDminus1Eprime1113888 1113889

2

Arep1 1113944

npc

nnepc+11113946

anepc

anec

Mn(a)ada

MD

π(2minusD)71197minus 71197D2

1113872 1113873 1113944

npc

nnepc+1

3KHcminusDn

4GDminus1Eprime1113888 1113889

2

Arep2 1113944

npc

nnepc+11113946

anl

anepc

Mn(a)ada

MD

(2minusD)1113944

npc

nnepc+1a

(2minusD)2nl minus 71197anec( 1113857

(2minusD)21113960 1113961a

D2nl

(48)

In this case the contact load of the joint interface is asfollows

Fr3 Fre + Frep1 + Frep2

Fre 1113944

npc

nnepc+11113946

anec

0fneMn(a)da

9MD(KH)3

16(3minusD) EprimeπGDminus1( 11138572 1113944

npc

nnepc+1cminus2 Dn

Frep1 28KYMD

2c2 minusD + 271197c2+1 minus 71197D2

1113872 1113873

middot 1113944

npc

nnepc+1

1π

3KHcminusDn

4GDminus 1Eprime1113888 1113889

2

(49)

When the second elastoplastic deformation occurs thenormal contact load of the joint interface is as follows

6 Advances in Materials Science and Engineering

Frep2 1113944

npc

nnepc+11113946

anl

anepc

fnep2prime Mn(a)da (50)

Substituting equations (31) and (38) in equation (46) wecan obtain

Frep2 2106 times 7119702544minusc4KYMD

2c4 minusD + 2

middot 1113944

npc

nnepc+1aminusc4neca

c4+1nl minus 71197c4minus(D2)+1

a1minus(D2)nec a

(D2)nl1113872 1113873

(51)

34 When the Frequency Exponent Belongs to npc lt nWhen the frequency exponent belongs to npc lt n elasticdeformation elastoplastic deformation or full plastic de-formation may take place in these asperities e actualcontact area of the joint interface can be evaluated as

Ar4 Are + Arep1 + Arep2 + Arp

Are 1113944

nmax

nnpc+11113946

anec

0Mn(a)ada

MD

2minusD1113944

nmax

nnpc+1anec

Arep1 1113944

nmax

nnpc+11113946

anepc

anec

Mn(a)ada

MD

2minusD71197minus 71197D2

1113872 1113873 1113944

nmax

nnpc+1anec

Arep2 1113944

nmax

nnpc+11113946

anpc

anepc

Mn(a)ada

MD

2minusD2053827minus 711971minus(D2)

middot 2053827D21113872 1113873 1113944

nmax

nnpc+1anec

Arp 1113944

nmax

nnpc+11113946

anl

anpc

Mn(a)ada

MD

2minusD1113944

nmax

nnpc+1a

(2minusD)2nl minus 2053827anec( 1113857

(2minusD)21113960 1113961a

D2nl

(52)

In this case the contact load of the joint interface is asfollows

Fr4 Fre + Frep1 + Frep2 + Frp

Fre 1113944

nmax

nnpc+11113946

anec

0fneMn(a)da

MDKH

(3minusD)π1113944

nmax

nnpc+1anec

Frep1 1113944

nmax

nnpc+11113946

anepc

anec

fnep1prime Mn(a)da

28KYMD

2c2 minusD + 271197c2+1 minus 71197D2

1113872 1113873 1113944

nmax

nnpc+1anec

Frep2 2106 times 7119702544minusc4KYMD

2c4 minusD + 2

middot 1113944

nmax

nnpc+1aminusc4neca

c4+1nl minus 71197c4minus(D2)+1

a1minus(D2)nec a

D2nl1113872 1113873

2106 times 7119702544minusc4KYMD

2c4 minusD + 2

middot 2053827c4+1 minus 2053827D2middot 71197c4minus(D2)+1

1113872 1113873

middot 1113944

nmax

nnpc+1anec

Frp 1113944

nmax

nnpc+11113946

anl

anpc

fnpMn(a)da

MHD

2minusD1113944

nmax

nnpc+1a

D2nl a

1minus(D2)nl minus 2053827anec( 1113857

1minus(D2)1113960 1113961

(53)

For all frequency exponents the total actual contact areaof the joint interface is

Ar Ar1 + Ar2 + Ar3 + Ar4 (54)

e total contact load of the joint interface is

Fr Fr1 + Fr2 + Fr3 + Fr4 (55)

e total real contact area and the total contact load in anondimensional form can be written as follows

Alowastr

Ar

Aa

Flowastr

Fr

AaE

(56)

where Aa is the nominal contact area and is given byAa L2L 1cnmin

Advances in Materials Science and Engineering 7

4 Results Analysis

In order to further analyze the above calculation results theparameters of equivalent joint interface are taken as shownin Table 1 [16]

Figure 2 shows the relation between all critical contactareas and frequency exponents of single asperity whenD 15 It can be seen from the figure that as for one definiteasperity when frequency exponent n is certain elastic criticalcontact area is minimum followed by the first elastoplasticcritical contact area and the second elastoplastic criticalcontact area is maximum With gradual increase of contactload the contact area increases e single asperity is firstlysubject to elastic deformation followed by the first elasto-plastic deformation the second elastoplastic deformation andfully plastic deformation successively which is consistent withtypical contact mechanics theory As for different asperitieswith increase of frequency exponent all critical contact areasdecrease correspondingly which shows that elastic criticalcontact area the first elastoplastic critical contact area and thesecond elastoplastic critical contact area are all related tofrequency exponent n

Figure 3 shows the relation curve between fractal di-mension D and critical frequency exponent n of asperityWhen fractal dimension is definite elastic critical frequencyexponent nec the first elastoplastic critical frequency ex-ponent nepc and the second elastoplastic critical frequencyexponent npc increase gradually As shown in Figure 3 whenDlt 106 nec nepc and npc are all negative As for asperitieswith minimum value of frequency exponent being 0 elasticdeformation elastoplastic deformation and fully plasticdeformation will all occur When D 113 nec and nepc arenegative and npc is positive At this time as for asperitieswith minimum value of frequency exponent being 0 elasticdeformation elastoplastic deformation will occur exceptfully plastic deformation

For D 15 G 25 times 10minus9 m H 55 times 109 Nm2 wecan obtain the elastic critical frequency exponent nec 32the first elastoplastic critical frequency exponent nepc 36and the second elastoplastic critical frequency exponentnpc 43 ese asperities whose frequency exponents rangefrom 20 to 32 are only under elastic deformation Elasticdeformation and the first elastoplastic deformation canoccur in these asperities whose frequency exponents rangefrom 33 to 36 Elastic deformation the first elastoplasticdeformation and the second elastoplastic deformation canoccur in these asperities whose frequency exponents rangefrom 37 to 43 When frequency exponents range from 43 to50 all deformations types can occur in these asperities

Figure 4 shows relation comparison diagram betweencontact load and contact area of single asperity with andwithout hardness change at the first elastoplastic stage ecomparison diagram is simulation result when n 33 It canbe seen from the figure that with gradual increase of contactarea with contact area of single asperity over 32 times 10minus13 m2contact load of the same asperity with hardness change willbe less than that without hardness change In addition as theamount of deformation increases the difference betweenthem tends to increase

Figure 5 shows relation comparison diagram betweencontact load and contact area of single asperity with andwithout hardness change at the second elastoplastic stagee comparison diagram is simulation result when n 37 Itcan be seen from the figure that when the deformation is

Table 1 e parameters of equivalent joint interface

Parameters ValuesEquivalent elastic modulus Eprime 72 times 1010 Nm2

Poissonrsquos ratio υ 017Initial hardness H 55 times 109 Nm2

Profile scale parameter G 25 times 10minus9 mFractal dimension D 1ltDlt 2Frequency exponent n 20sim50

20 25 30 35 40 45 5010ndash1510ndash1410ndash1310ndash1210ndash1110ndash1010ndash910ndash810ndash710ndash610ndash510ndash410ndash3

Criti

cal c

onta

ct ar

eas

of a

singl

e asp

erity

(am

2 )

Asperity levels (n)

Second elastoplastic critical contact areaFirst elastoplastic critical contact areaElastic critical contact area

Figure 2 e relationship between critical contact area and fre-quency exponent of a single asperity

10 11 12 13 14 15 16 17 18 19 20ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

0

20

40

Asp

erity

freq

uenc

y ex

pone

nts (n)

e fractal dimension (D)

Elastic critical frequency exponentsFirst elastoplastic critical frequency exponentsSecond elastoplastic critical frequency exponents

Figure 3 e relationship between fractal dimension D andcritical frequency exponent n of a single asperity

8 Advances in Materials Science and Engineering

definite contact load of the same asperity with hardnesschange will be less than that without hardness change Inaddition with increase of deformation amount the differ-ence between them tends to increase which is consistentwith the change trend at the first elastoplastic stage

Figure 6 shows the relation between limit mean geo-metric hardness and contact area (logarithm) of single as-perity at the first elastoplastic deformation stage Figure 6(a)shows relation curve that D 11 13 15 17 when n 34Figure 6(b) shows relation curve that n 32 33 34 35 whenD 15 It can be seen from Figure 6 that the limit meangeometric hardness of single asperity is related to contactarea fractal dimension and frequency exponent in the firstelastoplastic deformation stage e limit mean geometrichardness increases with increase of contact area When n is

definite the relation between limit mean geometric hardnessand contact area of asperity is related to fractal dimension De larger the D is the more obvious the relation curvebetween them changes when D is definite the relationbetween limit mean geometric hardness and contact area ofasperity is related to frequency exponent n e smaller n isthe more obvious the relation curve between them changes

Figure 7 shows the relation between limit mean geo-metric hardness and contact area (logarithm) of single as-perity at the second elastoplastic deformation stageFigure 7(a) shows the relation curve n 40 that D 11 1315 17 when n 40 Figure 7(b) shows the relation curvethat n 36 38 40 42 when D 15

During loading the contact area increases with the in-crease of deformation of a single asperity e ratio of de-formation to the natural height of the asperity is defined asthe pushing coefficient namely the pushing coefficientk ωnhn 0le kle 09 When fractal dimension is 15 we willresearch the relation between contact load and contact areaof single asperity with frequency exponent n being 30 35and 40 respectively during loading

When n 30 the asperity will only be subject to elasticdeformation During loading even the pushing coefficient k

is maximum no plastic deformation will occur e relationbetween contact area and contact load is fsima15 approxi-mately as shown in Figure 8(a)

As is shown in Figure 8(b) when n 35 elastic de-formation and the first elastoplastic deformation may takeplace in the asperity during loading When the pushingcoefficient k is less than 0247 the asperity will under elasticdeformation At this time the relation between contact areaand contact load is fsima15 approximately when the pushingcoefficient is over 0247 the first elastoplastic deformationoccurs At this time the relation between contact area andcontact load is fsima11093 approximately As is shown inFigure 8(c) when n 40 elastic deformation the firstelastoplastic deformation and the second elastoplastic

0000 0001 0002 0003 0004 0005Contact load of a single asperity in the

first elastoplastic deformation regime (fN)

00

12 times 10ndash12

10 times 10ndash12

80 times 10ndash13

60 times 10ndash13

40 times 10ndash13

20 times 10ndash13

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

Change of hardness is consideredChange of hardness is not considered

Figure 4 e relationship between contact load and contact area of single asperity in the first elastoplastic deformation stage

0000 0001 0002 0003 0004 00050

2 times 10ndash13

4 times 10ndash13

6 times 10ndash13

8 times 10ndash13

1 times 10ndash12

1 times 10ndash12

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

Contact load of a single asperity in the second elastoplastic deformation regime (f2N)

Change of hardness is not consideredChange of hardness is considered

Figure 5e relationship between contact load and contact area ofsingle asperity in the second elastoplastic deformation stage

Advances in Materials Science and Engineering 9

deformation may take place in the asperity during loadingWhen the pushing coefficient is greater than 01954 theasperity begins to enter the second elastoplastic de-formation the relation between contact area and contactload is fsima10977 approximately When n 45 and thepushing coefficient is greater than 0472 the asperity beginsto enter fully plastic deformation the relation betweencontact area and contact load is fsima approximately

Figure 9 shows that when the minimum frequency ex-ponent is 20 and the maximum value is 32 the actual contactarea of the joint interface increases with the increase of thetotal contact load and the relation between them isFlowastr simAlowast15

r approximately During the whole deformationprocess the joint interface appears to be of elastic property

Figure 10 shows the relation between dimensionless totalreal contact area and the dimensionless total contact loading

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash9

10ndash15

10ndash14

10ndash13

10ndash12

10ndash11

10ndash10

D = 11 n = 34D = 13 n = 34

D = 15 n = 34D = 17 n = 34

25 times 109 30 times 109 35 times 109 40 times 109 45 times 109 50 times 109 55 times 109

Limit mean geometric hardness in the firstelastoplastic deformation regime HG1(Nm2)

(a)C

onta

ct ar

eas o

f a si

ngle

aspe

rity

(am

2 )

10ndash10

10ndash15

10ndash14

10ndash13

10ndash12

10ndash11

n = 32 D = 15n = 33 D = 15

n = 34 D = 15n = 35 D = 15

25 times 109 30 times 109 35 times 109 40 times 109 45 times 109 50 times 109 55 times 109

Limit mean geometric hardness in the firstelastoplastic deformation regime HG1(Nm2)

(b)

Figure 6 e relationship between limit mean geometric hardness and contact for single asperity during the first elastoplastic deformationstage (a) n 34 11leDle 17 (b) D 15 32le nle 35

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash12

10ndash18

10ndash17

10ndash16

10ndash15

10ndash14

10ndash13

D = 11 n = 40D = 13 n = 40

D = 15 n = 40D = 17 n = 40

50 times 109 55 times 109 60 times 109 65 times 109 70 times 109 75 times 109

Limit mean geometric hardness in the secondelastoplastic deformation regime HG2(Nm2)

(a)

50 times 109 55 times 109 60 times 109 65 times 109 70 times 109 75 times 109

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash10

10ndash11

10ndash12

10ndash13

10ndash18

10ndash17

10ndash16

10ndash15

10ndash14

n = 36 D = 15n = 38 D = 15

n = 40 D = 15n = 42 D = 15

Limit mean geometric hardness in the secondelastoplastic deformation regime HG2(Nm2)

(b)

Figure 7 e relationship between limit mean geometric hardness and contact for single asperity during the second elastoplastic de-formation stage (a) n 40 11leDle 17 (b) D 15 36le nle 42

10 Advances in Materials Science and Engineering

of the joint interface when minimum value and maximumvalue of frequency exponent of asperities are respectively 20and 36When the pushing coefficient k is 01648 the asperitybegins to be subject to the first elastoplastic deformation andthe relation between dimensionless total real contact areaand the dimensionless total contact loading is FlowastrsimAlowast15

r

approximately e joint interface appears to be of elasticproperty As the load continues increasing when thepushing coefficient k is 05564 at this pointFlowastr gt 69023 times 10minus3 the relation between above load and areais approximately FlowastrsimAlowast11093

r e joint interface presentselastoplastic properties At this time the first elastoplasticdeformation takes place in these asperities whose frequencyexponents range 33sim36

Figure 11 shows the relation between dimensionless totalreal contact area and the dimensionless total contact loadingof the joint interface when minimum value and maximumvalue of frequency exponent of asperities are respectively 44and 50When the pushing coefficient k is 00621 the asperity

begins to be subject to the second elastoplastic deformationat this point 00016ltFlowastr lt 0023 and the relation betweendimensionless total real contact area and the dimensionlesstotal contact loading is FlowastrsimAlowast10977

r approximately Whenthe pushing coefficient k is 07076 the asperity begins to besubject to the fully plastic deformation at this pointFlowastr gt 0023 and the relation between dimensionless total realcontact area and the dimensionless total contact loading isapproximately FlowastrsimAlowastr

5 Conclusions

(1) e contact mechanical properties of a single as-perity on a joint interface are related to the frequencyexponent of the asperity while the frequency ex-ponent of an asperity is related to the fractal di-mension and the profile scale parameter In thispaper the critical frequency exponents of each de-formation stage of a single asperity are obtained and

000 005 010 015 020 025 030 03500

10 times 10ndash11

20 times 10ndash11

30 times 10ndash11

40 times 10ndash11

50 times 10ndash11

60 times 10ndash11

Contact load of a single asperity (fN)

n = 30

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

(a)

Contact load of a single asperity (fN)

n = 35

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

50

times10

ndash5

10

times10

ndash4

104 times 10ndash4

10 times 10ndash13

80 times 10ndash14

60 times 10ndash14

40 times 10ndash14

20 times 10ndash14

363 times 10ndash14

15

times10

ndash4

20

times10

ndash4

25

times10

ndash4

30

times10

ndash4

35

times10

ndash400

00

(b)

Contact load of a single asperity (fN)

n = 40

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

4 times 10ndash17

3 times 10ndash17

2 times 10ndash17

1 times 10ndash17

0

20

times10

ndash8

40

times10

ndash8

60

times10

ndash8

80

times10

ndash8

10

times10

ndash700

(c)

Figure 8 e relationship between contact load and contact area of a single asperity during loading

Advances in Materials Science and Engineering 11

the deformation characteristics of the asperity underdifferent frequency exponents are obtained

(2) e normal contact load of a single asperity in theelastoplastic stage is related to the hardness of thematerial When the material yields the hardness H isnot a constant but a function related to the amountof deformation In this paper the limit mean geo-metric hardness is introduced to express the normalcontact load of a single asperity in the first andsecond elastoplastic deformation stages consideringthe change of hardness

(3) e relationship between the contact load andcontact area of a single asperity in the first andsecond elastoplastic deformation stages considering

the change of material hardness and not consideringthe change of hardness is compared respectivelyWhen the contact area of the asperity is the same thecontact load with the change of hardness is smallerthan that without the change of hardness and thedifference between them increases with the increaseof deformation

(4) e limit mean geometric hardness of a single as-perity is related to the contact area fractal di-mension and frequency exponent during theelastoplastic deformation stage and the limit meangeometric hardness increases with the increase of thecontact area When the frequency exponent isconstant the relationship between the limit meangeometric hardness and the contact area of the as-perity is related to the fractal dimension e largerthe fractal dimension is the more obvious the re-lationship curve between them is When the fractaldimension is constant the relationship between thelimit mean geometric hardness of the asperity andthe contact area is related to the frequency exponente smaller the frequency exponent is the moreobvious the change of the curve is

(5) e relationship between dimensionless total realcontact area and the dimensionless total contactloading of joint interface at each deformation stageis obtained by introducing the pushing coefficientand the critical pushing coefficient exists Whenthe pushing coefficient exceeds this value someasperities begin to deform in the next stage andthe inflection point appears on the relationshipcurve

(6) e current model still has some limitations on theapplicable scope of materials and further researchand improvement are needed to make it more ap-plicable in the future

09

08

07

06

05

04

03

02

01

000000 0002 0004 0006 0008 0010 0012 0014

Nondimensional total contact load (Flowastr)

Non

dim

ensio

nal t

otal

cont

act a

rea (

Alowastr)

nmin = 20 nmax = 36

Figure 10 e relationship between dimensionless total realcontact area and the dimensionless total contact loading for20le nle 36

10

09

08

07

06

05

04

03

02

01Non

dim

ensio

nal t

otal

cont

act a

rea (

Alowastr)

000000 0005 0010 0015 0020 0025 0030 0035

Nondimensional total contact load (Flowastr)

nmin = 44 nmax = 50

Figure 11 e relationship between dimensionless total realcontact area and the dimensionless total contact loading for44le nle 50

10

09

08

07

06

05

04

03

02

01

Nondimensional total contact load (Flowastr1)

Non

dim

ensio

nal t

otal

real

cont

act a

rea (

Alowastr1

)

0000 10 times 10ndash3 20 times 10ndash3 30 times 10ndash3 40 times 10ndash3 50 times 10ndash3

nmin = 20 nmax = 32

Figure 9e relationship between dimensionless total real contactarea and the dimensionless total contact loading for 20le nle 32

12 Advances in Materials Science and Engineering

Nomenclature

ωn Interference of the asperityωnec Elastic critical interference of the asperityωnepc First elastoplastic critical interference of the

asperityωnpc Second elastoplastic critical interference of the

asperityan Contact area of the asperityanec Elastic critical contact area of the asperityanepc First elastoplastic critical contact area of the

asperityanpc Second elastoplastic critical contact area of the

asperityfne Normal load in the elastic deformation of a single

asperityfnec Normal contact load for ω ωnecfnep1 Normal contact load of a single asperity in the

first elastoplastic stagefnep2 Normal contact load of a single asperity in the

second elastoplastic stagefnp Normal contact load of a single asperity in the full

plastic deformation stagefnep1prime Normal contact load of a single asperity in the

first elastoplastic stage considering the change ofhardness

fnep2prime Normal contact load of a single asperity in thesecond elastoplastic stage considering the changeof hardness

HG1(an) Limit mean geometric hardness in the firstelastoplastic deformation stage

HG2(an) Limit mean geometric hardness in the secondelastoplastic deformation stage

nec Elastic critical frequency exponentnepc First elastoplastic critical frequency exponentnpc Second elastoplastic critical frequency exponentAr Actual contact area of the joint interfaceAr1 Actual contact area of the joint interface for

nmin lt nle necAr2 Actual contact area of the joint interface for

nec lt nle nepcAr3 Actual contact area of the joint interface for

nepc lt nle npcAr4 Actual contact area of the joint interface for

npc lt n

Fr1 Actual contact load of the joint interface fornmin lt nle nec

Fr2 Actual contact load of the joint interface fornec lt nle nepc

Fr3 Actual contact load of the joint interface fornepc lt nle npc

Fr4 Actual contact load of the joint interface fornpc lt n

pea(a) Average contact pressure of the asperity in elasticstage

pepa1(a) Average contact pressure of the asperity in thefirst elastoplastic deformation stage

pepa2(a) Average contact pressure of the asperity in thesecond elastoplastic stage

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (grant no 51275328) and the NaturalScience Foundation Project of Shanxi Province (grant no201601D011062)

References

[1] B Zhao S Zhang P Wang and Y Hai ldquoLoading-unloadingnormal stiffness model for power-law hardening surfacesconsidering actual surface topographyrdquo Tribology In-ternational vol 90 pp 332ndash342 2015

[2] J Greenwood and J Williamson ldquoContact of nominally flatsurfacesrdquo Mathematical Physical and Engineering Sciencesvol 295 no 1442 pp 299ndash319 1966

[3] J A Greenwood and J H Tripp ldquoe contact of twonominally flat rough surfacesrdquo Proceedings of the Institution ofMechanical Engineers vol 185 no 1 pp 625ndash633 1970

[4] W R Chang I Etsion and D B Bogy ldquoAn elastic-plasticmodel for the contact of rough surfacesrdquo Journal of Tribologyvol 109 no 2 pp 257ndash263 1987

[5] Y Zhao D M Maietta and L Chang ldquoAn asperity micro-contact model incorporating the transition from elastic de-formation to fully plastic flowrdquo Journal of Tribology vol 122no 1 pp 86ndash93 2000

[6] L Kogut and I Etsion ldquoElastic-plastic contact analysis ofasphere and a rigid flatrdquo Journal of Applied Mechanics vol 69no 5 pp 657ndash662 2002

[7] I Etsion Y Kligerman and Y Kadin ldquoUnloading of anelastic-plastic loaded spherical contactrdquo International Journalof Solids and Structures vol 42 no 13 pp 3716ndash3729 2005

[8] Y Kadin Y Kligerman and I Etsion ldquoUnloading an elastic-plastic contact of rough surfacesrdquo Journal of the Mechanicsand Physics of Solids vol 54 no 12 pp 2652ndash2674 2006

[9] Y Kadin Y Kligerman and I Etsion ldquoMultiple loading-unloading of an elastic-plastic spherical contactrdquo In-ternational Journal of Solids and Structures vol 43 no 22-23pp 7119ndash7127 2006

[10] A Majumdar and B Bhushan ldquoRole of fractal geometry inroughness characterization and contact mechanics of sur-facesrdquo Journal of Tribology vol 112 no 2 pp 205ndash216 1990

[11] A Majumdar and C L Tien ldquoFractal characterization andsimulation of rough surfacesrdquo Wear vol 136 no 2pp 313ndash327 1990

[12] AMajumdar and B Bhushan ldquoFractal model of elastic-plasticcontact between rough surfacesrdquo Journal of Tribology vol 113no 1 pp 1ndash11 1991

[13] S Wang and K Komvopoulos ldquoA fractal theory of the in-terfacial temperature distribution in the slow sliding regimePart I-elastic contact and heat transfer analysisrdquo Journal ofTribology vol 116 no 4 pp 812ndash822 1994

[14] S Wang and K Komvopoulos ldquoA fractal theory of the in-terfacial temperature distribution in the slow sliding regime

Advances in Materials Science and Engineering 13

Part II-multiple domains elastoplastic contacts and appli-cationsrdquo Journal of Tribology vol 116 no 4 pp 824ndash8321994

[15] Y Morag and I Etsion ldquoResolving the contradiction of as-perities plastic to elastic mode transition in current contactmodels of fractal rough surfacesrdquo Wear vol 262 no 5-6pp 624ndash629 2007

[16] H Tian X Zhong and C Zhao ldquoOne loading model of jointinterface considering elastoplastic and variation of hardnesswith surface depthrdquo Journal of Mechanical Engineeringvol 51 no 5 pp 90ndash104 2015

[17] Y Yuan Y Cheng K Liu et al ldquoA revised Majumdar andBushan model of elastoplastic contact between rough sur-facesrdquo Applied Surface Science vol 425 pp 1138ndash1157 2017

[18] R S Sayles and T R omas ldquoSurface topography as anonstationary random processrdquo Nature vol 271 no 5644pp 431ndash434 1978

14 Advances in Materials Science and Engineering

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Submit your manuscripts atwwwhindawicom

given according to literature [7] the contact load andcontact area of the asperity at this stage can be expressed as

fnp Han

an 2πRnωn(32)

In conclusion with the increase of load and deformationthe contact area of the same asperity increases graduallyie anec lt anepc lt anpc With the increase of the load andcontact area the asperity underwent elastic deformationfirst elastoplastic deformation second elastoplastic de-formation and full plastic deformation successively Underconstant load and deformation the actual contact area of theasperity is related to the radius of curvature at the vertex ofthe asperity

24 Asperityrsquos Frequency Exponent n When using W-Mfunction to describe the surface profile of an asperity theprofile function is related to the asperityrsquos frequency ex-ponent In other words the radius of curvature at the vertexof the asperity and the height of the asperity vary with thefrequency exponent when the load is constant According tothe equations (5)ndash(7) it was found that the value of hn Rnand ωnec correlated with the frequency exponent When thefrequency exponent is constant the deformation of theasperity is not greater than the height of the asperityunder the action of the load In order to obtain the criticalvalue of the frequency exponent we take hn ωnec ie(GDminus1c(2minusD)nec) (3KH4Eprime)2 middot (cminusnecDGDminus1)

e elastic critical frequency exponent can be obtainedas follows

nec intln 3KH4Eprime( 1113857

2middot G2(1minusD)1113960 1113961

2(Dminus 1)ln c

⎧⎨

⎩

⎫⎬

⎭ (33)

where int is the integer part of the value in the parenthesisSimilarly the first elastoplastic critical frequency expo-

nent can be obtained

nepc intln 6 3KH4Eprime( 1113857

2middot G2(1minusD)1113960 1113961

2(Dminus 1)ln c

⎧⎨

⎩

⎫⎬

⎭ (34)

e second elastoplastic critical frequency exponent canbe obtained

npc intln 110 3KH4Eprime( 1113857

2middot G2(1minusD)1113960 1113961

2(Dminus 1)ln c

⎧⎨

⎩

⎫⎬

⎭ (35)

From the above when the asperity frequency exponent isnmin lt nle nec elastic deformation only takes place in theseasperities under contact load When nec lt nle nepc elasticdeformation or the first elastoplastic deformation can takeplace in these asperities When nepc lt nle npc elastic de-formation the first elastoplastic deformation or the secondelastoplastic deformation can take place in these asperities and

full plastic deformation never occur When npc lt nle nmaxelastic deformation elastoplastic deformation or full plasticdeformation can take place in these asperities

3 Actual Contact Area and Normal ContactLoad of Joint Interface

According to reference [10] when the asperity frequencyexponent is n the area distribution density function of theasperity on the joint interface is defined as

nn(a) 12

D middotaD2nl

a(D+2)2 0lt ale anl 1ltDlt 2( 1113857 (36)

where anl represents the largest contact area when theasperityrsquos frequency exponent is n

In order to simplify equation (36) we define the areadistribution function of the asperity of any frequency ex-ponent as nn(a) Mn(a) According to reference [17] theactual contact area of joint interface is

Ar 1113944

nmax

nnmin

1113946anl

0nn(a)a da M 1113944

nmax

nnmin

1113946anl

0n(a)a da (37)

where M (al1113936nmaxnnmin

anl)(nmin le nle nmax al max anl1113864 1113865)

31 When the Frequency Exponent Belongs to nmin lt nle necWhen the frequency exponent belongs to nmin lt nle nec evenif these asperities are completely deformed only elasticdeformation will occur and anl lt anec In this case the actualcontact area of the joint interface is defined as Ar1

Ar1 1113944

nec

nnmin

1113946anl

0Mn(a)ada

MD

2minusD1113944

nec

nnmin

anl (38)

In this case the contact load of the joint interface is asfollows

Fr1 1113944

nec

nnmin

1113946anl

0fneMn(a)da (39)

Substituting equation (11) in equation (39) we canobtain

Fr1 MD

3minusD1113944

nec

nnmin

4Eπ12G(Dminus1)

3cminusDna32nl (40)

32 When the Frequency Exponent Belongs to nec lt nle nepcWhen the frequency exponent belongs to nec lt nle nepc forthe case anec lt anl le anepc elastic deformation or the firstelastoplastic deformation may take place in these asperitiesAt this point the actual contact area of the joint interfaceconsists of two parts the elastic deformation stage and thefirst elastoplastic deformation stage

Advances in Materials Science and Engineering 5

Ar2 Are + Arep1 (41)

Are 1113944

nepc

nnec+11113946

anec

0Mn(a)ada

MD

2minusD1113944

nepc

nnec+1a

(2minusD)2nec a

D2nl

(42)

For the determined frequency exponent the maximumactual contact area of the asperity appears at the maximumdeformation amount ωn where the maximum value of theelastic deformation phase ωn appears at ωnec whereuponformula (42) is simplified to

Are MD

2minusD1113944

nepc

nnec+1anec

MD

(2minusD)π1113944

nepc

nnec+1

3KHcminusDn

4G(Dminus 1)Eprime1113888 1113889

2

Arep1 1113944

nepc

nnec+11113946

anl

anec

Mn(a)ada MD

2minusD

middot 1113944

nepc

nnec+1a

(2minusD)2nl minus a

(2minusD)2nec1113960 1113961a

D2nl

(43)

e contact load is given by

Fr2 Fre + Frep1 (44)

Fre 1113944

nepc

nnec+11113946

anec

0fneMn(a)da

9MD(KH)3

16(3minusD) EprimeπGDminus1( 11138572

middot 1113944

nepc

nnec+1cminus2 Dn

(45)

Frep1 1113944

nepc

nnec+11113946

anl

anec

fnep1prime Mn(a)da (46)

Substituting equations (24) and (36) in equation (46) wecan obtain

Frep1 28KYMD

2c2 minusD + 21113944

nepc

nnec+1aminusc2neca

c2+1nl minus a

(2minusD)2nec a

D2nl1113960 1113961 (47)

33 When the Frequency Exponent Belongs to nepc lt nle npcWhen the frequency exponent belongs to nepc lt nle npc forthe case anepc lt anl le anpc elastic deformation the firstelastoplastic deformation or the second elastoplastic de-formation may take place in these asperities At this point

the actual contact area of the joint interface consists ofthree parts the elastic deformation stage the first elas-toplastic deformation stage and the second elastoplasticdeformation stage

Ar3 Are + Arep1 + Arep2

Are 1113944

npc

nnepc+11113946

anec

0Mn(a)ada

MD

(2minusD)π

middot 1113944

npc

nnepc+1

3KHcminusDn

4GDminus1Eprime1113888 1113889

2

Arep1 1113944

npc

nnepc+11113946

anepc

anec

Mn(a)ada

MD

π(2minusD)71197minus 71197D2

1113872 1113873 1113944

npc

nnepc+1

3KHcminusDn

4GDminus1Eprime1113888 1113889

2

Arep2 1113944

npc

nnepc+11113946

anl

anepc

Mn(a)ada

MD

(2minusD)1113944

npc

nnepc+1a

(2minusD)2nl minus 71197anec( 1113857

(2minusD)21113960 1113961a

D2nl

(48)

In this case the contact load of the joint interface is asfollows

Fr3 Fre + Frep1 + Frep2

Fre 1113944

npc

nnepc+11113946

anec

0fneMn(a)da

9MD(KH)3

16(3minusD) EprimeπGDminus1( 11138572 1113944

npc

nnepc+1cminus2 Dn

Frep1 28KYMD

2c2 minusD + 271197c2+1 minus 71197D2

1113872 1113873

middot 1113944

npc

nnepc+1

1π

3KHcminusDn

4GDminus 1Eprime1113888 1113889

2

(49)

When the second elastoplastic deformation occurs thenormal contact load of the joint interface is as follows

6 Advances in Materials Science and Engineering

Frep2 1113944

npc

nnepc+11113946

anl

anepc

fnep2prime Mn(a)da (50)

Substituting equations (31) and (38) in equation (46) wecan obtain

Frep2 2106 times 7119702544minusc4KYMD

2c4 minusD + 2

middot 1113944

npc

nnepc+1aminusc4neca

c4+1nl minus 71197c4minus(D2)+1

a1minus(D2)nec a

(D2)nl1113872 1113873

(51)

34 When the Frequency Exponent Belongs to npc lt nWhen the frequency exponent belongs to npc lt n elasticdeformation elastoplastic deformation or full plastic de-formation may take place in these asperities e actualcontact area of the joint interface can be evaluated as

Ar4 Are + Arep1 + Arep2 + Arp

Are 1113944

nmax

nnpc+11113946

anec

0Mn(a)ada

MD

2minusD1113944

nmax

nnpc+1anec

Arep1 1113944

nmax

nnpc+11113946

anepc

anec

Mn(a)ada

MD

2minusD71197minus 71197D2

1113872 1113873 1113944

nmax

nnpc+1anec

Arep2 1113944

nmax

nnpc+11113946

anpc

anepc

Mn(a)ada

MD

2minusD2053827minus 711971minus(D2)

middot 2053827D21113872 1113873 1113944

nmax

nnpc+1anec

Arp 1113944

nmax

nnpc+11113946

anl

anpc

Mn(a)ada

MD

2minusD1113944

nmax

nnpc+1a

(2minusD)2nl minus 2053827anec( 1113857

(2minusD)21113960 1113961a

D2nl

(52)

In this case the contact load of the joint interface is asfollows

Fr4 Fre + Frep1 + Frep2 + Frp

Fre 1113944

nmax

nnpc+11113946

anec

0fneMn(a)da

MDKH

(3minusD)π1113944

nmax

nnpc+1anec

Frep1 1113944

nmax

nnpc+11113946

anepc

anec

fnep1prime Mn(a)da

28KYMD

2c2 minusD + 271197c2+1 minus 71197D2

1113872 1113873 1113944

nmax

nnpc+1anec

Frep2 2106 times 7119702544minusc4KYMD

2c4 minusD + 2

middot 1113944

nmax

nnpc+1aminusc4neca

c4+1nl minus 71197c4minus(D2)+1

a1minus(D2)nec a

D2nl1113872 1113873

2106 times 7119702544minusc4KYMD

2c4 minusD + 2

middot 2053827c4+1 minus 2053827D2middot 71197c4minus(D2)+1

1113872 1113873

middot 1113944

nmax

nnpc+1anec

Frp 1113944

nmax

nnpc+11113946

anl

anpc

fnpMn(a)da

MHD

2minusD1113944

nmax

nnpc+1a

D2nl a

1minus(D2)nl minus 2053827anec( 1113857

1minus(D2)1113960 1113961

(53)

For all frequency exponents the total actual contact areaof the joint interface is

Ar Ar1 + Ar2 + Ar3 + Ar4 (54)

e total contact load of the joint interface is

Fr Fr1 + Fr2 + Fr3 + Fr4 (55)

e total real contact area and the total contact load in anondimensional form can be written as follows

Alowastr

Ar

Aa

Flowastr

Fr

AaE

(56)

where Aa is the nominal contact area and is given byAa L2L 1cnmin

Advances in Materials Science and Engineering 7

4 Results Analysis

In order to further analyze the above calculation results theparameters of equivalent joint interface are taken as shownin Table 1 [16]

Figure 2 shows the relation between all critical contactareas and frequency exponents of single asperity whenD 15 It can be seen from the figure that as for one definiteasperity when frequency exponent n is certain elastic criticalcontact area is minimum followed by the first elastoplasticcritical contact area and the second elastoplastic criticalcontact area is maximum With gradual increase of contactload the contact area increases e single asperity is firstlysubject to elastic deformation followed by the first elasto-plastic deformation the second elastoplastic deformation andfully plastic deformation successively which is consistent withtypical contact mechanics theory As for different asperitieswith increase of frequency exponent all critical contact areasdecrease correspondingly which shows that elastic criticalcontact area the first elastoplastic critical contact area and thesecond elastoplastic critical contact area are all related tofrequency exponent n

Figure 3 shows the relation curve between fractal di-mension D and critical frequency exponent n of asperityWhen fractal dimension is definite elastic critical frequencyexponent nec the first elastoplastic critical frequency ex-ponent nepc and the second elastoplastic critical frequencyexponent npc increase gradually As shown in Figure 3 whenDlt 106 nec nepc and npc are all negative As for asperitieswith minimum value of frequency exponent being 0 elasticdeformation elastoplastic deformation and fully plasticdeformation will all occur When D 113 nec and nepc arenegative and npc is positive At this time as for asperitieswith minimum value of frequency exponent being 0 elasticdeformation elastoplastic deformation will occur exceptfully plastic deformation

For D 15 G 25 times 10minus9 m H 55 times 109 Nm2 wecan obtain the elastic critical frequency exponent nec 32the first elastoplastic critical frequency exponent nepc 36and the second elastoplastic critical frequency exponentnpc 43 ese asperities whose frequency exponents rangefrom 20 to 32 are only under elastic deformation Elasticdeformation and the first elastoplastic deformation canoccur in these asperities whose frequency exponents rangefrom 33 to 36 Elastic deformation the first elastoplasticdeformation and the second elastoplastic deformation canoccur in these asperities whose frequency exponents rangefrom 37 to 43 When frequency exponents range from 43 to50 all deformations types can occur in these asperities

Figure 4 shows relation comparison diagram betweencontact load and contact area of single asperity with andwithout hardness change at the first elastoplastic stage ecomparison diagram is simulation result when n 33 It canbe seen from the figure that with gradual increase of contactarea with contact area of single asperity over 32 times 10minus13 m2contact load of the same asperity with hardness change willbe less than that without hardness change In addition as theamount of deformation increases the difference betweenthem tends to increase

Figure 5 shows relation comparison diagram betweencontact load and contact area of single asperity with andwithout hardness change at the second elastoplastic stagee comparison diagram is simulation result when n 37 Itcan be seen from the figure that when the deformation is

Table 1 e parameters of equivalent joint interface

Parameters ValuesEquivalent elastic modulus Eprime 72 times 1010 Nm2

Poissonrsquos ratio υ 017Initial hardness H 55 times 109 Nm2

Profile scale parameter G 25 times 10minus9 mFractal dimension D 1ltDlt 2Frequency exponent n 20sim50

20 25 30 35 40 45 5010ndash1510ndash1410ndash1310ndash1210ndash1110ndash1010ndash910ndash810ndash710ndash610ndash510ndash410ndash3

Criti

cal c

onta

ct ar

eas

of a

singl

e asp

erity

(am

2 )

Asperity levels (n)

Second elastoplastic critical contact areaFirst elastoplastic critical contact areaElastic critical contact area

Figure 2 e relationship between critical contact area and fre-quency exponent of a single asperity

10 11 12 13 14 15 16 17 18 19 20ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

0

20

40

Asp

erity

freq

uenc

y ex

pone

nts (n)

e fractal dimension (D)

Elastic critical frequency exponentsFirst elastoplastic critical frequency exponentsSecond elastoplastic critical frequency exponents

Figure 3 e relationship between fractal dimension D andcritical frequency exponent n of a single asperity

8 Advances in Materials Science and Engineering

definite contact load of the same asperity with hardnesschange will be less than that without hardness change Inaddition with increase of deformation amount the differ-ence between them tends to increase which is consistentwith the change trend at the first elastoplastic stage

Figure 6 shows the relation between limit mean geo-metric hardness and contact area (logarithm) of single as-perity at the first elastoplastic deformation stage Figure 6(a)shows relation curve that D 11 13 15 17 when n 34Figure 6(b) shows relation curve that n 32 33 34 35 whenD 15 It can be seen from Figure 6 that the limit meangeometric hardness of single asperity is related to contactarea fractal dimension and frequency exponent in the firstelastoplastic deformation stage e limit mean geometrichardness increases with increase of contact area When n is

definite the relation between limit mean geometric hardnessand contact area of asperity is related to fractal dimension De larger the D is the more obvious the relation curvebetween them changes when D is definite the relationbetween limit mean geometric hardness and contact area ofasperity is related to frequency exponent n e smaller n isthe more obvious the relation curve between them changes

Figure 7 shows the relation between limit mean geo-metric hardness and contact area (logarithm) of single as-perity at the second elastoplastic deformation stageFigure 7(a) shows the relation curve n 40 that D 11 1315 17 when n 40 Figure 7(b) shows the relation curvethat n 36 38 40 42 when D 15

During loading the contact area increases with the in-crease of deformation of a single asperity e ratio of de-formation to the natural height of the asperity is defined asthe pushing coefficient namely the pushing coefficientk ωnhn 0le kle 09 When fractal dimension is 15 we willresearch the relation between contact load and contact areaof single asperity with frequency exponent n being 30 35and 40 respectively during loading

When n 30 the asperity will only be subject to elasticdeformation During loading even the pushing coefficient k

is maximum no plastic deformation will occur e relationbetween contact area and contact load is fsima15 approxi-mately as shown in Figure 8(a)

As is shown in Figure 8(b) when n 35 elastic de-formation and the first elastoplastic deformation may takeplace in the asperity during loading When the pushingcoefficient k is less than 0247 the asperity will under elasticdeformation At this time the relation between contact areaand contact load is fsima15 approximately when the pushingcoefficient is over 0247 the first elastoplastic deformationoccurs At this time the relation between contact area andcontact load is fsima11093 approximately As is shown inFigure 8(c) when n 40 elastic deformation the firstelastoplastic deformation and the second elastoplastic

0000 0001 0002 0003 0004 0005Contact load of a single asperity in the

first elastoplastic deformation regime (fN)

00

12 times 10ndash12

10 times 10ndash12

80 times 10ndash13

60 times 10ndash13

40 times 10ndash13

20 times 10ndash13

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

Change of hardness is consideredChange of hardness is not considered

Figure 4 e relationship between contact load and contact area of single asperity in the first elastoplastic deformation stage

0000 0001 0002 0003 0004 00050

2 times 10ndash13

4 times 10ndash13

6 times 10ndash13

8 times 10ndash13

1 times 10ndash12

1 times 10ndash12

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

Contact load of a single asperity in the second elastoplastic deformation regime (f2N)

Change of hardness is not consideredChange of hardness is considered

Figure 5e relationship between contact load and contact area ofsingle asperity in the second elastoplastic deformation stage

Advances in Materials Science and Engineering 9

deformation may take place in the asperity during loadingWhen the pushing coefficient is greater than 01954 theasperity begins to enter the second elastoplastic de-formation the relation between contact area and contactload is fsima10977 approximately When n 45 and thepushing coefficient is greater than 0472 the asperity beginsto enter fully plastic deformation the relation betweencontact area and contact load is fsima approximately

Figure 9 shows that when the minimum frequency ex-ponent is 20 and the maximum value is 32 the actual contactarea of the joint interface increases with the increase of thetotal contact load and the relation between them isFlowastr simAlowast15

r approximately During the whole deformationprocess the joint interface appears to be of elastic property

Figure 10 shows the relation between dimensionless totalreal contact area and the dimensionless total contact loading

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash9

10ndash15

10ndash14

10ndash13

10ndash12

10ndash11

10ndash10

D = 11 n = 34D = 13 n = 34

D = 15 n = 34D = 17 n = 34

25 times 109 30 times 109 35 times 109 40 times 109 45 times 109 50 times 109 55 times 109

Limit mean geometric hardness in the firstelastoplastic deformation regime HG1(Nm2)

(a)C

onta

ct ar

eas o

f a si

ngle

aspe

rity

(am

2 )

10ndash10

10ndash15

10ndash14

10ndash13

10ndash12

10ndash11

n = 32 D = 15n = 33 D = 15

n = 34 D = 15n = 35 D = 15

25 times 109 30 times 109 35 times 109 40 times 109 45 times 109 50 times 109 55 times 109

Limit mean geometric hardness in the firstelastoplastic deformation regime HG1(Nm2)

(b)

Figure 6 e relationship between limit mean geometric hardness and contact for single asperity during the first elastoplastic deformationstage (a) n 34 11leDle 17 (b) D 15 32le nle 35

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash12

10ndash18

10ndash17

10ndash16

10ndash15

10ndash14

10ndash13

D = 11 n = 40D = 13 n = 40

D = 15 n = 40D = 17 n = 40

50 times 109 55 times 109 60 times 109 65 times 109 70 times 109 75 times 109

Limit mean geometric hardness in the secondelastoplastic deformation regime HG2(Nm2)

(a)

50 times 109 55 times 109 60 times 109 65 times 109 70 times 109 75 times 109

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash10

10ndash11

10ndash12

10ndash13

10ndash18

10ndash17

10ndash16

10ndash15

10ndash14

n = 36 D = 15n = 38 D = 15

n = 40 D = 15n = 42 D = 15

Limit mean geometric hardness in the secondelastoplastic deformation regime HG2(Nm2)

(b)

Figure 7 e relationship between limit mean geometric hardness and contact for single asperity during the second elastoplastic de-formation stage (a) n 40 11leDle 17 (b) D 15 36le nle 42

10 Advances in Materials Science and Engineering

of the joint interface when minimum value and maximumvalue of frequency exponent of asperities are respectively 20and 36When the pushing coefficient k is 01648 the asperitybegins to be subject to the first elastoplastic deformation andthe relation between dimensionless total real contact areaand the dimensionless total contact loading is FlowastrsimAlowast15

r

approximately e joint interface appears to be of elasticproperty As the load continues increasing when thepushing coefficient k is 05564 at this pointFlowastr gt 69023 times 10minus3 the relation between above load and areais approximately FlowastrsimAlowast11093

r e joint interface presentselastoplastic properties At this time the first elastoplasticdeformation takes place in these asperities whose frequencyexponents range 33sim36

Figure 11 shows the relation between dimensionless totalreal contact area and the dimensionless total contact loadingof the joint interface when minimum value and maximumvalue of frequency exponent of asperities are respectively 44and 50When the pushing coefficient k is 00621 the asperity

begins to be subject to the second elastoplastic deformationat this point 00016ltFlowastr lt 0023 and the relation betweendimensionless total real contact area and the dimensionlesstotal contact loading is FlowastrsimAlowast10977

r approximately Whenthe pushing coefficient k is 07076 the asperity begins to besubject to the fully plastic deformation at this pointFlowastr gt 0023 and the relation between dimensionless total realcontact area and the dimensionless total contact loading isapproximately FlowastrsimAlowastr

5 Conclusions

(1) e contact mechanical properties of a single as-perity on a joint interface are related to the frequencyexponent of the asperity while the frequency ex-ponent of an asperity is related to the fractal di-mension and the profile scale parameter In thispaper the critical frequency exponents of each de-formation stage of a single asperity are obtained and

000 005 010 015 020 025 030 03500

10 times 10ndash11

20 times 10ndash11

30 times 10ndash11

40 times 10ndash11

50 times 10ndash11

60 times 10ndash11

Contact load of a single asperity (fN)

n = 30

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

(a)

Contact load of a single asperity (fN)

n = 35

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

50

times10

ndash5

10

times10

ndash4

104 times 10ndash4

10 times 10ndash13

80 times 10ndash14

60 times 10ndash14

40 times 10ndash14

20 times 10ndash14

363 times 10ndash14

15

times10

ndash4

20

times10

ndash4

25

times10

ndash4

30

times10

ndash4

35

times10

ndash400

00

(b)

Contact load of a single asperity (fN)

n = 40

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

4 times 10ndash17

3 times 10ndash17

2 times 10ndash17

1 times 10ndash17

0

20

times10

ndash8

40

times10

ndash8

60

times10

ndash8

80

times10

ndash8

10

times10

ndash700

(c)

Figure 8 e relationship between contact load and contact area of a single asperity during loading

Advances in Materials Science and Engineering 11

the deformation characteristics of the asperity underdifferent frequency exponents are obtained

(2) e normal contact load of a single asperity in theelastoplastic stage is related to the hardness of thematerial When the material yields the hardness H isnot a constant but a function related to the amountof deformation In this paper the limit mean geo-metric hardness is introduced to express the normalcontact load of a single asperity in the first andsecond elastoplastic deformation stages consideringthe change of hardness

(3) e relationship between the contact load andcontact area of a single asperity in the first andsecond elastoplastic deformation stages considering

the change of material hardness and not consideringthe change of hardness is compared respectivelyWhen the contact area of the asperity is the same thecontact load with the change of hardness is smallerthan that without the change of hardness and thedifference between them increases with the increaseof deformation

(4) e limit mean geometric hardness of a single as-perity is related to the contact area fractal di-mension and frequency exponent during theelastoplastic deformation stage and the limit meangeometric hardness increases with the increase of thecontact area When the frequency exponent isconstant the relationship between the limit meangeometric hardness and the contact area of the as-perity is related to the fractal dimension e largerthe fractal dimension is the more obvious the re-lationship curve between them is When the fractaldimension is constant the relationship between thelimit mean geometric hardness of the asperity andthe contact area is related to the frequency exponente smaller the frequency exponent is the moreobvious the change of the curve is

(5) e relationship between dimensionless total realcontact area and the dimensionless total contactloading of joint interface at each deformation stageis obtained by introducing the pushing coefficientand the critical pushing coefficient exists Whenthe pushing coefficient exceeds this value someasperities begin to deform in the next stage andthe inflection point appears on the relationshipcurve

(6) e current model still has some limitations on theapplicable scope of materials and further researchand improvement are needed to make it more ap-plicable in the future

09

08

07

06

05

04

03

02

01

000000 0002 0004 0006 0008 0010 0012 0014

Nondimensional total contact load (Flowastr)

Non

dim

ensio

nal t

otal

cont

act a

rea (

Alowastr)

nmin = 20 nmax = 36

Figure 10 e relationship between dimensionless total realcontact area and the dimensionless total contact loading for20le nle 36

10

09

08

07

06

05

04

03

02

01Non

dim

ensio

nal t

otal

cont

act a

rea (

Alowastr)

000000 0005 0010 0015 0020 0025 0030 0035

Nondimensional total contact load (Flowastr)

nmin = 44 nmax = 50

Figure 11 e relationship between dimensionless total realcontact area and the dimensionless total contact loading for44le nle 50

10

09

08

07

06

05

04

03

02

01

Nondimensional total contact load (Flowastr1)

Non

dim

ensio

nal t

otal

real

cont

act a

rea (

Alowastr1

)

0000 10 times 10ndash3 20 times 10ndash3 30 times 10ndash3 40 times 10ndash3 50 times 10ndash3

nmin = 20 nmax = 32

Figure 9e relationship between dimensionless total real contactarea and the dimensionless total contact loading for 20le nle 32

12 Advances in Materials Science and Engineering

Nomenclature

ωn Interference of the asperityωnec Elastic critical interference of the asperityωnepc First elastoplastic critical interference of the

asperityωnpc Second elastoplastic critical interference of the

asperityan Contact area of the asperityanec Elastic critical contact area of the asperityanepc First elastoplastic critical contact area of the

asperityanpc Second elastoplastic critical contact area of the

asperityfne Normal load in the elastic deformation of a single

asperityfnec Normal contact load for ω ωnecfnep1 Normal contact load of a single asperity in the

first elastoplastic stagefnep2 Normal contact load of a single asperity in the

second elastoplastic stagefnp Normal contact load of a single asperity in the full

plastic deformation stagefnep1prime Normal contact load of a single asperity in the

first elastoplastic stage considering the change ofhardness

fnep2prime Normal contact load of a single asperity in thesecond elastoplastic stage considering the changeof hardness

HG1(an) Limit mean geometric hardness in the firstelastoplastic deformation stage

HG2(an) Limit mean geometric hardness in the secondelastoplastic deformation stage

nec Elastic critical frequency exponentnepc First elastoplastic critical frequency exponentnpc Second elastoplastic critical frequency exponentAr Actual contact area of the joint interfaceAr1 Actual contact area of the joint interface for

nmin lt nle necAr2 Actual contact area of the joint interface for

nec lt nle nepcAr3 Actual contact area of the joint interface for

nepc lt nle npcAr4 Actual contact area of the joint interface for

npc lt n

Fr1 Actual contact load of the joint interface fornmin lt nle nec

Fr2 Actual contact load of the joint interface fornec lt nle nepc

Fr3 Actual contact load of the joint interface fornepc lt nle npc

Fr4 Actual contact load of the joint interface fornpc lt n

pea(a) Average contact pressure of the asperity in elasticstage

pepa1(a) Average contact pressure of the asperity in thefirst elastoplastic deformation stage

pepa2(a) Average contact pressure of the asperity in thesecond elastoplastic stage

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (grant no 51275328) and the NaturalScience Foundation Project of Shanxi Province (grant no201601D011062)

References

[1] B Zhao S Zhang P Wang and Y Hai ldquoLoading-unloadingnormal stiffness model for power-law hardening surfacesconsidering actual surface topographyrdquo Tribology In-ternational vol 90 pp 332ndash342 2015

[2] J Greenwood and J Williamson ldquoContact of nominally flatsurfacesrdquo Mathematical Physical and Engineering Sciencesvol 295 no 1442 pp 299ndash319 1966

[3] J A Greenwood and J H Tripp ldquoe contact of twonominally flat rough surfacesrdquo Proceedings of the Institution ofMechanical Engineers vol 185 no 1 pp 625ndash633 1970

[4] W R Chang I Etsion and D B Bogy ldquoAn elastic-plasticmodel for the contact of rough surfacesrdquo Journal of Tribologyvol 109 no 2 pp 257ndash263 1987

[5] Y Zhao D M Maietta and L Chang ldquoAn asperity micro-contact model incorporating the transition from elastic de-formation to fully plastic flowrdquo Journal of Tribology vol 122no 1 pp 86ndash93 2000

[6] L Kogut and I Etsion ldquoElastic-plastic contact analysis ofasphere and a rigid flatrdquo Journal of Applied Mechanics vol 69no 5 pp 657ndash662 2002

[7] I Etsion Y Kligerman and Y Kadin ldquoUnloading of anelastic-plastic loaded spherical contactrdquo International Journalof Solids and Structures vol 42 no 13 pp 3716ndash3729 2005

[8] Y Kadin Y Kligerman and I Etsion ldquoUnloading an elastic-plastic contact of rough surfacesrdquo Journal of the Mechanicsand Physics of Solids vol 54 no 12 pp 2652ndash2674 2006

[9] Y Kadin Y Kligerman and I Etsion ldquoMultiple loading-unloading of an elastic-plastic spherical contactrdquo In-ternational Journal of Solids and Structures vol 43 no 22-23pp 7119ndash7127 2006

[10] A Majumdar and B Bhushan ldquoRole of fractal geometry inroughness characterization and contact mechanics of sur-facesrdquo Journal of Tribology vol 112 no 2 pp 205ndash216 1990

[11] A Majumdar and C L Tien ldquoFractal characterization andsimulation of rough surfacesrdquo Wear vol 136 no 2pp 313ndash327 1990

[12] AMajumdar and B Bhushan ldquoFractal model of elastic-plasticcontact between rough surfacesrdquo Journal of Tribology vol 113no 1 pp 1ndash11 1991

[13] S Wang and K Komvopoulos ldquoA fractal theory of the in-terfacial temperature distribution in the slow sliding regimePart I-elastic contact and heat transfer analysisrdquo Journal ofTribology vol 116 no 4 pp 812ndash822 1994

[14] S Wang and K Komvopoulos ldquoA fractal theory of the in-terfacial temperature distribution in the slow sliding regime

Advances in Materials Science and Engineering 13

Part II-multiple domains elastoplastic contacts and appli-cationsrdquo Journal of Tribology vol 116 no 4 pp 824ndash8321994

[15] Y Morag and I Etsion ldquoResolving the contradiction of as-perities plastic to elastic mode transition in current contactmodels of fractal rough surfacesrdquo Wear vol 262 no 5-6pp 624ndash629 2007

[16] H Tian X Zhong and C Zhao ldquoOne loading model of jointinterface considering elastoplastic and variation of hardnesswith surface depthrdquo Journal of Mechanical Engineeringvol 51 no 5 pp 90ndash104 2015

[17] Y Yuan Y Cheng K Liu et al ldquoA revised Majumdar andBushan model of elastoplastic contact between rough sur-facesrdquo Applied Surface Science vol 425 pp 1138ndash1157 2017

[18] R S Sayles and T R omas ldquoSurface topography as anonstationary random processrdquo Nature vol 271 no 5644pp 431ndash434 1978

14 Advances in Materials Science and Engineering

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Ar2 Are + Arep1 (41)

Are 1113944

nepc

nnec+11113946

anec

0Mn(a)ada

MD

2minusD1113944

nepc

nnec+1a

(2minusD)2nec a

D2nl

(42)

For the determined frequency exponent the maximumactual contact area of the asperity appears at the maximumdeformation amount ωn where the maximum value of theelastic deformation phase ωn appears at ωnec whereuponformula (42) is simplified to

Are MD

2minusD1113944

nepc

nnec+1anec

MD

(2minusD)π1113944

nepc

nnec+1

3KHcminusDn

4G(Dminus 1)Eprime1113888 1113889

2

Arep1 1113944

nepc

nnec+11113946

anl

anec

Mn(a)ada MD

2minusD

middot 1113944

nepc

nnec+1a

(2minusD)2nl minus a

(2minusD)2nec1113960 1113961a

D2nl

(43)

e contact load is given by

Fr2 Fre + Frep1 (44)

Fre 1113944

nepc

nnec+11113946

anec

0fneMn(a)da

9MD(KH)3

16(3minusD) EprimeπGDminus1( 11138572

middot 1113944

nepc

nnec+1cminus2 Dn

(45)

Frep1 1113944

nepc

nnec+11113946

anl

anec

fnep1prime Mn(a)da (46)

Substituting equations (24) and (36) in equation (46) wecan obtain

Frep1 28KYMD

2c2 minusD + 21113944

nepc

nnec+1aminusc2neca

c2+1nl minus a

(2minusD)2nec a

D2nl1113960 1113961 (47)

33 When the Frequency Exponent Belongs to nepc lt nle npcWhen the frequency exponent belongs to nepc lt nle npc forthe case anepc lt anl le anpc elastic deformation the firstelastoplastic deformation or the second elastoplastic de-formation may take place in these asperities At this point

the actual contact area of the joint interface consists ofthree parts the elastic deformation stage the first elas-toplastic deformation stage and the second elastoplasticdeformation stage

Ar3 Are + Arep1 + Arep2

Are 1113944

npc

nnepc+11113946

anec

0Mn(a)ada

MD

(2minusD)π

middot 1113944

npc

nnepc+1

3KHcminusDn

4GDminus1Eprime1113888 1113889

2

Arep1 1113944

npc

nnepc+11113946

anepc

anec

Mn(a)ada

MD

π(2minusD)71197minus 71197D2

1113872 1113873 1113944

npc

nnepc+1

3KHcminusDn

4GDminus1Eprime1113888 1113889

2

Arep2 1113944

npc

nnepc+11113946

anl

anepc

Mn(a)ada

MD

(2minusD)1113944

npc

nnepc+1a

(2minusD)2nl minus 71197anec( 1113857

(2minusD)21113960 1113961a

D2nl

(48)

In this case the contact load of the joint interface is asfollows

Fr3 Fre + Frep1 + Frep2

Fre 1113944

npc

nnepc+11113946

anec

0fneMn(a)da

9MD(KH)3

16(3minusD) EprimeπGDminus1( 11138572 1113944

npc

nnepc+1cminus2 Dn

Frep1 28KYMD

2c2 minusD + 271197c2+1 minus 71197D2

1113872 1113873

middot 1113944

npc

nnepc+1

1π

3KHcminusDn

4GDminus 1Eprime1113888 1113889

2

(49)

When the second elastoplastic deformation occurs thenormal contact load of the joint interface is as follows

6 Advances in Materials Science and Engineering

Frep2 1113944

npc

nnepc+11113946

anl

anepc

fnep2prime Mn(a)da (50)

Substituting equations (31) and (38) in equation (46) wecan obtain

Frep2 2106 times 7119702544minusc4KYMD

2c4 minusD + 2

middot 1113944

npc

nnepc+1aminusc4neca

c4+1nl minus 71197c4minus(D2)+1

a1minus(D2)nec a

(D2)nl1113872 1113873

(51)

34 When the Frequency Exponent Belongs to npc lt nWhen the frequency exponent belongs to npc lt n elasticdeformation elastoplastic deformation or full plastic de-formation may take place in these asperities e actualcontact area of the joint interface can be evaluated as

Ar4 Are + Arep1 + Arep2 + Arp

Are 1113944

nmax

nnpc+11113946

anec

0Mn(a)ada

MD

2minusD1113944

nmax

nnpc+1anec

Arep1 1113944

nmax

nnpc+11113946

anepc

anec

Mn(a)ada

MD

2minusD71197minus 71197D2

1113872 1113873 1113944

nmax

nnpc+1anec

Arep2 1113944

nmax

nnpc+11113946

anpc

anepc

Mn(a)ada

MD

2minusD2053827minus 711971minus(D2)

middot 2053827D21113872 1113873 1113944

nmax

nnpc+1anec

Arp 1113944

nmax

nnpc+11113946

anl

anpc

Mn(a)ada

MD

2minusD1113944

nmax

nnpc+1a

(2minusD)2nl minus 2053827anec( 1113857

(2minusD)21113960 1113961a

D2nl

(52)

In this case the contact load of the joint interface is asfollows

Fr4 Fre + Frep1 + Frep2 + Frp

Fre 1113944

nmax

nnpc+11113946

anec

0fneMn(a)da

MDKH

(3minusD)π1113944

nmax

nnpc+1anec

Frep1 1113944

nmax

nnpc+11113946

anepc

anec

fnep1prime Mn(a)da

28KYMD

2c2 minusD + 271197c2+1 minus 71197D2

1113872 1113873 1113944

nmax

nnpc+1anec

Frep2 2106 times 7119702544minusc4KYMD

2c4 minusD + 2

middot 1113944

nmax

nnpc+1aminusc4neca

c4+1nl minus 71197c4minus(D2)+1

a1minus(D2)nec a

D2nl1113872 1113873

2106 times 7119702544minusc4KYMD

2c4 minusD + 2

middot 2053827c4+1 minus 2053827D2middot 71197c4minus(D2)+1

1113872 1113873

middot 1113944

nmax

nnpc+1anec

Frp 1113944

nmax

nnpc+11113946

anl

anpc

fnpMn(a)da

MHD

2minusD1113944

nmax

nnpc+1a

D2nl a

1minus(D2)nl minus 2053827anec( 1113857

1minus(D2)1113960 1113961

(53)

For all frequency exponents the total actual contact areaof the joint interface is

Ar Ar1 + Ar2 + Ar3 + Ar4 (54)

e total contact load of the joint interface is

Fr Fr1 + Fr2 + Fr3 + Fr4 (55)

e total real contact area and the total contact load in anondimensional form can be written as follows

Alowastr

Ar

Aa

Flowastr

Fr

AaE

(56)

where Aa is the nominal contact area and is given byAa L2L 1cnmin

Advances in Materials Science and Engineering 7

4 Results Analysis

In order to further analyze the above calculation results theparameters of equivalent joint interface are taken as shownin Table 1 [16]

Figure 2 shows the relation between all critical contactareas and frequency exponents of single asperity whenD 15 It can be seen from the figure that as for one definiteasperity when frequency exponent n is certain elastic criticalcontact area is minimum followed by the first elastoplasticcritical contact area and the second elastoplastic criticalcontact area is maximum With gradual increase of contactload the contact area increases e single asperity is firstlysubject to elastic deformation followed by the first elasto-plastic deformation the second elastoplastic deformation andfully plastic deformation successively which is consistent withtypical contact mechanics theory As for different asperitieswith increase of frequency exponent all critical contact areasdecrease correspondingly which shows that elastic criticalcontact area the first elastoplastic critical contact area and thesecond elastoplastic critical contact area are all related tofrequency exponent n

Figure 3 shows the relation curve between fractal di-mension D and critical frequency exponent n of asperityWhen fractal dimension is definite elastic critical frequencyexponent nec the first elastoplastic critical frequency ex-ponent nepc and the second elastoplastic critical frequencyexponent npc increase gradually As shown in Figure 3 whenDlt 106 nec nepc and npc are all negative As for asperitieswith minimum value of frequency exponent being 0 elasticdeformation elastoplastic deformation and fully plasticdeformation will all occur When D 113 nec and nepc arenegative and npc is positive At this time as for asperitieswith minimum value of frequency exponent being 0 elasticdeformation elastoplastic deformation will occur exceptfully plastic deformation

For D 15 G 25 times 10minus9 m H 55 times 109 Nm2 wecan obtain the elastic critical frequency exponent nec 32the first elastoplastic critical frequency exponent nepc 36and the second elastoplastic critical frequency exponentnpc 43 ese asperities whose frequency exponents rangefrom 20 to 32 are only under elastic deformation Elasticdeformation and the first elastoplastic deformation canoccur in these asperities whose frequency exponents rangefrom 33 to 36 Elastic deformation the first elastoplasticdeformation and the second elastoplastic deformation canoccur in these asperities whose frequency exponents rangefrom 37 to 43 When frequency exponents range from 43 to50 all deformations types can occur in these asperities

Figure 4 shows relation comparison diagram betweencontact load and contact area of single asperity with andwithout hardness change at the first elastoplastic stage ecomparison diagram is simulation result when n 33 It canbe seen from the figure that with gradual increase of contactarea with contact area of single asperity over 32 times 10minus13 m2contact load of the same asperity with hardness change willbe less than that without hardness change In addition as theamount of deformation increases the difference betweenthem tends to increase

Figure 5 shows relation comparison diagram betweencontact load and contact area of single asperity with andwithout hardness change at the second elastoplastic stagee comparison diagram is simulation result when n 37 Itcan be seen from the figure that when the deformation is

Table 1 e parameters of equivalent joint interface

Parameters ValuesEquivalent elastic modulus Eprime 72 times 1010 Nm2

Poissonrsquos ratio υ 017Initial hardness H 55 times 109 Nm2

Profile scale parameter G 25 times 10minus9 mFractal dimension D 1ltDlt 2Frequency exponent n 20sim50

20 25 30 35 40 45 5010ndash1510ndash1410ndash1310ndash1210ndash1110ndash1010ndash910ndash810ndash710ndash610ndash510ndash410ndash3

Criti

cal c

onta

ct ar

eas

of a

singl

e asp

erity

(am

2 )

Asperity levels (n)

Second elastoplastic critical contact areaFirst elastoplastic critical contact areaElastic critical contact area

Figure 2 e relationship between critical contact area and fre-quency exponent of a single asperity

10 11 12 13 14 15 16 17 18 19 20ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

0

20

40

Asp

erity

freq

uenc

y ex

pone

nts (n)

e fractal dimension (D)

Elastic critical frequency exponentsFirst elastoplastic critical frequency exponentsSecond elastoplastic critical frequency exponents

Figure 3 e relationship between fractal dimension D andcritical frequency exponent n of a single asperity

8 Advances in Materials Science and Engineering

definite contact load of the same asperity with hardnesschange will be less than that without hardness change Inaddition with increase of deformation amount the differ-ence between them tends to increase which is consistentwith the change trend at the first elastoplastic stage

Figure 6 shows the relation between limit mean geo-metric hardness and contact area (logarithm) of single as-perity at the first elastoplastic deformation stage Figure 6(a)shows relation curve that D 11 13 15 17 when n 34Figure 6(b) shows relation curve that n 32 33 34 35 whenD 15 It can be seen from Figure 6 that the limit meangeometric hardness of single asperity is related to contactarea fractal dimension and frequency exponent in the firstelastoplastic deformation stage e limit mean geometrichardness increases with increase of contact area When n is

definite the relation between limit mean geometric hardnessand contact area of asperity is related to fractal dimension De larger the D is the more obvious the relation curvebetween them changes when D is definite the relationbetween limit mean geometric hardness and contact area ofasperity is related to frequency exponent n e smaller n isthe more obvious the relation curve between them changes

Figure 7 shows the relation between limit mean geo-metric hardness and contact area (logarithm) of single as-perity at the second elastoplastic deformation stageFigure 7(a) shows the relation curve n 40 that D 11 1315 17 when n 40 Figure 7(b) shows the relation curvethat n 36 38 40 42 when D 15

During loading the contact area increases with the in-crease of deformation of a single asperity e ratio of de-formation to the natural height of the asperity is defined asthe pushing coefficient namely the pushing coefficientk ωnhn 0le kle 09 When fractal dimension is 15 we willresearch the relation between contact load and contact areaof single asperity with frequency exponent n being 30 35and 40 respectively during loading

When n 30 the asperity will only be subject to elasticdeformation During loading even the pushing coefficient k

is maximum no plastic deformation will occur e relationbetween contact area and contact load is fsima15 approxi-mately as shown in Figure 8(a)

As is shown in Figure 8(b) when n 35 elastic de-formation and the first elastoplastic deformation may takeplace in the asperity during loading When the pushingcoefficient k is less than 0247 the asperity will under elasticdeformation At this time the relation between contact areaand contact load is fsima15 approximately when the pushingcoefficient is over 0247 the first elastoplastic deformationoccurs At this time the relation between contact area andcontact load is fsima11093 approximately As is shown inFigure 8(c) when n 40 elastic deformation the firstelastoplastic deformation and the second elastoplastic

0000 0001 0002 0003 0004 0005Contact load of a single asperity in the

first elastoplastic deformation regime (fN)

00

12 times 10ndash12

10 times 10ndash12

80 times 10ndash13

60 times 10ndash13

40 times 10ndash13

20 times 10ndash13

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

Change of hardness is consideredChange of hardness is not considered

Figure 4 e relationship between contact load and contact area of single asperity in the first elastoplastic deformation stage

0000 0001 0002 0003 0004 00050

2 times 10ndash13

4 times 10ndash13

6 times 10ndash13

8 times 10ndash13

1 times 10ndash12

1 times 10ndash12

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

Contact load of a single asperity in the second elastoplastic deformation regime (f2N)

Change of hardness is not consideredChange of hardness is considered

Figure 5e relationship between contact load and contact area ofsingle asperity in the second elastoplastic deformation stage

Advances in Materials Science and Engineering 9

deformation may take place in the asperity during loadingWhen the pushing coefficient is greater than 01954 theasperity begins to enter the second elastoplastic de-formation the relation between contact area and contactload is fsima10977 approximately When n 45 and thepushing coefficient is greater than 0472 the asperity beginsto enter fully plastic deformation the relation betweencontact area and contact load is fsima approximately

Figure 9 shows that when the minimum frequency ex-ponent is 20 and the maximum value is 32 the actual contactarea of the joint interface increases with the increase of thetotal contact load and the relation between them isFlowastr simAlowast15

r approximately During the whole deformationprocess the joint interface appears to be of elastic property

Figure 10 shows the relation between dimensionless totalreal contact area and the dimensionless total contact loading

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash9

10ndash15

10ndash14

10ndash13

10ndash12

10ndash11

10ndash10

D = 11 n = 34D = 13 n = 34

D = 15 n = 34D = 17 n = 34

25 times 109 30 times 109 35 times 109 40 times 109 45 times 109 50 times 109 55 times 109

Limit mean geometric hardness in the firstelastoplastic deformation regime HG1(Nm2)

(a)C

onta

ct ar

eas o

f a si

ngle

aspe

rity

(am

2 )

10ndash10

10ndash15

10ndash14

10ndash13

10ndash12

10ndash11

n = 32 D = 15n = 33 D = 15

n = 34 D = 15n = 35 D = 15

25 times 109 30 times 109 35 times 109 40 times 109 45 times 109 50 times 109 55 times 109

Limit mean geometric hardness in the firstelastoplastic deformation regime HG1(Nm2)

(b)

Figure 6 e relationship between limit mean geometric hardness and contact for single asperity during the first elastoplastic deformationstage (a) n 34 11leDle 17 (b) D 15 32le nle 35

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash12

10ndash18

10ndash17

10ndash16

10ndash15

10ndash14

10ndash13

D = 11 n = 40D = 13 n = 40

D = 15 n = 40D = 17 n = 40

50 times 109 55 times 109 60 times 109 65 times 109 70 times 109 75 times 109

Limit mean geometric hardness in the secondelastoplastic deformation regime HG2(Nm2)

(a)

50 times 109 55 times 109 60 times 109 65 times 109 70 times 109 75 times 109

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash10

10ndash11

10ndash12

10ndash13

10ndash18

10ndash17

10ndash16

10ndash15

10ndash14

n = 36 D = 15n = 38 D = 15

n = 40 D = 15n = 42 D = 15

Limit mean geometric hardness in the secondelastoplastic deformation regime HG2(Nm2)

(b)

Figure 7 e relationship between limit mean geometric hardness and contact for single asperity during the second elastoplastic de-formation stage (a) n 40 11leDle 17 (b) D 15 36le nle 42

10 Advances in Materials Science and Engineering

of the joint interface when minimum value and maximumvalue of frequency exponent of asperities are respectively 20and 36When the pushing coefficient k is 01648 the asperitybegins to be subject to the first elastoplastic deformation andthe relation between dimensionless total real contact areaand the dimensionless total contact loading is FlowastrsimAlowast15

r

approximately e joint interface appears to be of elasticproperty As the load continues increasing when thepushing coefficient k is 05564 at this pointFlowastr gt 69023 times 10minus3 the relation between above load and areais approximately FlowastrsimAlowast11093

r e joint interface presentselastoplastic properties At this time the first elastoplasticdeformation takes place in these asperities whose frequencyexponents range 33sim36

Figure 11 shows the relation between dimensionless totalreal contact area and the dimensionless total contact loadingof the joint interface when minimum value and maximumvalue of frequency exponent of asperities are respectively 44and 50When the pushing coefficient k is 00621 the asperity

begins to be subject to the second elastoplastic deformationat this point 00016ltFlowastr lt 0023 and the relation betweendimensionless total real contact area and the dimensionlesstotal contact loading is FlowastrsimAlowast10977

r approximately Whenthe pushing coefficient k is 07076 the asperity begins to besubject to the fully plastic deformation at this pointFlowastr gt 0023 and the relation between dimensionless total realcontact area and the dimensionless total contact loading isapproximately FlowastrsimAlowastr

5 Conclusions

(1) e contact mechanical properties of a single as-perity on a joint interface are related to the frequencyexponent of the asperity while the frequency ex-ponent of an asperity is related to the fractal di-mension and the profile scale parameter In thispaper the critical frequency exponents of each de-formation stage of a single asperity are obtained and

000 005 010 015 020 025 030 03500

10 times 10ndash11

20 times 10ndash11

30 times 10ndash11

40 times 10ndash11

50 times 10ndash11

60 times 10ndash11

Contact load of a single asperity (fN)

n = 30

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

(a)

Contact load of a single asperity (fN)

n = 35

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

50

times10

ndash5

10

times10

ndash4

104 times 10ndash4

10 times 10ndash13

80 times 10ndash14

60 times 10ndash14

40 times 10ndash14

20 times 10ndash14

363 times 10ndash14

15

times10

ndash4

20

times10

ndash4

25

times10

ndash4

30

times10

ndash4

35

times10

ndash400

00

(b)

Contact load of a single asperity (fN)

n = 40

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

4 times 10ndash17

3 times 10ndash17

2 times 10ndash17

1 times 10ndash17

0

20

times10

ndash8

40

times10

ndash8

60

times10

ndash8

80

times10

ndash8

10

times10

ndash700

(c)

Figure 8 e relationship between contact load and contact area of a single asperity during loading

Advances in Materials Science and Engineering 11

the deformation characteristics of the asperity underdifferent frequency exponents are obtained

(2) e normal contact load of a single asperity in theelastoplastic stage is related to the hardness of thematerial When the material yields the hardness H isnot a constant but a function related to the amountof deformation In this paper the limit mean geo-metric hardness is introduced to express the normalcontact load of a single asperity in the first andsecond elastoplastic deformation stages consideringthe change of hardness

(3) e relationship between the contact load andcontact area of a single asperity in the first andsecond elastoplastic deformation stages considering

the change of material hardness and not consideringthe change of hardness is compared respectivelyWhen the contact area of the asperity is the same thecontact load with the change of hardness is smallerthan that without the change of hardness and thedifference between them increases with the increaseof deformation

(4) e limit mean geometric hardness of a single as-perity is related to the contact area fractal di-mension and frequency exponent during theelastoplastic deformation stage and the limit meangeometric hardness increases with the increase of thecontact area When the frequency exponent isconstant the relationship between the limit meangeometric hardness and the contact area of the as-perity is related to the fractal dimension e largerthe fractal dimension is the more obvious the re-lationship curve between them is When the fractaldimension is constant the relationship between thelimit mean geometric hardness of the asperity andthe contact area is related to the frequency exponente smaller the frequency exponent is the moreobvious the change of the curve is

(5) e relationship between dimensionless total realcontact area and the dimensionless total contactloading of joint interface at each deformation stageis obtained by introducing the pushing coefficientand the critical pushing coefficient exists Whenthe pushing coefficient exceeds this value someasperities begin to deform in the next stage andthe inflection point appears on the relationshipcurve

(6) e current model still has some limitations on theapplicable scope of materials and further researchand improvement are needed to make it more ap-plicable in the future

09

08

07

06

05

04

03

02

01

000000 0002 0004 0006 0008 0010 0012 0014

Nondimensional total contact load (Flowastr)

Non

dim

ensio

nal t

otal

cont

act a

rea (

Alowastr)

nmin = 20 nmax = 36

Figure 10 e relationship between dimensionless total realcontact area and the dimensionless total contact loading for20le nle 36

10

09

08

07

06

05

04

03

02

01Non

dim

ensio

nal t

otal

cont

act a

rea (

Alowastr)

000000 0005 0010 0015 0020 0025 0030 0035

Nondimensional total contact load (Flowastr)

nmin = 44 nmax = 50

Figure 11 e relationship between dimensionless total realcontact area and the dimensionless total contact loading for44le nle 50

10

09

08

07

06

05

04

03

02

01

Nondimensional total contact load (Flowastr1)

Non

dim

ensio

nal t

otal

real

cont

act a

rea (

Alowastr1

)

0000 10 times 10ndash3 20 times 10ndash3 30 times 10ndash3 40 times 10ndash3 50 times 10ndash3

nmin = 20 nmax = 32

Figure 9e relationship between dimensionless total real contactarea and the dimensionless total contact loading for 20le nle 32

12 Advances in Materials Science and Engineering

Nomenclature

ωn Interference of the asperityωnec Elastic critical interference of the asperityωnepc First elastoplastic critical interference of the

asperityωnpc Second elastoplastic critical interference of the

asperityan Contact area of the asperityanec Elastic critical contact area of the asperityanepc First elastoplastic critical contact area of the

asperityanpc Second elastoplastic critical contact area of the

asperityfne Normal load in the elastic deformation of a single

asperityfnec Normal contact load for ω ωnecfnep1 Normal contact load of a single asperity in the

first elastoplastic stagefnep2 Normal contact load of a single asperity in the

second elastoplastic stagefnp Normal contact load of a single asperity in the full

plastic deformation stagefnep1prime Normal contact load of a single asperity in the

first elastoplastic stage considering the change ofhardness

fnep2prime Normal contact load of a single asperity in thesecond elastoplastic stage considering the changeof hardness

HG1(an) Limit mean geometric hardness in the firstelastoplastic deformation stage

HG2(an) Limit mean geometric hardness in the secondelastoplastic deformation stage

nec Elastic critical frequency exponentnepc First elastoplastic critical frequency exponentnpc Second elastoplastic critical frequency exponentAr Actual contact area of the joint interfaceAr1 Actual contact area of the joint interface for

nmin lt nle necAr2 Actual contact area of the joint interface for

nec lt nle nepcAr3 Actual contact area of the joint interface for

nepc lt nle npcAr4 Actual contact area of the joint interface for

npc lt n

Fr1 Actual contact load of the joint interface fornmin lt nle nec

Fr2 Actual contact load of the joint interface fornec lt nle nepc

Fr3 Actual contact load of the joint interface fornepc lt nle npc

Fr4 Actual contact load of the joint interface fornpc lt n

pea(a) Average contact pressure of the asperity in elasticstage

pepa1(a) Average contact pressure of the asperity in thefirst elastoplastic deformation stage

pepa2(a) Average contact pressure of the asperity in thesecond elastoplastic stage

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (grant no 51275328) and the NaturalScience Foundation Project of Shanxi Province (grant no201601D011062)

References

[1] B Zhao S Zhang P Wang and Y Hai ldquoLoading-unloadingnormal stiffness model for power-law hardening surfacesconsidering actual surface topographyrdquo Tribology In-ternational vol 90 pp 332ndash342 2015

[2] J Greenwood and J Williamson ldquoContact of nominally flatsurfacesrdquo Mathematical Physical and Engineering Sciencesvol 295 no 1442 pp 299ndash319 1966

[3] J A Greenwood and J H Tripp ldquoe contact of twonominally flat rough surfacesrdquo Proceedings of the Institution ofMechanical Engineers vol 185 no 1 pp 625ndash633 1970

[4] W R Chang I Etsion and D B Bogy ldquoAn elastic-plasticmodel for the contact of rough surfacesrdquo Journal of Tribologyvol 109 no 2 pp 257ndash263 1987

[5] Y Zhao D M Maietta and L Chang ldquoAn asperity micro-contact model incorporating the transition from elastic de-formation to fully plastic flowrdquo Journal of Tribology vol 122no 1 pp 86ndash93 2000

[6] L Kogut and I Etsion ldquoElastic-plastic contact analysis ofasphere and a rigid flatrdquo Journal of Applied Mechanics vol 69no 5 pp 657ndash662 2002

[7] I Etsion Y Kligerman and Y Kadin ldquoUnloading of anelastic-plastic loaded spherical contactrdquo International Journalof Solids and Structures vol 42 no 13 pp 3716ndash3729 2005

[8] Y Kadin Y Kligerman and I Etsion ldquoUnloading an elastic-plastic contact of rough surfacesrdquo Journal of the Mechanicsand Physics of Solids vol 54 no 12 pp 2652ndash2674 2006

[9] Y Kadin Y Kligerman and I Etsion ldquoMultiple loading-unloading of an elastic-plastic spherical contactrdquo In-ternational Journal of Solids and Structures vol 43 no 22-23pp 7119ndash7127 2006

[10] A Majumdar and B Bhushan ldquoRole of fractal geometry inroughness characterization and contact mechanics of sur-facesrdquo Journal of Tribology vol 112 no 2 pp 205ndash216 1990

[11] A Majumdar and C L Tien ldquoFractal characterization andsimulation of rough surfacesrdquo Wear vol 136 no 2pp 313ndash327 1990

[12] AMajumdar and B Bhushan ldquoFractal model of elastic-plasticcontact between rough surfacesrdquo Journal of Tribology vol 113no 1 pp 1ndash11 1991

[13] S Wang and K Komvopoulos ldquoA fractal theory of the in-terfacial temperature distribution in the slow sliding regimePart I-elastic contact and heat transfer analysisrdquo Journal ofTribology vol 116 no 4 pp 812ndash822 1994

[14] S Wang and K Komvopoulos ldquoA fractal theory of the in-terfacial temperature distribution in the slow sliding regime

Advances in Materials Science and Engineering 13

Part II-multiple domains elastoplastic contacts and appli-cationsrdquo Journal of Tribology vol 116 no 4 pp 824ndash8321994

[15] Y Morag and I Etsion ldquoResolving the contradiction of as-perities plastic to elastic mode transition in current contactmodels of fractal rough surfacesrdquo Wear vol 262 no 5-6pp 624ndash629 2007

[16] H Tian X Zhong and C Zhao ldquoOne loading model of jointinterface considering elastoplastic and variation of hardnesswith surface depthrdquo Journal of Mechanical Engineeringvol 51 no 5 pp 90ndash104 2015

[17] Y Yuan Y Cheng K Liu et al ldquoA revised Majumdar andBushan model of elastoplastic contact between rough sur-facesrdquo Applied Surface Science vol 425 pp 1138ndash1157 2017

[18] R S Sayles and T R omas ldquoSurface topography as anonstationary random processrdquo Nature vol 271 no 5644pp 431ndash434 1978

14 Advances in Materials Science and Engineering

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

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Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

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ate

ria

ls

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Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

Frep2 1113944

npc

nnepc+11113946

anl

anepc

fnep2prime Mn(a)da (50)

Substituting equations (31) and (38) in equation (46) wecan obtain

Frep2 2106 times 7119702544minusc4KYMD

2c4 minusD + 2

middot 1113944

npc

nnepc+1aminusc4neca

c4+1nl minus 71197c4minus(D2)+1

a1minus(D2)nec a

(D2)nl1113872 1113873

(51)

34 When the Frequency Exponent Belongs to npc lt nWhen the frequency exponent belongs to npc lt n elasticdeformation elastoplastic deformation or full plastic de-formation may take place in these asperities e actualcontact area of the joint interface can be evaluated as

Ar4 Are + Arep1 + Arep2 + Arp

Are 1113944

nmax

nnpc+11113946

anec

0Mn(a)ada

MD

2minusD1113944

nmax

nnpc+1anec

Arep1 1113944

nmax

nnpc+11113946

anepc

anec

Mn(a)ada

MD

2minusD71197minus 71197D2

1113872 1113873 1113944

nmax

nnpc+1anec

Arep2 1113944

nmax

nnpc+11113946

anpc

anepc

Mn(a)ada

MD

2minusD2053827minus 711971minus(D2)

middot 2053827D21113872 1113873 1113944

nmax

nnpc+1anec

Arp 1113944

nmax

nnpc+11113946

anl

anpc

Mn(a)ada

MD

2minusD1113944

nmax

nnpc+1a

(2minusD)2nl minus 2053827anec( 1113857

(2minusD)21113960 1113961a

D2nl

(52)

In this case the contact load of the joint interface is asfollows

Fr4 Fre + Frep1 + Frep2 + Frp

Fre 1113944

nmax

nnpc+11113946

anec

0fneMn(a)da

MDKH

(3minusD)π1113944

nmax

nnpc+1anec

Frep1 1113944

nmax

nnpc+11113946

anepc

anec

fnep1prime Mn(a)da

28KYMD

2c2 minusD + 271197c2+1 minus 71197D2

1113872 1113873 1113944

nmax

nnpc+1anec

Frep2 2106 times 7119702544minusc4KYMD

2c4 minusD + 2

middot 1113944

nmax

nnpc+1aminusc4neca

c4+1nl minus 71197c4minus(D2)+1

a1minus(D2)nec a

D2nl1113872 1113873

2106 times 7119702544minusc4KYMD

2c4 minusD + 2

middot 2053827c4+1 minus 2053827D2middot 71197c4minus(D2)+1

1113872 1113873

middot 1113944

nmax

nnpc+1anec

Frp 1113944

nmax

nnpc+11113946

anl

anpc

fnpMn(a)da

MHD

2minusD1113944

nmax

nnpc+1a

D2nl a

1minus(D2)nl minus 2053827anec( 1113857

1minus(D2)1113960 1113961

(53)

For all frequency exponents the total actual contact areaof the joint interface is

Ar Ar1 + Ar2 + Ar3 + Ar4 (54)

e total contact load of the joint interface is

Fr Fr1 + Fr2 + Fr3 + Fr4 (55)

e total real contact area and the total contact load in anondimensional form can be written as follows

Alowastr

Ar

Aa

Flowastr

Fr

AaE

(56)

where Aa is the nominal contact area and is given byAa L2L 1cnmin

Advances in Materials Science and Engineering 7

4 Results Analysis

In order to further analyze the above calculation results theparameters of equivalent joint interface are taken as shownin Table 1 [16]

Figure 2 shows the relation between all critical contactareas and frequency exponents of single asperity whenD 15 It can be seen from the figure that as for one definiteasperity when frequency exponent n is certain elastic criticalcontact area is minimum followed by the first elastoplasticcritical contact area and the second elastoplastic criticalcontact area is maximum With gradual increase of contactload the contact area increases e single asperity is firstlysubject to elastic deformation followed by the first elasto-plastic deformation the second elastoplastic deformation andfully plastic deformation successively which is consistent withtypical contact mechanics theory As for different asperitieswith increase of frequency exponent all critical contact areasdecrease correspondingly which shows that elastic criticalcontact area the first elastoplastic critical contact area and thesecond elastoplastic critical contact area are all related tofrequency exponent n

Figure 3 shows the relation curve between fractal di-mension D and critical frequency exponent n of asperityWhen fractal dimension is definite elastic critical frequencyexponent nec the first elastoplastic critical frequency ex-ponent nepc and the second elastoplastic critical frequencyexponent npc increase gradually As shown in Figure 3 whenDlt 106 nec nepc and npc are all negative As for asperitieswith minimum value of frequency exponent being 0 elasticdeformation elastoplastic deformation and fully plasticdeformation will all occur When D 113 nec and nepc arenegative and npc is positive At this time as for asperitieswith minimum value of frequency exponent being 0 elasticdeformation elastoplastic deformation will occur exceptfully plastic deformation

For D 15 G 25 times 10minus9 m H 55 times 109 Nm2 wecan obtain the elastic critical frequency exponent nec 32the first elastoplastic critical frequency exponent nepc 36and the second elastoplastic critical frequency exponentnpc 43 ese asperities whose frequency exponents rangefrom 20 to 32 are only under elastic deformation Elasticdeformation and the first elastoplastic deformation canoccur in these asperities whose frequency exponents rangefrom 33 to 36 Elastic deformation the first elastoplasticdeformation and the second elastoplastic deformation canoccur in these asperities whose frequency exponents rangefrom 37 to 43 When frequency exponents range from 43 to50 all deformations types can occur in these asperities

Figure 4 shows relation comparison diagram betweencontact load and contact area of single asperity with andwithout hardness change at the first elastoplastic stage ecomparison diagram is simulation result when n 33 It canbe seen from the figure that with gradual increase of contactarea with contact area of single asperity over 32 times 10minus13 m2contact load of the same asperity with hardness change willbe less than that without hardness change In addition as theamount of deformation increases the difference betweenthem tends to increase

Figure 5 shows relation comparison diagram betweencontact load and contact area of single asperity with andwithout hardness change at the second elastoplastic stagee comparison diagram is simulation result when n 37 Itcan be seen from the figure that when the deformation is

Table 1 e parameters of equivalent joint interface

Parameters ValuesEquivalent elastic modulus Eprime 72 times 1010 Nm2

Poissonrsquos ratio υ 017Initial hardness H 55 times 109 Nm2

Profile scale parameter G 25 times 10minus9 mFractal dimension D 1ltDlt 2Frequency exponent n 20sim50

20 25 30 35 40 45 5010ndash1510ndash1410ndash1310ndash1210ndash1110ndash1010ndash910ndash810ndash710ndash610ndash510ndash410ndash3

Criti

cal c

onta

ct ar

eas

of a

singl

e asp

erity

(am

2 )

Asperity levels (n)

Second elastoplastic critical contact areaFirst elastoplastic critical contact areaElastic critical contact area

Figure 2 e relationship between critical contact area and fre-quency exponent of a single asperity

10 11 12 13 14 15 16 17 18 19 20ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

0

20

40

Asp

erity

freq

uenc

y ex

pone

nts (n)

e fractal dimension (D)

Elastic critical frequency exponentsFirst elastoplastic critical frequency exponentsSecond elastoplastic critical frequency exponents

Figure 3 e relationship between fractal dimension D andcritical frequency exponent n of a single asperity

8 Advances in Materials Science and Engineering

definite contact load of the same asperity with hardnesschange will be less than that without hardness change Inaddition with increase of deformation amount the differ-ence between them tends to increase which is consistentwith the change trend at the first elastoplastic stage

Figure 6 shows the relation between limit mean geo-metric hardness and contact area (logarithm) of single as-perity at the first elastoplastic deformation stage Figure 6(a)shows relation curve that D 11 13 15 17 when n 34Figure 6(b) shows relation curve that n 32 33 34 35 whenD 15 It can be seen from Figure 6 that the limit meangeometric hardness of single asperity is related to contactarea fractal dimension and frequency exponent in the firstelastoplastic deformation stage e limit mean geometrichardness increases with increase of contact area When n is

definite the relation between limit mean geometric hardnessand contact area of asperity is related to fractal dimension De larger the D is the more obvious the relation curvebetween them changes when D is definite the relationbetween limit mean geometric hardness and contact area ofasperity is related to frequency exponent n e smaller n isthe more obvious the relation curve between them changes

Figure 7 shows the relation between limit mean geo-metric hardness and contact area (logarithm) of single as-perity at the second elastoplastic deformation stageFigure 7(a) shows the relation curve n 40 that D 11 1315 17 when n 40 Figure 7(b) shows the relation curvethat n 36 38 40 42 when D 15

During loading the contact area increases with the in-crease of deformation of a single asperity e ratio of de-formation to the natural height of the asperity is defined asthe pushing coefficient namely the pushing coefficientk ωnhn 0le kle 09 When fractal dimension is 15 we willresearch the relation between contact load and contact areaof single asperity with frequency exponent n being 30 35and 40 respectively during loading

When n 30 the asperity will only be subject to elasticdeformation During loading even the pushing coefficient k

is maximum no plastic deformation will occur e relationbetween contact area and contact load is fsima15 approxi-mately as shown in Figure 8(a)

As is shown in Figure 8(b) when n 35 elastic de-formation and the first elastoplastic deformation may takeplace in the asperity during loading When the pushingcoefficient k is less than 0247 the asperity will under elasticdeformation At this time the relation between contact areaand contact load is fsima15 approximately when the pushingcoefficient is over 0247 the first elastoplastic deformationoccurs At this time the relation between contact area andcontact load is fsima11093 approximately As is shown inFigure 8(c) when n 40 elastic deformation the firstelastoplastic deformation and the second elastoplastic

0000 0001 0002 0003 0004 0005Contact load of a single asperity in the

first elastoplastic deformation regime (fN)

00

12 times 10ndash12

10 times 10ndash12

80 times 10ndash13

60 times 10ndash13

40 times 10ndash13

20 times 10ndash13

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

Change of hardness is consideredChange of hardness is not considered

Figure 4 e relationship between contact load and contact area of single asperity in the first elastoplastic deformation stage

0000 0001 0002 0003 0004 00050

2 times 10ndash13

4 times 10ndash13

6 times 10ndash13

8 times 10ndash13

1 times 10ndash12

1 times 10ndash12

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

Contact load of a single asperity in the second elastoplastic deformation regime (f2N)

Change of hardness is not consideredChange of hardness is considered

Figure 5e relationship between contact load and contact area ofsingle asperity in the second elastoplastic deformation stage

Advances in Materials Science and Engineering 9

deformation may take place in the asperity during loadingWhen the pushing coefficient is greater than 01954 theasperity begins to enter the second elastoplastic de-formation the relation between contact area and contactload is fsima10977 approximately When n 45 and thepushing coefficient is greater than 0472 the asperity beginsto enter fully plastic deformation the relation betweencontact area and contact load is fsima approximately

Figure 9 shows that when the minimum frequency ex-ponent is 20 and the maximum value is 32 the actual contactarea of the joint interface increases with the increase of thetotal contact load and the relation between them isFlowastr simAlowast15

r approximately During the whole deformationprocess the joint interface appears to be of elastic property

Figure 10 shows the relation between dimensionless totalreal contact area and the dimensionless total contact loading

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash9

10ndash15

10ndash14

10ndash13

10ndash12

10ndash11

10ndash10

D = 11 n = 34D = 13 n = 34

D = 15 n = 34D = 17 n = 34

25 times 109 30 times 109 35 times 109 40 times 109 45 times 109 50 times 109 55 times 109

Limit mean geometric hardness in the firstelastoplastic deformation regime HG1(Nm2)

(a)C

onta

ct ar

eas o

f a si

ngle

aspe

rity

(am

2 )

10ndash10

10ndash15

10ndash14

10ndash13

10ndash12

10ndash11

n = 32 D = 15n = 33 D = 15

n = 34 D = 15n = 35 D = 15

25 times 109 30 times 109 35 times 109 40 times 109 45 times 109 50 times 109 55 times 109

Limit mean geometric hardness in the firstelastoplastic deformation regime HG1(Nm2)

(b)

Figure 6 e relationship between limit mean geometric hardness and contact for single asperity during the first elastoplastic deformationstage (a) n 34 11leDle 17 (b) D 15 32le nle 35

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash12

10ndash18

10ndash17

10ndash16

10ndash15

10ndash14

10ndash13

D = 11 n = 40D = 13 n = 40

D = 15 n = 40D = 17 n = 40

50 times 109 55 times 109 60 times 109 65 times 109 70 times 109 75 times 109

Limit mean geometric hardness in the secondelastoplastic deformation regime HG2(Nm2)

(a)

50 times 109 55 times 109 60 times 109 65 times 109 70 times 109 75 times 109

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash10

10ndash11

10ndash12

10ndash13

10ndash18

10ndash17

10ndash16

10ndash15

10ndash14

n = 36 D = 15n = 38 D = 15

n = 40 D = 15n = 42 D = 15

Limit mean geometric hardness in the secondelastoplastic deformation regime HG2(Nm2)

(b)

Figure 7 e relationship between limit mean geometric hardness and contact for single asperity during the second elastoplastic de-formation stage (a) n 40 11leDle 17 (b) D 15 36le nle 42

10 Advances in Materials Science and Engineering

of the joint interface when minimum value and maximumvalue of frequency exponent of asperities are respectively 20and 36When the pushing coefficient k is 01648 the asperitybegins to be subject to the first elastoplastic deformation andthe relation between dimensionless total real contact areaand the dimensionless total contact loading is FlowastrsimAlowast15

r

approximately e joint interface appears to be of elasticproperty As the load continues increasing when thepushing coefficient k is 05564 at this pointFlowastr gt 69023 times 10minus3 the relation between above load and areais approximately FlowastrsimAlowast11093

r e joint interface presentselastoplastic properties At this time the first elastoplasticdeformation takes place in these asperities whose frequencyexponents range 33sim36

Figure 11 shows the relation between dimensionless totalreal contact area and the dimensionless total contact loadingof the joint interface when minimum value and maximumvalue of frequency exponent of asperities are respectively 44and 50When the pushing coefficient k is 00621 the asperity

begins to be subject to the second elastoplastic deformationat this point 00016ltFlowastr lt 0023 and the relation betweendimensionless total real contact area and the dimensionlesstotal contact loading is FlowastrsimAlowast10977

r approximately Whenthe pushing coefficient k is 07076 the asperity begins to besubject to the fully plastic deformation at this pointFlowastr gt 0023 and the relation between dimensionless total realcontact area and the dimensionless total contact loading isapproximately FlowastrsimAlowastr

5 Conclusions

(1) e contact mechanical properties of a single as-perity on a joint interface are related to the frequencyexponent of the asperity while the frequency ex-ponent of an asperity is related to the fractal di-mension and the profile scale parameter In thispaper the critical frequency exponents of each de-formation stage of a single asperity are obtained and

000 005 010 015 020 025 030 03500

10 times 10ndash11

20 times 10ndash11

30 times 10ndash11

40 times 10ndash11

50 times 10ndash11

60 times 10ndash11

Contact load of a single asperity (fN)

n = 30

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

(a)

Contact load of a single asperity (fN)

n = 35

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

50

times10

ndash5

10

times10

ndash4

104 times 10ndash4

10 times 10ndash13

80 times 10ndash14

60 times 10ndash14

40 times 10ndash14

20 times 10ndash14

363 times 10ndash14

15

times10

ndash4

20

times10

ndash4

25

times10

ndash4

30

times10

ndash4

35

times10

ndash400

00

(b)

Contact load of a single asperity (fN)

n = 40

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

4 times 10ndash17

3 times 10ndash17

2 times 10ndash17

1 times 10ndash17

0

20

times10

ndash8

40

times10

ndash8

60

times10

ndash8

80

times10

ndash8

10

times10

ndash700

(c)

Figure 8 e relationship between contact load and contact area of a single asperity during loading

Advances in Materials Science and Engineering 11

the deformation characteristics of the asperity underdifferent frequency exponents are obtained

(2) e normal contact load of a single asperity in theelastoplastic stage is related to the hardness of thematerial When the material yields the hardness H isnot a constant but a function related to the amountof deformation In this paper the limit mean geo-metric hardness is introduced to express the normalcontact load of a single asperity in the first andsecond elastoplastic deformation stages consideringthe change of hardness

(3) e relationship between the contact load andcontact area of a single asperity in the first andsecond elastoplastic deformation stages considering

the change of material hardness and not consideringthe change of hardness is compared respectivelyWhen the contact area of the asperity is the same thecontact load with the change of hardness is smallerthan that without the change of hardness and thedifference between them increases with the increaseof deformation

(4) e limit mean geometric hardness of a single as-perity is related to the contact area fractal di-mension and frequency exponent during theelastoplastic deformation stage and the limit meangeometric hardness increases with the increase of thecontact area When the frequency exponent isconstant the relationship between the limit meangeometric hardness and the contact area of the as-perity is related to the fractal dimension e largerthe fractal dimension is the more obvious the re-lationship curve between them is When the fractaldimension is constant the relationship between thelimit mean geometric hardness of the asperity andthe contact area is related to the frequency exponente smaller the frequency exponent is the moreobvious the change of the curve is

(5) e relationship between dimensionless total realcontact area and the dimensionless total contactloading of joint interface at each deformation stageis obtained by introducing the pushing coefficientand the critical pushing coefficient exists Whenthe pushing coefficient exceeds this value someasperities begin to deform in the next stage andthe inflection point appears on the relationshipcurve

(6) e current model still has some limitations on theapplicable scope of materials and further researchand improvement are needed to make it more ap-plicable in the future

09

08

07

06

05

04

03

02

01

000000 0002 0004 0006 0008 0010 0012 0014

Nondimensional total contact load (Flowastr)

Non

dim

ensio

nal t

otal

cont

act a

rea (

Alowastr)

nmin = 20 nmax = 36

Figure 10 e relationship between dimensionless total realcontact area and the dimensionless total contact loading for20le nle 36

10

09

08

07

06

05

04

03

02

01Non

dim

ensio

nal t

otal

cont

act a

rea (

Alowastr)

000000 0005 0010 0015 0020 0025 0030 0035

Nondimensional total contact load (Flowastr)

nmin = 44 nmax = 50

Figure 11 e relationship between dimensionless total realcontact area and the dimensionless total contact loading for44le nle 50

10

09

08

07

06

05

04

03

02

01

Nondimensional total contact load (Flowastr1)

Non

dim

ensio

nal t

otal

real

cont

act a

rea (

Alowastr1

)

0000 10 times 10ndash3 20 times 10ndash3 30 times 10ndash3 40 times 10ndash3 50 times 10ndash3

nmin = 20 nmax = 32

Figure 9e relationship between dimensionless total real contactarea and the dimensionless total contact loading for 20le nle 32

12 Advances in Materials Science and Engineering

Nomenclature

ωn Interference of the asperityωnec Elastic critical interference of the asperityωnepc First elastoplastic critical interference of the

asperityωnpc Second elastoplastic critical interference of the

asperityan Contact area of the asperityanec Elastic critical contact area of the asperityanepc First elastoplastic critical contact area of the

asperityanpc Second elastoplastic critical contact area of the

asperityfne Normal load in the elastic deformation of a single

asperityfnec Normal contact load for ω ωnecfnep1 Normal contact load of a single asperity in the

first elastoplastic stagefnep2 Normal contact load of a single asperity in the

second elastoplastic stagefnp Normal contact load of a single asperity in the full

plastic deformation stagefnep1prime Normal contact load of a single asperity in the

first elastoplastic stage considering the change ofhardness

fnep2prime Normal contact load of a single asperity in thesecond elastoplastic stage considering the changeof hardness

HG1(an) Limit mean geometric hardness in the firstelastoplastic deformation stage

HG2(an) Limit mean geometric hardness in the secondelastoplastic deformation stage

nec Elastic critical frequency exponentnepc First elastoplastic critical frequency exponentnpc Second elastoplastic critical frequency exponentAr Actual contact area of the joint interfaceAr1 Actual contact area of the joint interface for

nmin lt nle necAr2 Actual contact area of the joint interface for

nec lt nle nepcAr3 Actual contact area of the joint interface for

nepc lt nle npcAr4 Actual contact area of the joint interface for

npc lt n

Fr1 Actual contact load of the joint interface fornmin lt nle nec

Fr2 Actual contact load of the joint interface fornec lt nle nepc

Fr3 Actual contact load of the joint interface fornepc lt nle npc

Fr4 Actual contact load of the joint interface fornpc lt n

pea(a) Average contact pressure of the asperity in elasticstage

pepa1(a) Average contact pressure of the asperity in thefirst elastoplastic deformation stage

pepa2(a) Average contact pressure of the asperity in thesecond elastoplastic stage

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (grant no 51275328) and the NaturalScience Foundation Project of Shanxi Province (grant no201601D011062)

References

[1] B Zhao S Zhang P Wang and Y Hai ldquoLoading-unloadingnormal stiffness model for power-law hardening surfacesconsidering actual surface topographyrdquo Tribology In-ternational vol 90 pp 332ndash342 2015

[2] J Greenwood and J Williamson ldquoContact of nominally flatsurfacesrdquo Mathematical Physical and Engineering Sciencesvol 295 no 1442 pp 299ndash319 1966

[3] J A Greenwood and J H Tripp ldquoe contact of twonominally flat rough surfacesrdquo Proceedings of the Institution ofMechanical Engineers vol 185 no 1 pp 625ndash633 1970

[4] W R Chang I Etsion and D B Bogy ldquoAn elastic-plasticmodel for the contact of rough surfacesrdquo Journal of Tribologyvol 109 no 2 pp 257ndash263 1987

[5] Y Zhao D M Maietta and L Chang ldquoAn asperity micro-contact model incorporating the transition from elastic de-formation to fully plastic flowrdquo Journal of Tribology vol 122no 1 pp 86ndash93 2000

[6] L Kogut and I Etsion ldquoElastic-plastic contact analysis ofasphere and a rigid flatrdquo Journal of Applied Mechanics vol 69no 5 pp 657ndash662 2002

[7] I Etsion Y Kligerman and Y Kadin ldquoUnloading of anelastic-plastic loaded spherical contactrdquo International Journalof Solids and Structures vol 42 no 13 pp 3716ndash3729 2005

[8] Y Kadin Y Kligerman and I Etsion ldquoUnloading an elastic-plastic contact of rough surfacesrdquo Journal of the Mechanicsand Physics of Solids vol 54 no 12 pp 2652ndash2674 2006

[9] Y Kadin Y Kligerman and I Etsion ldquoMultiple loading-unloading of an elastic-plastic spherical contactrdquo In-ternational Journal of Solids and Structures vol 43 no 22-23pp 7119ndash7127 2006

[10] A Majumdar and B Bhushan ldquoRole of fractal geometry inroughness characterization and contact mechanics of sur-facesrdquo Journal of Tribology vol 112 no 2 pp 205ndash216 1990

[11] A Majumdar and C L Tien ldquoFractal characterization andsimulation of rough surfacesrdquo Wear vol 136 no 2pp 313ndash327 1990

[12] AMajumdar and B Bhushan ldquoFractal model of elastic-plasticcontact between rough surfacesrdquo Journal of Tribology vol 113no 1 pp 1ndash11 1991

[13] S Wang and K Komvopoulos ldquoA fractal theory of the in-terfacial temperature distribution in the slow sliding regimePart I-elastic contact and heat transfer analysisrdquo Journal ofTribology vol 116 no 4 pp 812ndash822 1994

[14] S Wang and K Komvopoulos ldquoA fractal theory of the in-terfacial temperature distribution in the slow sliding regime

Advances in Materials Science and Engineering 13

Part II-multiple domains elastoplastic contacts and appli-cationsrdquo Journal of Tribology vol 116 no 4 pp 824ndash8321994

[15] Y Morag and I Etsion ldquoResolving the contradiction of as-perities plastic to elastic mode transition in current contactmodels of fractal rough surfacesrdquo Wear vol 262 no 5-6pp 624ndash629 2007

[16] H Tian X Zhong and C Zhao ldquoOne loading model of jointinterface considering elastoplastic and variation of hardnesswith surface depthrdquo Journal of Mechanical Engineeringvol 51 no 5 pp 90ndash104 2015

[17] Y Yuan Y Cheng K Liu et al ldquoA revised Majumdar andBushan model of elastoplastic contact between rough sur-facesrdquo Applied Surface Science vol 425 pp 1138ndash1157 2017

[18] R S Sayles and T R omas ldquoSurface topography as anonstationary random processrdquo Nature vol 271 no 5644pp 431ndash434 1978

14 Advances in Materials Science and Engineering

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

4 Results Analysis

In order to further analyze the above calculation results theparameters of equivalent joint interface are taken as shownin Table 1 [16]

Figure 2 shows the relation between all critical contactareas and frequency exponents of single asperity whenD 15 It can be seen from the figure that as for one definiteasperity when frequency exponent n is certain elastic criticalcontact area is minimum followed by the first elastoplasticcritical contact area and the second elastoplastic criticalcontact area is maximum With gradual increase of contactload the contact area increases e single asperity is firstlysubject to elastic deformation followed by the first elasto-plastic deformation the second elastoplastic deformation andfully plastic deformation successively which is consistent withtypical contact mechanics theory As for different asperitieswith increase of frequency exponent all critical contact areasdecrease correspondingly which shows that elastic criticalcontact area the first elastoplastic critical contact area and thesecond elastoplastic critical contact area are all related tofrequency exponent n

Figure 3 shows the relation curve between fractal di-mension D and critical frequency exponent n of asperityWhen fractal dimension is definite elastic critical frequencyexponent nec the first elastoplastic critical frequency ex-ponent nepc and the second elastoplastic critical frequencyexponent npc increase gradually As shown in Figure 3 whenDlt 106 nec nepc and npc are all negative As for asperitieswith minimum value of frequency exponent being 0 elasticdeformation elastoplastic deformation and fully plasticdeformation will all occur When D 113 nec and nepc arenegative and npc is positive At this time as for asperitieswith minimum value of frequency exponent being 0 elasticdeformation elastoplastic deformation will occur exceptfully plastic deformation

For D 15 G 25 times 10minus9 m H 55 times 109 Nm2 wecan obtain the elastic critical frequency exponent nec 32the first elastoplastic critical frequency exponent nepc 36and the second elastoplastic critical frequency exponentnpc 43 ese asperities whose frequency exponents rangefrom 20 to 32 are only under elastic deformation Elasticdeformation and the first elastoplastic deformation canoccur in these asperities whose frequency exponents rangefrom 33 to 36 Elastic deformation the first elastoplasticdeformation and the second elastoplastic deformation canoccur in these asperities whose frequency exponents rangefrom 37 to 43 When frequency exponents range from 43 to50 all deformations types can occur in these asperities

Figure 4 shows relation comparison diagram betweencontact load and contact area of single asperity with andwithout hardness change at the first elastoplastic stage ecomparison diagram is simulation result when n 33 It canbe seen from the figure that with gradual increase of contactarea with contact area of single asperity over 32 times 10minus13 m2contact load of the same asperity with hardness change willbe less than that without hardness change In addition as theamount of deformation increases the difference betweenthem tends to increase

Figure 5 shows relation comparison diagram betweencontact load and contact area of single asperity with andwithout hardness change at the second elastoplastic stagee comparison diagram is simulation result when n 37 Itcan be seen from the figure that when the deformation is

Table 1 e parameters of equivalent joint interface

Parameters ValuesEquivalent elastic modulus Eprime 72 times 1010 Nm2

Poissonrsquos ratio υ 017Initial hardness H 55 times 109 Nm2

Profile scale parameter G 25 times 10minus9 mFractal dimension D 1ltDlt 2Frequency exponent n 20sim50

20 25 30 35 40 45 5010ndash1510ndash1410ndash1310ndash1210ndash1110ndash1010ndash910ndash810ndash710ndash610ndash510ndash410ndash3

Criti

cal c

onta

ct ar

eas

of a

singl

e asp

erity

(am

2 )

Asperity levels (n)

Second elastoplastic critical contact areaFirst elastoplastic critical contact areaElastic critical contact area

Figure 2 e relationship between critical contact area and fre-quency exponent of a single asperity

10 11 12 13 14 15 16 17 18 19 20ndash140

ndash120

ndash100

ndash80

ndash60

ndash40

ndash20

0

20

40

Asp

erity

freq

uenc

y ex

pone

nts (n)

e fractal dimension (D)

Elastic critical frequency exponentsFirst elastoplastic critical frequency exponentsSecond elastoplastic critical frequency exponents

Figure 3 e relationship between fractal dimension D andcritical frequency exponent n of a single asperity

8 Advances in Materials Science and Engineering

definite contact load of the same asperity with hardnesschange will be less than that without hardness change Inaddition with increase of deformation amount the differ-ence between them tends to increase which is consistentwith the change trend at the first elastoplastic stage

Figure 6 shows the relation between limit mean geo-metric hardness and contact area (logarithm) of single as-perity at the first elastoplastic deformation stage Figure 6(a)shows relation curve that D 11 13 15 17 when n 34Figure 6(b) shows relation curve that n 32 33 34 35 whenD 15 It can be seen from Figure 6 that the limit meangeometric hardness of single asperity is related to contactarea fractal dimension and frequency exponent in the firstelastoplastic deformation stage e limit mean geometrichardness increases with increase of contact area When n is

definite the relation between limit mean geometric hardnessand contact area of asperity is related to fractal dimension De larger the D is the more obvious the relation curvebetween them changes when D is definite the relationbetween limit mean geometric hardness and contact area ofasperity is related to frequency exponent n e smaller n isthe more obvious the relation curve between them changes

Figure 7 shows the relation between limit mean geo-metric hardness and contact area (logarithm) of single as-perity at the second elastoplastic deformation stageFigure 7(a) shows the relation curve n 40 that D 11 1315 17 when n 40 Figure 7(b) shows the relation curvethat n 36 38 40 42 when D 15

During loading the contact area increases with the in-crease of deformation of a single asperity e ratio of de-formation to the natural height of the asperity is defined asthe pushing coefficient namely the pushing coefficientk ωnhn 0le kle 09 When fractal dimension is 15 we willresearch the relation between contact load and contact areaof single asperity with frequency exponent n being 30 35and 40 respectively during loading

When n 30 the asperity will only be subject to elasticdeformation During loading even the pushing coefficient k

is maximum no plastic deformation will occur e relationbetween contact area and contact load is fsima15 approxi-mately as shown in Figure 8(a)

As is shown in Figure 8(b) when n 35 elastic de-formation and the first elastoplastic deformation may takeplace in the asperity during loading When the pushingcoefficient k is less than 0247 the asperity will under elasticdeformation At this time the relation between contact areaand contact load is fsima15 approximately when the pushingcoefficient is over 0247 the first elastoplastic deformationoccurs At this time the relation between contact area andcontact load is fsima11093 approximately As is shown inFigure 8(c) when n 40 elastic deformation the firstelastoplastic deformation and the second elastoplastic

0000 0001 0002 0003 0004 0005Contact load of a single asperity in the

first elastoplastic deformation regime (fN)

00

12 times 10ndash12

10 times 10ndash12

80 times 10ndash13

60 times 10ndash13

40 times 10ndash13

20 times 10ndash13

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

Change of hardness is consideredChange of hardness is not considered

Figure 4 e relationship between contact load and contact area of single asperity in the first elastoplastic deformation stage

0000 0001 0002 0003 0004 00050

2 times 10ndash13

4 times 10ndash13

6 times 10ndash13

8 times 10ndash13

1 times 10ndash12

1 times 10ndash12

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

Contact load of a single asperity in the second elastoplastic deformation regime (f2N)

Change of hardness is not consideredChange of hardness is considered

Figure 5e relationship between contact load and contact area ofsingle asperity in the second elastoplastic deformation stage

Advances in Materials Science and Engineering 9

deformation may take place in the asperity during loadingWhen the pushing coefficient is greater than 01954 theasperity begins to enter the second elastoplastic de-formation the relation between contact area and contactload is fsima10977 approximately When n 45 and thepushing coefficient is greater than 0472 the asperity beginsto enter fully plastic deformation the relation betweencontact area and contact load is fsima approximately

Figure 9 shows that when the minimum frequency ex-ponent is 20 and the maximum value is 32 the actual contactarea of the joint interface increases with the increase of thetotal contact load and the relation between them isFlowastr simAlowast15

r approximately During the whole deformationprocess the joint interface appears to be of elastic property

Figure 10 shows the relation between dimensionless totalreal contact area and the dimensionless total contact loading

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash9

10ndash15

10ndash14

10ndash13

10ndash12

10ndash11

10ndash10

D = 11 n = 34D = 13 n = 34

D = 15 n = 34D = 17 n = 34

25 times 109 30 times 109 35 times 109 40 times 109 45 times 109 50 times 109 55 times 109

Limit mean geometric hardness in the firstelastoplastic deformation regime HG1(Nm2)

(a)C

onta

ct ar

eas o

f a si

ngle

aspe

rity

(am

2 )

10ndash10

10ndash15

10ndash14

10ndash13

10ndash12

10ndash11

n = 32 D = 15n = 33 D = 15

n = 34 D = 15n = 35 D = 15

25 times 109 30 times 109 35 times 109 40 times 109 45 times 109 50 times 109 55 times 109

Limit mean geometric hardness in the firstelastoplastic deformation regime HG1(Nm2)

(b)

Figure 6 e relationship between limit mean geometric hardness and contact for single asperity during the first elastoplastic deformationstage (a) n 34 11leDle 17 (b) D 15 32le nle 35

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash12

10ndash18

10ndash17

10ndash16

10ndash15

10ndash14

10ndash13

D = 11 n = 40D = 13 n = 40

D = 15 n = 40D = 17 n = 40

50 times 109 55 times 109 60 times 109 65 times 109 70 times 109 75 times 109

Limit mean geometric hardness in the secondelastoplastic deformation regime HG2(Nm2)

(a)

50 times 109 55 times 109 60 times 109 65 times 109 70 times 109 75 times 109

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash10

10ndash11

10ndash12

10ndash13

10ndash18

10ndash17

10ndash16

10ndash15

10ndash14

n = 36 D = 15n = 38 D = 15

n = 40 D = 15n = 42 D = 15

Limit mean geometric hardness in the secondelastoplastic deformation regime HG2(Nm2)

(b)

Figure 7 e relationship between limit mean geometric hardness and contact for single asperity during the second elastoplastic de-formation stage (a) n 40 11leDle 17 (b) D 15 36le nle 42

10 Advances in Materials Science and Engineering

of the joint interface when minimum value and maximumvalue of frequency exponent of asperities are respectively 20and 36When the pushing coefficient k is 01648 the asperitybegins to be subject to the first elastoplastic deformation andthe relation between dimensionless total real contact areaand the dimensionless total contact loading is FlowastrsimAlowast15

r

approximately e joint interface appears to be of elasticproperty As the load continues increasing when thepushing coefficient k is 05564 at this pointFlowastr gt 69023 times 10minus3 the relation between above load and areais approximately FlowastrsimAlowast11093

r e joint interface presentselastoplastic properties At this time the first elastoplasticdeformation takes place in these asperities whose frequencyexponents range 33sim36

Figure 11 shows the relation between dimensionless totalreal contact area and the dimensionless total contact loadingof the joint interface when minimum value and maximumvalue of frequency exponent of asperities are respectively 44and 50When the pushing coefficient k is 00621 the asperity

begins to be subject to the second elastoplastic deformationat this point 00016ltFlowastr lt 0023 and the relation betweendimensionless total real contact area and the dimensionlesstotal contact loading is FlowastrsimAlowast10977

r approximately Whenthe pushing coefficient k is 07076 the asperity begins to besubject to the fully plastic deformation at this pointFlowastr gt 0023 and the relation between dimensionless total realcontact area and the dimensionless total contact loading isapproximately FlowastrsimAlowastr

5 Conclusions

(1) e contact mechanical properties of a single as-perity on a joint interface are related to the frequencyexponent of the asperity while the frequency ex-ponent of an asperity is related to the fractal di-mension and the profile scale parameter In thispaper the critical frequency exponents of each de-formation stage of a single asperity are obtained and

000 005 010 015 020 025 030 03500

10 times 10ndash11

20 times 10ndash11

30 times 10ndash11

40 times 10ndash11

50 times 10ndash11

60 times 10ndash11

Contact load of a single asperity (fN)

n = 30

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

(a)

Contact load of a single asperity (fN)

n = 35

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

50

times10

ndash5

10

times10

ndash4

104 times 10ndash4

10 times 10ndash13

80 times 10ndash14

60 times 10ndash14

40 times 10ndash14

20 times 10ndash14

363 times 10ndash14

15

times10

ndash4

20

times10

ndash4

25

times10

ndash4

30

times10

ndash4

35

times10

ndash400

00

(b)

Contact load of a single asperity (fN)

n = 40

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

4 times 10ndash17

3 times 10ndash17

2 times 10ndash17

1 times 10ndash17

0

20

times10

ndash8

40

times10

ndash8

60

times10

ndash8

80

times10

ndash8

10

times10

ndash700

(c)

Figure 8 e relationship between contact load and contact area of a single asperity during loading

Advances in Materials Science and Engineering 11

the deformation characteristics of the asperity underdifferent frequency exponents are obtained

(2) e normal contact load of a single asperity in theelastoplastic stage is related to the hardness of thematerial When the material yields the hardness H isnot a constant but a function related to the amountof deformation In this paper the limit mean geo-metric hardness is introduced to express the normalcontact load of a single asperity in the first andsecond elastoplastic deformation stages consideringthe change of hardness

(3) e relationship between the contact load andcontact area of a single asperity in the first andsecond elastoplastic deformation stages considering

the change of material hardness and not consideringthe change of hardness is compared respectivelyWhen the contact area of the asperity is the same thecontact load with the change of hardness is smallerthan that without the change of hardness and thedifference between them increases with the increaseof deformation

(4) e limit mean geometric hardness of a single as-perity is related to the contact area fractal di-mension and frequency exponent during theelastoplastic deformation stage and the limit meangeometric hardness increases with the increase of thecontact area When the frequency exponent isconstant the relationship between the limit meangeometric hardness and the contact area of the as-perity is related to the fractal dimension e largerthe fractal dimension is the more obvious the re-lationship curve between them is When the fractaldimension is constant the relationship between thelimit mean geometric hardness of the asperity andthe contact area is related to the frequency exponente smaller the frequency exponent is the moreobvious the change of the curve is

(5) e relationship between dimensionless total realcontact area and the dimensionless total contactloading of joint interface at each deformation stageis obtained by introducing the pushing coefficientand the critical pushing coefficient exists Whenthe pushing coefficient exceeds this value someasperities begin to deform in the next stage andthe inflection point appears on the relationshipcurve

(6) e current model still has some limitations on theapplicable scope of materials and further researchand improvement are needed to make it more ap-plicable in the future

09

08

07

06

05

04

03

02

01

000000 0002 0004 0006 0008 0010 0012 0014

Nondimensional total contact load (Flowastr)

Non

dim

ensio

nal t

otal

cont

act a

rea (

Alowastr)

nmin = 20 nmax = 36

Figure 10 e relationship between dimensionless total realcontact area and the dimensionless total contact loading for20le nle 36

10

09

08

07

06

05

04

03

02

01Non

dim

ensio

nal t

otal

cont

act a

rea (

Alowastr)

000000 0005 0010 0015 0020 0025 0030 0035

Nondimensional total contact load (Flowastr)

nmin = 44 nmax = 50

Figure 11 e relationship between dimensionless total realcontact area and the dimensionless total contact loading for44le nle 50

10

09

08

07

06

05

04

03

02

01

Nondimensional total contact load (Flowastr1)

Non

dim

ensio

nal t

otal

real

cont

act a

rea (

Alowastr1

)

0000 10 times 10ndash3 20 times 10ndash3 30 times 10ndash3 40 times 10ndash3 50 times 10ndash3

nmin = 20 nmax = 32

Figure 9e relationship between dimensionless total real contactarea and the dimensionless total contact loading for 20le nle 32

12 Advances in Materials Science and Engineering

Nomenclature

ωn Interference of the asperityωnec Elastic critical interference of the asperityωnepc First elastoplastic critical interference of the

asperityωnpc Second elastoplastic critical interference of the

asperityan Contact area of the asperityanec Elastic critical contact area of the asperityanepc First elastoplastic critical contact area of the

asperityanpc Second elastoplastic critical contact area of the

asperityfne Normal load in the elastic deformation of a single

asperityfnec Normal contact load for ω ωnecfnep1 Normal contact load of a single asperity in the

first elastoplastic stagefnep2 Normal contact load of a single asperity in the

second elastoplastic stagefnp Normal contact load of a single asperity in the full

plastic deformation stagefnep1prime Normal contact load of a single asperity in the

first elastoplastic stage considering the change ofhardness

fnep2prime Normal contact load of a single asperity in thesecond elastoplastic stage considering the changeof hardness

HG1(an) Limit mean geometric hardness in the firstelastoplastic deformation stage

HG2(an) Limit mean geometric hardness in the secondelastoplastic deformation stage

nec Elastic critical frequency exponentnepc First elastoplastic critical frequency exponentnpc Second elastoplastic critical frequency exponentAr Actual contact area of the joint interfaceAr1 Actual contact area of the joint interface for

nmin lt nle necAr2 Actual contact area of the joint interface for

nec lt nle nepcAr3 Actual contact area of the joint interface for

nepc lt nle npcAr4 Actual contact area of the joint interface for

npc lt n

Fr1 Actual contact load of the joint interface fornmin lt nle nec

Fr2 Actual contact load of the joint interface fornec lt nle nepc

Fr3 Actual contact load of the joint interface fornepc lt nle npc

Fr4 Actual contact load of the joint interface fornpc lt n

pea(a) Average contact pressure of the asperity in elasticstage

pepa1(a) Average contact pressure of the asperity in thefirst elastoplastic deformation stage

pepa2(a) Average contact pressure of the asperity in thesecond elastoplastic stage

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (grant no 51275328) and the NaturalScience Foundation Project of Shanxi Province (grant no201601D011062)

References

[1] B Zhao S Zhang P Wang and Y Hai ldquoLoading-unloadingnormal stiffness model for power-law hardening surfacesconsidering actual surface topographyrdquo Tribology In-ternational vol 90 pp 332ndash342 2015

[2] J Greenwood and J Williamson ldquoContact of nominally flatsurfacesrdquo Mathematical Physical and Engineering Sciencesvol 295 no 1442 pp 299ndash319 1966

[3] J A Greenwood and J H Tripp ldquoe contact of twonominally flat rough surfacesrdquo Proceedings of the Institution ofMechanical Engineers vol 185 no 1 pp 625ndash633 1970

[4] W R Chang I Etsion and D B Bogy ldquoAn elastic-plasticmodel for the contact of rough surfacesrdquo Journal of Tribologyvol 109 no 2 pp 257ndash263 1987

[5] Y Zhao D M Maietta and L Chang ldquoAn asperity micro-contact model incorporating the transition from elastic de-formation to fully plastic flowrdquo Journal of Tribology vol 122no 1 pp 86ndash93 2000

[6] L Kogut and I Etsion ldquoElastic-plastic contact analysis ofasphere and a rigid flatrdquo Journal of Applied Mechanics vol 69no 5 pp 657ndash662 2002

[7] I Etsion Y Kligerman and Y Kadin ldquoUnloading of anelastic-plastic loaded spherical contactrdquo International Journalof Solids and Structures vol 42 no 13 pp 3716ndash3729 2005

[8] Y Kadin Y Kligerman and I Etsion ldquoUnloading an elastic-plastic contact of rough surfacesrdquo Journal of the Mechanicsand Physics of Solids vol 54 no 12 pp 2652ndash2674 2006

[9] Y Kadin Y Kligerman and I Etsion ldquoMultiple loading-unloading of an elastic-plastic spherical contactrdquo In-ternational Journal of Solids and Structures vol 43 no 22-23pp 7119ndash7127 2006

[10] A Majumdar and B Bhushan ldquoRole of fractal geometry inroughness characterization and contact mechanics of sur-facesrdquo Journal of Tribology vol 112 no 2 pp 205ndash216 1990

[11] A Majumdar and C L Tien ldquoFractal characterization andsimulation of rough surfacesrdquo Wear vol 136 no 2pp 313ndash327 1990

[12] AMajumdar and B Bhushan ldquoFractal model of elastic-plasticcontact between rough surfacesrdquo Journal of Tribology vol 113no 1 pp 1ndash11 1991

[13] S Wang and K Komvopoulos ldquoA fractal theory of the in-terfacial temperature distribution in the slow sliding regimePart I-elastic contact and heat transfer analysisrdquo Journal ofTribology vol 116 no 4 pp 812ndash822 1994

[14] S Wang and K Komvopoulos ldquoA fractal theory of the in-terfacial temperature distribution in the slow sliding regime

Advances in Materials Science and Engineering 13

Part II-multiple domains elastoplastic contacts and appli-cationsrdquo Journal of Tribology vol 116 no 4 pp 824ndash8321994

[15] Y Morag and I Etsion ldquoResolving the contradiction of as-perities plastic to elastic mode transition in current contactmodels of fractal rough surfacesrdquo Wear vol 262 no 5-6pp 624ndash629 2007

[16] H Tian X Zhong and C Zhao ldquoOne loading model of jointinterface considering elastoplastic and variation of hardnesswith surface depthrdquo Journal of Mechanical Engineeringvol 51 no 5 pp 90ndash104 2015

[17] Y Yuan Y Cheng K Liu et al ldquoA revised Majumdar andBushan model of elastoplastic contact between rough sur-facesrdquo Applied Surface Science vol 425 pp 1138ndash1157 2017

[18] R S Sayles and T R omas ldquoSurface topography as anonstationary random processrdquo Nature vol 271 no 5644pp 431ndash434 1978

14 Advances in Materials Science and Engineering

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

definite contact load of the same asperity with hardnesschange will be less than that without hardness change Inaddition with increase of deformation amount the differ-ence between them tends to increase which is consistentwith the change trend at the first elastoplastic stage

Figure 6 shows the relation between limit mean geo-metric hardness and contact area (logarithm) of single as-perity at the first elastoplastic deformation stage Figure 6(a)shows relation curve that D 11 13 15 17 when n 34Figure 6(b) shows relation curve that n 32 33 34 35 whenD 15 It can be seen from Figure 6 that the limit meangeometric hardness of single asperity is related to contactarea fractal dimension and frequency exponent in the firstelastoplastic deformation stage e limit mean geometrichardness increases with increase of contact area When n is

definite the relation between limit mean geometric hardnessand contact area of asperity is related to fractal dimension De larger the D is the more obvious the relation curvebetween them changes when D is definite the relationbetween limit mean geometric hardness and contact area ofasperity is related to frequency exponent n e smaller n isthe more obvious the relation curve between them changes

Figure 7 shows the relation between limit mean geo-metric hardness and contact area (logarithm) of single as-perity at the second elastoplastic deformation stageFigure 7(a) shows the relation curve n 40 that D 11 1315 17 when n 40 Figure 7(b) shows the relation curvethat n 36 38 40 42 when D 15

During loading the contact area increases with the in-crease of deformation of a single asperity e ratio of de-formation to the natural height of the asperity is defined asthe pushing coefficient namely the pushing coefficientk ωnhn 0le kle 09 When fractal dimension is 15 we willresearch the relation between contact load and contact areaof single asperity with frequency exponent n being 30 35and 40 respectively during loading

When n 30 the asperity will only be subject to elasticdeformation During loading even the pushing coefficient k

is maximum no plastic deformation will occur e relationbetween contact area and contact load is fsima15 approxi-mately as shown in Figure 8(a)

As is shown in Figure 8(b) when n 35 elastic de-formation and the first elastoplastic deformation may takeplace in the asperity during loading When the pushingcoefficient k is less than 0247 the asperity will under elasticdeformation At this time the relation between contact areaand contact load is fsima15 approximately when the pushingcoefficient is over 0247 the first elastoplastic deformationoccurs At this time the relation between contact area andcontact load is fsima11093 approximately As is shown inFigure 8(c) when n 40 elastic deformation the firstelastoplastic deformation and the second elastoplastic

0000 0001 0002 0003 0004 0005Contact load of a single asperity in the

first elastoplastic deformation regime (fN)

00

12 times 10ndash12

10 times 10ndash12

80 times 10ndash13

60 times 10ndash13

40 times 10ndash13

20 times 10ndash13

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

Change of hardness is consideredChange of hardness is not considered

Figure 4 e relationship between contact load and contact area of single asperity in the first elastoplastic deformation stage

0000 0001 0002 0003 0004 00050

2 times 10ndash13

4 times 10ndash13

6 times 10ndash13

8 times 10ndash13

1 times 10ndash12

1 times 10ndash12

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

Contact load of a single asperity in the second elastoplastic deformation regime (f2N)

Change of hardness is not consideredChange of hardness is considered

Figure 5e relationship between contact load and contact area ofsingle asperity in the second elastoplastic deformation stage

Advances in Materials Science and Engineering 9

deformation may take place in the asperity during loadingWhen the pushing coefficient is greater than 01954 theasperity begins to enter the second elastoplastic de-formation the relation between contact area and contactload is fsima10977 approximately When n 45 and thepushing coefficient is greater than 0472 the asperity beginsto enter fully plastic deformation the relation betweencontact area and contact load is fsima approximately

Figure 9 shows that when the minimum frequency ex-ponent is 20 and the maximum value is 32 the actual contactarea of the joint interface increases with the increase of thetotal contact load and the relation between them isFlowastr simAlowast15

r approximately During the whole deformationprocess the joint interface appears to be of elastic property

Figure 10 shows the relation between dimensionless totalreal contact area and the dimensionless total contact loading

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash9

10ndash15

10ndash14

10ndash13

10ndash12

10ndash11

10ndash10

D = 11 n = 34D = 13 n = 34

D = 15 n = 34D = 17 n = 34

25 times 109 30 times 109 35 times 109 40 times 109 45 times 109 50 times 109 55 times 109

Limit mean geometric hardness in the firstelastoplastic deformation regime HG1(Nm2)

(a)C

onta

ct ar

eas o

f a si

ngle

aspe

rity

(am

2 )

10ndash10

10ndash15

10ndash14

10ndash13

10ndash12

10ndash11

n = 32 D = 15n = 33 D = 15

n = 34 D = 15n = 35 D = 15

25 times 109 30 times 109 35 times 109 40 times 109 45 times 109 50 times 109 55 times 109

Limit mean geometric hardness in the firstelastoplastic deformation regime HG1(Nm2)

(b)

Figure 6 e relationship between limit mean geometric hardness and contact for single asperity during the first elastoplastic deformationstage (a) n 34 11leDle 17 (b) D 15 32le nle 35

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash12

10ndash18

10ndash17

10ndash16

10ndash15

10ndash14

10ndash13

D = 11 n = 40D = 13 n = 40

D = 15 n = 40D = 17 n = 40

50 times 109 55 times 109 60 times 109 65 times 109 70 times 109 75 times 109

Limit mean geometric hardness in the secondelastoplastic deformation regime HG2(Nm2)

(a)

50 times 109 55 times 109 60 times 109 65 times 109 70 times 109 75 times 109

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash10

10ndash11

10ndash12

10ndash13

10ndash18

10ndash17

10ndash16

10ndash15

10ndash14

n = 36 D = 15n = 38 D = 15

n = 40 D = 15n = 42 D = 15

Limit mean geometric hardness in the secondelastoplastic deformation regime HG2(Nm2)

(b)

Figure 7 e relationship between limit mean geometric hardness and contact for single asperity during the second elastoplastic de-formation stage (a) n 40 11leDle 17 (b) D 15 36le nle 42

10 Advances in Materials Science and Engineering

of the joint interface when minimum value and maximumvalue of frequency exponent of asperities are respectively 20and 36When the pushing coefficient k is 01648 the asperitybegins to be subject to the first elastoplastic deformation andthe relation between dimensionless total real contact areaand the dimensionless total contact loading is FlowastrsimAlowast15

r

approximately e joint interface appears to be of elasticproperty As the load continues increasing when thepushing coefficient k is 05564 at this pointFlowastr gt 69023 times 10minus3 the relation between above load and areais approximately FlowastrsimAlowast11093

r e joint interface presentselastoplastic properties At this time the first elastoplasticdeformation takes place in these asperities whose frequencyexponents range 33sim36

Figure 11 shows the relation between dimensionless totalreal contact area and the dimensionless total contact loadingof the joint interface when minimum value and maximumvalue of frequency exponent of asperities are respectively 44and 50When the pushing coefficient k is 00621 the asperity

begins to be subject to the second elastoplastic deformationat this point 00016ltFlowastr lt 0023 and the relation betweendimensionless total real contact area and the dimensionlesstotal contact loading is FlowastrsimAlowast10977

r approximately Whenthe pushing coefficient k is 07076 the asperity begins to besubject to the fully plastic deformation at this pointFlowastr gt 0023 and the relation between dimensionless total realcontact area and the dimensionless total contact loading isapproximately FlowastrsimAlowastr

5 Conclusions

(1) e contact mechanical properties of a single as-perity on a joint interface are related to the frequencyexponent of the asperity while the frequency ex-ponent of an asperity is related to the fractal di-mension and the profile scale parameter In thispaper the critical frequency exponents of each de-formation stage of a single asperity are obtained and

000 005 010 015 020 025 030 03500

10 times 10ndash11

20 times 10ndash11

30 times 10ndash11

40 times 10ndash11

50 times 10ndash11

60 times 10ndash11

Contact load of a single asperity (fN)

n = 30

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

(a)

Contact load of a single asperity (fN)

n = 35

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

50

times10

ndash5

10

times10

ndash4

104 times 10ndash4

10 times 10ndash13

80 times 10ndash14

60 times 10ndash14

40 times 10ndash14

20 times 10ndash14

363 times 10ndash14

15

times10

ndash4

20

times10

ndash4

25

times10

ndash4

30

times10

ndash4

35

times10

ndash400

00

(b)

Contact load of a single asperity (fN)

n = 40

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

4 times 10ndash17

3 times 10ndash17

2 times 10ndash17

1 times 10ndash17

0

20

times10

ndash8

40

times10

ndash8

60

times10

ndash8

80

times10

ndash8

10

times10

ndash700

(c)

Figure 8 e relationship between contact load and contact area of a single asperity during loading

Advances in Materials Science and Engineering 11

the deformation characteristics of the asperity underdifferent frequency exponents are obtained

(2) e normal contact load of a single asperity in theelastoplastic stage is related to the hardness of thematerial When the material yields the hardness H isnot a constant but a function related to the amountof deformation In this paper the limit mean geo-metric hardness is introduced to express the normalcontact load of a single asperity in the first andsecond elastoplastic deformation stages consideringthe change of hardness

(3) e relationship between the contact load andcontact area of a single asperity in the first andsecond elastoplastic deformation stages considering

the change of material hardness and not consideringthe change of hardness is compared respectivelyWhen the contact area of the asperity is the same thecontact load with the change of hardness is smallerthan that without the change of hardness and thedifference between them increases with the increaseof deformation

(4) e limit mean geometric hardness of a single as-perity is related to the contact area fractal di-mension and frequency exponent during theelastoplastic deformation stage and the limit meangeometric hardness increases with the increase of thecontact area When the frequency exponent isconstant the relationship between the limit meangeometric hardness and the contact area of the as-perity is related to the fractal dimension e largerthe fractal dimension is the more obvious the re-lationship curve between them is When the fractaldimension is constant the relationship between thelimit mean geometric hardness of the asperity andthe contact area is related to the frequency exponente smaller the frequency exponent is the moreobvious the change of the curve is

(5) e relationship between dimensionless total realcontact area and the dimensionless total contactloading of joint interface at each deformation stageis obtained by introducing the pushing coefficientand the critical pushing coefficient exists Whenthe pushing coefficient exceeds this value someasperities begin to deform in the next stage andthe inflection point appears on the relationshipcurve

(6) e current model still has some limitations on theapplicable scope of materials and further researchand improvement are needed to make it more ap-plicable in the future

09

08

07

06

05

04

03

02

01

000000 0002 0004 0006 0008 0010 0012 0014

Nondimensional total contact load (Flowastr)

Non

dim

ensio

nal t

otal

cont

act a

rea (

Alowastr)

nmin = 20 nmax = 36

Figure 10 e relationship between dimensionless total realcontact area and the dimensionless total contact loading for20le nle 36

10

09

08

07

06

05

04

03

02

01Non

dim

ensio

nal t

otal

cont

act a

rea (

Alowastr)

000000 0005 0010 0015 0020 0025 0030 0035

Nondimensional total contact load (Flowastr)

nmin = 44 nmax = 50

Figure 11 e relationship between dimensionless total realcontact area and the dimensionless total contact loading for44le nle 50

10

09

08

07

06

05

04

03

02

01

Nondimensional total contact load (Flowastr1)

Non

dim

ensio

nal t

otal

real

cont

act a

rea (

Alowastr1

)

0000 10 times 10ndash3 20 times 10ndash3 30 times 10ndash3 40 times 10ndash3 50 times 10ndash3

nmin = 20 nmax = 32

Figure 9e relationship between dimensionless total real contactarea and the dimensionless total contact loading for 20le nle 32

12 Advances in Materials Science and Engineering

Nomenclature

ωn Interference of the asperityωnec Elastic critical interference of the asperityωnepc First elastoplastic critical interference of the

asperityωnpc Second elastoplastic critical interference of the

asperityan Contact area of the asperityanec Elastic critical contact area of the asperityanepc First elastoplastic critical contact area of the

asperityanpc Second elastoplastic critical contact area of the

asperityfne Normal load in the elastic deformation of a single

asperityfnec Normal contact load for ω ωnecfnep1 Normal contact load of a single asperity in the

first elastoplastic stagefnep2 Normal contact load of a single asperity in the

second elastoplastic stagefnp Normal contact load of a single asperity in the full

plastic deformation stagefnep1prime Normal contact load of a single asperity in the

first elastoplastic stage considering the change ofhardness

fnep2prime Normal contact load of a single asperity in thesecond elastoplastic stage considering the changeof hardness

HG1(an) Limit mean geometric hardness in the firstelastoplastic deformation stage

HG2(an) Limit mean geometric hardness in the secondelastoplastic deformation stage

nec Elastic critical frequency exponentnepc First elastoplastic critical frequency exponentnpc Second elastoplastic critical frequency exponentAr Actual contact area of the joint interfaceAr1 Actual contact area of the joint interface for

nmin lt nle necAr2 Actual contact area of the joint interface for

nec lt nle nepcAr3 Actual contact area of the joint interface for

nepc lt nle npcAr4 Actual contact area of the joint interface for

npc lt n

Fr1 Actual contact load of the joint interface fornmin lt nle nec

Fr2 Actual contact load of the joint interface fornec lt nle nepc

Fr3 Actual contact load of the joint interface fornepc lt nle npc

Fr4 Actual contact load of the joint interface fornpc lt n

pea(a) Average contact pressure of the asperity in elasticstage

pepa1(a) Average contact pressure of the asperity in thefirst elastoplastic deformation stage

pepa2(a) Average contact pressure of the asperity in thesecond elastoplastic stage

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (grant no 51275328) and the NaturalScience Foundation Project of Shanxi Province (grant no201601D011062)

References

[1] B Zhao S Zhang P Wang and Y Hai ldquoLoading-unloadingnormal stiffness model for power-law hardening surfacesconsidering actual surface topographyrdquo Tribology In-ternational vol 90 pp 332ndash342 2015

[2] J Greenwood and J Williamson ldquoContact of nominally flatsurfacesrdquo Mathematical Physical and Engineering Sciencesvol 295 no 1442 pp 299ndash319 1966

[3] J A Greenwood and J H Tripp ldquoe contact of twonominally flat rough surfacesrdquo Proceedings of the Institution ofMechanical Engineers vol 185 no 1 pp 625ndash633 1970

[4] W R Chang I Etsion and D B Bogy ldquoAn elastic-plasticmodel for the contact of rough surfacesrdquo Journal of Tribologyvol 109 no 2 pp 257ndash263 1987

[5] Y Zhao D M Maietta and L Chang ldquoAn asperity micro-contact model incorporating the transition from elastic de-formation to fully plastic flowrdquo Journal of Tribology vol 122no 1 pp 86ndash93 2000

[6] L Kogut and I Etsion ldquoElastic-plastic contact analysis ofasphere and a rigid flatrdquo Journal of Applied Mechanics vol 69no 5 pp 657ndash662 2002

[7] I Etsion Y Kligerman and Y Kadin ldquoUnloading of anelastic-plastic loaded spherical contactrdquo International Journalof Solids and Structures vol 42 no 13 pp 3716ndash3729 2005

[8] Y Kadin Y Kligerman and I Etsion ldquoUnloading an elastic-plastic contact of rough surfacesrdquo Journal of the Mechanicsand Physics of Solids vol 54 no 12 pp 2652ndash2674 2006

[9] Y Kadin Y Kligerman and I Etsion ldquoMultiple loading-unloading of an elastic-plastic spherical contactrdquo In-ternational Journal of Solids and Structures vol 43 no 22-23pp 7119ndash7127 2006

[10] A Majumdar and B Bhushan ldquoRole of fractal geometry inroughness characterization and contact mechanics of sur-facesrdquo Journal of Tribology vol 112 no 2 pp 205ndash216 1990

[11] A Majumdar and C L Tien ldquoFractal characterization andsimulation of rough surfacesrdquo Wear vol 136 no 2pp 313ndash327 1990

[12] AMajumdar and B Bhushan ldquoFractal model of elastic-plasticcontact between rough surfacesrdquo Journal of Tribology vol 113no 1 pp 1ndash11 1991

[13] S Wang and K Komvopoulos ldquoA fractal theory of the in-terfacial temperature distribution in the slow sliding regimePart I-elastic contact and heat transfer analysisrdquo Journal ofTribology vol 116 no 4 pp 812ndash822 1994

[14] S Wang and K Komvopoulos ldquoA fractal theory of the in-terfacial temperature distribution in the slow sliding regime

Advances in Materials Science and Engineering 13

Part II-multiple domains elastoplastic contacts and appli-cationsrdquo Journal of Tribology vol 116 no 4 pp 824ndash8321994

[15] Y Morag and I Etsion ldquoResolving the contradiction of as-perities plastic to elastic mode transition in current contactmodels of fractal rough surfacesrdquo Wear vol 262 no 5-6pp 624ndash629 2007

[16] H Tian X Zhong and C Zhao ldquoOne loading model of jointinterface considering elastoplastic and variation of hardnesswith surface depthrdquo Journal of Mechanical Engineeringvol 51 no 5 pp 90ndash104 2015

[17] Y Yuan Y Cheng K Liu et al ldquoA revised Majumdar andBushan model of elastoplastic contact between rough sur-facesrdquo Applied Surface Science vol 425 pp 1138ndash1157 2017

[18] R S Sayles and T R omas ldquoSurface topography as anonstationary random processrdquo Nature vol 271 no 5644pp 431ndash434 1978

14 Advances in Materials Science and Engineering

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Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

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Analytical ChemistryInternational Journal of

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Polymer ScienceInternational Journal of

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Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Volume 2018

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Hindawiwwwhindawicom Volume 2018

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Journal of

Hindawiwwwhindawicom Volume 2018

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nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

deformation may take place in the asperity during loadingWhen the pushing coefficient is greater than 01954 theasperity begins to enter the second elastoplastic de-formation the relation between contact area and contactload is fsima10977 approximately When n 45 and thepushing coefficient is greater than 0472 the asperity beginsto enter fully plastic deformation the relation betweencontact area and contact load is fsima approximately

Figure 9 shows that when the minimum frequency ex-ponent is 20 and the maximum value is 32 the actual contactarea of the joint interface increases with the increase of thetotal contact load and the relation between them isFlowastr simAlowast15

r approximately During the whole deformationprocess the joint interface appears to be of elastic property

Figure 10 shows the relation between dimensionless totalreal contact area and the dimensionless total contact loading

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash9

10ndash15

10ndash14

10ndash13

10ndash12

10ndash11

10ndash10

D = 11 n = 34D = 13 n = 34

D = 15 n = 34D = 17 n = 34

25 times 109 30 times 109 35 times 109 40 times 109 45 times 109 50 times 109 55 times 109

Limit mean geometric hardness in the firstelastoplastic deformation regime HG1(Nm2)

(a)C

onta

ct ar

eas o

f a si

ngle

aspe

rity

(am

2 )

10ndash10

10ndash15

10ndash14

10ndash13

10ndash12

10ndash11

n = 32 D = 15n = 33 D = 15

n = 34 D = 15n = 35 D = 15

25 times 109 30 times 109 35 times 109 40 times 109 45 times 109 50 times 109 55 times 109

Limit mean geometric hardness in the firstelastoplastic deformation regime HG1(Nm2)

(b)

Figure 6 e relationship between limit mean geometric hardness and contact for single asperity during the first elastoplastic deformationstage (a) n 34 11leDle 17 (b) D 15 32le nle 35

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash12

10ndash18

10ndash17

10ndash16

10ndash15

10ndash14

10ndash13

D = 11 n = 40D = 13 n = 40

D = 15 n = 40D = 17 n = 40

50 times 109 55 times 109 60 times 109 65 times 109 70 times 109 75 times 109

Limit mean geometric hardness in the secondelastoplastic deformation regime HG2(Nm2)

(a)

50 times 109 55 times 109 60 times 109 65 times 109 70 times 109 75 times 109

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

10ndash10

10ndash11

10ndash12

10ndash13

10ndash18

10ndash17

10ndash16

10ndash15

10ndash14

n = 36 D = 15n = 38 D = 15

n = 40 D = 15n = 42 D = 15

Limit mean geometric hardness in the secondelastoplastic deformation regime HG2(Nm2)

(b)

Figure 7 e relationship between limit mean geometric hardness and contact for single asperity during the second elastoplastic de-formation stage (a) n 40 11leDle 17 (b) D 15 36le nle 42

10 Advances in Materials Science and Engineering

of the joint interface when minimum value and maximumvalue of frequency exponent of asperities are respectively 20and 36When the pushing coefficient k is 01648 the asperitybegins to be subject to the first elastoplastic deformation andthe relation between dimensionless total real contact areaand the dimensionless total contact loading is FlowastrsimAlowast15

r

approximately e joint interface appears to be of elasticproperty As the load continues increasing when thepushing coefficient k is 05564 at this pointFlowastr gt 69023 times 10minus3 the relation between above load and areais approximately FlowastrsimAlowast11093

r e joint interface presentselastoplastic properties At this time the first elastoplasticdeformation takes place in these asperities whose frequencyexponents range 33sim36

Figure 11 shows the relation between dimensionless totalreal contact area and the dimensionless total contact loadingof the joint interface when minimum value and maximumvalue of frequency exponent of asperities are respectively 44and 50When the pushing coefficient k is 00621 the asperity

begins to be subject to the second elastoplastic deformationat this point 00016ltFlowastr lt 0023 and the relation betweendimensionless total real contact area and the dimensionlesstotal contact loading is FlowastrsimAlowast10977

r approximately Whenthe pushing coefficient k is 07076 the asperity begins to besubject to the fully plastic deformation at this pointFlowastr gt 0023 and the relation between dimensionless total realcontact area and the dimensionless total contact loading isapproximately FlowastrsimAlowastr

5 Conclusions

(1) e contact mechanical properties of a single as-perity on a joint interface are related to the frequencyexponent of the asperity while the frequency ex-ponent of an asperity is related to the fractal di-mension and the profile scale parameter In thispaper the critical frequency exponents of each de-formation stage of a single asperity are obtained and

000 005 010 015 020 025 030 03500

10 times 10ndash11

20 times 10ndash11

30 times 10ndash11

40 times 10ndash11

50 times 10ndash11

60 times 10ndash11

Contact load of a single asperity (fN)

n = 30

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

(a)

Contact load of a single asperity (fN)

n = 35

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

50

times10

ndash5

10

times10

ndash4

104 times 10ndash4

10 times 10ndash13

80 times 10ndash14

60 times 10ndash14

40 times 10ndash14

20 times 10ndash14

363 times 10ndash14

15

times10

ndash4

20

times10

ndash4

25

times10

ndash4

30

times10

ndash4

35

times10

ndash400

00

(b)

Contact load of a single asperity (fN)

n = 40

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

4 times 10ndash17

3 times 10ndash17

2 times 10ndash17

1 times 10ndash17

0

20

times10

ndash8

40

times10

ndash8

60

times10

ndash8

80

times10

ndash8

10

times10

ndash700

(c)

Figure 8 e relationship between contact load and contact area of a single asperity during loading

Advances in Materials Science and Engineering 11

the deformation characteristics of the asperity underdifferent frequency exponents are obtained

(2) e normal contact load of a single asperity in theelastoplastic stage is related to the hardness of thematerial When the material yields the hardness H isnot a constant but a function related to the amountof deformation In this paper the limit mean geo-metric hardness is introduced to express the normalcontact load of a single asperity in the first andsecond elastoplastic deformation stages consideringthe change of hardness

(3) e relationship between the contact load andcontact area of a single asperity in the first andsecond elastoplastic deformation stages considering

the change of material hardness and not consideringthe change of hardness is compared respectivelyWhen the contact area of the asperity is the same thecontact load with the change of hardness is smallerthan that without the change of hardness and thedifference between them increases with the increaseof deformation

(4) e limit mean geometric hardness of a single as-perity is related to the contact area fractal di-mension and frequency exponent during theelastoplastic deformation stage and the limit meangeometric hardness increases with the increase of thecontact area When the frequency exponent isconstant the relationship between the limit meangeometric hardness and the contact area of the as-perity is related to the fractal dimension e largerthe fractal dimension is the more obvious the re-lationship curve between them is When the fractaldimension is constant the relationship between thelimit mean geometric hardness of the asperity andthe contact area is related to the frequency exponente smaller the frequency exponent is the moreobvious the change of the curve is

(5) e relationship between dimensionless total realcontact area and the dimensionless total contactloading of joint interface at each deformation stageis obtained by introducing the pushing coefficientand the critical pushing coefficient exists Whenthe pushing coefficient exceeds this value someasperities begin to deform in the next stage andthe inflection point appears on the relationshipcurve

(6) e current model still has some limitations on theapplicable scope of materials and further researchand improvement are needed to make it more ap-plicable in the future

09

08

07

06

05

04

03

02

01

000000 0002 0004 0006 0008 0010 0012 0014

Nondimensional total contact load (Flowastr)

Non

dim

ensio

nal t

otal

cont

act a

rea (

Alowastr)

nmin = 20 nmax = 36

Figure 10 e relationship between dimensionless total realcontact area and the dimensionless total contact loading for20le nle 36

10

09

08

07

06

05

04

03

02

01Non

dim

ensio

nal t

otal

cont

act a

rea (

Alowastr)

000000 0005 0010 0015 0020 0025 0030 0035

Nondimensional total contact load (Flowastr)

nmin = 44 nmax = 50

Figure 11 e relationship between dimensionless total realcontact area and the dimensionless total contact loading for44le nle 50

10

09

08

07

06

05

04

03

02

01

Nondimensional total contact load (Flowastr1)

Non

dim

ensio

nal t

otal

real

cont

act a

rea (

Alowastr1

)

0000 10 times 10ndash3 20 times 10ndash3 30 times 10ndash3 40 times 10ndash3 50 times 10ndash3

nmin = 20 nmax = 32

Figure 9e relationship between dimensionless total real contactarea and the dimensionless total contact loading for 20le nle 32

12 Advances in Materials Science and Engineering

Nomenclature

ωn Interference of the asperityωnec Elastic critical interference of the asperityωnepc First elastoplastic critical interference of the

asperityωnpc Second elastoplastic critical interference of the

asperityan Contact area of the asperityanec Elastic critical contact area of the asperityanepc First elastoplastic critical contact area of the

asperityanpc Second elastoplastic critical contact area of the

asperityfne Normal load in the elastic deformation of a single

asperityfnec Normal contact load for ω ωnecfnep1 Normal contact load of a single asperity in the

first elastoplastic stagefnep2 Normal contact load of a single asperity in the

second elastoplastic stagefnp Normal contact load of a single asperity in the full

plastic deformation stagefnep1prime Normal contact load of a single asperity in the

first elastoplastic stage considering the change ofhardness

fnep2prime Normal contact load of a single asperity in thesecond elastoplastic stage considering the changeof hardness

HG1(an) Limit mean geometric hardness in the firstelastoplastic deformation stage

HG2(an) Limit mean geometric hardness in the secondelastoplastic deformation stage

nec Elastic critical frequency exponentnepc First elastoplastic critical frequency exponentnpc Second elastoplastic critical frequency exponentAr Actual contact area of the joint interfaceAr1 Actual contact area of the joint interface for

nmin lt nle necAr2 Actual contact area of the joint interface for

nec lt nle nepcAr3 Actual contact area of the joint interface for

nepc lt nle npcAr4 Actual contact area of the joint interface for

npc lt n

Fr1 Actual contact load of the joint interface fornmin lt nle nec

Fr2 Actual contact load of the joint interface fornec lt nle nepc

Fr3 Actual contact load of the joint interface fornepc lt nle npc

Fr4 Actual contact load of the joint interface fornpc lt n

pea(a) Average contact pressure of the asperity in elasticstage

pepa1(a) Average contact pressure of the asperity in thefirst elastoplastic deformation stage

pepa2(a) Average contact pressure of the asperity in thesecond elastoplastic stage

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (grant no 51275328) and the NaturalScience Foundation Project of Shanxi Province (grant no201601D011062)

References

[1] B Zhao S Zhang P Wang and Y Hai ldquoLoading-unloadingnormal stiffness model for power-law hardening surfacesconsidering actual surface topographyrdquo Tribology In-ternational vol 90 pp 332ndash342 2015

[2] J Greenwood and J Williamson ldquoContact of nominally flatsurfacesrdquo Mathematical Physical and Engineering Sciencesvol 295 no 1442 pp 299ndash319 1966

[3] J A Greenwood and J H Tripp ldquoe contact of twonominally flat rough surfacesrdquo Proceedings of the Institution ofMechanical Engineers vol 185 no 1 pp 625ndash633 1970

[4] W R Chang I Etsion and D B Bogy ldquoAn elastic-plasticmodel for the contact of rough surfacesrdquo Journal of Tribologyvol 109 no 2 pp 257ndash263 1987

[5] Y Zhao D M Maietta and L Chang ldquoAn asperity micro-contact model incorporating the transition from elastic de-formation to fully plastic flowrdquo Journal of Tribology vol 122no 1 pp 86ndash93 2000

[6] L Kogut and I Etsion ldquoElastic-plastic contact analysis ofasphere and a rigid flatrdquo Journal of Applied Mechanics vol 69no 5 pp 657ndash662 2002

[7] I Etsion Y Kligerman and Y Kadin ldquoUnloading of anelastic-plastic loaded spherical contactrdquo International Journalof Solids and Structures vol 42 no 13 pp 3716ndash3729 2005

[8] Y Kadin Y Kligerman and I Etsion ldquoUnloading an elastic-plastic contact of rough surfacesrdquo Journal of the Mechanicsand Physics of Solids vol 54 no 12 pp 2652ndash2674 2006

[9] Y Kadin Y Kligerman and I Etsion ldquoMultiple loading-unloading of an elastic-plastic spherical contactrdquo In-ternational Journal of Solids and Structures vol 43 no 22-23pp 7119ndash7127 2006

[10] A Majumdar and B Bhushan ldquoRole of fractal geometry inroughness characterization and contact mechanics of sur-facesrdquo Journal of Tribology vol 112 no 2 pp 205ndash216 1990

[11] A Majumdar and C L Tien ldquoFractal characterization andsimulation of rough surfacesrdquo Wear vol 136 no 2pp 313ndash327 1990

[12] AMajumdar and B Bhushan ldquoFractal model of elastic-plasticcontact between rough surfacesrdquo Journal of Tribology vol 113no 1 pp 1ndash11 1991

[13] S Wang and K Komvopoulos ldquoA fractal theory of the in-terfacial temperature distribution in the slow sliding regimePart I-elastic contact and heat transfer analysisrdquo Journal ofTribology vol 116 no 4 pp 812ndash822 1994

[14] S Wang and K Komvopoulos ldquoA fractal theory of the in-terfacial temperature distribution in the slow sliding regime

Advances in Materials Science and Engineering 13

Part II-multiple domains elastoplastic contacts and appli-cationsrdquo Journal of Tribology vol 116 no 4 pp 824ndash8321994

[15] Y Morag and I Etsion ldquoResolving the contradiction of as-perities plastic to elastic mode transition in current contactmodels of fractal rough surfacesrdquo Wear vol 262 no 5-6pp 624ndash629 2007

[16] H Tian X Zhong and C Zhao ldquoOne loading model of jointinterface considering elastoplastic and variation of hardnesswith surface depthrdquo Journal of Mechanical Engineeringvol 51 no 5 pp 90ndash104 2015

[17] Y Yuan Y Cheng K Liu et al ldquoA revised Majumdar andBushan model of elastoplastic contact between rough sur-facesrdquo Applied Surface Science vol 425 pp 1138ndash1157 2017

[18] R S Sayles and T R omas ldquoSurface topography as anonstationary random processrdquo Nature vol 271 no 5644pp 431ndash434 1978

14 Advances in Materials Science and Engineering

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

of the joint interface when minimum value and maximumvalue of frequency exponent of asperities are respectively 20and 36When the pushing coefficient k is 01648 the asperitybegins to be subject to the first elastoplastic deformation andthe relation between dimensionless total real contact areaand the dimensionless total contact loading is FlowastrsimAlowast15

r

approximately e joint interface appears to be of elasticproperty As the load continues increasing when thepushing coefficient k is 05564 at this pointFlowastr gt 69023 times 10minus3 the relation between above load and areais approximately FlowastrsimAlowast11093

r e joint interface presentselastoplastic properties At this time the first elastoplasticdeformation takes place in these asperities whose frequencyexponents range 33sim36

Figure 11 shows the relation between dimensionless totalreal contact area and the dimensionless total contact loadingof the joint interface when minimum value and maximumvalue of frequency exponent of asperities are respectively 44and 50When the pushing coefficient k is 00621 the asperity

begins to be subject to the second elastoplastic deformationat this point 00016ltFlowastr lt 0023 and the relation betweendimensionless total real contact area and the dimensionlesstotal contact loading is FlowastrsimAlowast10977

r approximately Whenthe pushing coefficient k is 07076 the asperity begins to besubject to the fully plastic deformation at this pointFlowastr gt 0023 and the relation between dimensionless total realcontact area and the dimensionless total contact loading isapproximately FlowastrsimAlowastr

5 Conclusions

(1) e contact mechanical properties of a single as-perity on a joint interface are related to the frequencyexponent of the asperity while the frequency ex-ponent of an asperity is related to the fractal di-mension and the profile scale parameter In thispaper the critical frequency exponents of each de-formation stage of a single asperity are obtained and

000 005 010 015 020 025 030 03500

10 times 10ndash11

20 times 10ndash11

30 times 10ndash11

40 times 10ndash11

50 times 10ndash11

60 times 10ndash11

Contact load of a single asperity (fN)

n = 30

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

(a)

Contact load of a single asperity (fN)

n = 35

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

50

times10

ndash5

10

times10

ndash4

104 times 10ndash4

10 times 10ndash13

80 times 10ndash14

60 times 10ndash14

40 times 10ndash14

20 times 10ndash14

363 times 10ndash14

15

times10

ndash4

20

times10

ndash4

25

times10

ndash4

30

times10

ndash4

35

times10

ndash400

00

(b)

Contact load of a single asperity (fN)

n = 40

Con

tact

area

s of a

sing

le as

perit

y (a

m2 )

4 times 10ndash17

3 times 10ndash17

2 times 10ndash17

1 times 10ndash17

0

20

times10

ndash8

40

times10

ndash8

60

times10

ndash8

80

times10

ndash8

10

times10

ndash700

(c)

Figure 8 e relationship between contact load and contact area of a single asperity during loading

Advances in Materials Science and Engineering 11

the deformation characteristics of the asperity underdifferent frequency exponents are obtained

(2) e normal contact load of a single asperity in theelastoplastic stage is related to the hardness of thematerial When the material yields the hardness H isnot a constant but a function related to the amountof deformation In this paper the limit mean geo-metric hardness is introduced to express the normalcontact load of a single asperity in the first andsecond elastoplastic deformation stages consideringthe change of hardness

(3) e relationship between the contact load andcontact area of a single asperity in the first andsecond elastoplastic deformation stages considering

the change of material hardness and not consideringthe change of hardness is compared respectivelyWhen the contact area of the asperity is the same thecontact load with the change of hardness is smallerthan that without the change of hardness and thedifference between them increases with the increaseof deformation

(4) e limit mean geometric hardness of a single as-perity is related to the contact area fractal di-mension and frequency exponent during theelastoplastic deformation stage and the limit meangeometric hardness increases with the increase of thecontact area When the frequency exponent isconstant the relationship between the limit meangeometric hardness and the contact area of the as-perity is related to the fractal dimension e largerthe fractal dimension is the more obvious the re-lationship curve between them is When the fractaldimension is constant the relationship between thelimit mean geometric hardness of the asperity andthe contact area is related to the frequency exponente smaller the frequency exponent is the moreobvious the change of the curve is

(5) e relationship between dimensionless total realcontact area and the dimensionless total contactloading of joint interface at each deformation stageis obtained by introducing the pushing coefficientand the critical pushing coefficient exists Whenthe pushing coefficient exceeds this value someasperities begin to deform in the next stage andthe inflection point appears on the relationshipcurve

(6) e current model still has some limitations on theapplicable scope of materials and further researchand improvement are needed to make it more ap-plicable in the future

09

08

07

06

05

04

03

02

01

000000 0002 0004 0006 0008 0010 0012 0014

Nondimensional total contact load (Flowastr)

Non

dim

ensio

nal t

otal

cont

act a

rea (

Alowastr)

nmin = 20 nmax = 36

Figure 10 e relationship between dimensionless total realcontact area and the dimensionless total contact loading for20le nle 36

10

09

08

07

06

05

04

03

02

01Non

dim

ensio

nal t

otal

cont

act a

rea (

Alowastr)

000000 0005 0010 0015 0020 0025 0030 0035

Nondimensional total contact load (Flowastr)

nmin = 44 nmax = 50

Figure 11 e relationship between dimensionless total realcontact area and the dimensionless total contact loading for44le nle 50

10

09

08

07

06

05

04

03

02

01

Nondimensional total contact load (Flowastr1)

Non

dim

ensio

nal t

otal

real

cont

act a

rea (

Alowastr1

)

0000 10 times 10ndash3 20 times 10ndash3 30 times 10ndash3 40 times 10ndash3 50 times 10ndash3

nmin = 20 nmax = 32

Figure 9e relationship between dimensionless total real contactarea and the dimensionless total contact loading for 20le nle 32

12 Advances in Materials Science and Engineering

Nomenclature

ωn Interference of the asperityωnec Elastic critical interference of the asperityωnepc First elastoplastic critical interference of the

asperityωnpc Second elastoplastic critical interference of the

asperityan Contact area of the asperityanec Elastic critical contact area of the asperityanepc First elastoplastic critical contact area of the

asperityanpc Second elastoplastic critical contact area of the

asperityfne Normal load in the elastic deformation of a single

asperityfnec Normal contact load for ω ωnecfnep1 Normal contact load of a single asperity in the

first elastoplastic stagefnep2 Normal contact load of a single asperity in the

second elastoplastic stagefnp Normal contact load of a single asperity in the full

plastic deformation stagefnep1prime Normal contact load of a single asperity in the

first elastoplastic stage considering the change ofhardness

fnep2prime Normal contact load of a single asperity in thesecond elastoplastic stage considering the changeof hardness

HG1(an) Limit mean geometric hardness in the firstelastoplastic deformation stage

HG2(an) Limit mean geometric hardness in the secondelastoplastic deformation stage

nec Elastic critical frequency exponentnepc First elastoplastic critical frequency exponentnpc Second elastoplastic critical frequency exponentAr Actual contact area of the joint interfaceAr1 Actual contact area of the joint interface for

nmin lt nle necAr2 Actual contact area of the joint interface for

nec lt nle nepcAr3 Actual contact area of the joint interface for

nepc lt nle npcAr4 Actual contact area of the joint interface for

npc lt n

Fr1 Actual contact load of the joint interface fornmin lt nle nec

Fr2 Actual contact load of the joint interface fornec lt nle nepc

Fr3 Actual contact load of the joint interface fornepc lt nle npc

Fr4 Actual contact load of the joint interface fornpc lt n

pea(a) Average contact pressure of the asperity in elasticstage

pepa1(a) Average contact pressure of the asperity in thefirst elastoplastic deformation stage

pepa2(a) Average contact pressure of the asperity in thesecond elastoplastic stage

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (grant no 51275328) and the NaturalScience Foundation Project of Shanxi Province (grant no201601D011062)

References

[1] B Zhao S Zhang P Wang and Y Hai ldquoLoading-unloadingnormal stiffness model for power-law hardening surfacesconsidering actual surface topographyrdquo Tribology In-ternational vol 90 pp 332ndash342 2015

[2] J Greenwood and J Williamson ldquoContact of nominally flatsurfacesrdquo Mathematical Physical and Engineering Sciencesvol 295 no 1442 pp 299ndash319 1966

[3] J A Greenwood and J H Tripp ldquoe contact of twonominally flat rough surfacesrdquo Proceedings of the Institution ofMechanical Engineers vol 185 no 1 pp 625ndash633 1970

[4] W R Chang I Etsion and D B Bogy ldquoAn elastic-plasticmodel for the contact of rough surfacesrdquo Journal of Tribologyvol 109 no 2 pp 257ndash263 1987

[5] Y Zhao D M Maietta and L Chang ldquoAn asperity micro-contact model incorporating the transition from elastic de-formation to fully plastic flowrdquo Journal of Tribology vol 122no 1 pp 86ndash93 2000

[6] L Kogut and I Etsion ldquoElastic-plastic contact analysis ofasphere and a rigid flatrdquo Journal of Applied Mechanics vol 69no 5 pp 657ndash662 2002

[7] I Etsion Y Kligerman and Y Kadin ldquoUnloading of anelastic-plastic loaded spherical contactrdquo International Journalof Solids and Structures vol 42 no 13 pp 3716ndash3729 2005

[8] Y Kadin Y Kligerman and I Etsion ldquoUnloading an elastic-plastic contact of rough surfacesrdquo Journal of the Mechanicsand Physics of Solids vol 54 no 12 pp 2652ndash2674 2006

[9] Y Kadin Y Kligerman and I Etsion ldquoMultiple loading-unloading of an elastic-plastic spherical contactrdquo In-ternational Journal of Solids and Structures vol 43 no 22-23pp 7119ndash7127 2006

[10] A Majumdar and B Bhushan ldquoRole of fractal geometry inroughness characterization and contact mechanics of sur-facesrdquo Journal of Tribology vol 112 no 2 pp 205ndash216 1990

[11] A Majumdar and C L Tien ldquoFractal characterization andsimulation of rough surfacesrdquo Wear vol 136 no 2pp 313ndash327 1990

[12] AMajumdar and B Bhushan ldquoFractal model of elastic-plasticcontact between rough surfacesrdquo Journal of Tribology vol 113no 1 pp 1ndash11 1991

[13] S Wang and K Komvopoulos ldquoA fractal theory of the in-terfacial temperature distribution in the slow sliding regimePart I-elastic contact and heat transfer analysisrdquo Journal ofTribology vol 116 no 4 pp 812ndash822 1994

[14] S Wang and K Komvopoulos ldquoA fractal theory of the in-terfacial temperature distribution in the slow sliding regime

Advances in Materials Science and Engineering 13

Part II-multiple domains elastoplastic contacts and appli-cationsrdquo Journal of Tribology vol 116 no 4 pp 824ndash8321994

[15] Y Morag and I Etsion ldquoResolving the contradiction of as-perities plastic to elastic mode transition in current contactmodels of fractal rough surfacesrdquo Wear vol 262 no 5-6pp 624ndash629 2007

[16] H Tian X Zhong and C Zhao ldquoOne loading model of jointinterface considering elastoplastic and variation of hardnesswith surface depthrdquo Journal of Mechanical Engineeringvol 51 no 5 pp 90ndash104 2015

[17] Y Yuan Y Cheng K Liu et al ldquoA revised Majumdar andBushan model of elastoplastic contact between rough sur-facesrdquo Applied Surface Science vol 425 pp 1138ndash1157 2017

[18] R S Sayles and T R omas ldquoSurface topography as anonstationary random processrdquo Nature vol 271 no 5644pp 431ndash434 1978

14 Advances in Materials Science and Engineering

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

the deformation characteristics of the asperity underdifferent frequency exponents are obtained

(2) e normal contact load of a single asperity in theelastoplastic stage is related to the hardness of thematerial When the material yields the hardness H isnot a constant but a function related to the amountof deformation In this paper the limit mean geo-metric hardness is introduced to express the normalcontact load of a single asperity in the first andsecond elastoplastic deformation stages consideringthe change of hardness

(3) e relationship between the contact load andcontact area of a single asperity in the first andsecond elastoplastic deformation stages considering

the change of material hardness and not consideringthe change of hardness is compared respectivelyWhen the contact area of the asperity is the same thecontact load with the change of hardness is smallerthan that without the change of hardness and thedifference between them increases with the increaseof deformation

(4) e limit mean geometric hardness of a single as-perity is related to the contact area fractal di-mension and frequency exponent during theelastoplastic deformation stage and the limit meangeometric hardness increases with the increase of thecontact area When the frequency exponent isconstant the relationship between the limit meangeometric hardness and the contact area of the as-perity is related to the fractal dimension e largerthe fractal dimension is the more obvious the re-lationship curve between them is When the fractaldimension is constant the relationship between thelimit mean geometric hardness of the asperity andthe contact area is related to the frequency exponente smaller the frequency exponent is the moreobvious the change of the curve is

(5) e relationship between dimensionless total realcontact area and the dimensionless total contactloading of joint interface at each deformation stageis obtained by introducing the pushing coefficientand the critical pushing coefficient exists Whenthe pushing coefficient exceeds this value someasperities begin to deform in the next stage andthe inflection point appears on the relationshipcurve

(6) e current model still has some limitations on theapplicable scope of materials and further researchand improvement are needed to make it more ap-plicable in the future

09

08

07

06

05

04

03

02

01

000000 0002 0004 0006 0008 0010 0012 0014

Nondimensional total contact load (Flowastr)

Non

dim

ensio

nal t

otal

cont

act a

rea (

Alowastr)

nmin = 20 nmax = 36

Figure 10 e relationship between dimensionless total realcontact area and the dimensionless total contact loading for20le nle 36

10

09

08

07

06

05

04

03

02

01Non

dim

ensio

nal t

otal

cont

act a

rea (

Alowastr)

000000 0005 0010 0015 0020 0025 0030 0035

Nondimensional total contact load (Flowastr)

nmin = 44 nmax = 50

Figure 11 e relationship between dimensionless total realcontact area and the dimensionless total contact loading for44le nle 50

10

09

08

07

06

05

04

03

02

01

Nondimensional total contact load (Flowastr1)

Non

dim

ensio

nal t

otal

real

cont

act a

rea (

Alowastr1

)

0000 10 times 10ndash3 20 times 10ndash3 30 times 10ndash3 40 times 10ndash3 50 times 10ndash3

nmin = 20 nmax = 32

Figure 9e relationship between dimensionless total real contactarea and the dimensionless total contact loading for 20le nle 32

12 Advances in Materials Science and Engineering

Nomenclature

ωn Interference of the asperityωnec Elastic critical interference of the asperityωnepc First elastoplastic critical interference of the

asperityωnpc Second elastoplastic critical interference of the

asperityan Contact area of the asperityanec Elastic critical contact area of the asperityanepc First elastoplastic critical contact area of the

asperityanpc Second elastoplastic critical contact area of the

asperityfne Normal load in the elastic deformation of a single

asperityfnec Normal contact load for ω ωnecfnep1 Normal contact load of a single asperity in the

first elastoplastic stagefnep2 Normal contact load of a single asperity in the

second elastoplastic stagefnp Normal contact load of a single asperity in the full

plastic deformation stagefnep1prime Normal contact load of a single asperity in the

first elastoplastic stage considering the change ofhardness

fnep2prime Normal contact load of a single asperity in thesecond elastoplastic stage considering the changeof hardness

HG1(an) Limit mean geometric hardness in the firstelastoplastic deformation stage

HG2(an) Limit mean geometric hardness in the secondelastoplastic deformation stage

nec Elastic critical frequency exponentnepc First elastoplastic critical frequency exponentnpc Second elastoplastic critical frequency exponentAr Actual contact area of the joint interfaceAr1 Actual contact area of the joint interface for

nmin lt nle necAr2 Actual contact area of the joint interface for

nec lt nle nepcAr3 Actual contact area of the joint interface for

nepc lt nle npcAr4 Actual contact area of the joint interface for

npc lt n

Fr1 Actual contact load of the joint interface fornmin lt nle nec

Fr2 Actual contact load of the joint interface fornec lt nle nepc

Fr3 Actual contact load of the joint interface fornepc lt nle npc

Fr4 Actual contact load of the joint interface fornpc lt n

pea(a) Average contact pressure of the asperity in elasticstage

pepa1(a) Average contact pressure of the asperity in thefirst elastoplastic deformation stage

pepa2(a) Average contact pressure of the asperity in thesecond elastoplastic stage

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (grant no 51275328) and the NaturalScience Foundation Project of Shanxi Province (grant no201601D011062)

References

[1] B Zhao S Zhang P Wang and Y Hai ldquoLoading-unloadingnormal stiffness model for power-law hardening surfacesconsidering actual surface topographyrdquo Tribology In-ternational vol 90 pp 332ndash342 2015

[2] J Greenwood and J Williamson ldquoContact of nominally flatsurfacesrdquo Mathematical Physical and Engineering Sciencesvol 295 no 1442 pp 299ndash319 1966

[3] J A Greenwood and J H Tripp ldquoe contact of twonominally flat rough surfacesrdquo Proceedings of the Institution ofMechanical Engineers vol 185 no 1 pp 625ndash633 1970

[4] W R Chang I Etsion and D B Bogy ldquoAn elastic-plasticmodel for the contact of rough surfacesrdquo Journal of Tribologyvol 109 no 2 pp 257ndash263 1987

[5] Y Zhao D M Maietta and L Chang ldquoAn asperity micro-contact model incorporating the transition from elastic de-formation to fully plastic flowrdquo Journal of Tribology vol 122no 1 pp 86ndash93 2000

[6] L Kogut and I Etsion ldquoElastic-plastic contact analysis ofasphere and a rigid flatrdquo Journal of Applied Mechanics vol 69no 5 pp 657ndash662 2002

[7] I Etsion Y Kligerman and Y Kadin ldquoUnloading of anelastic-plastic loaded spherical contactrdquo International Journalof Solids and Structures vol 42 no 13 pp 3716ndash3729 2005

[8] Y Kadin Y Kligerman and I Etsion ldquoUnloading an elastic-plastic contact of rough surfacesrdquo Journal of the Mechanicsand Physics of Solids vol 54 no 12 pp 2652ndash2674 2006

[9] Y Kadin Y Kligerman and I Etsion ldquoMultiple loading-unloading of an elastic-plastic spherical contactrdquo In-ternational Journal of Solids and Structures vol 43 no 22-23pp 7119ndash7127 2006

[10] A Majumdar and B Bhushan ldquoRole of fractal geometry inroughness characterization and contact mechanics of sur-facesrdquo Journal of Tribology vol 112 no 2 pp 205ndash216 1990

[11] A Majumdar and C L Tien ldquoFractal characterization andsimulation of rough surfacesrdquo Wear vol 136 no 2pp 313ndash327 1990

[12] AMajumdar and B Bhushan ldquoFractal model of elastic-plasticcontact between rough surfacesrdquo Journal of Tribology vol 113no 1 pp 1ndash11 1991

[13] S Wang and K Komvopoulos ldquoA fractal theory of the in-terfacial temperature distribution in the slow sliding regimePart I-elastic contact and heat transfer analysisrdquo Journal ofTribology vol 116 no 4 pp 812ndash822 1994

[14] S Wang and K Komvopoulos ldquoA fractal theory of the in-terfacial temperature distribution in the slow sliding regime

Advances in Materials Science and Engineering 13

Part II-multiple domains elastoplastic contacts and appli-cationsrdquo Journal of Tribology vol 116 no 4 pp 824ndash8321994

[15] Y Morag and I Etsion ldquoResolving the contradiction of as-perities plastic to elastic mode transition in current contactmodels of fractal rough surfacesrdquo Wear vol 262 no 5-6pp 624ndash629 2007

[16] H Tian X Zhong and C Zhao ldquoOne loading model of jointinterface considering elastoplastic and variation of hardnesswith surface depthrdquo Journal of Mechanical Engineeringvol 51 no 5 pp 90ndash104 2015

[17] Y Yuan Y Cheng K Liu et al ldquoA revised Majumdar andBushan model of elastoplastic contact between rough sur-facesrdquo Applied Surface Science vol 425 pp 1138ndash1157 2017

[18] R S Sayles and T R omas ldquoSurface topography as anonstationary random processrdquo Nature vol 271 no 5644pp 431ndash434 1978

14 Advances in Materials Science and Engineering

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

Nomenclature

ωn Interference of the asperityωnec Elastic critical interference of the asperityωnepc First elastoplastic critical interference of the

asperityωnpc Second elastoplastic critical interference of the

asperityan Contact area of the asperityanec Elastic critical contact area of the asperityanepc First elastoplastic critical contact area of the

asperityanpc Second elastoplastic critical contact area of the

asperityfne Normal load in the elastic deformation of a single

asperityfnec Normal contact load for ω ωnecfnep1 Normal contact load of a single asperity in the

first elastoplastic stagefnep2 Normal contact load of a single asperity in the

second elastoplastic stagefnp Normal contact load of a single asperity in the full

plastic deformation stagefnep1prime Normal contact load of a single asperity in the

first elastoplastic stage considering the change ofhardness

fnep2prime Normal contact load of a single asperity in thesecond elastoplastic stage considering the changeof hardness

HG1(an) Limit mean geometric hardness in the firstelastoplastic deformation stage

HG2(an) Limit mean geometric hardness in the secondelastoplastic deformation stage

nec Elastic critical frequency exponentnepc First elastoplastic critical frequency exponentnpc Second elastoplastic critical frequency exponentAr Actual contact area of the joint interfaceAr1 Actual contact area of the joint interface for

nmin lt nle necAr2 Actual contact area of the joint interface for

nec lt nle nepcAr3 Actual contact area of the joint interface for

nepc lt nle npcAr4 Actual contact area of the joint interface for

npc lt n

Fr1 Actual contact load of the joint interface fornmin lt nle nec

Fr2 Actual contact load of the joint interface fornec lt nle nepc

Fr3 Actual contact load of the joint interface fornepc lt nle npc

Fr4 Actual contact load of the joint interface fornpc lt n

pea(a) Average contact pressure of the asperity in elasticstage

pepa1(a) Average contact pressure of the asperity in thefirst elastoplastic deformation stage

pepa2(a) Average contact pressure of the asperity in thesecond elastoplastic stage

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (grant no 51275328) and the NaturalScience Foundation Project of Shanxi Province (grant no201601D011062)

References

[1] B Zhao S Zhang P Wang and Y Hai ldquoLoading-unloadingnormal stiffness model for power-law hardening surfacesconsidering actual surface topographyrdquo Tribology In-ternational vol 90 pp 332ndash342 2015

[2] J Greenwood and J Williamson ldquoContact of nominally flatsurfacesrdquo Mathematical Physical and Engineering Sciencesvol 295 no 1442 pp 299ndash319 1966

[3] J A Greenwood and J H Tripp ldquoe contact of twonominally flat rough surfacesrdquo Proceedings of the Institution ofMechanical Engineers vol 185 no 1 pp 625ndash633 1970

[4] W R Chang I Etsion and D B Bogy ldquoAn elastic-plasticmodel for the contact of rough surfacesrdquo Journal of Tribologyvol 109 no 2 pp 257ndash263 1987

[5] Y Zhao D M Maietta and L Chang ldquoAn asperity micro-contact model incorporating the transition from elastic de-formation to fully plastic flowrdquo Journal of Tribology vol 122no 1 pp 86ndash93 2000

[6] L Kogut and I Etsion ldquoElastic-plastic contact analysis ofasphere and a rigid flatrdquo Journal of Applied Mechanics vol 69no 5 pp 657ndash662 2002

[7] I Etsion Y Kligerman and Y Kadin ldquoUnloading of anelastic-plastic loaded spherical contactrdquo International Journalof Solids and Structures vol 42 no 13 pp 3716ndash3729 2005

[8] Y Kadin Y Kligerman and I Etsion ldquoUnloading an elastic-plastic contact of rough surfacesrdquo Journal of the Mechanicsand Physics of Solids vol 54 no 12 pp 2652ndash2674 2006

[9] Y Kadin Y Kligerman and I Etsion ldquoMultiple loading-unloading of an elastic-plastic spherical contactrdquo In-ternational Journal of Solids and Structures vol 43 no 22-23pp 7119ndash7127 2006

[10] A Majumdar and B Bhushan ldquoRole of fractal geometry inroughness characterization and contact mechanics of sur-facesrdquo Journal of Tribology vol 112 no 2 pp 205ndash216 1990

[11] A Majumdar and C L Tien ldquoFractal characterization andsimulation of rough surfacesrdquo Wear vol 136 no 2pp 313ndash327 1990

[12] AMajumdar and B Bhushan ldquoFractal model of elastic-plasticcontact between rough surfacesrdquo Journal of Tribology vol 113no 1 pp 1ndash11 1991

[13] S Wang and K Komvopoulos ldquoA fractal theory of the in-terfacial temperature distribution in the slow sliding regimePart I-elastic contact and heat transfer analysisrdquo Journal ofTribology vol 116 no 4 pp 812ndash822 1994

[14] S Wang and K Komvopoulos ldquoA fractal theory of the in-terfacial temperature distribution in the slow sliding regime

Advances in Materials Science and Engineering 13

Part II-multiple domains elastoplastic contacts and appli-cationsrdquo Journal of Tribology vol 116 no 4 pp 824ndash8321994

[15] Y Morag and I Etsion ldquoResolving the contradiction of as-perities plastic to elastic mode transition in current contactmodels of fractal rough surfacesrdquo Wear vol 262 no 5-6pp 624ndash629 2007

[16] H Tian X Zhong and C Zhao ldquoOne loading model of jointinterface considering elastoplastic and variation of hardnesswith surface depthrdquo Journal of Mechanical Engineeringvol 51 no 5 pp 90ndash104 2015

[17] Y Yuan Y Cheng K Liu et al ldquoA revised Majumdar andBushan model of elastoplastic contact between rough sur-facesrdquo Applied Surface Science vol 425 pp 1138ndash1157 2017

[18] R S Sayles and T R omas ldquoSurface topography as anonstationary random processrdquo Nature vol 271 no 5644pp 431ndash434 1978

14 Advances in Materials Science and Engineering

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

Part II-multiple domains elastoplastic contacts and appli-cationsrdquo Journal of Tribology vol 116 no 4 pp 824ndash8321994

[15] Y Morag and I Etsion ldquoResolving the contradiction of as-perities plastic to elastic mode transition in current contactmodels of fractal rough surfacesrdquo Wear vol 262 no 5-6pp 624ndash629 2007

[16] H Tian X Zhong and C Zhao ldquoOne loading model of jointinterface considering elastoplastic and variation of hardnesswith surface depthrdquo Journal of Mechanical Engineeringvol 51 no 5 pp 90ndash104 2015

[17] Y Yuan Y Cheng K Liu et al ldquoA revised Majumdar andBushan model of elastoplastic contact between rough sur-facesrdquo Applied Surface Science vol 425 pp 1138ndash1157 2017

[18] R S Sayles and T R omas ldquoSurface topography as anonstationary random processrdquo Nature vol 271 no 5644pp 431ndash434 1978

14 Advances in Materials Science and Engineering

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom