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Scientific RepoRts | (2019) 9:7791 | https://doi.org/10.1038/s41598-019-44269-1
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Effect of elastic grading on fretting
wear
Emanuel Willert
1 , Andrey I. Dmitrie v 2,3 , Sergey G. Psakhie 2,3 & Valentin L. P opov
1,3
W e consider fretting wear in elastic frictional contact under influence of oscillations of small amplitude
and investigate the question, how wear damage can be influenced by the introduction of material
gradients. T o achieve a general understanding we restrict our consideration to media with a power -law
dependency of the elastic modulus on depth. In this case, a complete analytical solution can be found
for the final worn shape. In the limiting case of small fretting oscillations we obtain a simple, closed-
form asymptotic solution of the problem. W e find that the optimum grading depends on the oscillation
amplitude: for large amplitudes, the use of materials with a positive exponent decreases the wear
volume whilst for very small amplitudes the use of graded materials with slightly negative exponent
is beneficial. Especially interesting is the case of the Gibson-medium which may help avoiding both
fretting wear and fretting fatigue.
Fr etting w e ar an d fatigue rep resen t a considerab le and lon gstanding p rob lem in app lication s using frict ional
con tacts subjected to vibra tions a s e.g. fretting o f tubes in steam g enerato rs and h eat exch anger s 1 – 3 join ts in ortho -
paedics 4 , electr ical connector s 5 , and do vetail blade roo ts of gas turb ines 6 , 7 . The ph ysical reason f or the fret ting is
partia l sliding in the vicini ty of boundar y of a frictional con tac t. I t is d ue to the vanishing no rmal pres sure a t the
con tact b oundary in con tac ts with cur ved surfaces 8 . T ang ential osci llation s withou t slip wo uld cause a stres s sin-
gularity a t the con tac t boundary ; hence, f or an y finit e coefficient of friction, there will be some slip r egion in the
vicinity of the co ntact boundary . This partial slip and the r esulting frettin g wear can onl y b e pr e ven ted by usin g a
con tact with sharp edges. H owever , in this cas e, both normal an d t ang ential stresses will be singular at the bound -
ar y and oscillating str esses wi ll lead to f ret ting fatigue. Th us, ap plicatio ns with fric tional con tacts under vibra t ion s
sho u ld find an o ptimal path between S cy lla of fretting w ear and Chary bdis of fretting fa tigue.
A possible solu tion of the w ear/fatigue dilemma ma y be t he use of functionally graded ma teria ls (FGM). The
pr oblem of fret ting wear is d ue to the int er play o f normal and ta ngen ti al stresses. The distrib ution of the n or -
mal stresses is, ho wever , governed b y deep er parts of the ma teria l com pared with tang ential stress co mpon ents.
Changing the e lastic mo dul us of the surface lay er of the ma teria l or – mo re generally – in troducing m aterial
gradients co uld help solvin g the dilemma. FGM are fa irly commo n for m an y biological and na tural structures.
They hav e be en studied since the la te 1950s 9 , initially in con text of g eomechanics 10 and lat er in con text of variou s
engineering a pplica tions. I t has be en sho wn t ha t FGM can pr ovide better mecha nical pro p erties including w ear
and dam age resis tance than hom ogeneous ma terials 11 , 12 . In terest in FGM was enha nced by establi shing new man -
ufacturing techniq ues (3D prin t ing 13 ), which allo w man ufacturing of ma terials with basically arbi trar y spatial
distribu tion of mec hanical pro per ties. Ho wever , the p ossib ility of solvin g t he fret t ing wea r/fatigue dilemma by
using FGM has n ot been studied so far .
The necessary ingredients o f an y theor y of fretting w ear and fa tigue are sol utions o f normal and ta ngen ti al
con tact prob lems. Fo r FGM with power -la w and exponen tia l dependency of elastic moduli on dep th, t hese solu-
tions ha ve been fo und by Bo oker et al . 14 , 15 and Giann a ko p oulos & S uresh 16 , 17 . Sur esh et al . reported in a series of
p ap e r s 16 – 19 that the dama ge and fa ilure r esistance of graded glass/ceramic surfaces to n ormal and sliding co ntact
or im pact can b e chan ged significantly co mpar ed with the ceramic or glass co nstituen t. An analytical solu tion fo r
the con tact prob lem of a functionally graded coating wi t h ma terial pro p erties var y ing as a n exp onen tial func tion
was ob t ained in 20 . T wo-dimensio nal fr ictionless and frictional con tac t pr oblems ha ve been studied in 21 using the
linear m ulti-layer ed model. In 22 a uthors a pplied the linear m ulti-lay ered model for a nalysing the two-dimensio nal
fretting co ntact of functionally graded coat ed half-spaces. L iu et al . 23 use d a similar a ppr oach to stud y the f ret ting
1 T echnische Universität Berlin, 10623, Berlin, Germany . 2 Institute of Strength Physics and Materials Science SB RAS,
634055, T omsk, Russia. 3 National Research T omsk State University , 634050, T omsk, Russia. Correspondence and
requests for materials should be addressed to E. W . (email: e. [email protected] ) or V .L.P . (email: v .popov@tu-
berlin.de )
R eceived: 1 1 September 2018
A ccepted: 13 May 2019
P ublished: xx xx xxxx
opeN

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con tact of two elastic bodies under torsio n. W ang et al . in 24 suggested an efficien t modelling a ppr oach for sol ving
three-dimensio nal f ret ting con tac t in volving m ulti-la yered ma terials and functionally graded coatings.
Compa red with other p ublicatio ns in the field, her e we concen trat e our at ten t ion o n the final sha pe which
develops d ue to fretting w e ar an d inv estigate this sha pe for pow er-la w graded materials in the whole allowed
int e r va l of the exponen ts of the pow er law . W e tr y to find the a nswer to the q uestion, if it is possib le to reduce o r
even completel y ex clude frettin g wear by introd ucing gradients o f elastic moduli. T o achieve a g eneral qualita tive
understa nding we co nfine ourselv es to the case of axisymmetric con tacts. U nder these assum ption s it occurs to
be possible to pr ovide lar gely a nalytic s ol utions o f the fretting p roblem. W e will show tha t the character o f spatial
distribu tion of elas t ic pr opert ies essentially influen ces the limiting sha pe of the worn p rofile. B y correctly choos-
ing the ma terial gradient one ca n, depending on the frettin g amp litude, in dee d achieve a significan t reduction o f
wear a nd, under some cir cumstances, the wear -less beha viour .
The cen tra l idea of the a ppr oach use d in this paper to determine the limitin g worn pr ofile is ver y sim ple and
robust. In 25 , i t was shown tha t, if two bodies are pr essed against eac h other and sub jec ted to small-am plit ude
tangen tial osci llation s, the worn pr ofile develops in time t ending to some limi ting sha p e. I n t he fo l low-u p paper 26 ,
it was a rgued that the limi t ing sha pe as found in 25 i s a universal one determined solely by nor m al contac t proper -
ties of the medium a nd valid e ven fo r mul tiple-mode frettin g. Mo reov er , the final worn sh ape obtain ed by using
the method of dimen sionality r eduction (MDR) 27 wa s shown to be in exce llent agr e emen t with experimental
resul ts 26 . Ho wever , at tha t time, only the MDR v ersion a pplicab le to spatially hom ogeneous media was a vailable.
In the mean time, H eß 28 , base d on the general axisymmetric normal co ntact solutio n for pow er-la w elastic grading
by J in et al . 29 , has de velo p ed the MDR to incl ude funct ionally graded mat eri als. H ere we use the solu tions fo und
by H eß to analyse the limiting fret ting shape o f grade d ma teria ls.
Problem Formulation and R esults
Limiting profile shape in fretting wear of functionally graded bodies. The basic idea of determin-
ing the limi ting sha p e rem ains the same as in t he case of ho mogeneo us media. Let us first sta te the pr oblem and
the main ass umptio ns we accept in the p resent pa per . All deformatio ns shall be elastic, the contacting bodies
shall be elastica lly similar (to ens ure elastic decou pling) and the w e ar la w shall b e loca l. A par t from tha t we do
not as sume an y partic ular wear and friction laws b ut only mak e the following v er y gen eral assump tions abou t the
local wear rat e and fric tional fo rces:
a. W e ar sho uld only occur in the contact ar eas with non-va nishing re lative dis placement o f surfaces (thus
mere ly existence o f tangen tial stresses is not eno ugh for ini ti atin g the wear pr o cess).
b . W e ar sho uld o ccur only in the con tact areas with no n-vanishing p ress ure.
c. The law of friction (which does not necessarily ha ve to be the Coulomb la w) allows fo r t he exist ence of a
regio n of permanen t stick.
As a rgued in 26 , already these assum ptions una mbiguo usly determine the limiting w orn sha p e. F rom the exis t -
ence of a r egion of permanen t stick – ass umptio n (c) – it follo ws that in this r egion there will be no wear –
assum ption (a) – a nd that the sha pe of the inden ter in this region will coincide wi t h the initial non-w orn shape.
Outside of the r egion of permanen t stick, the wear can o nly vanish if the p ressur e reduces to zer o – assum ption
(b). This means, tha t the no-con tact conditio n must be fulfille d. H ow ever , as this form has t o b e achieved due
to we ar , the limiting p rofile o f the indent er must exactly co incide with the form o f the f ree sur face (no-pres sure
condi tion!) p roduced b y t he ini ti al inden ter sha pe inside the per manen t stick r egion. Th us, in the final stat e, in the
slip r egion, the inden ter and the elastic half space will be finally in the state o f “incipien t con tac t ” . This logic does
not depend o n w hether the ma teria l is hom ogeneous or h eterogeneo us, and it i s not confined to axi symmetr ic
con tact configuratio ns: The final wo r n sha pe is basica lly determined by the sol ution o f the normal con tac t pr ob-
lem under the action of the p rofile defined o nly in the p ermanen t stick regio n.
W e consider a co ntact of an axisymmetric rigid indent er with an elastic half-space, whose Y o ung’ s modulus
varies with depth acco rding to the po wer law
=− <<
Ez Ez k () ,1 1, (1 )
k
0

where z is the vertical co or dinat e measured fro m the surface, E 0 is a n arbi trar y p ositiv e consta nt a nd k is the expo -
nen t ( k = 0 corres p onds t o a homogen eous ma teria l). N ote, tha t the con tac t of two e lastic b odies do es not exhi bit
qualita tively diff eren t feat ures, altho ug h in this case k must be the same f or both mat eri als.
In e lastic con tac ts subjected to oscillation s with a small amp litude Δ u (0) , ther e generally exists a n inner area
of permanen t stick (with radi us c ) and the o uter ar ea (radius a ) where ther e is slip a t least during so me part of the
oscill atio n per iod.
As described above, the f orm of the limitin g pro file, given a permanen t stick radius c , is o nly dep ending o n the
s olut ion of th e nor mal contact pr oblem. I n 25 , it was sho wn that the limi ting worn pr ofile has a n espe cially simple
fo rm in t he “ MDR -space ” . The MDR -fo rmalism for axisymmetric no rmal con t acts of pow er-la w graded elastic
bodies is det ailed in the “ Methods ” s ection. The limiting p ro fi le in the “ MDR -space ” has the form 25
=










≤
<≤
∞
gx
gx xc

dc xa
()
() ,f or

,f or ,
(2 )

0
where
g x ()
0

is the equivalen t pr ofile corr esponding t o t he initial (no n-worn) sha pe and d the inden tatio n depth.
This fo rm gu aran ties that ther e is no wear in the ar ea of per manen t stick (because the sha p e is uncha nged for
≤ xc

and th us, accor ding to Eq. ( 20 ), fo r
≤ rc

) and tha t the pres sure ou tside the area o f p ermanen t stick van -

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ishes (acco rding to Eq . ( 21 ), w here w e hav e to insert
′= g 0

starting with x = c ). Mo reo ver , from Eq. ( 20 ) f ollows
that the diff erence between the worn a nd the non-wo rn pro file in t he 3D-doma in wil l be
∫
π
π
Δ= −= 









−
− <≤
∞ + ∞
fr fr fr k xd gx
rx xc ra () :( )( ) 2 co s 2
[( )]
()
d, fo r,
(3 )

c
r
k

k
0 0
22 1
whereas the limi t ing co ntact radius a fter the fretting has t o b e determined from the co ndition
Δ= =.
∞
fr a () 0

The to ta l wear vo lume is giv en by the in tegral
∫
π Δ= Δ
∞
Vf rr r 2( )d
(4 )
c
a

Let us consider a n initial non-wo rn pro file of the general power -l aw f orm
= fr Ar () , (5 )
n
0

with some co nstan t A and a positive expo nent n . The tra nsformed p rofile, acco rding t o Eq. ( 19 ) is given b y
κκ ==






+ 





gx nk Ax nk
nn k

() (, ), (, ):
2

B
2

,
1
2

,
(6 )

n
0
w he re
=
ΓΓ

Γ+
ab
ab

ab
B( ,) :
() ()

()
,
(7 )

is the com plete Beta f unction and Γ the Ga mma funct ion. W ith accoun t of the con dition
= dg a () (8 )
0 0

Eq. ( 3 ) gives the limiting sh ape in the worn a nnulus c < r < a ∞ :
π
π
π
π
Δ =− +



 











++ +







+ ++












× 

  



 










++ ++ + 







  
− 




++ ++ + 







 
+

++
fr
d
k
k
c
r
kk kc
r
k
nk
r
a
c
r
kn kn kc
r
kn kn k
() 1 2c os (/ 2)
(1 ) F 1
2 , 1
2 ; 3
2 ;
2c os (/ 2)
(1 )
F 1
2 , 1
2 ; 3
2 ;
F 1
2 , 1
2 ; 3
2 ;1 ,
(9 )
k

n
nk
1

1 2
2

2
0
1
1 2
2
2
1 2
where we u s ed the hyperg eometric func tion
∑
=
Γ+ Γ+ Γ

ΓΓΓ + ≤.
=
∞

ab cz
an bn c
ab cn
z
n

z F( ,; ;) :
() () ()
() () () !

,1
(10)
n
n

1 2
0

N ote, tha t the above r esults ar e independent o f the precise frict ion o r wear laws, as lo ng as the assum ption s
(a) – (c) stat ed in t he in troductory s ect ion ar e met. H ow e ver , t he con trol pa rameter in fret t ing is no t the radius c
of the permanen t stick zon e, but ra ther the am plit ude of the tang ential fretting oscillation, Δ u (0) . T o est abli sh a
rela tion between c and Δ u (0) , we ha ve to sta te a certain f riction law . U nder assump tion of Coulom b ’ s law in local
fo rmulatio n with the co efficient o f fr iction μ , the radius c o f the stick area is giv en by 30
μ
μ
=






 − Δ







 Δ≤ =. gc d Mu
d uu d
M
() 1, fo r:
(11)

(0 ) (0 ) ma x
(0 )
H ere M is the ra tio of tangen tial to normal stiffness – the oft en s o-calle d “ Mindlin ratio ” . As the rigid indenter
and the elas tic ha lf-space shall be elastical ly similar , M reduces to 31
= ++
. M
kk
2
(1 )(3 ) (12)

If
Δ> uu
(0 ) ma x
(0 )

, the con tact enter s the regime of full slidin g. Most tribological system s suffering fro m fretting
ar e subjected to small fretting a mpli tudes. The ar ea of local slip will in thes e cases b e small com pared to the co m-
plete co ntact ar ea and it is possib le to derive an asym pt otic closed-form solu tion fo r this limit from the a b ov e
resul ts. In troducing the small param eters

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εε
μ
= − = − ≈ Δ
∞ 
ac
c
ac
c
Mu
nd

:1 ,: 1,
(13)

a 0
(0 )

carr ying ou t exp ansio ns and n egle cting all terms of second or higher o rder in ε or ε a , leads to the rela tion
εε ≈
− k 3
2

,
(14)

a
which, in terestingly , is o nly dep ending o n the p ow er-la w grading and n ot on the inden ting pr ofile. The t otal wear
vol ume after the fret ting pr o cess is in this limit a pp ro ximate ly
π

ε Δ≈ −−−











 −.
−
V
nc d

kkk
k

k
8

(1 )(5 )(3 ) co s 2 [( 3) ]
(15)

a
k
2

2
5 2
Parabolic contact. L et us a pply the ob tained results t o t he “ generic case ” of parabolic co ntact with a radius
of cur vat ure R . The three-dimen sional un worn pr ofile in the vicinity o f t he con tact,
= fr
r
R

()
2

,
(16)

0
2

will readily give the equivalen t pr ofile
= + . gx x
Rk

()
(1 ) (17)

0
2

The initial con tact radius and the radi us of the permanen t stick ar ea are accor ding to Eqs ( 8 and 11 ) equal to
μμ
=+ =+





 − Δ 





 =−
Δ . ad Rk cd Rk Mu
d a Mu
d
(1 ), (1 )1 1
(18)

0
(0 )
0
(0 )
Figure 1 shows the limi ting pr ofile sha pe in normalized variables fo r differen t values o f k . The pro file has been
normalised fo r t he inden tation dep th and the radial co or dinat e for the initial con tact radius in the ho mogeneous
cas e,
Rd

. In these variab les the limiting pr ofile onl y dep ends o n k and the normalized frettin g amp litude , w hich
in Fig. 1 for illustra tion has been chosen to be
μ Δ= .. ud /( )0 5
(0 )

W e can easily recognize se veral fea tures: firs t t he con tact radius is incr easing with increasing k ; second the
slip a rea is pr opaga ting slo wer from the edge o f con tact inside with increasin g k , be cause the Min dlin ratio M is
decreasing wi t h k ; finally the worn vo lume seems to decrease with incr easing k . This, how e ver , is only a part of the
com plete p icture which is p resented in detail in Fig. 2(a,b) in form o f “wear ma ps ” . In these figures the to ta l wear
vol ume, no rmalize d fo r t he value in the ho mogeneou s case, as a funct ion o f the two rema ining paramet ers, the
exponen t k and the n ormalized f ret ting am plitude , is shown in co nt our isoline plo ts; the left hand side diagram
(a) gives the semi-analytic sol ution based on the exact eqs ( 4 and 9 ), and the right ha nd side figure (b) sho ws the
closed-form asym pto tic s olu tion ( 15 ).
Figure 1. Limiting p rofile sha pe after fret ting normalized for the inden tatio n depth as a function of the
normalized polar radi us for a para b olic inden ter with radiu s R for diff erent val ues of exponen t k of the pow er-
law grading. N ormalize d fretting a mp litude is 0.5. Blac k line denotes the un wo r n pr ofile.

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F or int ermedi a te and lar ge fretting a mpli tudes it i s obviou sly beneficial to use a grade d ma teria l with positive
exponen t k , i.e. a soft surface wi th a hard co re. F or nega tive k , i .e. har d surfaces, the wear vo lume is dras tic ally
increased. H ow e ver , for ver y small oscill a t ion a mpli tudes, it migh t be b eneficial to use a grading with slightly
nega tive k . N ote , nonetheless, tha t the con tact stresses in the case of negati ve elastic grading can be significan t ly
incr e ased if the har d surface is ver y thick and the soft co re is i s ola ted from carry ing a r elevant a moun t of the load.
This will accelerat e the wear pr ocess (although withou t chan ging the limiting p rofile if the inden tation dep th is
pr escr ibed) and ma y enhance fretting fa tigue.
Another s hape which is of in ter est for p ract ical ap plicatio ns is the cone-p rofile . As it t urns out, ther e are , how -
ever , no q ua lita tive differ ences bet ween the parabolic a nd conical case (we therefo re wo n ’ t sep ara tely p resent the
resul ts for co nica l con tact to avo id unnecessar y repetition), which indica tes, that the two f ollowing co nclusio ns
generally ho ld tr ue, independent o f the indent er shape:
• F or int ermedi a te and lar ge fretting a mpli tudes, the usage o f positive elastic grading (i .e. a soft surface with
a har der core) r educes wear; usage of nega tive elas tic g rading (ha rd surface with a softer co re) drastically
increases wear .
• F or small f ret ting am plitudes i t may be beneficial to use (slightly) n egative e lastic g rading.
Some remarks on the choice of control parameters. T o avo id confusion let u s add s ome co mmen ts on
the cho ice of fixed param eters in the considera tions a bove.
The limitin g pro file after frettin g wear normalized fo r the indenta tion depth will only depend on the expon ent
k of the elas t ic grading, the exponen t n of the power -l aw p ro file and the ratio
c a / 0

. This is clear ly indicat e d by Eq .
( 9 ), which has been derived under ver y general ass ump tions. H ow e ver , if instead of the inden tatio n depth d , for
exam ple, the no rmal for ce is pr escr ibed, the elastic grading will also influence d and the “w ear maps ” sho wn above
will lo ok differ ently . M or e ov er , they wi ll depend on mor e parameter s (e.g. the grading thickness), which is why
we chose the disp lacement-co ntro lle d form ulation. Th e chang es for fo rce-co ntro l can be e asily im plement ed
based on the normal con tact s ol ution fo r power -law graded elastic bodies.
As said befo re, the radi us c of the permanen t stick area is a ra ther imp rac tical choice as a co ntro l parameter ,
as the pr escr ibed variable to cha racterize t he tang ential loading in fretting in most cases will be t he tang ential
fretting a mp litude. T o incorporat e this, we had to assum e a cer tain fric tion la w , in vol ving the coefficient of fric -
tion (COF) μ . I n the results described above , t he COF was kep t cons tant (altho ugh it easily may be infl uenced
by the e lastic grading as well 32 ) because of tw o reasons: firs t, it is har d to find a syst ematic re lation between the
COF and the elas tic g rading an d s econdly w e want ed to study the infl uence of the elastic grading o n the limiting
sha pe after fretting w ear on a “ fair ” basis, which requir es the C OF to be same in the graded and in the ungraded
case. H owever , when app lying our r esults, one o f course has to co nsider that the COF ma y be altered b y the elastic
grading.
Some remarks on the possibility of wear -less behaviour . In the end we wo uld like to give some
rema rks on the in triguing question o f whether it is possible t o achieve “ no-wear ” conditio ns in frettin g contacts
via functional elastic grading. Fr om the basic ass umptio ns sta ted in the introd uc to r y section it is clea r , that this
inevitabl y requir es c = a , or in other terms, M = 0, i.e. the Mindlin ra tio of the elas t ic ma terial must va nish. This
condi tion can be fo rma lly fulfille d in the case for the Gibso n medium (i.e. k = 1 and ν = 0.5), w hich can be sho wn
by calculating the limi t
→ k 1

and
ν →. 05

in the general exp ression s for the no rmal and tang ential stiffness that
Figure 2. Cont our isoline plo ts of the total wo rn-off vol ume normalized fo r the value in the homog eneous
case for the fret ting wear of a pa rabolic indent er on a power -l aw graded elastic half-space as a function o f the
exponen t of elastic grading k a nd the normalized fretting a mpli tude. ( a ) Semi-analytic solu tion based on Eqs
( 4 ) and ( 9 ); b lack line denotes the tran sition to co mp lete sliding. ( b ) A sympt otic solutio n for small frettin g
am plit udes accordin g to Eq. ( 15 ).

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ha ve been publish ed by H eß & P opov 30 . W e stres s, however , that in this case the ass umptio n of elastic similari ty
used throughou t the presen t paper , is viola ted. N evert heless the tan gential con tact stiff ness due t o elastic coup ling
with the normal co ntact pr oblem should us ua lly still be sma ll.
N ote tha t the use of a Gibson-M e dium co uld b e a possible solu tion fo r both pro blems of fr etting wear a nd
fretting fa tigue as even the con tacts of sharp-edged pr ofiles with a Gi bs on-medi um do not ha ve an y stress co ncen -
tratio n in the vicinity of the con tact b ounda r y , which is the main r eas on fo r fretting fa tigue.
Discussion
In co nclusio n, we st udie d the final (station ar y) state o f f ret ting wear fo r a r igid indent er pres s ed into a function-
ally graded elastic half-space and sub je cted to tang ential osci llatio ns (the results o btained can stra ig h t fo r war dly
a pplied fo r differen t fretting m o des as well). W e pr esented an a nalytical s olu tion fo r the limiting inden ter sha pe
after the fret ting pr o cess, from which the w ear vol ume can be determined by sim ple numerical in tegration. F or
the limiting case o f ver y smal l oscill atio n am plitudes w e gav e a close d-fo rm asympt otic solutio n for the w e ar
vol ume. P arabolic an d conical con tac ts hav e be en studied in detail.
W e find that fo r int er mediat e and larg e f ret ting am plitudes, the usag e of elastic grading with a soft s ur face and
a har der cor e reduces wea r (w her e as grading with a ha rd surface an d a s ofter co re dras tic ally increases it) . For
ver y small fretting a mpli tudes the usage o f elastic grading with sligh tly p ositiv e exponent m ay be fav ourable . A
possible solu tion to co mplet ely av oid both fretting w ear and fret ting fatigue ma y be t he use of a Gi bs on-medi um.
As s tated befor e, our a nalysis is based on the assum ption o f purel y elastic con tact deforma t ion s. Ho wever , due
to the stres s concen tration a t the edge of the permanen t stick ar ea, t he immediat e vicinity of the stick r egion is
pr one to p lastic deforma tions. The eff ec t of these plastic defo rmation s on the fretting beha viour has been studied
in detail by H u et al . 33 . They found tha t plastic deforma tions m ay allow the wear scar t o con tinuousl y pro pagat e
in to the con tac t ar ea as t he stick-sli p boundar y extends in to the original stick a rea. The wear p rocess in this case
never ceases. N ote, ho wever , that incr easing values of the expo nent k r educe the stress sin gularity . Hence , the
usage o f FGM might allow to post p one o r even to com pletel y preven t this effect of plastici ty . This, no netheless, is
object of further research. M oreov er , in some a pplica tions the wear la w might be discon t in uous or no n-loca l (e.g.
due to the size o f the debris particles), which would r ender our abo ve findings ina pplica ble.
Methods
MDR -formalism for axisymmetric normal contacts of power -law graded bodies. L et us briefly
reca pit ulate the solu tion of the no rmal con t act pro blem with a graded material in the framewor k of MDR. N ote
that MD R pro vides an exact ref ormula tion of the full solutio n of the axisymmetric con tac t pr oblem in terms o f a
sim ple interp retatio n. No a pp roxim ation o r loss of inf ormatio n is inv olved. A ccor ding to the gen eral r ules of the
MDR, apart fro m the original t hree-dimensio nal axisymmetric pro file
fr ()

( r is the polar radius in the co ntact
plane), a m o dified pr ofile
g x ()

is defined as derived b y Heß 28 :
∫
=
′

−
−
−
gx x
fr

xr r ()
()

()
d,
(19)

k
x

k
1
0 22 1
where the p r ime deno tes the first deriva tive. The in verse transf orma tion reads 28
∫
π
π
= 








 − .
+
fr k xg x
rx x () 2 co s 2
()
()
d
(20)

r
k

k
0 22 1
The pr essur e distribution in the o riginal t hree-dimensio nal system can be calcu lat ed f ro m
∫
π
=− ′
− −
pr
c

xg x
xr x () ()
()
d,
(21)

N
r
a
k

k 22 1
where
c N

is a con stant dependin g on the ma teria l pr oper ties. If the rigid inden ter is pr essed into this elastic
half-space by a penetra tion depth d , the con tact radius a is determined b y t he con dition
= dg a () (22)

Eqs ( 19 , 21 and 22 ) solve the axisymm et ric normal con tact prob lem for po wer -law graded elastic bo dies.
Conclusions
W e studied the limiting p rofile sha pe in axisymmetr ic con tacts w ith pow er-la w elastic grading under
small-am plitude fr etting condi tions. I t has been shown in the lit eratur e that the st eady state (i .e. the limiting
pr ofile) exists a nd is universal with r espec t to the loading co ndition s, if the con tac t obeys t he limita tions o f the
Cat taneo-Mindlin ap pr oxima tion and if the w ear law is local. W e find that f or in termediate a nd large f ret ting
am plit udes, the usage of positi ve elastic grading (i.e. a so ft surface with a harder co re) r educes wear; us age o f
nega tive elastic grading (ha rd surface with a softer co re) drastically increases wear . Fo r small f ret ting am plitudes
it ma y be beneficia l to use (slightly) n egative e lastic grading.
Data A vailability
N o additional da ta (other than sta ted in the man uscript) was p roduced o r used for the pr epara tion of the ma n -
uscript.

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Acknowledgements
This resear ch was partia lly funded by the Deutsch e For schungsg emeinschaft (D FG, PO 810/53-1), by the T omsk
Sta te U niversi ty competitiv eness im pro vemen t pr ogramme, b y t he Deutscher Akademischer A usta uschdienst
(D AAD), and b y the Progra m for Basic Scientific Research o f the R AS on 2013–2020, P roject III.23.2.4.W e also
ackno wle dge su pport by the Open A ccess Publicatio n Fund o f TU B erlin.
Author Contributions
All au thors co ntribu ted to the man uscript. E.W ., S.G.P . an d V .L.P . pro vided analytical results, A.I.D . and E.W .
performed the num er ical calc ulation s. The man uscript was p repared b y V .L.P . and E.W .
Additional Information
C omp et ing Interes ts : The au t hor s declare no co mpeting in terests.
Publisher’ s not e: Spring er Na tur e remain s neutral with rega rd to j ur isdictional claims in p ublished ma ps and
insti tution a l affilia tions.

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