FACTA UN IVERSITATIS
Serie s: Mechanical Enginee ring Vol. 16, N o 1, 20 18, pp. 9 - 18
http s://doi.org/10.2 2190 /FUME 171121003W
© 2018 by Uni versi ty of Ni š, Serbia | C reativ e Com m on s Li cen ce: CC BY- NC - ND
Original scien tific pa per 
DUGDA LE-MAUGIS ADHESIVE NORMAL CONTACT OF
AXISYMMETRIC P OWER-LAW GRADED ELASTIC BODIES
UDC 539.3
Emanuel Willert
Ber l in Un i v e rs i t y of Tec h n ol og y , Ber l in , G er m an y
Abstra ct . A closed -form general analytic soluti on is presented for the adh esive norma l
contact of convex axisymm etric power-law graded elastic bodies using a Dugdale -
Maugis mo del fo r the a dhesive stress. The case o f sph erical conta cting bod ies is
studied in detail. The k nown JKR- and DMT-li mits can be derived from th e general
solutio n, whereas the transition between both can be captured introducin g a
generalized Tabo r parameter dependin g on th e materia l g rading . T he influen ce of th e
Tabo r parameter an d the ma terial grad ing is studied .
Key Word s : Adhesive Conta ct, Power -Law El astic Grad ing, Du gda le-Maug is Model,
Axial Symmetry, Method o f Dimensiona lity R edu ction
1. I NT RODUCT I ON
Pro pelled by the techn ological demand for versatile high -performance m aterials and
the study of b iological materials and contact solutions, living nature develop ed in several
circum stances, Functionally Gr aded Materials (FGM) , i.e. media w ith continuously
inhom o geneous mechanical p roper ties, have encountered a lot of scientific interest and
research in the p ast years. T he use of FGM is pro ven to b e po ssibly beneficial in p hy sical
[1] and b iological [ 2] ap plications. W hereas r igorous solutions for non -adhesive contact
prob lem s o f FGM , at least for some special form s of inhom o geneity , have been available
for quite a long tim e [3-5], the ad hesive contact o f FGM is still in the focus o f current
research [6-9]. T hese latter studies, n onetheless, on ly concern th e lim iting case of a
neglig ible range of the adhesive interaction, established by Johnson, Kendall and Rob erts
(JKR, [ 10 ]) in 1 971 . After Der jaguin, Muller and T opo rov (DMT , [ 11 ]) a few years later
presented a different theory of lo ng-range adhesive interactions giv ing a d iff erent result

Receive d Nove mber 21, 201 7 / Accepted January 11, 20 18
Corresponding author : Emanuel Wil ler t
Te chnisch e Univer sität B erl in, Sekr. C8 -4, Straße des 17. Jun i 13 5, D-10 623 Berlin
E-mail: e .wil ler t@tu -berlin.de

10 E . WI L LERT
fo r the critical pull -off force in a parab olic contact, a discussion started, w hich was only
finally r esolved by Maugis [ 12 ] , wh o – based on a m odel o f the ad hesive stress first
introduced by Dugdale [ 13 ] – w as able to show the transition b etw een w hat was pr oven
by T abo r [ 14 ] to be correc t d escriptions of limitin g cases. T he p resent pap er generalizes
Maugis’ solution for the adhesiv e contact of homogen eous spheres to arb itrary
axisy mm etric bod ies with elastic-grading in form of a po w er -law . As the contact prob lem
of interest can be ascribed to the frictionless, non-adhesive norm al contact of pow er -law
graded elastic materials a solution proce dure based on the Method o f Dimensionality
Reduction (MD R) can be ap plied.
2. G ENERA L A XI SYMMETRI C S OL UTI ON
We consider elastic grading of the Yo ung modulus E with dep th z in form of a po w er-
law :

0
0
( ) , 1 1.
k
z
E z E k
z

   



(1)
Thereb y constants E 0 and z 0 as w ell as Poisson ratio ν may b e different for the co ntacting
bodies. E xponent k , however, needs to be the same for both of them . As the exponent m ay
take po sitive or negative v alues, both soft surfaces w ith a hard co re and hard surfaces with
a soft core can b e studied.
It has been show n that the frictionless nor m al contact of ax isy m m etr ic pow er -law
graded elastic bod ies can be exactly mapped onto a plain co ntact of a rigid pr ofile g with
a one-dim ensional foundation of independent linear spr ings, each in distant Δx from each
oth er [ 15 , 16 ]. T hereby the equivalent plain pr ofile g = g ( x ) with in this m apping proce dure
called Method of Dimen sionality Reduction (MDR) can be calculated from th e
axisy mm etric gap f = f ( r ) betw een the non-deformed three-dimension al bo dies by the
integral transf orm:

1
2 2 1
0
()
( ) d .
()
x
k
k
fr
g x x r
xr



 


(2)
Stiff ness Δk z of a sing le spring at position x is given by the expression:

0
( ) ,
k
zN
x
k x c x
z

  



(3)
w ith:

1
22
12
1 01 2 02
11
:.
( , ) ( , )
N
c H k E H k E









(4)
Dim ensionless auxiliary fu nction H ca n be determ ined from exponent k and P oisson’s
ratio acco rding to:

Dugdale-Maugis A dhesive No rmal Contact of Ax isy mmetric Pow er-Law Gr aded Elastic Bodi es 11

2( 1 ) co s( / 2) ( 1 / 2)
( , ) : ,
( , ) ( , ) sin[ ( , ) / 2] [ ( 1 ) / 2]
k k k
Hk C k k k k

       
  
 

(5)
w ith:

1
2 [ (3 ( , )) / 2] [ (3 ( , )) / 2 ]
( , ) : (2 )
k k k k k
Ck k
   
 
      
 

(6)
and

( , ) : ( 1 ) 1 1
k
kk

 

  




(7)
and Gamm a function Γ .
Note that the spatial distribution of the spr ing stif fness in Eq. ( 3) obey s the sam e
power-law as the elastic grad ing. I f equivalent p rofile g is pr essed into the foundation o f
springs b y an indentation depth d the vertical spr ing disp lacemen t w 1D ( x ) in the area of
direct co ntact is elem entarily given by:

1D ( ) ( ) , , w x d g x x a   

(8)
w ith contact r adius a . Nor m al force F N as w ell as the lo cal distributions of pressure p and
relative d isplacemen t w in the o riginal three -dim ensional system can be calculated from
w 1D ( x ) accord ing to:

1D
0
1D
2 2 1
0
1D
2 2 1
0
( ) d ,
( )d
* ( ) ,
()
( )d
2 cos( / 2)
( ) .
()
k
N
N k
N k k
r
rk
k
c
F w x x x
z
c w x x
pr z xr
x w x x
k
wr rx










 
 




(9)
The seco nd of the se latter Eqs. ( 9) ca n be inverted to give:

0
1D 2 2 1
2 c os ( / 2) ( )d
( ) .
()
k
k
N x
zk rp r r
wx c rx



 


( 10 )
If we now assum e a Dugdale model of a co nstant adhesive stress σ 0 w ithin th e
adhesive zone w ith radius b :

a dh 0
( ) , , p r r b

  

( 11 )
the correspo nding displacem ents in th e MDR m odel ar e due to Eq. ( 10 ) given by :

2 2 1
00
1D ,adh
2 c os ( / 2)
( ) ( ) , .
( 1 )
k k
N
zk
w x b x x b
ck


   


( 12 )
Hence, the one-dim ensional displacemen ts in the Dug dale -Maug is adhesive contact are:

12 E . WI L LERT

1D 2 2 1
00
( ), ,
() 2 co s( / 2) ( ) , .
( 1 )
k k
N
d g x x a
wx zk b x a x b
ck


 

     
 


( 13 )
For th e th ree-dimen sional stresses to be finite at the edge o f direct contact these
displacements mu st be continuous at x = a , wh ich results in:

2 2 1
00
2 c os ( / 2)
( ) ( ) .
( 1 )
k k
N
zk
d g a b a
ck


  


( 14 )
The to tal external norm al force is due to the first of Eq s. (9 ) given by :

adh
0
0
1 2
2
adh 0 2 1
22
2 [ ( )] d ,
4 cos( / 2) 1 1 3
: F , ; ; ,
2 2 2
1
a k
N
N k
k
c
F d g x x x F
z
k a k k k a
Fb b
kb



  


  





 





( 15 )
w ith th e hy pergeometric fu nction:

21 0
( ) ( )
F ( , ; ; ) : , 1 ,
( ) !
Γ( )
( ) : .
Γ( )
n
nn
n n
n
ab z
a b c z z
cn
xn
x x







( 16 )
Radius b o f the ad hesive zone is not know n a priori but can be deter min ed from the
condition that the gap betw een the deformed surfaces at r = b has to equal the range h of
the adhesive stresses. As the gap betw een the deformed surfaces can b e easily ca lculated
from three-dim ensional relative displacemen t w , indentation d ep th d and axisy m metric
non-deform ed gap f , w e obtain the add itional relation

( ) ( ) . h w r b d f r b     

( 17 )
to close the eq uation system . E valuating Eq . ( 17 ) with the help of the third o f Eqs. (9) and
using the identity :

2 2 1
0
2 c os ( / 2) [ ( )] d ()
()
b k
k
k x d g x x d f b
bx



 



( 18 )
one obtains:

2 2 1
2 1
1 2
00 21
22
2 cos( / 2) [ ( )] d
()
3
cos ( 1 )
4 1 3
22
F , ; ; .
3 22
1
2
b k
k
a
k
kk
N
k x d g x x
h bx
kk
k
zb a k k a
k
k
cb
kb









  

  
   
  
   




   





 

 
 





( 19 )

Dugdale-Maugis A dhesive No rmal Contact of Ax isy mmetric Pow er-Law Gr aded Elastic Bodi es 13
Equations ( 14 ) , ( 15 ) and ( 19 ) completely solve the given contact prob lem. I n the
hom ogeneous case k = 0 they are reduced to the axisym metric generalization of Maugis’
results given very recently by Po pov et al. [ 17 ]. T he stresses in the area of dir ect contact
could theoretically be ca lculated inserting Eq. ( 13 ) into the second of Eq s. (9 ) .
3. T HE JKR L I MIT
It is of course possible to retrieve the known solution in the J KR limit o f adhesion
from the relatio ns d erived in the p revious sectio n. For this p urpose w e study the limit o f
neglig ible adhesion range h → 0, wh ereas the surf ace energy per unit area, Δγ = σ 0 h , is
kept constant. In this case the radius of the adhesiv e zone can be w ritten in the form :

( 1 ), ba



( 20 )
w ith a sm all p arameter ε . Using the lin earization:

( ) ( ) ( ) , g a x g a g a x

    

( 21 )
performing the integ ration and neglecting all term s o f high er than first order in ε leads to :

1 1
00
1
2
11
00
2
2 cos( / 2)
( ) ( 2 ) ,
1
2 ( ) 2 cos ( / 2)
cos ( 2 ) ( 2 ) .
21 ( 1 )
kk k
N
kk
kk
N
za
k
d g a kc
za
k d g a k
h kc
k

 




 


 

   
  


 
   

( 22 )
Hence,

1
2 1
00
2
0
2 c os ( / 2) (2 )
( 1 )
kk k
N
za
k
h c
k

 

 

 

( 23 )
and therefore:

1
0
2
( ) ,
k k
N
z
d g a a
c





( 24 )
w hich perfectly co incides w ith the know n solution in the JKR limit [8]. T he normal force
via the same m echanism is also red uced to the know n relation :

0
0
2 [ ( )] d .
a k
N
N k
c
F d g x x x
z



( 25 )
Note that Eq . ( 23 ) is actually independent of the profiles of the contacting bodies.
4. P A RABO L I C C ONTA C T
Let us now consider the specific case of p arabo lic co ntact w it h the r adius of curvature
R , i.e.:

2
( ) .
2
r
fr R


( 26 )

14 E . WI L LERT
The eq uivalent profile is accordingly:

2
( ) .
( 1 )
x
gx Rk
 

( 27 )
Thus, evaluating the general solution der ived above, the solution of the Dugdale-Maugis
adhesive norm al contact pro blem in case of power -law elastic grading is giv en by:

2 2 2 1
00
1
3 2
0
22
0
1
22
21
22
2 cos( / 2) ( ) ,
( 1 ) ( 1 )
4 4 cos( / 2)
( 1 ) (3 ) 1
1 1 3
1 F , ; ; .
2 2 2
k k
N
k
k
N
N k
k
zk
a
d b a
R k c k
c a k a
Fb
b
Rz k k k
a k k k a
bb







  

 
   
 
   




   
  


  

    
   





( 28 )
Radius b of the adhesive zone can be determined from the condition:

 
1 2
21 2
3
22
21 2
2 1
00
2
2 cos( / 2) 1 1 3
F , ; ; 1
1 2 2 2
4 cos( / 2) 1 3 5
F , ; ; 1
2 ( 1 )( 3 ) 2 2 2
3
cos 1
4 22
3
1
2
k
k
kk
N
k a k k k a
hd kb b
b k a k k k a
R k k b b
kk
k
zb
k
c
k












  

  



 





  

  









   
  
   
   
 
 
 

1 2
21 2
13
F , ; ; .
22
k
a k k a
k
b b







 



 





( 29 )
Introducing the norm alized variab les

: , : , : , : ,
N
c
c c c
F
d a b
d a F m
d a F a
   

( 30 )
w ith th e critical values in the JKR lim it under force -controlled bo undary conditions [6] :

2
2 2 2 3
0
1
2 2 2 3
0
( 1 ) (3 )
1 ,
( 1 )( 3 ) 8
( 1 ) (3 ) ,
8
3 ,
2
k k
c
N
k k
c
N
c
k k R z
k
d k k R c
k k R z
a c
k
FR






  

 
 

  
 


  

( 31 )

Dugdale-Maugis A dhesive No rmal Contact of Ax isy mmetric Pow er-Law Gr aded Elastic Bodi es 15
and the generalized T abor parameter for po w er-law elastic grading, i.e. the ratio of the
characteristic height of th e adhesive neck and the adhesion range:

:,
c
d
h


( 32 )
Eq uations ( 28 ) can be w ritten in the f orm:

2 1 2 1
3
32
21
1
21
22
3 16 cos( / 2) ( 1 ) ,
1 ( 1 )
2 1 4 cos( / 2) 1
()
1 1
1 1 1 3 1
1 F , ; ; .
2 2 2
kk
k
k
k
k k
d a a m
k k
kk
F a m a
k km
k k k
mm










   
 
 
    

 



  

    
  

    
   




( 33 )
The co mpatibility co ndition ( 29 ) in dimens ionless variables reads:

 
 
21
12
2
21
32
1
2
23
c os / 2
1 2 1 1 1 3 1
F , ; ; 1
1 2 2 2
( ) 4 1 1 3 5 1
cos F , ; ; ( 1 )(3 )
2( 1 ) 2 2 2 2
3
cos ( 1 )
32 22
3
( 1 ) ( 1 )
2
k
k
k
k k k k
d k mm
ma k k k k kk
k mm
kk
ma k
k
kk










  

  
 

 

    
   
    
   

    


   
   
   
   
 
 
 

21
12
1 1 3 1
F , ; ; ,
22
k
kk
k
mm






 







( 34 )
w hich in the hom ogeneous case co incides w ith Maugis’ solution [ 12 ] (Maugis uses a
sligh tly different scaling for norm alization ). T he JKR limit is given by the known
relations [ 17 ]:

1
JK R 2 2
3
JK R 3 2
3 4
,
11
2.
k
k
k
k
d a a
kk
F a a








( 35 )
As th e adhesive force in the DMT limit,

D M T
ad h 2, FR



( 36 )
is independent of the elastic co ntact proper ties (it is actually the force for the adhesive
contact of rigid spheres der ived by Bradley [ 18 ]), the DMT limit of Eqs. ( 33 ) read s:

16 E . WI L LERT

D M T 2
D M T 3
3 ,
1 4 .
3
k
k
da
k
F a
k


 



( 37 )
To illustrate ab ove findings and the influence of material grading Figs. 1 and 2 show
the implicitly defined force -indentation r elations as well as the respec tive JKR - and DMT
lim its for tw o different valu es of the power-law exponent k .

Fi g. 1 F orc e-i nde n ta t io n - cu rv es fo r th e D u g da le -Ma u g is ad h es iv e n orm al con tac t of pow e r-
la w g rade d el as tic sph ere s f or k = - 0.5 an d s ev era l v al ues of t he Tab or para m ete r Λ

Fi g. 2 F orc e-i nde n ta tio n- cu rv es f or t h e Du gd al e- Mau gi s a dh esi ve n or m al co n ta ct of pow er-
la w g rade d el as tic sph ere s f or k = 0 .5 and s eve ra l v alu es of th e Tab or pa ram ete r Λ

Dugdale-Maugis A dhesive No rmal Contact of Ax isy mmetric Pow er-Law Gr aded Elastic Bodi es 17
Note that the DMT lim it is only w ell-defined for p ositive indentation dep ths although
the branch with out direct contact (and therefore negative indentation depths) can b e seen
as its “natural” continu ation. To denote this slight distinction a sm all gap is left betw een
the DMT lim it and the curve with out dir ect contact in Fig. 1. Obviously the convergence
for high er values of the Tab or par ameter towards the J KR lim it is much faster for larger
values of k . For k = 0.5 there is alrea dy no noticeab le difference between the solution for
Λ = 1 and the JKR limit. Also the norm alized indentation d epths ar e getting m uch higher
for larger values of k . Interestingly , the critical pull -off forces in the JKR - and DMT lim it
are the same for k → 1 (as it w as po inted o ut already in [ 6 ]). In this case the left branch of
the JKR curve and the curve w ithout direc t contact will be practically in distingu ishable .
5. C ON CL USI ONS
Bas ed on th e MDR a clo s ed - f or m an al yt ic sol u t io n h as bee n obt ain ed f or th e Du gda l e-
Ma u g is adh esi ve no rm al con t ac t of arb it rary con ve x ax isy m m et ric , po w er- law g rade d el ast ic
bod ie s . As th e mo s t com m on an d proba bly m ost rele van t spe ci al cas e th e con tac t of
sp h er i ca l or para boli c bod i es ha s been stu die d in det ail . The com m on lim i ts fo r ver y la rg e
(JK R ) or ve ry sm al l (DM T) val ues of th e Tabor pa ra m et er ar e deri ved f ro m th e ge n er al
so l u ti on. In dim en sio n le ss va ri ab le s th e re la t io n s bet w een i n den tat ion dep th , con t ac t ra di i
an d no rm al fo rc e on ly depe n d on th e Tabo r par am et er and ex pon en t k of th e el as tic gra din g .
The reby th e con v er g en ce f or larg er va l u es of th e T abo r par am et er tow ar ds th e JKR lim i t is
f as ter f or h ig h er v alu es of k .
T he presented solution is of course based on strong contact -m echanical assumptions
(half-s pace hypothesis , absence of friction or roughness) and q uantitativ ely problematic
phy sical models (po w er-law grading with either infinitely stiff or infin itely soft surfaces,
Dugdale model for the adhesive stress); it is, however, to the author’s b est knowledge, the
only tool, to r igorously study t he influence of b oth material grad ing and ad hesion range in
a clo sed form, for exam ple in m icr o- or nano-applicatio ns, for w hich th e range of the
(adhesive) m o lecular forces becomes relevant. And although other m od els migh t seem
phy sically m ore appr opriate , they w ill p robab ly neither allow for analytic treatment nor
show a qualitatively different behavior.
For future w ork it would be interesting to compare the obtained analy tical results with
num erical or exper im ental fin dings.
R EF ERENCES
1. Suresh, S., 2001 , Graded Ma teria ls fo r Resista nce t o Co ntac t Deformat ion and Damage , Science, 29 2,
pp. 2447- 2451.
2. Scherg e, M., Go rb, S., 200 1, Biolo gica l Micro - and Nano -Tribo logy – Nat ure’s Sol utio ns , Springer,
Berl in Heidelberg .
3. Booker , J.R. , B alaam , N.P., Davis, E. H., 1985, The Beh aviou r o f a n Elast ic No n-Homogen eous Half -
Spac e. Part I – Line an d Point Loads , I nt ernational Journal fo r Numer ical and Analy tical Methods in
G eome c hani cs, 9(4), p p. 35 3 - 367.
4. G iannak opoulos, A.E., Suresh, S., 1997, Inden tati on of Solids with Gra dien ts i n Elastic Properti es:
Part I. Po int Fo rces , I nternational Journal of Solids and Stru ctures, 34 (19) , pp. 2357 - 2392.
5. G iannak opoulos, A.E., Suresh, S., 199 7, Indenta tio n o f Solid s wi th Gradients in Elastic Propert ies:
Part II. Axisymmetri c Inden tors , Inter nationa l Journal of Sol ids and Structures, 34 (19 ) , pp. 2393- 2428 .

18 E . WI L LERT
6. Chen, S., Yan, C., Zhang, P., Gao , H., 2009 , Me chan ics of a dhesi ve con tact o n a powe r -la w g raded
elasti c hal f-spa ce , Journal of the Mechanics an d Physics of Sol ids, 57 (9), pp . 14 37 - 1448.
7. G uo, X., Jin, F., G ao, H., 2011 , Mech anics o f non -sli ppin g ad hesive c onta ct on a power -law grad ed
elasti c hal f-spa ce . I n ternational Journal of Sol ids and Structu res, 48 (18), pp. 2 565 - 2575.
8. Jin, F., Guo, X., Zhang, W., 201 3, A Uni fied Treat ment o f Ax isy mmetric Adhesi ve C onta ct o n a Po wer -
Law Graded Elast ic Half -Sp ace . Journal of A pplied Mechanic s, 80 (6), 061 024.
9. L i u, Z., Mey er s, M.A., Zhang, Z., Rit chie, R.O., 20 17, Functio nal grad ients and hetero gene ities in
biol ogica l materia ls: Design pri ncipl es, fun c tio ns, and bioi nspired app licat ions , Prog ress in Ma terials
Science, 88, p p. 46 7 - 498.
10. Johnson, K.L . , Kendall, K., Roberts, A.D., 197 1, Surfac e En ergy and the Conta ct of El astic Soli ds .
Proce edings of the Roy a l Society of L ondon, Series A , 324, pp. 301 - 313.
11. De rjaguin, B.V., M ulle r, V.M. , Topo rov, Y.P., 1975 , Effect of co nta ct de formatio ns o n th e adh esion of
parti cle , Journal of Collo id and I nterface Scie nce, 53 (2 ), pp. 314 - 326.
12. Maugis, D., 19 92, Adhe sion o f sphe res: The JKR -DMT-transit ion u sing a Dugdal e mod e l . Journal of
Coll oid and I nterface Science, 150 (1), pp. 2 43 - 269.
13. Dugdale , D.S., 1960, Yielding of stee l sh eets conta inin g sl its , Journal of th e Mechan ics and Ph y sics of
Solids, 8(2), pp . 100 - 104.
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of el astica lly g raded materia ls , International Journal of Engineering Science, 104, p p. 20 - 33 .
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