On the use of DMT approximations in adhesive contacts, with remarks on random rough contacts

0.34 MB·

by  Michele Ciavarella

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1 On the use of DMT approximations in adhesive
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2 contacts, with remarks on random rough contacts

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a
J M.Ciavarella

6 Politecnico di BARI.
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V.le Gentile 182, 70125 Bari-Italy.

] Email [email protected]
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s January 17, 2017
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Abstract
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t The contact between rough surfaces with adhesion is an extremely
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m difficult problem, and the approximation of the DMT theory (to neglect
deformations due to attractive forces), originally developed for spherical
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d contact of very small radius, are receiving some new interest. The DMT
n approximationleadstoextremelylargeoverestimateoftheadhesiveforces
o inthecaseofsphericalcontact,exceptatpull-off. Forcylindricalcontact,
c theoppositetrendisfoundforlargercontactareas. Thesefindingssuggest
[ some caution in solving rough contacts with DMT models, unless the

1 Tabor parameter is really low. Further approximate models like that of
v Pastewka & Robbins’may beexplainedtowork only for acoincidence of
0 error cancellation in theirrange of parameters.
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Keywords: Adhesion, Maugis’ theory, rough surfaces, DMT theory, JKR
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4 theory
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.
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1 Introduction
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1
: The Derjaguin-Muller-Toporov (DMT) theory (Derjaguin et al., 1975, Muller
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et al., 1980, 1983), for the contact of elastic spheres with adhesion, has a long
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X history. After Bradley(1932)andDerjaguin(1934)obtainedthe adhesive force
r betweentworigid spheres,equalto2πRw,wherewistheworkofadhesion,and
a
R is the radius of the sphere, JKR (Johnson Kendall and Roberts 1971) devel-
opedatheoryforelasticspheres,assumingadhesiveforcesoccurentirelywithin
the contact area, obtaining 3/4 of the Bradley pull-off value, and hence the in-
dependenceonthe elasticmodulusraisedalongdebateabouttheacomparison
of the pull-off prefactor.
AsthemainattentioninthediatribebetweenJKRandDMTwaslimitedto
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thepull-offvalue,itisoftenbelievedthatDMTisthelimitforTaborparameter
(Tabor, 1977)
Rw2 1/3 Rl2 1/3 σ R 1/3
µ= = a = th 0 (1)
E 2∆r3 ∆r E l →
(cid:18) ∗ (cid:19) (cid:0) (cid:1) ∗ (cid:18) a(cid:19)
where ∆r is the rangeof attractionof adhesive forces,close to atomic distance,
and E the plane strain elastic modulus. Also, we have introduced the length
∗
l = w/E as an alternative measure of adhesion, and σ is the theoretical
a ∗ th
strength of the material. Now, while it is true that DMT predicts the Bradley
result for the force at pull-off also for elastic spheres, the DMT theories have
been much less compared with exact results, when considering the entire load-
displacement curves. In both DMT methods,
the attraction forces act exclusively outside the contact, and
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the repulsive forces only are responsible for deformation.
•
Then, in the DMT ”force method”:-
the force of adhesion can be simply obtained by integrating, according
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to Derjaguin’s approximation,the forces of facing elements outside of the
contact, separated by a gap which is given by Hertz theory.
Weshallconcentrateonthelatter(force)method,whichiswhatiscommonly
usedwhenDMTapproximationisconsideredinthegeneralizedcontextofrough
contact(seePersson&Scaraggi,2014). Inonelooksattheforceofadhesionnot
at pull-off, it decreases from 2πRw to πRw, in the ”thermodynamic method”1
while it increases in the ”force method”, as shown by (Muller et al., 1983),and
Pashley (1984). Pashley (1984) in particular notices that in the force method,
the adhesive force should be always larger than 2πRw, the value obtained for
a truncated rigid sphere independently on the contact radius, as the Hertzian
profile is closer to the flat surface than the rigid spherical profile.
Maugis (2000), in his Maugis-Dugdale analysis (which does not make the
DMT approximations) called the low µ end the ”DMT theory”, which in fact
is now the versionmostcommonly associatedwith DMT,and sometimescalled
DMT-M. In this version, the attractive forces are constant, and equal to the
pull-off value, 2πRw. This is indeed what comes out from DMT theory, but
only in the limit of µ=0: therefore, DMT is exact only in this limit case, and
for any finite value of µ, DMT theories give an errorwhich we shall estimate in
fact in details in the present paper as a function of the Tabor parameter, since
thepreviousestimatesof(Mulleretal.,1983),andPashley(1984)donotclarify
1Inthe”thermodynamic method”, the forceiscomputed bythe rateofchange ofsurface
energyasthesphereispressedwithapproachα,i.e. dWs/dα. Itturnsoutthatthe“thermo-
dynamic” method tends to give opposite error with respect to the ”force method”, and it is
alsomorecomplicatedtouse,soithasnotreceivedmuchattention.
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clearlythe roleofTabor parameter. Greenwood(2007)alsohas discussedmore
details of the DMT theory in the limit µ 0.
→
Butwe shallnotlimit ourselvestothe sphericalcontactcase,since this case
has been givenalreadymuch attention, and is only one special case. The DMT
approximationis gaining relevancemore recentlyagain,inthe contextofrough
contact,wherethereisalotofinterestinsimplifyingtheproblemsincetheJKR
assumptionleadstoverycomplicatedandhystereticbehaviour,whichsofar,has
notbeenincludedinaframeworkofanytheory,despitesomeattempts(Persson,
2002, Ciavarella, 2015). Moreover, as roughness at the small scales seems to
point to low values of Tabor parameter,the ”almost rigid” behaviour has some
fundamental interest. Persson & Scaraggi (2014) have indeed attempted using
the DMT approximations using the Persson’s theory for adhesionless contact,
andseemedtofindsomereasonableaccuracyatleastfortherangeofparameters
they observed. Also, Pastewka & Robbins (2014, PR in the following) make
some scaling predictions which seem to fit well some limited range of their
extensivefullnumericalsimulationsinvolvingatomisticsroughsolids. Wemade
afirstattempttodiscussPRfindingsinCiavarella(2016)wherewenoticedthat,
if PR were concerned with spherical contact, using the DMT approximation
with the additional simplification of using only the asymptotic first term in
the expansion of the gap outside the Hertzian contacts, they would find easily
largeerrors. But one limit to this estimate is that we assumedcircularcontact,
whereasPR calculationshows more like 2D fractalcontactarea,perhaps closer
to very elongated contacts like in 2D cylindrical contact — indeed, as we will
discussbelow,theyfindacharacteristicdiameterofthecontactindependenton
load, and loadonly affects the elongationof the contactarea. Therefore,in the
present note we develop a simple 2D line contact DMT model, we give more
detailsabouttheDMT limitforthesphere,andmakefurthercomparisonswith
the DMT rough contact results.
2 A 2D DMT-Maugis line contact model
For 2D contact with ”repulsive” diameter d = 2a, the full form of the gap
rep
outside the contact is
h(c) a c
= f (2)
a R a
(cid:16) (cid:17)
where c>a and (Johnson and Greenwood, 2008)
c 1 c c 2 c c 2
f = 1 log 1 (3)
a 2"ar a − − "a −r a − ##
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
whosefirsttermintheseriesexpansionnearc=aisf c = √8 a c 1 3/2,
as a 3 R a −
is used inthe PRversionof DMT method, as commented in(Ciavarella,2016),
(cid:0) (cid:1) (cid:0) (cid:1)
and in the later paragraph. We shall assume for the potential, a Maugis simple
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law. This will permit a direct comparison with the ”exact” Maugis solution
includingdeformationsinducedbytheadhesivestresses,givenbyBaneyandHui
(1997)Morrowand Lovell(2005)and Johnsonand Greenwood(2008),whereas
Jin et al. (2014) give a double Hertz solution which show that results will not
differ much with those with other choices of potential.
The pull-off force is not a simple multiple of Rw as with circular con-
tacts, but varies from P = √8Rσ w to P = 3 4πE Rw2 1/3 =
rigid th JKR 4 ∗
3 πE Rw2 1/3, so it depends on elastic modulus.
42/3 ∗ (cid:0) (cid:1)
(cid:0) (cid:1)
We define the following non-dimensional contact radius and load
a
a∗ = 2π1/3R2/3l1/3 (4)
a
P P
P = = (5)
∗ (πE∗Rw2)1/3 42/3PJKR/3
and accordingly the Hertz and JKR limits are found as (Johnson and Green-
wood, 2008)
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P =a (6)
H∗ertz ∗
PJ∗KR =a∗2−2√a∗ =PH∗ertz −2√a∗ (7)
whereas the Maugis-Dugdale model shows a smooth transition between the
Hertz and JKR limit — unlike the 3D case, where there is a transition from
Bradley rigid to JKR model. Notice that the rigid limit is subtle: while there
is a tendency to the Hertz regime, the actual pull-off in rigid limit is not zero.
Moving to a DMT force method estimate, setting the gap to ∆r gives
c R∆r
f = (8)
a a2
(cid:16) (cid:17)
and it is clear that the approximation is good until R∆r <1. In dimensionless
a2
notation, R∆r/a2 = 4π2/3a∗2µ −1.
Usingtheasymptoticterm,thelateraldistancedefiningthesizeofattractive
(cid:0) (cid:1)
region (which is composed of two strips of size d ) is
att
3 2 1/3
R∆r
2
d = (9)
att,asym
d
"(cid:0) rep(cid:1) #
and when contact radius is small, we require a correction from the solution
of (8), datt = c 1 = βdatt,asym. As we are using the Maugis potential, the
a a − a
attractive load is therefore simply the product of the theoretical strength and
the area of the adhesive strips, P = 2d w . Using (9), the attractive
DMT,att att∆r
load is obtained as
w 3 3 2 1 w 3
3 3 3 3 3 3 2 2
P = 2 β d = 2 β R ∆r (10)
DMT,att − att ∆r − 2 2a ∆r
(cid:16) (cid:17) (cid:18) (cid:19) (cid:16) (cid:17)
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Using (4, 5) and Tabor parameter (1),
µ 1/3
PD∗MT,att ≃−β a (11)
∗
(cid:16) (cid:17)
TheresultsoftheDMTtheoryarepresentedinFig.1a(dashedlines)forµ=
0.05,0.25,1,5, together with the Maugis solution of Johnson and Greenwood
(2008) which we take as reference as ”exact”. In Fig.1b, we compare the JG
Maugis solutionwith the further approximationof taking only the first term in
thegapprofile,whichclearlyleadstoseriouserrorsevenatlowTaborparameter.
WeshallexplainwhyinthePRuseofthisfurtherapproximation,theerrorwas
probably balanced by another approximation.
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a*
2.5
JKR HΜ>5L
2.0
Μ=0.05, 0.25, 1, 5 1.5
1.0
Hertz
0.5
P*
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
(a)
a*
2.5
2.0 JKR
Μ=0.05,0.25,1,5
1.5
1.0
Hertz
0.5
P*
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Fig.1 Line contact. Comparison of Maugis (a) ”exact” solution (solid blue
line), with the DMT model with(a) full shape ofthe gap(dashedline), or with
the first term of the gap (b). The black (red) solid line are Hertz (JKR) limits.
The arrow shows increasing values of Tabor parameter µ=0.05,0.25,1,5.
It is clear that the DMT theory gives a reasonable result only for µ < 0.05
as at µ = 0.25, the error is already significant, of the order of 20% at pull-off.
Errorsbecomelargeatµ>1,particularlyatpull-off,aslargerthan100%. This
is clearer from fig.2, where the pull-off values are plotted.
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-P *
min
5
4
DMT
3
2
Maugis (exact)
1
0 Μ
0 1 2 3 4 5
Fig.2 Line contact. Comparison of pull-off values for Maugis ”exact” solution
(solid blue line), with the DMT model with full shape of the gap(dashed line).
3 The spherical DMT model
Asthiscaseisclassical,DMThasbeencomparedwithJKRandothermodelsin
alargenumberofpapers. However,thecomparisonismostlydoneforthevalue
at pull-off, where of course the DMT model gives the correct Bradley result
for µ = 0: only perhaps (Muller et al., 1983), Pashley (1984), and Greenwood
(2007)discuss more details of DMT at compressions largerthan zero, and even
they, do not fully clarify the error as a function of Tabor parameter.
We shall assume for the potential, a Maugis simple law, consistently to
the line contact of the previous paragraph. We have already discussed a fur-
ther simplified form of this model in (Ciavarella, 2016) inspired by PR paper
(Pastewka & Robbins, 2014), namely using the first term asymptotic form of
thegapfunctionoutsideofthe contact,andofcomputingtheareaofattraction
by multiplying the perimeter of the contact by a d size of attractive region.
att
As we already discussed the very strong effect of both these approximationson
the DMT solution, we shall try here to give a fair assessment of a full DMT
model, and we do not make such further assumptions.
Therefore, we introduce the following dimensionless notation for the ap-
proach, and the load in spherical contact
δ =δ/(µ∆r) ; P =P/(πRw) (12)
We know the exprebssion of the gap oubtside the contact area (Muller et al.,
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1983), Pashley (1984), and Greenwood (2007)
h r a r 2 r 2 r2
a = 1 2 arctan 1 (13)
(cid:0)a (cid:1) πR"r(cid:16)a(cid:17) − −(cid:18) −(cid:16)a(cid:17) (cid:19) ra2 − #
Imposing the gap is equal to the range of attractive forces, gives the size of
attractive region. Writing the size of the region using the first order expansion
2/3
of the gapis possible in closed form, giving d = drep 3πR∆r , and
att,asym 2 2√2d2
(cid:18) rep(cid:19)
the correct solution with the full expression of the gap requires a numerical
correction for small contact areas, which is d = β d . We write β to
att s att,asym s
avoid confusion with the equivalent coefficient in line contact. For the circular
geometry Hertz theory gives δ = d2 /(4R) and hence, using the notation of
rep
PR, d for the repulsive contact diameter, and d for the size of attractive
rep att
region (not a diameter), we have
π 2 2 w
P = (d +2β d ) d
att −4 rep s att − rep ∆r
h i
Normalizing with (12), we have
2/3 2
3π
P = µδ 1+β 1 (14)
att s
−  (cid:18)8√2µδ(cid:19) ! − 
b b 
The result can be compared with the Maubgis solution for µ 0 (which
→
indeed is the ”DMT” limit for µ 0), and the JKR theory (Johnson, et al.,
2 →
1971)
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Pµ=0 = δ3/2 2 (15)
3π −
Pb = 4 δb3/2 1 3π 4/3 1 β δµ (16)
DMT s
3π − 4(cid:18)8√2µδ(cid:19) − !
b b 1/2 b3/2
PJKR =P0 1.1 δ δ0 +b0.43 δ δ0 (17)
− − −
(cid:16) (cid:17) (cid:16) (cid:17)
where δ0 = −43πb2/3,P0 =b−5/6 arebthebJKR values abt publl-off in displacement
control.
Fig.b3reportstheDbMTsolution,withthecorrectMaugissolutionforvarious
values of µ, and immediately it appears that all of the DMT solutions are
incorrectly below the limit for µ 0, while the correct Maugis solutions are
→
all above this limit — indeed, already at µ =0.05 the Maugis solution is quite
distinct from the limit solution µ 0 (which, adding to the confusion, Maugis
→
callsthe DMT limit!). Not shownis alsothat the correctionfor the full formof
the gap is necessary at small indentations, as otherwise there is an additional
spurious increase of adhesive forces estimate.
2JKRispresentedinacurvefittedforminordertobeeasilyused.
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Fig.4 reports the values of the pull-off estimate with the DMT method, as
compared with the Maugis solution.
`
P
`
∆
-2 -1 Mau1gis 2 3 4 5
-1
-2
DMT Μ= 0.05, 0.25, 1, 5
-3
Fig.3 Spherical contact. Comparison of Maugis ”exact” solution (solid blue
line), with the DMT model with full shape of the gap (dashed line). The black
and red solid line are µ=0 (what Maugis calls the DMT limit), and JKR
limits. The arrow shows increasing values of Tabor parameter
µ=0.05,0.25,1,5.
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`
-P
min
3.5
3.0
DMT
2.5
2.0
1.5
Maugis (exact)
1.0
0.5
0.0 Μ
0 1 2 3 4 5
Fig.4. Spherical contact. Comparison of pull-off values for Maugis ”exact”
solution (solid blue line), with the DMT model with full shape of the gap
(dashed line).
4 DMT for rough contacts
These simple estimates suggest that the order of error in using the DMT ap-
proximation in actually solving rough adhesive contact problems, like done by
Persson&Scaraggi(2014)forexample,couldbeverylarge,unlessTaborparam-
eter is quite low, and the body very close to ”rigid”. Incidentally, this means a
conditionstricterthanjustTaborparameterlowerthan,say,0.2atthesmallest
scale in the model, since at the large scale, the contact may result hysteretic,
and the DMT model would lead to very large errors. This may be in practice
a very strong condition and would therefore limit significantly the use of such
methodology. In the particular case, the Persson-Scaraggimethod, which relies
on use of additional approximations in solving the adhesiveless contact and in-
tegrating for the adhesive force, doesn’t lead to simple analytical results so it
is also important to check, in the end, if there is a significant advantage with
respect to a full numerical solution of the problem, like done by Pastewka and
Robbins (2014).
SpeakingofPR,theyinterpretedtheir”exact”resultswithsomeverysimple
estimates based on DMT approximation, and further simplifications. One may
wonderwhyPRobtainedareasonablefitoftheirdata,withassumingfirstorder
expansion of the gap function, which for example would lead to completely
wrong results for very small contact area (see Fig.1b). In particular, this is
surprising since their Tabor parameter are said to be of the order of 1, and
10