Contact mechanics of and Reynolds flow through saddle points: On the coalescence of contact patches and the leakage rate through near-critical constrictions

1.8 MB·English

by  Wolf B. Dapp

epl draft
Contact mechanics of and Reynolds flow through saddle points
On the coalescence of contact patches and the leakage rate through near-critical constrictions
Wolf B. Dapp1 and Martin H. Mu¨ser1,2
1 Supercomputing Centre Ju¨lich, Institute for Advanced Simulation, FZ Ju¨lich, 52425 Ju¨lich, Germany
2 Lehrstuhl fu¨r Materialsimulation, Universit¨at des Saarlandes, 66123 Saarbru¨cken, Germany
5
1
0
2
PACS 46.55.+d–Tribology and mechanical contacts
n PACS 68.35.-p–Solid surfaces and solid-solid interfaces: structure and energetics
a
J Abstract – We study numerically local models for the mechanical contact between two solids

6 with rough surfaces. When the solids softly touch either through adhesion or by a small normal
2 loadL,contactonlyformsatisolatedpatchesandfluidscanpassthroughtheinterface. Whenthe
loadsurpassesathresholdvalue,Lc,adjacentpatchescoalesceatacriticalconstriction,i.e.,near
]
points where the interfacial separation between the undeformed surfaces forms a saddle point.
t
f This process is continuous without adhesion and the interfacial separation near percolation is
o
fully defined by scaling factors and the sign of L −L. The scaling factors lead to a Reynolds
s c
. flow resistance which diverges as (L −L)−β with β = 3.45. Contact merging and destruction
t c
a nearsaddlepointsbecomesdiscontinuouswheneithershort-rangeadhesionorspecificshort-range
m repulsionareaddedtothehard-wallrepulsion. Theseresultsimplythatcoalescenceandbreak-up
- of contact patches can contribute to Coulomb friction and contact aging.
d
n
o
c
[

1 Introduction. – When two nominally rough solids thegapopens,whileintheorthogonal,in-planedirection,
v arepressedagainsteachother,theirsurfacestendtotouch the gap is closed, because the interfacial separation of the
0 microscopically only at isolated points [1,2]. Simple mod- undeformed surfaces decreases in that direction. Thus,
9
els assume that real contact between two rough surfaces studying the contact mechanics of saddle points entails
2
canbedecomposedintosingle-asperity,Hertzian-likecon- the analysis of critical constrictions and that of contact
6
0 tacts. However,bothexperiment[3,4]andlarge-scalesim- patch merging. Here, we investigate not only the con-
. ulations [5–9] have revealed that the majority fraction of tact mechanics of (near-) critical constrictions, including
1
contiguouscontactpatchesappeartobefractalandtore- a scaling analysis of the gap topography near the percola-
0
5 sultfrommany(formerly)single-asperitycontacts. While tion point, but also address the pertaining Reynolds flow
1 the latter are well studied, less is known about the way with and without adhesion.
v: in which they merge [10–13], in particular when adhesion To simulate the contact mechanics of an isolated sad-
i is present. For example, it is unclear if adhesive contact- dlepoint,wereinvestigaterathersimplemodels[10,11,13]
X
patch coalescence and break-up happen discontinuously. for the (combined) surface roughness and for the interac-
r Contact-patch coalescence also plays a role for seals: it tions between the surfaces. Yet, in addition to the hard-
a
has been argued that their leakage rate is determined to wall repulsion commonly assumed infinitely short-ranged
a significant degree from the topology of the last criti- in continuum mechanics, we also study the effects of
cal constriction [14], i.e., the neighborhood of the point, finite-range attraction [15] and repulsion [16] between the
which, upon increasing load or adhesion, is the first point surfaces. Our motivation for additional repulsion stems
to interrupt a percolating non-contact path. from recent simulations, in which appropriately designed
In this work, we study local models for the merging of cohesive-zonemodelsqualitativelyreproducedthesurface
contact patches, or, in the context of seals, the contact interactionsmediatedbyastronglywettingfluidinducing
mechanics of critical constrictions. The local gap topog- an effective negative surface energy of finite range [16].
raphy of a critical constriction is such that the interfacial Thelength-scaleoftheseadditionalsurfaceinteractionsis
separation of the undeformed surfaces is (close to) a sad- nevertheless short enough for interfacial interactions not
dle point. More precisely, near the percolation point, the toactfarawayfromacontactline,i.e.,ourTaborparam-
gap is very small. Parallel to a just-blocked fluid channel, eters [15] are (cid:29)1.
p-1
W. B. Dapp et al.
Models and Methods. – We assume linear elastic- 2saddle- saddle- 2
ity and the small-slope approximation so that roughness point mgaapx max point
gap
canbemappedtoarigidsubstrateandtheelasticcompli-
max
ancetoaflatcounterbody. Theeffectivecontactmodulus 1 saddle- gap 1
point
is used to define the unit of pressure, i.e., E∗ = 1. Elas-
ticity is treated with Green’s function molecular dynam-
ics (GFMD) [17] and the continuum version of the stress-
0 0
displacementrelationinFourierspace,σ˜(q)=qE∗u˜(q)/2, 0 0 1 2 0
whereqisawavevectorandq itsmagnitude. Simulations
are run in a force-controlled fashion. After the external Fig. 1: Elastic displacements at zero load for the hexago-
load has changed by a small amount, all degrees of free- nal,square,andtriangularsubstratelattice,fromlefttoright.
dom are relaxed until convergence is attained. Dark colors represent small displacements, that is, points of
contact. These are enveloped by yellow halos indicative of lo-
Threemodelsforthe(combined)heightprofilesareem-
calized, adhesive necks. Thin black lines connect substrate
ployed. Each profile has roughness only at a single wave-
heightmaxima,whilethinwhitelinesconnectsubstrateheight
length λ, a root-mean-square height fluctuation of h ,
0 minima or troughs, which are marked by white points. Grey
anda root-mean-squaregradient of2πh /λ, whichshould
0 starsreflectsaddlepoints. Onlyselectedlinesbetweennearest-
be small enough for the small-slope approximation to be
neighbor saddle points are drawn (thick dotted lines).
valid. In other words, all three lattices yield identical
angle-averaged spectra at non-zero wave numbers. First,
we consider a square lattice the surface energy of flat, non-polar, and chemically pas-
sivated surfaces. Finite-range repulsion is modeled with
(cid:18) (cid:19) (cid:18) (cid:19)
h (x,y) 2πx 2πy
sq =2+cos +cos , (1) a different functional form than adhesion, namely with
h0 λ λ γ(x,y)=−γ exp{−g(x,y)2/2ρ2}. The detailed rationale
0
forthesefunctionalformsisgivenelsewhere[16]. Inbrief:
for which the simulation cell dimensions along x and y
the exponential adhesive model mimics the large Tabor
direction are chosen to coincide with the wavelength λ of
number limit quite efficiently while allowing one to deter-
the height undulation, that is L = L = λ. The two
x y
minecontactareainanunambiguousmanner. TheGaus-
remaining lattices are triangular and hexagonal:
sian repulsive interactions can simulate the effect of a last
(cid:114) (cid:40) (cid:32)√ (cid:33)
h (x,y) 2 3 3πx (cid:16)πy(cid:17) layer, which needs to be squeezed out before two surfaces
tl = +2cos cos
h 3 2 λ λ touch.
0
Figure 1 conveys an impression of our models. It shows
(cid:18) (cid:19)(cid:27)
2πy
+cos (hexagonal lattice)(2) the elastic displacement for the investigated symmetries
λ in case of adhesion and zero external load. The dark ar-
h (x,y) (cid:114)27 h (x,y) eas show the locations of small displacement and thereby
hl = − tl (triangular lattice)(3)
the structure of initial contact patches. The halos around
h 2 h
0 0
contactpointsreflectbulgesnearthecontactlinethatare
forwhichthedimensionso√ftheperiodicallyrepeatedsim- wellknownfromsingle-asperitycontactswithshort-range
ulation cells are Lx =4λ/ 3 and Ly =2λ. This shape is adhesion. One can also recognize that the displacements
relatively close to a square so that discretization correc- areratherconstantneartheheightminima. Thelatterlie
tions [18] are almost isotropic in our GFMD calculations, on lattices dual to the height maxima for the considered
which decomposes the surfaces into 2n×2n elements. cases. For example, the height minima (or gap maxima)
In each model, the height offset is set to yield a min- formatriangularlatticewhenthesubstrateheightslieon
imum height of zero. Other key geometrical attributes a hexagonal lattice, and vice versa. Saddle points form
(cid:112) (cid:112)
are: Maximum√height( 27/2h0, 4h0, 27/2h0), mean a kagome lattice in these two cases. For the square lat-
(cid:112) (cid:112)
trough depth ( 6 h0, 2 h0, 2/3 h0), skewness (− 2/3, tice, not only height maxima but also minima and saddle
(cid:112)
0, 2/3),andexcesskurtosis(-1/2,-3/4,-1/2),eachtime points each form a square lattice.
in the order hexagonal, square, and triangular lattice. Flow through the interface is described by Reynolds
Two surfaces interact with a hard-wall constraint, i.e., thin-film equation, which assumes a local conductivity
they are not allowed to overlap. In addition we assume that scales as the inverse third power of the gap. We use
cohesive zone models for the finite-range interactions be- the hypre package [19] to solve the sparse linear system
tweenthesurfaces. Itisγ =−γ exp{−g/ρ}forattractive that the discretized Reynolds equation can be expressed
0
forces,whereg =g(x,y)isthelocalgaporinterfacialsep- as. We employ the solvers supplied with hypre using the
aration. Typical values for both γ and ρ are small num- CG (conjugate gradient), or GMRES (generalized mini-
0
bersintheappropriateunitsystem,e.g.,γ =0.2E∗h2/λ mal residual) methods [20], each preconditioned using the
0 0
and ρ = 0.04 h . To give one possible realization: If PFMG method, which is a parallel semicoarsening multi-
0
E∗ were 1 GPa, h = 10 nm, and λ = 1 µm, then gridsolver[21]. Ourin-housecodeisMPI-parallelizedand
0
γ =20 mN/m, which reflects the order of magnitude for uses HDF5 for I/O.
0
p-2
Contact mechanics of saddle points
0.5 full contact
1.0
hard-wall
only
ea 0.4 ea
ar ar percolated
ct ct 0.4 contact
onta 0.3 adhesive repulsive onta hexagonal square triangular
c c
e e 0.2
v 0.2 v
ati ati
el el
r r
0.1 0.1 percolated
non contact
0.0
0.0 0.5 1.0 1.5 2.0 0.1 1 10
load adhesion
Fig. 2: Relative contact area as a function of load for the Fig. 3: Relative contact area as a function of surface energy
square-lattice substrate and different finite-range interactions. γ forthethreeinvestigatedmodels. Theinstabilitiesatlower
0
Red (blue) lines refer to increasing (decreasing) external load values of γ are related to percolation transitions. The single-
0
as marked by arrows. Exponential adhesion (solid lines) and wavelength, triangular substrate circumvents the percolated,
Gaussianrepulsion(dashedlines)showtheestablishedhystere- partial contact regime by transitioning directly from isolated
ses at small contact area related to contact formation. Unless contact patches to full contact.
solidsinteractsolelybyhard-wallrepulsion,additionalhystere-
ses appear at relative contact areas near 0.4. They reflect the
contact-patch-coalescence and break-up instabilities. The precise values in the purely load-driven case with
only hard-wall repulsion are a =0.17826(11) (triangular
p
lattice), 0.40185(6) (square lattice), and 0.67323(1)
Results. – Acentralquestionaddressedinthisstudy (hexagonal lattice). Interestingly, the average of these
is whether the merging of two contact patches happens numbers (0.418) is close to the percolation threshold
continuously or discontinuously. Figure 2 confirms previ- identified for self-affine, isotropic, randomly-rough and
ous findings [10,13] that percolation is continuous when non-adhesive bodies (0.425) [22]. In the adhesion-driven
surfaces interact solely with hard-wall repulsion. How- case, the percolation of contact is triggered at relative
ever, oncefinite-rangerepulsion(withthefunctionalform contacts slightly below the given values of a and ends
p
defined in the method section) or short-range attraction at values slightly above a . The reverse transitions from
p
are added, contact patches and likewise fluid channels co- partial but coalesced to isolated contact patches also
alesce and break up through instabilities. The behavior is occur near the respective values for a , though shifted to
p
qualitativelysimilarforthetworemainingmodels,despite slightly smaller values.
large quantitative differences.
Whileonlyperipheraltothiswork,webrieflydiscussthe
Contacts also coalesce discontinuously when driven en- transition to full contact. In the purely force-driven case,
tirely by adhesion. This is revealed in Figure 3. Despite full contact occurs above critical mean pressures of p =
fc
similar conditions — identical adhesion and surface spec- πE∗t¯/λ, where t¯is the mean trough depth (stated in the
tra—thequantitativedifferencesbetweenthethreemod- model section). This relation is readily obtained from the
els are clearly borne out: The hexagonal (triangular) lat- single-wavelength, full-contact solution [10]. (In the orig-
√
tice has by far the largest (smallest) propensity to form a inal work, an additional factor of 2 was included, as the
√
coalesced or percolated contact area. This is because its wavelengthwasdefinedtobeλ/ 2.) Thepurelyadhesion-
peaks are rather blunt or obtuse (pointed or acute) and driven percolation roughly follows the force-driven perco-
the ridges connecting the peaks have small (large) cur- lation, e.g., the ratio γ /γ is largest for the hexagonal
fc p
vature. However, the hexagonal (triangular) lattice has (135)andsmallestforthetriangular(1)model. Pertinent
the smallest tendency to go into full contact, because its ratios for the critical forces in the absence of adhesion are
troughs are rather deep (shallow) and the curvature nor- 31 (hexagonal) and 1.4 (triangular). Here and in the fol-
mal to the ridges are large (small) in magnitude. In the lowing, we do not address the reverse transition from full
triangular lattice, the propensity to form full contact is to partial contact for adhesive surfaces, because it would
even so large that partial, coalesced contact is not even correspond to a Griffith-like fracture problem lacking an
metastable at finite adhesion and zero load. The square initial crack. As such, for our and related adhesive laws,
model is somewhere in between the two extreme cases. full contact becomes unstable at tensile pressure (cid:46) γ /ρ.
0
While adhesion-free/load-driven and adhesion- The other adhesion-related results presented here barely
driven/load-free percolation are qualitatively different in depend on the precise choice of ρ as long as ρ(cid:28)h0.
that the former is continuous and the latter is discon- The above results reveal a quite significant influence of
tinuous, they happen at similar relative contact areas. local structure on prefactors. Some of the differences can
p-3
W. B. Dapp et al.
2 2
2
7
6
1 1 1
e 5
n
li
0 0 act 0 fdliorwection 4gap
0 0 1 20 cont 3
-1
Fig.4: Gaps(incolor,wherebrightcolorsindicatelargegaps 2
(a) (b)
anddarkcolorsmallgaps)andcontact(white)neartheperco-
lationthresholdforthethreeinvestigatedgeometries. Systems -2 1
are arranged in the same way as in Figure 1.
0
-4 -2 0 2 4 -4 -2 0 2 4
distance from saddle point
be rationalized from the contact and gap geometries near
the percolation threshold, which are depicted in Figure 4
Fig. 5: (a) Contact line shape for normal loads below (blue),
for simple hard-wall repulsion. In coarse-grained repre-
at (green), and, above (red) the critical load. (b) Gap on
sentations the impression is conveyed that, depending on the symmetry axis as a function of the distance from the sad-
geometry, opening angles can range from acute to obtuse. dle point. Different symbols represent different saddle-point
Locally, however, allinvestigatedsaddlepointsaresim- geometriesanddifferentcolorsdifferentreducedloadsl,asin-
ilar in that they sit at points of inversion symmetry, the dicatedinthecaption. Solidareasandlinesareobtainedfrom
curvatures of the substrate heights parallel to the fluid- analytical approximations to the contact shape. At a given
channel (h(cid:48)(cid:48)) are negative and those in orthogonal, in- value of l, distances are rescaled according to equation (4).
||
Unitsarechosensuchthatgapheight,length,andwidthare1
plane direction (h(cid:48)(cid:48)) are positive. One can therefore ex-
⊥ for |l|=0.01 in the square model.
pect that the solutions for gap and displacement are (lo-
cally) similar for the different symmetries, except for pre-
and scaling factors. In addition, percolation is continuous
cally adopt the contact line shapes from Figure 5 when
in the adhesion-free case, thereby constituting a critical
viewedwithsufficientmagnification. Panel(b)showsthat
point. Thus, there should exist (quasi-) universal func-
the gap can also be collapsed onto universal functions.
tions for the various fields in the vicinity of the critical
The numerical estimates for the exponents were ob-
point, e.g., for the gap, on which we focus exemplarily.
tained as follows: We first identified L as accurately as
Following the ideas of the scaling hypothesis, the gap c
possible. Wethenransimulationsforl=−0.002,−0.005,
should then satisfy
−0.01, and −0.02. By analyzing how the gap at the sad-
(cid:18) x y (cid:19) dle point scales with l, the exponent ζ was determined to
g(l,x,y)=|l|ζg± , (4)
|l|χ |l|υ be ζ = 1.2±0.01. The widths of the open channels were
determined from the same set of simulations. The con-
where l ≡(L−L )/L is a reduced load, g±(...) denotes tact line positions, r , resulted from fitting the relation
c c √ c
two master functions depending on the sign of l, while χ, F(r = 0,r ) ∝ r −r to the forces exerted by the
|| ⊥ ⊥ c
υ, and ζ are universal scaling exponents. To superimpose substrateontotheelasticbodynearthecontactline. Here,
the gap functions, not only for different values of l but thesaddlepointsatatr =0andr =0inthelocalcoor-
|| ⊥
alsofordifferentmodels—ormoregenerallyspeakingfor dinate system. This fitting procedure allowed us to deter-
different curvature ratios η ≡ −h(cid:48)(cid:48)/h(cid:48)(cid:48) characterizing the minethewidthoftheopenchannelwithsub-discretization
|| ⊥
geometry of the saddle point — appropriate unit systems resolution. All data was consistent with υ = 0.45±0.01.
for x, y, and z need to be defined. For example, for the To determine the length of blocked channels, simulations
squarelattice,orη =1,areasonablechoiceistodefinethe wererunforpositivereducedloadsandthesameabsolute
minimum width and minimum height of the open channel values of l as before. The pressure profile on the symme-
as1forl=−0.01justlikethelengthoftheblockedchan- try axis (r ,r = 0) followed a Hertzian pressure profile
|| ⊥
nelforl=0.01. Forothergeometries,orη (cid:54)=1,theproper very accurately, which allowed us to determine the length
definition of units then depends on η and also on the sign of the contact line on the symmetry axis to high precision
of l. and ultimately to ascertain that χ=0.6±0.01. Since the
Numerically, we see equation (4) to hold and demon- pressure profile on the contact ridges appears to be iden-
strate this in Figure 5. Specifically, in panel (a), contact tical to that of a Hertzian contact, we believe that there
lines obtained for different reduced loads and different ge- exists an analytical solution, in which case, we would ex-
ometries(exemplarilytriangularandsquare)collapseonto pecttheexponentstobesimplerationalnumbers—asin
their master curve that solely depends on the sign of l. theHertziancontactproblem. Wethereforespeculatethat
Thus, all (near-) critical constrictions from Figure 4, lo- χ=3/5, υ =9/20, and ζ =6/5 are the exact exponents.
p-4
Contact mechanics of saddle points
L
transition, flowceasesandstartsdiscontinuouslywiththe
0.0 0.5 1.0 1.5 2.0 2.5
closing or opening of the constrictions once adhesion is
considered.
-1
10
Ourpresentanalysisdoesnotincludeplasticity,because
-3 it will mostly affect the points in earliest contact, namely
10
j the peak heights where the local pressures are highest.
10-4 hex tri We are mostly concerned with the areas with the most
squ tenuous contact, and lowest contact pressures. Lateral
10-5 plastic material flow from the peaks might slightly alter
the detailed shape of the saddle point but such effects are
5
3.4 0.4 tri beyond the scope of the present work.
(L-L)c0.2 hex squ flow uaTtihone mofosttheprpoobwleemr alatiwc adpopwrnoxtiomathtieonnsanfoorsctohpeiccovnictiinn--
~j / 0.0 ity of the final constriction are most likely that we ne-
0.001 0.01 0.1
L-L glecttoconsider(i)theexistenceofasliplength(andthe
c
associated deviation from a Poisseuille flow profile), (ii)
confinement-induced viscosity change (via the orientation
Fig.6: Top: Fluidflowthroughtheinterfaceasafunctionof
load. Solid lines refer to γ =0 and dashed lines to γ =0.05. of molecules), as well as (iii) the non-smoothness of sur-
0 0
The latter have end points, which are indicated by symbols. faces at nano-scales (where the continuum approximation
Bottom left: Normalized fluid flow as a function of L −L, breaks down).
c
inthenon-adhesivecase. Bottom right: Visualizationofthe However, we note that shear-thinning does not come
Reynolds thin-film current density in the critical region. into play for the flow analysis, not even very close to the
percolationthreshold. Thisisbecausethemaximumshear
rate of a Poisseuille flow through a constriction is propor-
Knowledge of the three exponents ζ, χ, and υ allows
tionaltothegapheightdividedbythechannellength. As
one to deduce the power law with which Reynolds flow
a consequence, the maximum shear rate disappears with
is suppressed as the load is increased toward L . Since
c lζ−χ with ζ − χ ≈ 0.6, which we see confirmed by our
the fluid-pressure gradient is non-negligible only near the
simulation results.
critical constriction, the resistance to fluid flow is propor-
tional to the length of the channel. Moreover, it is in-
Conclusions. – In this study we find that adjacent
versely proportional to the channel width and, according
contact patches merge or break up discontinuously when
to the Reynolds equation, the third power of the channel
finite-range interactions act in addition to hard-wall re-
height. Thus, the resistance scales as
pulsion. This result has two main implications: First,
in most cases, contact patches cannot be separated by
R(L)∝(L −L)−β (5)
c a marginal distance or joined via arbitrarily thin ridges,
which might explain why classical experiments [3] visual-
with
izingthecontactbetweentwonominallyflatsolidsaswell
β =3ζ+υ−χ, (6)
as recent computer simulations including adhesion [25]
andanumericalvalueofβ =3.45. Figure6confirmsthese showedclearlyseparatedcontactpatches,whilelarge-scale
expectations for the investigated adhesion-free surfaces: simulationsofnon-adhesivesolids[5–9]usuallynecessitate
Nearthepercolationthreshold,allthreesystemdisappear excessivelyhighresolutiontodeterminetheconnectedness
with the same power law, even if prefactors may differ of contact patches. Second, the contact-patch coalesence
substantially. The exponent of β = 3.45 is reproduced andbreak-upoccurthroughfirst-orderinstabilitiesimply-
quite accurately by the flow simulations. In the critical ing multistability. Thus, coalesence and break-up dynam-
regime, the (scaled) current density always looks like the ics constitute a dissipation mechanism for Coulomb fric-
flow shown in the bottom right of Figure 6. tion as any other instability induced by the relative slid-
The absolute flow through the contacts changes dra- ing motion of two solids [26,27]. Moreover, multistability
matically with the saddle point geometry even for fixed entails thermal aging, which could be perceived as bulk
height spectra, in parts, of course, because the percola- relaxation and lead, for example, to a non-negligible time
tion point is so sensitive to the geometry. In other words, dependence of the leak rate of seals near percolation.
the leakage rate is strongly dependent on the curvature In reference to Persson’s contact mechanics [2] and his
ratio η or the skewness s of the height distribution, po- single-junctionleak-ratetheory[14],wealsofindtwomain
tentiallyevenmoresothanpreviouslyrecognized[23],see implications. First, we demonstrate that the contact me-
also [24]. Specifically, the single-wavelength models have chanics of surfaces with single-wavelength roughness and
(cid:112) (cid:112)
η = (4,1,1/4) and s = (− 2/3,0, 2/3) for hexagonal, their leakage rate are highly sensitive to the way in which
square, and triangular lattices, respectively. In addition, local maxima and minima are arranged. For example,
justasthecontactareaisdiscontinuousatthepercolation ourtriangularandhexagonalmodelssealatverydifferent
p-5
W. B. Dapp et al.
normal loads and relative contact areas, although the two [8] Campan˜a´ C., Persson B. N. J. and Mu¨ser M. H., J.
geometries have identical height spectra. It might be pos- Phys.: Condens. Matter, 23 (2011) 085001.
sibletoincorporatesucheffects—effectivelyrepresenting [9] Putignano C., Afferrante L., Carbone G. and
acorrelationofthephasesfordifferenth˜(q)atfixed|q|— G.Demelio, J. Mech. Phys. Sol., 60 (2012) 973.
[10] Johnson K., Greenwood J. and Higginson J., Int. J.
into the theory via appropriate generalizations. Second,
Mech. Sci., 27 (1985) 383.
intheabsenceofadhesion,wefindthatthelengthandthe
[11] Krithivasan V. and Jackson R. L., Tribol. Lett., 27
widthofaconstrictionarenotsimilarinmagnitudeasas-
(2007) 31.
sumed in Ref. [14]. Instead they disappear with different
[12] ScaraggiM.,CarboneG.,PerssonB.N.J.andDini
power laws as Lc is approached. D., Soft Matter, 7 (2011) 10395.
When adhesion is absent, the contact mechanics of [13] Yastrebov V. A., Anciaux G. and Molinari J.-F.,
“well-behaved” saddle points shows universal behavior Tribol. Lett., 56 (2014) 171.
near the percolation point. There, the gap is entirely de- [14] LorenzB.andPerssonB.N.J.,EPL,86(2009)44006.
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pressureprofileonanewly-formedcontactridgealongthe [17] Campan˜a´ C.andMu¨serM.H.,Phys.Rev.B,74(2006)
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[18] Dapp W. B., Prodanov N. and Mu¨ser M. H., Tribol.
contact. We therefore see the possibility for closed-form
Lett., 53 (2014) 433.
analytical solutions of the contact mechanics of a saddle
[19] Falgout R., Jones J. and Yang U., The design and
point, even if formulating the boundary conditions could
implementation of hypre, a library of parallel high perfor-
be difficult.
mance preconditioners presented at Numerical Solution
Finally, we find for idealized conditions frequently as- of Partial Differential Equations on Parallel Computers
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c
(1996) 145.
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intheimmediatevicinityofthepercolationpoint[14],the
M. H., Phys. Rev. Lett., 108 (2012) 244301.
leakrateofasealthatispressedagainstarandomlyrough
[23] Lorenz. B. and Persson B. N. J., EPL, 90 (2010)
macroscopic solid would depend on load with the same
38002.
powerlawastheisolatedconstriction. Despitethisknowl- [24] Yastrebov V. A., Anciaux G. and Molinari J.-F.,
edge it is quite difficult to predict the absolute current Int. J. Solids Struct., 52 (2015) 83.
because the detailed topography of the last constriction [25] Pastewka L. and Robbins M. O., Proc. Natl. Acad.
has a tremendous effect on prefactors but is usually un- Sci., 111 (2014) 3298.
known. Still, the percolation exponent would result from [26] Prandtl L., Z. Angew. Math. Mech., 8 (1928) 85.
localcontactmechanicsandnot,ascommonlyassumedin [27] Mu¨ser M. H., Phys. Rev. Lett., 89 (2002) 224301.
the theory of percolation [28], from the disorder at large [28] Stauffer D.andAharony A.,An Introduction to Per-
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length scales. Whether the classical view is reestablished
once adhesion is included remains to be seen.
∗∗∗
We thank the Ju¨lich Supercomputing Centre for com-
puting time.
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