Effects of stretching on the frictional stress of rubber
aa r X i v : . [ c ond - m a t . s o f t ] J un Effects of stretching on the frictional stress of rubber.
Antoine Chateauminois, Danh-Toan Nguyen and Christian Fr´etigny ∗ Soft Matter Sciences and Engineering Laboratory (SIMM),PSL Research University, UPMC Univ Paris 06, Sorbonne Universits,ESPCI Paris, CNRS, 10 rue Vauquelin, 75231 Paris cedex 05, France (Dated: June 2, 2017)In this paper, we report on new experimental results on the effects of in-plane surface stretchingon the friction of Poly(DiMethylSiloxane) (PDMS) rubber with smooth rigid probes. Friction-induced displacement fields are measured at the surface of the PDMS substrate under steady-statesliding. Then, the corresponding contact pressure and frictional stress distributions are determinedfrom an inversion procedure. Using this approach, we show that the local frictional stress τ isproportional to the local stretch ratio λ at the rubber surface. Additional data using a triangularflat punch indicate that τ ( λ ) relationship is independent on the contact geometry. From frictionexperiments using pre-stretched PDMS substrate, it is also found that the stretch-dependence ofthe frictional stress is isotropic, i.e. it does not depend on the angle between stretching and slidingdirections. Potential physical explanations for this phenomenon are provided within the frameworkof Schallamach’s friction model. Although the present experiments are dealing with smooth contactinterfaces, the reported τ ( λ ) dependence is also relevant to the friction of statistically rough contactinterfaces, while not accounted for in related contact mechanics models. PACS numbers: 46.50+d Tribology and Mechanical contacts; 62.20 Qp Friction, Tribology and HardnessKeywords: Contact, Rubber, Elastomer, stretching, neo-Hookean
I. INTRODUCTION
In many practical situations, soft solids such as elas-tomers, gels or biological tissues experience mechanicalloading. These systems being very easily deformed, eventhe lightest stresses can induce strain level well beyondthe small strain hypothesis, which remains an open issuefor the description of their mechanical behaviour. Suchsituations are especially encountered in many contact ex-periments involving rigid probes where, as a consequenceof the finite size of the contacting bodies, high in-planestrains are invariably experienced at the periphery of thecontact. As an example, one can cite the friction of rub-ber with spherical glass probes which was found to resultin local strain higher than 30%. [1, 2] Another exam-ple is the peeling of soft adhesive tapes where large ten-sile stresses are combined with localized friction processeswithin the regions located at the peeling front. [3, 4]As far as one is concerned with the friction of rubbermaterials, one do not a priori expect friction to be af-fected by stretching by virtue of the liquid-like natureof these systems well above their glass transition tem-perature. However, some scarce experimental observa-tions tend to suggest the opposite. In a couple of papersdealing with pre-stressed rubbers strips, [5, 6] Barquinsand co-workers concluded that tensile stretching affectrolling friction with rigid cylinders. Although controver-sial, [7] their explanation was based on the hypothesisof a decrease in adhesion energy of the stretched rub-ber. They also showed that pre-stretching could affectsliding friction but in a regime complicated by the occur- ∗ [email protected] rence Schallamach detachment waves. [8] More recently,Yashima et al [9] observed that frictional shear stresswithin smooth contact between silicone rubber and glassspherical probes could depend on contact size or on thecurvature of the contact interface, an effect which couldtentatively be related to difference in contact-inducedsurface strains.More generally, these overlooked issues pertain to anylocal physical description of rubber friction, especiallywith statistically rough surfaces. In such systems, indi-vidual micro-asperity contacts occur locally on a rubbersurface which is known to experience finite stretch gra-dient at the scale of the macroscopic contact. [1, 2] How-ever, the effects of such finite strains on micro-contactsshape and stresses are largely overlooked in current meanfield descriptions of rough contacts (see e.g. [10]), al-though they may affect the prediction of the actual con-tact area and the associated frictional forces.In the present study, we report on new experimen-tal evidences of a dependence of frictional stress on sur-face stretch within macroscopic sliding contacts betweena smooth silicone rubber and rigid probes. From opti-cal contact imaging methods, we determine both surfacedisplacement and stress fields with a space resolution ofabout 10 µ m. Using this approach, we show that the lo-cal frictional stress is proportional to the local in-planesurface stretch independently on contact pressure, slidingvelocity and contact geometry. Additional experimentsusing pre-stressed silicone strips also show that the in-crease in frictional stress with stretch ratio is isotropic,i.e. it is insensitive to the orientation of the sliding di-rection with respect to the stretch direction. Potentialexplanations for this phenomenon are discussed withinthe framework of the Schallamach’s model of rubber fric-tion. II. EXPERIMENTAL DETAILS
Friction experiments are carried out using either un-stretched or uniaxially stretched Poly(DiMethylSiloxane)(PDMS) substrates in contact with smooth glass lenses.Contacts with the unstretched PDMS specimens areinvestigated using a custom-built device allowing tomeasure optically friction-induced surface displacementfields within the contact zone with sub-micrometeraccuracy and a space resolution of about 10 µ m. Asdetailed in reference, [2] surface displacements aremeasured from the deformation of a square network ofsmall cylindrical holes stamped on the PDMS surfaceby means of standard soft lithography techniques Then,a numerical inversion procedure allows to retrieve thecorresponding contact stress distribution while takinginto account the geometrical and material non linearitiesassociated with the occurrence of finite strains within thecontact (see Nguyen et al [2] for further details). Theseexperiments are carried out under a constant appliednormal force (between 1.4 and 3.3 N) and at imposedsliding velocity (between 0.1 and 1 mm s − ) using BK7plano-convex glass lenses with radii of curvature rangingfrom 5.2 to 25.9 mm.Additionally, some experiments with unstretchedPDMS specimens are performed using a triangu-lar flat punch which was micro-machined from aPoly(MethylMetAcrylate) (PMMA) block. The contact-ing face of the punch has the shape of an isoceles righttriangle with two 6 mm edges. In order to minimizestress concentration at the edge of the contact, thecorners of the triangular punch are rounded with a300 µ m radius of curvature.Using another dedicated set-up, friction experimentsusing uniaxially pre-stretched PDMS substrates arealso carried out in order to investigate whether a bulktensile stretching could result in an anisotropy in thefrictional shear stress. As schematically described inFig. 1, a PDMS strip is stretched between two grips ona loading frame which is fixed on two crossed motorizedtranslation stages which allow to vary the angle betweenstretching and sliding directions. Contact is achievedunder an imposed displacement condition using a glasslens with a radius of curvature of 5.2 mm. The frictionforce along the sliding direction is measured usinga custom-built optical load sensor. It consists in athin (1 mm) layer of a silicone elastomer crosslinkedbetween a flat glass slide and a glass disk 20 mmin diameter on top of which the glass lens is glued.The internal face of the glass disk in contact with thesilicone rubber is marked with a network of cylindricalposts (diameter 20 µ m, center-to-center spacing 70 µ m)using conventional SU8 resin lithography. It is easilydetected optically as a result of the mismatch betweenthe refractive index of the SU8 resin and the siliconematerial. The displacement of this markers’ networkunder shear is measured by digital image correlationtechniques and translated into a frictional force through FIG. 1: Schematic description of the custom-built experi-ments for friction measurements on uniaxially pre-stretchedPDMS substrates. A stretched PDMS strip (a) is clampedon two crossed linear translation stages (b) allowing to varythe sliding direction with respect to the stretching axis. Con-tact with a plano-convex glass lens (c) is achieved under animposed normal displacement condition using a manual lin-ear translation stage (d). A CMOS camera (e) allows contactvisualization trough the transparent PDMS substrate. Thefrictional force along the sliding axis is measured using a cus-tom made sensor consisting in a silicone disk (f) enclosed be-tween a patterned (with SU8 resin pillars) glass disk (g) anda glass plate. After stiffness calibration, the force is deter-mined from sub-pixel measurements of the displacement ofthe pattern using a CMOS camera (h). The inset shows twosuperimposed images of the pillars pattern taken before andafter the application of a frictional force. appropriate calibration. The geometrical confinement ofthe incompressible silicone layer was intended to preventtilting motions during shearing. In addition, the stiffnessof the load sensor can be tuned by playing with the shearmodulus of the silicone elastomer. Based on a workby Palchesko et al , [11] this can be achieved by mixingin various weight ratios commercially available Sylgard184 and 527 PDMS (Dow Chemicals, Midland, MI).Here, mixing of Sylgard 184 and Sylgard 527 in a 40:60weight ratio resulted in an isotropic lateral stiffness of15.3 10 N m − .Sylgard 184 PDMS is used as an elastomer substratefor all friction experiments. The silicon monomer andthe hardener are mixed in a 10:1 weight ratio andcrosslinked at 70 ◦ C for 48 hours. Contact experi-ments with unstretched PDMS are carried out using15 × ×
60 mm specimens marked with a squarenetwork of small cylindrical holes (diameter 20 µ m,depth 2 µ m and center-to-center spacing 400 µ m) inorder to measure the surface displacement field understeady-state sliding conditions.Friction experiments with uniaxially stretched PDMSsubstrates are performed using 5 × ×
100 mm strips.The stretch ratio of the PDMS substrate in the contactregion is measured optically from the deformation of aholes square network (diameter 20 µ m, depth 5 µ m andcenter-to-center spacing 80 µm ) at the surface of therubber specimen.In all the experiments to be reported, the contact con-ditions ensured the achievement of semi-infinite contactconditions ( i.e. the ratio of the substrate thickness tothe contact radius is larger than ten [12]). III. STRETCH AND STRESSES WITHINCONTACT AREA
Fig. 2 shows a typical example of a measured sur-face displacement field during steady-state friction of aspherical glass probe on a PDMS substrate. In this rep-resentation, in-plane displacements components u x and u y (where y is the sliding axis and x is perpendicular)are mapped in Lagragian coordinates, i.e. space coordi-nates refer to the deformation state prior to the appli-cation of lateral contact loading. The edge of the con-tact area is delimited in the same Lagrangian coordinatesby a dotted line in this figure. The main displacementcomponent is obviously along the sliding direction, thetransverse displacements being about ten times lower.The quadrupolar symmetry of u x field reflects the occur-rence of Poisson’s effect: the PDMS surface is compressed(resp. stretched) along the sliding direction at the lead-ing (resp. trailing) edge of the contact, which results inan expansion (resp. contraction) in the transverse direc-tion.local surface stretching can be estimated from a spacederivative of this displacement field. Fig. 3, shows thedistribution of the longitudinal and transverse stretchratio defined as λ y = 1 + ∂u y ∂y and λ x = 1 + ∂u x ∂x , re-spectively. Transverse surface stretch λ x appears to berestricted to a very narrow band at the contact periph-ery and its magnitude is much lower than that of thelongitudinal stretch λ y . The later exhibits a clear gradi-ent along the sliding direction, from compression at theleading edge of the contact to traction at the trailingedge. Noticeably, the maximum and minimum values ofthe longitudinal stretch ratio (about 1.27 and 0.83, re-spectively for the contact conditions under considerationin the figure) fall well beyond the small strain hypoth-esis. More precisely, previously reported tensile testingdata for the used silicone rubber [13] indicate that surfacestrains experienced by the substrate during steady statefriction fall within the neo-Hokean range. Unless oth-erwise specified, all the stress distributions determinedherein from the inversion of displacement fields were thusobtained using a neo-Hookean description of the mechan-ical response of PDMS.Fig. 4 details the characteristic contact pressure and mm -40-2002040 x - mm FIG. 2: Measured displacement field during steady-state slid-ing of a glass lens (radius of curvature R = 9 . v = 0 . − , appliednormal load P = 1 . u y along the slid-ing direction; bottom: displacement u x perpendicular to thesliding direction. The PDMS substrate is moved from bottomto top with respect to the fixed glass lens. Displacements aremapped in Lagragian coordinates, i.e. relative to the equilib-rium state before the application of lateral displacement. Thedotted lines delimits the edge of the contact area using thesame Lagragian coordinates. Note that the magnitude of u x is much lower than that of u x . frictional shear stress distributions which are obtainedfrom the inversion of measured surface displacementfields under steady-state friction with a smooth sphericalprobes. Here, according to an Eulerian description, werefer to Cauchy stress components, i.e. to equilibriumstresses expressed in the deformed space. As detailedin Fig. 5, contact stress distribution exhibits a classicalHertzian shape which do not deserve further comments.In what follows, we focus instead on the frictional stress λ y λ x FIG. 3: Measured stretch ratios along ( λ y , top) and perpen-dicular ( λ x , bottom) to the sliding direction in Lagragian co-ordinates (same experimental conditions as for Fig. 2). ThePDMS substrate is moved from bottom to top with respectto the fixed glass lens. field distribution which, at first sight, seems constant forthe smooth contact interface under consideration.However, a careful examination of the correspondingstress field reveals a clear gradient of the frictional stressalong the sliding direction with a minimum at the leadingedge of the contact and a maximum at the trailing edge(Fig. 6, bottom). Such a feature was systematically pre-served whatever the contact pressure, the sliding velocityand the radius of curvature of the glass lens. A com-parison of stress cross-sections perpendicular to (Fig. 6,top) and along (Fig. 6, bottom) the sliding direction in-dicates that the observed changes in frictional stress arenot correlated to the contact pressure distribution. Theycan neither be explained from a simple viscoelastic ef-fect which would induce a stress gradient in the opposite M P a M P a FIG. 4: Contact pressure and frictional shear stress distri-bution deduced from the inversion of a measured displace-ment field (same experimental conditions as for Fig. 2). ThePDMS substrate is moved from bottom to top with respectto the fixed glass lens. Space coordinates are relative to thedeformed state, i.e. stresses correspond to Cauchy’s stressesexpressed in Eulerian coordinates. The white lines delimitsthe edge of the contact area in the same deformed space. direction: shear stress would be higher in the less re-laxed state encountered at the leading contact edge thanin the more relaxed regions at the trailing edge. In ad-dition, one can also mention that the frequency of theglass transition of the selected PDMS substrate at roomtemperature is about 10 Hz, [14] while the characteris-tic strain frequency of the contact, v/a (where v is thesliding velocity and a the contact radius) is no more than10 Hz.Interestingly, it turns out that the frictional stressgradient is rather correlated to the value of the local lon-gitudinal surface stretch ratio achieved within the con-tact. An example of this correlation is shown in Fig. 7 C on t a c t p r e ss u r e p ( M P a ) -3 -2 -1 0 1 2 3Space coordinate (mm) FIG. 5: contact pressure profiles across the contact zone andperpendicular to the sliding direction for increasing values ofthe applied normal load P ( v = 0 . − , R = 9 . P = 1 . P = 2 . P = 3 . where the local frictional stress τ has been reported as afunction of the local stretch ratio λ = λ y for a set of datapoints taken within the contact area ( λ values were trans-ferred to Eulerian coordinates for that purpose). Fromthis example, it turns out that the local frictional shearstress is linearly increasing from about 0.34 to 0.43 MPawhen the local stretch ratio is increasing from 0.85 to1.15.As shown in Fig. 8, this linear increase in the frictionalstress with stretch ratio is preserved whatever the radiusof the spherical probe, the applied contact force and thesliding velocity within the ranges under consideration.For the shake of clarity, only τ ( λ ) data taken along con-tact cross-sections parallel to the sliding direction werereported as blue lines in this figure. In order to accountfor slight changes in the average frictional stress betweendifferent PDMS substrates and for different sliding veloc-ities, the local frictional stress data τ where normalizedwith respect to the value τ corresponding to λ = 1, i.e.to the unstretched state achieved at the middle of thecontact area. From this plot, it emerges that the localfrictional stress is always proportional to the local stretchratio, i.e. τ = λτ .Noticeably, the τ ( λ ) relationship is found to be in-dependent on contact geometry. This is demonstratedby measurements using a flat triangular punch insteadof a spherical probe. The corresponding stretch ratioand frictional shear stress fields are shown in Fig. 9 withthe sliding direction oriented along the largest edge ofthe triangular punch. Here again, a stretch gradient isevidenced along the sliding direction. Stress amplifica-tion at the vicinity of the edge of the flat punch is foundto result in stretch ratios which are significantly higher(typically 1.45 a the trailing edge of the contact) thanfor spherical probes. As a result, inversion using a neo-Hookean behaviour law for the PDMS proved to be in-accurate and the numerical inversion of the displacement F r i c t i ona l s t r e ss τ ( M P a ) -3 -2 -1 0 1 2 3Space coordinate (mm) F r i c t i ona l s t r e ss τ ( M P a ) -3 -2 -1 0 1 2 3Space coordinate (mm) FIG. 6: Shear stress profiles across the contact zone, bothperpendicular to (top) and along (bottom) the sliding direc-tion for increasing values of the applied normal load P (sameexperimental conditions as for Fig. 5). Red: P = 1 . P = 2 . P = 3 . field was carried out using Yeoh’s hyperelastic model [15]which is able to account for the strengthening of the rub-ber response at high stretch ratios ( λ > . τ /τ ( λ ) representation(red line in Fig. 8), it turns out that the stretch depen-dence of the frictional stress is exactly the same as forspherical probes. Moreover, this is verified whatever theorientation of the facets of the punch with respect to thesliding direction (the red and green lines in Fig. 8 cor-respond to sliding directions parallel and perpendicular,respectively, to the large edge of the triangular punch).As a conclusion, it emerges that the local frictional stressof the PDMS substrate sliding against a smooth rigid sur-face is proportional to the local value of the stretch ratio,independently of the geometry of the contact area, of thesliding velocity and of the applied contact pressure. FIG. 7: Local shear stress versus local stretch ratio within asliding contact ( P = 3 . v = 0 . − , R = 9 . τ / τ λ FIG. 8: Normalized local frictional stress τ /τ versus localstretch ratio λ ( τ is the measured local frictional stress for λ = 1). Blue lines: spherical probes with 1 . < P < . . < R < . . < v < − . Red line:isoceles triangular flat punch with the sliding direction alongthe larger edge ( P = 1 . v = 0 . − ); green line:same with the sliding direction along the height of the isoce-les triangle. For the shake of clarity, data were taken alongcontact cross-sections parallel to the sliding direction. IV. FRICTION ON A PRE-STRETCHEDSUBSTRATE
Some additional insights into this dependence are pro-vided by experiments where the stretch state of the rub-ber network is varied by superimposing a bulk tensilestretch λ p to the contact induced deformation. Suchexperiments especially allow to investigate whether the λ y FIG. 9: Longitudinal stretch ratio (top) and frictional shearstress (bottom) fields within a sliding contact between aPDMS substrate and a triangular flat punch ( v = 0 . − , P = 1 . stretch dependence of the frictional stress depends onthe orientation of the sliding direction with respect tothe stretch direction. As fully detailed and accountedfor in a companion paper [13], normal contact between aspherical probe and a stretched rubber substrate is char-acterized by an elliptical shape of the contact with themajor axis perpendicular to the stretching direction. Asan example, a normal contact image on a pre-stretched( λ p = 1 .
24) PDMS is show in Fig. 10(a) where the ratioof the major to the minor axis of the elliptical contact is1.12.
FIG. 10: Images of the contact area between a pre-stretched( λ p = 1 .
24) PDMS substrate and a spherical probe ( R =5 . v =0 . − ) with (b) θ = 0, (c) θ = − π/ θ = − π/ θ is the angle between the stretching and sliding direc-tions. The stretching direction is along the vertical directionand the displacement of the PDMS substrate with respect tothe fixed glass lens is indicated by an arrow. Under steady-state sliding, strong changes in this ini-tially elliptical shape of the contact area are observedwhich depend on the angle θ between the sliding andstretching directions (Fig. 10). However, the associatedfrictional stress is observed to be independent on the slid-ing direction. Here, we just consider the average value ofthe frictional stress τ defined as the ratio of the frictionalforce F to the actual contact area A measured understeady state sliding. When the normalized value τ /τ ofthis averaged shear stress (where τ is the average shearstress of the unstretched rubber substrate) is reported asun function of the pre-stretch ratio λ p (Fig. 11), we ob-tain exactly the same linear relationship as for the local τ ( λ ) analysis carried out using un-stretched substrates.Moreover, this dependence is found to be insensitive tothe orientation of the sliding direction with respect tothe stretch direction, despite a strong anisotropy in thecontact shape. V. DISCUSSION
From a local analysis of rubber friction with smoothrigid bodies, we showed that the local frictional shearstress is proportional to in-plane stretching of the elas- τ / τ λ p FIG. 11: Normalized average frictional stress τ/τ as a func-tion of the tensile stretch ratio λ p of the PDMS substrate ( τ and τ are the average frictional stress measured for λ p = 1and λ p = 1, respectively).( ◦ ) θ = 0, ( • ) θ = π/
2, ( ⋄ ) θ = π/ tomer surface. To our best knowledge, such a dependencehas never been reported and its physical origin remainsunclear. In order to get additional insights into this phe-nomenon, experiments were also carried out on uniaxi-ally stretched substrates which showed that the stretch-dependence of friction is independent on sliding directiondespite a strong anisotropy in the contact shape understeady-state sliding. It therefore turns out that this phe-nomenon is a generic feature of rubber friction.For the contact conditions under consideration, a weak,logarithmic, dependence of the frictional stress on veloc-ity has previously been reported.[14] As a consequence,one cannot explain the observed changes in the local fric-tional stress from a consideration of the variations in localsliding velocity which result from the non-homogeneousdeformation of the PDMS substrate: as a result of com-pressive (resp. tensile) surface strains at the leading(resp. trailing) edges of the contact, the actual slidingvelocity at the contact interface is effectively lower (resp.higher) than the driving velocity imposed to the PDMSsubstrate. Here, the resulting variations in sliding veloc-ity within the contact area (about ± σ can be calculated in the form σ ≈ N k B T ζ ln V uV ∗ (1)where V is the steady velocity, N represents the totalnumber of surface sites to which the polymer can bondand ζ is an activation length, u = e − W/k B T with W the free energy difference between the bounded and un-bounded states of the bonding site. V ∗ = k B T /τ ζM where M represents the stiffness of the polymer chainand τ = τ e E/kT with E the bond activation energy.This regime can be shown to exist if u << u < V /V ∗ < N is pro-portional to the stretch ratio. However, for a liquid-likematerial such a rubber, one do not expect the chain den-sity to vary with the extension of the surface. One shouldtherefore postulate that surface stretching brings to therubber surface some polar bonding sites which would oth-erwise remain buried within the bulk network and un-available for pinning to the glass surface. Some analogiesbetween such a mechanism and chemical modifications ofPDMS surface may interestingly be drawn. As detailedin refs., [21, 22] exposing the PDMS surface to an airor oxygen plasma introduces silanol (Si-OH) groups andmakes the surface hydrophilic. If the treated surface isleft in contact with air, surface rearrangements progres-sively occur that bring new hydrophobic groups to thesurface to lower the surface free energy and increase thewetting angle with water.Alternately, the observed stretch-dependence of fric-tion could be considered within a continuum mechanicsframework. If friction is still assumed to result from con-centrated point forces acting on the rubber surface atsome length scale, one could consider the effect of fi-nite substrate strains on such forces. This can be ac-counted for by considering the formulation of surface Green’s function for a stretched substrate which was re-cently derived by He [23] for neo-Hookean materials. Bytaking a concentrated force oriented to the axis 1, TheGreen’s coefficient G ( x ) provides the displacement ofsurface points situated along the same axis. For a stretchratio λ oriented along x , G reads G ( x , ,
0) = λ πµx (2)This expression is similar to that derived for linear elas-ticity, expect that the shear modulus µ is replaced by anapparent modulus equal to µ/λ . This result can be ra-tionalized by considering that the above expression wasderived using incremental deformation theory with a neo-Hookean consitutive law characterized by a decrease inthe tangent modulus as a function of λ . However, it can-not not simply account for the experimental observationunless we speculate on the occurrence of some sliding het-erogeneities, such as localized stick-slip events, at someintermediate length scale below the experimental spaceand time resolution. Finite strain effects such as thosedescribed by the above neo-Hookean Green’s tensor couldthen be involved. In its spirit, such an approach wouldbe reminiscent of spring-block models used to describethe propagation of slip fronts during sliding. [24] Its val-idation would clearly require further experimental andtheoretical investigations which are beyond the scope ofthis work.Although we provided some tentative explanations forthe observed stretch-dependence of friction, it still needto be rationalized within the framework of a theoreticalmodel. However, it must be emphasised that any theo-retical description of the experimental results should ac-count for the isotropy of the effect of substrate stretchingon the frictional response VI. CONCLUSION
In this study, we have shown that local frictional stresswithin smooth contacts between silicone rubber and rigidprobes is proportional to the in-plane surface stretchof the elastomer substrate. Additionally, the observedincrease in friction with stretch ratio is found to beisotropic, i.e. it does not depend on the relative orien-tation of sliding and stretching. To our best knowledge,such an effect has never been reported. Although the de-termination of its physical origin requires further eluci-dation, it is found to be very robust and independent oncontact geometry. This overlooked stretch-dependenceof rubber friction pertains to many different situations.As mentioned in the introduction, frictional contact withstatistically rough surfaces are clearly an example wherefrictional stresses achieved locally at micro-contact scalewill be affected by surface strains induced at the macro-scopic contact scale. Interestingly, such surface strainsare a consequence of the finite size of the contact. Ittherefore turns out that the stretch-dependence of fric-tion introduces a coupling between the macroscopic andmicroscopic contact length scales which is clearly not ac-counted for in existing rough contact mechanics theoriesdealing with extended - infinite - contact interfaces.The observed stretch-dependence of rubber friction alsopertains to the contact response of bio-mimetic, fibrillaradhesives where friction occurs locally on slender elas-tomer fibrils which can be easily elongated well beyondlinear regime.
VII. ACKNOWLEDGMENTS
We gratefully acknowledge Chung Yuen Hui, AnandYagota, Johan Le Chenadec and St´ephane Roux for very stimulating discussions about this paper. We areespecially indebted to A. Jagota for suggesting us apotential effect of stretching on the availability of polarsites on the PDMS surface. The authors also wish tothank Alexis Prevost for his kind help in the realizationof the triangular punch. [1] M. Barquins. Sliding friction of rubber and schallamachwaves- a review.
Materials Science and Engineering ,73:46–63, 1985.[2] D.T. Nguyen, P. Paolino, M-C. Audry, A. Chateaumi-nois, C. Fr´etigny, Y. Le Chenadec, M. Portigliatti, andE. Barthel. Surface pressure and shear stress field withina frictional contact on rubber.
Journal of Adhesion ,87:235–250, 2011.[3] B. Newby and M.K. Chaudhury. Effect of interfacial slip-page on viscoelastic adhesion.
Langmuir , 13:1805?1809,1997.[4] B.Z. Newby, M.K. Chaudhury, and H.R. Brown. Macro-scopic evidence of the effect of interfacial slippage on ad-hesion.
Science, New Series , 269:1407–1409, 1995.[5] M Barquins and E. Felder. Influence of pre-strains on theadhesion of natural rubber.
Compte rendus de l’Acad´emiedes Sciences S´erie II , 313:303–306, 1991.[6] M Barquins and S. H´enaux. Effects of pre-strains on therolling speed of rigid cylinders on stretched plane andsmooth sheets of natural rubber.
Comptes Rendus del’Acad´emie des Sciences. S´erie II , 317:1141–1147, 1993.[7] C. Gay. Does stretching affect adhesion?
InternationalJournal of Adhesion and Adhesives , 20:387–393, 2000.[8] M. Barquins, R. Courtel, and D. Maugis. Friction onstretched rubber.
Wear , 38:385–389, 1976.[9] S. Yashima, V. Romero, E. Wandersman, C. Fr´etigny,M.K. Chaudhury, A. Chateauminois, and A.M. Prevost.Normal contact and friction of rubber with model ran-domly rough surfaces.
Soft Matter , 11:871 – 881, 2015.[10] M. Scaraggi and BNJ Persson. General contact me-chanics theory for randomly rough surfaces with appli-cation to rubber friction.
Journal of Chemicals Physics ,143:224111, 2015.[11] R.N. Palchesko, L. Zhang, Y. Sun, and A.W. Fein-berg. Development of polydimethylsiloxane substrateswith tunable elastic modulus to study cell mechanobiol-ogy in muscle and nerve.
PLoS ONE , 7(12):e51499, 2012.[12] E. Gacoin, A. Chateauminois, and C. Fr´etigny. Mea-surement of the mechanical properties of polymer filmsgeometrically confined within contacts.
Tribology Letters ,21(3):245–252, 2006. [13] C. Fr´etigny and A. Chateauminois. Contact of a sphericalprobe with a stretched rubber substrate. to appear inPhyiscal Review E , 2017.[14] D. T. Nguyen, S. Ramakrishna, C. Fr´etigny, N. D.Spencer, Y. Le Chenadec, and A. Chateauminois. Fric-tion of rubber with surfaces patterned with rigid spheri-cal asperities.
Tribology Letters , 49(1):135–144, January2013.[15] O. Yeoh. Some forms of the strain energy function forrubber.
Rubber Chemistry and Technology , 66:754–771,1993.[16] A. Schallamach. A theory of dynamic rubber friction.
Wear , 6:375–382, 1963.[17] Y.B. Chernyack and A.I. Leonov. On the theory of ad-hesive friction of elastomers.
Wear , 108:105–138, 1986.[18] A.I. Leonov. On the dependence of friction force on slid-ing velocity in the theory of adhesive friction of elas-tomers.
Wear , 141:137–145, 1990.[19] A.K. Singh and V.A. Juvekar. Steady dynamic frictionat elastomer-hard solid interface: a model based on pop-ulation balance of bonds.
Soft Matter , 7:10601, 2011.[20] Katherine Vorvolakos and Manoj K Chaudhury. The ef-fects of molecular weight and temperature on the kineticfriction of silicone rubbers.
Langmuir , 19(17):6778–6787,2003.[21] J.N. Lee, C. Park, and G.M. Whitesides. solvent compat-ibility of poly(dimethylsiloxane)-based microfluidic de-vices.
Analytical Chemistry , 75:6544–6554, 2003.[22] J.M.K. Ng, I. Gitlin, A.D. Stroock, and G.M. Whitesides.Components for integrated poly(dimethylsiloxane) mi-crofluidic systems.
Electrophoresis , 23:3461–3473, 2002.[23] L.H. He. Elastic interaction between force dipoles ona stretchable substrate.
Journal of the Mechanics andPhysics of Solids , 56(10):2957 – 2971, 2008.[24] D.S. Amundsen, J.K. Tromborg, K. Thogersen,E. Katzav, A. Malte-Sorenssen, and J. Scheibert. Steady-state propagation speed of rupture fronts along one-dimensional frictional interfaces.