Contact of a spherical probe with a stretched rubber substrate
aa r X i v : . [ c ond - m a t . s o f t ] M a r Contact of a spherical probe with a stretched rubber substrate
Christian Fr´etigny and Antoine Chateauminois ∗ Soft Matter Science 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: March 27, 2018)We report on a theoretical and experimental investigation of the normal contact of stretched neo-Hookean substrates with rigid spherical probes. Starting from a published formulation of surfaceGreen’s function for incremental displacements on a pre-stretched, neo-Hookean, substrate (L.H. Lee
J. Mech. Phys. Sol. (2008) 2957-2971), a model is derived for both adhesive and non-adhesivecontacts. The shape of the elliptical contact area together with the contact load and the contactstiffness are predicted as a function of the in-plane stretch ratios λ x and λ y of the substrate. Thevalidity of this model is assessed by contact experiments carried out using an uniaxally stretchedsilicone rubber. for stretch ratio below about 1.25, a good agreement is observed between theoryand experiments. Above this threshold, some deviations from the theoretical prediction are inducedas a result of the departure of the mechanical response of the silicone rubber from the neo-Hokeeandescription embedded in the model. PACS numbers: 46.50+d Tribology and Mechanical contacts; 62.20 Qp Friction, Tribology and HardnessKeywords: Contact, Rubber, Elastomer, stretching, neo-Hookean
Introduction
Contact of soft solids such as elastomers, gels or biolog-ical tissues with rigid probes pertains to many practicalsituations including, as an example, the determinationof the mechanical properties of these objects using in-dentation methods. Such systems being very easily de-formed, contact stresses are often superimposed to bulkstresses which fall beyond the limit of linear elastic de-scriptions. As an example, one can cite the local frictionof smooth rubbers surfaces with statistically rough rigidbodies. At the scale of the macroscopic contact, the fi-nite sizes of the contacting bodies induce in-plane surfacestrains which can easily exceed 0.2 can under the action ofa frictional stress [1, 2]. At the microscopic scale, this im-plies that single micro-asperity contacts occur locally ona pre-stretched rubber surface. The effects of such finitestrains on micro-contacts shape and stresses are largelyoverlooked in current contact mechanics description ofrough contacts, although they may affect the predictionof the actual contact area and the associated frictionalforces.From a theoretical perspective, contact problems onsoft rubber substrates subjected to finite strains havebeen essentially handled within the framework of theinfinitesimal deformation theory developed by Biot [3],Green and co-workers [4, 5]. In these approaches,contact-triggered infinitesimal deformations are superim-posed upon finite deformations due to pre-stress. Earlysolutions along these lines include the work by Dhali-wal [6, 7] and co-workers who handled the problem of thecontact of rigid axisymmetric probes with a neo-Hookean ∗ [email protected] body under a state of uniform biaxial stretching. Us-ing the solution established by Dhaliwal and Singh [7],Yang derived analytical expressions for the relation be-tween contact stiffness, contact area, elastic constants,and finite stretch [8]. Here again, the theory deals withthe axisymmetrical indentation of a neo-Hoohean solidunder uniform bi-axial stretching. Additional solutionsfor plane-strain contacts with hyperelastic half spaceswere also derived by Brock [9, 10] which incorporateanisotropic frictional situations.These problems were experimentally addressed by Bar-quins and co-workers who carried out a series of con-tact experiments involving spherical or cylindrical probesand natural rubber sheets under a state of either uni-axial or uniform biaxial tensile strains [11–14]. Theseworks were carried out with the objective of investigat-ing the effects of a pre-stretch on the formation of Schal-lamach waves [11], on rolling friction [12] and on adhe-sion [13, 14]. They especially evidenced the anisotropyinduced by uniaxial tensile stretching which result in thedevelopment of elliptical contact shapes (with sphericalprobes) and in anistotropic friction forces. These experi-ments were revisited later on by Gay [15] which assumedthat a superposition principle can be applied to the re-sponse of the rubber to both the initial stretching and thedeformation due to the rigid probe. This superpositionbeing performed in Lagrangian coordinates, Gay devel-oped an argument stating that the indentation stage of anuniaxially pre-stretched substrate can be assimilated tothe indentation of an elastic half-space by an ellipsoidalindenter which indeed account for the elliptical contactshape.In the present study, we develop a more general ap-proach of the contact problem of a pre-stretched neo-Hookean substrate with a rigid spherical probe. It isbased on the Green’s function of the solid which describesits response to a point force. Knowledge of this functionallows one to calculate the response of the solid to anarbitrary force distribution as the weighted sum of pointforce responses. In a recent paper, Biggins et al [16] de-veloped a linear theory with perfect volume conservationfor the Green’s function which enforce the constraint ofisochoric deformations exactly. This approach is foundto remain valid until strains become geometrically largebut it is only generalizable to 2-D or axisymmetric situa-tions. Here, we make use of the expressions of the Green’sfunction for a neo-Hookean substrate which were recentlyderived by He [17, 18] within the framework of incremen-tal strain theory in order to develop a contact model ableto handle non axial stretch situations. Solutions are pro-vided for the contact shape, load and contact stiffnesswhich include the effects of adhesion. This model is val-idated by contact experiments between a spherical glassprobe and a silicone substrate under various extent ofuni-axial stretching. Experimental details
A commercially available transparentPoly(DiMethylSiloxane) silicone (PDMS Sylgard184, Dow Corning, Midland, MI) is used as an elastomersubstrate. The silicon monomer and the hardener aremixed in a 10:1 weight ratio and crosslinked at 70 ◦ C for 48 hours. In order to accurately monitor the level ofsurface stretching in the contact zone, a square networkof small cylindrical holes (diameter 20 µ m, depth 5 µ mand center-to-center spacing 80 µm ) is stamped on thePDMS surface by means of standard soft lithographytechniques. Once imaged in transmission with a whitelight, the pattern appears as a network of dark spotswhich are easily detected. Full details regarding thedesign and fabrication of PDMS substrates are providedin [1].The stretching behavior of the PDMS rubber wasdetermined using a conventional tensile testing machine(Instron 5565) equipped with an optical extensome-ter. Dog-bone shaped specimens with a gage length28x4x2 mm were loaded at an imposed cross-headspeed of 0.5 mm s − up to a stretch ratio of 1.6. Theresulting nominal stress σ versus stretch ratio λ responseis shown in Fig. 1. Data up to λ = 1 .
25 (i.e. before theoccurrence of significant strain hardening) have beenfitted using a neo-Hookean model σ = 2 C (cid:18) λ − λ (cid:19) , (1)with C = 0 .
526 MPa. The resulting fit is reported as ablack line in Fig. 1.Contact experiments are carried out using PDMS sub-strates 5 × ×
100 mm and a plano-convex BK7 glasslens with a radius of curvature of 5.2 mm (Melles Griot,France). A schematic of the custom set-up is shown inFig. 2. The lens indenter is fixed to a vertical trans-lation stage (Microcontrole, UMR 8.25) by means of a N o m i na l t en s il e s t r e ss s ( M P a ) FIG. 1: (Color online) Nominal tensile stress as a function ofstretch ratio of the PDMS rubber (cross-head speed: 0.5 mms − λ < . C =0 .
526 MPa. double cantilever. An optical fiber (Philtec, Model D25)mounted on the vertical stage, allows to measure withsub-micrometer resolution the deflection of the bladesduring the indentation process. A mirror is located onthe cantilever tip which provides a reflecting surface forthe displacement sensor. Then, from a knowledge of thecalibrated stiffness of the cantilever (11.7 N m − ), theapplied normal load can be determined with mN accu-racy from the measured deflection of the cantilever. Allthe experiments are carried out with normal loads lessthan 150 mN. These load range ensures the achievementof semi-infinite contact conditions ( i.e. the ratio of sub-strate thickness to contact radius is larger than ten [19]).In order to vary the extent of adhesive forces between sur-faces, some experiments were carried out with the contactfully immersed within a droplet of deionized water.A zoom lens mounted on a CMOS camera (Photon-Phocus, MV1024E) records 1024 x 1024, 8 bits images ofthe contact region through the transparent PDMS sub-strate. When stretched, the PDMS substrate is fixedbetween two grips and the stretch ratio is measured op-tically from the deformation of the dot pattern at thesurface of the rubber specimen. Indentation experimentsare carried out using a step-by-step loading procedure.At each load step, a contact image is recorded after en-suring that the contact size is no longer evolving due toadhesive effects. Contact model
We consider the normal contact between a rigid spher-ical probe with radius R and a neo-Hookean elastomersubstrate (shear modulus µ ) which undergoes a uniform,finite pre-stretch λ x , λ y along the two orthogonal direc- FIG. 2: Schematic of the custom-built indentation set-up. Aspherical indenter (a) is fixed to a vertical translation stage(b) by means of a cantilever with two flexible arms (c). Duringthe application of normal contact loading, a measurement ofthe deflection of the cantilever by means of an optical fiber(d) and a reflecting surface (e) allows to determine the appliednormal load on the PDMS substrate (f). Images of the contactregion are recorded with a microscope and a camera (g). tions x and y along the surface plane. In order to estab-lish the contact equations, we take advantage of the workby He [17] who derived a surface Green function for incre-mental displacements on a pre-stretched incompressiblesubstrate obeying a neo-Hookean constitutive law. Us-ing this approach, the characteristics of the contact arededuced below under the assumptions that the contact isfrictionless and that the coupling between normal pres-sure and lateral displacements - which does not cancel,contrarily to the elastic case - has negligible effects onthe results.Guided from experimental results, we assume that thecontact area is elliptic (semi-major axis a and b along thedirections x and y respectively). We further suppose thatthe normal stress at the surface of the substrate can bederived from a one-dimensional function through scalingalong both axis, i.e. a function p ( u ) can be defined whichcancels for u > σ zz canbe expressed as σ zz ( r ) = p r x a + y b ! . (2)It is shown below that a stress distribution in thisform may indeed give rise to paraboloidal vertical dis-placements with cylindrical symmetry on a pre-stretchsubstrate; it can thus be a solution to the small indenta-tion problem of a sphere. Normal displacements on thepre-stretched substrate can be accounted for by using theappropriate Green tensor coefficient G zz , which can bewritten in the two-dimensional Fourier transform space as [17] G zz ( k ) = 16 λ x λ y K ( η + k ) kω , (3)where η = λ x λ y q λ x k x + λ y k y , (4) ω = η + η k + 3 ηk − k , (5)and K is a reduced modulus often used in contact me-chanics [20]; for incompressible materials, K = µ where µ is the shear modulus. Normal displacements u z ( r ) are deduced from the inverse Fourier transform u z ( r ) = 14 π Z Z G zz ( k ) ˆ σ zz ( k ) e − i kr dk x dk y , (6)where ˆ σ zz ( k ) is the Fourier transform of the normalstress, which can be expressed using the similarity prop-erty as ˆ σ zz ( k ) = Z Z p r x a + y b ! e i kr dxdy (7)= ab Z udu Z π p ( u ) e ikγu cos v dv (8)= 2 πab Z p ( u ) J ( kγu ) u du (9)= 2 πab ˆ σ ( kγ ) , (10)where γ = q a ¯ k x + b ¯ k y , (11) k = q k x + k y . (12) J ( . ) is the Bessel function of order 0 and the compo-nents of the unit vector k − k along the axis x and y arenoted ¯ k x and ¯ k y respectively. Then, Fourier transformof the normal stress, ˆ σ ( k ), is expressed from the Hankeltransform of the one-dimensional stress function p ( u ).The polar angles of k and r are noted respectively β and θ . Normal displacements can be expressed as u z ( r ) = ab λ x λ y πK Z π ¯ η + 1¯ ω dβ Z ∞ ˆ σ ( kγ ) e − ikr cos( β − θ ) dk (13)where ¯ η = λ x λ y q λ x cos β + λ y sin β , (14)¯ ω = ¯ η + ¯ η + 3¯ η − . (15)As the function to be back transformed G zz ( k ) ˆ σ zz ( k ) isinvariant through the change k → − k , one can drop theimaginary term in the integral and express u z ( r ) = ab λ x λ y πK Z π ¯ η + 1¯ ω dβ Z ∞ ˆ σ ( kγ ) cos ( kr cos ( β − θ )) dk (16)which includes a cosine transform. As expected fromthe Fourier-Hankel-Abel (FHA) cycle [21], expressingstress as an Hankel transform and using Eq. (6.671.2)in ref. [22], inner integral in the RHS of Eq. 16 can bewritten as an Abel transform: I = Z ∞ ˆ σ ( kγ ) cos ( kr cos ( β − θ )) dk (17)= Z p ( u ) udu Z ∞ J ( kγu ) cos ( kr cos ( β − θ )) dk (18)= 1 γ Z r | cos( β − θ ) | γ up ( u ) q u − r cos ( β − θ ) γ du (19)when r | cos ( β − θ ) | ≤ γ and I = 0 when r | cos ( β − θ ) | ≥ γ . Defining the Abel transform of the stress function for0 ≤ s ≤ H ( s ) = Z s up ( u ) √ u − s du (20)and H ( s ) = 0 otherwise, the vertical displacement canbe expressed as follows u z ( r ) = ab λ x λ y πK Z π ¯ η + 1¯ ωγ H (cid:18) r | cos ( β − θ ) | γ (cid:19) dβ . (21)The function H (cid:16) r | cos( β − θ ) | γ (cid:17) does not cancel if r | cos ( β − θ ) | ≤ q a cos β + b sin β . (22)It can be shown that this condition is fulfilled, what-ever the angle β is, for all the points r inside the contactarea. Indeed, condition (22) reads D ≤ D = r cos ( β − θ ) − (cid:0) a cos β + b sin β (cid:1) . (23)For a given point ( r, θ ) situated at the contact edge, theparametric representation of the ellipse implies the exis-tence of an angle α which verifies r cos θ = a cos α and r sin θ = b sin α . Then r cos ( β − θ ) = ( a cos α cos β + b sin α sin β ) (24)and thus, for this point, D = − ( a sin α cos β − b cos α sin β ) ≤ . (25)For points situated in the contact area, the condition isa fortiori fulfilled.We now define two anisotropy parameters, ϕ and ψ which characterize the stretch state and the eccentricityof the elliptic contact area, respectively: ϕ = λ x − λ y λ x + λ y ; ψ = a − b a + b , (26)¯ η = λ x λ y r λ x + λ y p ϕ cos 2 β , (27) γ = c p ψ cos 2 β , (28) c = r a + b . (29) In the following, we note δ the normal displacementof the apex of the spherical indenter below the substrateplane. It will be assumed that the contact size is muchsmaller than the curvature radius of the indenter and thatin-plane displacements can be neglected as compared tonormal displacements. In such a situation, normal dis-placements within the contact area can be assumed toobey an axisymmetrical parabolic dependence to the dis-tance from the apex. Accordingly, they can be expressedas δ − r R = 8 λ x λ y πK abc Z π ¯ η + 1¯ ω ¯ γ H (cid:18) rc | cos ( β − θ ) | ¯ γ (cid:19) dβ , (30)where ¯ γ = √ ψ cos 2 β . This expression constitutes alinear integral equation for the function H , of the firstkind with constant limits of integration. A polynomialsolution can be found [23] in the form H ( s ) = 3 K λ x λ y cab (cid:18) δC − c s RC (cid:19) , (31) C m = 1 π Z π ¯ η + 1¯ ω ¯ γ (cid:18) | cos ( β − θ ) | ¯ γ (cid:19) m dβ , (32)with m = 0 ,
2. Stress distribution is retrieved using in-version of the Abel transform Eq. (20): p ( u ) = − π u ddu Z u sH ( s ) √ s − u ds (33)= 2 π (cid:20) H (1) √ − u − Z u H ′ ( s ) √ s − u ds (cid:21) , (34)where H ′ ( s ) = dH ( s ) /ds . Normal load is obtained byintegration of the stress function Eq. (2) expressed by(33): P = Z Z p r x a + y b ! dxdy (35)= 2 πab Z p ( u ) u du (36)= 4 ab Z H ( s ) ds (37)When adhesion is neglected, normal stress is not singu-lar at the contact edge and thus H (1) = 0. In the John-son, Kendall and Roberts (JKR) adhesion theory [24],adhesion induces a stress singularity in this region, i.e. H (1) = 0. Both situations are discussed in the followingsections. It can be noticed that H function is very similarto the auxiliary function defined by Sneddon to describethe contact of an axisymmetric punch on a flat [25], whichwas generalized to the adhesive case [26–28]. Non-adhesive contact
When H (1) = 0, it comes from Eqs. (31) and (34) thatthe normal stress can be derived from Eq. (2) using thefunction p ( u ) = 3 K πRC λ x λ y c ab p − u , (38)which exhibits the classical Hertzian shape. To be admis-sible, the stress field derived from this function must giverise to axi-symmetrical displacements within the contactarea. From Eq. (31), it comes that the condition that C does not depend on the orientation θ is sufficient to fulfilthis requirement. Expressingcos ( β − θ ) = 12 (1 + cos 2 β cos 2 θ ) + 12 sin 2 β sin 2 θ , (39)the second right hand term, with a factor sin 2 β , gives avanishing contribution to the integral (32) due to symme-try reasons: the integrand changes sign when β → π − β .One thus obtains C = 12 π Z π ¯ η + 1¯ ω ¯ γ (1 + cos 2 β cos 2 θ ) dβ . (40)In order to obtain an isotropic result, the contributionof the cos 2 θ term to the integral should vanish, i.e., usingthe symmetry properties of the integrand, Z π ¯ η + 1¯ ω ¯ γ cos 2 β dβ = 0 . (41)For a given stretch state, the functions ¯ η and ¯ ω are deter-mined. The solution ψ of this equation, easily obtainedusing numerical integration, is noted ψ s . For this solu-tion, we define C s = 2 π Z π ¯ η + 1¯ ω ¯ γ s dβ , (42) C s = 4 π Z π ¯ η + 1¯ ω ¯ γ s dβ , (43)¯ γ s = p ψ s cos 2 β . (44)From the condition (41) and the definition of ¯ γ , it can beverified that C s = 2 C s . Furthermore, exchanging thestretches λ x and λ y gives a solution ψ s which is opposite(the role of the axis is inverted), but the value of C s remains unchanged.The contact ellipticity ρ s is ρ s = ba = s − ψ s ψ s . (45)It is independent of the normal load and of the curva-ture radius of the indenter. The penetration depth, δ , isobtained from the condition H (1) = 0 in Eq. (31): δ = c R (46)= a R ρ s b R ρ − s H ( s ) = 3 K λ x λ y C s R c ab (cid:0) − s (cid:1) (49)The normal load P is obtained from (37) as P = 1 λ x λ y C s c KR (50)It can also be expressed using the semi-axis lengths as P = (cid:0) ρ s (cid:1) λ x λ y C s a KR (51)= (cid:0) ρ − s (cid:1) λ x λ y C s b KR (52)and the load-penetration law reads P = KR δ λ x λ y C s . (53)The contact stiffness, S , which is known to be affectedby initially stressed state [8], can be obtained by dif-ferentiating the previous expression with respect to thepenetration depth. Expressing the result in terms of thecontact size parameter c , one obtains S = 32 Kcλ x λ y C s . (54)Eqs. (50), (46) and (54) are similar to the correspondingones for the Hertzian contact on an un-stretched sub-strate, provided that the contact radius and the reducedmodulus are respectively replaced with an averaged con-tact size, c , and by the quantity K (cid:0) λ x λ y C s (cid:1) − .From Eqs. (51) and (52), at a given stretch state, bothsemi-axis lengths are observed to follow a Hertzian-likedependence on the normal load, but with different ef-fective moduli. This result could have been anticipatedusing dimensional analysis (see Appendix A).When the substrate is un-stretched, one obtains a cir-cular contact area: ϕ = 0 in (41) leads to ψ s = 0. Thus ρ s = 1 and, as ¯ η = 1, one obtains C s = 1. The classicalresults for Hertzian contacts are retrieved from expres-sions (51), (52), (47) and (48). For equiaxially stretchedsubstrate, the contact remains circular too. In this case,however, as ¯ η = λ , if λ is the stretch and a the contactradius, P = a K ′ ( λ ) R (55) δ = a R , (56)where K ′ ( λ ) = λ + λ + 3 λ − λ ( λ + 1) K . (57)Thus, in the case of equiaxial stretch, the contact isHertzian, with a stretch-dependent effective modulus. Itcan be verified that it is higher (resp. lower) than the sub-strate elastic modulus when there is traction (resp. com-pression) in the contact plane. These results are in agree-ment with the analysis of adhesive contact of a sphere ona equiaxially stretched substrate [18] when adhesion isneglected.
Adhesive contact
When H (1) = 0, normal stress can be derived fromEqs. (31) and (34) as p ( u ) = 2 π H (1) √ − u + 3 K πRC λ x λ y c ab p − u . (58)A similar form was postulated in the case of adhesivecontact of a sphere on an equiaxially stretched substrateby He and Dong [18]. In their analysis, the parameter c would represent the actual radius of the circular contactarea. In line with the JKR description of adhesion, theabove expression represents the superposition of a rigidcylindrical punch displacement and of a non-adhesive in-dentation. The constant H (1) may be determined froman appropriate estimation of the adhesion energy [18, 20].Alternatively, Maugis and Barquins [26, 27] used a frac-ture mechanics argument jointly with Griffith criterion tofix the value of the constant. In the following, a methodequivalent to the former one is used.It should be noticed first that the isotropy conditionfor the displacements in the contact area is the sameas for the non-adhesive case. Indeed, the second termin Eq. (58) induces normal displacements with a squaredependence with the radius while the first one only con-tributes to a rigid displacement of the points within thecontact area. The later is equivalent to the stress fieldassociated with a cylindrical punch and does not add anyin-plane anisotropy. Then, as in the non-adhesive case,the value ψ s for the anisotropy parameter ψ insures theisotropy of the displacements. It follows that adhesionhas no effect on the eccentricity of the contact area.From eq. (31), the constant H (1) is related to theindentation depth by H (1) = 3 K C s λ x λ y cab (cid:18) δ − c R (cid:19) . (59)Stress function can be written as p ( u ) = 3 K πC s λ x λ y cab " δ − c R √ − u + 2 c R p − u . (60) Indentation induced stored elastic energy is U el = 12 Z Z (cid:18) δ − x + y R (cid:19) p r x a + y b ! dxdy (61)= ab Z π dθ Z (cid:18) δ − s a cos θ + b sin θ R (cid:19) p ( s ) s ds (62)= Kc R C s λ x λ y (cid:0) R δ − Rc δ + 3 c (cid:1) . (63)Under equilibrium conditions, the relation (cid:0) ∂U el ∂A (cid:1) δ = w holds, with A = πab . As anisotropy of the contact shapeis kept constant, (cid:18) ∂U el ∂A (cid:19) δ = p ρ s √ πρ s a (cid:18) ∂U el ∂c (cid:19) δ (64)and thus w = p ρ s ρ s a √ K πC s λ x λ y (cid:18) δ − c R (cid:19) . (65)Integration of the normal stress gives P = Kc C s λ x λ y (cid:18) δ − c R (cid:19) . (66)Defining P , the load corresponding to the same contactarea in the non-adhesive case (50) as P = Kc C s λ x λ y R = (cid:0) ρ s (cid:1) λ x λ y C s a KR , (67)one obtains ( P − P ) πa K = wρ s C s λ x λ y r ρ s . (68)The form of this relation is equivalent to the JKR ex-pression and it constitutes its generalization to te case ofan incompressible stretched substrate. Here again, theunstretched situation corresponding to JKR model is re-trieved by letting C s = 1 , ρ s = 1.For equiaxially stretched substrate, the previous ex-pression reads ( P − P ) πa K ′ ( λ ) = w , (69)where P = a K ′ ( λ ) R (70)and K ′ ( λ ) = λ + λ + 3 λ − λ ( λ + 1) K . (71)The usual JKR relation is obtained, with an effectivemodulus which depends on the stretch ratio. An equiva-lent expression was obtained in ref. [18].
Experimental results
An example of a contact area picture is shown in Fig. 3for a stretch ratio λ =1.32 and a normal load P =100 mN.The elliptical shape of the contact is clearly evidenced
250 (cid:181)m
FIG. 3: Elliptical contact shape recorded for a stretch ratio λ =1.32 and a contact load P =100 mN (the stretching isapplied along the vertical direction). with the major axis of the ellipse perpendicular to thestretch direction. Consistently with a numerical analysisof Eq. (45) with λ x = λ, λ y = λ − , the aspect ratio islarger than one ( b > a ). Fig. 4 shows the changes in thecontact ellipticity ρ s = b/a as a function of the appliedcontact load for a stretch ratio λ = λ x = 1 .
19. con-sistently with the prediction for a non adhesive contact(Eqs (41) and (45)), it turns out that the measured as-pect ratio is nearly constant over the whole investigatedload range. This feature was preserved for all the stretchratios under consideration.The dependence of the contact shape on the stretchratio is further examined in Figures (5) and (6) where a , b and their ratio ρ s are reported as a function of λ . Inthese figures, the black solid lines correspond to theoret-ical predictions. For stretch ratios less than about 1.2,experimental data are found to be in good accordancewith model. Above this threshold, deviations from thetheoretical predictions can be attributed to departure ofthe PDMS mechanical behavior from the Neo-Hookeandescription which is embedded in the model (cf Fig. (1)).The effects of adhesion on the stretch dependence ofthe contact ellipticity were considered by carrying outsome of the experiments with the contact fully immersedin a droplet of deionized water. As indicated in Ap-pendix A, this resulted in a decrease in the adhesionenergy of the unstretched substrate from 27 mJ m − to5 mJ m − . A comparison between experiments carriedout both in air and in water (Fig. (6)) show that the ρ s ( λ )relationship is insensitive to such a change in adhesion. C on t a c t e lli p t i c i t y r s C on t a c t r ad i u s ( m x - ) FIG. 4: Ellipticity ratio ρ s = b/a of the contact area as afunction of the applied contact load for a stretch ratio λ =1 . • ) the major and ( ◦ )the minor semi-axis lengths of the elliptical contact. a / a , b / a FIG. 5: Semi-axis lengths ( ◦ ) a and b ( • ) of the elliptical con-tact as a function of the stretch ratio λ . Solid lines correspondto the theoretical prediction of Eqn. (51) and (52). As stressed in the theoretical section, the isotropy condi-tion for the displacements in the contact area is enforcedby both adhesive and non-adhesive contacts. As a con-sequence adhesion does not add any in plane anisotropyand the ellipticity remains load-independent, consistentlywith experimental observations.The load dependence of the semi-axis lengths was fur-ther examined in the light of Eqs. (51) and (52) whichpredict a linear dependence of a and b on F . In Fig. (7),the measured values of a and b are reported as a func-tion of applied load for two values of the stretch ratio( λ = 1 .
19 and λ = 1 . a ( F ) and b ( F ) relationships is indeed observed. Thevanishing intercept of the linear fits indicates that adhe-sive effects can be neglected in such a representation. Itcan also be noted that the semi-axis length a for a given C on t a c t e lli p t i c i t y r s FIG. 6: Ellipticity ratio ρ s = b/a of the contact area as afunction of the stretch ratio λ . ( ◦ ) contact in air; ( • ) contactin water. Solid line: theoretical prediction of Eq. (41). load is only marginally affected by the stretch ratio dif-ference while a more pronounced effect is observed forthe semi-axis length b .According to Eqns. (51) and (52), the slopes of theselinear relationships provide an estimate of the ’effective’moduli K aeff and K beff along the major and minor axisof the contact ellipse which can be defined as follows K aeff = K (cid:0) ρ s (cid:1) λ x λ y C s , (72) K beff = K (cid:0) ρ − s (cid:1) λ x λ y C s . (73)For λ = 1 .
19, linear fits to data provides K aeff /K = 1 . K aeff /K = 0 .
79, in relatively good agreement withEqns. (51) and (52) which predict K aeff /K = 1 .
20 and K aeff /K = 0 .
86. Conversely, the deviation from the neo-Hokeean behaviour results in a significant departure fromthe theory for λ = 1 . K aeff valuewhich is under-predicted by about 25%. Concluding remarks
The present model was derived for a spherical probeand it should be mentioned that it cannot directly be ex-tended to axisymmetrical punch shapes other than spher-ical or cylindrical. As an example, for a conical shapewith semi-angle α , assuming a stress function similar to(2) imply to solve an integral equation which is equivalentto Eq. (30): δ − r tan α = 8 λ x λ y πK abc Z π ¯ η + 1¯ ω ¯ γ H (cid:18) rc | cos ( β − θ ) | ¯ γ (cid:19) dβ (74) a , b ( m x - ) FIG. 7: Cube of the semi-axis lengths a (open symbols) and b (filled symbols) of the elliptical contact as a function of theapplied load. Circles: λ = 1 .
19, squares: λ = 1 .
32. Solid linescorrespond to linear regression fits.
The solution H ( s ) is linear and can be accepted if thecoefficient C = 1 π Z π ¯ η + 1¯ ω ¯ γ | cos ( β − θ ) | dβ (75)can be made independent of θ by a proper choice of theparameter ψ . This is clearly the case when the substrateis isotropically stretched ( λ x = λ y , φ = 0): in this case ψ = 0 ( a = b ) is a solution. However, in the general casewhere λ x = λ y , considering that the Fourier series of | cos ( β − θ ) | = ∞ X m =0 A m cos (2 m ( β − θ )) (76)and noting that (¯ η + 1) ¯ ω − ¯ γ − is a function of cos 2 β ,the coefficient can be written as C = ∞ X m =0 B m cos 2 mθ , (77)where B m = A m π Z π ¯ η + 1¯ ω ¯ γ cos (2 mβ ) dβ . (78)The invariance with respect to θ of the coefficientwould require that the Fourier series (77) is constant,or, equivalently, B m = 0 for m = 1 , , ... . In turn thiswould require (¯ η + 1) ¯ ω − ¯ γ − to be constant, which isimpossible when ϕ = 0 (or λ x = λ y ). This argumentalso applies for punch profiles ∼ r p when p is an oddinteger. When p is even, a finite number of harmonicscontribute. However p = 2 (the spherical shape) is theonly case where the cancellation of a single harmonicsof the function is sufficient, leading to the exact solutionpresented in this paper.Though exact solution cannot be found for an arbitrarypunch profile, approximate solutions may be derived con-sidering that the difference between actual contact andan elliptic shape is not expected to be large. In the caseof the conical indenter, for instance, we enforce that thevalues of the coefficient C (Eq. (75)) should be identi-cal along the directions x and y . Alternatively, we canenforce that the leading anisotropic term of the Fourierexpansion (77), B , is zero: Z π ¯ η + 1¯ ω ¯ γ cos (2 β ) dβ = 0 (79)For an uniaxially stretched substrate, numerical calcu-lations shows that both approximations are in very goodagreement and that a/b ≃ λ within few percents when λ < .
5. Once the ellipticity parameter is determined, ex-pressions relating normal load and displacement or con-tact size are obtained in a way similar to the case of thespherical indenter. To be more specific, above discussionwas dealing with the conical punch but it can be extendedto any axisymmetric indenter with a power law profile.Some additional comments are also in order regard-ing elastic contact theories dealing with general Hertzianelliptical contacts. In this context, Johnson and Green-wood developed an approximate theory for adhesive el-liptical contacts by expressing that the stress intensityfactor remains almost constant all around the contact pe-riphery [29]. They concluded that eccentricity varies withthe load. In the case of an adhesive stretched substrate,above results indicate that the contact area for a sphereis elliptical, but its eccentricity is load-independent andidentical to the Hertzian case. The stress intensity factorvaries along the contact edge, reflecting the anisotropy ofthe pre-stretched substrate properties upon incrementaldisplacements.In conclusion, we have shown that contact area of a un-deformable sphere on a stretched elastomeric substratehas an elliptic shape. Its eccentricity is completely de-termined by the in-plane stretch of the substrate. In par-ticular, it neither depends on the applied load, nor on thecurvature radius of the sphere or on adhesion properties.
Acknowledgments
The authors wish to thank L. Olanier for his help inthe design and in the realization of the contact device.
Appendix A: Dimensional analysis of the stretchedcontact
According to Eqs. (51) and (51), both semi-axis lengthsare observed to follow a Hertzian-like dependence on thenormal load at a given stretch state, but with different effective moduli. This result could have been anticipatedusing dimensional analysis. It can be observed that thesemi-axis length, a , is determined when the modulus, K ,the normal load, P , the sphere radius, R , and the stretchstate, λ x , λ y are fixed. Forming non-dimensional num-bers, these parameters must be linked as PKa = f a (cid:16) aR , λ x , λ y (cid:17) (A1)Now, keeping the contact area constant, if all normaldisplacements are multiplied by a number χ , by linear-ity of the stress-strain problem described by the Greentensor, the normal load is also multiplied by χ . From sim-ple geometrical considerations, it correspond to a contactproblem for a sphere with a radius divided by χ . Thus χ PKa = f a (cid:16) χ aR , λ x , λ y (cid:17) (A2)which should hold for any values of χ . The function f a is thus homogeneous and of the first degree with respectto a/R . One may deduce P = a KR g a ( λ x , λ y ) . (A3)Similar results can be obtained for the other axis and forthe penetration depth. One have also ba = g ( λ x , λ y ) g ( λ y , λ x ) . (A4)which expresses that the contact shape anisotropy is con-trolled by the stretch anisotropy only. It can be noticedthat, in the un-stretched situation, previous considera-tions, allow to recover the Hertzian expressions up to aconstant numerical factor. Appendix B: Adhesive contact of the unstretchedsubstrate
The adhesive contact between the unstretched ( λ = 1)PDMS substrate and the glass lens was examined withinthe framework of the JKR theory [24]. The followinglinearised form of the relationship between the contactradius a and the applied load F was considered a / R = 1 K Fa / + r πwK . (B1)The slope of this linearised relationship thus providesthe value of the reduced modulus K while the adhesiveenergy w is deduced from the intercept. In Fig. (8),experimental contact results have been reported usingthis representation for contacts either in air or fully im-mersed in a droplet of deionized water. As expected,linear relationships are obtained in both case with thesame slope, i.e. the same value of the reduced modulus( K = 3 . ± .
02 MPa). On the other hand, a decrease in0 a / / R ( m / x - ) (N m -3/2 ) FIG. 8: Linearised plot giving the contact radius a versus theapplied normal force F for an unstretched ( λ = 1) PDMS sub-strate. ( ◦ ) contact in air, ( • ) contact immersed in deionizedwater. Solid lines are linear regression fits. the intercept is observed for contact in air which reflectsa decrease in the adhesion energy from w = 27 mJ/m (air) to w = 5 mJ/m (water) as a result of the screen-ing of van der Walls forces between surfaces by the watermolecules. [1] 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.[2] D.T. Nguyen, E. Wandersman, A. Prevost,Y. Le Chenadec, C. Fretigny, and A. Chateauminois.Non Amontons-Coulomb local friction law of randomlyrough contact interfaces with rubber.
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