The nucleation and propagation of solitary Schallamach waves
Koushik Viswanathan, Anirban Mahato, Srinivasan Chandrasekar
TThe nucleation and propagation of solitary Schallamach waves
Koushik Viswanathan, ∗ Anirban Mahato, and Srinivasan Chandrasekar
Center for Materials Processing and TribologyPurdue University, West Lafayette, IN 47907-2023 (Dated: November 25, 2016)
Abstract
We isolate single Schallamach waves — detachment fronts that mediate inhomogeneous slidingbetween an elastomer and a hard surface — to study their creation and dynamics. Based onmeasurements of surface displacement using high–speed in situ imaging, we establish a Burgersvector for the waves. The crystal dislocation analogues of nucleation stress, defect pinning andconfigurational force are demonstrated. It is shown that many experimentally observed features canbe quantitatively described using a conventional model of a dislocation line in an elastic medium.We also highlight the evolution of nucleation features such as surface wrinkles, with consequencesfor interface delamination.
PACS numbers: 82.35.Gh, 81.40.Pq, 83.60.Uv, 83.80.Va ∗ E-mail: [email protected] a r X i v : . [ c ond - m a t . m t r l - s c i ] J a n . INTRODUCTION While the phenomenological study of friction between solid surfaces has a long history [1],it was not until the mid-twentieth century that the microscopic aspects were first probed [2].Subsequently, significant attention has been devoted to the fundamental microscale mecha-nisms underlying the phenomena of static and dynamic friction [3–5]. Simple universal lawssuch as those of Amontons and Coulomb, though used extensively, frequently remain unsat-isfactory [6]. For example, under extreme sliding conditions, the friction depends on the areaof contact [7]. Likewise, at low sliding velocities, sliding friction is non–trivially dependenton velocity [8] and normal load history [9], resulting in the occurence of inhomogeneousmodes of sliding with localized slip.A soft adhesive elastomer sliding on a smooth surface is a model system that exhibitsinhomogeneous sliding modes, while also capturing the physics underlying processes of prac-tical and industrial interest [10]. At a length scale of a few hundred micrometers and lowrelative sliding velocity ( < ∼
10 mm/s), motion between the two surfaces does not occur ho-mogeneously, but via the propagation of ‘waves of detachment’, also known as Schallamachwaves [11–13]. Under similar conditions, sometimes another inhomogeneous mode of slid-ing, called the self–healing slip pulse is also observed [14]. Schallamach waves have beenlikened to crystal dislocations [15] or rucks in carpets [16] [17]. Such comparisons have onlybeen qualitative; however, they have helped rationalize some observed features such as thelocality of surface slip and existence of a nucleation stress.Motivated by these considerations, we have further explored, using experiments, thesimilarity between a single Schallamach wave and a dislocation line in an elastic medium.For this purpose, a long adhesive contact was established between an elastomer and a solidsurface, enabling observation of solitary Schallamach waves (wave pulses). This provides asuitable framework in which to study their characteristics quantitatively. High–speed in situ imaging was used to capture their nucleation and propagation, at resolution of ∼ µ m.Improved observations of the intrinstic features of isolated Schallamach waves shouldhelp better understand inhomogeneous interfacial sliding phenomena prevalent in earthquakeruptures [18], polymer friction [4, 5] and locomotion of soft–bodied animals [19]. It couldalso shed new light on the relation between Schallamach waves and the self–healing slippulse mode of sliding [14]. 2he paper is organized as follows. Details of the experimental setup are provided inSec. II, followed by high–speed photographic observations of Schallamach wave nucleationand propagation (Sec. III). Using image analysis techniques, we obtain quantitative informa-tion about the individual wave properties that help explain various features of the nucleationand propagation stages. The results are analyzed in detail in Sec.IV, along with a discus-sion of why inhomogeneous sliding modes occur. The principal findings are summarized inSec. V. II. EXPERIMENTAL DETAILS
The adhesive contact used is that between an uncoated plano–convex lens, made of syn-thetic glass (Edmund Optics) and the elastomer PolyDiMethylSiloxane (PDMS, Sylgard184 from Dow Corning). A schematic of the experimental setup is shown in Fig. 1. Theelastomer and the lens are optically transparent. The contact region is illuminated by a120 W halogen lamp and observed by a microscope (Nikon Optiphot) mounted in frontof a high–speed camera (PCO dimax). This system was used to image the contact regionat framerates of 5000 − . − . µ m perpixel, depending on the microscope lens used. Normal and tangential forces in sliding weremeasured using a piezoelectric dynamometer (Kistler).Two different plano–convex lens geometries were used for the experiments — a sphericallens of radius R s = 5 mm and a cylindrical lens of radius R = 16.25 mm and length L =25 mm. Sample images of the contact region with the cylindrical lens (Fig. 1 (right, top))and the spherical lens (Fig. 1 (right, bottom)). In both cases, the size of the contact regionwas maintained constant, between experiments. For the cylindrical lens geometry, the anglesubtended by the contact region at the lens axis was around 1 . ◦ ; the lens curvature causeda shift in the image of (cid:39) µ m, which was, however, not resolvable by the imaging system.Upon contact, the cylindrical lens formed a long aspect–ratio adhesive ‘channel’ in which topropagate solitary Schallamach waves. From the high speed observations, this was found tobe most conducive for the production of single wave pulses at the interface. Characteristicsof wave nucleation were studied using the spherical lens geometry, because the resultingfinite contact region allowed complete observation of the contact edges.The PDMS elastomer sample was prepared by mixing a base (vinyl-terminated poly-3imethylsiloxane) with a curing agent (methylhydrosiloxane–dimethylsiloxane copolymer)in the ratio 10:1 by weight. The resulting mixture was cured for 12 hours at 60 ◦ C. ThePDMS was cast in a mold into slab type specimens, with dimensions of L = 70 mm × H = 22 mm in the xy plane (see Fig. 1). L is the length in the sliding direction. The thick-ness of the slab was 12 mm. For experiments requiring a longer sliding length, a slab with L = 90 mm was used. The Young’s modulus and Poisson’s ratio for PDMS were estimatedfrom reported shear [20] and bulk modulus [21] values for Sylgard 184 as E (cid:39)
800 kPa and ν = 0 .
45 respectively. The sample was mounted on a linear slide which could impose slidingvelocities of 10 µ m/s – 20 mm/s. The camera and the indenter (lens) were fixed on rigidsupports as shown in Fig. 1 (left).The starting contact size was kept constant in all the experiments by maintaining theapplied initial normal load at 35 mN (spherical lens) and 55 mN (cylindrical lens). Thedynamics of the interface maybe expected to vary with the normal load.The high–speed image sequences obtained in the experiments were analyzed to obtaindisplacements for each pixel between successive images. This is done by assuming thatthe image intensity is convected with the physical velocity field. Since the framerate is keptconstant, the inter–frame displacement is proportional to the instantaneous velocity. A briefdescription of the image processing methods is provided in Appendix A. Once the velocitiesof every pixel are obtained for each image, specific surface properties are determined byfollowing a set of predetermined ‘virtual’ tracer points. These are pixel locations in the firstimage (with perfect adhesive contact) spaced 2 pixels apart, along the horizontal contactmid–line ( x = 0) in the middle of the contact region. Their positions ( x i , y i ) are alteredbetween successive frames, using the local velocity field determined for that particular frame.All of the results are presented and discussed in the elastomer rest–frame. It is in this sensethat the ‘leading’ and ‘trailing’ edges of the lens are defined. III. OBSERVATIONS
Interfacial slip via Schallamach waves consists of two main stages — nucleation of a waveand its propagation through the interface. These are studied using the spherical and cylin-drical lens geometries respectively. The former is chosen due to the small circular contactzone, which enables easier observation. The latter removes the effects of contact geome-4ry, thereby allowing us to observe the intrinsic propagation characteristics of individualSchallamach wave pulses.
A. Nucleation of Schallamach waves
Schallamach waves nucleate due to a buckling instability of the elastomer surface [11,12]. A prototypical nucleation event is shown in Fig. 2 (top row) with the correspondingschematic side–view in Fig. 2 (bottom row). Initially, in Fig. 2(a), the spherical lens andthe elastomer surface are in adhesive contact. When a relative sliding velocity is imposed,the elastomer free surface ahead of the lens is compressed, causing it to buckle (Fig. 2(b)).This compression results from a combination of the applied tangential force and adhesionat the interface.In order to maintain a constant sliding velocity v s , continued application of the tangentialforce is necessary. The van der Waals force between the surfaces causes the elastomer toreattach to the lens, at point B in Fig. 2(c). An air pocket (region A B ) is thus trappedinside the contact region. The presence of a strong shear stress gradient causes this pocketto traverse the length of the contact region in the form of a single Schallamach wave, as seenin Fig. 2(d). The region BC in the wake of the wave is now again in adhesive contact. Thesewaves hence travel from the leading edge of the lens to the trailing edge i.e. in a directionopposite to the imposed sliding velocity v s .A prominent feature of Fig. 2 is the wrinkle pattern on the surface (region A A ) accom-panying the wave [22]. This is seen in the movie M1 (supplemental material [23]) and shownin Fig. 3. These wrinkles are also compression–induced features, akin to the formation ofsulci [24], and have important consequences for wave propagation. The average spacingbetween adjacent wrinkles gives the pattern wavelength. The initial value of the wavelength λ , as measured from the images, is 18 µ m, see Fig. 3 (left). Upon further application ofshear, even though the two surfaces remain adhered at the interface (Figs. 2(b) and 2(c)),there is an increase in the compressive stress on the elastomer free surface. The patternwavelength correspondingly increases to λ = 40 µ m (Fig. 3 (right)), with is approximatelytwice the initial value λ . There is an accompanying increase in wrinkle amplitude.For low sliding velocity ( v s ∼ B. Propagation through the interface
Once a solitary Schallamach wave is nucleated, it traverses the contact region, due toa stress gradient. The propagation characteristics of the nucleated wave pulse are bestobserved in the cylindrical contact. The initial contact region resembles a long, thin adhesive‘channel’. The length of the contact region L = 2 . a ∼ A ) with the Schalla-mach wavefront clearly demarcated (edge B ). The wave itself is seen as a depression (region C ) due to the trapped air pocket. The surface wrinkles (eg. at point D ) are also visible.Once the solitary Schallamach wave has passed, readhesion between the surfaces is incom-plete, leaving small stationary residual air pockets (like at point E ). In movie M2 [23], itis seen that these air pockets form exactly over the free surface wrinkle, pattern owing toincreased strain concentration in the wrinkles. Such wrinkles, formed during wave nucle-ation, were found to cause significant interface delamination after the passage of successiveSchallamach wave pulses.The result of passage of a single Schallamach wave is also brought out by its effect on6he surface displacement of a dirt particle. Fig. 5(a) shows perfect adhesive contact betweenthe cylinder lens and elastomer surface, with a dirt particle (point A ) on the elastomersurface. As in Fig. 4, the leading front of the incoming wave pulse is denoted by B . Whenthe wave propagates through the observed region (Fig. 5(b)), the wrinkles C again resultin incomplete readhesion in the wake (point D ). The dirt particle A is displaced from itsoriginal position as the wave pulse passes over it. A small air pocket is also left around it inthe process (point E in Fig. 5(c)). From the initial and final positions of the particle at A in Fig. 5(a), it is clear that relative motion between the surfaces has occured intermittentlyand only due to wave passage, see movie M2 [23]. This cycle is repeated when a secondwave ( F in Fig. 5(d)) is nucleated and traverses the contact region. Since the v s in Fig. 5 islarger than in Fig. 4, more air pockets remain trapped in the wake of the wave. The contactcondition is hence changed for subsequent wave nucleation and the shape of the followingwave pulses is altered, as in Fig. 5(d).An interesting phenomenon is observed during propagation past stationary dirt particlesattached to the lens, as shown in Fig. 6. Here, the wavefront approaches a static dirtparticle A in the contact region and gets ‘pinned’, causing a bend in the wave profile (point B ). As the front moves away from the particle, the wave regains its original profile, leavinga residual air pocket C around the particle. This resembles the motion of dislocations incrystals past static obstacles (solute particles), leaving behind so-called Orowan loops [26].For dislocations in metals, this is known to lead to the Fisher, Hart and Pry (FHP) effect[27]. Under dilute solute particle concentration, the dislocation line is bent by the obstaclesand it eventually relaxes its shape. C. Properties of a solitary Schallamach wave
In studying the propagation of a single Schallamach wave, three different velocities mustbe distinguished — the imposed (remote) sliding velocity v s , the local material velocity v p and the velocity v w of propagation of the Schallamach wave. In each sliding experiment, v p and v w could, in principle, vary along the contact.Standard image analysis techniques were applied to the high–speed image sequences ofan isolated wave to obtain the material velocity v p ( x, y ) for each pixel ( x, y ) in an imageframe (See Appendix A for details). By tracking a set of horizontal ‘virtual’ tracer points7 x i , y i ) lying in the initally perfect contact region, the relative inter–frame displacement ofthe surfaces during wave propagation was obtained. This is shown for four different valuesof v s in Fig. 7(a). The graph shows a distinct jump, implying that relative motion occursonly due to wave passage; the surfaces are otherwise stationary and in perfect contact. Ananalogous situation prevails during the irreversible displacement (slip) caused by motion ofan edge dislocation on a crystal glide plane. In this case, the displacement magnitude isgiven by the dislocation Burgers vector. Similarly, the displacement jump in Fig. 7(a) canbe associated with a Burgers vector b for the solitary Schallamach wave. It is clear fromFig. 7(a) that | b | = 255 µ m and is independent of v s . It is determined, for a given contactgeometry, by the substrate material properties. Both of these characteristics are also truefor the dislocation Burgers vector [26]. The value of | b | for a Schallamach wave depends onthe contact dimensions, and hence, on the normal load.The velocity magnitudes | v p ( x i , y i ) | of each of the material tracer points ( x i , y i ) may nowbe assembled in the form of a space–time diagram, as shown in Fig. 7(b). The x -axis is theinitial location of the material tracer points and y -axis denotes time. The pixel color valuesdenote | v p ( x i , y i ) | of each tracer point for a particular time slice. The leading edge of theSchallamach wave pulse is represented by the line AB and the trailing edge by CD . Theslopes of these lines are equal, giving a wave group velocity v w = 110 mm/s and v w /v s (cid:39) v s is increased, v w also increasesbut v w /v s appears to reduce a little. For the range of v s used, this ratio was between 35and 50. Note that along the wave pulse profile, v w is different from the local phase velocity | v p ( x i , y i ) | (=50 −
140 mm/s). It is interesting that both v w and v p are much smaller thanthe shear wave velocity for PDMS ( ∼
15 m/s).
D. Motion of multiple Schallamach waves
The tangential force F T on the elastomer was measured during sliding and is shown (inblue (dark grey)) in Fig. 8(a). This force provides a measure of the shear stress at theinterface. Since the experiments were performed under velocity control, the force varieswith time. The measured value of the normal force, F N (in green (light grey)), is also seento oscillate with time in Fig. 8(a). The force data were correlated with the high–speed8mage data to confirm that only one wave pulse traversed the contact region at a given time.It is seen from Fig. 8(a) that prior to wave nucleation, F T builds up due to adherence ofthe two surfaces. There is a critical tangential force F c (cid:39) . A ), at which asingle Schallamach wave is nucleated and begins to propagate (point B ). The correspondingcritical interfacial shear stress is 6 . (cid:15) c (cid:39) .
03. This is somewhat larger thanthe shear strain due to slip | b | /h (cid:39) . C ). This cycle thenrepeats with another wave nucleation event. Incomplete readhesion at the interface, causedby the surface wrinkles, results in a reduction of the critical force ( F c ) in the cycles thatfollow the passage of the first wave pulse (point A ).Each Schallamach wave pulse produces the same amount of slip ( cf. Fig. 7(a)) irrespectiveof the sliding velocity. The interface accommodates the imposed v s by changing the frequency n with which waves are nucleated. The value of n may be obtained from the force trace orthe image sequences, both of which are correlated. Fig. 8(b) shows the variation of n with v s . The critical velocity v c = 150 µ m/s for sliding by Schallamach waves is also marked inthe figure. The frequency n is seen to vary linearly with v s over a large range except verynear v c — if the best fit line in Fig. 8(b) is extended to n = 0, it intersects the v s axis ata small negative value. Schallamach waves were observed at velocities very close to, butabove, the value v c . The dependence of n on v s was found to be qualitatively independentof the contact geometry used. IV. ANALYSIS AND DISCUSSION
The high–resolution measurements provide a basis for analyzing nucleation and propaga-tion characteristics of Schallamach waves using simple physical models. Based on the forcemeasurements, we also briefly discuss the stability of homogeneous sliding.
A. Wrinkle pattern during wave nucleation
The free surface wrinkles, ahead of the lens, result from compression of the elastomer.Correspondingly, an analysis of surface instabilities in a compressed elastic half–space [28]9hows that at a critical compression ratio ∼ .
5, the surface is unstable to perturbations ofall wavelengths. Hence this cannot explain the observed wavelength pattern.However, an analysis of the nucleation stage may be guided by the observed similarity ofthe surface wrinkle pattern in Fig. 3 with that seen in compressed elastic films on soft sub-strates [29]. Furthermore, the change in pattern wavelength observed during wave nucleationat large v s , see Fig. 3, appears very similiar to that seen in the longitudinal compressionof such a thin film [25]. Motivated by this similarity, we obtain an estimate of the wrinkleamplitude using the model of an elastic thin film on an elastomer substrate, where the elasticproperties of the film and the substrate are identical.For such a system, the wavelength λ of the first–appearing wrinkle pattern on the freesurface, upon compression, is [30] λ = 2 π (cid:18) B (1 + ν )(3 − ν ) E s (1 − ν ) (cid:19) / (1)where E s , B, h, ν are Young modulus of the substrate and the film’s bending modulus, thick-ness and Poisson ratio respectively. Using B = h E s / (12(1 − ν )), λ = 18 µ m from ourobservations (Fig. 3) and ν = 0 .
45 for PDMS, we obtain h ∼ µ m, which gives an equivalent‘film’ thickness. A crucial feature is that even though the properties of the thin film andthe elastomer substrate are set to be the same, the model used here does not simplify to aregular half–space (Biot’s instability). This is because geometric nonlinearity is included inthe deformation of the ‘film’, while the substrate is assumed to experience small displace-ment gradients. This approximation is hence consistent, only if the strains are confined toa thin surface layer, as the small value of h a posteriori indicates — if h were comparableto the elastomer slab height, the surface would prefer to stretch instead of forming wrinklesand buckling, due to energy considerations.The period doubling with increased compression (Fig. 3(right)) reinforces the thin filmanalogy [25]. As the compression is increased, the critical compression ratio at which thesecond (subharmonic) wavelength appears is, to a first approximation, determined entirelyby ν [25]. For ν = 0 .
45, this occurs at a critical compression ratio δ (cid:39) .
42. Correspondingly,the amplitude of the wrinkle pattern is
A > ∼ h . With further compression, this amplitudeincreases with δ . The large amplitude wrinkles that result in residual air pockets in Fig. 5must have an amplitude larger than the thickness A (cid:29) h , hence corresponding to large δ . The increased amplitude of the larger wavelength wrinkles on the free surface causes10ncomplete readhesion after the passage of a single Schallamach wave. This is seen bycomparing images of the contact region in the two cases — low v s (smaller amplitude,wavelength) in Fig. 4 (top, right) and high v s (larger amplitude, wavelength) in Fig. 5(c)— as well as movie M1 [23]. It is clear that the number of trapped interfacial air pocketsis much greater in the latter case, resulting from an increase in wrinkle amplitude duringnucleation. Motion of multiple Schallamach waves hence causes siginificant degradation tothe adhesive interface. The value of δ obtained from this thin film model depends sensitivelyon ν , for ν values between 0.45 and 0.49, while the resulting h and A remain roughly thesame.The model of a thin elastic film on an elastomer substrate thus appears to capture keyaspects of the mechanics of nucleation of a single Schallamach wave. This analogy hencesuggests that by suitable surface treatment (e.g., surface texturing, pattern impregnation,exposure to ozone) of a very thin (pre-determined) surface layer of thickness h , Schallamachwaves can be suppressed. It must be noted, however, that the thin film analogy is based onlyon wavelength observations and is not fully physically justified. In practice, it is likely thata crease forms on the elastomer surface due to some localized imperfection, as suggested forinstance in Ref. [31]. B. Comparison with dislocations and critical stress for propagation
The observed propagation characteristics of solitary Schallamach waves help expand onthe analogy between wave propagation and crystal dislocation glide. Firstly, Schallamachwaves are nucleated at a critical stress (point A , Fig. 8(a)), similar to crystal dislocations.This stress is the compression required for buckling to occur on the elastomer free surface.Secondly, slip at the interface determines an equivalent Burgers vector b for the Schallamachwave ( cf. Fig. 7(a)) with | b | independent of v s . This b can be obtained from surfacedisplacement measurements and shares key characteristics with its dislocation counterpart.Thirdly, when Schallamach waves encounter static dirt particles in the contact region, theyare pinned, leaving behind an air pocket separating the two surfaces (see Fig. 6). This is akinto the well–known pinning of a dislocation line by a solute particle and the resulting residualdislocation loop [26]. Finally, the nature of the driving force on the wave pulse is similar tothe ‘configurational’ Peach–Koehler force on a dislocation — both the solitary Schallamach11ave and an elastic dislocation translate only because their constituent material points movecollectively.For the elastic dislocation model, the (Peach–Koehler) force on the discontiunuity line,arising from applied stress σ , is given by [26] F = ˆ t × ( σ · b ) (2)A typical profile of a single Schallamach wave is reproduced in Fig. 9. An equivalent dis-location line, subtending an angle 2 θ is superimposed over the wave. This line is acted onby a stress state with non-zero components σ yz = σ zy = τ and σ zz = σ N , applied at thecontact interface. The resulting surface displacement determines the Burgers vector b inthe y direction.If the discontinuity line is displaced by δ r = ∆ x ˆ x ± ∆ y ˆ y (for the top and bottomsegments), the change in potential energy is given by δV = ( F · δ r ) L = τa | b | cos θ (sin θ ∆ x +sin θ ∆ y ), with L the length of the discontinuity line. For the discontinuity to move, thechange in the potential energy of the medium must be provided physically by the hysteresisin peeling and readhering of the elastomer surface in the contact region. This energy balanceprovides an estimate of the critical shear stress needed to propagate a single wave pulse.Assuming ∆ x (cid:28) a , the critical force F cs required to move a wave pulse is hence given by F cs = a L c ∆ W | b | (3)where ∆ W is the adhesion hysteresis, 2 a and L c are the contact width and length respec-tively. Using | b | = 255 µ m, 2 a = 1 mm, L c = 2 . W (cid:39) for PDMS [32], F cs is estimated to be 0 . F c (cid:39) . F cs or thewave exits the contact region; the latter occurs in Fig. 8(a). During propagation, a decreasein the normal force F N (green (light grey) in Fig. 8(a)) results, due to a change in contactsize when the wave traverses the contact region.12 . Slip accumulation in the contact Interfacial displacement only results from the passage of single wave pulses; hence thedislocation model can be used to obtain an expression for the strain rate. The interfacialshear strain (cid:15) in a time interval ∆ t is due to the passage of n ∆ t parallel dislocation lines.This is given by (cid:15) = v s ∆ th = | b | ∆ A V ( n ∆ t ) (4)where h is the height of the elastomer sample, ∆ A the area swept by a single wave in time∆ t , n is the wave generation frequency and V = 2 a h L c . The second equality in Eq. 4above follows from the expression for shear strain due to glide of a single dislocation [26].As seen in the experiments, | b | is constant for each wave pulse and also independent of v s .In conjunction with Eq. 4, this shows that n ∝ v s , consistent with the experimental resultsof Fig. 8(b). The relation for the strain rate ( (cid:15)/ ∆ t ) above resembles the Orowan equationfor dislocation glide, which relates strain rate in the glide plane to dislocation motion [26]. D. Why inhomogeneous sliding?
Relative motion between two surfaces via propagation of Schallamach waves is part of alarger class of inhomogeneous interface motions constituting stick–slip behavior. We havealready mentioned the self–healing slip pulse observed in other sliding systems [14]. Aninvestigation of forces at the interface provides insight into why such inhomogeneous modesoccur.We use the coordinate system shown in Figs. 1 and 9. In this reference, the interfacebetween the lens and elastomer forms part of the xy plane. Before v s is imposed, the lensand elastomer are in adhesive contact. The corresponding normal force F N introduces anormal pressure distribution p ( x, y ) on the elastomer surface. Due to the tangential force F T (in the y -direction), a shear stress q ( x, y ) also acts on the elastomer surface within thecontact region. Typically, the pressure distribution p ( x, y ) is altered upon the applicationof F T , but this change is expected to be atmost a few percent [10] and can be neglected.We define µ ( x, y ) = q ( x, y ) / | p ( x, y ) | along the interface and let µ = F T /F N . The sizeof the contact region — diameter of contact circle for spherical lens and width of contactfor cylindrical lens — is 2 a . K = 4 E/ E = − ν E + − ν E for the elastic properties13 , E , ν , E of the elastomer and lens respectively. R is the radius of the lens and 2 L isthe length of the cylinder lens. The exact expression for µ ( x, y ) will depend on the contactgeometry.When there is no adhesion between the two contacting bodies, the corresponding ratio µ ( x, y ) tends to infinity at the edges of the contact. If a Coulomb friction model is assumed,then this implies relative slip locally near the outer edge of the contact zone. Hence a centralstick region is postulated inside the contact zone, surrounded by a slip region towards itsperiphery [10, 33]. This cannot be done for an adhesive contact, because the pressuredistribution is singular inside the contact, even in the absence of a tangential force. [34]The pressure distribution under static adhesive contact is axisymmetric when F T = 0,given by [10, 35, 36] p ( ξ ) = − π a ( √ − ξ ) (cid:104) F N a − Ka R (3 ξ − (cid:105) sphere lens − π a ( √ − ξ ) (cid:104) F N L − πKa R (2 ξ − (cid:105) cylinder lens (5)where ξ = r/a for sphere lens and ξ = x/a for cylinder lens, and 0 ≤ | ξ | ≤ F T is applied, if the entire contact region moves together, i.e. without relative slip,then the tangential stress on the surface is [10] q ( ξ ) = F T π a ( (cid:112) − ξ ) − / sphere lens F T πaL ( (cid:112) − ξ ) − / cylinder lens (6)In the second expression above for the cylinder lens, the shear stress q ( ξ ) correspondingto a displacement u y along the axis of the cylinder ( y -axis, see Fig. 1), is obtained from thesingular integral equation ∂u y ∂x = − πG (cid:90) a − a q ( s ) x − s ds (7)for u y = constant = δ , i.e. no relative slip between the surfaces.The ratio µ ( ξ ) = q ( ξ ) /p ( ξ ) is given by µ ( ξ ) /µ = | π (1 − Ka F N R (3 ξ − | − sphere lens | − π KLa F N R (2 ξ − | − cylinder lens (8)This expression for µ ( ξ ) /µ has a singularity at ξ c = (cid:0) + F N R Ka (cid:1) / and ξ c = (cid:0) + F N R πKLa (cid:1) / for the spherical and cylinder lens geometries, respectively. This ξ c corresponds to the point14t which the normal stress p ( ξ ) changes sign from compressive to tensile inside the contactregion. It is interesting to note that ξ c is independent of the applied tangential force F T andalways lies within the contact region 0 ≤ ξ ≤
1. Even if the singularity inherent in Eq. 8is replaced by a large finite value, the fact that no relative homogeneous slip occurs at theinterface ensures that, locally, the static friction coefficient µ > µ ( ξ ).The implication of the above expression for µ ( ξ ) can be stated as follows, followingRef. [9]. For frictionless contact between ‘sufficiently’ dissimilar materials, as with the lens–elastomer system in our experiments, the generalized Rayleigh wave [37] does not exist. Inthis case, there is a finite value of µ ( < ∼
1) above which homogeneous sliding is unstable toperturbations of all wavelengths.Assuming a Coulomb model in the experiments and using the force values in Fig. 8(a), theeffective static friction coefficient, µ = F T /F N , is very high. Prior to nucleation, the surfacesare stationary even when F T (cid:29) F N , until at F T = F c , a Schallamach wave is nucleated.This relaxes the stresses, thereby lowering the value of µ . One can thus infer that, for thelens–elastomer system, homogeneous interfacial sliding is unstable for large values of µ .Since µ ( ξ ), for | ξ | < ξ c in Eq. 8 occurs inside the contact region(and not at the edge, as in the purely elastic case), interfacial stick and slip regions cannotexist. This also rules out partial homogeneous interfacial motion. Thus, inhomogeneoussliding modes are very likely to occur in cases involving adhesion. V. CONCLUSIONS
The nucleation and propagation of isolated Schallamach waves in an adhesive elastomercontact has been studied in situ using high–speed imaging. These two phases of inhomoge-neous sliding were observed and characterized using spherical and cylindrical lens contacts.The former enabled observation of nucleation features such as the formation of wrinklesand their subsequent evolution. The latter was conducive for isolating and analyzing thedynamics of solitary Schallamach waves. Based on characterization of the individual waveproperties, a Burgers vector, analogous to dislocations, was established. Pinning of Schal-lamach waves by static dirt particles and existence of critical nucleation force were alsodemonstrated, all of which have analogues in dislocation glide. Simple analytical consider-15tions of the contact stresses also provide clues as to why inhomogeneous sliding modes viaSchallamach waves may be preferred in adhesive contact systems.
ACKNOWLEDGEMENTS
This work was supported in part by US Army Research Office grant W911NF-12-1-0012and NSF grants CMMI 1234961 and 1363524. Insightful discussions with Dr. J. Hanna andDr. D. P. Holmes (Virgina Tech.); and Dr. N. K. Sundaram (Indian Institute of Science) aregratefully acknowledged. We thank the referees for constructive feedback on the manuscript.
Appendix A: Image processing methods
The high–speed image sequence obtained from the experiments were analyzed in orderto obtain pixel–level velocities. Even though the elastomer appears transparent, very smallopaque features (such as embedded minute dust particles) always exist, providing contrastfor tracking purposes. The estimated inter–frame pixel displacements are proportional tothe local pixel velocity v p ( x, y ) because the time between frames is constant.In order to do this, the local image intensity I ( x ) at each image point x ≡ ( x, y ) isapproximated locally by a quadratic polynomial, i.e. I ( x ) = x T Ax + b T x + c , with coefficients A , b , c determined by a weighted least squares fit to intensity values in the neighborhoodof x . These coefficients are computed for each pixel in the image. If the image intensity isconvected with the velocity field, I ( x + d , t + dt ) = I ( x , t ) for a pixel x at time t , translatedby d to the next frame dt seconds later; d ( x ) is obtained by comparing these intensities.From a practical point of view, this results in significant noise in the displacement field.This is overcome by assuming that d is slowly varying and performing a weighted averageover pixels in a window. Furthermore, to minimize estimation error for large displacements, a priori estimates are used for each frame by performing the displacement estimation atmultiple length scales [38]. The displacement fields are first calculated at a coarse scale, forlarge blocks of the image. Subsequently, the chunk size is reduced and the coefficients aboveare iteratively calculated with information from the previous scale as an a priori estimate.This method hence works even for large displacements. The entire scheme was implementedby combining custom code with functions from the OpenCV code library [39].16or the analyses reported here, a window size of 15 pixels was found to give the besttradeoff between noisy data and a blurred velocity field. Three successive scales were usedfor the a priori estimate and at each scale the image size was halved. To refine the displace-ment estimate, the algorithm was iterated 3 times at each scale. For estimating A , b , c , aneighborhood size of 5 pixels was found to give good results for the least squares fit. [1] F. P. Bowden and D. Tabor, Friction: An Introduction to Tribology (RE Krieger PublishingCompany, 1982).[2] M. E. Merchant, Journal of Applied Physics , 230 (1940); F. P. Bowden and D. Tabor,Nature , 197 (1942).[3] E. Gerde and M. Marder, Nature , 285 (2001).[4] S. M. Rubinstein, G. Cohen, and J. Fineberg, Nature , 1005 (2004).[5] D. P. Vallette and J. P. Gollub, Physical Review E , 820 (1993).[6] T. Baumberger and C. Caroli, Advances in Physics , 279 (2006).[7] E. Orowan, Proceedings of the Institution of Mechanical Engineers , 140 (1943);T. Von Karman, Zeitschrift f¨ur Angewandte Mathematik und Mechanik , 130 (1925).[8] B. N. J. Persson, Physical Review B , 104101 (2001).[9] K. Ranjith and J. R. Rice, Journal of the Mechanics and Physics of Solids , 341 (2001);C. Caroli, Physical Review E , 1729 (2000).[10] K. L. Johnson, Contact Mechanics (Cambridge University Press, 1987).[11] A. Schallamach, Wear , 301 (1971).[12] M. Barquins, Materials Science and Engineering , 45 (1985).[13] C. J. Rand and A. J. Crosby, Applied Physics Letters , 261907 (2006).[14] See T. Baumberger, C. Caroli, and O. Ronsin, Physical Review Letters , 075509 (2002)and references therein.[15] J. H. Gittus, Philosophical Magazine , 317 (1975); G. A. D. Briggs and B. J. Briscoe,Philosophical Magazine A , 387 (1978).[16] D. Vella, A. Boudaoud, and M. Adda-Bedia, Physical Review Letters , 174301 (2009);J. M. Kolinski, P. Aussillous, and L. Mahadevan, ibid . , 174302 (2009).[17] The analogy between carpet rucks and dislocations causing slip in crystals has often been ttributed in the recent literature to E. Orowan. However, it appears to have been proposedby Bragg — see the article by W. Lomer in D. C. Phillips and J. M. Thomas, Selections andReflections: The legacy of Sir Lawrence Bragg (Science Reviews Limited, 1990), pp. 115 –118.[18] T. H. Heaton, Physics of the Earth and Planetary Interiors , 1 (1990).[19] E. R. Trueman, Locomotion of Soft-Bodied Animals (Elsevier, 1975).[20] J. E. Mark,
Polymer Data Handbook (Oxford University Press, 2009).[21] H. Schmid and B. Michel, Macromolecules , 3042 (2000).[22] The only report of a reference to wrinkles that we are aware of is A. A. Koudine and M. Bar-quins, Journal of Adhesion Science and Technology , 951 (1996).[23] See Supplemental Material at URL for more details and movies M1, M2.[24] E. Hohlfeld and L. Mahadevan, Physical review letters , 105702 (2011).[25] F. Brau, H. Vandeparre, A. Sabbah, C. Poulard, A. Boudaoud, and P. Damman, NaturePhysics , 56 (2010).[26] F. R. N. Nabarro, Theory of Crystal Dislocations (Clarendon Press, 1967).[27] A. H. Cottrell,
The Mechanical Properties of Matter (John Wiley and Sons, 1964).[28] M. A. Biot, Applied Scientific Research, Section A , 168 (1963).[29] E. Cerda and L. Mahadevan, Physical Review Letters , 074302 (2003).[30] J. Groenewold, Physica A: Statistical Mechanics and its Applications , 32 (2001).[31] Y. Cao and J. W. Hutchinson, Proceedings of the Royal Society A: Mathematical, Physicaland Engineering Science , rspa20110384 (2011).[32] M. K. Chaudhury, T. Weaver, C. Y. Hui, and E. J. Kramer, Journal of Applied Physics ,30 (1996).[33] R. D. Mindlin, Journal of Applied Mechanics (1949).[34] The tangential loading of adhesive spherical contacts was discussed, using an energy approach,by Savkoor and BriggsA. R. Savkoor and G. A. D. Briggs, Proceedings of the Royal Societyof London. A. Mathematical and Physical Sciences , 103 (1977).[35] K. L. Johnson, K. Kendall, and A. D. Roberts, Proceedings of the Royal Society of London.A. Mathematical and Physical Sciences , 301 (1971).[36] M. Barquins, The Journal of Adhesion , 1 (1988).[37] J. Weertman, Journal of the Mechanics and Physics of Solids , 197 (1963).
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FIGURES
FIG. 1. (Color online) Schematic of the experimental setup, with reference coordinates. Images onthe right show the contact regions for the cylindrical (top) and spherical (bottom) lens geometries.Scale bar corresponds to 200 µ m. a) (b) (c) (d) FIG. 2. (Color online) Four frames from a high–speed sequence showing the nucleation of Schalla-mach waves (top row) with a schematic side–view (bottom row). (a) Circular contact region beforeapplication of tangential force. (b) Change in shape of contact region and accompanying bucklinginstability the initiates wave nucleation. (c) The surfaces readhere ahead of the lens (point B ). (d)A single Schallamach wave pulse travels through the contact region. v s = 20 mm/s, spherical lens.FIG. 3. Wrinkles accompanying the nucleated wave. (Left) Initial pattern, with wrinkle wavelength λ = 18 µ m. (Right) Increase in wrinkle wavelength to λ = 40 µ m due to continued applicationof tangential stress. IG. 4. (Color online) Schallamach wavefront and its propagation features. (Top row) Sequenceof images showing propagation of a solitary Schallamach wave. The full length of the contactregion is about 20X the length shown in the images. (Bottom) 3D intensity plot, derived from theimages, showing various features on the elastomer surface. A - Initial adhesive contact betweenthe surfaces, B - front of Schallamach wave, C - Extent of trapped air pocket comprising the wave, D - Wrinkles on the surface and E - Incomplete readhesion after wave passage. Wave velocity v w (cid:39)
110 mm/s. v s = 2 . a) (b) (c) (d) FIG. 5. (Color online) Propagation of single Schallamach wave pulses in an adhesive contact. Thewaves retain their profile over long distances. The length of the contact region is 2.5 cm. Scale barcorresponds to 250 µ m; Wave velocity v w (cid:39)
380 mm/s. v s = 10 mm/s, cylinder lensFIG. 6. Pinning of a single Schallamach wave by static dirt particles. The wavefront approachesa single dirt particle A (top row), and is bent by it (point B , middle row). The wave then regainsits original shape, while leaving behind an air pocket C around the dirt particle. D i s p l ace m e n t ( µ m ) |b| (a)(b) FIG. 7. (Color online) Burgers vector and group velocity of a Schallamach wave pulse. (a) Meansurface displacement due to a single Schallamach wave, for various values of v s ; the magnitude ofthe jump denotes the | b | of the wave. (b) Space–time diagram showing local velocity | v p | for pointson surface, v s = 2 . AB and CD denote the front and rear of the wave pulse. Cylinderlens. F o r ce ( N ) F T F N C A BA (a) G e n e r a ti on fr e qu e n c y ( w a v e s / s ) DataBest fit Vc (b) FIG. 8. (Color online) Normal ( F N ) and tangential ( F T ) forces, and generation frequency ( n ) forSchallamach waves. (a) Time–variation of F T (blue or dark grey) and F N (green or light gray)for v s = 1 mm/s. Each oscillation of F T represents the propagation of a single wave. Criticalforce for nucleation F C is marked by point A . (b) Dependence of n on v s . The critical velocity forSchallamach wave formation is v c = 150 µ m/s. Cylinder lens.FIG. 9. (Color online) Geometry of equivalent line during propagation of Schallamach wave. 2 a isthe width of the contact region. t denotes tangent vector along the discontinuity line, which hasinclination θ . b is the Burgers vector. Cylinder lens.is the Burgers vector. Cylinder lens.