Strong dynamical effects during stick-slip adhesive peeling
Marie-Julie Dalbe, Stéphane Santucci, Pierre-Philippe Cortet, Loïc Vanel
aa r X i v : . [ c ond - m a t . s o f t ] N ov Strong dynamical effects during stick-slip adhesive peeling
Marie-Julie Dalbe, ab St´ephane Santucci, a Pierre-Philippe Cortet, c and Lo¨ıc Vanel b (Dated: November 14, 2013)We consider the classical problem of the stick-slip dynamics observed when peeling a roller adhesive tape at a constant velocity.From fast imaging recordings, we extract the dependencies of the stick and slip phases durations with the imposed peelingvelocity and peeled ribbon length. Predictions of Maugis and Barquins [in Adhesion 12 , edited by K.W. Allen, Elsevier ASP,London, 1988, pp. 205–222] based on a quasistatic assumption succeed to describe quantitatively our measurements of the stickphase duration. Such model however fails to predict the full stick-slip cycle duration, revealing strong dynamical effects duringthe slip phase.
Everyday examples of adhesive peeling are found in applica-tions such as labels, stamps, tape rollers, self-adhesive envelopsor post-it notes. During the peeling of those adhesives, a dy-namic instability of the fracture process corresponding to ajerky advance of the peeling front and called “stick-slip” mayoccur. This stick-slip instability has been an industrial concernsince the 1950’s because it leads to noise levels above the lim-its set by work regulations, to adhesive layer damage and/orto mechanical problems on assembly lines. Nowadays this in-stability is still a limiting factor for industrial productivity dueto the limitations of generic technical solutions applied to sup-press it, such as anti-adhesive silicon coating.From a fundamental point of view, the stick-slip instabilityof adhesive peeling is generally understood as the consequenceof an anomalous decrease of the fracture energy G ( v p ) of theadhesive-substrate joint in a specific range of peeling front ve-locity v p . Indeed, when the peeling process also involvesa compliance between the point where the peeling velocity isimposed and the fracture front, this decreasing fracture energynaturally leads to oscillations of the fracture velocity v p aroundthe mean velocity V imposed by the operator. Often, it is sim-ply the peeled ribbon elasticity which provides a complianceto the system. From a microscopic perspective, such anoma-lous decrease of the fracture energy G ( v p ) (correctly defined forstable peeling only) could correspond (but not necessarily) totransition from cohesive to adhesive failure or between twodifferent interfacial failure modes. More fundamentally, thisdecrease of the fracture energy has been proposed to be the con-sequence of the viscous dissipation in the adhesive material. De Gennes further pointed out the probable fundamental roleof the adhesive material confinement (which was evidenced ex-perimentally in ref. 3) in such viscoelastic theory. Since then, ithas however appeared that a model based on linear viscoelastic- a Laboratoire de Physique de l’ENS Lyon, CNRS and Universit´e de Lyon,France b Institut Lumi`ere Mati`ere, UMR5306 Universit´e Lyon 1-CNRS, Universit´ede Lyon, France. c Laboratoire FAST, CNRS, Univ. Paris Sud, France.M.-J. Dalbe, E-mail : [email protected]; S. Santucci, E-mail :[email protected]; P.-P. Cortet, E-mail : [email protected];L. Vanel, E-mail : [email protected] ity solely cannot be satisfactory and that the role of creep, largedeformations and temperature gradient in the adhesive materialis important (refs. 11–14 and references therein).Experimentally, the stick-slip instability was first charac-terized thanks to peeling force measurements which revealedstrong fluctuations in certain ranges of peeling velocity.
Since then, it has also been studied through indirect measure-ments of the periodic marks left on the tape or of theemitted acoustic noise.
Thanks to the progress in highspeed imaging, it is now possible to directly access the peel-ing fracture dynamics in the stick-slip regime.
In the late 1980’s, Barquins and co-workers, performeda series of peeling experiments of a commercial adhesive tape(3M Scotch R (cid:13) V and for var-ious lengths of peeled ribbon L . For the considered adhesive,the velocity range for which stick-slip was evidenced, thanksto peeling force fluctuations measurements, was shown to be0 . < V < . − . In a subrange of unstable peeling ve-locity 0 . < V < .
65 m s − , the authors succeeded to ac-cess the stick-slip cycle duration thanks to the post-mortem de-tection of periodic marks left on the tape by stick-slip events.Moreover, they managed to model quantitatively the measuredstick-slip period, assuming the fracture dynamics to remain aquasistatic problem during the stick phase and backing on mea-surements of the stable branch of the fracture energy G ( v p ) atlow peeling velocities below the instability onset.In this article, we revisit these experiments by studying thestick-slip dynamics during the peeling of a roller adhesive tapeat an imposed velocity. The principal improvement comparedto Barquins’s seminal work is that, thanks to a high speed cam-era coupled to image processing, we are able to access the dy-namics of the peeling fracture front. We focus on the study ofthe duration of the stick-slip cycle and its decomposition intostick and slip events, which data are inaccessible through othertechniques. We present experimental data of the stick and slipdurations for a wide range of imposed peeling velocity V andfor different peeled ribbon lengths L . We show that the modelproposed by Barquins and co-workers describes the evolu-tion of the duration of the stick phase, but fails to predict theduration of the whole stick-slip cycle due to unexpectedly longslip durations.1 eelingpoint adhesive tape rollermounted on a pulley windingcylinder PSfrag replacements R θ α θ m β L V ℓ F Figure 1 (Color online) Schematic view of the experimental setup.The angles a and b are oriented clockwise and counterclockwise respectively. Roller diameter: 40 mm < R <
58 mm, roller and tapewidth: b =
19 mm.
In this section, we describe briefly the experimental setupwhich has already been presented in details in a recent work. We peel an adhesive tape roller (3M Scotch R (cid:13) V using a servo-controlled brushless motor (see Fig. 1). The experiments havebeen performed at a temperature of 23 ± ◦ and a relative hu-midity of 45 ± b =
19 mm, itsthickness e = m m and its Young modulus E = .
26 GPa.Each experiment consists in increasing the winding veloc-ity from 0 up to the target velocity V . Once the velocity V is reached, it is maintained constant during two seconds be-fore decelerating the velocity back to zero. When stick-slipis present this 2-second stationary regime of peeling providessufficient statistics to compute well converged stick-slip meanfeatures. We have varied the imposed velocity V from 0 . . − for different values of the peeled tape length between L = .
08 and 1 .
31 m. During an experiment, the peeled tapelength L (Fig. 1) is submitted to variations, due to the stick-slipfluctuations and to slow oscillations of the peeling point angularposition, which however always remain negligible compared toits mean value (less than 0.3%). Thanks to a force sensor (Interface R (cid:13) SML-5) on the holdermaintaining the pulley, we are able to measure the mean valueof the force F transmitted along the peeled tape during one ex-periment. When peeling is stable, we compute the strain energyrelease rate G from the mean value of the force F , following thetraditional relation for the peeling geometry G = Fb ( − cos q ) + Ee (cid:18) Fb (cid:19) ≃ Fb , (1)for a peeling angle q ≃ ◦ (see Fig. 1). The quantity G cor-responds to the amount of mechanical energy released by thegrowth of the fracture by a unit surface. The right-hand term ofeqn (1) finally simply takes into account the work done by the operator but discards the changes in the elastic energy storedin material strains (term ( F / b ) / Ee in eqn (1)) which arenegligible here. Indeed, the maximum encountered force inour experiments is typically of about 2 N, which gives F / b ≈
100 J m − , to be compared to ( F / b ) / Ee ≈ .
12 J m − .In the context of elastic fracture mechanics, the condition fora fracture advance at a constant velocity v p is a balance betweenthe release rate G and a fracture energy G ( v p ) required to peela unit surface and accounting for the energy dissipation nearthe fracture front. When the fracture velocity v p approachesthe Rayleigh wave velocity v R , G ( v p ) also takes into accountthe kinetic energy stored in material motions which leads to adivergence when v p → v R . In our system, the strain energyrelease rate G , computed through eqn (1), therefore stands as ameasure of the fracture energy G ( v p ) when the peeling is stableonly, i.e. when v p is constant. We will nevertheless compute G for the experiments in the stick-slip regime for which thepeeling fracture velocity v p ( t ) is strongly fluctuating in time.In such a case, G cannot be used as a measure of a fractureenergy: it is simply the time average of the peeling force F inunits of G .In Fig. 2, we plot G as a function of the imposed peelingvelocity V for three different peeled tape lengths L . Whenthe peeling is stable, the peeling force is nearly constant intime, whereas it fluctuates strongly when stick-slip instabilityis present. The standard deviation of these fluctuations is rep-resented in Fig. 2 with error bars. Large error bars are indicativeof the presence of stick-slip.Between V = . − and V = . ± .
03 m s − , weobserve that G = F / b increases slowly with V and that its tem-poral fluctuations are nearly zero, revealing that the peeling isstable. This increasing branch G ( V ) is therefore a measure ofthe adhesive fracture energy G ( v p = V ) = G ( V ) for V < . ± .
03 m s − . Our results are compatible with the data reportedby Barquins and Ciccotti for the same adhesive tape (seeFig. 2). However, they explored a much larger range of veloci-ties in this stable branch of peeling, down to V = − m s − .Using both series of measurements, it is reasonable to modelthe stable peeling branch with a power law, G ( V ) = aV n , with n = .
146 and a = . ± .
03 m s − < V . − ,we observe that the measured value of G ( V ) decreases with V .This tendency, which was already observed in previous experi-ments, is accompanied with the appearance of temporal fluc-tuations which are the trace of the stick-slip instability. Fromthese data, we can estimate the onset of the instability to be V a = . ± .
03 m s − . The measured decreasing branch of G ( V ) for V > V a appears as a direct consequence of the anoma-lous decrease of the fracture energy at the origin of the insta-bility. It is important to note that the measured mean value of G = F / b is nearly independent of the length of peeled ribbon L . This result is natural in the stable peeling regime but was apriori unknown in the stick-slip regime.Barquins and Ciccotti succeeded to measure a second stablepeeling branch for V >
19 m s − . This increasing branch con-stitutes a measure of the peeling fracture energy G ( v p = V ) = G ( V ) in a fast and stable peeling regime. In ref. 8, this branchis inferred to exist for velocities even lower than V =
19 m s − ,2 −6 −5 −4 −3 −2 −1 PSfrag replacements G ( J m − ) V (m s − ) L = 0 .
08 m L = 0 .
48 m L = 1 .
18 mData from Barquins et al. [8] G = 137 V . G = 6 . − V . G , G , V , − V , − − V a V a V , V , G , G , G a Figure 2 (Color online) Mean value of the peeling force F , in units ofstrain energy release rate G = F / b , as a function of V for 3 differentpeeled tape lengths L . Stars report the data of Barquins and Ciccotti for the same adhesive. Solid line is a power law fit G = V . ofthe data in the low velocity stable branch. Errorbars represent thestandard deviation of the force fluctuations during one experiment. although it was not possible to measure it. Backing on the dataof ref. 6 for a very close adhesive, one can however guess thatthe local minimum value of G ( V ) , corresponding to a velocityin the range 2 . − < V <
19 m s − , would be bounded by G , = < G < G , =
33 J m − . The local dynamics of the peeling point is imaged using a highspeed camera (Photron FASTCAM SA4) at a rate of 20 000 fps.The recording of each movie is triggered once the peeling hasreached a constant average velocity V ensuring that only thestationary regime of the stick-slip is studied. Through directimage analysis, the movies allow access to the curvilinear po-sition of the peeling point ℓ a ( t ) = R a in the laboratory frame(with a the angular position of the peeling point and R the rollerdiameter, a > d b / dt ( t ) in the laboratory frame (where b isthe unwrapped angular position of the roller, b > ℓ b ( t ) = R b ). We finally compute numerically the curvilinearposition ℓ p ( t ) = ℓ b ( t ) + ℓ a ( t ) and velocity v p ( t ) = d ℓ p / dt ofthe peeling point in the roller reference frame.The curvilinear position of the peeling point ℓ a ( t ) in the lab-oratory frame is actually estimated from the position of thepeeled ribbon at a small distance 0 . ± .
05 mm from the peel-ing fracture front on the roller surface. We therefore do not de-tect strictly the peeling fracture front position but a very closequantity only. This procedure can consequently introduce somebias in our final estimation of the fracture front velocity v p ( t ) .This bias is notably caused by the changes in the radius of cur-vature of the tape at the junction with the substrate which aredue to the force oscillations in the peeled tape characteristicsof the stick-slip instability. Such effect actually biases the mea-surement toward larger velocities during the stick phase and −3 PSfrag replacements t ′ h T ss i (ms) V V a t (s) ℓ p ( mm ) ℓ p ( mm ) v p ( m s − ) v p ( m s − ) ℓ p (m) ℓ p (m)(a)(b)00.511.522.50.330.3350.340.3450.350.35530.36245250255260265 (a) (b)(c) (d) Figure 3 (Color online) (a) Peeling point position ℓ p ( t ) in the rollerreference frame for an experiment performed at V = .
55 m s − and L = .
47 m. The dashed line shows ℓ p = V t , with V the averagepeeling velocity. (b) Corresponding phase averaged peeling pointposition as a function of t ′ h T ss i (see main text). (c) and (d)Corresponding instantaneous (c) and phase averaged (d) peeling pointvelocity v p . The dashed horizontal lines show the average peelingvelocity V and the continuous horizontal lines show 3 V a . In (b) and(d), the vertical lines show the transitions between the stick( v p < V a ) and the slip ( v p > V a ) phases. lower velocities during the slip phase. Another effect that leadsto uncertainties on velocity measurement is the emission of atransverse wave in the peeled tape when the fracture velocityabruptly changes at the beginning and at the end of slip phases.Figs. 3(a) and (c) represent the fracture position ℓ p ( t ) andvelocity v p ( t ) as a function of time for a typical experimentperformed at V = .
55 m s − and L = .
47 m. In these figures,we observe alternate phases of slow –stick phase– and fast –slipphase– peeling which are the signature of the stick-slip motion.These large velocity fluctuations are quite regular in terms ofduration and to a lesser extent in terms of amplitude at leastat the considered peeling velocity. Our general data analysisfurther consists in the decomposition of the signal of instan-taneous peeling velocity v p ( t ) into stick-slip cycles by settingthe beginning of each cycle at times t n ( n denoting the n th cycle)when v p ( t n ) = V and dv p / dt ( t n ) <
0. From this data, we extractthe duration T ss of each stick-slip cycle for which we define arescaled time t ′ = ( t − t n ) / T ss . We further compute the phaseaveraged evolution of the peeling fracture velocity v p ( t ′ ) from t ′ = V and peeled tape length L the typical fracture velocityevolution during a stick-slip cycle getting rid of intrinsic fluc-tuations of the stick-slip period. In Figs. 3(b) and (d), we showthe phase averaged position and velocity profiles, correspond-ing to Figs. 3(a) and (c) respectively, as a function of t = t ′ h T ss i ( h i denotes the ensemble averaged value over all the cycles inone experiment).From these phase averaged velocity profiles, we define, foreach experimental condition V and L , stick events as continu-ous periods during which v p ( t ) < V a and slip events as con-tinuous periods during which v p ( t ) > V a . According to the3odel of Barquins et al. a natural threshold in order to sep-arate the stick and slip phases is the onset of the instability V a (as defined in Fig. 2). However, as discussed previously, dueto the procedure used for the detection of the peeling point, ourmeasurement of the fracture velocity can be affected by biasescaused by the variation the tape curvature at the peeling pointand by the propagation of transverse waves in the tape. Theeffect of the later can be observed in Fig. 3(d) in the early stageof the stick phase. In order to avoid taking into account thevelocity biases in the decomposition of the stick-slip cycle, wechose for the threshold separating the stick and slip phases avalue little larger the “theoretical” threshold V a , that is to say3 V a .Finally, as we have shown recently in ref. 21, when the peel-ing velocity V is increased, low frequency pendular oscillationsof the peeling angle q develop. Due to a dependence of thestick-slip instability onset with the mean peeling angle, theseoscillations lead to intermittencies in the stick-slip dynamicsfor peeling velocities V > . − . We therefore exclude theexperiments with V > . − in the sequel. For the studiedexperiments, we have a mean peeling angle h q i = ± ◦ withslow temporal variations in the range D q = ± ◦ during oneexperiment. From the signal of peeling point position ℓ p ( t ) (see Fig. 3(a)),we define the stick-slip amplitude A ss as the distance travelledby the fracture during a stick-slip cycle. In Fig. 4, we report thisamplitude A ss for each stick-slip event as a function of the cor-responding stick-slip period T ss , for all events in 6 different ex-periments. These data gather close to the curve A ss = V T ss . Thelarge spread of the data along the curve A ss = V T ss reflects thestatistics of the stick-slip cycle amplitude and duration whichcould be due for instance to adhesive heterogeneities. On thecontrary, the dispersion of the data around the curve A ss = V T ss is much smaller. It actually estimates the discrepancy betweenthe imposed velocity V and the averaged fracture velocity foreach stick-slip cycle. The observed small discrepancy actuallyboth traces back measurement errors on the instantaneous frac-ture velocity and intrinsic fluctuations of the dynamics.In Fig. 4, one can already see that the statistically averagedvalues of A ss and T ss increase with L for a given peeling ve-locity V . In the following, we will focus on the study of thestatistical average h T ss i of the duration of the stick-slip oscil-lation and its decomposition into stick and slip phases with inmind the aim of testing the description of Barquins, Maugis andco-workers. There is no need to study the averaged stick-slipamplitude h A ss i since it is univocally related to h T ss i through h A ss i = V h T ss i .In Fig. 5(a), we plot the mean stick-slip duration T ss as afunction of V for three different lengths L of the peeled rib-bon. The data corresponds to the average h T ss i and the er-ror bars to the standard deviation of the statistics of T ss overall the stick-slip events in each experiment. In the following,since we will consider the averaged values only, we will skip PSfrag replacements A ss ( mm ) T ss (ms) L = 0 .
47 m L = 1 .
31 m V = 0 .
30 m s − V = 0 .
55 m s − V = 1 .
00 m s − Figure 4 (Color online) Stick-slip amplitude A ss as a function ofstick-slip period T ss for each stick-slip cycle in 6 differentexperiments with L = .
47 and 1 .
31 m and V = .
30, 0 .
55 and1 .
00 m s − . The lines represent the curves A ss = V T ss . PSfrag replacements T ss ( m s ) (ms) V (m s − ) L = 0 .
47 m L = 0 .
97 m L = 1 .
31 m (a)(b) L = 0 .
47 m T stick T slip T ss PSfrag replacements T ss (ms) ( m s ) V (m s − ) L = 0 .
47 m L = 0 .
97 m L = 1 .
31 m (a) (b) L = 0 .
47 m T stick T slip T ss Figure 5 (Color online) Average stick-slip cycle duration T ss as afunction of the average peeling velocity V , for different lengths of thepeeled ribbon L . (b) Average stick-slip, stick and slip durations asfunction of the average peeling velocity for L = .
47 m. Each datapoint corresponds to the average and each error bar to the standarddeviation of the statistics over one experiment. PSfrag replacements T s t i c k / T s l i p (ms) V (m s − ) L = 0 .
47 m L = 0 .
97 m L = 1 .
31 m (a)(b) T stick T slip T ss Figure 6 (Color online) T stick / T slip vs. V for 3 different L . Each datapoint corresponds to the average and each error bar to the standarddeviation of the statistics over one experiment. the brackets h i . At first sight, it appears that, within the er-ror bars, the stick-slip duration T ss is stable over the major partof the explored range of peeling velocity V . One can how-ever note that, independently of L , T ss tends to decrease with V for V V c = . ± . − . Such behavior is compati-ble with the observations of Barquins et al. but appears hereover a rather limited velocity range. The characteristic velocity V c = . ± . − above which T ss is nearly constant seemsnot to depend strongly on the length of the peeled ribbon L .In Fig. 5(b), we show the mean durations of stick and slipevents, T stick and T slip respectively, as a function of the im-posed peeling velocity V for the experiments performed withthe peeled length L = .
47 m. Interestingly, we observe thatthe stick and slip phases evolve differently with V : the stickduration decreases with V , while the slip duration increasesover the whole explored range of V . In consequence, theratio T stick / T slip , presented in Fig. 6, decreases with V from T stick / T slip ∼ ± T stick / T slip ∼ . ± .
2. Such be-havior of T stick / T slip appears to be very little dependent on L according to Fig. 6. For V > . ± .
05 m s − , T stick / T slip becomes smaller than 1, meaning that the slip phase is longerthan the stick one. Our data therefore show that it is not possi-ble to neglect the slip duration compared to the stick durationin general. In this section, we compare our experimental data with themodel proposed by Barquins, Maugis and co-workers in refs.5,6. This model is based on measurements of the stable branchof the fracture energy G ( v p ) for low peeling velocities belowthe instability onset V a , and on the following assumptions: • During the stick phase, the equilibrium between the in-stantaneous energy release rate G = F / b and the fractureenergy G ( v p ) (of the low velocity stable branch) is stillvalid dynamically, i.e. G ( t ) = G ( v p ( t )) . • The peeled ribbon remains fully stretched during the peel-ing, which means G = Fb = EeL u , (2)where u is the elongation of the tape of Young modulus E and thickness e . • The slip duration is negligible compared to the stick dura-tion.Backing on these hypothesis, it is possible to derive a predictionfor the stick-slip duration T ss . Introducing the inverted function v p = G − ( G ) and noting that du / dt = V − v p (see next para-graph and ref. 21), eqn (2) leads to the dynamical relation dGdt = EeL ( V − G − ( G )) , (3)which can be integrated over the stick phase to get T stick = LEe Z G a G dGV − G − ( G ) . (4) G a is the maximum value of G ( v p ) at the end of the “slow”stable branch G slow ( v p ) . G is the minimum value of G ( v p ) atthe beginning of the “fast” stable branch G fast ( v p ) (see Fig. 2)and is assumed to be also the value of G at which the stick phasestarts on the slow branch after a slip phase.In this model, the ribbon is assumed to remain taut during thewhole stick-slip cycle. In order to challenge the validity of thishypothesis, let us estimate the evolution of the elongation u ( t ) of the tape as a function of time. If we note P ( t ) the peelingpoint position and M the point where the peeled tape is winded,we can define the quantity u ( t ) as the difference between thedistance | MP ( t ) | and the length of the peeled tape in the un-strained state. If u ( t ) is positive, this quantity indeed measuresthe elongation of the tape as in eqn (2), whereas it measures theexcess of slack tape if it is negative. Following ref. 21, one canshow that u ( t ) = u + Z t ( V − v p ( t )) dt − cos q Z t ( R ˙ b − v p ( t )) dt . (5)Since in our experiments the peeling angle q is close to 90 ◦ andthe roller rotation velocity R d b / dt sticks to the imposed peel-ing velocity V to a precision always better than ± . wefinally have u ( t ) ≃ u + R t ( V − v p ( t )) dt . The elongation/slack u ( t ) increases of D u = R T stick ( V − v p ( t )) dt during the stickphase and decreases of the same amplitude D u = − R T ss T stick ( V − v p ( t )) dt during the slip phase. This compensation is ensured bythe fact the averaged velocity over the stick-slip cycle matchesthe imposed velocity V , i.e. R T ss ( V − v p ( t )) dt =
0, and is validwhether or not the tape remains always taut during the stick-slipcycle.To test the relevance of the hypothesis of a tape always intension, one can actually compare the increase/decrease D u ofthe quantity u ( t ) during the stick/slip phase to the one predictedby the quasistatic model of Barquins and co-workers D u theo = LEeb ( F a − F ) = LEe ( G a − G ) , (6)5or an always taut tape. Throughout our data, the relative dis-crepancy ( D u theo − D u ) / D u is typically less than 15% whichconfirms the relevance of the assumption of a tape in tensionduring the whole stick-slip cycle.An equivalent but more instructive way to test the model ofBarquins and co-workers is to integrate numerically eqn (4) andcompare it to experimental measurements of stick duration. Todo so, we use the fit of the data of energy release rate G ( V ) of Fig. 2, i.e. G ( V ) = G slow ( V ) = aV n , with n = .
146 and a = G is affected by a significant uncertaintyin our data. We will therefore use two different guesses corre-sponding to the limit values introduced at page 3 (see G , and G , in Fig. 2). These values of G correspond to two limit val-ues of the fracture velocity at the beginning of the stick phase: V , = − m s − measured in another adhesive but with aclose behavior, and V , = . × − m s − which is an up-per limit for V according to the data of Fig. 2.In the insert of Fig. 7(b), we report the measured data for T ss / L as a function of V for three different lengths L as well asthe predictions of eqn (4) with V , (solid line) and V , (dashedline). The model appears compatible with the experimentaldata only for a marginal range of very low peeling velocities.Once V > . − , the measured values of T ss / L indeed de-viates more and more from the theoretical prediction. A firstnatural explanation for this discrepancy is that the assumptionof a negligible slip duration T slip (barely verified for low ve-locities for which 0 . < T slip / T stick < .
5) becomes more andmore false as V is increased (see Fig. 6).In Fig. 7(b) we therefore directly plot T stick / L as a functionof V , along with the prediction (4). One can note that the the-oretical predictions using the two limit guesses for V are notvery different. A first interesting result is that the stick durationappears, to the first order, proportional to the peeled tape length L as evidenced by the reasonable collapse of the data T stick / L on a master curve, which is compatible with the analytical pre-diction of the model (4). But more importantly, we observe thatfor the range of velocity explored, the model for T stick , whichdo not use any adjustable parameter, reproduces very well theexperimental data.Obviously, one can consider an equivalent quasistationaryapproximation during the slip phase in order to predict theslip duration using G − ( G ) instead of G − ( G ) in eqn (4).Here, G − ( G ) corresponds to the inverse of the energy frac-ture G = G fast ( v p ) in the fast and “stable” peeling regime ofFig. 2. The integration using the model of the fast branch G fast ( V ) = . × − V . (see Fig. 2) however leads to valuesof T slip always 2 orders of magnitude smaller than the exper-imental values as evidenced in Fig. 7(a). It is however worthnoting that the collapse of the data T slip / L for the different L shows that T slip increases nearly linearly with L . In this paper, we report experiments of a roller adhesive tapepeeled at a constant velocity focusing on the regime of stick-slip instability. From fast imaging recordings, we extract the
PSfrag replacements T stick /L (ms m − ) T s l i p / L ( m s m − ) T ss /L (ms m − ) (ms) V (m s − ) V (m s − ) L = 0 .
47 m L = 0 .
97 m L = 1 .
31 m(a)(b)
Model 1Model 2 T slip T ss A ss (mm) T ss /L (ms m − )10 − − PSfrag replacements T s t i c k / L ( m s m − ) T slip /L (ms m − ) T ss / L ( m s m − ) (ms) V (m s − ) V (m s − ) L = 0 .
47 m L = 0 .
97 m L = 1 .
31 m(a) (b)
Model 1Model 2 T slip T ss A ss (mm) T ss /L (ms m − )10 − − Figure 7 (Color online) (a) T slip / L , (b) T stick / L and T ss / L (insert) vs.V for 3 different L . Each data point corresponds to the average andeach errorbar to the standard deviation of the statistics over oneexperiment. In (a), the curve close to the x -axis represents thetheoretical prediction for a quasistationnary slip phase. In (b), thelines show the predictions of eqn (4) with V a = .
10 m s − and V , = − m s − (solid line) or V , = . × − m s − (dashedline). V and peeled ribbon length L .The stick phase duration T stick of the stick-slip oscillationsis shown to be nearly proportional to the peeled tape length L and to decrease with the peeling velocity V . These datamoreover appear in quantitative agreement with the predic-tions of a model proposed by Barquins, Maugis and co-workersin refs. 5,6 which do not introduce any adjustable parameter.This successful comparison confirms the relevance of the twomain assumptions made in the model: (i) the tape remains intension during the whole stick-slip cycle; (ii) the principle ofan equilibrium between the instantaneous energy release rate G ( t ) = F ( t ) / b and the fracture energy G ( v p ( t )) , as measuredin the steady peeling regime, is valid dynamically during thestick phase.Describing the peeling dynamics as a function of time t bythe knowledge of the fracture velocity v p ( t ) and of the force F ( t ) = b G ( t ) in the peeled tape, the considered model furtherassumes that the system jumps instantaneously, at the end ofthe stick phase, from the “slow” stable branch to the “fast” sta-ble branch of the steady fracture energy G = G ( v p ) and theninstantaneously backward from the “fast” branch to the “slow”branch at the end of the slip phase. In such a framework, re-producing the assumptions (i) and (ii) for the slip phase leadsto a prediction for the slip duration. We have shown that thisprediction is at least hundred times smaller than the slip phaseduration T slip measured in our experiments. We actually re-port that, contrary to what is finally proposed in refs. 5,6, theslip duration T slip cannot be neglected compared to the stickone T stick , since it is at best 4 times smaller, and becomes evenlarger than T stick for V > . ± .
05 m s − .These last experimental results account for the existence ofstrong dynamical effects during the slip phase which can there-fore not be described by a quasistatic hypothesis. These dy-namical effects could be due to the inertia of the ribbon close tothe fracture front. Some models also predict a strong influenceof the roller inertia. Notably, thanks to numerical compu-tation, De and Ananthakrishna have shown that for certainvalues of the roller inertia, the slip phase could consist in sev-eral jumps from the “fast stable” branch to the “slow stable”branch in the ( v p , G = G ( v p )) diagram. Such a process wouldcertainly produce a longer slip time than expected in the frame-work of Barquins’s model. It would be most interesting to con-front our experimental observations to the predictions of thismodel, based on a detailed set of dynamical equations and ad-hoc assumptions made on the velocity dependence of G . How-ever, such a comparison is not straightforward in our currentsetup since we do not have the temporal and spatial resolutionsto detect such eventual fast oscillations. Besides, in order toobtain a quantitative comparison, measurement of the instanta-neous peeling force F ( t ) is required but it remains a challenge. Acknowledgments
This work has been supported by the French ANR throughGrant “STICKSLIP” No. 12-BS09-014. We thank Costantino Creton and Matteo Ciccotti for fruitful discussions and MatteoCiccotti for sharing data with us.
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