Int J Fract (2017) 204:175–190DOI 10.1007/s10704-016-0171-1

ORIGINAL PAPER

In-situ measurement of the large strain responseof the fibrillar debonding region during the steadypeeling of pressure sensitive adhesives

Richard Villey · Pierre-Philippe Cortet ·Costantino Creton · Matteo Ciccotti

Received: 22 July 2016 / Accepted: 11 November 2016 / Published online: 23 November 2016© Springer Science+Business Media Dordrecht 2016

Abstract The debonding of pressure sensitive adhe-sives (PSA) is a classical example of the difficult andunsolved issue of fracture in soft viscoelastic confinedmaterials. The presence of a complex debonding regionwhere the adhesive undergoes cavitation and the verylarge strain of a spontaneously formed fibrillar net-work has defied many modeling attempts over the past70years. We present here a novel technique to pro-vide an accurate measurement of the local large strainresponse of the fibrillar debonding region during thesteady-state peeling of a well known commercial adhe-sive over a wide range of peeling velocity and angle.The technique is based on high resolution imaging ofthe debonding region during peeling and is coupled toa cohesive zone modeling of the adhesive interactionbetween the flexible tape backing and the rigid sub-strate. The resulting database provides a strong ground

Electronic supplementary material The online version ofthis article (doi:10.1007/s10704-016-0171-1) containssupplementary material, which is available to authorized users.

R. Villey · C. Creton · M. CiccottiLaboratoire SIMM, ESPCI Paris, CNRS, PSL ResearchUniversity, 10 rue Vauquelin, Paris, France

R. Villey · C. Creton · M. Ciccotti (B)Laboratoire SIMM, Université Pierre et Marie Curie,Sorbonne-Universités, 10 rue Vauquelin, Paris, Francee-mail: matteo.ciccotti@espci.fr

R. Villey · P.-P. CortetLaboratoire FAST, CNRS, Univ. Paris-Sud,Université Paris-Saclay, 91405 Orsay Cedex, France

for validating and further developing models (Villey etal. in Soft Matter 11:3480–3491, 2015) aiming to cap-ture the effects of both geometry and non-linear adhe-sive rheology on the exceptional adherence energy ofPSAs.

Keywords Pressure sensitive adhesive · Large strain ·Peeling · Cohesive zone · Adherence energy

1 Introduction

Peeling is a widespread testing method to character-ize the adherence properties of all sorts of thin bondedfilms, certainly due to its easiness of implementa-tion and quantitative evaluation (Kendall 1975; ISO8510-1 1990; ISO 8510-2 2006; Creton and Ciccotti2016). However, the weak transposability of peelingtest results to predict the adherence performance of agiven adhesive joint under different loading geometriessuch as shear lap, mode I cleavage between rigid beamsor probe tack is an important concern for applications.The difficulties of providing a clear and robustmechan-icalmodel and interpretation of the peeling strength of agiven material layer or joint have focused many effortssince the 50’s and no clear picture is available to coverall applications in a comprehensive way (Kaelble 1959,1960, 1965;Yarusso 1999;Barquins andCiccotti 1997;Amouroux et al. 2001; Villey et al. 2015; Derail et al.1997, 1998; Gent and Hamed 1977; Gent and Petrich1969).

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Despite the apparent simplicity of the peeling geom-etry at the macroscopic scale and the unambiguousmeasurement of the injected external work during thetest, the structure, applied deformation and stress fieldsof the debonding region are quite complex and dra-matically change when considering different materials.Moreover, when considering the peeling of soft filmssuch as most adhesive tapes—which are the object ofthe present investigation—these are invariably backedby a thin layer made of a stiffer material in order tolimit stretch at a macroscopical scale. The adhesivematerial in the debonding region is thus subject to aprogressive loss of confinement and consequently to avariable degree of stress triaxiality as it is debonded.These position dependent stress and strain fields arecoupled to the local bending profile of the backingand are related in a very sensitive way to the macro-scopic angle of application of the peeling force, lead-ing to variable damage scenarios. For example, theadhesive debonding region during the peeling of com-mon office tapes undergoes a complex process of cav-itation and fibrillation at the microscopic scale suchas illustrated in Fig. 1, which leads to a still poorlyunderstood transition between the very stiff oedometricresponse preceding cavitation and the almost uniaxialresponse in the unconfined fibrillar region (Villey et al.2015). The formation of the complex fibrillar structureis very similar to that observed in probe tack exper-

Fig. 1 Model geometry of a peeling experiment. The extremityof the peeled tape is pulled at a velocity V−→eθ tilted of an angle θ

with respect to the substrate (along−→ex ), which is itself translatedat velocity V−→ex . −→F is the peeling force at the pulling extremityof the tape. During peeling, the adhesive layer is stretched overa region called the “debonding region”, of curvilinear length lsalong the tape backing. The adhesivematerial, of initial thicknessa0, is strained and forms fibrils up to a maximum length a f atthe debonding front. We note ly the elevation of the tape backingat the end of the debonding region and lx the projection of ls onthe substrate

iments, which present stress-elongation curves char-acterized to the first order by a strain-rate dependentstress plateau up to a limited elongation before debond-ing (Creton and Ciccotti 2016). While comparativestudies between the uniaxial elongational behavior andprobe tack experiments have been carried out for sev-eral types of PSA (Lindner et al. 2006; Deplace 2009;Chiche et al. 2005) , the development of such a fib-rillar pattern during the propagation of a peeling frontis more complex since it is strongly coupled with thelocal bending of the backing, implying variable degreeof confinement and level of local stress (Villey et al.2015).

The aim of the present work is to develop new toolsto finely characterize the damage mechanisms at thescale of the debonding region during steady-state peel-ing of PSA, combining real-time microscopic imag-ing of the debonding region and a sound modelingof the mechanical interaction between the bending ofthe elastic backing and the deformation of the fibril-lar network, which will be treated as an effective non-linear cohesive zone. A careful compromise betweenthe image resolution on such a small region and the rel-evant mechanical information coming from the presentknowledge on the structure and behavior of the fib-ril network will be the guide to the identification of aminimal yet rich set of local parameters of the debond-ing region, namely the local radius of curvature of thebacking, the effective average stress applied by the fib-rils on the backing, the average strain rate of the fibrils,their maximum extension and tilt angle before debond-ing.

In this work we investigate the variations of theselocal parameters via a series of tests on a well knowncommercial tape over a large range of peeling angleand velocity, in order to discuss the soundness of theestimated parameters in light of the known rheologicalbehavior of such an adhesive and of visual compar-isons with the imaging of the fibrils. The robustnessand reproducibility of the estimated parameters repre-sent a real experimental breakthrough and become akey to the sound modeling of the adherence proper-ties of peeled tapes as a function of the large strainrheology of the adhesive material and of the sur-face properties of the adherends (Villey et al. 2015).These enriched parameters can also be used to under-stand the relationships between the peeling and thetack properties of each adhesive (Creton and Ciccotti2016).

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In-situ measurement of the large strain response 177

2 Microscope filmed peeling experiments

Peeling experiments are performed at room tem-perature (23 ± 2 ◦C) on a commercial PSA (3MScotch® 600)1 frequently studied in the literature (Bar-quins and Ciccotti 1997; Amouroux et al. 2001; Cortetet al. 2007, 2013; Dalbe et al. 2014a, b, 2015; Villeyet al. 2015). The adhesive tape is peeled from a flatbar by an Instron testing machine (model 3343), whichrecords the peeling force F while imposing a constantpull-out velocity V ∈ [3 : 3000] µm s−1. The bar, tiltedof an angle θ ∈ [30◦ : 150◦] with respect to the pullaxis of the testing machine, is mounted on a translationstage: it is translated at the same velocity V , resultingin a steady peeling at constant angle θ (see Fig. 1).

These experiments give access to the peeling forceF(θ, V ) from which we compute the energy releaserate G associated to the peeling of a unit surface

G = F

b(1 − cos θ) , (1)

where b = 19 mm is the tape width (see Fig. 1). Thisexpression of G accounts for the work of the operator,but discards the contribution associated to the elasticenergy stored in the peeled tape backing elongation(Kendall 1975), which is never larger than 1% of G inthe reported experiments.

Experiments are performed in a range of velocityV where peeling is steady, i.e. where no stick-slipdynamical instability is observed [stick-slip appears atlarger velocities for this PSA (Dalbe et al. 2014b)]. Inthese conditions the energy release rate G defines anadherence energy Γ (θ, V ) that can also be consideredas an effective fracture energy for the peeling of thewhole adhesive joint (backing, adhesive and substrate)at velocity V and angle θ

G = Γ (θ, V ). (2)

The substrate of the adhesive tape to be peeled ismade of a first layer of the same PSA tape, which hasbeen applied on the peeling bar following the proto-col described in Villey et al. (2015). The substrate tapelayer is changed after each peeling experiment in orderto preserve the integrity of its release coating. Thisprocedure enables comparison with previous measure-ments made by peeling directly from the commercial

1 This PSA is made of an acrylic adhesive layer of thicknessa0 � 20µmcoated on aUPVCbacking of comparable thickness,Young’s modulus E � 2.9 GPa and width b = 19 mm.

roller (Barquins and Ciccotti 1997; Amouroux et al.2001; Cortet et al. 2007, 2013; Dalbe et al. 2014a). Italso ensures a moderate level of adhesion (F ∈ [0.3 :4] N, Γ ∈ [20 : 80] J/m2), which results in interfacialadhesive failure only,with no residuals on the substrate,even at peeling velocities V as low as few µm s −1.

The shape of the tape backing profile close to thedebonding region (where the adhesive is deformedbefore debonding) is monitored during the peelingexperiments using a 1624 × 1228 pixels2 cameraequipped with a microscope objective (frame ratebetween 5 and 30 fps, resolution between 0.55 and 2.2px/µm). For each frame of the movies (such as thoserepresented in Figs. 2, 3), the outer profile of the tapebacking ismeasured using a binarization algorithm thatdetects the interface between the dark background andthe illuminated tape. The tape backing profile does notsignificantly change with time during steady peeling(see the movie in Online Resource 1). We can thusaverage this profile over the entire movie (several hun-dreds of frames), which removes detection imperfec-tions (related e.g. to the presence of micrometric dusts)and eventually results in a significant increase of thesignal-to-noise ratio. This average profile is reportedin yellow in Figs. 2 and 3.

In these two figures, we also report the local angleα made by the averaged profile with the substrate asa function of the curvilinear abscissa s along the tapebacking in the reference frame of the laboratory (seeFig. 4), related by

ds−→t = ds cos(α)

−→ex + ds sin(α)−→ey

= dx −→ex + dy −→ey . (3)

3 Simple elastica model

We first compare the experimental profiles to the the-oretical expression of an inextensible elastica beam(Love 1944) with a simple clamping boundary con-dition at s = s0

α(s)= θ − 4 arctan

[tan

(θ

4

)exp

(− s−s0

r

)], (4)

with

r =√

E I

F(5)

the buckling length of the tape backing of bendingmod-ulus E I submitted to a force F , which is the typical

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178 R. Villey et al.

Fig. 2 a–e Pictures of thedebonding region duringpeeling experiments atV = 100 µm s−1 anddifferent peeling angles θ .The profile of the tapebacking (reported in yellow)is obtained from an averageover several hundreds ofdetected profiles (one foreach frame of the movie).This profile is compared tothe classical elastica modelof Eq. 4, represented ingreen. We also report thebest fit of the profile usingthe model of elastica with acohesive zone presented inSect. 4 (red dashed lines).This model has two fitparameters, namely thelimits of the cohesive zone(CZ), which are highlightedby red circles. In f, werepresent the same data as ina–e in curvilinearcoordinates (see Eq. 3),showing the slope α of thetape backing as a functionof curvilinear abscissa s.s = 0 is set at the beginningof the fitted cohesive zones

(a)

(b)

(c) (f)

(e)

(d)

value of the curvature radius of the backing close to thedebonding region. We actually fit the α(s) experimen-tal profiles outside of the debonding region (sufficientlyfar from the last visible fibrils) using s0 and r as fittingparameters. The result is reported in green in Figs. 2and 3 where the elastica profiles are extrapolated downto the substrate. The matching between experimentaland elastica profiles outside of the debonding region isexcellent (see Fig. 2f) and, since we measure the peel-ing force F independently, the fit of the buckling lengthr provides a measurement of the bending modulus E Iof the tape backing for the portion of the tape beingpeeled.2

2 We find that this modulus changes significantly between dif-ferent rollers, and even between distant portions of the sameroller, with an average value E I = (3.7 ± 1.7) 10−8 N m2.

As it can be observed in Figs. 2 and 3, the theo-retical profiles of the simple elastica model of Eq. 4clearly do not match the experimental ones in thedebonding region. In particular, the profile of the sim-ple elastica model presents a curvature discontinu-

Footnote 2 continuedThis value agrees with the estimate that can be made from theexpression E I = Ebh3/12, where E is the Young’s modulusand h the thickness of the tape backing, using an independenttensile test giving Ebh = 1100± 200 N and microscopic obser-vations of the thickness providing h = 20 ± 5µm. An attentivestudy reveals that the variations in bendingmodulus E I can actu-ally be attributed to small variations of the tape backing thick-ness of amplitude ±3 µm, since E I ∝ h3. In any case, in thepresent study the bending modulus E I will enter the problemonly through the parameter r = √

E I/F , which will be directlymeasured for each experiment through elastica fits of the peeledtape outside of the debonding region.

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In-situ measurement of the large strain response 179

(a)

(b)

(c) (h)

(e)

(d) (f)

(g)

Fig. 3 a–g Pictures of the debonding region during peelingexperiments at θ = 90◦ and different peeling velocities. Thecolor and symbol code is the same as in Fig. 2. In h, we show

the profiles from a–g in curvilinear coordinates (see Eq. 3), withs = 0 taken at the beginning of the fitted cohesive zones

ity at the junction with the substrate where s = s0,with a maximum curvature at s = s+

0 . On the con-trary, the experimental profiles present no curvaturediscontinuity and the maximum curvature is remotefrom the beginning of the debonding region (yet stillinside it). It is important to note that, in Figs. 2f and

3h, the origin of the curvilinear abscissa s of theexperimental profiles has been set by fitting the pro-file of the backing by an elastica model with a cohe-sive zone, as described in Sect. 4, and not by simplevisual observation of the fibrillated region from theside.

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180 R. Villey et al.

Fig. 4 Cohesive zone model of the debonding region expressedin the reference frame of the laboratory, where the peeling profileis steady. s = x = 0 is set at the beginning of the cohesive zone.

The tape is undeformed at x ≤ 0 and submitted to a force−→F far

from this undeformed zone. The adhesive response is modelledby a uniform stress σ̄ for 0 ≤ s ≤ ls . We assume this stress tofollow the fibrils orientation, given by the angle ϕ

The reason for the observed discrepancy with thesimple elastica model is obviously related to the exis-tence of forces applied on the tape backing by thestretched adhesive material in the debonding region,which have not been taken into account. Note that theelastic extension of the tape backing (neglected in thesimple elastica model) cannot account for the above-mentioned discrepancies, since the tape extension isvery small in our experiments. This can be verified bycomparing the tape extension εdx = dL of a portiondx of the tape to its vertical deflection dy due to bend-ing: with F ≤ 4N and Ebh � 1100N, the tape relativeextension ε is always smaller than 0.4% (in the worstcase), which can be safely neglected compared to thetape deflection due to bending tan α = dy/dx as soonas α is larger than 1◦.

4 A model of elastica with a cohesive zone

In this section we account for the stress distributionσ(s) applied to the tape backing in the debondingregion in terms of a cohesive zone coupled to the inex-tensible elastica model. As a first approximation weconsider a uniform effective stress σ̄ in the debond-ing region, which corresponds to a traction-separationcurve that is constant up to a maximum elongationat debonding, where it drops to zero. The relevanceof such an approximation relies on the fact the adhe-sive material experiences a relative stretching of sev-eral hundred percents in the majority of the debond-ing region (see Figs. 2, 3), which for confined soft

incompressible materials is always associated with fib-rillation. This loading condition is very similar to theprobe-tack test, which for typical PSA provides stress-strain curves dominated by a large plateau for the stress(Lakrout et al. 1999; Creton et al. 2009; Creton andCiccotti 2016). One should note that, because of cavi-tation and fibrillation, σ̄ does not represent a true localstress, but the average force per unit area acting on theboundaries of the adhesive material (both for peelingand probe-tack experiments).

Since the adhesive presents a fibrillar structure inthe majority of the debonding region, it is reasonableto represent the stress applied to the tape backing as avector that is locally oriented along the fibrils (and withconstant modulus σ̄ ). Moreover, since the fibrils do notappear to slide on the tape backing nor on the substrate(before debonding), a fibril attachedon the tape backingat curvilinear abscissa s should also be attached to thesubstrate at position x = s.3 This enables to computethe fibrils length a(s) = √

(s − x(s))2 + (a0 + y(s))2

and the angle ϕ(s) = arctan(a0+y(s)s−x(s)

)that the fib-

rils form with the substrate (see Fig. 4). Note thatx(s) = ∫ s

0 cos (α(t)) dt and y(s) = ∫ s0 sin (α(t)) dt

are the horizontal and vertical coordinates of the pointalong the tape at curvilinear abscissa s. In the initialpart of the debonding region (before fibrillation) theforces applied by the adhesive are essentially verticaland ϕ remains a valid estimate of the stress orientation,even if fibrils are not present, since the tape backing isonly slightly bent so that s � x and ϕ � 90◦.

Our model can then be described by the followingsystem

E Iα′′−→ez + −→t ∧ −→

R = −→0 ,

−→R ′(0 < s < ls) = σ̄b

[− cos(ϕ)−→ex + sin(ϕ)

−→ey],

−→R (s ≥ ls) = −→

F ,

α(s < 0) = 0, (6)

where ′ is the derivative with respect to s and−→R (s)

is the internal force, applied by the half-part s̃ > s ofthe tape on the other half-part s̃ < s. The first line ofthis system is the classical beam flexure equation (Love1944). The second line accounts for the presence of the

3 The validity of this assumption is limited to the inextensibletape approximation of our model, that follows the argumentsprovided in Sect. 3.

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In-situ measurement of the large strain response 181

uniform external stress σ̄ in the cohesive zone betweens = 0 and s = ls (in curvilinear abscissa). The tworemaining lines represent the boundary conditions. Acombination of the first three equations of (6) gives themaster equation of our model

E Iα′′ + F sin(θ − α)

− σ̄b∫ ls

ssin [α(s) + ϕ(t)] dt = 0. (7)

Equation 7 is valid in the cohesive zone s ∈ [0; ls] andshould be associated with the description of the rest ofthe tape backing by

α = 0, for s ≤ 0,

E Iα′′ + F sin(θ − α) = 0, for s ≥ ls, (8)

where the first line represents the absence of flexurein the bonded part of the backing out of the debondingregion, while the last line is the simple elastica equationin the freestanding part of the backing.

Integration of the second line of (8) gives

α(s ≥ ls) = θ − 4 arctan

[X exp

(− s − ls

r

)], (9)

where the constant X has to be determined from thecontinuity relations at the end of the cohesive zone s =ls

α(l−s ) = α(l+s ) = θ − 4 arctan(X),

α′(l−s ) = α′(l+s ) = 4X

r(1 + X2

) . (10)

The continuity of the curvature α′ is equivalent tothe absence of point torque at s = ls . Its derivative α′′is actually also continuous at the end of the cohesivezone, because no point force is applied there. The latestcontinuity relation is however already imposed by thecontinuity of α associated to Eq. 7 in which the termmultiplying σ̄ disappears at s = ls .

Similarly, continuity relations at s = 0 are

α(0−) = α(0+) = 0,

α′(0−) = α′(0+) = 0. (11)

Using Eq. 7 and the boundary conditions (10) and(11) one can compute4 the value taken by the energyrelease rate G = (1 − cos θ) F/b during steady-statepropagation according to this model

G = σ̄(a f − a0

). (12)

G is actually equal to the work (per unit peeled sur-face) of the cohesive forces during the stretching of theadhesive from its initial thickness a0 up to the maxi-mum fibril length a f = a(s = ls) at debonding. Thisrelation, which tells that the work of the peeling force−→F is fully used to stretch the adhesive material, actu-ally simply accounts for the energy conservation in thebending deformation of the tape backing during steady-state peeling. Note that, in contrast, in the simple elas-tica model of Eq. 4 with no cohesive zone, bendingenergy conservation can only be enforced thanks to theintroduction of a curvature discontinuity at s = 0 thatacts as an energy sink

G = E I

2b

[α′(0+)2 − α′(0−)2

].

4 Multiplying Eq. 7 by α′ and integrating from 0 to ls yields

8FX2

(1+X2

)2 +F [cos(4 arctan X) − cos θ ]

− σ̄b∫ ls

0

[cos (α(s))

∫ ls

ssin (ϕ(t)) dt

]α′(s)ds

− σ̄b∫ ls

0

[sin (α(s))

∫ ls

scos (ϕ(t)) dt

]α′(s)ds = 0.

After trigonometric simplifications of cos(4 arctan(X)) and inte-gration by parts of the two integrals, we get

F

b(1 − cos θ) = σ̄

∫ ls

0{sin (α(s)) sin (ϕ(s))

+ [1 − cos (α(s))] cos (ϕ(s))} ds.

The integrand can be rewritten as

sin ϕ sin α + (1 − cosα) cosϕ = (y + a0)dyds + (s − x) ds−dx

ds√(y + a0)2 + (s − x)2

= da(s)

ds

where a(s) = √(y + a0)2 + (s − x)2 is the length of the fibril

attached to the tape backing at curvilinear position s. This finallyleads to Eq. 12. One can also demonstrate Eq. 12 by computingthe work associated to a small advance of the peeling front (seeOnline Resource 2).

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182 R. Villey et al.

We can finally rewrite the whole system describingthe tape profile in the cohesive zone (0 ≤ s ≤ ls), withthe boundary conditions

α(0) = 0,

α(ls) = θ − 4 arctan(X),

α′(ls) = 4X

r(1 + X2

) , (13)

associated to the master equation

r2α′′ + sin(θ − α) − 1 − cos θ

a f − a0

∫ ls

ssin [α(s)

+ϕ(t)] dt = 0, (14)

where the maximum fibril length a f is given by

a f =√(∫ ls

0(1 − cosα(s))ds

)2

+(a0 +

∫ ls

0sin α(s)ds

)2

.

(15)

Note that, since Eq. 12 was used in Eq. 14 to replaceσ̄ , one of the initial boundary conditions has becomeredundant: we have therefore removed the α′(0) = 0boundary condition from the system.

This system contains an integro-differential equa-tion with derivatives up to the second order, plus oneunknown, X , requiring therefore three boundary con-ditions, which are provided by Eq. 13. We can solvethis problem for different sets of values for θ , r , a0and ls , the only parameters it depends on. Note thatthis problem does not depend explicitly on σ̄ anymore.However, σ̄ can be retrieved via Eq. 12, using a f (ls)determined from the tape profile and the factG is deter-mined by F and θ (see Eq. 1), which can be easily andprecisely measured.

To solve the set of Eqs. 13 and 14, we normalize alllengths by r andwe sample the interval s/r ∈ [0 : ls/r ]over N points (typically several hundreds to thou-sands). We replace α′′ by its midpoint method esti-mate, andα(1), α(N−1) andα(N )by their expressionsaccording to (13).We then use theMatlab fsolve routineto find the values of X and α(2) . . . α(N −2) that min-imize the set of N − 2 equations obtained from Eq. 14.All lines in this system contain the term sin(θ − α),which is of order of magnitude 0.1–1, enabling anobjective criterion for the minimization procedure: thesystem is considered to be solvedwhen all lines are verysmall compared to 0.1 (in practice when the maximum

(a)

(b)

(c)

Fig. 5 Theoretical profiles of elasticas with cohesive zones ofdifferent sizes ls (the ends of which are shown by red disks)and for different peeling angles θ . We normalize the curvilinearabscissa s by the typical radius of curvature r = √

E I/F . Theconsidered values for the ratio ls/r are typical of our experiments(Figs. 1, 2)

term is smaller than 10−6).We use as initial guesses forX and α(i) the analytical solutions of the approxima-tion of Eq. 14 for small angles α (see Online Resource2).

Figure 5 reports numerical solutions of the systemof Eqs. 13 and 14 for peeling angles θ and cohesivezone size ratios ls/r typical of our experiments. Asexpected, the solution converges towards the simpleelastica profile Eq. 4 at small ls/r values. We can alsoobserve that the curvature (slope of α(s)) goes from 0at s = 0 up to a maximum included in the cohesivezone, before decreasing back towards 0 at s = ∞.

Computing the dimensional profile α(s) (and thusthe y(x) profile by integration) eventually requires toknow r and ls , as well as the curvilinear abscissa s0

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In-situ measurement of the large strain response 183

along the tape at which the debonding region begins(i.e. the starting point of the local reference frame tocompute the curvilinear abscissa, taken as 0 for thesake of simplicity in Eqs. 6–14). For each experimen-tal profile of the tape backing, r = √

E I/F is firstobtained from the fit of α(s) outside of the debondingregion by the simple elastica prediction (9) (leavingthe constant X as an unused parameter at this step).The whole experimental profile α(s) (including thedebonding region) can finally be fitted by the systemof Eqs. (13, 14) using ls and s0 as the only fitting para-meters.5

5 Results and discussion

5.1 Comparison between model and experimentalprofiles

The fitting procedure described in Sect. 4 has been firstsystematically applied to all the individual frames of theexperimentalmovies, for all couples of control parame-ters (θ, V ).

Both the radius of curvature r and the size of thecohesive zone ls , which are the two valuable raw prod-ucts of the fitting procedure, are found to be very stableduring each movie (see the Online Resource 1), corre-sponding to the peeling of typically several hundredsof µm to several cm, depending on V . More precisely,we observe typical slow variations of 5–15% for thesetwo lengths during apeeling experiment.Besides,whenlarge portions of fibrils debond at once, the tape backingprofile and thus the fitted value of ls do not change sig-nificantly (2% for the strongest event at time t = 4.6sin the movie in the Online Resource 1, with a cumu-lated variation in ls of 13% along the entire movie).On the other hand, the extent of the region where thefibrils are observed on the visible side presents a scat-ter of ±20%. This last observation is related to the factthe only visible fibrils in the movie are the ones onthe side of the tape observed by the camera, while thetape backing profile is the result of the influence of thedebonding region over the whole tape width b. Thiseventually makes our fitting procedure a more accurateand representative tracer of the geometry of the debond-ing region over the whole tape width than the optical

5 Oncewe knowα(s), ls and s0 , a f can be determined by Eq. 15.Curvilinear abscissa is eventually redefined as s → s − s0 afterthe fit, in order to set the beginning of the cohesive zone to s = 0.

observations of the fibrils on the side view, which rep-resent a limited set of fibrils that are close to the sideof the tape. A detailed analysis reveals that the averagesize of the debonding region observed on the tape sidepresents a bias going from −20 to +20% with respectto the cohesive zone length extracted from the fittingof the backing profile for θ ranging from 30◦ to 120◦.6The extent of the debonding region on the tape sidetends to be shorter than at the center of the tape whenthe peeling angle is less than 90◦ and longer for peelingangles larger than 90◦. This bias reveals an interestingedge effect that is not at the core of the present inves-tigation. In the rest of the manuscript we will focus onthe estimations derived from the bending profile, whichare not affected by this edge effect.

Because of the small fluctuations of the detected tapebacking profiles (due in particular to dusts on the tape ortape imperfections) and of the lengths r and ls extractedfrom them, we have applied as a second step the fit-ting procedure to profiles averaged over all images ina movie, which enables to considerably increase thesignal-to-noise ratio. The results of this fitting proce-dure is in excellent agreement with the experimentalprofiles, as can be seen in Figs. 2 and 3, including in thedebonding region. This agreement confirms the rele-vance of themodel of elastica combinedwith a constantstress cohesive zone, which catches the main featuresof the tape profile using only few mechanical ingredi-ents : a mean stress σ̄ oriented along the fibrils over aregion of length ls .

5.2 Cohesive stress and maximum fibril extension

In the model developed in Sect. 4, energy dissipa-tion during peeling occurs because of the loss of thework expended to stretch the adhesive material up todebonding, which brings the following expression forthe effective fracture energy of peeling

Γ = σ̄(a f − a0

). (16)

This work done against the cohesive stress σ̄ is par-tially dissipated by viscosity during the stretching ofthe adhesive, the rest being stored in the adhesive elas-tic energy and eventually lost at debonding by “elas-tic hysteresis” (Villey et al. 2015). We remark that the

6 The bias gets larger for the largest angle θ = 150◦, which willbe discussed later.

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184 R. Villey et al.

transient kinetic energy developed by the fibrils afterdebonding is also dissipated by viscosity during thefibrils collapse, but in any case this evolution is notcoupled to the peeling mechanics since the broken fib-rils are not bridging the backing to the substrate any-more.

In this section we use the results of the fitting of theexperimental tape profiles to access the two parameterscontrolling energy dissipation during peeling (accord-ing to the cohesive zone model): (i) the average stressσ̄ in the cohesive zone and (ii) the fibrils extensiona f −a0 at the onset of debonding, which are reported inFig. 6.

Figure 6a reports the evolution of the adherenceenergyΓ (θ, V ) = σ̄ (a f −a0) as a function of the peel-ing angle θ for peeling velocity V = 0.1 mm s−1. Asreported in previous experiments on the same adhesive-substrate joint (Villey et al. 2015), the adherence energyis found to be a regularly increasing function of thepeeling angle θ . Assuming the cohesive zone modeldeveloped in Sect. 4 and considering Fig. 6b, c, thisangular dependency seems to be mainly due to thefibril elongation a f − a0 that is found to be clearlyincreasing with θ , while the average stress in the cohe-sive zone σ̄ is displaying weaker trends. If all anglesare considered, the average stress decreases by a fac-tor 1.7 while the fibril elongation increases by a factor3.7.

We should remark that when limiting to the anglesup to 120◦, σ̄ is almost constant and all the increasein Γ is due to the increase in a f − a0. In fact, thecase of θ = 150◦ presents some peculiarities com-pared to all other experimental conditions studied inthis paper: at this large angle (see Fig. 2e), the detectedsizes ls of the cohesive zone seem systematically larger(by typically 30–60%) than the visible fibrils region(keeping in mind that those may not be representativeof all fibrils along the tape width). Moreover, a care-ful examination of the α(s) profiles at this large anglereveals a systematic overshoot of 1◦ to 2◦ in the localangle (α > θ locally, a feature too small to be visi-ble in Fig. 2). Such an overshoot can be attributed tothe occurrence of some plastic deformation in the tapebacking, which should add a residual curvature to thetape profile. If plasticity happens, it should indeed beat large peeling angles (Gent and Hamed 1977; Derailet al. 1997, 1998), where the tape curvature is large.This is consistent with a simple everyday life test: anoffice tape remains twisted after being peeled when

peeling proceeds at a large angle. However, the valuesfor a f − a0 and σ̄ obtained at θ = 150◦ remain of thesame order as the ones obtained at smaller angles, andthe fitted profiles remain in good agreement with theexperimental ones even at this large angle: this indicatesthat the results our model produces at very large peel-ing angles may be conserved, while considered withprecaution.

Figure 6d–f report the same three quantities, Γ , σ̄

and a f − a0, as a function of the peeling velocity Vfor θ = 90◦. Γ and σ̄ follow power laws of V withclose exponents (0.20 and 0.27 respectively): contraryto the angular dependency, we observe that most of theincrease of the adherence energy Γ = σ̄

(a f − a0

)with the peeling velocity is due to the increase ofthe cohesive zone stress σ̄ and that it is only slightlyaffected by a weak decrease of the fibril extensiona f − a0 with V . Moreover, the values reported for σ̄

in Fig. 6b, e are consistent with what is expected forthe plateau stress of such acrylic-based PSA (Lindneret al. 2004, 2006).

In the present work, we have studied only twoline cuts (V = 0.1 mm s −1 and θ = 90◦) in theexperimental parameters plane (θ, V ). Reference Vil-ley et al. (2015) reports measurements of the peelingadherence energy Γ for the same PSA over a widerrange of parameters (θ, V ). The data reported in Vil-ley et al. (2015) actually revealed that the dependen-cies of Γ with V and θ were almost separable, i.e.that Γ (θ, V ) � f (θ) g(V ). In light of our presentexperiments, this separability can be justified by thefact that for this adhesive the dependence of σ̄ onthe angle θ is barely detectable, and in any case isweak compared to the one of a f (θ), and converselythat the decreasing trend of a f (V ) is weak com-pared to the increasing trend of σ̄ (V ), so that we canwrite

Γ (θ, V ) = σ̄ (θ, V ) × [a f (θ, V ) − a0

],

≈ σ̄ (V ) × [a f (θ) − a0

]. (17)

In addition to the average stress and the maximumelongation, another key rheological parameter can beextracted from our treatment of peeling tests: the aver-age strain rate

¯̇ε = 1

ls

∫ ls

0ε̇(s)ds = V

lsln

(a f

a0

). (18)

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In-situ measurement of the large strain response 185

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 6 Adherence energy Γ , average stress σ̄ in the cohesivezone and fibril elongation at debonding a f −a0 for peeling exper-iments at V = 0.1 mm s−1 and different peeling angles θ (a–c)and at θ = 90◦ and different peeling velocities V (d–f). Γ is

measured through the peeling force F , using Eqs. 1 and 2. a f isobtained from the fit of the tape backing profiles by Eq. 15. σ̄ isfurther obtained using Eq. 16

atwhich the adhesivematerial is strained in the debond-ing region (ε̇(s) is the local logarithmic strain).7 Fig-ure 7a shows that ¯̇ε is increasing by a factor less than 1.8

7 The evaluationof the average strain rate requires the calculationof the temporal dependence of the length a(t) of a typical fibrilduring the peeling. We should thus operate a change of referenceframe from the one of the laboratory (where the bending profileis steady and identified by our measurements of α(s)) to the localreference frame of the substrate. The evolution of the bendingprofile in time in the local reference frame should be described byanother slope function β(u, t) expressed in terms of the materialcurvilinear abscissa u. Since the substrate is moving at the samevelocity as the peeling velocity V , the two bending functionsare related by α(u + V t) = β(u, t). From the logarithmic strain

ε(s) = ln (a(s)/a0) = ln(√

(s − x)2 + (y + a0)2)

− ln(a0) of

the fibril attached at position s in the laboratory reference frame,we can then compute the local strain rate as ε̇(s) = 1

a(s)dads

∂s∂t =

Va(s)

dads , which after integration results in Eq. 18.

for θ ranging from 30◦ to 150◦ for V = 0.1 mm s −1,a factor comparable to the variations of the local strainrate ε̇(s) over the cohesive zone. In parallel, we seein Fig. 7b that ¯̇ε increases by more than three ordersof magnitude with V increasing from 3 · 10−6 to3 · 10−3 m s−1 for θ = 90◦. This last dependencyis well accounted for by a nearly linear power law¯̇ε ∼ V 1.06. The fact the relationship between ¯̇ε andV is almost linear means that the factor ln(a f /a0)/ lsin Eq. 18 is almost independent of the peeling velocity.This result is related to the fact the lengths ls and a f fol-low very small exponent power law of V (ls ∝ V−0.08

and a f ≈ a f − a0 ∝ V−0.07). In fact, lx and ly alsofollow power laws of V with very close small expo-nents (−0.08 and −0.075 respectively): the cohesivezone is self-similar to the first order, with just a weakdownscaling when V increases.

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186 R. Villey et al.

(a)

(b)

Fig. 7 Average strain rate ¯̇ε in the cohesive zone, obtained fromthe fitted tape backing profiles (using the model of Sect. 4) andfrom Eq. 18. ¯̇ε is represented a as a function of θ for V =0.1 mm s −1 and b as a function of V for θ = 90◦. The error baris the standard deviation of ε̇(s) in the cohesive zone

Since one can expect a direct influence of the strainrate of the adhesive material on the level of stress, wealso report in Fig. 8 σ̄ as function of ¯̇ε for all stud-ied couples of control parameters (θ, V ). These datareveal that σ̄ nearly follows a power law ¯̇ε1/4 whenV is varied, except for the data acquired at θ = 150◦and V = 0.1 mm s −1, which depart from all otherdata presented in this paper, probably due to incipientplastification of the backing as discussed before. Forthe other angles studied at this same velocity, the influ-ence of changing θ does not dramatically move awaythe data from the main power law, in agreement withthe limited influence of θ on σ̄ and on ¯̇ε (compared tothe one of V ). Therefore, one can say that the typicalstress in the debonding region σ̄ is to the first order con-trolled by the value of the mean strain rate ¯̇ε(θ, V ) tothe power ∼1/4, at least for peeling angles up to 120◦.At these angles, the role of θ in the relation betweenσ̄ and ¯̇ε remains unresolved, but it is undoubtedly ofsecond order.

A last parameter interesting to look at is the angle

ϕ f = arctan(a0+lyls−lx

)between the last fibril of length

a f and the substrate, as reported in Fig. 9. We shouldhighlight here that the values we report for ϕ f resultfrom the fit of the tape profile by the cohesive zone

Fig. 8 Average stress σ̄ (from Figs. 6b, e) as a function of theaverage strain rate ¯̇ε (from Fig. 7) in the debonding region, forall the tested couples (θ, V ) of control parameters

model and are not obtained from direct observationsof the fibrils inclination (which would nevertheless beclosely matching for θ ≤ 120◦). We find that ϕ f isremarkably constant with V (at least at θ = 90◦),which is consistent with the self-similar downscalingof the cohesive zone with increasing V . In parallel, ϕ f

decreases from about 80◦ at θ = 30◦ down to approx-imatively 50◦ for the highest peeling angle θ = 150◦(where the angle of the visible side fibrils with thesubstrate is around 65 − 75◦, a value sensibly higherthan the one obtained with our model fit, which dis-plays some limitations at very large peeling angles,as abovementioned). These features suggest a corre-lation between the maximum fibril elongation a f − a0and their inclination ϕ f , which we call the “angle atdebonding” of the fibrils.

Understanding the debonding criterion for a highlystretched soft fibril from a substrate is clearly a keyto modeling the adherence energy Γ (V, θ) of PSAs.While this remains a very subtle unsolved problemproblem (Yarusso 1999; Villey et al. 2015; Creton andCiccotti 2016) and is out of the scope of the presentwork, the availability of our measurements of a f , ¯̇εand ϕ f is very promising for future investigations oncustom PSAwith controlled rheology and surface con-ditions.

5.3 Beyond the cohesive zone model

The good agreement between the imaged tape pro-files and the fits suggests that extracting more infor-mation from these profiles than an average stress inthe debonding region may not be an easy task. Obtain-ing more refinements in the stress distribution indeedimplies to look at the minute differences between the

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In-situ measurement of the large strain response 187

Fig. 9 “Angle atdebonding” ϕ f between thesubstrate and the longestfibril (about to debond), afor V = 0.1 mm s −1 anddifferent peeling angles θ

and b for θ = 90◦ anddifferent peeling velocities

V . ϕ f = arctan(a0+lyls−lx

)is

obtained through the fit ofexperimental profiles by themodel of Sect. 4

(a)

(b)

fitted profiles and the experimental ones.Notice that theexact stress distribution can theoretically be accessedby computing the fourth derivative dy(4)/dx4 of thetape profile (Love 1944; Niesiolowski and Aubrey1981). However, since the typical size of the debond-ing region is in the hundreds of µm range, even get-ting a spatial stress distribution with a few resolvedvalues is difficult: getting the changes in dy(4)/dx4

with a resolution as low as several tens of µm wouldrequire a defectless tape (no dusts or scratches) at theµm scale, which is hardly reachable in practice. Thesame issue led Niesiolowski and Aubrey to upscalethe size of their adhesive tape in order to apply thismethod [using derivatives of the tape profile up to thefourth order to compute the forces acting on the tapebacking, see appendix 3 in Niesiolowski and Aubrey(1981)]. Eventually, it seems that only a few averagestress values can be inferred at best from thewhole tapeprofile.

A thorough examination of the minute differencesbetween the experimental profiles and their fit by ourmodel does nevertheless reveal one systematic featurethat can be used for further refinements, enabling toobtain more information about the actual stress distri-bution. A zoom at the very beginning of the debondingregion (see Fig. 10) indeed reveals a small zone wherethe tape displacement normal to the substrate y(s) isnegative, corresponding to a local vertical compressionof the adhesive layer.

Fig. 10 The inset represents the zoom of the basal region (bluerectangular box) of an experimental profile (θ = 90◦, V =100 µm s−1) after averaging over 540 frames. The entire profilewas fitted using the cohesive zone (CZ) model of Eqs. 13 and 14(red profile, with the CZ beginning at s = 0); the zone where thebacking displacement y corresponds to an adhesive deformationsmaller than 25% (i.e. |y| < a0/4 = 5 µm) was fitted using thelinear elastic foundation model of Eq. 19, providing an estimateof λβ (and thus of Y ) as a fitting parameter

This can be explained by considering that, beforereaching a plateau value σ̄ , the stress in the adhesivehas to build up from zero. The simplest model for thisincrease consists in considering the linear elastic behav-

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188 R. Villey et al.

iour of the adhesive at small strain, as proposed byKaelble (1959; 1960), which results in such a compres-sion zone before the adhesive is significantly stretched.Kaelble actually modeled the adhesive layer by a linearrate-dependent elastic foundation which, in our case,can be relevant to describe the initial part of the debond-ing region (before cavitation and fibrillation) where theadhesive is not too much strained. In this zone, Kaelblepredicts the following tape profile:

y = y0ex

λβ

[cos

(x

λβ

)+ K sin

(x

λβ

)],

with λβ = 4

√4E Ia0Yb

,

and K = 1 − λβ sin θ

r√2 − 2 cos θ − 0.5h cos θ + λβ sin θ

.

(19)

In this equation, h is the backing thickness and Y isthe adhesive effective tensile modulus [larger than theadhesive Young’s modulus because of incompressibil-ity and confinement, see discussion in Villey et al.(2015)].

We should remark that in Kaelble’s model the posi-tion x = 0 corresponds to the location where the nor-mal stress σ(x) = Y y/a0 in the adhesive reaches amaximum value σ0, which is also the location of adhe-sive debonding when σ0 reaches a critical value [we donot discuss the shear stress distribution here becauseit is not relevant for the peeling angles discussed inthis paper, see Villey et al. (2015)]. The same equation19 remains however valid and can be combined withour cohesive zone model if we assume that the posi-tion x = 0 corresponds to the transition between thelinear elastic zone (where the normal stress builds upin the confined adhesive) and the cohesive zone (wherethe stress reaches saturation in the fibrillated adhesive),i.e. that Y y0/a0 = σ̄ .

A first refinement of our model could thus simplyconsist in combining an elastic foundation with a cohe-sive zone, where in the foundation (−∞ < s ≤ 0) theadhesive response is linear elastic (σ(x) = σ(s) =Y y/a0) and where in the cohesive zone (0 ≤ s ≤ ls) itis plastic-like (σ(s) = σ̄ ). In such a model, the adher-ence energy is given by

Γ = a0σ̄ 2

2Y+ σ̄

(a f − a0 − y0

), (20)

which is simply the sum of the work to first stretchthe adhesive in the elastic foundation and then in thecohesive zone up to debonding.

Such a model combining the elastic foundation andthe cohesive zone certainly offers a better descriptionof the stress distribution within the adhesive, but itsrelevance to explain energy dissipation during peelingshould be further examined. To do so, we estimate theimportance of the elastic foundation contribution to theadherence energy in Eq. 20 compared to the contribu-tion of the cohesive zone. We thus need estimates ofσ̄ and Y , the former being given by fitting the wholeprofile with the model of Sect. 4 and the latter by fit-ting the detected profile in the small strain zone byEq. 19: in this zone, the characteristic length λβ of thespatially damped oscillation of the tape profile can befitted, giving access to Y = bλ4β/4E Ia0. Practically,the tape profile was fitted using Eq. 19 with K , y0 andλβ as independent fitting parameters, as it is done forexample in Fig. 10.

The systematic fitting of the averaged experimentalprofiles provides measurements of λβ , which reveal Yvalues in the hundreds of kPa to severalMPa range: thisis larger than the several tens of kPa expected for theunconfined adhesive (Amouroux et al. 2001), but it canbe explained by the confinement effect: an incompress-ible material uniaxially stretched from an initial con-fined geometry has a stiffer response than in an uncon-fined stretching test.

The contribution of the elastic foundation in theoverall energy budget is slightly decreasing withθ and globally increasing with V . More precisely,a0σ̄ 2/(2YΓ ) < 5% in all our experiments, exceptat V = 0.3 mm s−1 where it is around 8% and atV = 3 mm s−1 where it is around 30%.

This means that the experiments presented in thispaper correspond to the case where the stress plateaudominates the dissipation in the stretched adhesive, andthat the elastic foundation zone can safely be neglectedboth in the energy budget and when estimating thelevel of the stress plateau, except maybe at the highesttested peeling velocity V = 3 mm s−1. The model ofSect. 4 is however expected to become less relevant athigher velocities, where the maximum fibrils extensionmay become small enough so that the contribution ofthe elastic foundation zone becomes important in theoverall work used to stretch the adhesive. The morerefined model briefly described in this section couldtherefore be very useful to extract relevant local rheo-

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In-situ measurement of the large strain response 189

logical parameters and to explain energy dissipation atlarger peeling velocities than in the present investiga-tion, in order to give an overall complete picture of thepeeling mechanics of soft confined layers.

6 Conclusion

In this work, we measure experimentally the shape ofthe adhesive tape backing during steady peeling of atypical commercial PSA at controlled peeling angle θ

and velocity V with sufficient resolution to describethe debonding region. We model the tape backing byan elastica submitted to a constant force F at the peelingend and to a uniform cohesive stress σ̄ in the debond-ing region, where the adhesive material is stretchedand forms fibrils. We make the reasonable assump-tion that this cohesive stress is oriented in the fibrilsdirection. Fitting the experimental profiles with sucha model allows to extract the typical stress σ̄ in thedebonding region as well as the extension a f − a0 ofthe adhesivematerial at debonding,which are the effec-tive rheological ingredients that determine the energydissipated per peeled surface area Γ = σ̄

(a f − a0

)in

this model. This fitting procedure also allows to com-pute the strain rate ε̇ imposed to the adhesive fibrils andits average ¯̇ε in the debonding region. This model thusenables to use peeling as a sort of local rheological testgiving access to the average stress, the average strainrate and the maximum extensibility of the adhesive inits complex fibrillar debonding region for a given setof control parameters (θ, V ).

The model profiles fit properly the experimentalones down to the micrometric scale. We explain thisagreement by the fact the model average stress σ̄ is nottoo far from the actual stress distribution in the debond-ing region, since the stress-extension behavior of typ-ical PSA is known from probe tack tests (which has aquite similar local loading configuration) to exhibit alarge stress plateau, with similar typical stress values atcorresponding strain rates (Lindner et al. 2004, 2006).Moreover, except for very large peeling angles, the sizeof the cohesive zone and thus the fibrils extension isclose to what can be observed by directly looking atthe visible fibrils on the side of the tape. This is the firsttime to our knowledge that a curve displaying the typi-cal stress and maximum strain in the adhesive debond-ing region during peeling versus the peeling angle andvelocity (or versus the typical strain rate) is obtained

and interpreted in terms of large strain rheology of thefibrillated adhesive material.

This use of peeling as a local rheological test ofthe complex debonding region should provide a use-ful tool to link the adhesive formulation to the desireduser properties (i.e. adherence energy curves Γ (V, θ))through the intermediate parameters that are the mater-ial large strain rheological behaviour and fibril debond-ing conditions. While further studies on different for-mulations with well controlled large strain rheologiesand interface properties should be conducted to com-pletely examine this chain of influences, the paramet-ric study in this paper already examines the influenceof material’s large-strain rheology on energy dissipa-tion and thus resistance to peeling. Our study suggeststhat energy dissipation is linked to the peeling veloc-ity V mainly through the strain-rate dependency ofthe average stress within the adhesive, which seemsto be insensitive to the peeling angle θ , except maybeat the highest peeling angle, where other features suchas plasticity in the tape backing are likely to occur.Conversely, the fibril elongation at debonding a f − a0increases with θ , but only slowly decreases with V inthe explored velocity range. Finally and more specu-latively, the angular dependency of a f (and thus themain angular dependency of Γ ) could come from achange in the stress configuration at the fibrils extremi-ties due to the changing directionϕ f inwhich the fibrilsare pulled, shifting the critical elongation at which therelevant debonding criterion [stress or energy density,see discussion in Yarusso (1999)] is reached. In otherwords, our results suggest that longer fibril extensionscan be reached before debonding and thus more energycan be dissipated when these fibrils are pulled with anangle ϕ f smaller than 90◦ with respect to the substrateon which they are attached, provided the fibril attach-ment does not slide on the substrate (sliding being notobserved in the experiments presented in this paper).

Finally and more generally, a consistent model toexplain energy dissipation during peeling needs threeingredients: the equations of stress and displacementtransmission between the operator and the zone whereenergy is dissipated; the effective rheological behaviorof the adhesive in this zone; and a local debonding cri-terion. The modeling strategy presented in this paperaccounts for the first point. We use a simplified ver-sion of the local adhesive rheology (a constant effec-tive stress depending on the strain rate) to account forthe second point, a simplification allowed by the large

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190 R. Villey et al.

stress plateau exhibited by adhesives during probe tacktests. As for the third ingredient, ourmodel gives accessto observables that can shed some light on it, namelythe elongation a f − a0 and orientation ϕ f of the fibrilabout to debond and the typical strain rate experiencedby the fibrils.

Acknowledgements We thank B. Bresson and L. Olanier fortheir help in the conception of the experiments. We thank E.Barthel, M.-J. Dalbe, S. Santucci, L. Vanel, and D. Yarusso forfruitful discussions. This work has been supported by the FrenchANR through Grant #12-BS09-014.

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