Bending to kinetic energy transfer in adhesive peel front micro-instability

TL;DR

This paper investigates a microscopic instability during adhesive tape peeling, revealing how bending energy converts into kinetic energy and establishing a scaling law relating amplitude and period of the instability.

Contribution

It provides the first detailed experimental and theoretical analysis of micro-instability dynamics in adhesive peel fronts, linking bending energy to kinetic energy transfer.

Findings

Amplitude scales with period as A_mss ∝ T_mss^{1/3}

Bending modulus influences instability characteristics

Elastic bending energy converts into kinetic energy during peeling

Abstract

We report an extensive experimental study of a detachment front dynamics instability, appearing at microscopic scales during the peeling of adhesive tapes. The amplitude of this instability scales with its period as $A_{\text{mss}} \propto T_{\text{mss}}^{1/3}$, with a pre-factor evolving slightly with the peel angle $\theta$, and increasing systematically with the bending modulus $B$ of the tape backing. Establishing a local energy budget of the detachment process during one period of this micro-instability, our theoretical model shows that the elastic bending energy stored in the portion of tape to be peeled is converted into kinetic energy, providing a quantitative description of the experimental scaling law.

Equations

$$E_{B-r}+E_{S-g}=\Gamma A_{\text{mss}}+E_k,$$

$$A_{\text{mss}}={\left[\frac{2B/\mu }{1-\cos \theta }\right]}^{1/6}{d}^{1/3}{T}_{\text{mss}}^{1/3}.$$

Full text

The periodic velocity oscillations of the detachment front during the peeling of adhesive tapes constitute an archetypal example of a dynamical rupture instability. This stick-slip motion leads to a screechy sound that everyone has experienced, when peeling-off packing tape. However, despite a large number of studies Gardon1963 ; Aubrey1969 ; Racich1975 ; Barquins1986 ; Maugis1988 ; Barquins1997 ; Ryschenkow1996 ; Gandur1997 ; Ciccotti2004 ; De2004 ; Cortet2007 ; De2008 , this instability is not fully understood and still causes industrial problems, with deafening noise levels, and damages to both adhesives and peel systems.

The effective fracture energy of adhesive-substrate joints can decrease over certain ranges of peel front velocity Aubrey1969 ; Racich1975 ; Barquins1986 ; Maugis1988 . In such unstable condition, for which less energy is required for the crack front to grow faster, a transition from a quasi-static rupture mode to a dynamic one occurs, as for frictional interfaces Baumberger1994 ; Rubinstein2004 ; Svetlizky2014 . During the rapid slip phases, the dynamical mode of failure is likely to give rise to small scale spatio-temporal front instabilities Fineberg1999 . Indeed, ultra-fast imaging could unveil that the peel front locally advances by steps in the main peel direction as a result of the propagation of a dynamic fracture kink in the transverse direction, at spatio-temporal scales much smaller than the macroscopic stick-slip Thoroddsen2010 ; Marston2014 ; Dalbe2015 : the kink occurs periodically at ultrasonic frequencies with an amplitude of a few hundred microns, not only during the slip phase of the macro-instability, but also, for imposed peel velocities in a finite range beyond the macro-stick-slip domain where the peeling is regular at macroscopic scales Dalbe2015 .

Interestingly, this micro-instability of the peel front characterized by the side-ways propagation of fracture kinks share similarities with other physical processes, as for instance the local contact lines dynamics on textured surfaces Gauthier2013 , or the dislocations motion in the yielding of crystalline materials Dislocations . While it was shown that those transverse cracks are accompanied by cycles of load and release of the elastic bending energy stored in the tape backing in the vicinity of the peel front Dalbe2015 , the physical origin of the micro-instability and its interaction with the macroscopic one remains an open issue.

In this Letter, we provide a detailed experimental study of this micro-instability, varying systematically the peeled length $L$, the peel angle $\theta$, the lineic mass $\mu$ and bending modulus $B$ of the ribbon, over a wide range of driving peel velocities $V$. Thanks to a large data statistics, we show that the micro-instability amplitude scales with its period as $A_{\text{mss}}\propto T_{\text{mss}}^{1/3}$, with a pre-factor that increases with the bending modulus of the tape backing. We demonstrate that the bending elastic energy of the ribbon released during each micro-slip is converted into kinetic energy, allowing a quantitative prediction of this scaling law.

We peel a 3M Scotch${}^{\mbox{\scriptsize{\textregistered}}}$ 600 adhesive tape from a transparent flat substrate by winding its extremity at a constant velocity $V$ with a brushless motor. Changing the relative position of the substrate and peeling motor, we can easily vary the peel angle $\theta$ and ribbon length $L$. The peeling of this tape (a polyolefin blend backing b=19\leavevmode\nobreak\mm wide, $e=34\leavevmode\nobreak\ \mu$m thick, tensile modulus E=1.41\leavevmode\nobreak\GPa, density \mu=8.10^{-4}\leavevmode\nobreak\kg/m) coated with a 15 $\mu$m synthetic acrylic adhesive layer has been widely studied Barquins1986 ; Maugis1988 ; Barquins1997 ; Thoroddsen2010 ; Cortet2013 ; Marston2014 ; Dalbe2014 ; Dalbe2015 ; Dalbe2014b ; Dalbe2016 . In contrast with those studies, for each experiment, using a scalpel, we carefully extract two layers of the adhesive tape from its original roller. We attach them on a transparent plate and then, perform the peeling at the interface between those two layers. The release side of the adhesive tape backing constitutes the top part of the substrate of our peel experiments. Thanks to the homogeneous properties of the commercial roller adhesive layer, this protocol improves the reproducibility of our experiments, by avoiding a pre-peeling which may damage the adhesive.

A Photron SA5 fast camera with a macro-lens images a small portion of the peel front through the transparent substrate with a resolution of 9.8 $\mu$m/pixel, at a rate of $175\,000$ or $300\,000$ fps, for $640\times 56$ px2 and $832\times 24$ px2 images respectively. Analyzing the grey levels of each image, we extract the detachment front longitudinal position $x$ at a given transverse position $z$. During an acquisition, this front typically advances of a few mm, so that $L$ and $\theta$ can be considered constant (varying less than $5\%$). A sketch of the setup can be found in supplemental material together with typical recorded videos.

In the driving velocity range $[V_{a},V_{d}]$ thresholdmacro , for which the peel force decreases, the detachment front displays the classical stick-slip instability, with regular velocity oscillations at the millisecond timescale, and millimetric slips related to cycles of loading and release of the stretching energy stored in the whole peeled tape. The oscillating front velocity is thus different from the driving velocity $V$ imposed at the extremity of the peeled tape, which is the control parameter of the experiment. When the mean front velocity (measured at the ms timescale) becomes larger than $v_{a}\simeq 1$ m/s DeZotti2018 , the peel front advances by the propagation of fracture kinks, across the tape width, in the transverse direction $z$, at very high velocities, from $650$ m s*-1* up to $900$ m s*-1* Dalbe2015 . Those dynamic transverse cracks occur during the slip phase of the macro-instability, but also, permanently as shown in Fig. 1, for driving velocities above $V_{d}$, the disappearance threshold of the macro-instability. This micro-instability finally disappears as well, when the mean peel front velocity is above $v_{d}\simeq 20$ m/s Dalbe2015 . Figure 1 gives a typical example of the local front position time series for an experiment at $V=1.8$ m/s, $L=50$ cm, $\theta=90^{\mathrm{o}}$. While the average front velocity measured at the ms timescale is equal to the driving velocity $V$ (regular peeling without macro-stick-slip), at shorter timescales, we observe a staircase dynamics with sudden jumps of amplitude $A_{\text{mss}}\simeq 170$ $\mu$m separated by periods of rest of $T_{\text{mss}}\simeq 100$ $\mu$s.

From the peel front temporal evolution measured for each experiment, we could detect several thousands of micro-stick-slip events. For each of them, we extract the period of rest $T_{\text{mss}}$ preceding a micro-slip of amplitude $A_{\text{mss}}$. In Fig. 2 (top), we display $A_{\text{mss}}$ as a function of $T_{\text{mss}}$, averaged in logarithmic bins. We gather here data of numerous experiments: for a fixed peel angle $\theta=90^{\mathrm{o}}$ and several peeled ribbon length $L$ (inset), and changing this angle $\theta$ while keeping $L$ fixed to $50$ cm (main panel). We clearly observe that $A_{\text{mss}}$ increases with $T_{\text{mss}}$, following a power-law scaling with an exponent close to $1/3$, independent of the peeled length $L$ (inset). On the other hand, both the range and pre-factor of the scaling law evolve slightly with the peel angle (inset, bottom panel), but seemingly not the power-law exponent.

In order to evaluate the role of the elastic bending energy, stored locally in the vicinity of the peel front, we have studied the impact of the ribbon bending modulus $B$. Superimposing up to 4 layers of the 3M Scotch${}^{\mbox{\scriptsize{\textregistered}}}$ 600 rigid backing (cleansed of its adhesive coating) with a rigid glue (using $e_{L}=26\pm 10$ $\mu$m layers of Loctite${}^{\mbox{\scriptsize{\textregistered}}}$ 406), we could increase the bending modulus of our adhesive tape by 2 orders of magnitude. Indeed, considering that the n-layer backing of thickness $h_{n}=ne+(n-1)e_{L}$ has the same tensile modulus $E$ than the 3M Scotch${}^{\mbox{\scriptsize{\textregistered}}}$ 600 ribbon, its bending modulus can be estimated as $B=Eh_{n}^{3}b/12$. As a result, we find a systematic increase in the micro-slips size $A_{\text{mss}}$ with $B$, as shown in Fig. 3 (inset), preserving nevertheless the $T_{\text{mss}}^{1/3}$ scaling.

We discuss now the evolution of the adhesive tape close to the peel front, in order to obtain a local energy balance of the detachment process during a microscopic stick-slip cycle (per unit width of the peel front). During the micro-stick phase, the adhesive layer is stretched, while the ribbon is bent locally at the vicinity of the static detachment front. We assume that a micro-slip of size $A_{\text{mss}}$ occurs suddenly, when the glue is stretched up to a critical length $d$. During this fast interfacial crack propagation, both the elastic bending energy of the ribbon $E_{B-r}$ and the stretching energy of the glue $E_{S-g}$ stored during the micro-stick phase are released. Therefore, we can write the local energy balance during a micro-stick-slip:

[TABLE]

where $\Gamma$ is the effective fracture energy of the adhesive-substrate joint, and, $E_{k}$ corresponds to the excess of elastic energy released, converted into kinetic energy that the tape locally acquires. Such energy balance is similar to the one proposed by Mott Mott1948 , generalizing Griffith’s criterion Griffith1920 to describe dynamic rupture processes.

The effective fracture energy $\Gamma$ has been theoretically related to the energy needed to deform the adhesive layer up to a critical strain at which it debonds Kaelbe1964 ; Gent1969 ; Villey2015 . Recent experiments with polyacrylate adhesives have shown that this energy is indeed proportional to the integral of the non-linear rate-dependent stress-strain curve of the confined glue Chopin2018 . The critical elongation of the glue at debonding $d$ is thus a crucial parameter in the determination of the peeling energy $\Gamma$ (the value of $d$ and its dependencies with material, geometrical and dynamical parameters is still an open issue Chopin2018 ; Villey2017 ; Creton2016 ). In this context, we can consider that the energy accumulated in the deformation of the glue $E_{S-g}$ is completely dissipated when it debonds and a micro-slip of size $A_{\text{mss}}$ occurs, i.e. $\Gamma A_{\text{mss}}=E_{S-g}$. As a consequence, the local energy balance (1) leads to the transfer of the ribbon bending energy $E_{B-r}$ into an increase of kinetic energy $E_{k}$ that the tape locally gains during a micro-slip, $E_{B-r}=E_{k}$.

Assuming that the tape is bent over a length scale equal to the micro-slip size $A_{\text{mss}}$, just before its triggering at the critical elongation of the glue $d$, with a radius of curvature $R_{0}\simeq A_{\text{mss}}^{2}/2d$, the order of magnitude of the bending energy released during a micro-slip writes $E_{B-r}\simeq\frac{1}{2}BA_{\text{mss}}/R_{0}^{2}\simeq 2Bd^{2}/A_{\text{mss}}^{3}$. In our model, the micro-slip duration is fixed by the time scale for the release of this local bending energy $E_{B-r}$, controlled, a priori, by the bending waves period $\tau=\frac{\lambda^{2}}{2\pi}\sqrt{\frac{\mu}{B}}$ Landau of wavelength $\lambda=A_{\rm mss}$. Considering a single tape layer with $\lambda\sim 150-200\leavevmode\nobreak\ \mu$m gives a time scale $\tau\sim 0.3-0.6\leavevmode\nobreak\ \mu$s, much smaller than the micro-stick-slip period $T_{\rm mss}$, in agreement with our measurements. Therefore, in this approach, the fracture kink transverse propagation proceeds at the group velocity $v_{g}=2\lambda/\tau$ of bending waves of wavelength $\lambda=A_{\rm mss}$. The estimated velocities ($650\rightarrow 900$ m/s) are also in excellent agreement with the experimental reported values Dalbe2015 .

On the other hand, the increase of kinetic energy that the tape locally gains during a micro-slip is $E_{k}=\frac{1}{2}\mu A_{\text{mss}}\left(A_{\text{mss}}/T_{\text{mss}}\right)^{2}2(1-\cos\theta)$. In this formula, the factor $2(1-\cos\theta)$ results from the motion of the tape just beyond the curved region, which is a combination of a translation in the direction $\theta$ at $V_{\text{mss}}=A_{\text{mss}}/T_{\text{mss}}$ and a translation at the same velocity in the direction of the peel front motion Dalbe2016 . Finally, the transfer of bending to kinetic energy $E_{B-r}=E_{k}$ allows to link amplitude and period of the micro-stick-slips:

[TABLE]

The predictions of Eq. (2) are in excellent agreement with our measurements: the power-law exponent of $1/3$ between $A_{\text{mss}}$ and $T_{\text{mss}}$, the independence with the peeled length $L$ and the weak dependence with the peel angle $\theta$ and ribbon bending modulus $B$ through the $1/6$ exponent reproduce the various measured dependencies reported in Figs. 2 and 3. Indeed, normalizing the amplitude $A_{\text{mss}}$ by the pre-factor $\left[2B/(\mu(1-\cos\theta))\right]^{1/6}$, in Eq. (2), which accounts for the changes in mass and bending modulus of the ribbon, as well as the different peel angle used, tends to gather the data on a master curve. The data collapse is particularly convincing for the samples with different backing thickness as shown in Fig. 3. Moreover, following Eq. (2), we could fit the large amount of data reported in Fig. 2, for the various experiments performed at different $(L,\theta)$ with only one backing, to extract the free parameter $d$, corresponding to the critical elongation at which the glue detaches, $d=5$ $\mu$m. Such order of magnitude is compatible with our direct side observations of the adhesive tape unstable peeling. Strikingly, with this single value for $d$, we can finally describe quantitatively our various experiments with samples of different lineic mass $\mu$ and bending modulus $B$, as shown by the dashed line reported in Fig. 3.

To conclude, thanks to an extensive experimental study in addition to a careful preparation of adhesive-substrate joints with the exploration of several tape backing bending modulus, we have been able to unveil the precise characteristics of the detachment front micro-stick-slip dynamics, appearing when peeling an adhesive tape at high velocities. A local energy balance of the detachment process shows that the elastic bending energy stored in the tape region that will detach during the micro-slip is converted into a kinetic energy increase of the peeled tape during a micro-stick-slip cycle. Our model allows a quantitative description of the observed scaling-law linking amplitudes and periods of the micro-instability, and in particular its dependency with the peeling angle, as well as with the bending modulus and lineic mass of the ribbon. This energy transfer arising from the assumption of the complete dissipation of the energy stored in the deformation of the adhesive layer when it detaches, by elastic hysteresis, highlights that the rapid micro-slip corresponds to a specific dynamic rupture mode of propagation of the detachment front. In our scenario, the elastic stretching energy stored in the whole peeled ribbon during the stick phase of the macro-instability and released during the macro-slip phase constitutes an energy reservoir for the micro-stick-slip cycles and more precisely for the reloading of the tape local bending during the micro-sticks. The release of the stretching energy therefore proceeds by quanta of elastic bending energy of the ribbon close to the peel front.

Nevertheless, the physical origin of the kinked detachment front propagation in the direction transverse to the main peeling direction still needs to be uncovered. A possible explanation could come from a local enhancement of the mechanical energy release rate, which has been shown AddaBedia2006 ; Vilmin2010 to be at the origin of the elastic fingering instability when peeling quasi-statically a confined elastomer Ghatak2000 .