## Abstract

The debonding of pressure sensitive adhesives (PSA) is a classical example of the difficult and unsolved issue of fracture in soft viscoelastic confined materials. The presence of a complex debonding region where the adhesive undergoes cavitation and the very large strain of a spontaneously formed fibrillar network has defied many modeling attempts over the past 70 years. We present here a novel technique to provide an accurate measurement of the local large strain response of the fibrillar debonding region during the steady-state peeling of a well known commercial adhesive over a wide range of peeling velocity and angle. The technique is based on high resolution imaging of the debonding region during peeling and is coupled to a cohesive zone modeling of the adhesive interaction between the flexible tape backing and the rigid substrate. The resulting database provides a strong ground for validating and further developing models (Villey et al. in Soft Matter 11:3480–3491, 2015) aiming to capture the effects of both geometry and non-linear adhesive rheology on the exceptional adherence energy of PSAs.

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## Notes

This PSA is made of an acrylic adhesive layer of thickness \(a_0 \simeq 20~\upmu \)m coated on a UPVC backing of comparable thickness, Young’s modulus \(E\simeq 2.9\) GPa and width \(b=19\) mm.

We find that this modulus changes significantly between different rollers, and even between distant portions of the same roller, with an average value \(EI=(3.7 \pm 1.7)~10^{-8}\) N m\(^2\). This value agrees with the estimate that can be made from the expression \(EI=Ebh^3/12\), where

*E*is the Young’s modulus and*h*the thickness of the tape backing, using an independent tensile test giving \(Ebh = 1100 \pm 200\) N and microscopic observations of the thickness providing \(h=20\pm 5\,\upmu \)m. An attentive study reveals that the variations in bending modulus*EI*can actually be attributed to small variations of the tape backing thickness of amplitude \(\pm 3\) µm, since \(EI\propto h^3\). In any case, in the present study the bending modulus*EI*will enter the problem only through the parameter \(r=\sqrt{EI/F}\), which will be directly measured for each experiment through elastica fits of the peeled tape outside of the debonding region.The validity of this assumption is limited to the inextensible tape approximation of our model, that follows the arguments provided in Sect. 3.

Multiplying Eq. 7 by \(\alpha '\) and integrating from 0 to \(l_s\) yields

$$\begin{aligned} \frac{8FX^2}{\left( 1+X^2\right) ^2}&+F\left[ \cos (4\arctan {X})-\cos \theta \right] \\&-\bar{\sigma } b \int _0^{l_s}\left[ \cos \left( \alpha (s)\right) \int _s^{l_s}\sin \left( \varphi (t)\right) dt\right] {\alpha '(s)} ds\\&-\bar{\sigma } b \int _0^{l_s}\left[ \sin \left( \alpha (s)\right) \int _s^{l_s}\cos \left( \varphi (t)\right) dt\right] {\alpha '(s)} ds=0. \end{aligned}$$After trigonometric simplifications of \(\cos (4\arctan (X))\) and integration by parts of the two integrals, we get

$$\begin{aligned} \frac{F}{b}\left( 1-\cos \theta \right)&=\bar{\sigma }\int _0^{l_s}\left\{ \sin \left( \alpha (s)\right) \sin \left( \varphi (s)\right) \right. \\&\quad \left. +\,\left[ 1-\cos \left( \alpha (s)\right) \right] \cos \left( \varphi (s)\right) \right\} ds. \end{aligned}$$The integrand can be rewritten as

$$\begin{aligned} \sin \varphi \sin \alpha +\left( 1-\cos \alpha \right) \cos \varphi&=\frac{\left( y+a_0\right) \frac{dy}{ds}+\left( s-x\right) \frac{ds-dx}{ds}}{\sqrt{(y+a_0)^2+(s-x)^2}}\\&=\frac{da(s)}{ds} \end{aligned}$$where \(a(s)=\sqrt{(y+a_0)^2+(s-x)^2}\) is the length of the fibril attached to the tape backing at curvilinear position

*s*. This finally leads to Eq. 12. One can also demonstrate Eq. 12 by computing the work associated to a small advance of the peeling front (see Online Resource 2).Once we know \(\alpha (s)\), \(l_s\) and \(s_0\) , \(a_f\) can be determined by Eq. 15. Curvilinear abscissa is eventually redefined as \(s\rightarrow s-s_0\) after the fit, in order to set the beginning of the cohesive zone to \(s=0\).

The bias gets larger for the largest angle \(\theta = 150^\circ \), which will be discussed later.

The evaluation of the average strain rate requires the calculation of the temporal dependence of the length

*a*(*t*) of a typical fibril during the peeling. We should thus operate a change of reference frame from the one of the laboratory (where the bending profile is steady and identified by our measurements of \(\alpha (s)\)) to the local reference frame of the substrate. The evolution of the bending profile in time in the local reference frame should be described by another slope function \(\beta (u,t)\) expressed in terms of the material curvilinear abscissa*u*. Since the substrate is moving at the same velocity as the peeling velocity*V*, the two bending functions are related by \(\alpha (u+Vt)=\beta (u,t)\). From the logarithmic strain \(\varepsilon (s)=\ln \left( a(s)/a_0\right) =\ln \left( \sqrt{(s-x)^2+(y+a_0)^2}\right) -\ln (a_0)\) of the fibril attached at position*s*in the laboratory reference frame, we can then compute the local strain rate as \(\dot{\varepsilon }(s)=\frac{1}{a(s)}\frac{da}{ds}\frac{\partial s}{\partial t}=\frac{V}{a(s)}\frac{da}{ds}\), which after integration results in Eq. 18.

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## Acknowledgements

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

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Villey, R., Cortet, PP., Creton, C. *et al.* In-situ measurement of the large strain response of the fibrillar debonding region during the steady peeling of pressure sensitive adhesives.
*Int J Fract* **204**, 175–190 (2017). https://doi.org/10.1007/s10704-016-0171-1

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DOI: https://doi.org/10.1007/s10704-016-0171-1