# Intermittent Peel Front Dynamics and the Crackling Noise in an Adhesive Tape

Jagadish Kumar1 Rumi De2 G. Ananthakrishna1 1Materials Research Centre, Indian
Institute of Science, Bangalore 560012, India
2 Department of Materials and Interfaces, Weizmann Institute of Science,
Rehovot 76100, Israel

###### Abstract

We report a comprehensive investigation of a model for peeling of an adhesive
tape along with a nonlinear time series analysis of experimental acoustic
emission signals in an effort to understand the origin of intermittent peeling
of an adhesive tape and its connection to acoustic emission. The model
represents the acoustic energy dissipated in terms of Rayleigh dissipation
functional that depends on the local strain rate. We show that the nature of
the peel front exhibits rich spatiotemporal patterns ranging from smooth,
rugged and stuck-peeled configurations that depend on three parameters,
namely, the ratio of inertial time scale of the tape mass to that of the
roller, the dissipation coefficient and the pull velocity. The stuck-peeled
configurations are reminiscent of fibrillar peel front patterns observed in
experiments. We show that while the intermittent peeling is controlled by the
peel force function, the model acoustic energy dissipated depends on the
nature of the peel front and its dynamical evolution. Even though the acoustic
energy is a fully dynamical quantity, it can be quite noisy for a certain set
of parameter values suggesting the deterministic origin of acoustic emission
in experiments. To verify this suggestion, we have carried out a dynamical
analysis of experimental acoustic emission time series for a wide range of
traction velocities. Our analysis shows an unambiguous presence of chaotic
dynamics within a subinterval of pull speeds within the intermittent regime.
Time series analysis of the model acoustic energy signals is also found to be
chaotic within a subinterval of pull speeds. Further, the model provides
insight into several statistical and dynamical features of the experimental AE
signals including the transition from burst type acoustic emission to
continuous type with increasing pull velocity and the connection between
acoustic emission and stick-slip dynamics. Finally, the model also offers an
explanation for the recently observed feature that the duration of the slip
phase can be less than that of the stick phase.

###### pacs:

83.60.Df, 05.45.-a, 05.45.Tp, 62.20.Mk

##  I Introduction

Adhesive tapes are routinely used in a variety of situations including daily
usage as stickers, in packing and sealing. Yet, day-to-day experiences like
intermittent peeling of an adhesive tape and the origin of the accompanying
audible noise have remained ill understood. This may be partly attributed to
the fact that adhesion is a highly interdisciplinary subject involving diverse
but interrelated physical phenomena such as intermolecular forces of
attraction at the interface, mechanics of contact, debonding and rupture,
visco-plastic deformation and fracture Kendall00 , and frictional dissipation
which operates during peeling Kendall00 ; Urbakh04 ; Persson . Yet another
reason is that most information on adhesion is obtained from quasistatic or
near stationary conditions. Apart from scientific interest, understanding the
intermittent peel or the stick-slip process has relevance to industrial
applications as well. For example, optimizing production schedules that
involve pasting or peeling of an adhesive tapes at a rapid pace in an assembly
line requires a good understanding of stick-slip dynamics. Moreover, insight
into the time dependent and dynamical aspects of adhesion is expected to be
important in design of adhesives with versatile properties required in variety
of applications, in understanding the mechanisms leading to the failure of
adhesive joints as also in understanding biologically relevant systems such as
the geckoJagota07  or reorientation dynamics of cells Rumi07 .

Adhesion tests are essentially fracture tests designed to study adherence of
solids and generally involve normal pulling off and peeling. Such experiments
can be performed under quasistatic or near-stationary and nonequilibrium
conditions as well. The latter kind of experiments demonstrate the rate
dependence of adhesive properties. It is this rate dependence and the inherent
nonlinearity that leads to a variety of instabilities. These kinds of peeling
experiments are comparatively easy to setup in a laboratory. Moreover, the set
up also allows one to record unusually long force waveforms and AE signals
that should be helpful in extracting useful information on the nonlinear
features of the system.

One type of peeling experiment that yields dynamical information is carried
out with an adhesive tape mounted on a roller subjected to a constant pull
velocity MB ; CGVB04 . Peeling experiments have also been performed under
constant load conditions BC97 ; CGVB04 . At low pull velocities, the velocity
of the contact point $v$ keeps pace with the imposed velocity $V$. The same is
true at high velocities as well. However, there is an intermediate regime of
traction velocities where the peeling is intermittent. Peeling in this regime
is accompanied by a characteristic audible noise MB ; BC97 ; CGVB04 . It must
be stressed that these two stable dissipative branches refer to stationary
branches. Even so, the stick-slip dynamics observed in the intermediate region
of pull velocities has been attempted by assuming an unstable branch
connecting the two stable branches. The strain energy release rate shows a
power law for low velocities with an exponent around 0.3. The high velocity
branch also shows a power law but with a much higher exponent value of about
5.5 MB . The low velocity branch is known to arise from viscous dissipation
and that at high velocity corresponds to fracture. These studies report a
range of wave forms starting from saw tooth, sinusoidal or even irregular wave
form that has been termed ’chaotic’ MB . More recently, the dynamics of the
peel point has been imaged as well Cortet .

Stick-slip processes are usually observed in systems subjected to a constant
response where-in the force developed in the system is measured by dynamically
coupling the system to a measuring device. The phenomenon is experienced
routinely, for example, while writing with chalk piece on a black board,
playing violin or walking down a staircase with the hand placed on the hand-
rail. A large number of studies on stick-slip dynamics have been reported in
systems ranging from atomic length scales, for instance, stick-slip observed
using atomic force microscope soc04  to geological length scales like the
stick-slip of tectonic plates causing earthquakes BK ; BKC . A few well known
laboratory scale systems are - sliding friction Heslot ; Urbakh04 ; Persson
and the Portevin-Le Chatelier (PLC) effect PLC ; GA07 , a kind of plastic
instability observed during tensile deformation of dilute alloys GA07 ; Anan04
, to name only two. Most stick-slip processes are characterized by the system
spending a large part of the time in the stuck state and a short time in the
slip state. This feature is observed both in experiment and in models. See for
instance Heslot ; soc04 ; BK . A counter example where the time spent in the
stuck is less than that in the slip state (observed at high applied strain
rates) is the PLC effect PLC . These studies show that while the physical
mechanisms that operate in different situations can be quite varied Persson ,
in general, stick-slip results from a competition among the inherent internal
relaxational time scales GA07 ; Anan04  and the applied time scale. In the
case of peeling, one identifiable internal relaxation time scale is the
viscoelastic time scale of the adhesive. Other relevant time scales that may
be operative need to be included for a proper description of the dynamics. All
stick-slip systems are governed by deterministic nonlinear dynamics.

Models that attempt to explain the dynamical features of stick-slip systems
use the macroscopic phenomenological negative force response (NFR) feature as
an input although the unstable region is not accessible. This is also true for
models dealing with the dynamics of the adhesive tape including the present
work. In this context, it must be stated that there is no microscopic theory
that predicts the negative force-velocity relation in most stick-slip
situations except in the case of the PLC effect where we have provided a
dynamical interpretation of the negative strain rate sensitivity of the flow
stress Anan82 ; Rajesh  (see below). There are several theoretical attempts to
model the stick-slip process observed during peeling of an adhesive tape.
Maugis and Barquin MB , were the first to write down a model set of equations
suitable for the experimental situation and to carry out approximate dynamical
analysis. These equations were later modified and a dynamical analysis of
these equations was reported HY1 . However, the stick-slip oscillations were
not obtained as a natural consequence of the equations of motion HY2 ; CGB .
Indeed, these equations are singularRumi04 . Subsequently, we devised a
special algorithm to solve these differential algebraic equations (DAE) HLR ;
Rumi04 . This algorithm allows for dynamical jumps across the two stable
branches of the peel force function. This was followed by converting the DAE
into a set of nonlinear ordinary differential equations (ODE) by including the
missing kinetic energy of the stretched tape thereby lifting the singular
nature of the DAE Rumi04 ; Rumi05 . Apart from supporting dynamical jumps, the
ODE model exhibits rich dynamical features. However, all these studies discuss
only contact point dynamics while the tape has a finite width. The ODE model
has been extended to include the spatial degrees of freedom that is crucial
for describing the dynamics of the peel front as also for understanding the
origin of acoustic emission Rumiprl ; Jagdish08 .

Acoustic emission is commonly observed in an unusually large number of systems
such as seismologically relevant fracture studies of rock samples Scholz68a ;
Lockner96 ; Sam92 , martensite transformation vives ; rajeevprl ; kalaprl ,
micro-fracturing process Petri94 , volcanic activity Diodati , collective
dislocation motion Miguel ; Weiss  etc. The general mechanism attributed to AE
is the abrupt release of the stored potential energy although the underlying
mechanisms triggering AE are system specific. The nondestructive nature of the
AE technique has been useful in tracking the microstructural changes during
the course of deformation by monitoring the AE signals. For instance, it is
used in fracture studies of rock samples Lockner96  and more recently, a
similar approach has been used in understanding collective behavior of
dislocations Weiss . In both these cases, multiple transducers are used to
locate the hypocenters through an inversion process of arrival times Lockner96
; Weiss . In the latter case, by analysing the dislocations sources generating
AE signals, the study establishes the fractal nature of the collective motion
of dislocations. (In contrast to these dynamical studies, most studies on AE
Petri94 ; Diodati ; Miguel  are limited to compiling the statistics of the AE
signals in an effort to find experimental realizations of self-organized
criticality Bak .) However, in the case of peeling, using multiple transducers
is far from easy and only a single transducer is used leading to scalar time
series. In such situations, dynamical information is traditionally recovered
using nonlinear time series analysis GP ; HKS ; KS . However, a major
difficulty arises in the present case due to a high degree of noise present
and the associated difficulties involved in curing the noise content.

Despite large number of experimental investigations and to a lesser extent
model studies, several issues related to intermittent peeling and the
associated acoustic emission remain ill understood. For instance, there are no
models (even in the general area of stick-slip) which show that the duration
of the stick phase can be equal to or even less than that of the slip phase
Cortet , a feature which is quite unlike conventional stick-slip dynamics.
From a dynamical point of view, this is also suggestive of the existence of at
least three time scales. The model represents the acoustic energy in terms of
the Rayleigh dissipation functional that depends on the local strain rate of
the peel front and thus is sensitive to the nature of the peel front dynamics.
While preliminary results of the model Rumiprl ; Jagdish08  based on a small
domain of parameters were encouraging, no systematic study of the influence of
all the relevant time scales on the dynamics of the peel front was carried
out. In particular, while the nature of experimental AE signals changes with
the traction velocity, the study of the influence of pull speed on internal
relaxational mechanisms, the consequent peel front dynamics and its
relationship with the acoustic energy was not studied either.

The principal objective of the present study is to understand the various
contributing mechanisms to the intermittent peel process and its connection to
acoustic emission. The objective is accomplished by carrying out a systematic
study of the influence of the three internal relaxational time scales namely
the two inertial time scales of the tape mass and the roller inertia, and
dissipative time scale of the peel front. In particular, we report the
influence of the experimentally relevant pull velocity (covering the entire
range) on the peel front dynamics. These studies show that the model exhibits
rich spatiotemporal peel front patterns (including the stuck-peeled
configurations that mimic fibrillar patterns seen in experiments) arising due
to the interplay of the three time scales. Consequently, varied patterns of
model acoustic signals are seen. Another consequence of the inclusion of the
three time scales is that it explains the recent observation that the duration
of the slip phase can be larger than that of the stick-phase Cortet .
Interestingly, the model studies show that it is possible to establish a
correspondence between the various types of model acoustic energy profiles
with certain peel front patterns. More importantly, the study shows that even
as the acoustic energy dissipated is the spatial average of the local strain
rate, it can be noisy suggesting the possible deterministic origin of the
experimental acoustic signals. Here, we report a detailed analysis of the
statistical and dynamical analysis of the experimental AE signals. The study
shows that while the intermittent peeling is controlled by the peel force
function, acoustic emission is controlled by the dynamics of the peel front
patterns that determine the local strain rate. This coupled with a comparative
study of a comprehensive nonlinear time series analysis (TSA) of the
experimental AE signals for a wide range of traction velocities supplemented
by a similar study on the model acoustic energy time series provides
additional insights into the connection between AE signals and stick-slip
dynamics. In particular, the model displays the recently observed experimental
feature that the duration of the slip phase can be more than that of the stick
phase with increase in the pull velocity. Finally, the model studies together
with the dynamical analysis of the model acoustic signal provide a dynamical
explanation for the changes in the nature of the experimental AE signal in
terms of the changes in the peel front patterns.

Figure 1:  (a) (Color online) A schematic representation of the experimental
setup. (b) Plot of $\phi(v^{s})$ as a function of $v^{s}$.

##  II The model

A typical experimental set up consists of an adhesive tape mounted on a
roller. The tape is pulled at a constant pull velocity using a motor. A
schematic representation of the set up is shown in Fig. 1(a). The axis of the
roller passes through the point O into the plane of the paper. The drive motor
is positioned at O′. Let the distance between $O$ and $O^{\prime}$ be denoted
by $l$. $P$ is the contact point on the peel front $PQ$. Let the peeled length
of the tape $PO^{\prime}$ be denoted by $L$. Several geometrical features can
be discussed using a projection on to the plane of the paper. Let the angle
between the tangent to the contact point $P$ and $PO^{\prime}$ be denoted by
$\theta$ and the angle $\angle{POO^{\prime}}$ by $\alpha$. Then, from the
geometry of the Fig. 1(a), we get $L\ {cos}\,\theta=-l\ {sin}\,\alpha$ and $L\
{sin}\,\theta=l\ {cos}\,\alpha-R$ where $R$ is the diameter of the roller
tape. Let the local velocity of the peel point be denoted by $v$ and the
displacement (from a uniform stuck state) of the peel front by $u$. Then, the
pull velocity has to satisfy

$V=v+\dot{u}+R\ {\rm cos}\ \theta\ \dot{\alpha}.$ (1)

As the peel front has a finite width, we define the corresponding quantities
along the peel front coordinate $y$ (i.e., along the contact line) by
$v(y),\theta(y)$ and $\alpha(y)$. Then as the entire tape width is pulled a
constant velocity, the above constraint generalizes to

$\displaystyle{1\over b}\int^{b}_{0}\big{[}V-v(y)-\dot{u}(y)-R\ \
\dot{\alpha}(y)\ \ {\rm cos}\ \theta(y)\big{]}dy=0,$ (2)

where $b$ is the width of the tape. However, we are interested in the
deformation of the peel front of the adhesive, which is a soft visco-elastic
material. For the purpose of modeling, while we shall ignore the viscoelastic
nature of the adhesive, we recognize its low modulus, i.e., we assume an
effective spring constant $k_{g}$ (along the contact line) whose value is much
smaller than the spring constant of the tape material $k_{t}$. This also
implies that the force along $PO^{\prime}$ equilibrates fast and therefore the
integrand in Eq. (2) can be assumed to vanish for all $y$. Thus, the above
equation reduces to Eq. (1).

The present model is an extension of the ODE model for the contact point
dynamics Rumi05 . The ODE model already contains information on the inertial
time scale of the tape mass that allows for dynamical jumps across the two
branches of the peel force function. The extension involves introducing the
Rayleigh dissipation functional to deal with acoustic emission apart from
introducing the spatial degrees of freedom. The equations of motion for the
contact line dynamics are derived by writing down the relevant energy terms
consisting of the kinetic energy, potential energy and the energy dissipated
during the peel process. The total kinetic energy $U_{k}$ is the sum of the
rotational kinetic energy of the roller tape and the kinetic energy of the
stretched part of the tape. This is given by

$U_{K}={1\over 2}\int^{b}_{0}\xi\big{[}\dot{\alpha}(y)+{v(y)\over
R}\big{]}^{2}dy+{1\over 2}\int^{b}_{0}\rho\big{[}\dot{u}(y)\big{]}^{2}dy.$ (3)

Here, $\xi$ is the moment of inertia per unit width of the roller tape and
$\rho$ the mass per unit width of the tape. The total potential energy $U_{p}$
consists of the contribution from the displacement of the peel front due to
stretching of the peeled tape and possible inhomogeneous nature of the peel
front. This is given by

$U_{P}={1\over 2}\int^{b}_{0}{k_{t}\over b}\Big{[}u(y)\Big{]}^{2}dy+{1\over
2}\int^{b}_{0}{k_{g}b}\Big{[}{\partial u(y)\over\partial y}\Big{]}^{2}dy.$ (4)

The peel process always involves dissipation. Indeed, the peel force function
with the two stable branches, one corresponding to low velocities and another
at high velocity arises from two different dissipative mechanisms. Apart from
this, there is an additional dissipation that arises from the rapid rupture of
the peel front which in turn results in the accelerated motion of local
regions of the peel front. We consider this accelerated motion of the local
slip as the source responsible for the generation of acoustic signals Rumiprl
. Any rapid movement also prevents the system from attaining a quasistatic
equilibrium which in turn generates dissipative forces that resist the motion
of the slip. Such dissipative forces are modeled by the Rayleigh dissipation
functional that depends on the gradient of the local displacement rate Land .
Indeed, such a dissipative term has proved useful in explaining the power law
statistics of the AE signals during martensitic transformation vives ;
rajeevprl ; kalaprl  as also in explaining certain AE features in fracture
studies of rock sample Rumiepl . Then, the total dissipation can be written as
the sum of these two contributions

${\cal R}={1\over b}\int^{b}_{0}\int f(v(y))dvdy+{1\over
2}\int^{b}_{0}{\Gamma_{u}\over b}\Big{[}{\partial\dot{u}(y)\over\partial
y}\Big{]}^{2}dy,$ (5)

where $f(v)$ physically represents the peel force function assumed to be
derivable from a potential function $\Phi(v)=\int f(v)dv$ (see Ref. Rumi05 ).
We denote the second term in Eq. (5) by ${\cal R}_{ae}$ which is identified
with the energy dissipated in the form of AE. In the context to plastic
deformation, the acoustic energy arising from the abrupt motion of
dislocations is given by ${\cal R}_{ae}\propto\dot{\epsilon}^{2}(r)$, where
$\dot{\epsilon}(r)$ is the local plastic strain rate Rumiepl . Following this,
we interpret ${\cal R}_{ae}$ as the energy dissipated in the form of AE
signals. Note that $\frac{\partial\dot{u}}{\partial y}$ is the local strain
rate of the peel front. As for the first term in Eq. (5), the form of the peel
force function we use is given by

$f(v)=402v^{0.34}+171v^{0.16}+68e^{(v/7.7)}-369.65v^{0.5}-2.$ (6)

We stress here that as we are interested in the generic properties of the
peeling process, the exact form of the peel force function used here is not
important as long as major experimental features like the magnitude of the
jump in the velocity across the two branches, the range of values of the
measured peel force function, in particular the values at the maximum and
minimum, are captured.

As can be seen from Eq. (3), there are two time scales; one corresponding to
the inertia of the tape mass and the other due to the roller inertia. In
addition, there is a third time scale, namely the dissipative time scale in
Eq. (5) (second term). Thus, there are three internal relaxational time scales
in the model. Apart from this, there is also a time scale due to the pull
speed. Then the nature of the dynamics is determined by an interplay among all
these time scales.

It is more convenient to deal with scaled quantities. Consider introducing
basic length and time scales which will be used to rewrite all the energy
terms in scaled form. A natural choice for a time like variable is
$\tau=\omega_{u}t$ with $\omega_{u}^{2}={k_{t}/(b\ \rho)}$. In a similar way,
we introduce a basic length scale defined by $d=f_{max}/k_{t}$, where
$f_{max}$ is the value of $f(v)$ at $v_{max}$ on the left stable branch. We
define scaled variables by $u=Xd=X(f_{max}/k_{t})$, $l=l^{s}d$, $L=L^{s}d$ and
$R=R^{s}d$. The peel force function $f$ can be written as
$\phi(v^{s})=f(v(v^{s}))/f_{max}$. Here $v^{s}=v/v_{c}\omega_{u}d$ and
$V^{s}=V/v_{c}\omega_{u}d$ are the dimensionless peel and pull velocities
respectively with $v_{c}=v_{max}/\omega_{u}d$ representing the dimensionless
critical velocity at which the unstable branch starts. Using this we can
define a few relevant scaled parameters $C_{f}=(f_{max}/k_{t})^{2}(\rho/\xi)$,
$k_{0}=k_{g}b^{2}/(k_{t}a^{2})$,
$\gamma_{u}=\Gamma_{u}\omega_{u}/(k_{t}a^{2})$, and $y=ar$, where $a$ is a
unit length variable along the peel front. The parameter $C_{f}$ is a measure
of the relative strengths of the inertial time scale of the stretched tape to
that of the roller, $k_{0}$ the relative strengths of the effective elastic
constant of the adhesive to that of the tape material and $\gamma_{u}$ the
strength of the dissipation coefficient. Then, the scaled local form of Eq.
(1) takes the form

$\dot{X}=(V^{s}-v^{s})v_{c}+R^{s}\ {l^{s}\over L^{s}}\ ({sin}\ \alpha)\
\dot{\alpha}.$ (7)

In terms of the scaled variables, the scaled kinetic energy $U^{s}_{K}$ and
scaled potential energy $U^{s}_{P}$ can be respectively written as

$\displaystyle U^{s}_{K}$ $\displaystyle=$ $\displaystyle{1\over
2C_{f}}\int^{b/a}_{0}\Big{[}\Big{(}\dot{\alpha}(r)+{v_{c}v^{s}(r)\over
R^{s}}\Big{)}^{2}+C_{f}{\dot{X}}^{2}(r)\Big{]}dr,$ (8) $\displaystyle
U^{s}_{P}$ $\displaystyle=$ $\displaystyle{1\over
2}\int^{b/a}_{0}\Big{[}X^{2}(r)+k_{0}\Big{(}{\partial X(r)\over\partial
r}\Big{)}^{2}\Big{]}dr.$ (9)

The total dissipation in the scaled form is

${\cal R}^{s}=R_{f}^{s}+R_{ae}={1\over
b}\int^{b/a}_{0}\Big{[}\int\phi(v^{s}(r))dv^{s}+{\gamma_{u}\over
2}\Big{(}{\partial\dot{X}(r)\over\partial r}\Big{)}^{2}\Big{]}dr.$ (10)

The first term on the right hand side is the frictional dissipation arising
from the peel force function. The scaled peel force function, $\phi(v^{s})$,
can be obtained by using the scaled velocities in Eq. (6). The nature of
$\phi(v^{s})$ is shown in Fig. 1(b). Note that the maximum occurs at
$v^{s}=1$. We shall refer to the left branch AB as the ‘stuck state’ and the
high velocity branch CD as the ’peeled state’. The second term on the right
hand side denotes the scaled form of the acoustic energy dissipated.

The Lagrange equations of motion in terms of the generalized coordinates
$\alpha(r),\dot{\alpha}(r),X(r)$ and $\dot{X}(r)$ are

$\displaystyle{d\over d\tau}\left({\partial{\cal
L}\over{\partial\dot{\alpha}(r)}}\right)-{\partial{\cal
L}\over{\partial\alpha(r)}}+{\partial{\cal
R}^{s}\over{\partial\dot{\alpha}(r)}}$ $\displaystyle=0$ , (11)
$\displaystyle{d\over d\tau}\left({\partial{\cal
L}\over{\partial\dot{X}(r)}}\right)-{\partial{\cal L}\over{\partial
X}(r)}+{\partial{\cal R}^{s}\over{\partial\dot{X}(r)}}$ $\displaystyle=$
$\displaystyle 0.$ (12)

Using this, we get the equations of motion as

$\displaystyle\ddot{\alpha}$ $\displaystyle=$
$\displaystyle-{v_{c}\dot{v}^{s}\over R^{s}}-C_{f}R^{s}{{l^{s}\over
L^{s}}\,{sin}\,\alpha\over(1+{l^{s}\over L^{s}}\,{sin}\,\alpha)}\phi(v^{s}),$
(13) $\displaystyle\ddot{X}$ $\displaystyle=$
$\displaystyle-X+k_{0}{\partial^{2}X\over\partial
r^{2}}+{\phi(v^{s})\over(1+{l^{s}\over
L^{s}}\,{sin}\,\alpha)}+\gamma_{u}{\partial^{2}\dot{X}\over\partial r^{2}}.$
(14)

However, Eqs. (13, 14) should satisfy the constraint Eq. (7). This consistency
can be imposed by using the theory of mechanical systems with constraints ECG
. This leads to an equation for the acceleration variable $\dot{v}^{s}(r)$
obtained by differentiating Eq. (7) and using Eqs. (14),

$\displaystyle\dot{v}^{s}$ $\displaystyle=$
$\displaystyle\big{[}-{\ddot{X}}+{R^{s}l^{s}\over
L^{s}}\big{(}\dot{\alpha}^{2}(cos\alpha-{R^{s}l^{s}}({sin\alpha\over
L^{s}})^{2}\big{)}$ (15) $\displaystyle+$ $\displaystyle
sin\alpha{\ddot{\alpha}}\big{)}\big{]}/v_{c}.$

These Eqs. (7, 13) and (15) constitute a set of nonlinear partial differential
equations that determine the dynamics of the peel front. They have been solved
by discretizing the peel front on a grid of N points and using an adaptive
step size stiff differential equations solver (MATLAB package). We have used
open boundary conditions appropriate for the problem. The initial conditions
were drawn from the stuck configuration (i.e., the values are from the left
branch of $\phi(v^{s})$) with a small spatial inhomogeneity in $X$ such that
they satisfy Eq. (7) approximately. The system is evolved till a steady state
is reached before the data is accumulated.

The nature of the dynamics depends on the pull velocity $V^{s}$, the
dissipation coefficient $\gamma_{u}$ and $C_{f}$. We have carried out detailed
studies of the dynamics of the model over a wide range of values of these
parameters keeping other parameters fixed at $R^{s}=0.35$, $l^{s}=3.5$,
$k_{0}=0.1$ ($k_{t}=1000$ N/m) and $N=50$ (in units of the grid size). Larger
system size $N=100$ is used whenever necessary.

##  III Time series analysis of experimental AE signals

One of the objectives is to carry out statistical and nonlinear time series
analysis of experimental AE signals associated with the jerky peel process
with a view to understand the results on the basis of model studies. Acoustic
emission data files were obtained from peel experiments under constant
traction velocity conditions that cover a wide range of values from $0.2$ to
$7.6$ cm/s Ciccotti07 . Signals were recorded at the standard audio sampling
frequency of $44.1$ kHz (having $6$ kHz band width) using a high quality
microphone. They were digitized and stored as $16$ bit signals in raw binary
files. There are 38 data files each containing approximately $1.2\times
10^{6}$ points. The AE signals are noisy as in most experiments on AE.

Two characteristic features of low dimensional chaos are the existence of a
strange attractor with self similar properties quantified by a fractal
dimension (or equivalently the correlation dimension) and sensitivity to
initial conditions quantified by the existence of a positive Lyapunov
exponent. Given the equations of motion, these quantities can be directly
calculated. However, when a scalar time series is suspected to be a projection
from a higher dimensional dynamics, they are traditionally analyzed by using
embedding methods that attempt to recover the underlying dynamics. The basic
idea is to unfold the dynamics through a phase space reconstruction of the
attractor by embedding the time series in a higher dimensional space using a
suitable time delayPackard ; GP . Consider a scalar time series measured in
units of sampling time $\Delta t$ defined by $[x(k),k=1,2,3,\cdots,N]$. Then,
we can construct $d-$dimensional vectors defined by
$\vec{\xi}_{k}=[x(k),x(k+\tau),\cdots,x(k+(d-1)\tau)];\,\,k=1,\cdots,[N-(d-1)\tau]$.
The delay time $\tau$ suitable for the purpose is either obtained from the
autocorrelation function or from mutual information HKS . Once the
reconstructed attractor is obtained, the existence of converged values of
correlation dimension and a positive exponent is taken to be a signature of
the underlying chaotic dynamics. In real systems, most experimental signals
contain noise which in this case is high. There are several methods designed
to cure the noise component HKS ; KS ; KS1 ; GHKSS ; CH . Usually, the cured
data sets are then subjected to further analysis.

The correlation integral defined as the fraction of pairs of points
$\vec{\xi}_{i}$ and $\vec{\xi}_{j}$ whose distance is less than $r$, is given
by

$C(r)=\frac{1}{N_{p}}\sum_{i,j}\Theta(r-|\vec{\xi}_{i}-\vec{\xi}_{j}|),$ (16)

where $\Theta(\cdots)$ is the step function and $N_{p}$ the number of vector
pairs summed. A window is imposed to exclude temporally correlated points HKS
. The method provides equivalence between the reconstructed attractor and the
original attractor. It has been shown that a proper equivalence is possible if
the time series is noise free and long Ding . For a self similar attractor
$C(r)\sim r^{\nu}$, where $\nu$ is the correlation dimension GP . Then, as $d$
is increased, one expects to find a convergence of the slope $dlnC(r)/dlnr$ to
a finite value in the limit of small $r$. However, in practice, the scaling
regime is found at intermediate length scales due to the presence of noise.

The existence of a positive Lyapunov exponent is considered as an unambiguous
quantifier of chaotic dynamics. However, the presence of superposed noise
component, which in the present case is high, poses problems. In principal the
noise component can be cured and then the Lyapunov exponent calculated HKS ;
KS ; KS1 ; GHKSS ; CH . Here, we use an algorithm that does not require
preprocessing of the data; it is designed to average out the influence of
superposed noise. The algorithm, which is an extension of Eckmann’s algorithm,
has been shown to work well for reasonably high levels of noise in model
systems as well as for short time series. The method has been used to analyze
experimental time series as well (for details, see Ref. Anan99 ; Noro01 ).

In the conventional Eckmann’s algorithm Eckmann , a sequence of tangent
matrices are constructed that connect the initial small difference vector
$\vec{\xi}_{i}-\vec{\xi}_{j}$ to evolved difference vectors
$\vec{\xi}_{i+k}-\vec{\xi}_{j+k}$, where $k$ is the propagation time. In the
algorithm, the number of neighbors used is small typically min$[2d,d+4]$
contained in a spherical shell of size $\epsilon_{s}$. A simple modification
of this is to use those neighbors falling between an inner and outer radii
$\epsilon_{i}$ and $\epsilon_{0}$ respectively. Then, the inner shell
$\epsilon_{i}$ is expected to act as a noise filter. However, so few neighbors
will not be adequate to average out the noise component superposed on the
signal. Thus, the modification we effect is to allow more number of neighbors
so that the noise statistics is sampled properly. (See for details Anan99 ;
Noro01 .) As the sum of the exponents should be negative for a dissipative
system, we impose this as a constraint. In addition, we also demand the
existence of stable positive and zero exponents (a necessary requirement for
continuous time systems like AE) over a finite range of shell sizes
$\epsilon_{s}$. As a cross check, we have also calculated the correlation
integral and Lyapunov spectrum using the TISEAN package as well HKS .

##  IV Dynamics of the peel front

A systematic study of the dynamics of the model is essential to understand the
influence of the various parameters on the spatiotemporal dynamics of the peel
front, its connection to intermittent peeling and to the accompanying acoustic
emission. From Eq. (10), it is clear that the acoustic energy $R_{ae}$ is the
spatial average of the local strain rate. As the peel front patterns determine
the nature of acoustic energy, a detailed study of the dependence of the
patterns on the relevant parameters and on the pull velocity should help us to
get insight into AE generation process during peeling.

###  IV.1 General considerations on time scales and parameter values

We begin by making some general observation about the various parameters and
their influences. The dynamics of the model is sensitive to the three time
scales (reduced from four due to scaling) determined by the parameters
$C_{f}$, $\gamma_{u}$ and $V^{s}$. $C_{f}$ is related to the ratio of inertial
time of the tape mass to that of roller inertia (see below). The dissipation
parameter $\gamma_{u}$ reflects the rate at which the local strain rate
relaxes. The pull velocity $V^{s}$ determines the duration over which all the
internal relaxations are allowed to occur. The range of $C_{f}$ is determined
by the allowed values of the tape mass $m$ and the roller inertia $I$.
Following our earlier studies, we vary $I$ from $10^{-5}$ to $10^{-2}$ and $m$
from 0.001 to 0.1. Thus, $C_{f}$ can be varied over a few orders of magnitude
keeping one of them fixed. For model calculations, the dissipation parameter
is varied from 0.001 to 1. (However, an order of magnitude estimate shows that
$\gamma_{u}<<1$, see below.) The range of $V^{s}$ of interest is determined by
the instability domain which is from 1 to $\sim 12$ as shown in Fig. 1(b).

To appreciate the influence of inertial time scale of tape mass parameterized
by $C_{f}$, consider the low mass limit of the ODE model Rumi05  which has
been shown to lead to the DAE model equations Rumi04 . In this limit, the
velocity jumps across the two branches of the peel force function are abrupt
with infinite acceleration. However, finite tape mass introduces an additional
time scale that leads to jumps in $v^{s}$ to occur over a finite time scale
which in turn the magnitude of the velocity jumps. Indeed, the phase space
trajectory need not jump to the high velocity branch of $\phi(v^{s})$, as we
shall see. This can be better appreciated by considering the ODE model (that
ignores the spatial degrees of freedom). Consider the relevant ODE model
equations Rumi05  (in unscaled form).

$\displaystyle{\ddot{\alpha}}$ $\displaystyle=$
$\displaystyle-\frac{\dot{v}}{R}+\frac{R}{I}\frac{cos\,\theta}{(1-cos\,\theta)}f(v),$
(17) $\displaystyle m{\ddot{u}}$ $\displaystyle=$
$\displaystyle\frac{1}{(1-cos\,\theta)}[f(v)-ku(1-cos\,\theta)],$ (18)

where $\alpha$ is shown in Fig. 1(a) and $u$ the displacement of the contact
point. $m$ is mass of the tape and $k$ the spring constant of the tape. From,
Eqs. ( 17, 18), two inertial time scales can be identified, one corresponding
to the roller inertia $\Omega_{\alpha}=(Rf/I)^{1/2}$ and another to that of
the tape mass $\Omega_{u}=(k/m)^{1/2}$. (Note that $k$ in Eq. (18) of the ODE
model corresponds to $k_{t}$ in the present model.) Thus, $C_{f}$ in present
model is directly related to the ratio of these two inertial time scales.
Differentiating Eq. (1), we get

${\dot{v}}+{\ddot{u}}+R\,{\ddot{\alpha}}\,cos\,\theta=R\,{\dot{\alpha}}\,{\dot{\theta}}\,sin\theta.$
(19)

Eq. (18) is the force balance equation. In the limit $m\rightarrow 0$, we have
the algebraic constraint $f(v)=F(t)(1-cos\,\theta(t))$. Differentiating this
equation shows that $\dot{v}$ diverges at points of maximum and minimum of the
peel force function $f(v)$. This demonstrates that in the low mass limit, the
orbits jump to the high velocity branch abruptly. Now consider Eq. (19) that
relates the acceleration of the peel point ($\dot{v}$), acceleration of the
displacement $u$ i.e., $\ddot{u}$ and $\ddot{\alpha}$. This again is basically
a force balance equation as can be seen by multiplying the equation by the
tape mass $m$. As the right hand side is small, any increase in one of these
acceleration variables implies a decrease in the other variables. As low mass
limit implies infinite acceleration of the peel front ($\dot{v}$) across the
peel force function, finite mass implies the velocity jumps across the peel
force function is reduced. It is worthwhile to note that the effect of
inertial time scale causing jumps across the unstable branch to occur at a
finite time scale is a general feature. This has been recognized and
demonstrated experimentally in the context of the PLC effect SF85 .

Figure 2: (Color online a, b, e, f) (a, b) Snapshots for $C_{f}=7.88$,
$V^{s}=1.48$, and $\gamma_{u}=1.0$. (a) a smooth peel front with small
amplitude high frequency oscillations due to finite roller inertia and (b) a
smooth peel front. (c) Phase plot for an arbitrary spatial point on the peel
front. Bold line represents $\phi(v^{s})$. (d) Model acoustic energy plot. (e,
f) Snapshots for $C_{f}=7.88$, $V^{s}=1.48$, and $\gamma_{u}=0.1$. (Rugged
peel front and the onset of stuck-peeled configuration.)

Now consider estimating the order of magnitude of the dissipative time scale.
The unscaled dissipation parameter $\Gamma_{u}$ is related to the fluid shear
viscosity $\eta$ Land  and thus an order of magnitude estimate can be
obtained. Typical values of $\eta$ for adhesives at low shear rates is $\sim
1000-10000$ Pa.s. As stress is directly related to shear viscosity,
$\Gamma_{u}$ can be estimated using typical dimensions of the peel front. It
has been shown that deformed peel front dimension is about 100 $\mu m$
Dickinson ; Yamazaki , the thickness of the adhesive is $\sim$ 50 $\mu m$ and
the width of the peel front $\sim 20$ mm (width of the tape). It is easy to
show that $\Gamma_{u}\sim 10^{-3}-10^{-2}$ J.s. Thus, the range of
$\gamma_{u}$ is $\sim 10^{-3}-10^{-4}$ taking $\eta\sim 1000$ Pa.s. As some of
the numbers used are material dependent, this is just an order of magnitude
estimate. For model studies, the range of $\gamma_{u}$ is taken to be from 1
to 0.001. However, we will not discuss the results for $\gamma_{u}=0.001$ as
these are similar to 0.01.

Within the scope of the model, the model acoustic energy given by
$R_{ae}(\tau)={1\over 2}\gamma_{u}\sum_{i}(\dot{X}_{i+1}-\dot{X}_{i})^{2}$ (in
the discretized form) depends on the nature of the local displacement rate.
Based on this relation, some general observations can be made on the nature of
$R_{ae}$ and its dependence on the peel front dynamics. From Eq. (14), high
$\gamma_{u}$ implies that the coupling between neighboring sites is strong and
hence the local dynamics at one spatial location has no freedom to deviate
from that of its neighbor. Thus, the displacement rate at a point on the peel
front cannot differ from that of its neighbor. For the same reason, low
$\gamma_{u}$ implies weak coupling between displacement rates on neighboring
points on the peel front which therefore can differ substantially. This
clearly should lead to significantly more inhomogeneous peel velocity profile.
Based on the above arguments, high $\gamma_{u}$ should lead to smooth peel
front and consequently sharp bursts in the model acoustic energy $R_{ae}$ that
occurs during jumps between the two branches of $\phi(v^{s})$. In contrast,
when $\gamma_{u}$ is small, $R_{ae}$ should be high as also spread out in
time. However, as the exact nature of the peel front pattern is sensitive to
the values of $C_{f}$, pull velocity $V^{s}$ and $\gamma_{u}$, the nature of
$R_{ae}$ depends on all the three time scales. Indeed, one should expect that
the more rapidly the peel front patterns change with time, the noisier the
model acoustic energy should be. This is one feature that we hope to compare
with experimental acoustic signals.

###  IV.2 Results of the model

We have carried out extensive studies on the nature of the dynamics for a wide
range of values of the parameters stated above. The peel front dynamics is
analyzed by recording the velocity-space-time patterns of the peel front, the
phase plots in the $X^{s}-v^{s}$ plane for an arbitrary spatial point on the
peel front and the model acoustic energy dissipated $R_{ae}$. (Unless
otherwise stated, these plots refer to steady state dynamics after all the
transients have died out.) Here, we present a few representative solutions for
different sets of parameters within the range of interesting dynamics. Our
analysis shows that while the nature of the dynamics results from competing
influences of the three time scales, the dissipation parameter $\gamma_{u}$
appears to have a significant influence on the spatiotemporal dynamics of the
peel front.

####  IV.2.1  Case (i), $C_{f}=7.88$ \- high (low) tape mass, low (high)
roller inertia

Figure 3: (Color online) (a-d) Snapshots during the peeling process for
$C_{f}=7.88,V^{s}=1.48$, and $\gamma_{u}=0.01$ : (a) highly rugged peel front
when the system is on the left branch of $\phi(v^{s})$, (b) the onset of peel
process, (c) stuck-peeled configuration and (d) resulting nearly uniform
peeled state.

Given a value of $C_{f}$ there is a range of values of $(m,I)$. In this case,
the set of values are: $(0.1,10^{-3}),(0.01,10^{-4})$ and $(0.001,10^{-5})$.
The dissipation coefficient is varied from $\gamma_{u}=1$ to 0.01. For high
$\gamma_{u}=1.0$, only smooth peeling is seen independent of the magnitude of
the pull velocity. The peel front switches between the low and high velocity
branches of the peel force function $\phi(v^{s})$. Plots of the smooth nature
of the entire peel front are shown in Figs. 2(a, b) for $V^{s}=1.48$. Figure
2(a) shows the nature of the peel front when the system is on the low velocity
branch of $\phi(v^{s})$, i.e., the local velocities of all spatial elements
follow the AB branch of $\phi(v^{s})$. The small amplitude synchronous high
frequency oscillation of the entire peel front results from the roller
inertia. (Compare the values of $v^{s}$ in the two figures.) A phase plot in
the $X^{s}-v^{s}$ plane for an arbitrary point on the peel front is shown in
Fig. 2(c). The small amplitude oscillation of the peel front shown in Fig.
2(a) corresponds to the velocity oscillations in the phase plot (Fig. 2(c)).
As high $C_{f}$ implies relatively low values of $I$, it can be shown (on
lines similar to Ref. Rumi04 ) that the orbit sticks to the stationary
branches (slow manifold) of the peel force function $\phi(v^{s})$ jumping
between the branches only at the limit of stability typical of relaxation
oscillations. The corresponding model acoustic energy $R_{ae}(\tau)$ shows a
sequence of small amplitude spikes corresponding to the small amplitude
oscillations arising from the roller inertia [Fig. 2(d)] followed by large
bursts that occur at regular intervals. The bursts result from the peel front
jumping from the stuck to the peeled state and back. Note that the duration of
the bursts are short compared to duration between them.

However, as we decrease $\gamma_{u}$ to $0.1$ keeping $V^{s}=1.48$, we observe
rugged and stuck-peeled configurations. The rugged pattern is seen when the
system is on the AB branch of $\phi(v^{s})$. Even so, on reaching the limit of
stability, the entire contact line peels nearly at the same time as shown in
Fig. 2(e). But once it jumps to the high velocity branch CD of $\phi(v^{s})$,
the peel front that has nearly uniform peel velocity commensurate with that of
the right branch of $\phi(v^{s})$ becomes unstable and breaks up into stuck
and peeled segments as shown in Fig. 2(f). The width of these segments
increases in time with a concomitant decrease in the magnitude of the velocity
jumps of peeled segments, eventually the entire peel front goes into a stuck
state. Then, the cycle restarts with the peel front switching between the
rugged and stuck-peeled (SP) states. The phase plot is similar to that for
$\gamma_{u}=1$ again sticking to the slow manifold. The model acoustic energy
dissipated $R_{ae}$ is also similar to that for $\gamma_{u}=1$ except that the
large bursts are comparatively broader as should be expected due to presence
of stuck-peeled configurations that contribute to large changes in the local
velocity.

Figure 4: (Color online a,e) (a, b) Snapshot of a stuck-peeled configuration
and model acoustic energy respectively for $C_{f}=7.88$, $\gamma_{u}=0.01$ and
$V^{s}=2.48$. (c, d) Phase plot and model acoustic energy plot respectively
for $C_{f}=7.88$, $V^{s}=4.48$, and $\gamma_{u}=0.01$. (e) Snapshot of long
lived stuck-peeled configuration for $C_{f}=7.88$, $V^{s}=5.48$, and
$\gamma_{u}=0.01$ and (f) the associated model acoustic energy.

As we decrease $\gamma_{u}$ to 0.01, the observed patterns are similar to
those for $\gamma_{u}=0.1$ but the sequence of the peel front patterns is
different. Starting with a low velocity configuration that is even more rugged
compared to that for $\gamma_{u}=0.1$ as shown in Fig. 3(a), the peel process
starts with a small stuck segment getting peeled [Fig. 3(b)]. There after,
several stuck segments peel out leading to a stuck-peeled pattern as shown in
Fig. 3(c), eventually, the entire peel front peels-out leaving a nearly
uniform peeled state as shown in Fig. 3(d) (with a velocity commensurate with
the high velocity branch of $\phi(v^{s})$). This is again destabilized with
some segments of the peel front getting stuck as in the case of
$\gamma_{u}=0.1$ (similar to Fig. 2(f)). The number of such stuck segments
increases with time, eventually the whole peel front goes into a stuck state.
The cycle restarts. The phase plot is similar to $\gamma_{u}=0.1$ and 1.
Indeed, for a given $C_{f}$, independent of $\gamma_{u}$ the phase plot
changes only when $V^{s}$ is increased. But $R_{ae}$ shows broader bursts
compared to $\gamma_{u}=0.1$ as the corresponding stuck-peeled configurations
last longer. Even so, the duration of the SP configurations in a cycle is
short, i.e., the duration of the bursts is short compared to the duration
between them.

Now we consider the influence of increasing the pull velocity (keeping $C_{f}$
fixed at 7.88) which in turn should leave less time for internal relaxational
mechanisms to operate. Intuitively one should expect that some patterns
observed for low $V^{s}$ may not be seen for higher values of $V^{s}$.
$\gamma_{u}=1$ case is uninteresting for the reasons stated above. But,
reducing $\gamma_{u}$ to 0.1 does provide some degree of freedom for the local
dynamics to operate at each point. Even so, for $V^{s}=2.48$, the peel front
switches between a SP configuration with most segments momentarily in the
stuck state (similar to Fig. 3(b)) and a configuration that has several stuck-
peeled segments (similar to Fig. 3(c)). The corresponding $X^{s}-v^{s}$ phase
plot shows that the orbit jumps moves slightly beyond the upper value of
$\phi(v^{s})$ and jumps back from the right branch even before reaching the
minimum of $\phi(v^{s})$ ( not shown). As we decrease $\gamma_{u}$ to 0.01,
the rugged configuration seen for $V^{s}=1.48$ is no longer seen and only SP
configurations are observed as shown in Fig. 4(a). The SP configurations are
dynamic in the sense, segments that are in stuck state at one time become
unstuck at a later time and vice versa. For this case
($C_{f}=7.88,V^{s}=2.48,\gamma_{u}=0.01$), these rapid changes occur over a
short time scale. Consequently, the model acoustic energy is quite noisy as
shown in Fig. 4(b) but has a noticeable periodic component. The points of
minima correspond to configurations that have fewer peeled segments compared
to those near the peak of $R_{ae}$. The phase plot in the $X^{s}-v^{s}$ plane
is limited to the upper part of $\phi(v^{s})$. Even as the phase plots for any
two spatial points look similar, there is a phase difference. For instance, at
any given time, the phase point of stuck segment will be on the left branch
while that for peeled point will be on the right branch.

As we increase $V^{s}$ to $4.48$ (keeping $C_{f}$ at $7.88$), there is even
lesser time for peel front inhomogeneities to relax and thus, we observe a
smooth peeling for $\gamma_{u}=1$ and as also for $0.1$. As we decrease
$\gamma_{u}$ to $0.01$, we see only SP patterns (not shown but similar to Fig.
4(a)). The corresponding $X^{s}-v^{s}$ phase plot for an arbitrary point on
the peel front shown in Fig. 4(c) is confined to the top of $\phi(v^{s})$. The
corresponding $R_{ae}$ is noisy and irregular as shown in Fig. 4(d). However,
when we increase $V^{s}$ to $5.48$, initially, one does observe the patterns
switching between rugged and SP configurations. If we wait long enough, we
observe only SP configurations that are different from those for lower
$V^{s}$. In this case, the stuck and peeled segments are long lived. A top
view of the SP pattern is shown in Fig. 4(e). The phase space orbit in the
$X^{s}-v^{s}$ plot is pushed beyond the upper limit of $\phi(v^{s})$. The
energy dissipated is quite regular (but aperiodic) unlike that for lower pull
velocities as shown in Fig. 4(f). This regularity is clearly due to the long
lived nature of these SP configurations. The long lived nature of the SP
configurations for high pull velocity is a general feature, i.e., the duration
over which the stuck segments remain stuck (peeled segments remain peeled)
increases as we increase the pull velocity. The dynamics is no longer
interesting beyond $V^{s}=6.48$ as only smooth peeling is seen.

Figure 5: (Color online a, b) (a, b) Snapshots of stuck-peeled configurations
for $C_{f}=0.788,V^{s}=1.48$, and $\gamma_{u}=1.0$. Note that (a) has fewer
peeled segments compared to (b). (c) Phase plot for an arbitrary point on the
peel front, and (d) model acoustic energy plot.

Figure 6: (Color online a, c) (a, b) Snapshot of a stuck-peeled configuration
for $C_{f}=0.788,V^{s}=1.48$, and $\gamma_{u}=0.1$ and model acoustic energy
plot respectively. (c, d) Plot of ‘an edge of peeling’ configuration for
$C_{f}=0.788,V^{s}=1.48$, and $\gamma_{u}=0.01$ and the corresponding model
acoustic energy.

Figure 7: (Color online a, b) (a, b) Snapshots during peel process for
$C_{f}=0.788,V^{s}=2.48$, and $\gamma_{u}=1.0$. (c) Model acoustic energy
plot. (d) Phase plot for an arbitrary spatial point on the peel front. Bold
line represents $\phi(v^{s})$.

####  IV.2.2  Case (ii), $C_{f}=0.788$ \- high (and low) tape mass, high (and
intermediate) roller inertia

For this value of $C_{f}$, the allowed set of values of $(m,I)$ are
$(0.1,10^{-2}),(0.01,10^{-3})$ and $(0.001,10^{-4})$. The dynamics is more
interesting for this case as there is a scope for competition among the three
time scales.

We first study the dynamics keeping $V^{s}=1.48$ and varying the dissipation
parameter. For $\gamma_{u}=$ 1.0, the uniform nature of the peel front seen
for $C_{f}=7.88$ disappears and even for short times, stuck-peeled
configurations are seen. The peel front patterns stabilize to stuck-peeled
configurations as shown in Figs. 5(a, b). As can be seen these SP patterns
have only a few stuck or peeled segments with moderate velocity jumps and
smooth variation along the peel front unlike the SP configurations discussed
earlier. (Note that the SP configuration in Fig. 5(b) has more stuck segments
compared to Fig. 5(a).) The moderate velocity jumps can be understood by
noting that the phase space orbit never visits the high velocity branch of
$\phi(v^{s})$ as can be seen from Fig. 5(c). It is interesting to note that
the trajectory stays close to the unstable branch of $\phi(v^{s})$ even after
attempting to jump from the low velocity branch. Such orbits are reminiscent
of canard type solutions Canard . The trajectory is irregular and is
suggestive of spatiotemporal chaotic nature of the peel front. The energy
dissipated $R_{ae}$ shown in Fig. 5(d) is continuous and irregular due to the
dynamic SP pattern as should be expected, but there is a noticeable periodic
component. The rough periodicity of $R_{ae}$ can be traced to fact that the
peel front configurations switch between patterns with more stuck segments and
less stuck segments. (From the number shown on $x$-axis, Fig. 5(a) can be
identified with minimum and Fig. 5(b) with the peak of $R_{ae}$ in Fig. 5(d).
See the marked arrows as well.)

As we decrease $\gamma_{u}$ to 0.1, the SP configurations observed have more
stuck and peeled segments compared to $\gamma_{u}=1$ (compare Fig. 6(a) with
Fig. 5 (a)). However, the magnitude of the velocity jumps remains moderate as
in the previous case. This is again due to the fact that the orbit never
visits the high velocity branch of $\phi(v^{s})$. (Recall that given a value
of $C_{f}$, the phase plot remains the same for different $\gamma_{u}$ values
as long as $V^{s}$ is fixed). Indeed, for this value of $C_{f}=0.788$, the
orbit never jumping to the high velocity branch is a consequence of finite
inertia of the tape mass compared to that of the roller inertia as discussed
earlier. For this case, the model acoustic energy $R_{ae}$ is also irregular
and continuous as shown in Fig. 6(b) with a noticeable periodic component.
Now, if we decrease $\gamma_{u}$ further to 0.01, the peel front pattern
displays increased number of stuck and peeled segments with each stuck segment
having only a few contiguous stuck points as can be seen from Fig. 6(c). Note
also that there is a large dispersion in the magnitudes of the velocity jumps
of the peeled segments even as the largest one is significantly smaller than
the value of CD branch of $\phi(v^{s})$. As can be seen from the Fig. 6(c),
even though the pattern is dynamic, the segments that are stuck are barely so.
Thus, the configuration shown in Fig. 6(c) gives the feeling of a critically
poised state. The corresponding $X^{s}-v^{s}$ phase plot (similar to that
shown in Fig. 5(c)) is irregular and possibly suggestive of spatiotemporal
chaotic nature of the peel front. The acoustic energy $R_{ae}$ is very
irregular without any trace of periodicity as shown in Fig. 6(d).

Figure 8: (Color online a, b, d, e) (a, b) Snapshots during peel process
$C_{f}=0.788,V^{s}=4.48$, and $\gamma_{u}=1.0$. (c) Phase plot for an
arbitrary spatial point on the peel front. Bold line represents $\phi(v^{s})$.
(d, e) Snapshots during peel process $C_{f}=0.788,V^{s}=4.48$, and
$\gamma_{u}=0.01$.

We now consider the influence of increasing the pull velocity $V^{s}$. As we
increase $V^{s}$ to 2.48, the spatiotemporal patterns seen for
$\gamma_{u}=1.0$, 0.1 and 0.01 are slightly different from those for
$V^{s}=1.48$. For $\gamma_{u}=1.0$, the peel process goes through a cycle of
configurations shown in Figs. 7(a, b). It is clear that Fig. 7(a) has more
segments in the stuck state while Fig. 7(b) is the usual kind of SP
configuration except that the stuck and peel segments are fewer. For this
case, the stuck and peeled segments last longer than those for $V^{s}=1.48$.
The corresponding $R_{ae}$ for each $\gamma_{u}$ exhibits noisy bursts
overriding a periodic component. A typical plot for $\gamma_{u}=1$ is shown in
Fig. 7(c). From the time labels as also the arrows shown, the minima and
maxima in $R_{ae}$ can be identified with Figs. 7(a, b) respectively. The
orbit in the $X^{s}-v^{s}$ plane moves into regions much beyond the values
allowed by $\phi(v^{s})$ as is clear from Fig. 7(d). The phase plots for
$\gamma_{u}=0.1$ and 0.01 are similar to this case.

For $\gamma_{u}=0.1$ also, the peel front pattern goes through a cycle of
stuck-peeled configurations (with more stuck and peeled segments than for
$\gamma_{u}=1.0$) and stuck segments (similar to Fig. 7(a)). Yet, the energy
dissipated $R_{ae}$ is similar to Fig. 7(c) for $\gamma_{u}=1.0$ which is
surprising considering that there are more stuck and peeled segments compared
to $\gamma_{u}=1$ case. This can be traced long lived of the stuck or peeled
configurations that hardly change over a cycle (as in the case of
$C_{f}=7.88,V^{s}=5.48,\gamma_{u}=0.01$, see Fig. 4(e)). The peel process is
similar even for $\gamma_{u}=0.01$.

As we increase the peel velocity to 4.48, the influence of this time scale on
the peel front pattern is discernable even for $\gamma_{u}=1.0$. The
spatiotemporal patterns of the peel front switches sequentially from nowhere
stuck configuration shown in Fig. 8(a) to stuck-peeled configuration with few
stuck and peeled segments shown in Fig. 8 (b). Note that there are very few
stuck and peeled segments. The corresponding $R_{ae}$ exhibits noisy periodic
pattern similar to Fig. 7(c) for $V^{s}=2.48$. The $X^{s}-v^{s}$ phase plot in
Fig. 8(c) shows that the orbit can move much beyond the values allowed by
$\phi(v^{s})$. As we decrease $\gamma_{u}$ to 0.1, the nowhere stuck
configuration [Fig. 8(a)] is replaced by a partly stuck, partly peeled
configuration and a SP configuration. For $\gamma_{u}=1.0$ case ( Fig. 8(c)),
the $X^{s}-v^{s}$ phase plot is slightly different as the orbit makes several
loops before it jumps to low velocity branch without visiting the high
velocity branch of $\phi(v^{s})$. The nature of $R_{ae}$ is still noisy and
periodic similar to Fig. 7(c). As we decrease $\gamma_{u}$ to 0.01, the peel
process goes through SP configurations shown in Figures. 8(d, e). Note that
Fig. 8(e) has large dispersion in the magnitude of velocity jumps of the
peeled segments compared to that in Fig. 8(d). It is worth emphasizing that
the increase in the number of stuck and peeled segments with decrease in
$\gamma_{u}$ is a general feature. Despite the higher number of stuck and
peeled segments, $R_{ae}$ for $\gamma=0.01$ is similar to that for
$\gamma_{u}=1.0$ as these peel front configurations are long lived which again
is a general feature observed at high pull velocities. Finally, it should be
stated that for $C_{f}=0.788$, in general the velocity variation along the
peel front is much more smooth compared to other values of $C_{f}$. The
dynamics is uninteresting beyond $V^{s}=7.48$ as only smooth peeling is seen.

Figure 9: (Color online b, c) (a) $R_{ae}$ as a function for
$C_{f}=0.00788,V^{s}=1.48$, and $\gamma_{u}=1$. (b) a stuck-peeled
configuration with more stuck segments compared to (c) where nearly equal
number of stuck and peeled segments for $C_{f}=0.00788,V^{s}=1.48$, and
$\gamma_{u}=0.01$. (d) The corresponding phase plot for an arbitrary spatial
point along the peeling front. Bold line represents $\phi(v^{s})$. (e)
Corresponding $R_{ae}(\tau)$ as a functions of time and (f) substructure of
(e).

####  IV.2.3  Case (iii), $C_{f}=0.00788$, low tape mass and high roller
inertia

For this value of $C_{f}$, there is just one set of values of tape mass and
roller inertia, namely, $m=0.001$ and $I=0.01$. As the tape mass is low, this
also corresponds to the DAE type of solutions for each spatial point. Thus,
the velocity jumps between the two branches of the peel force function will
always be abrupt with the roller inertia playing a major role in allowing the
orbits to jump between the branches of the peel force function as demonstrated
earlier Rumi04 .

Figure 10: (Color online a, b) (a, b) Snapshots during peel process for
$C_{f}=0.00788,V^{s}=2.48$, and $\gamma_{u}=0.01$. (c) Corresponding model
acoustic energy plot.

Figure 11:  (a) (Color online) Snapshot during peel process
$C_{f}=0.00788,V^{s}=4.48$, and $\gamma_{u}=0.01$. (b) Model acoustic energy
plot. (c) Phase plot for an arbitrary spatial point on the peel front for
$C_{f}=0.00788$, $V^{s}=4.48$, and $\gamma_{u}=0.01$. Bold line represents
$\phi(v^{s})$

Consider the influence of the dissipation parameter $\gamma_{u}$ keeping
$V^{s}=1.48$. For $\gamma_{u}=1$, peeling is uniform and thus the whole peel
front switches between the two branches of the peel force function. The
acoustic energy shows a bunch of seven double spikes that appear at regular
interval as shown in Fig. 9(a). (The number of spikes is correlated with the
number of cascading loops seen in the $X^{s}-v^{s}$ phase plot, see below.) As
we decrease $\gamma_{u}$ to 0.01, the peel front goes through a cycle of
patterns with only few peeled segments and those with large number of stuck-
peeled segments as shown in Figs. 9(b) and (c) respectively. The phase plot in
the $X^{s}-v^{s}$ plane of an arbitrary point on the peel front jumps between
the $AB$ to $CD$ branches of $\phi(v^{s})$. As shown in Fig. 9(d), in a cycle,
the trajectory starting at the highest value of $\phi(v^{s})$ stays on $CD$
for a significantly shorter time compared to that on the left branch. The
orbit then cascades down through a series of back and forth jumps between the
two branches of $\phi(v^{s})$. (For $V^{s}=1.48$, independent of $\gamma_{u}$
value, the nature of the phase plot is the same with seven loops.)

The corresponding model acoustic energy consists of rapidly fluctuating time
series with an overall convex envelope of bursts separated by a quiescent
state as shown in Fig. 9(e). (Contrast this with Fig. 9(a) for
$\gamma_{u}=1$.) From the time labels in Figures. 9(b, c), both configurations
belong to the region within the bursts [Fig. 9(e)]. To understand this complex
pattern of bursts in $R_{ae}$ we have looked at the fine structure of each of
these bursts along with the evolution of the associated configurations. One
such plot is shown in Fig. 9(f) which shows that fine structure consists of
seven bursts within each convex envelope. These seven bursts can be correlated
with the seven loops in the phase plot shown in Fig. 9(d). The time interval
marked LM in the phase plot corresponds largely to stuck configuration (not
shown) and hence can be easily identified with the quiescent region in
$R_{ae}$. Following the peel front patterns continuously, it is possible to
identify the sequence of configurations that leads to the substructure shown
in $R_{ae}$ [Fig. 9(f)]. For instance, the loop marked PQRST in the
$X^{s}-v^{s}$ plot corresponds to the burst between P and T in Fig. 9(f).
During this period, the configuration at P is largely in the stuck state (as
in Fig. 9(b)) which gradually evolves with more and more segments peeling out
[Fig. 9(c)] as the trajectory moves from $P\rightarrow Q\rightarrow
R\rightarrow S$. As the number of stuck and peeled segments reaches a maximum,
$R_{ae}$ reaches the peek region. Then, during the interval corresponding to S
to T the number of peeled segments decreases abruptly. Thereafter, the next
cycle of configurations (corresponding to the next loop in the phases plot)
ensues.

As we increase the pull velocity, the peel front is smooth for
$\gamma_{u}=1.0$ as also for 0.1 for the entire range of pull speeds. However,
for $\gamma_{u}=0.01$, as we increase $V^{s}$ to 2.48, the peel process goes
through a cycle of SP configurations shown in Figs. 10(a, b). Note that there
is a large dispersion in the jump velocities as is clear from Fig. 10(a). The
corresponding $R_{ae}$ shows rapidly fluctuating triangular envelope of bursts
with no quiescent region seen for $V^{s}=1.48$ case. This is shown in Fig.
10(c). The corresponding phase plot is similar to Fig. 9(d) but has twelve
loops. In addition, the value of the upper loop extends far beyond that
allowed by $\phi(v^{s})$. We also see a fine structure similar to that in Fig.
9(e). As in the previous case, it is possible to identify configurations that
correspond to minima and near the maxima of $R_{ae}$. As we increase $V^{s}$
further to 4.48, only SP configurations are seen. The energy dissipated
$R_{ae}$ shows continuous bursts overriding a sawtooth form as shown in Fig.
11(b). The $X^{s}-v^{s}$ phase plot shows large excursions way beyond the peel
force function values as shown in Fig. 11(c).

A general comment may be relevant regarding large excursions of the trajectory
in the phase plot $X^{s}-v^{s}$ as we increase the pull velocity. This is
easily explained for the low $C_{f}$ (low tape mass, high roller inertia). It
is clear from Eq. (18) that $m\rightarrow 0$ we have
$F(t)\sim\frac{f(v)}{(1+sin\,\alpha(t))}$. As $\alpha(t)$ can takes on
positive and negative values, one can see $F_{max}$ and $F_{min}$ are
determined by minimum (negative) and maximum values of $\sin{\alpha}$ as
argued in Rumi04 . It is possible to extend this argument to finite tape mass
case. Finally it must be stated that the dynamics is no longer interesting
beyond $V^{s}=7.48$.

Figure 12: Largest Lyapunov exponents of the model for $C_{f}=0.788$ and $\gamma_{u}=0.01$: (a) $V^{s}=1.48$ , (b) $V^{s}=2.48$ and (c) $V^{s}=4.48$. $C_{f}$ |  $V^{s}$ |  $\gamma_{u}$ |  $LLE$
---|---|---|---
$7.88$ |  $2.48$ |  $0.01$ |  $0.110$
|  $4.48$ |  $0.01$ |  $0.102$
$0.788$ |  $1.48$ |  $1.00$ |  $0.095$
|  |  $0.10$ |  $0.120$
|  |  $0.01$ |  $0.148$
$0.788$ |  $2.48$ |  $1.00$ |  $0.068$
|  |  $0.10$ |  $0.090$
|  |  $0.01$ |  $0.139$
$0.788$ |  $4.48$ |  $1.00$ |  $0.028$
|  |  $0.10$ |  $0.030$
|  |  $0.01$ |  $0.035$
$0.00788$ |  $1.48$ |  $0.01$ |  $0.105$
|  $2.48$ |  $0.01$ |  $0.180$
|  $4.48$ |  $0.01$ |  $0.224$
Table 1: Largest Lyapunov exponent for the model for various parameter values.
For all $C_{f}$, the LLE reaches a near value zero for $V^{s}=5.48$.

####  IV.2.4 Spatiotemporal Chaotic Dynamics

As discussed above, there are several sets of parameter values for which the
phase plots are irregular which may suggest the possibility of spatiotemporal
chaotic dynamics. To verify this, we have calculated the largest Lyapunov
exponent (LLE) for all the cases using the model equations. (The transient
solutions for the first 2000 time units have been ignored for calculating the
LLE.) Figure 12 shows a plot of the largest Lyapunov exponent for
$C_{f}=0.788$ and $\gamma_{u}=0.01$ for various pull speeds. The value of LLE
for $V^{s}=5.48$ is close to zero (not displayed in the figure). As can be
seen, the LLE is positive being largest for $V^{s}=1.48$ decreasing to near
zero value for $V^{s}=5.48$. Table 1 shows the values of LLE for various
parameter values for which the spatiotemporal dynamics has been detected. From
this we conclude that the dynamics of the peel front is spatiotemporally
chaotic for a range of parameters values.

##  V  Analysis of AE signals

###  V.1 Statistical Analysis of AE signals

$V$ cm/s |  $m_{A1}$ |  $m_{A2}$ |  $\nu$ |  $\lambda_{1}$ |  $D_{ky}$
---|---|---|---|---|---
$1.0$ |  $...$ |  $2.00$ |  NC |  NC |  NC
$1.6$ |  $...$ |  $2.15$ |  NC |  NC |  NC
$3.0$ |  $0.31$ |  $2.26$ |  NC |  NC |  NC
$3.8$ |  $0.30$ |  $2.75$ |  $2.80$ |  $1.70$ |  $2.94$
$5.0$ |  $0.32$ |  $3.00$ |  $2.70$ |  $1.73$ |  $2.96$
$6.2$ |  $0.27$ |  $3.00$ |  $2.55$ |  $1.54$ |  $2.84$
$7.4$ |  $0.30$ |  $2.99$ |  NC |  NC |  NC
Table 2: Statistical and dynamical invariants for the experimental AE signals
for typical traction velocities. The second and third columns show power law
exponents $m_{A1}$ and $m_{A2}$ corresponding to small and large amplitudes.
When only a single power law is seen, $m_{A2}$ is the exponent value. Fourth
to sixth columns list the values of the correlation dimension $\nu$, the
largest exponent $\lambda_{1}$ and Kaplan-Yorke dimension $D_{ky}$ obtained
from dynamical analysis of the AE signals. NC corresponds to nonchaotic
dynamics where we did not find any convergence of the correlation dimension.

The acoustic emission data obtained from experiments are fluctuating and noisy
only within the domain where the peel process is intermittent form $0.2$ to
$7.6$ cm/s. As known from early experiments MB , force wave forms change as
the traction velocity is increased. Correspondingly, the nature of the AE
signals also change with the traction velocity. At low traction velocities,
the AE signals have a burst like character appearing at nearly regular
intervals separated by oscillatory decay of the amplitudes. These bursts are
correlated with the stick-slip events. With increasing traction velocity, the
bursts become increasingly irregular and continuous. Examples of burst and
continuous type of AE time series are shown in Figures. 13(a, b).

As shown in the previous section, the nature of the model AE signal depends on
parameter values. In general $R_{ae}$ can be of noisy burst type overriding a
periodic component, continuous and irregular, rapidly fluctuating triangular
envelope of bursts or simply a set of spikes. Clearly, interesting cases for
comparison with the experimental AE signals are those where $R_{ae}$ is
continuous and noisy. Simplest quantity to compare is the nature of the model
acoustic signal with the energy of the experimental AE signal (i.e., square
the amplitude). Figures 13(c, d) show a comparison between model acoustic
signal for $C_{f}=7.88,\gamma_{u}=0.01$ and $V^{s}=1.48$ and energy of the
experimental signal for $V=0.4$ cm/s. Both show burst type emission. As
another example Figs. 13(e, f) show respectively the continuous model signal
(for $C_{f}=7.88,\gamma_{u}=0.01$ and $V^{s}=4.48$) and experimental acoustic
energy for $V=6.4$ cm/s.

Figure 13:  (a, b) Raw AE signal for $V=1.6$ cm/s and 7.6 cm/s respectively.
(c) Burst like model acoustic energy plot for $C_{f}=7.88,V^{s}=1.48$ and
$\gamma_{u}=0.01$. (d) Burst like experimental acoustic energy for $V=0.4$
cm/s. (e) Continuous model acoustic energy plot for $C_{f}=7.88,V^{s}=4.48$
and $\gamma_{u}=0.01$. (f) Continuous experimental acoustic energy for $V=6.4$
cm/s.

Given an experimental time series, the simplest statistical quantity to
compute is the statistics of events. The definition of events depends on the
physical situation, which in the case of the AE signal may be the time
interval between the bursts of AE, the amplitude of bursts etc. Indeed the
former has been computed CGVB04 . Here we compute the distribution of the
amplitudes of the AE signals. The difference between the maximum and next
minimum, denoted by $\Delta A$ can be taken to be a measure of the amplitude
of the AE signal. (In experiments, it is measured by setting a cut-off and
measuring all amplitudes larger than the cut-off.) We have computed the
distribution of the amplitudes $D(\Delta A)$ for all the 38 data files for
pull velocities starting from $0.2$ to $7.6$ cm/s. Surprisingly, we find only
power law distributions for all the data files, i.e., $D(\Delta A)\sim\Delta
A^{-m_{A}}$; we do not find peaked distributions. For small traction
velocities, we find a single power law crossing over to a two stage power law
for high traction velocities. A typical single power law distribution for
$V=1.6$ cm/s is shown in Fig. 14(a) with an exponent $m_{A}=2.15$. Figure
14(b) shows a two stage power law for high velocity $V=5.0$ cm/s. The exponent
values are $m_{A}=0.32$ and $3.0$ respectively for small and large amplitude
regimes. The transition from a single to two stage power law distribution
occurs with the deviation for small values seen in Fig. 14(b) becoming more
dominant with increase in the pull velocity. A two stage power law (over one
order of magnitude range) is first observed for $V=3.0$ cm/s. The exponent
values are functions of the pull velocity. Table 2 shows the exponent values
for a selected set of pull velocities. Even though statistical features are
easy to calculate, they are sufficiently discriminating. The analysis will be
useful while comparing the cured data files as also with the statistics of
model acoustic energy signals.

These results may be compared with the statistics of the amplitude of the
model energy bursts $R_{ae}$, i.e., from the maximum to the next minimum.
Denoting $\Delta R_{ae}$ to be the amplitude of $R_{ae}(\tau)$, let $D(\Delta
R_{ae})$ be the distribution of the amplitude of $R_{ae}$. For
$C_{f}=0.00788$, only $\gamma_{u}=0.01$ case is interesting (see Fig. 10(d)
corresponding to $V^{s}=2.48$). To determine $D(\Delta R_{ae})$, we use long
time series (typically $\sim 10^{5}$ points in units of the integration step).
For this case, we find a two stage power law as shown in Fig. 14(d) i. e.,
$D(\Delta R_{ae})\sim\Delta R_{ae}^{-m_{E}}$ with the exponents $m_{E}\sim
0.60$ and 2.0 for the small and large amplitude regimes respectively. Figure
14(c) shows an example of a single power law for $C_{f}=0.788,\gamma_{u}=0.1$
and $V^{s}=1.48$. The exponent value is $m_{E}=0.7$. To compare, we note that
the experimental time series refers to the amplitude of the AE signals while
the model signal $R_{ae}$ is the energy. Thus, the two exponents are related
through $m_{E}=(m_{A}+1)/2$. Using the value of $m_{E}$ shown in Fig. 14(d),
we obtain $m_{A1}=0.2$ for the exponent corresponding to small amplitude
regime (of the AE signal) and $m_{A2}=3.0$ for large amplitudes. Clearly, the
values are in reasonable agreement with the exponent values for the
experimental signals [Fig. 14(b)]. Unlike the experimental signal where
scaling regime is good for all values of the pull velocity, for the model
acoustic energy $R_{ae}$, the distributions for $\Delta R_{ae}$ show a power
law statistics (of at least one order of scaling regime) only for a certain
sets of parameter values. Table 3 lists the exponent values wherever the power
law distribution is seen. For high and low values of $C_{f}$ we find two stage
power law distributions. However, for the intermediate $C_{f}$( 0.788), only a
single stage power law is found. It is interesting to note that the power law
generated here is purely of dynamical origin.

Figure 14:  (a) Plot of a single stage distribution of the amplitudes of the uncured data for V = 1.6 cm/s. (b) Plot of a two stage power law distribution of the amplitudes of the uncured data for V = 5.0 cm/s. (c) A plot of a single stage power law distribution for the magnitudes of acoustic energy dissipated for $C_{f}=0.788,\gamma_{u}=0.01$ and $V^{s}=1.48$. (d) A plot of a two stage power law distribution for the magnitudes of model acoustic energy for $C_{f}=0.00788,\gamma_{u}=0.01$ and $V^{s}=2.48$. $C_{f}$ |  $V^{s}$ |  $\gamma_{u}$ |  $m_{E1}$ |  $m_{E2}$ |  $\nu$ |  $\lambda_{1}$ |  $D_{ky}$
---|---|---|---|---|---|---|---
$7.88$ |  $2.48$ |  $0.01$ |  $0.45$ |  $2.10$ |  $2.40$ |  $1.05$ |  $2.90$
|  $4.48$ |  $0.01$ |  $0.65$ |  $1.97$ |  $2.35$ |  $0.46$ |  $2.74$
$0.788$ |  $1.48$ |  $1.00$ |  $0.55$ |  $...$ |  $2.45$ |  $1.53$ |  $2.74$
|  |  $0.10$ |  $0.70$ |  $...$ |  $2.49$ |  $1.80$ |  $2.77$
|  |  $0.01$ |  $0.65$ |  $...$ |  $2.15$ |  $1.85$ |  $2.45$
$0.788$ |  $2.48$ |  $1.00$ |  $1.00$ |  … |  2.45 |  1.50 |  2.48
|  |  $0.10$ |  $0.90$ |  … |  2.54 |  1.78 |  2.59
|  |  $0.01$ |  $0.67$ |  … |  $2.45$ |  $1.59$ |  $2.86$
$0.788$ |  $4.48$ |  $1.00$ |  $0.70$ |  … |  2.55 |  1.48 |  2.86
|  |  $0.10$ |  $0.60$ |  … |  2.35 |  1.57 |  2.53
|  |  $0.01$ |  $0.72$ |  … |  $2.50$ |  $1.50$ |  $2.52$
$0.00788$ |  $1.48$ |  $0.01$ |  $0.74$ |  $2.0$ |  NC |  NC |  NC
|  $2.48$ |  $0.01$ |  $0.60$ |  $2.0$ |  $2.20$ |  $0.32$ |  $2.40$
|  $4.48$ |  $0.01$ |  $0.75$ |  $2.0$ |  $2.70$ |  $0.13$ |  $2.76$
Table 3: Statistical and dynamical quantities for the model acoustic signal.
Columns 4 and 5 show the power law exponents. Columns 6 to 8 list the
correlation dimension, positive exponent and Lyapunov dimension respectively.

###  V.2 Dynamical Analysis of AE time series

In Section IV, we showed that the peel front patterns for several sets of
parameters are spatiotemporally chaotic. More importantly, the model acoustic
energy is quite irregular even as it is of dynamical origin. This suggests the
possibility that the experimental AE signals could be chaotic. However, often
time series have undesirable systematic component, which needs to be removed
from the original data. For instance, in the PLC effect, the stress-strain
time series has an overall increasing stress arising from the work hardening
component of the stress Anan99  which needs to be subtracted. In the present
case, the experimental data for high pull velocities does show a background
variation. A simple way of eliminating this background component is to use a
window averaging and subtract this component from the raw data. Moreover, as
stated in the introduction, the experimental AE data are quite noisy and
therefore it is necessary to cure the data (using standard noise reduction
techniques HKS ) before subjecting them to further analysis. Simple visual
checks for the existence of chaos such as phase plots, power spectrum etc.
have been carried out. We have also used singular value decomposition, false
neighbor search etc. Figures 15(a, b) show the raw and cured data respectively
for $V=5.0$ cm/s. Clearly, the dominant features of the time series are
retained except that small amplitude fluctuations are reduced or washed out
HKS . Statistical features like the distribution function for the amplitude of
the AE signals, power spectrum etc. are not altered. For instance, the two
stage power law distribution for the amplitude of AE signals for the raw data
(shown in Fig. 14(b)) is retained except that the exponent value for the small
amplitude regime is reduced from $0.32$ to $0.24$ without altering the
exponent corresponding to large amplitudes. This reduction is understandable
as small amplitude fluctuations are affected during curing.

The cured data are used to calculate the correlation dimension for all the
data files. However, for calculating the Lyapunov spectrum using our
algorithm, raw data is adequate as our algorithm is designed to process noisy
data. (In contrast, calculating the Lyapunov spectrum using the TISEAN package
requires the cured data). To optimize the computational time, all our
calculations are carried out using one fifth of each data set as each file
contains large number of points $\sim 10^{6}$ points, and there are 38 data
sets.

Figure 15:  (a, b) Raw and cured AE signal respectively for $V=5.0$ cm/s. (c)
Correlation integral for pull velocity $3.8$ cm/s for $d=9$ to $13$. Dashed
lines are guide to eye. (d) Lyapunov spectrum for the same data file. (e)
Correlation integral for pull velocity $5.0$ cm/s from $d=7$ to $10$. Dashed
lines are guide to eye. (f) Lyapunov spectrum of the AE signals for traction
velocity $5.0$ cm/s.

Figure 16:  (a) Correlation integral of $R_{ae}(\tau)$ for
$C_{f}=0.00788,\gamma_{u}=0.01$ and $V^{s}=2.48$ with $\nu=2.2$ ($d=5$ to
$8$). Dashed lines are guide to eye.(b) The Corresponding Lyapunov spectrum
for $R_{ae}$.

Typical autocorrelation time is about four units in sampling time. However,
using a smaller value of $\tau=1$, we have calculated the correlation integral
$C(r)$ for all the data files. Converged values of correlation dimension are
seen only in the region of pull velocities in the subinterval 3.8 to 6.2 cm/s.
A log-log plot of $C(r)$ for the pull velocity $3.8$ cm/s is shown in Fig.
15(c) for $d=9$ to $13$. A scaling regime of more than three orders of
magnitude is seen with $\nu\sim 2.80\pm 0.05$. This is at the beginning of the
chaotic window.

We have calculated the Lyapunov spectrum using our algorithm. The Lyapunov
spectrum for $V=3.8$ cm/s is shown in Fig. 15(d). (The outer shell radius is
kept at $\epsilon_{o}=0.065$.) Note that the second exponent is close to zero
as should be expected of continuous flow systems. Using the spectrum, we have
calculated the Kaplan-Yorke dimension (also called Lyapunov dimension)
$D_{ky}$ using the relation
$D_{ky}=j+\frac{\sum_{i=1}^{j}\lambda_{i}}{|\lambda_{j+1}|};\sum_{i=1}^{j}\lambda_{i}>0;\sum_{i=1}^{j+1}\lambda_{i}<0$.
The value so obtained in each case should be consistent with that obtained
from the correlation integral. For the case shown in Fig. 15(d) we get
$D_{ky}=2+1.70/1.81=2.94$ consistent with $\nu=2.80$. (Typical error bars on
the first three Lyapunov exponents are $\pm 0.01,\pm 0.005$ and $\pm 0.05$.
Thus the errors in $D_{ky}$ values are $\pm 0.05$.)

As an example of converged value of correlation dimension near the upper end
of the chaotic domain, a log-log plot of $C(r)$ for $V^{s}=5.0$ cm/s is shown
in Fig. 15(e) with $\nu=2.73\pm 0.05$ for $d=7$ to 10. Again, the scaling
regime is seen to be nearly three orders of magnitude. The Lyapunov spectrum
for the data file is shown in Fig. 15(f). The calculated Lyapunov dimension
from the spectrum is $D_{ky}=2+1.73/1.80=2.96$ which is again consistent with
$\nu=2.73$.

The values of $\nu$ for all the files are found to be in the range $2.55$ to
$2.85\pm 0.05$ as can be seen from Table 2. We have calculated the Lyapunov
spectrum for the full range of traction velocities and we find (stable)
positive and zero exponents only in the region 3.8 to 6.2 cm/s, consistent
with the range of converged values of $\nu$ as can be seen from Table 2. The
corresponding values of $D_{ky}$ are in the range of $2.7$ to $3.0$. We have
also calculated the Lyapunov spectrum using the TISEAN package using cured
files. The $D_{ky}$ values obtained from the TISEAN package are uniformly
closer to the $\nu$ values, typically $D_{ky}=\nu+0.1$. Finally, we note that
the positive exponent decreases toward the end of the chaotic domain (6.2
cm/s). These results (see Table 2) show unambiguously that the underlying
dynamics responsible for AE during peeling is chaotic in a mid range of pull
speeds.

In order to compare the low dimensional chaotic nature of the experimental AE
signals with the model acoustic signal, we have analyzed the low dimensional
dynamics of $R_{ae}(\tau)$ using the embedding procedure after subtracting the
periodic component.

We have computed the correlation dimension and Lyapunov spectrum for the
entire instability domain. A log-log plot of the $C(r)$ is shown in Fig. 16(a)
for $d=5$ to 8. The convergence over more than three orders of magnitude is
clear. The value of $\nu=2.20\pm 0.05$. For this file, we find stable positive
and zero exponents for a range of $\epsilon_{o}$ values. A plot of the
spectrum for $C_{f}=0.00788,\gamma_{u}=0.01$ and $V^{s}=2.48$
($\epsilon_{o}=0.08$) is shown in Fig. 16(b). Using this we get
$D_{ky}=2+0.32/0.77=2.4$ which is again consistent with $\nu=2.2\pm 0.02$.

We have calculated both correlation dimension and Lyapunov spectrum of
$R_{ae}$ for a range of values of the parameters. For each $C_{f}$, we find
converged values of $\nu$ and $D_{ky}$ within a window of pull speeds.
Generally, the range of $\nu$ is between 2.15 to 2.70 while $D_{ky}$ is in the
range 2.4 to 2.90. Table 3 shows the values of correlation dimension and
$D_{ky}$ for various sets of parameter values. It is interesting to note that
the magnitude of the largest exponent for the model AE signal also decreases
as we increase the pull velocity, a feature displayed by the experimental time
series as well.

##  VI Summary and Conclusions

In summary, the present investigation is an attempt to understand the origin
of the intermittent peeling of an adhesive tape and its connection to acoustic
emission. At the conceptual level, we have established a relationship between
stick-slip dynamics and the acoustic energy, the latter depends on the local
strain rate Land  which in turn is controlled by the roughness of the peel
front. As the model is fully dynamical, one basic result that emerges is that
the model acoustic energy is controlled by the nature of spatiotemporal
dynamics of the peel front. Further, even as the model acoustic emission is a
dynamical quantity, the nature of $R_{ae}$ turns out to be quite noisy
depending on the possible interplay of different time scales in the model.
Thus, the highly noisy nature of the experimental signals need not necessarily
imply stochastic origin of AE signal; instead, they could be of deterministic
origin. This motivated us to carry out a detailed analysis of statistical and
dynamical features of the experimental AE signals. Despite the high noise
content, we have been able to demonstrate the existence of finite correlation
dimension and positive Lyapunov exponent for a window of pull speeds. The
Kaplan-Yorke dimension (for various traction velocities) calculated from the
Lyapunov spectrum is consistent with the value obtained from the correlation
integral. Thus, the analysis establishes unambiguously the deterministic
chaotic nature of the experimental AE signals. Interestingly, the largest
Lyapunov exponent shows a decreasing trend toward the end of the chaotic
window, a feature displayed by the model acoustic signal as well. The work
also addresses the general problem of extracting dynamical information from
noisy AE signals. A similar analysis of the model acoustic energy shows that
$R_{ae}$ is chaotic for a range of parameter values. More importantly, several
qualitative features of the experimental AE signals such as the statistics of
the signals and the change from burst to continuous type with increase in the
pull velocity are also displayed by $R_{ae}$. The observed two stage power law
distribution for the experimental AE signals [Fig. 14(b)] is reproduced by the
model [Fig. 14(d)]. It must be emphasized that this power law distribution for
the amplitudes is completely of dynamical origin. This result should be of
general interest in the context of dynamical systems as there are very few
models that generate power laws purely from dynamics. The only other example
known to the authors is that of the PLC effect where the amplitude of the
stress drops shows a power law distribution within the context of the
Ananthakrishna model Anan04 .

The spatiotemporal patterns of the peel front are indeed rich and depend on
the interplay of the three time scales. Although, the nature of spatiotemporal
patterns is quite varied, they can be classified as smooth synchronous,
rugged, stuck-peeled and even nowhere stuck patterns. As expected on general
consideration of dynamics, rich patterns are observed for the case when all
the three time scales are of similar magnitude (illustrated for
$C_{f}=0.788$). All spatiotemporal patterns, except the smooth synchronous
peel front are interesting. As a function of time, the nature of the peel
front can go through a specific sequence of these patterns (depending on the
parameter values). The most interesting pattern is the stuck-peeled
configuration which is reminiscent of fibrils observed in experiments
Dickinson ; Urahama ; Yamazaki . Even among the SP configurations, there are
variations, for example, rapidly changing, long lived, edge of peeling etc.
Despite the varied range of patterns, a few general trends of the influence of
the parameters on the peel front patterns are worth noting. First, in general
the number of stuck and peeled segments increases as $\gamma_{u}$ is
decreased. Second, as the pull velocity is increased, the rapidly varying
stuck-peeled configurations observed at low pull velocities become long lived.
The dynamical signature of these two parameters are reflected in the nature of
the phase space orbit. For instance, given a value of $C_{f}$ and
$\gamma_{u}$, the nature of the phase space orbit changes only when $V^{s}$ is
increased which allows the orbit to move way beyond the values of
$\phi(v^{s})$.

The study of the model shows that while the intermittent peeling is controlled
by the peel force function, the dynamics of the peel front is influenced by
all the three time scales. This together with the dynamical analysis of the
experimental acoustic emission signals establishes that deterministic dynamics
is responsible for AE during peeling. The various sequences of peel front
patterns and their time dependences lead to quite varied model acoustic
signals. These can be classified as bunch of spikes, isolated bursts occurring
at near regular intervals, continuous bursts with an overall envelope
separated by quiescent state, continuous bursts overriding a near periodic
triangular form, irregular waveform overriding a periodic component, and
continuous irregular type. Interestingly, our studies show that there is a
definite correspondence between the model acoustic energy and the nature of
peel front patterns even though $R_{ae}(\tau)$ is the spatial average of the
local strain rate. Despite this, two distinguishable time scales in
$R_{ae}(\tau)$ can be detected, one corresponding to short term fluctuations
and another corresponding to overall periodic component. The short term
fluctuations can be readily identified when the model acoustic signal is
fluctuating without any background component (see for example Fig. 6(d)).
These rapid changes in $R_{ae}$ arise due fast dynamic changes in the SP
configurations. The minimum in $R_{ae}$ corresponds to the situation where the
average velocity jumps of the SP configurations are smaller compared to that
at the preceding maximum. In contrast, the overall periodicity in
$R_{ae}(\tau)$ (for instance see Fig. 7(c) among many other cases) can be
identified with the changes in the peel front patterns that occur over a cycle
in the phase plot. The minima in the $R_{ae}(\tau)$ corresponds to the peel
patterns where more segments are in stuck state than in the peeled state while
the maxima corresponds to more stuck and peeled segments (see Figs. 7(a),
(b)). The corresponding phase plot usually goes through a cycle of visits
between the low and high velocity branches.

Often, however, the nature of the model acoustic energy signal can be
complicated as in the case of low tape mass ($C_{f}=0.00788$). Even in such
cases, some insight is possible. This is aided by the analysis of the
corresponding phase plot. For example for the low $C_{f}$ case where the
roller inertia plays an important role in the dynamics, the rapidly
fluctuating acoustic energy has an overall triangular envelope [Fig. 9(e)]. On
an expanded scale, the convex envelope consists of seven local peaks [Fig.
9(f)]. Each of these is generated when the various peel front segments make
abrupt jumps between the two branches of the peel force function. Note that
the phase space orbit has seven loops in this case [Fig. 9(d)]. The general
identification of the minima in $R_{ae}$ with patterns that have more stuck
segments than peeled segments still holds. Similarly, the maxima in $R_{ae}$
usually correspond to the presence of large number of stuck-peeled
configurations.

The above correspondence between the model acoustic energy and the peel front
patterns provides insight into the transition from burst to continuous type of
AE seen in experiments as a similar transition from burst type to continuous
type is also seen in the model acoustic energy (for large $C_{f}=7.88$ ). At
low pull velocities, the peel front goes through a cycle of patterns where
most segments of the peel front (or the entire peel front) spends substantial
time in the stuck state switching to stuck-peeled configuration. As the
duration of the SP configuration is short and velocity bursts are large,
$R_{ae}(\tau)$ is of burst type [Fig. 13(c)]. With increasing pull velocity,
only dynamic stuck-peeled configurations are seen which in turn leads to
continuous AE signals [Figures. 4(a, b)]. This coupled with time series
analysis of the model acoustic signal shows that the associated positive
Lyapunov exponent decreases with increase in the traction velocity. This is
precisely the trend observed for experimental signals as well. Thus, the
decreasing trend of the largest Lyapunov exponent can be attributed to the
peel front breaking up into large number of small segments providing insight
into stick-slip dynamics and its connection to the AE process.

The present study has relevance to the general area of stick-slip dynamics. As
mentioned earlier, models for stick-slip dynamics use negative force-drive
rate relation. In such models, the phase space orbit generally sticks to the
slow manifold (stable branches) of the force-drive rate function. This leads
to clearly identifiable stick and slip phases, the former lasting much longer
than the latter. However, recent work on imaging the peel point dynamics
Cortet  shows that the ratio of the stick phase to the slip phase, is about
two or even less than unity for high peel velocities. While all the known
models of the peel process predict that the duration of the stick phase is
longer than that of the slip phase, our model displays the experimentally
observed feature. This feature emerges in the model due to the interplay of
the three time scales aided by incomplete relaxation of the relevant modes.
Our studies show that only for low pull velocity and high $C_{f}$ do we
observe the stick phase lasting much longer than the slip phase. As the pull
velocity is increased, and for all other parameter values, we find that the
duration of the slip phase (peel velocity being larger than unity) is nearly
the same as or less than that of the stick phase (peel velocity less than
unity). Further, the present model provides an example of the richness of
spatiotemporal dynamics arising when more than two time scales are involved.
In this context, we emphasize that the introduction of the Rayleigh
dissipation functional to model the acoustic energy is crucial for the
richness of the spatiotemporal peel front patterns. It is important to note
that this kind of dissipative term is specific to spatially extended systems
as it represents relaxation of neighboring points on the peel front.

The present study has relevance to time dependent issues of adhesion. For
instance, apart from the fact that the time series analysis addresses the
general problem of extracting dynamical information from noisy AE signals, it
may have relevance to failure of adhesive joints and composites that are
subject to fluctuating loads. The failure time can be estimated by calculating
the Lyapunov spectrum for the AE signals. If the largest Lyapunov exponent is
positive, the inverse of the exponent should give an estimate of the time
scale over which the failure can occur and hence could prove to be useful in
predicting failure of joints. The present study should also help to optimize
production schedules in peeling tapes.

Finally, several features of the present study are common to the PLC effect
even though the underlying mechanism is very different. In this case the
repeated occurrence of stress drops during constant strain rate deformation
PLC ; GA07 ; GA07a , are associated with the formation and possible
propagation of dislocation bands that are visible to the naked eye. The
phenomenon occurs only in a window of applied strain rates. The instability is
attributed to the pinning and unpinning of dislocations from solute
atmosphere, yet, the dominant feature underlying the instability is the
negative strain rate sensitivity of the flow stress that has two stable
branches separated by an unstable branch. Clearly, these features are similar
to the occurrence of the peel instability within a window of pull velocities
and the existence of unstable branch in the peel force function. Further, the
Ananthakrishna (AK) model for the PLC instability predicts that the stress
drops should be chaotic in a subinterval of the instability domain Anan82 .
This prediction has been verified subsequently through the analysis of
experimental stress-strain curves obtained from single and polycrystals Anan93
; Anan95 ; Anan99 ; Bhaprl . This feature is again similar to the existence of
chaotic dynamics observed in a mid range of pull velocities in the peeling
problem, both in experiment and in the model. In the case of the PLC effect,
one finds that the positive Lyapunov exponent characterizing the stress-time
series decreases toward the end of chaotic window, both in experiments and in
the AK model. Again this feature is also seen in the present peel model as
also in experimental AE signals. Dynamically, in the case of the AK model for
the PLC effect the decreasing trend of the positive Lyapunov exponent has been
shown to be a result of a forward Hopf bifurcation (HB) followed by a reverse
HB Anan04 . In the case of peeling problem as well, the instability begins
with a forward HB followed by a reverse HB. Finally, in the PLC effect (both
in experiments and the AK model), as in the peeling problem, the duration of
the slip phase can be longer than that of the stick phase with increasing
drive rate. As many of these features are common to two different systems, it
is likely that these are general features in other stick-slip situations with
multiple time scales that are limited to a window of drive rates with multiple
participating time scales.

A few comments may be in order about the model, in particular about the
parameters that are crucial for the dynamics. While the agreement of several
statistical and dynamical features of the model ( for several sets of
parameter values) with the experimental AE series is encouraging, it would be
interesting to verify model results for other sets of parameters. For
instance, it is clear that the roller inertia and the inertia of the tape mass
are experimentally assessable parameters. Thus, the influence of these two
inertial time scales can in principal be studied in experiments. However,
conventional experiments have been performed keeping these parameters fixed,
presumably, as there has been no suggestion that the dynamics can be sensitive
to these variables. It would be interesting to verify the predicted dynamical
changes in the AE signals as a function of these two parameters. As for the
influence of the dissipation parameter $\gamma_{u}$, the range of physically
reasonable values of $\gamma_{u}$ is expected to be small ($10^{-4}$ to
$10^{-3}$) as argued. Interestingly, the region of low $\gamma_{u}$ is indeed
the region where both statistics and dynamical features compare well with that
of the experiments. However, within the scope of the model, the visco-elastic
properties of the adhesive have been modeled using an effective spring
constant. (This kind of assumption is common to studies in adhesion and
tackiness etc Gay .) However, it is possible to include this feature as well.
Finally, it must be stated that features that critically depend on thickness
of the film and its visco-elastic properties such the shape of peel front are
beyond the scope of the present model.

Acknowledgment: The authors wish to thank Professor M. Ciccotti for providing
the AE data. GA acknowledges support from BRNS Grant No.
$2005/37/16/BRNS,2007/36/62-BRNS/2564$, and Raja Ramanna Fellowship.

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