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Journal of Applied Mechanics
1
Adhesion Asymmetry in Peeling of Thin
Films with Homogeneous Material
Properties: A Geometry-Inspired Design
Paradigm
Ahmed Ghareeb
Civil and Environmental Engineering, University of Illinois at Urbana-Champaign
2119 Newmark Civil Engineering Lab, 205 N. Mathews Ave, Urbana, IL 61801
ghareeb2@illinois.edu
Ahmed Elbanna
1
Civil and Environmental Engineering, University of Illinois at Urbana-Champaign
2219 Newmark Civil Engineering Lab, 205 N. Mathews Ave, Urbana, IL 61801
elbanna2@illinois.edu
ABSTRACT
Peeling of thin films is a problem of great interest to scientists and engineers. Here, we study the peeling
response of thin films with non-uniform thickness profile attached to a rigid substrate through a planar
homogeneous interface. We show both analytically and using finite element analysis that patterning the
film thickness may lead to direction-dependent adhesion such that the force required to peel the film in
one direction is different from the force required in the other direction, without any change to the film
material, the substrate interfacial geometry, or the adhesive material properties. Furthermore, we show
that this asymmetry is tunable through modifying the geometric characteristics of the thin film to obtain
higher asymmetry ratios than reported previously in the literature. We discuss our findings in the broader
context of enhancing interfacial response by modulating the bulk geometric or compositional properties.
Keywords
Interfacial adhesion; Adhesion Asymmetry; Peeling Force;
1
Corresponding author.
Journal of Applied Mechanics
2
1. INTRODUCTION
Understanding the mechanics of adhesion of thin films to rigid substrates
continues to be a topic of extensive research in science and engineering. Films attached
to substrates are very common in many engineering and biological systems such as solar
panels [1], integrated circuits [2], flexible electronics [3], packing and adhesive tapes [4],
coatings [5], and medical tapes [6]. The bonding strength of the interface is usually
quantified by the adhesion energy [7], defined as the amount of energy required to
create a unit area of fractured surface. In many applications, strong bonding between
the film and the substrate is required to ensure system reliability. However, other
properties may be desirable such as having asymmetric response when peeling the thin
film from one direction versus the other.
Asymmetry in the peeling of thin films is the dependence of peeling force,
adhesion energy, or both on the direction or the axis of peeling. Asymmetry may be
utilized to achieve reversible control of adhesion, which is the ability to change adhesion
strength from weak to strong modes, through changing the direction of peeling. This
feature is critical to many existing and potential engineering systems, including but not
limited to, climbing robots, medical tapes, and stamps for transfer printing [8].
In the past few decades, extensive research has been conducted to better
understand various aspects of the peeling mechanism of thin films in terms of both the
adhesion strength and energy [7, 9]. Several studies have focused on improving the
bond strength and adhesion energy by changing the mechanical properties of the
interface, through the modification of interface topology and roughness, to alter the
Journal of Applied Mechanics
3
conditions of nucleation and propagation of peel front. [10–13]. This pointed to the
possible role of adherent or interface heterogeneities in controlling the peeling
response and opened new opportunities for utilizing these heterogeneities to improve
the adhesion [14–21].
The peeling asymmetry has also been realized through a variety of techniques
such as using non-uniform distribution of adhesive strength [22], adding arc patterns on
the interface [18], changing the geometry of the interface through adding arrays of
tilted microfibrils [23], and adding arrays of nano-fabricated polymer pillars coated with
a thin layer of a synthetic polymer [24] inspired by geckos [25]. Other studies included
using a duplex attachment pad composed of two geometrically identical wedge shaped
blocks with different modulus of elasticity [26], or using shape memory polymer
surfaces with geometrically asymmetric micro-wedge arrays for reversible dry adhesives
[27]. However, none of these studies have demonstrated the possibility of inducing
adhesion asymmetry in the peeling response using solely changes in the thickness of the
film.
In this paper, we focus on studying the peeling response of homogeneous thin
films, with patterned thickness, from rigid homogeneous substrates. We show using a
semi-analytical approach and finite element analysis that varying the film thickness may
lead to asymmetry in the peeling response. Furthermore, we show that this asymmetry
is tunable and can reach higher values than that reported in several previous studies
[18, 22, 27]. We also investigate the dependence of the asymmetry ratio on various
model parameters.
Journal of Applied Mechanics
4
2. RESULTS
2.1 Theoretical Prediction for Adhesion Asymmetry in Thin films with Variable
Thickness
We follow the analytical procedure developed by Xia et al. [16] for the analysis of
adhesion in heterogeneous thin films. Fig. 1-a shows a schematic diagram of the non-
uniform strip under consideration here. The strip is modeled as an inextensible Euler-
Bernoulli beam. This assumption holds for moderate or large peel angles [9]. The strip is
discretized into N segments, having the same elastic modulus and Poisson’s ratio
, but each one has a different thickness , and hence different bending rigidity
where is the second moment of area of the
segment, and is the strip width. The strip thickness is described by a sawtooth
function with thickness varying between and as shown in Fig. 1-b.
The strip profile is uniquely described by a function of where is the angle
between the tangent to the strip and the horizontal plane at discretization node , and
is the arc length from the origin point along the strip to node . The film is perfectly
bonded to the rigid substrate up to the point where , and is being peeled by a
force with peeling angle . The potential energy of the idealized beam is given by:
(1)
Where is the adhesion energy per unit length of the interface between the strip and
the substrate. Rate-independent adhesion energy is assumed in this study similar to
some prior work [16]. However, the qualitative nature of the results is not expected to
Journal of Applied Mechanics
5
change if rate dependence is assumed as long as inertia effects are dominated by
damping forces. We also assume a smooth homogeneous interface and thus the value
of is taken to be spatially constant. Heterogeneities in , for example due to surface
roughness, may lead to adhesion asymmetry on its own [18] and thus we elect not to
include it in this study to critically evaluate the role of film geometry alone on the
effective adhesion.
The condition of equilibrium of the strip is that the first variations of Eq. 1
vanishes. As shown in Xia et al. [16] this yields the following relation for the peeling
force and the thin film elastic line:
(2)
And
(3)
Where . Eq. 2, 3 are solved numerically to
determine the force F and the slope angles . The solution scheme and the details
of the discretization process are outlined in the methods section.
Since the thickness profile of the strip is sawtooth as shown in Fig. 1-b, the
normal stresses are redistributed at the locations where the thickness is changing
abruptly leading to a smoother transition in the stress profile. The effective thickness
for resisting bending deformation is thus smaller than the local thickness of the
strip in these regions and the two thickness measures coincide only after a horizonal
Journal of Applied Mechanics
6
transition distance as shown in Fig. 1-c. This horizontal distance is estimated using the
finite element model under the same boundary conditions as the theoretical model, i.e.
the film is perfectly bonded to the rigid substrate up to the crack front, and is found to
be nearly equal to the step change in the thickness. We use the effective thickness
profile to calculate the rigidities of segments when numerically integrating Eq. 2, 3.
We normalize the peak force with respect to the peeling force of a
homogenous strip given by [7]. Furthermore, we follow Xia et
al. [16] and introduce the following length scale:
(4)
Where is the effective rigidity of the strip, and is the adhesion energy. This length
scale represents the length of a strip of stiffness subjected to a moment generated
by a force proportional to the adhesion energy . We will use this length scale to
normalize the period . For a strip with varying thickness, we define the effective rigidity
as .
We use the following parameters in the numerical integration of the model
unless otherwise stated: Young’s modulus = 1 GPa, and Poisson’s ratio = 0.35 and
fracture energy = 35 J/m2 [4]. The peeling angle is kept constant at 90 degrees.
The results of the semi-analytical model are shown in Fig. 2. The results in Fig. 2-
a,b show that peeling force required to advance the peeling front in direction 1
(Forward peeling) is higher than that required for direction 2 (Backward peeling). We
Journal of Applied Mechanics
7
call the ratio between the maximum force in each direction the Asymmetry ratio. For
forward peeling, the peel front stops at the location of the sudden increase in the
thickness. The force increases till it reaches a peak value, then the peel front propagates
with a decrease in force until the peeling front reaches the next sudden increase in
thickness. However, for backward peeling, the force increases gradually while the peel
front propagates in the direction of increasing thickness. The force then drops as the
thickness is abruptly reduced. For both directions, the peak force is higher than that of
a homogenous strip. The sudden increase in the normalized peak force in the case of
forward propagation is similar to the case of a strip with piecewise thickness, i.e. strip
with uniform high thickness segment followed by low thickness segment [16]. However,
in the case of piecewise thickness distribution, the peeling behavior is symmetric, and
the peak force is the same in both peeling directions.
To verify the predictions of the analytical model, we use the finite element
software package ABAQUS [28] to simulate a quasi-static peeling test of a thin film with
the same saw tooth profile and model parameters as used in the analytical model. The
FEM model is composed of three main parts: (i) a rigid base plate that represents the
substrate, (ii) a thin film with varying thickness, and (iii) a zero-thickness cohesive layer
joining the film and the substrate to represent the adhesion between the two
components. The details of the model setup are included in the methods section.
Journal of Applied Mechanics
8
The length on the interface between crack initiation and total separation defines
the cohesive zone length , which depends on the film thickness and a characteristic
length that depends on the bulk material properties and the cohesive zone model
parameters [29]. The characteristic length of the cohesive law is given by [30]:
(5)
Where is young’s modulus of the bulk material, is the cohesive model energy
release rate, and is the critical cohesive stress. This is another important length scale
that controls the adhesion response as we will discuss later.
Figure 2-a,b show the normalized force vs the normalized peel front position,
whereas Fig. 2-c,d show the normalized force vs the normalized vertical peel
displacement. The finite element results are plotted along with the theoretical results
using the effective thickness. Both the finite element and the theoretical model
solutions share the same trend and match well in the middle part of the unit cells as
shown in Fig. 2-a,b . Some discrepancy exists at the location of sudden change in
thickness due to the complicated stress redistribution at this zone. In addition, the finite
element normalized force vs peel displacement plots shown in Fig. 2-c,d for both the
forward and backward propagation show a snap-back effect due to the nature of the
displacement controlled loading we adopt in the finite element model which does not
allow the tip displacement to decrease. The hatched areas in Fig. 2-c,d represent the
energy dissipated due to this snap-back instability. The effective energy release rate
Journal of Applied Mechanics
9
calculated from the finite element model for this case is for forward
propagation and for Backward propagation.
Figure 2-e,f show the maximum normalized peak force as a function of the
normalized period for the theoretical model using the actual thickness, the theoretical
model using the effective thickness, and the finite element results for both forward and
backward peeling and thickness ratio of 0.5. For forward propagation, the semi-
analytical model using the actual thickness is completely off the other two curves. The
sudden change in thickness leads to higher peak force than the gradual change. The
semi-analytical model using the effective thickness, and the finite element results have
good agreement for large normalized periods. However, as the period approaches the
value of the bending length scale , the discrepancy increases. Similarly, for
backward peeling, the solution for the three models show good agreement at large
values of normalized period. Both the semi-analytical model using the effective
thickness and the finite element results show non-monotonic dependence of the peak
force in the backward peeling direction on the normalized period. This may be
attributed to the changes in the slope of the film free surface as follows.
For a similar problem of a tapered cantilever beam with mode I adhesive
interfacial fracture, the critical strain energy release rate is given by
[31] where, is the adhesion energy, is the critical load, is
the width of the specimen, and is the compliance rate change with respect to
crack length . For a constant adhesion energy , the relation between the critical load
and the compliance rate change with respect to crack length is . Qiao
Journal of Applied Mechanics
10
et al [32] showed analytically that for a tapered cantilever beam with linearly increasing
thickness, increases with reducing the slope of the tapered beam. Hence,
decreasing the slope, through increasing the period length for the same thickness ratio,
leads to higher compliance rate change with respect to crack length and hence lower
critical load.
2.2 Tunability of the Adhesion Asymmetry
Here, we show using the finite element model results that the adhesion
asymmetry is tunable and investigate its dependence on different model parameters
including: the thickness ratio, the normalized period, and the adhesive characteristic
length.
Figure 3-a shows the relation between the thickness ratio and the normalized
peak force for both forward and backward peeling for three different period lengths to
the strip maximum thickness ratios . Decreasing the ratio between the minimum
and maximum thickness leads to an increase in the peak force for both forward and
backward peeling. However, the rate of increase is higher for the forward peeling and
thus the asymmetry ratio increases as shown in Fig. 3-b. As the thickness ratio
approaches 1, the peak force for both forward and backward peeling approaches that of
a uniform homogenous strip and thus the asymmetry ratio goes to 1 (symmetric
behavior).
Figure 3-c shows the relation between the normalized peak force and the
normalized period for both forward and backward peeling and two different thickness
ratios. For forward peeling, the normalized peak force has a value of 1 as the normalized
Journal of Applied Mechanics
11
period approaches 0, which represent the case of homogenous uniform strip. The
normalized peak force increases with increasing the normalized period, approaching a
maximum value that depends on the thickness ratio. For backward peeling, the peak
force has a value of 1 at small normalized period values. The normalized force increases
with increasing the normalized period till it reaches a maximum value that depends on
the thickness ratio, then it decreases with increasing the period and asymptotically
approach 1. This non-monotonic trend may be explained as follows. As the period
becomes small enough, of the order of the thickness difference or smaller, the
bending stresses are concentrated in a strip of thickness (i.e. the smaller thickness)
and thus the effect of thickness variation decreases and the response approaches that
of a strip with uniform thickness. Longer periods enable stresses to be distributed across
the full strip thickness, except within the transition length around the step change in
thickness, and thus the effect of thickness variation becomes more pronounced. This
non-monotonic behavior is not captured by the semi-analytical model using the
apparent thickness as the peak force keeps increasing with decreasing the normalized
period as shown in Fig. 2-f. The semi-analytical model, however, correctly predicts this
non-monotonicity when the effective thickness is used.
Since the film thickness is varying in our study, the cohesive zone length changes
with the peel front location. To demonstrate the interplay between the properties of
the cohesive law and the film variable thickness, we instead use the characteristic length
scale since it depends only on the cohesive law parameters and the film material
properties. Fig. 3-e shows that increasing the cohesive interface characteristic length
Journal of Applied Mechanics
12
relative to the bending length scale reduces the effect of thickness variation in the
case of forward peeling. When the cohesive zone size increases, the effect of changing
the thickness is smoothed leading to homogenous like behavior. Similar results have
been recently reported for peeling of thin films with material heterogeneities, where
the peel force enhancement due to heterogeneities in the film material properties
decreases with increasing the cohesive zone length [33]. Fig. 3-f shows that with
decreasing , the asymmetry ratio increases and approaches the theoretical
estimate for this choice of parameters. The characteristic length may be increased by
using adhesives with higher intrinsic adhesion energy or lower intrinsic strength. Thus,
the asymmetry may be maximized for interfaces that are intrinsically strong and brittle.
We note that the asymmetry ratios reported in Fig. 3-b,d approach a value of 7.
This ratio exceeds what has been reported previously in the literature [18, 22, 27] as we
will discuss further in the discussion section. The asymmetry ratio may be further
increased through tuning the geometric parameters (e.g. by further decreasing the
thickness ratio or increasing the normalized period ).
2.3 The Energetic Underpinning of Adhesion Asymmetry
To gain further insights into the details of adhesion response asymmetry, we
investigate the change in strain energy of the strip at steady state peeling for both
forward and backward peeling directions. The strip strain energy is normalized by the
strain energy of a homogenous strip having a uniform thickness . Fig. 4 shows
that the strain energy in the strip depends on the peeling direction.
Journal of Applied Mechanics
13
For forward peeling, the strip requires a high force for the peel front to suddenly
propagate from the low thickness to high thickness. Hence, the strain energy in the strip
increases at this zone then it starts decreasing while the peel front propagates in the
direction of decreasing thickness. For backward peeling, the force continues to increase,
and the strip gains more energy while the peel front propagates in the direction of
increasing thickness. After that, the energy starts decreasing gradually due to the stress
redistribution that occurs near the location of the sudden drop in thickness. The results
suggest that the peeling force enhancement in the forward peeling direction is due to
the rapid variation in the stored elastic energy as the peeling front crosses under
segments with different thicknesses.
3. DISCUSSION
The primary result of this paper is that for a thin homogeneous strip adhered to
a rigid substrate, patterning the strip thickness leads to asymmetry in the peeling
behavior without changing the interface adhesive properties. If compared with a strip
having a uniform thickness, a strip with patterned thickness shows enhanced adhesion
strength that is higher in one direction than the other. The asymmetry ratio increases
with (a) increasing the difference between the thickness at the beginning and the end of
the period, (b) increasing the period length with respect to the length scale of bending,
and (c) reducing the cohesive zone length to bending length scale ratio.
The asymmetric behavior is attributed to the difference in the strain energy
required to peel the strip in the two directions. When the peel front propagates
suddenly from low thickness to high thickness zones in forward peeling, the rapid
Journal of Applied Mechanics
14
variation in stored elastic energy leads to significant enhancement in the force. This
enhancement is due to high fraction of the external work exerted to increase the stored
energy in the strip to bend the suddenly thicker and hence stiffer region. This is similar
to the findings of Xia et al [16] on peeling of heterogeneous films with patterned elastic
moduli. However, our current work shows that in a film with patterned thickness this
mechanism may also lead to adhesion asymmetry. In backward peeling, as the peel
front propagates in the direction of increasing thickness, it also requires higher strain
energy and hence higher external work and force. The enhancement in the forward
propagation is higher than that of the backward propagation leading to asymmetry. The
location of the peak force for forward propagation is at the sudden increase in
thickness, whereas for the backward propagation is found to be in the fourth quarter of
the period.
We have used a semi-analytical model based on inextensible Euler-Bernoulli
beam to analyze the peeling problem. The assumption of inextensibility is acceptable for
moderate and high peeling angles [9]. However, for soft strips with low peeling angle,
the effect of extensibility may become important [9]. One limitation of using one
dimensional model in the analytical approach is that we do not allow for adhesion
asymmetry in the transverse direction. It is thus implied that the peeling out of plane is
uniform and follow the Rivlin model [7]. However, the semi analytical model serves to
prove that introducing designed irregularities by patterning the thickness may lead to
adhesion enhancement and asymmetry, and the same concept may be applied to
Journal of Applied Mechanics
15
peeling of two-dimensional sheets by patterning the thickness in two orthogonal
directions. This will be the focus of a future investigation.
The direction dependency in peeling has been explored in previous studies
where the adhesion strength may be made dependent on the axis of the pull or the
peeling direction [18, 22–24, 27]. Direction dependency enables changing adhesion
strength from weak to strong modes based on peeling direction which could be a
desirable feature in many systems, such as climbing robots, medical tapes, and stamps
for transfer printing [8]. However, most, if not all, of the previous studies dealing with
direction dependency and reversible control of adhesion leverages changes in the
geometry of the interface or the material properties of the adhesive layer, or require
external factor to stimulate the different peeling modes, such as temperature [34].
In this paper, we propose, for the first time to the best of our knowledge, a
simple and efficient way to introduce asymmetry by patterning the thickness of the thin
strip for the same bulk material, and interface properties. We have achieved asymmetry
ratios close to 7 which exceeds what have been reported previously in the literature. For
example, in the work of Xia it el [18], the reported asymmetry ratio using interfacial arc
patterns was around 1.5. In Hsueh and Bhattacharya [22], the reported maximum
asymmetry ratio through optimizing adhesion strength distribution was around 2.5. In
the work of Seok et al [27] using micro-wedge array surface, the reported asymmetry
ratio was around 5. However, the studied mechanism in Seok et al [27] requires
temperature changes. By controlling parameters such as thickness ratio or the
periodicity of the thin film in our design, the asymmetry effect may be further increased.
Journal of Applied Mechanics
16
The geometric parameters to tune asymmetry in our proposed pattern are easily
controlled due to the recent advances in manufacturing techniques and 3D printing [35]
and thus provide an attractive passive pathway for achieving high adhesion asymmetry
ratios in a variety of applications.
An indirect result of the theoretical model is that adhesion asymmetry may be
also achieved in a thin film with uniform thickness by using a periodic distribution of
elastic modulus gradient along the longitudinal direction. While it is not unexpected that
adhesion will be asymmetric in peeling of a film with monotonic gradient in its elastic
properties, it is not very clear that this should be the case for periodic variations. This is
because in the latter case, the homogenized elastic properties of a unit cell will be
direction independent. In particular, a Bloch-wave analysis for the unit cell (not shown
here), assuming linear elasticity, yields a symmetric dispersion relation. One value of the
theoretical model is that it suggests the asymmetry in adhesion persists even in the case
of a film with periodic gradient elasticity in the longitudinal direction. We note,
however, that our proposed approach in varying film rigidity through patterning
thickness is possibly easier, from a manufacturing perspective, than producing gradients
in elastic properties. Furthermore, since rigidity has a linear dependence on the
modulus of elasticity but cubic dependence on film thickness, it is possible to obtain
large asymmetry ratios from small variations in thickness.
In this work, we focus on the asymmetry of the peak force required to peel a film
with patterned thickness, but we note that it is also possible to describe adhesion in
terms of the total work required to peel off the film which is given by the area under the
Journal of Applied Mechanics
17
force displacement curve. For low peeling velocities considered in this study, the
asymmetry in the peak force was found to be larger than the asymmetry in the peel
work. We expect the peel work for both forward and backward peeling to increase with
increasing the rate of peeling, even for adhesives with rate-independent adhesion
energy. When the peel front propagates under the sudden change in thickness, waves
are radiated due to the snap-back effect noticed in the finite element results in Fig. 2-
c,d and affect the peel work. The peeling rate effect will further increase if the film
material or the adhesive properties are rate-dependent. The effect of rate dependence
on the asymmetry of both peel force and peel work will be the focus of future work.
This work also highlights the role of stiffness variation in material design
particularly related to problems in fracture or adhesion. Heterogeneities in general,
through patterning the bulk geometry, introducing voids, introducing patterned cuts, or
adding soft or hard inclusion may result in materials with enhanced fracture toughness
[36], wave propagation properties [37], negative Poisson’s ratio [38], or improved
adhesion [38, 39], among other material properties. This opens new pathways in
designing and fabricating metamaterials with optimized and enhanced material
priorities.
Recently, there has been an increased interest in non-reciprocal behavior in the
mechanics and physics communities especially in the context of wave propagation. The
possibility of breaking the wave propagation symmetry or obtaining one-way
propagation is highly desirable in many technological applications [41]. Although our
current work focuses only on the peeling strength, future extensions may find
Journal of Applied Mechanics
18
connections to problems in wave propagation. It will be potentially interesting to
investigate the propagation of interfacial waves that may develop during dynamic
peeling and whether they will show non-reciprocal response or not. This research may
also be extended to investigation of surface wave propagation in films with patterned
thickness and examination of their non-reciprocal response.
Future extension of this study may include exploration of inertia effects and
dynamic peeling, studying the effect of different thickness profiles on the peeling
response, and optimizing the film shape for the highest possible asymmetry ratio. It may
also include investigating the effect of patterning thickness on peeling of 2D sheets.
4. METHODS
4.1 Analytical Solution: The solution Scheme
We follow the analytical procedure developed by Xia et al. [16] for the analysis of
adhesion in heterogeneous thin film. The strip is being peeled by a force with peeling
angle . The force is given by:
(6)
And the tip displacement is given by:
(7)
which is measured relative to a reference position at . There are two
boundary conditions:
(8)
Journal of Applied Mechanics
19
The second boundary condition is due to the absence of applied moment at The
negative superscript denotes the limit from the left. The potential energy of the system
is given by:
(9)
Where is the adhesion energy per unit length of the interface between the strip and
the substrate. We assume that both the material and the adhesive are rate-independent
at low peeling velocities. In addition, we assume a smooth homogenous interface, and
thus take as a constant value. Substituting Eq. 6, 7, and assuming a constant bending
rigidity within each discretized segment , the potential
energy reduces to Eq. 1.
The condition for equilibrium is that the first variation of this equation with
respect to and should vanish for any values of and consistent with the
boundary conditions. Making use of this and the slope continuity at each discretization
point , Eq. 2, 3 are obtained. The detailed derivation is
shown in Xia et al. [16].
Equations 2, 3 are solved numerically to determine the unknowns (the
force and the slope angles . The solution scheme is as follows:
1. For a given peel front position , where , and the segments
rigidity , make an initial guess for .
2. Calculate from Eq. 2 using the current .
Journal of Applied Mechanics
20
3. For to , calculate from using Eq. 3. We have chosen trapezoid
method for solving the initial value problem where , and:
(10)
This is a nonlinear equation that is solved for .
4. Recalculate using the updated .
5. Check the relative error between the previous and the updated value of . If the
error is greater than a defined tolerance, go to step 3. Otherwise, go to step 6. The
tolerance is set to
6. Calculate the tip displacement for the current peel front position. Advance the peel
front position and repeat.
We have discretized every unit cell into a large enough number of segments to ensure
that the linearly varying strip thickness is accurately approximated by a series of
constant thickness intervals. We have chosen the segment width to be at least
where is the width of the periodic unit cell of the strip. Further mesh refinement has
negligible effect on the solution.
4.2 Finite Element Analysis: Model Setup
We use the finite element software package ABAQUS [28]. The model setup is
shown in Fig. 5. We conduct Implicit dynamic analysis with slow peeling rate to
represent a quasi-static peeling test under displacement-controlled boundary conditions
without considering gravity loads. The dimensions of the base plate are chosen in a way
that the boundaries have no effect on the results. The lower edge of the base plate is
Journal of Applied Mechanics
21
restrained from movement in both directions. The base plate is tilted to have an angle
of with the vertical upward direction. The edge of the strip is pulled upwards with a
constant rate.
To model the cohesive interface, we implement a zero-thickness cohesive
element. We adopt an intrinsic bilinear cohesive law composed of a linear elastic part
up to the critical cohesive stress , followed by linear degradation that evolves from
crack initiation to complete failure. The analytical expression for the cohesive law is
given by:
(11)
Where is penalty stiffness of the cohesive law, is the failure (total separation)
normal displacement, and is the ratio between the critical and the failure normal
displacements. value may be determined by selecting proper values of and .
The total area under the curve is the cohesive fracture energy , and the length on
which stress changes from (Crack initiation) to 0 (total separation) defines the
cohesive zone length .
We have chosen the strip longitudinal dimension to have at least 10 periods, and
all results are obtained after reaching steady state peeling. The strip is meshed using 2D
plane strain quadrilateral elements and the cohesive interface is meshed using zero
thickness cohesive elements. The cohesive elements size is chosen to be at
Journal of Applied Mechanics
22
most to avoid solution jump that may result in divergence or global oscillations [42]. We
have conducted a mesh sensitivity analysis to insure adequate accuracy.
FUNDING
This research has been supported by funds from Campus Research Board of University
of Illinois Urbana Champaign (RB17043), and the National Science Foundation grants
(CMMI-Award Number 1435920) and (CAREER Award Number 1753249).
Journal of Applied Mechanics
23
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Figure Captions List
Fig. 1
Peeling of a thin strip with patterned thickness from a rigid substrate: (a)
Idealized longitudinal profile of the strip modeled as an Euler-Bernoulli
beam, and (b) the thin strip profile showing the actual thickness of the
strip, and (c) representation of the strip profile based on the effective
thickness concept allowing for a gradual redistribution of the bending
stresses over a transition length at locations of step change in thickness.
The forward and backward peeling directions are also highlighted.
Fig. 2
Comparison between the theoretical models and finite element results:
(a), (b) the normalized forces vs. normalized peel front position for
forward and backward peeling directions, , . (c),
(d) the normalized forces vs. normalized vertical peel displacement for
forward and backward peeling directions, , . The
hatched areas represent the energy dissipated due to the snap-back
instability. (e), (f) the normalized peak forces vs. for forward and
backward peeling and .
Fig. 3
Dependence of peeling force and adhesion asymmetry on different
model parameters: (a), (b) Effect of the ratio of minimum to maximum
thicknesses of the strip on the normalized peak forces and the
asymmetry ratio for different values of period length for both peeling
directions. The asymmetry ratio increases as the thickness ratio
Journal of Applied Mechanics
27
decreases and the period length increases. (c), (d) Effect of the ratio of
period to the length scale of the bending on the normalized peak forces
and the asymmetry ratio for two values of thickness ratio and both
peeling directions. The peak force for peeling in the forward direction
monotonically increases as the normalized period increases. The peak
force for peeling in the backward direction shows a non-monotonic
dependence on the normalized period. The asymmetry ratio increases
with the increase in the normalized period. (e), (f) Effect of the ratio of
cohesive law characteristic length to the length scale of the bending on
the normalized peak forces and the asymmetry ratio for ,
for both peeling directions. The normalized peak force
significantly decreases in forward peeling direction as the cohesive
length scale increases relative to the length scale of bending. The
asymmetry ratio increases, and approaches its theoretical limit, as the
cohesive length scale decreases relative to the length scale of the
bending. The results in this figure are obtained using the finite element
model unless otherwise mentioned.
Fig. 4
Change in strain energy at steady state peeling: The normalized strain
energy of the strip vs. the normalized peel front location for both
forward and backward propagation, , and . The
results in this figure are obtained using the finite element model.
Journal of Applied Mechanics
28
Fig. 5
Model setup for the finite element analysis. A homogenous thin film with
saw tooth thickness profile is peeled from a rigid substrate. The peeling
angle is .
Journal of Applied Mechanics
29
Fig. 1: Peeling of a thin strip with patterned thickness from a rigid substrate: (a)
Idealized longitudinal profile of the strip modeled as an Euler-Bernoulli beam, and (b)
the thin strip profile showing the actual thickness of the strip, and (c) representation of
the strip profile based on the effective thickness concept allowing for a gradual
redistribution of the bending stresses over a transition length at locations of step
change in thickness. The forward and backward peeling directions are also highlighted.
Journal of Applied Mechanics
30
(a) Forward Peeling (b) Backward Peeling
(e) Forward Peeling (f) Backward Peeling
(c) Forward Peeling (d) Backward Peeling
Fig. 2: Comparison between the theoretical models and finite element results: (a), (b)
the normalized forces vs. normalized peel front position for forward and backward
peeling directions, , . (c), (d) the normalized forces vs. normalized
vertical peel displacement for forward and backward peeling directions, ,
Journal of Applied Mechanics
31
. The hatched areas represent the energy dissipated due to the snap-back
instability. (e), (f) the normalized peak forces vs. for forward and backward peeling
and .
Journal of Applied Mechanics
32
Fig. 3: Dependence of peeling force and adhesion asymmetry on different model
parameters: (a), (b) Effect of the ratio of minimum to maximum thicknesses of the strip
on the normalized peak forces and the asymmetry ratio for different values of period
length for both peeling directions. The asymmetry ratio increases as the thickness ratio
(c)
(d)
(a)
(b)
(e)
(f)
Journal of Applied Mechanics
33
decreases and the period length increases. (c), (d) Effect of the ratio of period to the
length scale of the bending on the normalized peak forces and the asymmetry ratio for
two values of thickness ratio and both peeling directions. The peak force for peeling in
the forward direction monotonically increases as the normalized period increases. The
peak force for peeling in the backward direction shows a non-monotonic dependence
on the normalized period. The asymmetry ratio increases with the increase in the
normalized period. (e), (f) Effect of the ratio of cohesive law characteristic length to the
length scale of the bending on the normalized peak forces and the asymmetry ratio for
, for both peeling directions. The normalized peak force significantly
decreases in forward peeling direction as the cohesive length scale increases relative to
the length scale of bending. The asymmetry ratio increases, and approaches its
theoretical limit, as the cohesive length scale decreases relative to the length scale of
the bending. The results in this figure are obtained using the finite element model
unless otherwise mentioned.
Journal of Applied Mechanics
34
Fig. 4: Change in strain energy at steady state peeling: The normalized strain energy of
the strip vs. the normalized peel front location for both forward and backward
propagation, , and . The results in this figure are obtained using the
finite element model.
Journal of Applied Mechanics
35
Fig. 5: Model setup for the finite element analysis. A homogenous thin film with saw
tooth thickness profile is peeled from a rigid substrate. The peeling angle is .