FUME 7300

FACTA UNIVERSITATIS

Series: Mechanical Engineering

https://doi.org/10.22190/FUME210101019C

Original scientific paper

ON THE AFFERRANTE-CARBONE THEORY

OF ULTRATOUGH TAPE PEELING

Michele Ciavarella1, Robert M. McMeeking 2, 3, 4, 5, Gabriele Cricrì6

1Politecnico di BARI. DMMM Department, Bari, Italy
2Materials Department, University of California, USA

3Department of Mechanical Engineering, University of California, USA
4School of Engineering, University of Aberdeen, King’s College, Scotland, UK

5INM – Leibniz Institute for New Materials, Germany
6Department of Industrial Engineering (DII), Università di Napoli Federico II, Napoli, Italy

Abstract. In a simple and interesting theory of ultratough peeling of an elastic tape

from a viscoelastic substrate, Afferrante and Carbone find that there are conditions for

which the load for steady state peeling could be arbitrarily large in steady state

peeling, at low angles of peeling - what they call "ultratough" peeling (Afferrante, L.,

Carbone, G., 2016, The ultratough peeling of elastic tapes from viscoelastic substrates,

Journal of the Mechanics and Physics of Solids, 96, pp.223-234). Surprisingly, this

seems to lead to toughness enhancement higher than the limit value observed in a very

large crack in an infinite viscoelastic body, possibly even considering a limit on the

stress transmitted. The Afferrante-Carbone theory seems to be a quite approximate,

qualitative theory and many aspects and features of this "ultratough" peeling (e.g.

conformity with the Rivlin result at low peel angles) are obtained also through other

mechanisms (Begley, M.R., Collino, R.R., Israelachvili, J.N., McMeeking, R.M., 2013,

Peeling of a tape with large deformations and frictional sliding, Journal of the

Mechanics and Physics of Solids, 61(5), pp. 1265-1279) although not at “critical

velocities”. Experimental and/or numerical verification would be most useful.

Key Words: Soft Materials, Peeling, Adhesion, Viscoelastic Materials

1. INTRODUCTION

Adhesion of two objects can arise due to chemical bonding as by a glue, or by

physical interaction such as by van der Waals or electrostatic attraction [1]. Various tests

have been devised to probe the strength and toughness of adhesive bonds, including the

Received January 01, 2021 / Accepted February 12, 2021

Corresponding author: Michele Ciavarella

DIMeG - Politecnico di BARI, V.le Japigia 182, Politecnico di Bari, 70125 Bari, Italy

E-mail: Mciava@poliba.it

2 M. CIAVARELLA, R.M. MCMEEKING, G. CRICRÌ

peeling of a tape from a substrate or from another tapelike feature [2, 3]. Our schematic of

the former version of the test is illustrated in Fig. 1. Rivlin [2] provided a formula for the

relationship between the effective adhesion energy, Γ, and the force, P, required to peel

an inextensible tape from an elastic substrate. The energy of adhesion as the work per unit

area of bond requires to progressively separate the objects from each other. Kendall [4, 5]

then augmented the formula by accounting for the effect of the strain energy of an elastic

tape. The importance of adhesion, and its science and technology, can be confirmed by

perusal of the references provided by Afferrante and Carbone [6], and by the reviews by

Creton and Ciccotti [7] and Long et al. [8].

Fig. 1 Geometry of the peeling of an elastic tape from a viscoelastic substrate for the AC

theory

Viscoelastic effects are known to occur in the peeling of polymer and organic

materials, e.g. Kendall [2, 3] presents such data and attributes the effect to viscoelastic

behavior in the separation process. Such viscoelastic behavior is addressed in an

extensive literature (e.g. [9–12]), suggesting steady state critical crack growth during

peeling occurs with an enhanced apparent work of adhesion according to the Gent-Schultz

law

0 0



 
  

  
(1)

where Γ is the apparent or effective work of adhesion, Γ0 is its reference or datum value at

a very slow rate of propagation, ν is the rate of peeling, and  
1

T
ka



 in which k and n

On the Afferrante-Carbone Theory of Ultratough Tape Peeling 3

(0 < n < 1) are (or are supposed to be) materials constants, and
n

T
a is the WLF factor for

unifying results from various temperatures (Williams, Landel and Ferry [13]).

Gent and Schultz [10] describe the reference value, Γ0, as the thermodynamic work of

fracture, and attribute the enhancement in Eq. (2) to viscoelastic dissipation. A model

incorporating such a point of view was developed by de Gennes [14], described as the

“viscoelastic trumpet,” with different mechanisms of deformation and adhesive rupture

dominating at different scales at different rates of crack or peel front propagation.

However, de Gennes [14] finds that the resulting highest effective work of adhesion for

very rapid crack or peel front propagation is finite instead of unlimited as in Eq. (1), and

is given by

0 0

 





(2)

where E∞ is the instantaneous modulus and E0 is the relaxed, or equilibrium modulus.

Note that the ratio in Eq. (2) can span 3 or 4 orders of magnitude [12].

The velocity dependence of Eqs. (1) and (2) was, at first, difficult to rationalize as

cracks and peel fronts in linear viscoelastic materials admit an inverse square root stress

singularity, implying that the high strain rate modulus should always dominate at the

crack and peel front tip, indicating a lack of velocity dependence [15]. However,

introduction of a Barenblatt or Maugis-Dugdale cohesive zone at the crack tip enabled the

identification of velocity dependence for crack propagation [16-20], as then the effective

modulus at the crack tip depends on the propagation rate. The resulting model predicts the

relationship in Eq. (2) for Mode I crack propagation (i.e. tension), where, in this case, Γ∞

is defined to be
2

0
/K E


, where K∞ is the Mode I stress intensity factor at high rates of

crack propagation. Similarly,
2

0 0 0
/K E  .

One can expect an interplay among cohesive zone behavior, peel rate, viscoelastic

dissipation and specimen sizescale. Such an effect was explored in finite element

modeling by Rahulkumar et al. [21] for T-peeling of a viscoelastic polymer strip. They

found that the apparent adhesion energy at low and high rates of peeling effectively obeys

Eq. (2), but with an intermediate propagation rate where viscoelastic dissipation

additionally enhances the apparent adhesion energy. As a result, the relationship between

the apparent adhesion energy and the peel rate is non-monotonic, in contrast to Eq. (1).

Afferrante and Carbone [6] (henceforth AC) address a similar issue but for an elastic tape

being peeled from a viscoelastic substrate as in Fig. 1. The substrate is considered to be

semi-infinite, the tape thickness is L and the peel angle is θ. To model the transfer of

loads to the substrate, AC assumed that they could be treated as uniform tractions applied

over a width L at the peel front, i.e. a traction normal to the substrate surface plus a shear

traction. The substrate surface is traction free elsewhere, allowing use of the viscoelastic

correspondence principle for the substrate. The AC model therefore has features that

make it similar to a cohesive zone model with uniform cohesive tractions, and has a size

scale, the elastic tape thickness, that can introduce an interplay between the peel rate and

viscoelastic dissipation in the substrate. In the next section we identify some aspects of

the peeling behavior in the AC model.

4 M. CIAVARELLA, R.M. MCMEEKING, G. CRICRÌ

2. AFFERRANTE-CARBONE MODEL

For an elastic tape of modulus E and a viscoelastic substrate with a single relaxation

time τ0, (e.g. the standard model of linear viscoelasticity), the AC equation (14) [6]

provides

 
2

0 0

1
1 cos

E P P
f

EL E L EL EL





       
        

     
(3)

where P is the load per unit out-of-plane width of the tape and fν is a function of

exponential integrals of its argument ντ0/L, see the AC equation (15) [6]. Note that Eq. (3)

is written as an expression for the intrinsic adhesion energy, Γ0. The function fν is zero for

both very slow peeling rate, 0  , and at extreme rates of peeling,  . Otherwise, fν

is positive, i.e. consistent with viscoelastic dissipation from which it arises, and has a

maximum at an intermediate rate of peeling that depends weakly on E∞/E0 if E∞/E0>15.

At very low peel rates, and very high peel rates, with fν 0 , the result in Eq. (3) is

simply Kendall’s [4], or Rivlin’s [2] if the strain energy term containing P2 is neglected,

as it can be for high peel angles. This behavior is expected as the system behaves purely

elastically at very low and very high rates of peeling with negligible viscoelastic

dissipation. As a result, the formula in Eq. (3) at very slow and very fast rates of peeling

shows that in these cases the intrinsic adhesion energy is the only contributor to the work

of adhesion. However, the intrinsic adhesion energy may be rate dependent, as in the case

where a soft, lightly cross-linked polymer is being used as an adhesive and its crazing and

fibrillation can lead to viscoelastic behavior within the adhesive.

The result in Eq. (3) is a quadratic equation for P/EL with discriminant

 
2 0 0

4 1
1 cos

E
f

EL E L





   
      

  
(4)

and, therefore, real solutions for P/EL exist only for 0  . The solutions are

1 cos

2
1

EEL
f

E L






  


 
  

 

(5)

and we accept only positive real solutions. We deduce that the apparent adhesion energy,

Γ, the sum of the thermodynamic adhesion energy and the contribution of viscoelastic

dissipation, is given by

 
2 2

0 0

1
1 cos

P P E P
f

EL EL EL EL E L EL





       
          

      
(6)

Note that the first result on the right-hand side of Eq. (6) is the normalized energy

release rate as given by Kendall [4]. We observe, therefore, that, according to the AC

model, the Kendall [4] formula for the energy release rate also provides the effective

adhesion energy, as expressed by the first equality of Eq. (6).

From Eq. (6) we obtain the ratio

On the Afferrante-Carbone Theory of Ultratough Tape Peeling 5

0 0 0

1
EL E P

f
E L EL



   
    

    
(7)

where P is obtained from Eq. (5) when there are real, positive solutions for it.

For illustration, we consider the case where E/E0=1.5 and E∞/E0=10, Γ0/EL=10
-4 as

also used by AC. In Fig. 2a, we present the result for Γ/Γ0 from Eq. (7) where the lower

positive real result for P/EL from Eq. (5) is utilized. It can be seen that the result for

ϑ=π/64=2.8° can have an enhancement of the apparent adhesion energy that can be very

large. Although not shown in Fig. 2a, the result for Γ/Γ0-1 for ϑ=π/64=2.8
° can be as high

as ≈ 140. There are no real solutions for this angle in the range where the results in black

are not visible in the plot. This outcome is much larger than E∞/E0=10 as predicted by de

Gennes [14] for the ratio of the apparent adhesion energy for very fast peeling compared

to very slow peeling. The enhancement of the apparent adhesion energy is thus non-

monotonic and has a maximum at peeling rates for which ντ0/L ≈ 1.

Fig. 2 Enhancement (Γ/Γ0-1) for the AC peeling theory shown as a function of the peeling

velocity : (a) lower load solution (b) higher load solution, both for E/E0=1.5, Γ0/EL=10
-4,

E∞/E0=10. Peel angles are θ = π/64, π/8, π/4, π/2 (respectively black, blue, red and green).

6 M. CIAVARELLA, R.M. MCMEEKING, G. CRICRÌ

Note also that for peel angles equal to ϑ = π/8 = 22.5° and above the enhancement of

the apparent adhesion energy is negligible. We note that we find negligible enhancement

of the apparent adhesion energy for peel angles greater than approximately 8° for the

parameters used to plot Fig. 2a. As noted above, the effective adhesion energy is then

essentially equal to the intrinsic adhesion energy, Γ0, and is so over the whole range of

peel rates. As also noted above, Γ0 itself may be rate dependent, leading to a higher peel

force at fast rates of peeling than at low rates. However, it is then not obvious that Eq. (2)

will predict the ratio of effective adhesion energies at fast peel rates versus slow rates, and

the behavior in terms of effective adhesion energy may well be given by Eq. (1). If the

intrinsic adhesion energy lacks rate dependence, then the peel force for peel angles equal

to or greater than 22.5° will be insensitive to the rate of peeling, in disagreement with

Eqs. (1) and (2). For example, if van der Waals forces are in control of adhesion,

according to the AC model, the peel force can be expected to be insensitive to the rate of

peeling for peel angles greater than or equal to 22.5°.

As noted by AC, it is possible that the higher magnitude, 2nd solution of Eq. (5) is

relevant during peeling and that the system can be forced into this regime. To explore this

possibility, in Fig. 2b we use the higher magnitude, 2nd solution from Eq. (5) to evaluate

and plot the result from Eq. (7) for the same parameters used for Fig. 2a. We plot only

some of the lower values we obtain as many of the values are extremely high for the

parameters utilized. Given that, for peel angles greater than or equal to 22.5° the 2nd,

higher magnitude solution gives an apparent adhesion energy that is at least 3 orders of

magnitude greater than the lower magnitude solution, we conclude that it is unlikely that,

for the parameters used in Fig. 2, that the 2nd, higher magnitude solution is relevant to

peeling. There are no real solutions in Fig. 2b for the ϑ = π/64 case.

We also find that the high apparent adhesion energy regime of peeling in the AC

model depends on the ratio E/E0. As noted above, when E/E0 = 1.5, such high apparent

adhesion energy behavior occurs for peel angles below 8°. For E/E0 > 1.5 this range of

ultratough adhesion becomes larger.

3. DISCUSSION

Compared to other widely used models, like the Gent-Schultz law (1), and the

‘viscoelastic trumpet’ developed by de Gennes, the AC point of view appears to be, like

them, an approximate solution. Indeed, the first two give results that are essentially

qualitative, and whose validity depends on experimental databases; this point is clearly

stated, e.g., by Saulnier et al. [22] in the case of the viscoelastic trumpet. On the other

hand, the AC model is developed following the mathematical consequences of an energy

balance imposed through a well-defined, though inexact, physical model. Notwithstanding

the great difference between the respective frameworks, the AC results appear to be

qualitatively similar to the outcomes of Saulnier et al. [22]. In fact, in [22] the energy

balance is based on a singular stress field directly derived from the correspondence

principle, where such a singular stress is completely absent in AC. More specifically, both

models predict that the viscous contribution to the detachment energy rate, Γ(ν), is zero

for very low or very high peel rates, whereas it can be very large at an intermediate rate.

At this intermediate rate the AC results seem to be in contrast with the experimental

On the Afferrante-Carbone Theory of Ultratough Tape Peeling 7

outcomes from peel experiments in geometries similar to Fig. 1 with viscoelastic

components.

It seems quite likely that the AC model is oversimplified as the interaction between

the tape and the substrate is modeled as a uniform traction within a constant interaction

length L, whatever the traction angle ϑ, the tape and substrate stiffness and the tape

thickness. To explore this point, let us consider two extreme conditions ϑ = π/2 and ϑ = 2

(see Fig. 3). In the first case, the AC model neglects the tape flexural stiffness (although if

the tape is very thin its flexural stiffness will be negligible so this is not always a

problem), and in the second one, there is an issue about the interaction zone length: the

cohesive zone length could be much smaller than L and it might be larger than L. Also, if

it is friction that determines the interaction length frictional slip zone might be larger than

L and it might be smaller than L depending on the friction shear stress. If it is sticking

friction, the interaction zone will be very small. On the other hand, Eq. (7) shows clearly

that the interaction length L is a very important parameter in the evaluation of the viscous

contribution to the apparent adhesion energy.

For these reasons, one should consider the AC predictions to be only approximate and

qualitative, especially for low peel angles. Indeed, it seems very unlikely that ‘ultratough

peeling’, even if it exists, can be captured with such simplistic assumptions as are utilized

by AC.

Fig. 3 Interface interaction for the AC model for ϑ = π/2 and ϑ = 0 (above) with a more

realistic one shown below

Even if the AC result is a straightforward consequence of the energy balance (and a

poorly designed model), the interpretation of Eq. (7) as a ‘toughness’ may be misleading,

notwithstanding its coherence with many widely accepted formulations. This implies that,

in some conditions, the power absorbed by the viscous substrate in steady state peeling is

unlimited, which is physically an unlikely result. Instead of identifying toughness, as is

normally what is done using Kendall [5], in the AC case one can speak unambiguously of

8 M. CIAVARELLA, R.M. MCMEEKING, G. CRICRÌ

peel load and, ultimately, of ‘ultra-high-strength’ peeling. While AC provides a result that

we find to be counter-intuitive, unfortunately we cannot yet present a better model. Thus,

we can neither prove our claims nor disprove AC.

There is a substantial difference between the AC and other models cited. This

situation may partially be attributed to the fact that the volume in which viscous energy is

dissipated in the AC model is infinite. A more detailed model, consistent with physical

behavior of the interface interaction between the tape and the substrate would help deeper

investigation.

4. CONCLUSIONS

We have discussed the fact that the "ultratough peeling" theory of Afferrante and

Carbone [6] leads to two possible equilibrium solutions for the load at a given peeling

speed. The low load solution in some regimes produces very high toughness

enhancement, possibly already much larger even than that expected in a crack in an

infinite system. More problematic appears to be the high load solution which, even

considering a limit stress carried by tape and substrate, will not avoid some solutions to

have unbounded toughness enhancement. In addition, viscoelastic effects seem to induce

an on-off mechanism, where for large angles of peeling it predicts solutions extremely

close to the purely elastic Kendall case, while below a threshold angle, the toughening

abruptly rises to very high levels, and the "ultratough" peeling regime appears. It seems

very likely that this strange result is due to approximations present within the model, even

if it is quite sophisticated from a mathematical point of view. Features of this "ultratough"

peeling (e.g. conformity with the Rivlin result at low peel angles) are obtained also

through other mechanisms [23] although not at “critical velocities”.

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