## Abstract

In part I of this work, we presented a theory for adhesionless contact of a pressurized neo-Hookean plane-strain membrane to a rigid substrate. Here, we extend our theory to include adhesion using a fracture mechanics approach. This theory is used to study contact hysteresis commonly observed in experiments. Detailed analysis is carried out to highlight the differences between frictionless and no-slip contact. Membrane detachment is found to be strongly dependent on adhesion: for low adhesion, the membrane ‘pinches-off’, whereas for large adhesions, it detaches unstably at finite contact (‘pull-off’). Expressions are derived for the critical adhesion needed for pinch-off to pull-off transition. Above a threshold adhesion, the membrane exhibits bistability, two stable states at zero applied pressure. The condition for bistability for both frictionless and no-slip boundary conditions is obtained explicitly.

## 1. Introduction and motivation

Adhesion is important in many engineering applications such as the performance of coatings, labelling, laminates and biomedical devices. For example, cells must adhere well to tissue scaffolds, whereas weak adhesion is desirable for coronary stents. To control and to characterize adhesion, it is necessary to devise testing methods that are highly sensitive to weak interactions and easily carried out on different substrates. The high compliance of membranes allows for intimate contact, even for substrates that are not perfectly flat, such as tissues. Thus, the membrane–contact adhesion tests used by [1–5] provide an attractive combination of sensitivity and substrate versatility. A typical experimental set-up is illustrated in figure 1*a*–*c*. A membrane is clamped to the edge of a cylindrical tube, which is suspended at a distance above a flat substrate. An external pressure is applied by pumping air into the tube, inflating the membrane (stage 1, free inflation). Contact is established in stage 2 (increasing contact). Once the desired contact coverage is achieved, the pressure is reduced to detach the membrane from the substrate (stage 3, decreasing contact).

In part I of this study [6], we provided the solution for the adhesionless contact problem. Experiments [5–10] have shown that the adhesion energy associated with stage 2 (increasing contact) is invariably much smaller than stage 3 (decreasing contact), even for materials that exhibit very little viscoelasticity. For this reason, it is possible to neglect adhesion in stage 2. However, adhesion must be considered in stage 3 where contact is decreasing. In this part of the work, we extend our theory to include adhesive contact. We use our theory to study a typical adhesion test where stage 2 is adhesionless, whereas stage 3 is governed by a constant work of adhesion.

As pointed out in part I, the role of friction on the contact mechanics of membranes is poorly understood. In this work, we carried out detailed analysis for two limiting cases: frictionless and no-slip adhesive contact. Wrinkling or buckling of the membrane near the contact line is often observed in axis-symmetric geometry during stage 3 [11]. Wrinkling is caused by the membrane tension in the hoop direction becoming compressive. This is one of the motivations to study plane-strain membranes because we anticipate this geometry suppresses wrinkling. We found that for no-slip contact, wrinkling is suppressed, even at very large adhesion energies. Our results suggest that a more robust membrane adhesion test can be designed using long rectangular (plane-strain) membranes.

The layout of the article is as follows. §2 summarizes relevant results in part I. The formulation of adhesive contact is given in §3. Based on these results, simulations are carried out to model a test where stage 2 is adhesionless, whereas stage 3 is governed by a constant work of adhesion. The results and discussions are presented in §4. §5 summarizes the key findings of this work.

## 2. Geometry, constitutive model and free inflation

Consider the plane-strain deformation of an infinitely long elastic membrane of initial width 2*a* subjected to uniform *excess* pressure (i.e. pressure difference) *p*. The undeformed membrane is flat and has uniform initial thickness *h*_{0}. It is suspended at a distance of *d* above a rigid flat substrate and is clamped at the edges O_{1} and O_{2}. Figure 2 shows a cross section of the membrane in contact with a flat rigid substrate, with the contact zone occupying the strip . Symmetry allows us to consider *x*≥0. Deformation carries a material point on the membrane with material coordinate *ρ* to (*x*,*z*) (figure 2). The half-length of the contact strip is denoted by *c*, and its material coordinate is denoted by *ρ**. The portion of the membrane not in contact will be labelled as free standing. Figure 2 shows that the slope of the deformed membrane varies from *θ*_{0} at the contact edge to *θ*_{m} at the clamp.

As shown in part I (eqn (2.1) of [6]), equations in part I will be denoted by ‘eqn’ hereafter), the free-standing membrane is a circular arc with uniform tension *T* and radius *R*=*T*/*p*. The membrane is described by an incompressible neo-Hookean model where the true tension *T* of a material element is related to its in-plane stretch ratio *λ* by
2.1where *μ* is the small strain shear modulus of the material.

The exact solution of the inflated membrane before contact (stage 1) is given in part I [6]. The inflated membrane profile is a circular arc and its pressure *p*, stretch ratio *λ*, and apex deflection *δ* can be expressed in terms of a single parameter: the angle at the clamp edge *θ*_{m} (see eqns (3.8), (3.3) and (3.7), respectively). The relation between variables (e.g. *p* and *δ*) can be obtained by eliminating this parameter. For a given initial separation *d*, contact will occur when *δ*=*d*. As shown in part I eqn (4.1*b*), the normalized pressure *β*_{c}=*ap*_{c}/*μh*_{0} required to make point contact is
2.2where .

## 3. Adhesion and energy release rate

To account for adhesion, we use a fracture mechanics approach where the space between the free-standing membrane and the substrate is viewed as an external crack that propagates inwards (outwards) as the contact area decreases (increases) [4,7,12–14]. The energy release rate of this crack, *G*, is defined as the negative rate of change of the potential energy of the system *E* (per unit thickness in the out-of-plane direction) with respect to the crack surface,
3.1The analytical expression for *G* turns out to be the same for both frictionless and no-slip cases (details of the derivation are given in the electronic supplementary material),
3.2awhere *λ*_{out}, *λ*_{in}, *T*_{out} and *T*_{in} are the stretch and in-plane tension in the outer and inner portions of the membrane, respectively. For no-slip contact, the stretch ratio in the contact zone is *non-uniform*; hence *λ*_{in} in (3.2a) must be replaced by the stretch ratio just inside the contact edge, that is, by . The condition for detachment or crack growth is
3.2bwhere *w*_{ad} is the effective work of adhesion of the interface or simple adhesion energy. Note that (3.2a) differs from the expression given by Kendall [15] because, in our system, elastic strain energy is stored both behind and ahead of the crack front, whereas in Kendall's peel test, the membrane ahead of the crack front is unstretched.

### (a) Frictionless interface

The stretch ratio inside the contact zone is still given by eqn (4.2). However, owing to adhesion, the free-standing portion of the membrane makes a non-zero angle *θ*_{0} with the substrate; as a result, the outer stretch ratio (eqn (4.3)) has to be modified to
3.3As *θ*_{0}>0, the membrane tension is not continuous across the contact edge so eqn (4.4) is no longer true and should be replaced by (3.4). Denoting the tension in the contact zone by *T*_{in} and outside by *T*_{out}, the force balance at the contact edge in the horizontal direction shows that
3.4Eqns (4.5) and (4.7) are modified to
3.5aand
3.5bFor given *a*, *d* and *p*, equation (2.1), eqn (4.2) and equations (3.4) and (3.5*a*,*b*), together with the equilibrium equation (eqn (2.1)) and constitutive law (eqn (2.5)) constitute eight equations with nine unknowns (*c*, *ρ**, *λ*_{out}, *λ*_{in}, *T*_{out}, *T*_{in}, *θ*_{0}, *R*, *θ*_{m}). The extra equation to complete the calculation is the energy balance condition for quasi-static receding contact, equations (3.2*a*,*b*). These nine equations can be reduced to two nonlinear algebraic equations in two unknowns (details are given in appendix B), which are solved in Matlab using a nonlinear root finding algorithm.

### (b) No-slip interface

Owing to the presence of friction at the interface, the tension and stretch ratio are no longer uniform inside the contact region, thereby rendering eqn (4.2) invalid. In addition, the force balance equation (3.4) is no longer applicable because of friction. As a result, we need to solve the problem by incrementally removing a small element at the contact line. The idea is similar to the incremental scheme developed in part I. A detailed description of the numerical procedure is presented in appendix A.

## 4. Simulation of adhesion test

Here, we present the simulation results of an adhesion test where stage 2 (increasing contact) is adhesionless and stage 3 (decreasing contact) is governed by a constant work of adhesion *w*_{ad}. In stage 2, we use our theory in part I [6]. To simulate stage 3, we use our theory presented above. Note that our formulation is still valid if both increasing and decreasing contacts were adhesive, with different work of adhesions. All simulations are carried out using no-slip and frictionless boundary conditions.

The following normalized variables are used in the rest of the paper:

### (a) Frictionless interface

Figure 3 plots the pressure contact relationship of a frictionless membrane for different adhesion energies. As we pressurize an undeformed membrane, it first undergoes free inflation (dotted line) until the pressure is sufficiently large (given by (2.2)) to make point contact with the substrate. Upon further increase in pressure, the membrane contact increases monotonically (solid line). The assumption of adhesionless contact during stage 2 implies that the pressure–contact curves are identical for stages 1 and 2.

In general, if the adhesion energies for the contact and detachment phases are different, then the contact edge will be pinned during the initial phase of unloading. For example, in our case, the energy release rate during the contact phase is zero (adhesionless), and continuity implies that the energy release rate must increase from zero as the pressure is reduced at maximum contact. The energy release rate eventually reaches the work of adhesion when the pressure is reduced to a critical value. Before this value is reached, there is not sufficient energy to detach the interface. As a result, the contact length remains unchanged during this period. Such *contact pinning* or *crack trapping* has been observed in experiments [5,11,16]. It is indicated by the horizontal dashed line in figure 3.

After achieving the maximum contact, pressure is reduced to decrease the contact length and eventually detach the membrane from the substrate. For small adhesion energies (e.g. see in figure 3), contact hysteresis is very small as the unloading curve almost retraces the loading curve. The region of contact decreases *stably all the way down to a line* before detaching (referred to as ‘pinch-off’ [17]). Membrane profiles at pinch-off for two small adhesion energies ( and 0.05) are shown in figure 4. As shown in figure 3, the pressure at pinch-off is always lower than the critical pressure needed to bring the membrane into contact (). As we increase adhesion, contact hysteresis becomes more pronounced, and the detachment pressure decreases (). It is interesting to note that the membrane profile approaches a planar wedge as the pinch-off pressure approaches zero.

For adhesion energies , the pinch-off mode of detachment is suppressed, and the membrane detaches from the substrate unstably with a *finite* contact length (e.g. see (black) in figure 3). This spontaneous detachment without the contact region shrinking all the way to a line is referred to as ‘pull-off’ (membrane profiles at pull-off for and 1.5 are shown in figure 4*a*). If we reduce the pressure beyond this pull-off pressure, then there is no solution to the governing equations, and the membrane jumps out of contact. Our numerical results show that pull-off occurs when the slope at the clamped edge (*θ*_{m}) is *exactly zero*. The pull-off pressures and contact lengths for and 1.5 are identified in figure 3 by points *A*_{1} and *B*_{1}, respectively. They correspond to the minima of pressure: . The critical adhesion at which we see a transition from pinch-off to pull-off depends only on the initial membrane–substrate separation *d* (for *d*=0.5, this happens at ), and will be discussed in detail later. At high adhesion energies, our simulations reveal that there is an unstable branch of the pressure–contact curve. In figure 4*b*, *A*_{1}*A*_{2}, *B*_{1}*B*_{2} and *C*_{1}*C*_{2} denote the unstable branches for three different values of adhesion.

As a membrane cannot support compression, it is necessary to ensure that the membrane tension is everywhere positive (and hence, *λ*>1 using (2.1)). Furthermore, for the case of *frictionless contact*, (3.4) indicates that the contact angle *θ*_{0}≤*π*/2, otherwise, *T*_{in} is compressive and the membrane inside the contact zone will wrinkle. Our simulations showed that wrinkling of the membrane in frictionless contact is possible for very high adhesion energies. In figure 4*b*, the pluses indicate normalized pressures and contact lengths where wrinkling can occur (*T*_{in}<0). For the case of , the membrane detaches via unstable pull-off and never wrinkles. As the adhesion is increased, there is a possibility of wrinkling, but the membrane detaches unstably before this threshold can be reached. However, for very high adhesion , the membrane wrinkles in the stable branch of the contact length versus pressure curve. For these cases, our model cannot be used to predict detachment. As a reference, for a 1 μm thick polydimethylsiloxane (PDMS) membrane (*μ*≈10^{6} Pa), corresponds to an adhesion energy of *w*_{ad}=3 J m^{−2}, at least 60 times the surface energy of PDMS. For a plane-strain membrane, the normalized *out-of-plane* true tension is ; therefore, plane-strain neo-Hookean membranes always wrinkle simultaneously in the in-plane and out-of-plane directions and occur when the stretch ratio is less than 1. This is different from an axisymmetric membrane test where the hoop stress becomes compressive first, leading to wrinkling.

### (b) Effect of initial separation

Figure 5*a*,*b* shows contact versus pressure curves for two different initial separations. We observe that for both low and high adhesion, a *larger* *pressure drop* is required to peel-off a membrane suspended *closer* to the substrate, irrespective of the maximum contact length achieved during inflation. It is important to note that the area under the contact length–pressure loop is not a good measure of the hysteresis, which is the energy dissipated by the system. In figure 6, we plot the pressure as a function of the change in the area enclosed by the membrane (area enclosed is taken to be zero for the undeformed configuration). For an incompressible pressurizing fluid, the area under this plot represents the energy dissipated on a loading–unloading cycle. Comparing figures 5 and 6, we can see that even though the pressure–contact ‘loop’ is larger for a membrane suspended closer to the substrate, it does not imply that it has more energy dissipation.

### (c) Effect of friction

In the previous subsections §4*a*,*b*, we have presented the results for a frictionless membrane. Here, we present the results for a no-slip membrane to study the effect of friction. The solution for frictionless and no-slip contact cases shares many common qualitative features, for example, detachment via pinch-off for low adhesion and via unstable pull-off for high adhesion. However, as plotted in figure 7*a*, higher pressures are required for both increasing and decreasing contact for no-slip contact; this happens irrespective of adhesive strength. This result is not unexpected as peeling every small increment of membrane requires an extra pressure drop (when compared with frictionless contact), as there is no energy contribution from the rest of the membrane stuck in the contact region.

In contrast to a frictionless interface, the contact angle for a no-slip membrane can exceed *π*/2 (membrane is inverted inwards as in figure 7*b*) without inducing compression. This can be seen by balancing the horizontal forces at the contact edge (figure 7*b*, inset), where both the in-plane tensions, *T*^{+} and *T*^{−} are aiding each other, but they are balanced by the friction force. In our simulation, we do not find wrinkling of a no-slip membrane, even for very large adhesion energies. This can be explained by noting that the membrane in contact is stretched because of the no-slip condition, whereas the stretch ratio in the free-standing part must be greater than unity because of geometry.

In figure 8, we demonstrate how friction affects the pull-off parameters (contact length and load). At any fixed adhesion and initial separation, a lower (or more negative) pressure is needed to pinch-off/pull-off a no-slip membrane from the substrate. The contact length at pull-off is also higher for no-slip contact. These differences increase with adhesion.

### (d) Bistability

Our interest in bistability is motivated by the recent experiments of Nadermann *et al.* [18]. They demonstrated that a continuous PDMS membrane supported by small posts can be switched between two metastable states. In the first state, the membrane is flat, and is shown to have strong adhesion and friction, whereas in the second state, the membrane collapses onto the substrate between posts, resulting in a surface with a periodic array of bumps, with much reduced adhesion and friction. This structure can be switched mechanically between these two states repeatedly, thus providing a means for active control of surface mechanical properties such as adhesion and friction.

Figures 3, 5 and 7 reveal the presence of multiple solutions in our system. Specifically, for any separation *d*, there exists a critical adhesion energy *w*_{b} above which the membrane becomes *bistable*, i.e. there exists a stable configuration with a finite contact length when no pressure is applied (the other stable configuration at zero pressure is the undeformed flat membrane). For instance, figure 3 shows that for . The point *J*_{0} in figure 3 indicates the flat configuration for the membrane at zero pressure, i.e. the undeformed membrane. The second (non-trivial) stable configurations for , 0.5 and 1.5 are denoted by *J*_{1}, *J*_{2} and *J*_{3} respectively. In the following discussion, we denote all variables at the onset of bistability using the subscript ‘b’. Our goal is to derive an analytical criterion to predict the onset of bistability. As bistability is affected by the presence of friction at the interface, we consider these cases separately.

#### (i) Onset of bistability for a frictionless interface

Substituting *p*=0 in (eqn (2.1)) and noting that tensions in the membrane (*T*_{out} and *T*_{in}) cannot vanish, we deduce that the radius of curvature has to be singular, i.e.
4.1Using (3.5a), we can deduce that for the system to have a bounded solution in the presence of this singularity, we need a competing limit: *θ*_{m}→*θ*_{b}=*θ*_{0}. Substituting these limits in (3.5*a*,*b*), we obtain
4.2Equations (4.1) and (4.2) demonstrate that the membrane configuration at the onset of bistability has a unique geometric trait: its shape is a planar wedge (already seen in simulations: figure 4*a*).

Substitution of (3.5a) and (4.2) in (3.3) gives us a simplified expression for outer stretch,
4.3The inner stretch cannot be calculated directly using (eqn (4.2)) because both *c* and *ρ** vanish at this point. Instead, we use the equilibrium condition at the edge (3.4), along with the constitutive relation (2.1), resulting in
4.4Solving (4.4) determines (*λ*_{in})_{b} as a function of *d*. Once (*λ*_{in})_{b} and (*λ*_{out})_{b} are determined, we substitute them into the energy balance equation (3.2*a*,*b*) to obtain the relation between adhesive energy and separation. A plot of as a function of *d* is shown in figure 9.

#### (ii) No-slip interface

For a no-slip membrane, the analysis is similar to the frictionless case above because both cases share eqn (2.1) and equations (2.1), (3.3) and (3.5*a*,*b*). Hence, we obtain
4.5Additionally, the no-slip condition ensures that *λ*^{−} at the onset of bistability is the same as the apex stretch ratio at point contact (substitute eqn (4.1*a*) in eqn (3.3)). Thus,
4.6We determine the relation between and *d* by substituting the stretch ratios and *θ*_{b} in the energy balance condition (3.2*a*,*b*). This relationship is shown in figure 9.

We observe that the critical adhesion for bistability is practically unaffected by friction for all separations *d*≤*a*. For large initial separations, a frictionless membrane is found to have a higher threshold for exhibiting bistability. It should be noted that at the onset of bistability (*w*_{ad}=*w*_{b}), membrane detachment always happens via pinch-off. However, for adhesion values greater than *w*_{b}, both pinch-off and pull-off are possible.

### (e) Analytical condition for membrane pull-off

We show that the energy release rate at pull-off reduces to Kendall's well-known formula [15] for no-slip contact. We have not been able to obtain an expression for frictionless contact due to slipping of the membrane on the substrate during receding contact.

#### (i) Pull-off on a no-slip interface

Detailed calculations are given in the electronic supplementary material. Here, we briefly summarize our argument. At pull-off, the pressure is at its minimum, any further reduction in contact area would require the pressure to increase and the membrane goes into unstable contact (figure 7*a*),
4.7Our numerical simulations show that the membrane clamp angle is zero (i.e. the membrane is locally flat at the clamped boundary) and the outer stretch is at its minimum at pull-off, that is,
4.8Substituting (4.7) and (4.8) in (3.3) and (3.5*a*,*b*), we obtain
4.9and
4.10Equation (4.10) states that the stretch ratio at pull-off is continuous across the contact edge, which simplifies the energy balance equation (3.2*a*,*b*) to Kendall's result for membrane peeling,
4.11where *T*=*T*^{+}=*T*^{−} because of (4.10).

#### (ii) Transition from pinch-off to pull-off

The membrane switches from pinch-off to pull-off when and satisfies a certain relation, which will be established below. For a fixed separation , we vary the adhesion to obtain the critical adhesion *w*_{t} at which the pinch-off mode switches to pull-off. As the pinch-off to pull-off transition has the properties of both detachment modes, we obtain *w*_{t} analytically by solving the governing equations in conjunction with conditions of both pinch-off,
4.12and pull-off,
4.13For a *no-slip interface*, we substitute (4.7)–(4.10) along with (4.12) in eqn (2.1) and equations (2.1), (3.3), (3.5*a*,*b*) and (3.2*a*,*b*) to obtain (details are given in the electronic supplementary material),
4.14where . For a *frictionless interface*, we cannot find a closed-form expression for *w*_{t}, although the contact edge slope at detachment at the transition is still given by . The relation between and *d* for this case is found numerically (details are given in the electronic supplementary material) and is shown in figure 10, together with the no-slip result (4.14).

## 5. Conclusion

We model the plane-strain membrane contact adhesion test using a large strain, nonlinear constitutive description of the membrane. Adhesive interaction between the membrane and substrate is modelled using an energy balance approach. Contact hysteresis is modelled by assuming adhesionless contact in stage 2 (increasing contact) with a constant adhesion energy in stage 3 (decreasing contact). We study in detail the effect of friction (frictionless and no-slip) on the adhesion and contact mechanics.

The main findings are the following.

— Higher pressure/suction needs to be applied to achieve the same contact length for no-slip contact. The effect of friction is small for small contact, which justifies the neglect of friction in previous work.

— Differences in adhesion energies cause pinning of the contact edge during unloading. Contact starts to recede once the pressure drops below a threshold. This pressure drop increases with adhesion energy. A no-slip membrane requires a larger drop in pressure to unpin the contact (figure 7

*a*).— For high adhesion, the membrane can stick to the substrate at zero pressure, creating two stable configurations (bistability). Our calculation shows that friction has little effect on the onset of bistability.

— Detachment occurs by two mechanisms: by receding the contact stably to a line (‘pinch-off’) or by developing instability at some finite contact length (‘pull-off’). The former occurs at low adhesion values and the latter at higher adhesion. We establish analytical results for pull-off and the transition from pinch-off to pull-off.

— Frictionless membranes can develop a wrinkling instability at very high adhesion. We do not observe wrinkling in no-slip membranes.

Our analysis can be extended readily to constitutive laws other than the neo-Hookean model. It can be also extended to include inter-surface interactions, such as infinite range potentials [19–21] and the finite cohesive zone model [22,23]. Our model does not account for any bulk dissipation of energy such as viscoelasticity. This limitation should be kept in mind when comparing our results with experimental data.

## Funding statement

This work was supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under award DEFG02-07ER46463.

## Appendix A. Numerical solution scheme for no-slip boundary condition

**(a) Increasing contact**

The stretch ratio (and tension) is non-uniform inside the contact zone, and hence we use the superscript ‘–’ to denote the value of a quantity just inside the contact zone. The membrane not in contact with the substrate still obeys the equilibrium relations (eqns (2.1) and (2.2)) of the free-standing membrane and has uniform stretch and tension. Nevertheless, we denote the stretch and tension in the free-standing portion of the membrane using the superscript ‘+’ from here on. In the absence of adhesion, the slope is zero at the contact edge, and therefore the stretch ratio is continuous across the contact edge.

Compared with the frictionless case, we are now short of one equation, since eqn (4.2) is no longer valid. However, eqns (2.1), (2.5), (4.3), (4.5) and (4.7) are still valid. The stretch inside the contact zone is determined incrementally as follows. We denote quantities at the *i*th step by the subscript *i*, for example, where *i*=0 corresponds to the initiation of contact, e.g. . Assuming in the *i*th step are known, we seek these quantities in the (*i*+1)th step. This is done by incrementing the material coordinate of the contact edge by a small amount *Δρ** to obtain the new material coordinate . Using the continuity of *λ* at the contact edge, the increment in the contact radius is given by
A1where is the stretch ratio at the contact edge (and in the free-standing part of the membrane) in the *i*th step. We can now solve eqns (4.3), (4.5) and (4.7) simultaneously to determine , (*θ*_{m})_{i+1} and *R*_{i+1}. The pressure *p*_{i+1} is then calculated using the equilibrium condition (part 1–1) and the constitutive law (eqn (2.5)). The procedure is stopped after we reach the desired contact radius, say at the *N*th step. The material coordinate and the contact width *c*_{i} as well as the inner stretch ratio for all *i* are stored in a vector of length 3*N*.

**(b) Decreasing contact**

We start unloading in the (*N*+1)th step. We decrease the contact length by the same increment *Δc*_{i} as during increasing contact. For example, in step *N*+1, we decrease the contact length by *c*_{N}−*c*_{N−1}. Because of the no-slip condition, the quantities , and *c*_{N} do not change during unloading, and these quantities are stored as mentioned above. The unknown quantities for this stage are: and pressure *p*_{N+1}. We can solve for these unknowns using equations eqn (2.1) and equations (2.1), (3.3), (3.5*a*,*b*) and (3.2*a*,*b*). This procedure is then repeated with the next decrement *c*_{N−1}−*c*_{N−2} and so on. Similar to the frictionless case, the no-slip case is also expected to exhibit contact pinning during the peeling process owing to contact hysteresis.

## Appendix B. Simplification of frictionless adhesive contact problem

Given any contact length *c*, we seek to calculate the applied pressure *β* required. We introduce two intermediate variables,
B1Combining (3.5*a*,*b*), *x*_{1} is completely determined in terms of *c*,
B2We can rewrite eqn (4.2) and equations (3.3) and (3.5b) to express the stretches and the radius of curvature in terms of two unknowns *x*_{2} and *ρ**,
B3Substituting (B1), (B2) and (B3) along with the constitutive law (eqns (2.4) and (2.5)) into (3.4) and (3.2*a*,*b*) gives us a system of two nonlinear algebraic equations in two unknowns: *x*_{2} and *ρ**. After solving this system, we can calculate the pressure using the membrane equilibrium condition (eqn (2.1)).

- Received June 26, 2013.
- Accepted August 29, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.