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ResearchCite this article: Srivastava A, Hui C-Y. 2013Large deformation contact mechanics of apressurized long rectangular membrane. II.Adhesive contact. Proc R Soc A 469: 20130425.http://dx.doi.org/10.1098/rspa.2013.0425

Received: 26 June 2013Accepted: 29 August 2013

Subject Areas:mechanical engineering, mechanics

Keywords:adhesion, large deformation, membrane,neo-Hookean, friction, contact mechanics

Author for correspondence:Chung-Yuen Huie-mail: ch45@cornell.edu

Electronic supplementary material is availableat http://dx.doi.org/10.1098/rspa.2013.0425 orvia http://rspa.royalsocietypublishing.org.

Large deformation contactmechanics of a pressurizedlong rectangular membrane.II. Adhesive contactAbhishek Srivastava and Chung-Yuen Hui

Field of Theoretical and Applied Mechanics, Sibley School ofMechanical and Aerospace Engineering, Cornell University, Ithaca,NY 14853, USA

In part I of this work, we presented a theory foradhesionless contact of a pressurized neo-Hookeanplane-strain membrane to a rigid substrate. Here,we extend our theory to include adhesion using afracture mechanics approach. This theory is usedto study contact hysteresis commonly observed inexperiments. Detailed analysis is carried out tohighlight the differences between frictionless and no-slip contact. Membrane detachment is found to bestrongly dependent on adhesion: for low adhesion, themembrane ‘pinches-off’, whereas for large adhesions,it detaches unstably at finite contact (‘pull-off’).Expressions are derived for the critical adhesionneeded for pinch-off to pull-off transition. Above athreshold adhesion, the membrane exhibits bistability,two stable states at zero applied pressure. Thecondition for bistability for both frictionless and no-slip boundary conditions is obtained explicitly.

1. Introduction and motivationAdhesion is important in many engineering applicationssuch as the performance of coatings, labelling, laminatesand biomedical devices. For example, cells mustadhere well to tissue scaffolds, whereas weak adhesionis desirable for coronary stents. To control and tocharacterize adhesion, it is necessary to devise testingmethods that are highly sensitive to weak interactionsand easily carried out on different substrates. The highcompliance of membranes allows for intimate contact,even for substrates that are not perfectly flat, such astissues. Thus, the membrane–contact adhesion tests usedby [1–5] provide an attractive combination of sensitivityand substrate versatility. A typical experimental set-up

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

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pressure tube

substrate substrate substrate

membrane

2a

p p p

2cd

(a) (b) (c)

Figure 1. Schematic of a membrane adhesion experiment. (a) For applied pressure p less than the critical pressure needed forcontact, themembrane undergoes free inflation (stage 1). (b) For pressure greater than the critical value, themembranemakescontact with the substrate (stage 2). (c) Once the maximum desired contact is achieved, the pressure is reduced to peel themembrane off (stage 3). (Online version in colour.)

is illustrated in figure 1a–c. A membrane is clamped to the edge of a cylindrical tube, which issuspended at a distance above a flat substrate. An external pressure is applied by pumping airinto 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 todetach 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 (increasingcontact) is invariably much smaller than stage 3 (decreasing contact), even for materials thatexhibit 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 ofthe work, we extend our theory to include adhesive contact. We use our theory to study a typicaladhesion test where stage 2 is adhesionless, whereas stage 3 is governed by a constant workof adhesion.

As pointed out in part I, the role of friction on the contact mechanics of membranes is poorlyunderstood. In this work, we carried out detailed analysis for two limiting cases: frictionless andno-slip adhesive contact. Wrinkling or buckling of the membrane near the contact line is oftenobserved in axis-symmetric geometry during stage 3 [11]. Wrinkling is caused by the membranetension 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 forno-slip contact, wrinkling is suppressed, even at very large adhesion energies. Our results suggestthat 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 formulationof adhesive contact is given in §3. Based on these results, simulations are carried out to modela test where stage 2 is adhesionless, whereas stage 3 is governed by a constant work ofadhesion. The results and discussions are presented in §4. §5 summarizes the key findings ofthis work.

2. Geometry, constitutive model and free inflationConsider the plane-strain deformation of an infinitely long elastic membrane of initial width 2asubjected to uniform excess pressure (i.e. pressure difference) p. The undeformed membrane is flatand has uniform initial thickness h0. It is suspended at a distance of d above a rigid flat substrateand is clamped at the edges O1 and O2. Figure 2 shows a cross section of the membrane in contactwith a flat rigid substrate, with the contact zone occupying the strip x ∈ (−c, c), z = 0, |y| < ∞.Symmetry allows us to consider x ≥ 0. Deformation carries a material point on the membranewith material coordinate ρ to (x, z) (figure 2). The half-length of the contact strip is denoted by c,

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(x,z)p d

O

O2O1

xsubstrate

C

2a

2c

z

R

rr*

q

qm

q0

R

Figure 2. Geometry of a membrane in contact with the substrate. The undeformed membrane is shown as a dashed line andthe deformed membrane as a solid curve. The substrate is assumed to be rigid.

and its material coordinate is denoted by ρ∗. The portion of the membrane not in contact will belabelled as free standing. Figure 2 shows that the slope of the deformed membrane varies from θ0at the contact edge to θm 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. Themembrane is described by an incompressible neo-Hookean model where the true tension T of amaterial element is related to its in-plane stretch ratio λ by

T = μh0

λ

(λ2 − 1

λ2

), (2.1)

where 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 deflectionδ can be expressed in terms of a single parameter: the angle at the clamp edge θm (see eqns (3.8),(3.3) and (3.7), respectively). The relation between variables (e.g. p and δ) can be obtained byeliminating this parameter. For a given initial separation d, contact will occur when δ = d. Asshown in part I eqn (4.1b), the normalized pressure βc = apc/μh0 required to make point contact is

βc = θc − sin4 θc

θ3c

, (2.2)

where θc = 2 tan−1(d/a).

3. Adhesion and energy release rateTo 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 thiscrack, G, is defined as the negative rate of change of the potential energy of the system E (per unitthickness in the out-of-plane direction) with respect to the crack surface,

G = − lim�c→0

�E�c

. (3.1)

The 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),

G = Tout

(λout

λin− cos θ0

)− h0

λin[W(λout) − W(λin)] , (3.2a)

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where λout, λin, Tout and Tin are the stretch and in-plane tension in the outer and inner portionsof the membrane, respectively. For no-slip contact, the stretch ratio in the contact zone is non-uniform; hence λin in (3.2a) must be replaced by the stretch ratio just inside the contact edge, thatis, by λ− ≡ limx→c− λin. The condition for detachment or crack growth is

G = wad, (3.2b)

where wad is the effective work of adhesion of the interface or simple adhesion energy. Notethat (3.2a) differs from the expression given by Kendall [15] because, in our system, elastic strainenergy is stored both behind and ahead of the crack front, whereas in Kendall’s peel test, themembrane ahead of the crack front is unstretched.

(a) Frictionless interfaceThe 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 θ0 with the substrate; as aresult, the outer stretch ratio (eqn (4.3)) has to be modified to

λout = R(θm − θ0)a − ρ∗ . (3.3)

As θ0 > 0, the membrane tension is not continuous across the contact edge so eqn (4.4) is no longertrue and should be replaced by (3.4). Denoting the tension in the contact zone by Tin and outsideby Tout, the force balance at the contact edge in the horizontal direction shows that

Tout cos θ0 − Tin = 0. (3.4)

Eqns (4.5) and (4.7) are modified to

R(sin θm − sin θ0) = (a − c) (3.5a)

andR(− cos θm + cos θ0) = d. (3.5b)

For given a, d and p, equation (2.1), eqn (4.2) and equations (3.4) and (3.5a,b), together with theequilibrium equation (eqn (2.1)) and constitutive law (eqn (2.5)) constitute eight equations withnine unknowns (c, ρ∗, λout, λin, Tout, Tin, θ0, R, θm). The extra equation to complete the calculationis the energy balance condition for quasi-static receding contact, equations (3.2a,b). These nineequations can be reduced to two nonlinear algebraic equations in two unknowns (details aregiven in appendix B), which are solved in Matlab using a nonlinear root finding algorithm.

(b) No-slip interfaceOwing to the presence of friction at the interface, the tension and stretch ratio are no longeruniform inside the contact region, thereby rendering eqn (4.2) invalid. In addition, the forcebalance equation (3.4) is no longer applicable because of friction. As a result, we need to solvethe problem by incrementally removing a small element at the contact line. The idea is similar tothe incremental scheme developed in part I. A detailed description of the numerical procedure ispresented in appendix A.

4. Simulation of adhesion testHere, we present the simulation results of an adhesion test where stage 2 (increasing contact) isadhesionless and stage 3 (decreasing contact) is governed by a constant work of adhesion wad. Instage 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 frictionlessboundary conditions.

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free inflationincreasing contactcontact pinningdecreasing contact

0.5

0.05

0.00

1

0

0

0.01

0.02

0.03

0.1 0.2 0.3 0.4

B1

w- ad= 1.5

A1

J1

J2

J3

J0

norm

aliz

ed c

onta

ct, c

-

normalized pressure, b

0–2 –1 0 1 2 3 4 5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

b p2 b

p1 bc

Figure 3. Plot of contact length variationwith applied pressure for a frictionlessmembrane for four different adhesion energies.Arrows indicate the path traversed in a loading–unloading cycle. Owing to contact hysteresis, there are four distinct parts of thecurve: free inflation (dotted), increasing contact (solid), contact pinning (dashed) anddecreasing contact (dashed-dotted). Plotsfor four different adhesion energies during unloading are shown: w̄ad = 0.001, 0.05, 0.5 and 1.5. βp

1 and βp2 are the pinch-off

pressures for the low adhesion cases: w̄ad = 0.001 and w̄ad = 0.05, respectively. A1 and B1 mark the pull-off points for highadhesion cases: w̄ad = 0.5 and w̄ad = 1.5, respectively. (Online version in colour.)

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

c̄ = ca

, d̄ = da

, R̄ = Ra

, ρ̄∗ = ρ

a, W̄ = W

μ, T̄ = T

μh0and w̄ad = wad

μh0.

(a) Frictionless interfaceFigure 3 plots the pressure contact relationship of a frictionless membrane for different adhesionenergies. As we pressurize an undeformed membrane, it first undergoes free inflation (dottedline) until the pressure is sufficiently large (given by (2.2)) to make point contact with thesubstrate. Upon further increase in pressure, the membrane contact increases monotonically (solidline). The assumption of adhesionless contact during stage 2 implies that the pressure–contactcurves are identical for stages 1 and 2.

In general, if the adhesion energies for the contact and detachment phases are different, thenthe 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 thatthe 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 toa 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 cracktrapping has been observed in experiments [5,11,16]. It is indicated by the horizontal dashed linein figure 3.

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–1.0 –0.5x-0 0.5 1.0 –4 –3 –2 –1 0

unloading curve

wrinklingunstable branch

1 2normalized pressure, b

w- ad= 3

w- ad= 1.5w-ad = 0.001

w-ad = 0.05w-ad = 0.5

w-ad = 1.5w- ad

= 0.5

norm

aliz

ed c

onta

ct, c

-

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.90.50adhesionless

0.25z-

0

C1

C2

B1

B2

A1

A2

(a) (b)

Figure 4. (a–b) Frictionlessmembrane receding contact plots for intermediate separation d̄ = 0.5. These results are obtainedwhen we inflate to a maximum contact length of c̄ = 0.9.

After achieving the maximum contact, pressure is reduced to decrease the contact lengthand eventually detach the membrane from the substrate. For small adhesion energies (e.g. seew̄ad = 0.001 in figure 3), contact hysteresis is very small as the unloading curve almost retracesthe 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(w̄ad = 0.001 and 0.05) are shown in figure 4. As shown in figure 3, the pressure at pinch-off isalways lower than the critical pressure needed to bring the membrane into contact (βp

1 , βp2 < βc).

As we increase adhesion, contact hysteresis becomes more pronounced, and the detachmentpressure decreases (βp

2 < βp1 ). It is interesting to note that the membrane profile approaches a

planar wedge as the pinch-off pressure approaches zero.For adhesion energies w̄ad > 0.2, the pinch-off mode of detachment is suppressed, and the

membrane detaches from the substrate unstably with a finite contact length (e.g. see w̄ad = 0.5(black) in figure 3). This spontaneous detachment without the contact region shrinking all theway to a line is referred to as ‘pull-off’ (membrane profiles at pull-off for w̄ad = 0.5 and 1.5 areshown in figure 4a). If we reduce the pressure beyond this pull-off pressure, then there is nosolution to the governing equations, and the membrane jumps out of contact. Our numericalresults show that pull-off occurs when the slope at the clamped edge (θm) is exactly zero. Thepull-off pressures and contact lengths for w̄ad = 0.5 and 1.5 are identified in figure 3 by points A1and B1, respectively. They correspond to the minima of pressure: dβ/dc̄ = 0. The critical adhesionat 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 w̄ad ≈ 0.2), 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 4b, A1A2, B1B2 and C1C2 denote the unstable branches for three differentvalues of adhesion.

As a membrane cannot support compression, it is necessary to ensure that the membranetension is everywhere positive (and hence, λ > 1 using (2.1)). Furthermore, for the case offrictionless contact, (3.4) indicates that the contact angle θ0 ≤ π/2, otherwise, Tin is compressiveand the membrane inside the contact zone will wrinkle. Our simulations showed that wrinklingof the membrane in frictionless contact is possible for very high adhesion energies. In figure 4b,the pluses indicate normalized pressures and contact lengths where wrinkling can occur (Tin < 0).For the case of w̄ad = 0.5, the membrane detaches via unstable pull-off and never wrinkles.As the adhesion is increased, there is a possibility of wrinkling, but the membrane detachesunstably before this threshold can be reached. However, for very high adhesion (w̄ad ∼ 3),

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

norm

aliz

ed c

onta

ct, c

-

10 2 3 4 5 –2 0 2 4 6

free inflation

d-=

0.9

d- =0.

9

d-=

0.5

d- = 0.5

increasing contactcontact pinningdecreasing contact

free inflationincreasing contactcontact pinningdecreasing contact

normalized pressure, b normalized pressure, b

(a) (b)

Figure 5. Contact length variation with pressure in a frictionlessmembrane for (a) small adhesion (w̄ad = 0.05) and (b) largeadhesion (w̄ad = 1.5). For both cases, two different initial separation values are plotted: d̄ = 0.5 (thin lines) and d̄ = 0.9(thick lines).

the membrane wrinkles in the stable branch of the contact length versus pressure curve. Forthese cases, our model cannot be used to predict detachment. As a reference, for a 1 µmthick polydimethylsiloxane (PDMS) membrane (μ ≈ 106 Pa), w̄ad = 3 corresponds to an adhesionenergy of wad = 3 J m−2, at least 60 times the surface energy of PDMS. For a plane-strainmembrane, the normalized out-of-plane true tension is T̄2 = (λ2 − 1)/λ3; therefore, plane-strainneo-Hookean membranes always wrinkle simultaneously in the in-plane and out-of-planedirections and occur when the stretch ratio is less than 1. This is different from an axisymmetricmembrane test where the hoop stress becomes compressive first, leading to wrinkling.

(b) Effect of initial separationFigure 5a,b shows contact versus pressure curves for two different initial separations. We observethat for both low and high adhesion, a larger pressure drop is required to peel-off a membranesuspended closer to the substrate, irrespective of the maximum contact length achieved duringinflation. It is important to note that the area under the contact length–pressure loop is nota 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 incompressiblepressurizing 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 moreenergy dissipation.

(c) Effect of frictionIn the previous subsections §4a,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 solutionfor 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 7a, higher pressures are required for both increasing and decreasing contactfor no-slip contact; this happens irrespective of adhesive strength. This result is not unexpected

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5

4

3

2

1

0

–1

–2

0 0.1

d-= 0.9, w -ad = 1.5

d-= 0.5, w -ad = 1.5

B

A

0.2 0.3 0.4 0.5 0.6 0.7

norm

aliz

ed p

ress

ure,

b

normalized enclosed area change, D A-

= D A/pa2

Figure 6. Applied pressure versus enclosed area change plot of a frictionlessmembrane for a high adhesion case (w̄ad = 1.5).The solid curve denotes the case of moderate separation (d̄ = 0.5) and the dotted curve denotes large separation (d̄ = 0.9).A and B denote the points where the membrane pulls-off from the substrate. Arrows indicate the path traversed in a loading–unloading cycle.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

norm

aliz

ed c

onta

ct, c

-

–2 0 2 4 6 8 –1 –0.5 0x-

0.5

f

1.0

frictionless w-ad = 0.05

frictionless w-ad = 0.05

frictionless w-ad = 1.5

frictionless w-ad = 1.5

no-slip w-ad = 0.05

no-slip w-ad = 0.05

no-slip w-ad = 1.5

no-slip w-ad = 1.5

normalized pressure, b

0.50

0.25

T+

T–

z-

0

(a) (b)

Figure 7. Comparison of frictionless and no-slip contact conditions for two different adhesion values for d̄ = 0.5.Contact length–pressure curves are shown in (a) and the corresponding membrane profiles at detachment in (b). The plotsfor different adhesion values (for both frictionless and no-slip separately) overlap in the loading stage because the contactis assumed to be adhesionless during increasing contact. Arrows in (a) indicate the path traversed in a loading–unloadingcycle. Inset in (b) shows the free-body diagram of the contact edge where f is the friction per unit depth acting on thecontact edge.

as peeling every small increment of membrane requires an extra pressure drop (when comparedwith frictionless contact), as there is no energy contribution from the rest of the membrane stuckin the contact region.

In contrast to a frictionless interface, the contact angle for a no-slip membrane can exceedπ/2 (membrane is inverted inwards as in figure 7b) without inducing compression. This canbe seen by balancing the horizontal forces at the contact edge (figure 7b, inset), where both the

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0 1 2 3 4 10 2 3 4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.01

0

–1

–2

–3

–4

–5

–6

–7

–8

normalized work of adhesion, w-ad normalized work of adhesion, w-ad

no-slip d-= 0.5

no-slip d-= 0.9

frictionless d-= 0.5

frictionless d-= 0.9

no-slip d-= 0.5

no-slip d-= 0.9

frictionless d-= 0.5

frictionless d-= 0.9

cont

act l

engt

h at

pul

l-of

f, c- p

pull-

off

pres

sure

, bp

(a) (b)

Figure 8. Pull-off parameters for a rectangular membrane. Results are shown for two different initial separation values: d̄ =0.5 (thin lines) and d̄ = 0.9 (thick lines). For each separation, plots are shown for both no-slip (solid) and frictionless (dashed)boundary conditions. (a) Contact length at pull-off and (b) pressure at pull-off.

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 adhesionenergies. This can be explained by noting that the membrane in contact is stretched because of theno-slip condition, whereas the stretch ratio in the free-standing part must be greater than unitybecause of geometry.

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

(d) BistabilityOur interest in bistability is motivated by the recent experiments of Nadermann et al. [18]. Theydemonstrated that a continuous PDMS membrane supported by small posts can be switchedbetween two metastable states. In the first state, the membrane is flat, and is shown to have strongadhesion and friction, whereas in the second state, the membrane collapses onto the substratebetween posts, resulting in a surface with a periodic array of bumps, with much reduced adhesionand 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 adhesionand friction.

Figures 3, 5 and 7 reveal the presence of multiple solutions in our system. Specifically, forany separation d, there exists a critical adhesion energy wb above which the membrane becomesbistable, i.e. there exists a stable configuration with a finite contact length when no pressureis applied (the other stable configuration at zero pressure is the undeformed flat membrane).For instance, figure 3 shows that w̄b ∼ 0.045 for d̄ = 0.5. The point J0 in figure 3 indicatesthe flat configuration for the membrane at zero pressure, i.e. the undeformed membrane. Thesecond (non-trivial) stable configurations for w̄ad = 0.05, 0.5 and 1.5 are denoted by J1, J2 andJ3 respectively. In the following discussion, we denote all variables at the onset of bistabilityusing the subscript ‘b’. Our goal is to derive an analytical criterion to predict the onset ofbistability. As bistability is affected by the presence of friction at the interface, we consider thesecases separately.

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(i) Onset of bistability for a frictionless interface

Substituting p = 0 in (eqn (2.1)) and noting that tensions in the membrane (Tout and Tin) cannotvanish, we deduce that the radius of curvature has to be singular, i.e.

Rb → ∞. (4.1)

Using (3.5a), we can deduce that for the system to have a bounded solution in the presence ofthis singularity, we need a competing limit: θm → θb = θ0. Substituting these limits in (3.5a,b), weobtain

θm = θ0 ≡ θb = tan−1(

da

). (4.2)

Equations (4.1) and (4.2) demonstrate that the membrane configuration at the onset of bistabilityhas a unique geometric trait: its shape is a planar wedge (already seen in simulations: figure 4a).

Substitution of (3.5a) and (4.2) in (3.3) gives us a simplified expression for outer stretch,

(λout)b = sec(θb) =√

1 +(

da

)2. (4.3)

The inner stretch cannot be calculated directly using (eqn (4.2)) because both c and ρ∗ vanish atthis point. Instead, we use the equilibrium condition at the edge (3.4), along with the constitutiverelation (2.1), resulting in

(λout)b − (λout)−3b

(λout)b= ((λin)b − (λin)−3

b ). (4.4)

Solving (4.4) determines (λin)b as a function of d. Once (λin)b and (λout)b are determined, wesubstitute them into the energy balance equation (3.2a,b) to obtain the relation between adhesiveenergy and separation. A plot of w̄b 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 casesshare eqn (2.1) and equations (2.1), (3.3) and (3.5a,b). Hence, we obtain

λ+b = sec(θb) =

√1 +

(da

)2. (4.5)

Additionally, the no-slip condition ensures that λ− at the onset of bistability is the same as theapex stretch ratio at point contact (substitute eqn (4.1a) in eqn (3.3)). Thus,

λ−b = λapex = 2 tan−1(d/a)

sin[2 tan−1(d/a)

] . (4.6)

We determine the relation between w̄b and d by substituting the stretch ratios and θb in the energybalance condition (3.2a,b). This relationship is shown in figure 9.

We observe that the critical adhesion for bistability is practically unaffected by friction for allseparations d ≤ a. For large initial separations, a frictionless membrane is found to have a higherthreshold for exhibiting bistability. It should be noted that at the onset of bistability (wad = wb),membrane detachment always happens via pinch-off. However, for adhesion values greater thanwb, both pinch-off and pull-off are possible.

(e) Analytical condition for membrane pull-offWe 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 toslipping of the membrane on the substrate during receding contact.

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1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

criti

cal a

dhes

ion

for

bist

abili

ty, w

- b

normalized initial separation, d-

0.5 1.0 1.5 2.0

bistability

bistability

no

no-slipfrictionless

Figure 9. Critical value of adhesion for onset of bistability as a function of initial separation betweenmembrane and substrate.

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

Detailed calculations are given in the electronic supplementary material. Here, we brieflysummarize our argument. At pull-off, the pressure is at its minimum, any further reduction incontact area would require the pressure to increase and the membrane goes into unstable contact(figure 7a),

dβ

dc̄= 0. (4.7)

Our numerical simulations show that the membrane clamp angle is zero (i.e. the membrane islocally flat at the clamped boundary) and the outer stretch is at its minimum at pull-off, that is,

θm = 0 anddλ+

dc̄= 0. (4.8)

Substituting (4.7) and (4.8) in (3.3) and (3.5a,b), we obtain

dR̄dc̄

= 0,dθ0

dc̄= 0 (4.9)

and

λ+ = λ− = −R̄θ0

(1 − ρ̄∗). (4.10)

Equation (4.10) states that the stretch ratio at pull-off is continuous across the contact edge, whichsimplifies the energy balance equation (3.2a,b) to Kendall’s result for membrane peeling,

T̄(1 − cos θ0) = w̄ad, (4.11)

where 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 d̄ and w̄ad satisfies a certain relation,which will be established below. For a fixed separation d̄, we vary the adhesion to obtain the

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9

8

7

6

5

4

3

2

1

0

norm

aliz

ed a

dhes

ion

for

tran

sitio

n, w-

t

normalized initial separation, d-0.5 1.0 1.5 2.0

pull-off pinch-off

no-slipfrictionless

Figure 10. Critical adhesion for transition from pinch-off to pull-off as a function of initial separation.

critical adhesion wt at which the pinch-off mode switches to pull-off. As the pinch-off to pull-offtransition has the properties of both detachment modes, we obtain wt analytically by solving thegoverning equations in conjunction with conditions of both pinch-off,

c̄ = ρ̄∗ = 0, (4.12)

and pull-off,dβ

dc= 0 and θm = 0. (4.13)

For a no-slip interface, we substitute (4.7)–(4.10) along with (4.12) in eqn (2.1) and equations (2.1),(3.3), (3.5a,b) and (3.2a,b) to obtain (details are given in the electronic supplementary material),

w̄t = 2

(θ0t

sin(θ0t)−(

θ0t

sin(θ0t)

)−3)

sin2(

θ0t

2

), (4.14)

where θ0t = 2 tan−1 d̄. For a frictionless interface, we cannot find a closed-form expression for wt,although the contact edge slope at detachment at the transition is still given by θ0t = 2 tan−1(d̄).The relation between w̄t and d for this case is found numerically (details are given in the electronicsupplementary material) and is shown in figure 10, together with the no-slip result (4.14).

5. ConclusionWe model the plane-strain membrane contact adhesion test using a large strain, nonlinearconstitutive description of the membrane. Adhesive interaction between the membrane andsubstrate is modelled using an energy balance approach. Contact hysteresis is modelled byassuming adhesionless contact in stage 2 (increasing contact) with a constant adhesion energyin stage 3 (decreasing contact). We study in detail the effect of friction (frictionless and no-slip) onthe 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 offriction in previous work.

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— 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 dropincreases with adhesion energy. A no-slip membrane requires a larger drop in pressure tounpin the contact (figure 7a).

— For high adhesion, the membrane can stick to the substrate at zero pressure, creating twostable configurations (bistability). Our calculation shows that friction has little effect onthe 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 formeroccurs at low adhesion values and the latter at higher adhesion. We establish analyticalresults for pull-off and the transition from pinch-off to pull-off.

— Frictionless membranes can develop a wrinkling instability at very high adhesion. We donot observe wrinkling in no-slip membranes.

Our analysis can be extended readily to constitutive laws other than the neo-Hookean model. Itcan 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 dissipationof energy such as viscoelasticity. This limitation should be kept in mind when comparing ourresults 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 contactThe stretch ratio (and tension) is non-uniform inside the contact zone, and hence we use thesuperscript ‘–’ to denote the value of a quantity just inside the contact zone. The membrane notin 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 andtension in the free-standing portion of the membrane using the superscript ‘+’ from here on. Inthe absence of adhesion, the slope is zero at the contact edge, and therefore the stretch ratio iscontinuous across the contact edge.

Compared with the frictionless case, we are now short of one equation, since eqn (4.2) is nolonger valid. However, eqns (2.1), (2.5), (4.3), (4.5) and (4.7) are still valid. The stretch inside thecontact zone is determined incrementally as follows. We denote quantities at the ith step by thesubscript i, for example, pi, ρ∗

i , ci . . . , where i = 0 corresponds to the initiation of contact, e.g. (p0 =pc, ρ∗

0 = c0 = 0). Assuming pi, ρ∗i , ci . . . in the ith step are known, we seek these quantities in the

(i + 1)th step. This is done by incrementing the material coordinate ρ∗i of the contact edge by a

small amount �ρ∗ to obtain the new material coordinate ρ∗i+1 = ρ∗

i + �ρ∗. Using the continuity ofλ at the contact edge, the increment in the contact radius is given by

�ci ≈ λ−i �ρ∗, (A 1)

where λ−i (= λ+

i ) is the stretch ratio at the contact edge (and in the free-standing part ofthe membrane) in the ith step. We can now solve eqns (4.3), (4.5) and (4.7) simultaneouslyto determine λ+

i+1(= λ−i+1), (θm)i+1 and Ri+1. The pressure pi+1 is then calculated using the

equilibrium condition (part 1–1) and the constitutive law (eqn (2.5)). The procedure is stoppedafter we reach the desired contact radius, say at the Nth step. The material coordinate ρ∗

i and thecontact width ci as well as the inner stretch ratio λ−

i for all i are stored in a vector of length 3N.

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(b) Decreasing contactWe start unloading in the (N + 1)th step. We decrease the contact length by the same increment�ci as during increasing contact. For example, in step N + 1, we decrease the contact length bycN − cN−1. Because of the no-slip condition, the quantities ρ∗

N , λ−N and cN do not change during

unloading, and these quantities are stored as mentioned above. The unknown quantities for thisstage are: λ+

N+1, RN+1, (θm)N+1, (θ0)N+1, T+N+1 and pressure pN+1. We can solve for these unknowns

using equations eqn (2.1) and equations (2.1), (3.3), (3.5a,b) and (3.2a,b). This procedure is thenrepeated with the next decrement cN−1 − cN−2 and so on. Similar to the frictionless case, theno-slip case is also expected to exhibit contact pinning during the peeling process owing tocontact hysteresis.

Appendix B. Simplification of frictionless adhesive contact problemGiven any contact length c, we seek to calculate the applied pressure β required. We introducetwo intermediate variables,

x1 = θm + θ0 and x2 = θm − θ0. (B 1)

Combining (3.5a,b), x1 is completely determined in terms of c,

x1 = 2 tan−1(

d(a − c)

). (B 2)

We can rewrite eqn (4.2) and equations (3.3) and (3.5b) to express the stretches and the radius ofcurvature in terms of two unknowns x2 and ρ∗,

λin = c/ρ∗, R = d2 sin(x1/2) sin(x2/2)

and λout = dx2

2(a − ρ∗) sin(x1/2) sin(x2/2). (B 3)

Substituting (B 1), (B 2) and (B 3) along with the constitutive law (eqns (2.4) and (2.5)) into (3.4) and(3.2a,b) gives us a system of two nonlinear algebraic equations in two unknowns: x2 and ρ∗. Aftersolving this system, we can calculate the pressure using the membrane equilibrium condition(eqn (2.1)).

References1. Shanahan ME. 2000 Adhesion of a punch to a thin membrane. C.R. Acad. Sci. Ser. IV Phys. 1,

517–522. (doi:10.1016/S1296-2147(00)00147-5)2. Shanahan MER. 1995 A variant of JKR test: a slightly inflated balloon. C.R. L Acad. Sci. Ser. II

Mech. Phys. Chim. Astron. 321, 259–264.3. Shanahan MER. 1997 A novel test for the appraisal of solid/solid interfacial interactions.

J. Adhes. 63, 15–29. (doi:10.1080/00218469708015211)4. Wan KT. 2001 Adherence of an axisymmetric flat punch on a thin flexible membrane. J. Adhes.

75, 369–380. (doi:10.1080/00218460108029611)5. Flory AL, Brass DA, Shull KR. 2007 Deformation and adhesive contact of elastomeric

membranes. J. Polym. Sci. B, Polym. Phys. 45, 3361–3374. (doi:10.1002/Polb.21322)6. Srivastava A, Hui C-Y. 2013 Large deformation contact mechanics of long rectangular

membranes. I. Adhesionless contact. Proc. R. Soc. A 469, 20130424 (doi:10.1098/rspa.2013.0424)7. Maugis D, Barquins M. 1978 Fracture mechanics and the adherence of viscoelastic solids.

J. Phys. D 11, 1989–2023. (doi:10.1088/0022-3727/11/14/011)8. Bhushan B. 2001 Fundamentals of tribology and bridging the gap between the macro and

micro/nanoscales, p. 320. Berlin, Germany: Springer.9. Shull KR. 2002 Contact mechanics and the adhesion of soft solids. Mater. Sci. Eng. R, Rep. 36,

1–45. (doi:10.1016/S0927-796X(01)00039-0)10. Vajpayee S, Hui C-Y, Jagota A. 2008 Model-independent extraction of adhesion energy from

indentation experiments. Langmuir 24, 9401–9409. (doi:10.1021/la800817x)

on May 21, 2018http://rspa.royalsocietypublishing.org/Downloaded from

15

rspa.royalsocietypublishing.orgProcRSocA469:20130425

..................................................

11. Laprade EJ, Long R, Pham JT, Lawrence J, Emrick T, Crosby AJ, Hui C-Y, Shull KR. 2013Large deformation and adhesive contact studies of axisymmetric membranes. Langmuir 29,1407–1419. (doi:10.1021/la303810d)

12. Johnson KL, Kendall K, Roberts AD. 1971 Surface energy and the contact of elastic solids. Proc.R. Soc. Lond. A 324, 301–313. (doi:10.1098/rspa.1971.0141)

13. Nadler B, Tang T. 2008 Decohesion of a rigid punch from non-linear membrane undergoingfinite axisymmetric deformation. Int. J. Non-Linear Mech. 43, 716–721. (doi:10.1016/j.ijnonlinmec.2008.03.006)

14. Long R, Shull KR, Hui C-Y. 2010 Large deformation adhesive contact mechanics of circularmembranes with a flat rigid substrate. J. Mech. Phys. Solids 58, 1225–1242. (doi:10.1016/j.jmps.2010.06.007)

15. Kendall K. 1975 Thin-film peeling-the elastic term. J. Phys. D 8, 1449–1452. (doi:10.1088/0022-3727/8/13/005)

16. Glassmaker NJ, Jagota A, Hui CY, Noderer WL, Chaudhury MK. 2007 Biologically inspiredcrack trapping for enhanced adhesion. Proc. Natl Acad. Sci. USA 104, 10 786–10 791.(doi:10.1073/pnas.0703762104)

17. Wong M-F, Duan G, Wan K-T. 2007 Adhesion-delamination mechanics of a prestressedrectangular film adhered onto a rigid substrate. J. Appl. Phys. 101, 024903. (doi:10.1063/1.2422775)

18. Nadermann N, Ning J, Jagota A, Hui C-Y. 2010 Active switching of adhesion in a film-terminated fibrillar structure. Langmuir 26, 15 464–15 471. (doi:10.1021/la102593h)

19. Wu J-J. 2006 Numerical analyses on elliptical adhesive contact. J. Phys. D 39, 1899–1907.(doi:10.1088/0022-3727/39/9/027)

20. Li G, Wan K-T. 2010 Delamination mechanics of a clamped rectangular membrane in thepresence of long-range intersurface forces: transition from JKR to DMT limits. J. Adhes. 86,335–351. (doi:10.1080/00218460903482531)

21. Jin C, Jagota A, Hui C-Y. 2011 An easy-to-implement numerical simulation method foradhesive contact problems involving asymmetric adhesive contact. J. Phys. D 44, 405303.(doi:10.1088/0022-3727/44/40/405303)

22. Dugdale DS. 1960 Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8, 100–104.(doi:10.1016/0022-5096(60)90013-2)

23. Duan G, Wan K-T. 2007 Analysis of one-dimensional and two-dimensional thin film ‘pull-in’ phenomena under the influence of an electrostatic potential. J. Appl. Mech. 74, 927–934.(doi:10.1115/1.2722311)

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