## Abstract

A theoretical and numerical analysis of a long rectangular nonlinear viscoelastic membrane in adhesionless contact with a rigid substrate is presented. Computations are carried out for the case where material behaviour is given by Christensen's large deformation viscoelastic law. The effect of friction, loading and unloading rates, hold time and material viscosity on the deformation, energy dissipation and detachment load is investigated in detail. Wrinkling of the membrane is found for a subset of the parameter space. Our results show that energy dissipation and detachment load depend on the contact length achieved at peak pressure, in addition to the rate of loading and unloading. Our numerical results also reveal that there can be situations where maximum dissipation does not correspond to maximum detachment load.

## 1. Introduction and motivation

Deformation and contact of thin films play an important role in engineering and biosciences, touching a diverse range of areas such as biomedical applications, coatings and laminates, MEMS design and graphene devices. There have been various theoretical and experimental studies on the membrane contact problem for elastic systems [1–12]. Many of these works address situations where membrane deflections are small, so that small strain theory can be applied to simplify the analysis. However, in many experiments, membrane strains can be sufficiently large [13,14] so that large deformation theory based membrane contact models [15–19] need to be used to provide an accurate description of the phenomenon.

Many polymer and biological membranes exhibit viscoelastic behaviour that cannot be adequately described by elasticity models. Viscoelasticity can significantly affect the deflection and adhesion strength of membranes, as shown by recent experimental works [13,14,20]. The coupling of viscoelastic dissipation, interfacial adhesion and large deformation makes the interpretation of test results extremely difficult. However, to the best of our knowledge, there has been limited work on the theoretical modelling of the viscoelastic membrane problem. Evans & Hochmuth [21] and Wineman [22–25] have presented rigorous analyses of circular nonlinear viscoelastic membranes inflated by pressure. The membranes in these studies are allowed to inflate freely without contacting any external boundaries. More germane to this work are the recent studies by Nguyen *et al.* [26,27]. The main focus of these papers is to determine the deformation of a nonlinear viscoelastic membrane indented by a spherical [26] and a flat circular indenter [27], respectively. Contact between the indenter and the membrane is assumed to frictionless and adhesionless. In this work, we consider a simpler geometry, i.e. a long rectangular film suspended above a rigid substrate and inflated by pressure. A major advantage of using this geometry is that the membrane segment not in contact has uniform stresses (discussed later) and hence the analytical equations describing the viscoelastic behaviour are much more manageable. In contrast to Nguyen *et al.*, our primary interest is on how pull-off force and energy dissipation depends on the viscoelastic properties as well as on the loading and unloading history. Additionally, we study the influence of friction by comparing the two extreme cases: frictionless and no-slip contact boundary.

We modelled the film as a membrane with negligible bending stiffness, an assumption that is justified when deflection is much greater than the initial film thickness (see [28]). This condition is easily satisfied in contact experiments with sufficiently thin films (e.g. [14]). The motivation for the use of rectangular geometry also stems from the circumferential wrinkling observed in circular membranes [14] and the tendency of rectangular membranes to resist wrinkling [19,29,30]. However, as noted by an anonymous reviewer, experiments using a circular membrane are much easier to carry out.

The layout of the paper is as follows. Sections 1 and 2 define the geometry of the problem and introduce the nonlinear constitutive relation for the membrane. The solution of a freestanding membrane inflated by pressure is presented in §3. The contact problem is formulated in §4 and numerical results and discussion of these results are given in §5. A summary is presented in §6.

## 2. Geometry

A rectangular viscoelastic membrane of initial width 2*a* is inflated by a uniform excess pressure *p*(*t*). The membrane has uniform initial thickness *h*_{0} and is assumed to be very long in the out of plane direction; so deformation is independent of the out of plane coordinate. Figure 1 shows a cross section of a deformed state. Before loading, the membrane is stress free and suspended by the clamps O_{1} and O_{2} at a distance *d* above a rigid flat substrate. The undeformed membrane lies on the *x*-axis (*z*=0). A material point on the membrane is completely specified by its initial position denoted by *ρ*. This point is carried to (*x*(*t*),*z*(*t*)) as the membrane deforms. The half-length of the contact region is denoted by *c*(*t*). The material coordinate associated with the edge of contact or contact line is denoted by *ρ**(*t*). Inside the contact zone, the membrane is perfectly flat against the substrate. Outside the contact zone, the membrane has a uniform radius of curvature *R*(*t*) with its slope varying from *θ*_{0}(*t*) at the contact line to *θ*_{m}(*t*) at the clamp (see discussion in §4).

The in-plane deformation is completely specified by the stretch ratio of a material point λ(*t*) on the cross section. We denote the in-plane tension by *T*; note that *T* has units of force per unit length. As the deformation is symmetrical about the *z*-axis, we limit our attention to *x*∈[0,*a*].

If the pressure is insufficient to bring the membrane in contact, that is, when the apex displacement *δ*(*t*)<*d*, the deformed membrane has a circular arc profile, as required by force balance.

## 3. Constitutive model

For mathematical simplicity, we use a large deformation, incompressible, viscoelastic constitutive model proposed by Christensen [31,32] to describe membrane deformation. According to this model, the Cauchy stress *σ*_{ij} is related to the deformation gradient tensor *F*_{ij} by
*p* is the pressure required to enforce incompressibility, *g*_{0} is a time-independent material constant, *δ*_{ij} is the Kronecker delta, **I** is the identity tensor, *g*_{1} is material function that varies with time and *E*_{KL} is the Lagrangian strain tensor, i.e. **E**=(**F**^{T}**F**−** I**)/2.

For a plane strain membrane, it is only necessary to relate the in-plane stretch ratio λ to the in-plane true line tension *T*=*σ*_{11}*h*_{0}/λ. Using (2.1) and the plane strain assumption results in
*g*_{0} and *g*_{1}(*t*) can be deduced by studying the limit λ(*t*)→1 which corresponds to small strain viscoelasticity theory. The material parameters *g*_{0} and *g*_{1}(*t*) were found to be related to the small strain shear relaxation modulus *μ*(*t*) by
*μ*_{0} is the instantaneous small strain modulus and *t*_{r} is the relaxation time. Substituting (3.3) and (3.4) in (3.2) and applying integration by parts, the in-plane line tension can be written as (details are provided in the electronic supplementary material)
*t*→0. The stress relaxation curves predicted by (3.6) for three different step stretch loadings are shown in figure 2.

## 4. Solutions for the freestanding part of the membrane

In this paper, any section of the membrane that is not in contact is called *freestanding*. The geometry and equilibrium equations for the freestanding viscoelastic membrane are independent of the material property and can be found in Srivastava & Hui [18]. Here, we state these results. Force balance requires that, the radius of curvature *R*(*t*) and tension *T*(*t*) is spatially uniform outside the contact zone and are related by [18]
*R*.

In particular, before the membrane came into contact, the relation between the apex displacement *δ*, the stretch ratio λ and radius *R* are [18]
*θ*_{m}(*t*) is the slope of the deformed membrane at *x*=*a*. The in-plane line tension *T*(*t*), stretch ratio λ(*t*), apex displacement *δ*(*t*), clamp slope *θ*_{m}(*t*) and the radius of curvature *R*(*t*) can be obtained as a function of loading history *p*(*t*) by solving equations (4.1)–(4.4) along with the constitutive relation (3.5).

## 5. Membrane in contact

Unlike the purely elastic case, the critical pressure needed for making first contact (*p*_{c}) depends on the pressure history. Subsequent motion of the contact line is driven by viscoelastic relaxation and applied pressure. During the loading stage, they aid each other and hence the contact region grows faster than the purely elastic case (with the instantaneous modulus). During unloading, viscoelastic relaxation will slow the receding of the contact line. As noted in §3, the membrane outside the contact zone is freestanding and is an arc of a circle. In addition to the viscoelastic material behaviour, the membrane profile is also governed by conditions of contact—namely adhesion and friction. In this study, we only consider adhesionless contact between the membrane and substrate. The absence of adhesion imposes the condition that the slope at the contact edge (*θ*_{0}(*t*)) is always zero. To assess the influence of friction, we model two limiting cases: frictionless and no-slip. Note that irrespective of the contact condition, the membrane segment that in contact with the substrate continues to relax as the contact is growing (or receding).

### (a) Frictionless interface

For frictionless contact, the stretch ratio is uniform both inside (|*x*|<*c*(*t*)) and outside (|*x*|>*c*(*t*)) the contact region (denoted as λ_{in}(*t*) and λ_{out}(*t*), respectively). The geometry is identical to that for the elastic membrane [18], specifically, the stretch ratios are given by
*θ*_{m}
*p*(*t*) is given, these equations can be solved using a nonlinear numerical root finding algorithm in Matlab [33]. The contact length is then determined using (5.5).

### (b) No-slip interface

Even though most existing works consider frictionless contact, no slip contact is probably a better approximation of the actual contact mechanics in experiments. For a no-slip interface, the analysis is much harder since the stretch ratio as well as the tension is no longer uniform inside the contact zone and (5.1), (5.3) and (5.6) are not applicable. Instead, the stretch ratio increases from the centre of contact (*x*=0) to the contact edge (*x*=*c*(*t*)). Because of the no-slip condition, the stretch ratio of a membrane segment in contact is fixed at its contact value and it undergoes stress relaxation until it is peeled-off. This *locking* of the stretch leads to membrane segments inside and outside the contact zone undergoing different stress and stretch ratio histories. Therefore, the *tension is no longer continuous* at the contact edge during membrane peeling (decreasing contact). On the contrary, while making (increasing) contact, the material points just inside and outside the contact edge would have undergone the same stretch history thereby ensuring tension continuity. In the absence of adhesion, the energy release rate must be zero for both decreasing and increasing contact. As shown by Srivastava & Hui [19], this condition requires *θ*_{0}=0 and continuity of stretch ratio at the contact edge. Owing to continuity of the stretch ratio, discontinuity in tension does not violate the zero energy release rate requirement for adhesionless contact.

A simple thought experiment illustrating the tension discontinuity during peeling is shown in figure 3. The deformation at two distinct material points A and B on the membrane are tracked as we increase the pressure on the membrane up to a certain limit followed by reduction until the membrane completely detaches (figure 3*a*). A is chosen such that it goes into contact at *t*=*t*_{2} and stays in contact at a fixed stretch *b*. Up till *t*=*t*_{2}, both points have the same stretch history and hence have the same tension (as dictated by (3.5)). For *t*≥*t*_{2}, the stretch at A remains fixed at *t*=*t*_{4}). At *t*=*t*_{4}, adhesionless contact requires *T*^{−}=*T*_{A}(*t*_{4})≠*T*_{B}(*t*_{4})=*T*^{+}.

The no-slip contact case is solved numerically using an incremental scheme similar to those employed by Srivastava & Hui [19] and Long *et al.* [16]. This scheme has been tailored individually for the increasing or decreasing contact stages and uses the stored stretch ratio values (locked-in) in the contact region. Stretch continuity has been imposed for both stages. Details of the concept and application of our scheme are presented in the electronic supplementary material.

## 6. Results and discussion

Numerical results are presented for a ramp like pressure loading history. Specifically, the pressure is increased at a constant rate *k*_{1} until it reaches its maximum value *p*_{max}; after which it is held constant for the time-interval *t*_{hold}. At the end of hold period, the pressure is decreased at a constant rate *k*_{2}. Also, to reduce the number of parameters in the numerical calculations and results, we introduce the following normalized variables:
*a* and all time variables are normalized with the relaxation time *t*_{r}. Tension is normalized by *k*_{1} and *k*_{2} are normalized by

### (a) Free inflation

For a viscoelastic material, the loading rate has a strong effect on the evolution of membrane shape. Figure 4*a* plots the apex deflection of a freestanding membrane (i.e. before contact) loaded linearly at different rates. As expected, for very fast loading, figure 4*a* shows that the deflection is almost identical to that of an elastic membrane with instantaneous modulus *μ*_{0}. As we reduce the loading rate, the behaviour deviates from the instantaneous response and the membrane deforms more for the same applied pressure. For very slow loading, the deflection-pressure graph approaches the response of an elastic material with the relaxed modulus

If the pressure on the free membrane is increased at a constant rate and then held constant (as shown in figure 4*b*), the membrane will eventually relax and the apex deflection asymptotically approaches the static deflection of an elastic membrane with modulus

### (b) Membrane contact (frictionless)

When the applied pressure is large enough, the membrane makes contact with the substrate (this pressure is denoted as *p*_{c}) and any further pressure increase causes the contact length to increase. In this section, we consider frictionless contact where the membrane is allowed to slip freely on the substrate resulting in a uniform stretch ratio in the contact zone.

#### (i) Effect of modulus ratio

We first examined the effect of modulus ratio *m* by fixing the relaxed modulus *μ*_{0}, the pressure required to bring the membrane into contact increases. The special case of *m*=1 represents a purely elastic solid, where the loading and unloading curves lie on top of each other. Figure 5 shows that the larger the modulus ratio *m*, the larger the deviation between the loading and unloading curve. However, these deviations do not truly measure energy dissipation, as discussed in (§5.3).

Figure 5 shows that as pressure is reduced, the membrane contact region eventually reduces to a line (*c*=0) followed by detachment from the substrate. This type of detachment is referred to as ‘pinch-off’ (as opposed to ‘pull-off’ mode of detachment in adhesive systems where detachment may occur at a *finite* contact) [34].

#### (ii) Effect of unloading rate

The mechanics of peeling is strongly influenced by the pressure loading history. Consider first the case of fast loading, followed by unloading at different rates. Figure 6 shows the contact–pressure plot for a membrane which is loaded a fast rate

For fast unloading (*reaches zero before the membrane can detach via pinch-off*. According to (4.1), a negative pressure results in a negative tension (i.e. compression). As a membrane, by definition, has no bending stiffness, it cannot resist compression and we shall call this wrinkling. In this case, the membrane formulation is no longer applicable. To study the behaviour under compression, it is necessary to use a nonlinear plate theory where bending and membranes stresses are both accounted for. We will not be discussing the actual mechanics of wrinkling in this paper and the term wrinkling is used to denote the likelihood of a buckled structure. In the following figures, the point at which wrinkling occurs (i.e. tension becomes zero) will be labelled by an asterisk (*).

If we reduce the unloading rate further, e.g. *m*=5 and

Note that at the slowest unloading rate

The previous results are for fast loading, we now consider slow loading *μ*_{0}, making it harder to peel.

Comparison of figures 6 and 7 shows that the membrane responds *asymmetrically to loading/unloading*: fast loading followed by slow unloading leads to stable detachment, whereas slow loading followed by a fast unloading causes contact line trapping

#### (iii) Effect of hold time

So far, the membrane is unloaded immediately after it reaches the desired maximum pressure. However, common experience suggests that some hold time is required to establish intimate contact and this often results in a larger peel force. It is therefore interesting to study the effect of ‘hold’ time (*t*_{hold}) for different loading and unloading combinations, as shown in figure 8*a*,*b* for a material with *m*=5.

The results in figure 8*a*,*b* show that *holding produces the same effect as slow loading* (shown in figure 7) because in both cases the membrane is allowed to relax. Even though loading was done at a fast rate, due to holding, slow unloading (*a* (*t*_{hold}=*t*_{r}) and 8*b* (*t*_{hold}=20*t*_{r}). For the former, the membrane is only partially relaxed before unloading. As a result, during slow unloading, there is still significant contact increase before it begins to reduce. Also, even though fast and intermediate unloading rates lead to wrinkling, there is a significant decrease in contact length before that happens.

For the case of very large hold time (compared to the characteristic relaxation time *t*_{r}, figure 8*b*), the membrane is completely relaxed when unloading starts. Hence, even for the slowest unloading rate (*b* is almost identical to the pinch-off force for an elastic membrane with the relaxed modulus (figure 6). This confirms one of our common experiences that peeling slowly reduces the peel force. However, for the cases of fast and intermediate unloading, we see contact trapping and eventually wrinkling. This behaviour is very pronounced in the case of long hold times followed by very fast unloading where the resistance to peeling is high (e.g. *b*).

Both these cases (along with the plots in the previous subsection) demonstrate contact trapping and wrinkling occurs when the membrane is unloaded rapidly in the relaxed state. As mentioned above, the situation is not symmetrical, in the sense that if the membrane is unloaded slowly in the unrelaxed state (e.g. fast loading followed by slow unloading), contact trapping and wrinkling does not occur.

#### (iv) No-slip contact

Figure 9 shows the contact length versus pressure plots for the frictionless and no-slip cases for three different cases: fast loading (*t*_{hold}=20*t*_{r}. For each case, we consider two different rates (*c*≈0.8*a*), the difference between the frictionless and no-slip conditions is not significant. The main difference is that for full friction (no-slip), the same applied pressure gives smaller contact length compared with the frictionless case. The explanation is simple: because of the no-slip condition, the portion of membrane in contact with the substrate cannot stretch once it is in contact so it cannot contribute to the increase in contact length.

The above results suggested that it is a good approximation to use the frictionless contact condition as long as the contact length is less than 0.5*a*. This condition is usually satisfied in a typical test. However, for very large contact, our previous analysis of an elastic membrane [18] showed that there are substantial differences between the no slip and the frictionless boundary conditions. For example, for elastic membranes, there exists a finite maximum pressure beyond which no solution exists for frictionless contact. As the elastic contact solution is a special case of viscoelastic contact (e.g. fast loading/unloading), it may be necessary to account for friction in experiments involving very large contact. In the following sections, we shall assume that contact is not so large so that the frictionless boundary condition remains meaningful.

### (c) Energy dissipation and detachment from substrate

Two important practical quantities are the pressure required for detachment and the energy dissipation (hysteresis) in a loading–unloading cycle. However, the pressure–contact length plots presented above do not provide enough information to determine the amount of energy dissipation [19]. The simplest way to determine the amount of dissipation is to imagine that the membrane is pressurized by an incompressible fluid. The energy spent by the loading machine in one complete loading–unloading cycle (i.e. load at the rate *c*_{peak}. In general, *c*_{peak} is not the maximum contact length since contact can continue to grow during reduction of pressure due to relaxation. In figure 10, *c*_{peak}=0.5*a*. In the first case (*c*_{peak} is different.

One often is interested in the ease (or difficulty) with which a membrane can be peeled off. For adhesive elastic membranes, the absolute value of the pinch-off pressure is a convenient measure for the ease of detachment. For viscoelastic membrane without adhesion, a reasonable measure for ease of detachment is to specify how much *reduction* in pressure would be needed to detach the membrane. If only a small amount of unloading is needed to detach the membrane, it can be qualified as an easy to peel system. The drop in pressure needed for detachment (denoted by Δ*p*_{detach}) is the difference between the maximum applied pressure, *p*_{max} and the pressure at which the membrane detaches *p*_{detach} (cases of wrinkling are also included in the broad term ‘detachment’). Note that for stable detachment Δ*p*_{detach}=*p*_{max}−*p*_{detach}≤*p*_{max}, whereas if the membrane wrinkles *p*_{detach}=0→Δ*p*_{detach}=*p*_{max}−*p*_{detach}=*p*_{max}. All the following simulations are carried out for frictionless membranes.

We first discuss the sensitivity of the dissipation and detachment pressure drop to *a*,*b* shows this behaviour, respectively. We observe that the energy dissipated and Δ*p*_{detach} both increase monotonically with *p*_{detach} increase faster for higher

Next, we consider the influence of loading parameters and material properties on energy dissipation and detachment pressure drop. Figure 12 plots the energy dissipation (figure 12*a*) and detachment pressure drop (figure 12*b*) against different unloading rates for both fast (*m*=5 and the membrane is loaded up to the contact length

Figure 12*b* shows that for fast loading *p*_{detach} increases with the unloading rate until a critical unloading rate (

Comparing figure 12*a*,*b*, we see that the amount of dissipation and detachment pressure drop are closely linked and follow similar trends. Specifically, for the same contact length at peak pressure, a loading unloading cycle that dissipates large amount of energy is also very difficult to detach—requiring a large pressure drop. However, this statement applies only if we consider the continuous portion of the curves in figure 12*a*,*b* before wrinkling. For instance, in the fast loading case (*a*,*b*), both dissipation and detachment pressure drop are increasing in the segment *a*,*b* (indicated by arrow) corresponds to the maximum detachment pressure drop but its dissipation is about half of the maximum.

So far we have focused on how loading history affects membrane deformation by keeping the viscoelastic strength *m* in dissipation and pinch-off (figure 13). We simulated the loading and unloading cycle with different modulus ratios *a*). As the modulus ratio is increased, dissipation also increases. This is expected because a membrane with higher *m* would be more viscous and dissipate more energy for pressure profiles with the same

The pressure drop required for pinch-off also increases monotonically with increasing *m* (figure 13*b*). This can also be attributed to the increasing viscosity in the material which causes the membrane to relax more and hence needing a bigger pressure drop for pinch-off.

## 7. Conclusion

We have presented a theoretical and numerical study of the contact mechanics of a nonlinear viscoelastic rectangular membrane loaded by uniform pressure. Our model accounts for large deformation as well as frictionless and no-slip boundary conditions. We have discussed the effect of rates of loading and unloading on the pressure contact relationship as well as on the viscous dissipation. The main conclusions of this study are:

(1) Adhesionless viscoelastic membranes can wrinkle in a load/unload cycle. Wrinkling does not imply maximum energy dissipation.

(2) Response to the pressure history with regards to loading and unloading rates is asymmetric: In the sense that fast loading–slow unloading leads to stable detachment via pinch-off, whereas slow loading–fast unloading causes contact edge trapping and wrinkling. The response is also asymmetric with regards to energy dissipated in a cycle.

(3) Joint strength (pinch-off pressure drop) and energy dissipation is strongly affected by the loading and unloading rates and modulus ratio.

(4) To peel off a given amount of membrane in contact, maximum dissipation does not necessarily imply maximum detachment pressure drop. On the contrary, low dissipation and low detachment pressure drop are strongly correlated.

(5) Energy dissipation and detachment pressure drop increase monotonically with the viscosity of the material (i.e. the modulus ratio) for membranes with the same contact length at peak pressure.

In items (3–5), the dissipation and detachment pressure drop for different pressure histories are compared for *the same contact length c*_{peak} *at (different) peak pressure*. Instead, if we compare these quantities for histories with the *same peak pressure* (hence different *c*_{peak}), the results are quite different (detailed plots given in the electronic supplementary material). For example, (i) a slowly pressurized membrane can be more or equally difficult to detach than a fast loaded membrane (unlike figure 12*b*). (ii) Detachment pressure drop for fast loading and slow unloading is insensitive to the modulus ratio *m* (unlike figure 13*b*).

To avoid making the analyses and simulations too complicated, we have based our analysis on the viscoelastic model proposed by Christensen. Our analysis can be readily adapted to more general viscoelastic constitutive laws. Also, we have not included the effect of adhesion on the contact mechanics. Undoubtedly, adhesion is important in some applications and will be considered in our future work.

## Acknowledgements

C.Y.H. acknowledges the support of the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under award no. DE-FG02–07ER46463. A.S. was partially supported by the same award.

- Received July 9, 2014.
- Accepted August 27, 2014.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.