## Abstract

In part I of this work, we study adhesionless contact of a long rectangular elastic membrane with a rigid substrate. Our model is based on finite strain theory and is valid for arbitrarily large deformations. Both frictionless and no-slip contact conditions are considered. Exact closed form solutions are obtained for frictionless contact. For small contact, the differences between these two contact conditions are small. However, significant differences occur for large contacts. For example, frictionless contact predicts a maximum pressure (and contact region) beyond which there is no solution; while the no-slip model places no restriction on both quantities. The effect of adhesion will be considered in part II of this work.

## 1. Introduction

Recent interest in adhesion of soft materials to hard substrates has led to the development of thin-film-based contact tests [1–5]. A typical test consists of a circular film clamped to the edge of a cylindrical tube. A flat substrate is placed underneath the undeformed film at a separation of *d*. An external pressure *p* is applied by pressurizing the tube, deforming the film and creating contact with the substrate [5]. Such test systems have much larger mechanical compliance than the popular Johnson–Kendall–Roberts [6] test and thus are more sensitive to the interfacial adhesion. A closely related test is the Bulge test [7–9], where the free inflation of a membrane is used to determine its elastic properties. However, as pointed out by Vlassak & Nix [7], axisymmetric geometries are plagued by problems such as buckling, and sensitivity to loading and small variations in film dimensions. They suggested that a better method is to use long rectangular membranes. We will further motivate the advantages of using long rectangular geometry over axisymmetric for adhesion test in part II of this work. Note that as the membrane undergoes free inflation in a Bulge test, it can be considered as a prelude to contact.

There have been many studies on the adhesive contact mechanics of thin films [3,4,10–12]. In these works, the thin film was modelled as a plate with bending deformation (e.g. section 2 of [10,13]), a membrane with stretching deformation [4,11,12] or a combination of both deformation modes [14,15]. Adhesion was modelled using Griffith’s critical energy release rate criterion or by imposing an adhesive traction between the contacting surfaces [16]. These works constitute a library of models to describe thin film adhesion under different geometrical assumptions.

The theoretical studies mentioned above are based on the assumption that the deflection of a thin film is much smaller than its dimension (e.g. radius *a*). However, in many experiments, the deformation of the membrane can be very large, with contact radius approaching the undeformed radius of the film [5]. For these cases, small strain solutions are no longer applicable and a large deformation membrane theory is needed to capture the contact mechanics. Since equations describing large deformation are nonlinear, it is extremely difficult to obtain analytical solutions. Nadler & Tang [17] studied the effect of large deformation on the detachment of thin circular membranes from rigid substrates. They numerically solved the deformation of an initially un-stretched circular membrane made of a neo-Hookean material. A more complete formulation and analysis of this problem was later solved by Rong *et al.* [18], who obtained an explicit expression for the strain energy release rate and addressed the effect of pretension in the film.

An important modelling issue that is rarely addressed in contact mechanics studies of thin films is friction between the contacting surfaces. Most studies implicitly assume frictionless contact, while in practice, a no-slip boundary condition is more appropriate. Recently, Peng & Chen [19] studied the effect of friction on the peeling of a prestressed film. More germane to our analysis is the work of Rong *et al.* [18] who show that the adhesive contact mechanics of the membrane depends on the friction boundary condition. In this work, we study in detail these two limiting cases and illustrate the significance of friction in thin film contact mechanics.

In part I of this work, we consider adhesionless contact. Results for adhesive contact will be given in part II [20] of this work. The layout of part I is as follows. Section 2 begins with a description of the geometry. This is followed by a derivation of the equilibrium equations for a membrane loaded by external pressure. A large strain constitutive model which is general enough to model any isotropic *elastic* membrane subjected to arbitrarily large deformation is introduced. In this work, we have chosen to employ the neo-Hookean model to simplify the analysis. The first stage of membrane loading: free inflation before contact, is covered in §3. The analysis here also applies to the plane strain bulge tests. In §4, solutions for both frictionless and no-slip contact are presented and discussed. In §5, we summarize the key findings of the study.

## 2. Geometry, balance laws and constitutive model

### (a) Geometry

Following Rong *et al.* [18], the thin film is modelled as a membrane with negligible bending stiffness. For a clamped film under uniform pressure, this assumption is justified when the maximum deflection *d* is greater than the film thickness *h*_{0} (see Timoshenko & Woinowsky-Krieger [21]), a condition that is easily satisfied in practically all contact experiments. For example, in the experiments of Laprade *et al.* [22], the film thickness is approximately 1 *μ*m, whereas the maximum deflection is approximately 1–3 mm.

Consider 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 thickness *h*_{0}. Figure 1 shows a cross-section of the geometry during the loading/unloading of the membrane. The undeformed membrane is suspended at a distance *d* above a rigid flat substrate and is clamped at the edges O_{1} and O_{2}. The membrane geometry is symmetric about the *z*-axis because of uniform loading and hence either half would suffice for analysis.

As shown in figure 1, the deformation carries a material point on the membrane with material coordinate *ρ* to *x*. 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. The slope of the deformed membrane varies from *θ*_{0} (=0) at the contact edge to *θ*_{m} at the clamp. In §2*b*, we will show that the free standing membrane is a circular arc with radius *R*.

### (b) Equilibrium equations for a free standing plane strain membrane loaded by pressure

In this section, we derive the equilibrium equations of any portion of the membrane that is not in contact (free standing membrane). Figure 1 shows the free body diagram of a small segment of the deformed membrane AB. This element can always be reoriented (without any loss of generality), so that the tangent to the deformed membrane at A is horizontal, i.e. *θ*=0. At the end B, the tangent then makes an angle d*θ* with respect to the horizontal. Force balance in the vertical direction shows that
2.1which is the Laplace’s equation for pressure difference across an interface with tension *T*. Note that d*T* and d*R* d*θ* are higher-order terms which can be neglected for d*θ*→0. Balance of forces in the horizontal direction implies that
2.2Thus, the tension in the membrane must be *constant*. Since the applied pressure is uniform, the radius of curvature of the membrane must be constant too, independent of the constitutive behaviour of the membrane. Hence the deformed shape of the free standing membrane is an arc of a circle with radius *R*. Note that the centre of this circle is an unknown.

### (c) Constitutive model

For plane strain deformation, the out-of-plane stretch ratio is 1 and the in-plane stretch ratio is denoted by *λ*. Therefore, the in-plane tension *T* is a function of the stretch *λ* only. In this paper, the membrane is assumed to be isotropic, incompressible and hyperelastic. For this case, the elastic energy density *W* depends only on the first and second invariants, *I*_{1} and *I*_{2} of the left Cauchy–Green tensor [18]. For a plane strain membrane, the tension is
2.3If the membrane deforms as an idealized rubber (or a neo-Hookean solid, for a physical motivation of this model, see Flory [23]), then
2.4where *μ* is the small strain shear modulus and is related to the small strain Young’s modulus *E* by *E*=3 *μ*. For this case, (2.3) reduces to
2.5It should be noted that, in plane strain, the neo-Hookean model is equivalent to Mooney–Rivlin (see Bower [24]).

## 3. Free inflation

The membrane is said to undergo free inflation before contact. For this case, *c*=0 and the centre of the circle lies on the *z*-axis in figure 1. According to (2.2), the tension is spatially uniform; as a result, the stretch ratio *λ* of the deformed membrane is also uniform and is given by
3.1where *θ*_{m} is the clamped angle defined in figure 1. The radius *R* is related to the initial half-length *a* by
3.2Combining (3.1) and (3.2), the stretch ratio is
3.3The stretch ratio can also be computed by finding the ratio of the length of a material line element (d*ρ*) before and after deformation (*R*d*θ*), that is,
3.4where *ρ* is the position of a material point in the reference (undeformed) configuration (see figure 1). Equating (3.4) and (3.1), we found, using *θ*=0 at *ρ*=0,
3.5With respect to the coordinate system (*x*,*z*), the membrane profile is
3.6where we have used equations (3.5) and (3.2). Membrane profiles during various stages of free inflation are shown in figure 2.

The maximum deflection, *δ* occurs at *θ*=*θ*_{m} and is
3.7Equation (3.7) shows that the deflection becomes unbounded as *θ*_{m}→*π*, thus the maximum angle at the clamped edge cannot exceed *π* in free inflation.

It must be noted that equations (3.1)–(3.7) are independent of constitutive model. However, a constitutive model is needed to relate the maximum deflection to the applied pressure. Since the deflection and the stretch ratio are uniquely determined by the clamped angle *θ*_{m} (see (3.3) and (3.7)), we need to find only the relation between *θ*_{m} and the applied pressure. For a neo-Hookean membrane, this relation is obtained by substituting (2.1), (3.2) and (3.3) in (2.5), resulting in
3.8where *β* is the normalized pressure. Once *θ*_{m} is found by solving (3.8), the dependence of *δ*/*a* on the normalized pressure *β*=*ap*/*μh*_{0} can be found using (3.7). For small deflections *θ*_{m}≈0, hence we expand the sine function in (3.8) and tangent function in (3.7) about *θ*_{m}=0 and obtain the earlier result of Vlassak & Nix [7]:
3.9aFor large stretch ratios, (3.3) implies that which simplifies (3.8) and (3.7) to give:
3.9bAccording to (3.9b), *δ* becomes unbounded as the normalized pressure or the clamp angle approaches *π*.

Figure 3 plots the normalized maximum deflection *δ*/*a* versus the normalized pressure *β*. The small strain approximation of Vlassak & Nix [7], (3.9a) as well as the large deflection behaviour (3.9b) is also plotted in the same figure as a comparison.

It is seen that the small strain solution as reported by Vlassak & Nix [7] underestimates the deflection and is only accurate for normalized pressures less than 0.1. The large deflection approximation given by (3.9b) is accurate for *β*≥2.

## 4. Membrane in contact

### (a) Condition of point contact

For a given initial separation *d*, contact will occur when *δ*=*d*. Using (3.7), this condition is
4.1awhere *θ*_{c}<*π* is the angle at the clamped edge at point contact. Once *θ*_{c} is found using (4.1a), the critical pressure required for point contact, *β*_{c}=*ap*_{c}/*μh*_{0} can be determined using (3.8), i.e.
4.1b

### (b) Frictionless contact

Figure 1 shows the geometry of the membrane in contact with a flat rigid substrate, with the contact zone occupying the strip *x*∈(−*c*,*c*), *z*=0, . Because of symmetry, we only need to consider *x*≥0. As shown earlier, the tension *T* of the membrane outside the contact zone is constant and the membrane shape in this region is a circular arc with radius *R* and is related to the applied pressure *p* by (2.1). The angle made by the membrane at the contact edge, *θ*_{0} is continuous and equals zero for adhesionless contact since the absence of adhesion implies that there cannot be a vertical force acting on the contact edge.

The frictionless condition implies that membrane in contact is stretched uniformly, with stretch ratio
4.2Since the tension is uniform in the free standing portion of the membrane, the stretch ratio is uniform too and is given by
4.3In the absence of any friction or adhesion, the balance of forces at the contact edge requires that the tensions in the inner and outer segments of the membrane be the same. This implies *λ*_{in}=*λ*_{out} so that the stretch ratio is spatially uniform. Equating (4.2) and (4.3) determine the material coordinate of the contact edge, *ρ**, i.e.
4.4The radius of curvature of the free standing portion can be found from geometry in figure 1, which for *θ*_{0}=0 is
4.5Substituting (4.5) into (4.4) gives the location of the contact edge in the reference configuration:
4.6The contact length can be related to the maximum deflection *d* using geometry and (4.5), i.e.
4.7Equation (4.7) implies that *c*=*a* when *θ*_{m}=*π*. Substituting (4.7) into (4.5) allows us to find the unknown radius *R* in terms of the clamp angle *θ*_{m}, i.e.
4.8Using (4.2), (4.6) and (4.7), the stretch ratio *λ* can be expressed solely in term of *θ*_{m}. After simplification, it becomes
4.9The tension in the membrane is related to the pressure and the clamp angle by substituting (4.8) into (2.1) resulting in
4.10For a given pressure, the clamp angle *θ*_{m} can be determined by substituting (4.9) and (4.10) in the constitutive model (2.5); this results in:
4.11with *λ* given by (4.9). Since the contact length and the stretch is a monotonically increasing function of *θ*_{m} for all *θ*_{m}∈(0,2*π*) (see (4.7) and (4.9)), they are uniquely determined by *θ*_{m}. Therefore, the pressure contact length relationship can be determined by varying *θ*_{m} between and 2*π* in (4.7) and (4.11). Alternatively, it is possible to express the pressure as a function of contact length, using the identities
4.12along with (4.9) and (4.11), i.e.
4.13awhere
4.13bFigure 4 plots the contact length versus the applied pressure for frictionless contact for two initial separations: *d*_{1}=0.5*a* and *d*_{2}=0.9*a*. For the rest of this paper, the following normalizations will be used to present numerical results:
4.14Examination of (4.13a) shows that, for any given *d*/*a*, the contact length versus pressure curve has two branches. Specifically, there exists two contact lengths for any normalized pressure between 2*π* and , where is the maximum pressure where (Point P_{3} in figure 4). There is no solution if the applied pressure is greater than . The upper branch of the *c* versus *p* curve is unstable since the contact length increases with decreasing pressure and therefore cannot be observed in experiments. The contact length in the upper branch becomes unbounded (corresponding to infinite stretch ratio) as *β*→2*π*. Therefore, the maximum stable contact length that can be achieved in frictionless contact occurs at and will be denoted by .

Unlike free inflation, the stretch ratio and the tension in a membrane in contact are *bounded* at *θ*_{m}=*π* which occurs when *c*=*a* (Point P_{2} in figure 4). The stretch ratio at *c*=*a* is given by (4.9), i.e.
4.15aand the corresponding normalized pressure *β** is found using (4.11)
4.15bNote that for *θ*_{m}>*π* the contact length *c* exceeds *a*. The normalized pressures and *β** are functions of *d*/*a* only and this dependence is shown in figure 5. As expected, for all *d*/*a*. According to (4.15b), and .

Figure 5 shows that both and *β** decrease as the initial separation between the membrane and the substrate is increased. However, is always greater than *β**, thus implying that the maximum allowed pressure on the membrane is encountered only for *c*>*a* (i.e. for clamp slope greater than *π*) irrespective of the separation *d*. On the other hand, maximum stable contact length increases monotonically with separation *d* and grows without bound as *d* approaches infinity. On the lower end, is bounded and approaches *c**=1.

Figure 4 shows an interesting result. The contact versus pressure curves for different initial separations *d* cross each other (e.g. point A). Before the cross-over, the pressure required to bring the membrane into contact increases with *d*. This behaviour is *reversed* after the cross-over point. Later, we will show that this is also the case for the no-slip condition. This result seems to contradict the expectation that since tension is higher in the membrane with the larger separation, a greater pressure is required to achieve the same amount of contact. This argument is flawed since the pressure is not solely determined by the tension. Indeed, (2.1) states that the pressure depends on the ratio *T*/*R*. Since for the same contact length, a membrane suspended farther from the substrate would have a larger radius of curvature, it is possible for the pressure to be smaller despite a higher tension.

Membrane profiles subjected to different pressures (including *β*=*β**, ) are shown in figure 6 for the case of *d*/*a*=0.5. Note that P_{1}, P_{2} and P_{3} lie on the stable branch of the solution curve, whereas P_{4} is unstable and hence the corresponding membrane profile cannot be achieved experimentally in a pressure controlled test.

### (c) No-slip contact

The mechanics of contact must be modified when the membrane is not allowed to slip on the substrate. The no-slip condition dictates that the stretch ratio must increase from the centre of contact (*x*=*a*) to the contact edge (*x*=*c*). As a result, (4.2) is not valid and a solution can only be obtained numerically. In our numerical scheme, we increment the contact length by small steps and determine the change in pressure. Details of the numerical implementation are given in the appendix A of part II [20].

Numerical results for no-slip condition are shown in figure 7 where the contact length is plotted against the applied pressure for two different initial separations. In contrast to frictionless contact, the contact length increases monotonically with the applied pressure; as a result, there is no upper bound for pressure. To see that pressure can increase without bound as contact length goes to infinity, we note that (2.1), (2.5), (4.3), (4.5) and (4.7) are still valid for no-slip contact. For very large contact, (4.7) implies that *θ*_{m}→2*π*. Combining (2.5), (4.3) and (2.1), we found
4.16where we have used the approximation *λ*_{out}≫1. Since the contact length is very large in comparison with *a*, *ρ**→*a* in this limit implying that pressure grows without bound. The distribution of frictional force in the contact region can be obtained numerically by calculating the difference of membrane tensions at the centre point of the contact region and the contact edge. As we store the stretch values at each incremental step, this is a straightforward calculation.

A feature that is shared by both contact conditions is that the contact versus pressure curves for different initial separations *d* cross each other (point B in figure 7). Before the cross-over, the pressure required to bring the membrane into contact increases with *d*. This behaviour is *reversed* after the cross-over point. Figure 8 shows the membrane profiles for different applied pressures (points Q_{1}, Q_{2}, Q_{3} and Q_{4} in figure 7) for *d*/*a*=0.5. These pressures are chosen to be the same as the frictionless contact case (P_{1}, P_{2}, P_{3} and P_{4} in figure 6).

To demonstrate the differences between frictionless and no-slip contact, contact length versus pressure curves are plotted for both contact conditions at two different separations: and in figure 9. As long as contact length is *small*, i.e. *c*/*a*<0.25, there is little difference between these two contact conditions and *friction plays a negligible role*. For larger contact, the frictionless contact condition underestimates the pressure required to bring the membrane into contact. The two models gave entirely different prediction for very large contact. For example, the frictionless contact model predicts a maximum pressure () and a maximum stable contact length (), while the no-slip model places no upper bound on both quantities.

## 5. Conclusion

We have presented a rigorous nonlinear hyperelastic model capable of handling arbitrarily large deformations to study the contact mechanics of a rectangular membrane and a rigid substrate. We study in detail the effect of friction (frictionless and no-slip) on adhesionless membrane contact. Key findings of the study are:

— Apex deflection during free inflation of a neo-Hookean membrane grows without bound as both normalized pressure and clamp angle approach

*π*. By contrast, a membrane in contact can have clamp angles greater than*π*.— Small deflection and small strain theory underestimates the deflection of free membranes and is accurate for normalized applied pressures less than 0.2.

— For small and intermediate contacts, a membrane suspended farther from the substrate requires higher pressure to achieve the same contact length. This behaviour is reversed for large contacts.

— For small contacts,

*c*/*a*<0.3, the effect of friction is small; this result justifies earlier analyses based on small strain and deflection theory where friction is neglected. However, for large contacts, friction significantly affects the contact–pressure relationship and no-slip membranes require a substantially higher applied pressure to achieve the same contact length.— The effect of friction can be significant for very large deformations. For frictionless membranes, there exists a maximum pressure beyond which there is no solution. This maximum pressure decreases with increasing separation

*d*whereas the corresponding maximum half contact length increases with*d*. was found to be always greater than the undeformed half-length (*a*) of the membrane. By contrast, for a no-slip membrane, contact length can increase indefinitely with applied pressure.

The numerical results in this paper are based on a neo-Hookean membrane. It should be noted that the neo-Hookean model in plane strain deformation is identical to the Mooney–Rivlin model, and these models tend to underestimate the amount of strain hardening at large strains. However, it is straightforward to carry out the analysis for other strain energy density functions. For example, for frictionless contact, only (4.11) needs to be changed (replaced by (2.3)).

As long as contact is adhesionless, our model applies for both loading and unloading and hence the pressure–contact length curves for both these stages overlap each other as shown in figures 4 and 7 (the arrows in the diagram denote the loading and unloading path). However, this behaviour is rarely observed in experiments. In typical experiments the loading and unloading curves do not coincide (hysteresis) [5,22,25]. This is because while adhesionless contact is usually a good approximation for making contact, it is not a good approximation for breaking contact [26–28]. Adhesive contact will be addressed in part II of this work.

## 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.

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

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