## Abstract

This paper presents the free and constrained inflation of a pre-stretched hyperelastic cylindrical membrane and a subsequent constrained deflation. The membrane material is assumed as a homogeneous and isotropic Mooney–Rivlin solid. The constraining soft cylindrical substrate is assumed to be a distributed linear stiffness normal to the undeformed surface. Both frictionless and adhesive contact are modelled during the inflation as an interaction between the dry surfaces of the membrane and the substrate. An adhesive contact is modelled during deflation. The free and constrained inflation yields governing equations and boundary conditions, which are solved by a finite difference method in combination with a fictitious time integration method. Continuity in the principal stretches and stresses at the contact boundary is dependent on the contact conditions and inflation–deflation phase. The pre-stretch has a counterintuitive softening effect on free and constrained inflation. The variation of limit point pressures with pre-stretch and the occurrence of a cusp point is shown. Interesting trends are observed in the stretch and stress distributions after the interaction of the membrane with soft substrate, which underlines the effect of material parameters, pre-stretch and constraining properties.

## 1. Introduction

Constrained inflation of membranes exhibits some interesting behaviour owing to material and geometrical nonlinearities, constraining properties and contact conditions. The application of inflated membranes is of fundamental importance in a number of scientific studies ranging from plastic manufacturing processes to medical engineering, such as thermoforming, blow moulding, biological membranes, cell-to-cell adhesion, balloon angioplasty, etc. Discussions on the mechanics and applications of membranes in inflatable structures, plastic manufacturing and biomechanics can be found in literature [1–8].

Various problems related to the stretching and inflation of hyperelastic membranes of different geometries have been addressed in the past by analytical and numerical solution techniques [2,9–21]. The functional optimization of membranes has been presented [22]. The interaction of the material and geometrical nonlinearities in inflated membranes can lead to some interesting and counterintuitive effects. Recently, Patil & DasGupta [23] demonstrated a stretch-induced softening/stiffening phenomenon during free inflation of unstretched and pre-stretched hyperelastic circular flat membranes. The geometry-dependent scaling behaviour of the limit point pressure is uncovered by Tamadapu *et al*. [24]. Over the past four decades, many researchers have studied the membrane contact problems, largely with frictionless contact, in various contexts and for different geometries [25–29]. Recently, Patil & DasGupta [30] studied nonlinear frictionless constrained inflation of a pre-stressed flat circular membrane against an elastic cone, and proposed a semi-analytical method to solve the resulting constrained inflation problem.

The association of adhesion with friction is well known, as noted by Johnson [31]. Specifically, the contact of elastomers is best modelled using dry adhesive friction theory, as advocated by Savkoor [32], owing to their highly elastic behaviour, viscoelastic properties and the extremely high wear resistance, which disassociates wear from friction. Through their plastic junction theory of friction, Bowden & Tabor [33] put forward the basic ideas that adhesion is the main source of friction between two dry bodies, and that surface roughness plays only a secondary role. Researchers [34] explicitly mention that the Amontons–Coulomb law is not applicable to elastomeric friction on a smooth solid surface, which is essentially adhesive in nature. They explain the adhesive friction of elastomers through the formation and breakage of adhesive linking chains, which bind the polymeric body to a solid surface as a stationary stochastic process.

For adhesive contact problems, continuum theories considering elastic deformation are presented by Johnson *et al.* (JKR) [35] and Derjaguin *et al.* [36]. Of these two, the JKR theory is the most popular theory for adhesion measurement of thin films in contact. Adhesion can be modelled either using Griffith’s critical energy release rate criterion, or by imposing an adhesive traction between the contacting surfaces. To separate two adherends, an external pulling force (‘pull-off force’) or high debonding energy (measured in terms of energy release rate) is required [37]. Recently, Long & Hui [38] pointed out a jump discontinuity in meridional stretch and a continuous circumferential stretch at the contact boundary, when studying the detachment of a membrane from a rigid, adhesive substrate. Long *et al*. [39] mentioned that adhesion is a process of making and breaking contact, and applied a fracture mechanics approach to study adhesion between membranes and flat rigid substrates. They also derived an exact expression for the energy release rate in terms of local variables and a contact angle for a delaminating membrane at various pressures. The expressions were used by [40,41] to study peeling of long rectangular membranes. Recently, Patil *et al*. [42] studied the contact mechanics of an inflated circular flat membrane against a soft adhesive substrate and a peeling of contact upon deflation. They proposed a correction to previous energy release rate calculations, and showed that a numerical approach to the energy release rate is more accurate than the analytical one. The ‘pinch-off’ and ‘pull-off’ effect during detachment of membrane from a rigid substrate has been shown by [38,41,43]. Johnson *et al*. [35] proved experimentally a ‘jump-in-instability’ phenomenon, when the distance between membrane and rigid punch falls below a certain critical distance. However, similar to reference [43], we assume a pressure-sensitive adhesive contact, which does not possess any long-range force field, and therefore does not cause ‘jump-in instability’.

Understanding the inflation mechanics of a cylindrical membrane in balloon angioplasty is of paramount importance. Many possibilities exist to analyse it in order to optimize the stenting procedure based on patient-specific needs. The balloon angioplasty is well studied in recent years with the help of finite-element analysis and experimental methods with elaboration on the stenting procedure [44–47]. Gaser & Holzapfel [48] studied balloon angioplasty by considering an elastic–non-elastic behaviour for the artery by simulating balloon–artery interaction with a point-to-surface strategy. However, Holzapfel *et al*. [49] developed a model of angioplasty which is based on the magnetic resonance imaging and mechanical testing. As cylindrical membranes are used for widening narrowed or obstructed arteries in angioplasty, an extensive study is a demanding task. The free inflation of a cylindrical membrane has been studied by researchers such as [14] and [50,51], whereas confined inflation of a cylindrical membrane in relevance to blow moulding is studied in [4,5]. In a number of research works in the past, the membrane contact problem with rigid frictionless surfaces has been considered. However, contacts with no-slip and adhesion on soft surfaces have been scarcely discussed. Adhesion on soft surfaces is more complex than contact problems involving hard surfaces, because the surface conforms (relaxes) with inflation (deflation). In the case of an inflated membrane interacting with a soft adhesive substrate, changes in contact conditions with inflation and deflation have a further level of complexity owing to material and geometrical nonlinearities. The comparative study of unstretched and pre-stretched cylindrical membranes during free and constrained inflation is not available in the literature. This motivates us to present a study on free and constrained inflation of cylindrical membranes with emphasis on contact conditions and constraining properties. This work can thereby serve as a simple model for balloon angioplasty.

In this work, we study the contact mechanics of a cylindrical hyperelastic membrane, without and with pre-stretch, inflated against a soft adhesive cylindrical surface along with free inflation. The soft cylindrical surface is modelled as a linear stiffness distribution directed normal to the undeformed surface, while it is assumed that the substrate is stiffer in the axial direction than in the radial direction. The membrane is modelled as a homogeneous and isotropic Mooney–Rivlin solid (with the neo-Hookean solid as a special case). The adhesive contact is modelled as a perfectly sticking contact without slip. We consider the adhesive and frictionless contact during inflation and adhesive contact during deflation. The inflation and deflation processes are assumed to be isothermal. The equilibrium equations are obtained by a variational formulation for free and constrained inflation (or deflation) problems. The set of algebraic equations are obtained from differential equations by a finite difference method (FDM). These algebraic equations are solved by a fictitious time integration method (FTIM) proposed in [52]. The FTIM eliminates computation of a Jacobian matrix. In addition, it is insensitive to initial guess, thereby simplifying the complexity of the highly nonlinear algebraic equations. We have studied the effects of material parameters and pre-stretch on the free and constrained inflation of cylindrical membranes, which underlines some interesting behaviour of inflated membranes.

## 2. Problem formulation

### (a) Study set-up

Consider a homogeneous, isotropic, hyperelastic cylindrical membrane of initial radius *R*_{o}, length *L*_{o} and constant initial thickness *h*_{o} with open ends. Both ends of the membrane are then subjected to uniform axial traction to pre-stretch it. The end sections of the cylindrical membrane are free to contract during pre-stretching, but are then fixed through a rigid flat circular disc of the same radius as the pre-stretched membrane. The deformed radius, length and thickness of the pre-stretched membrane are *R*_{f},*L*_{f} and *h*_{f}, respectively, as shown in figure 1. In general, subscript *o* denotes parameters of the unstretched membrane and subscript *f* those of the pre-stretched membrane. For an initially unstretched membrane, *R*_{f}=*R*_{o} and *L*_{f}=*L*_{o}. The radius *R*_{f} and length *L*_{f} is taken to be same for cylindrical membranes with and without pre-stretching, for better comparison.

As the membrane is inflated to a specified pressure, it successively comes into contact with a soft, cylindrical constraining surface placed at a radial distance of *S*. As the membrane is then subsequently deflated, the portion of membrane in adhesive contact remains (figure 1*c*). The soft cylindrical surface is modelled as a radial linear stiffness distribution *K* directed normal to the undeformed surface. The axial stiffness of the substrate is modelled by a constraint.

### (b) Kinematics of deformation

For a cylindrical membrane upon pre-stretching under an action of uniform axial traction, the principal stretches in meridional, circumferential and thickness directions are
*ζ*(*ω*) and *ν*(*ω*), where *ω* is an independent axial material coordinate, and *ω*∈[0,*L*_{o}]. A material point P, initially at a position P_{1} with coordinates (*R*_{o},*ϕ*,*ω*), moves to the position P_{2} (*R*_{f},*ϕ*,*ω*_{p}) upon axial stretching, and then to P_{3} (*ζ*,*ϕ*,*ν*) after inflation (and then to P_{4} (*ζ*,*ϕ*,*ν*) upon subsequent deflation). Here, *ζ*(*ω*)=*R*_{f}+*U*(*ω*)=*δ*_{2}*R*_{o}+*U*(*ω*), and *ν*(*ω*)=*ω*_{p}(*ω*)+*W*(*ω*)=*δ*_{1}*ω*+*W*(*ω*) represent the total radial and axial coordinates, respectively. With meridional and circumferential pre-stretches *δ*_{1} and *δ*_{2}, the incremental radial and axial displacements owing to inflation are *U*(*ω*) and *W*(*ω*). The principal stretch ratios of the membrane along meridional, circumferential and thickness directions are obtained as, respectively,
_{,ω}=∂(⋅)/∂*ω*, and *h*_{o} are, respectively, the deformed and undeformed thicknesses of the membrane.

### (c) Material strain energy

In this work, we consider the incompressible, homogeneous and isotropic two parameter Mooney–Rivlin hyperelastic material model for the membrane. The strain energy density function over undeformed geometry for such a material is given by *C*_{1} and *C*_{2} are the material parameters, and _{1}λ_{2}λ_{3}=1, and the axisymmetry assumption, the total elastic potential energy can be expressed as

For the axisymmetric conditions, the Cauchy principal stress resultants *T*_{1} (meridional) and *T*_{2} (circumferential) in the original configuration are obtained from the strain energy density function, as [53]
*α*=*C*_{2}/*C*_{1} is sometimes referred to as the strain hardening parameter.

As the pre-stretched membrane before inflation is subjected to axial traction only, the circumferential Cauchy stress resultant *T*_{2f} reduces to zero (see equation (2.4))
*δ*_{1}*δ*_{2}*δ*_{3}=1), it is clear that

### (d) Potential energy of the gas

The potential energy of the inflating gas with over-pressure *P* may be written as

### (e) Potential energy of the soft substrate

The cylindrical soft substrate is modelled as a linear stiffness distribution *K*. The stiffness is considered to be directed perpendicularly to the undeformed surface, i.e. in the radial direction in figure 1. As the membrane is inflated against the substrate, the region of contact grows and the substrate deforms. On the other hand, when the membrane is deflated, the substrate relaxes, whereas the region of contact does not change (as no peeling is considered during deflation). In the following, we consider these two cases separately. The potential energy stored in the deformed substrate during the inflation phase may be expressed as
*ζ*≥*S*. During deflation, with no peeling, we consider the potential energy of the elastic cylindrical substrate in the form
*ω*_{s} and *ω*_{e} represent the axial material coordinates at start and end of the contact in the range [0,*L*_{o}], achieved at the end of the inflation phase.

### (f) Equations of equilibrium

The total potential energy for the system is now given by
*U*(*ω*) and *W*(*ω*). Thus, the potential energy functional is of the form *δΠ*=0, the equations of equilibrium can be written formally as

It is evident that equations (2.11)–(2.12) form a two-point boundary value problem (TPBVP). In the rest of the paper, all quantities are non-dimensionalized using the definitions
*z*∈[0,*l*_{f}/*δ*_{1}] is the non-dimensional axial (length-wise) coordinate of the unstretched membrane. We also define a stretched axial coordinate denoted by *Z*=*δ*_{1}*z*, where *Z*∈[0,*l*_{f}].

## 3. Contact conditions and solution procedures

In the following, we first consider the inflation problem with frictionless and adhesive contact conditions against a soft cylindrical substrate. The deflation problem with adhesive contact without peeling is also discussed. Adhesion is modelled as a perfectly sticky contact between the dry surfaces of the membrane and substrate, giving no-slip contact conditions in the contact zone. In the final section, a solution procedure for free inflation is elaborated.

### (a) Constrained inflation with frictionless contact conditions

The TPBVP described by equations (2.11)–(2.12) is discretized by an FDM, under the assumption of frictionless contact during the inflation phase. The independent coordinate *z* is discretized with *n* intervals along the axial direction, which gives *n*−1 internal nodal points in addition to the endpoints. The first and second derivatives of *u* and *w* are discretized as
*n*−1) nodal points, we thereby obtain 2(*n*−1) algebraic equations after discretization. These algebraic equations are nonlinearly coupled, and solved numerically by the FTIM [52]. The method eliminates the need for the Jacobian matrix, and is insensitive to initial guesses, as it simplifies the complexity of highly nonlinear algebraic equations. After discretization, equations (2.11) become a set of algebraic equations in non-dimensional variables in the radial and axial directions, respectively,
*i* of *u* and *w* represents a nodal number starting from *i*=0 at *z*=0. The set of algebraic equations are converted into a fictitious initial value problem with fictitious time *t* and numerical stability coefficient *v* as follows
*F*_{1i}=0 and *F*_{2i}=0 are the fixed points of equation (3.5), which are the displacement values at the nodal points. We may use a fourth-order Runge–Kutta method on equations (3.5) by starting from a chosen initial condition for assumed pressure and fixed value of *k* and *s*. For the next pressure step, the previous solution is taken as initial guess. The Runge–Kutta method starts from *t*=0 and is repeated until convergence is reached. The convergence of the solution, at the steps *k* and *k*+1 of the Runge–Kutta method, is found by the Euclidean norm of increments going towards zero
*ϵ* is a given convergence criterion. After convergence, the solution for assumed pressure is obtained. For our numerical treatment with FTIM, we choose numerical stability coefficient *v*=0.02 and fictitious time step

### (b) Constrained inflation with adhesive contact condition

The TPBVP described by equations (2.11)–(2.12) is discretized by the FDM, under the assumption of an adhesive contact condition during the inflation phase. In this case, the contact area depends upon the history of contact and has to be determined incrementally by imposing a kinematic constraint on all material points in the contact zone. The first contact is assumed to occur at the centre of the deformed membrane *z*=*l*_{f}/(2*δ*_{1}) owing to axial symmetry. This is considered as the zeroth step ( *j*=0).

The solution of the ( *j*+1)th step with pressure value *p*_{j+1} is obtained using the following procedure.

For the *i*th nodal point if (*ρ*_{i})_{j}>*s*, then (*w*_{i})_{j+1}=(*w*_{i})_{j}, which maintains sticky contact between the dry surfaces of membrane and soft substrate.

The discretization of equations and a solution procedure, similar to the one in §3*a*, is used to obtain the displacement field variables at nodal points.

### (c) Constrained deflation with adhesive contact condition

The TPBVP described by equations (2.11)–(2.12) is discretized by the FDM, under the assumption of adhesive contact during the deflation phase, without peeling between the membrane and the soft substrate. The axial coordinates (*η*_{s})_{0}, (*η*_{e})_{0} of contact starting and ending, with corresponding material coordinates (*z*_{s})_{0} and (*z*_{e})_{0} at the starting step *j*=0, obtained at the end of the inflation process with adhesive contact, are assumed to remain fixed throughout. In this case, kinematic constraints are imposed on all material points in the contact zone. The deflation solution (*j*+1)th step is obtained from the following procedure.

In the interval *z*=(*z*_{s})_{0} to (*z*_{e})_{0}; for all nodal point (*w*_{i})_{j+1}=(*w*_{i})_{0}, which maintains sticky contact between the dry surfaces of the membrane and soft substrate as well as the fixed contact area.

The discretization of equations and a solution procedure, similar to the above §3*a*, is used to obtain the displacement field variables at nodal points.

### (d) Free inflation

For free inflation, *k*=0 and the TPBVP described by equations (2.11)–(2.12) is discretized by the FDM. As a neo-Hookean, membrane encounters a limit point in free inflation, the pressure does not increase monotonically. Therefore, the solution procedure described in the preceding sections, which is based on pressure as controlling variable, is not able to pass the limit point. To overcome this limitation, we add one global equation into the system and make pressure a depending variable of the applied radial displacement (*u*_{c}), which is applied at the axial centre *z*=*l*_{f}/2*δ*_{1}. This gives the equation
*n*+1)/2 of *u* represents the nodal number of the mid node at *z*=*l*_{f}/(2*δ*_{1}). The solution for free inflation is straightforward and simple compared with the constrained inflation, as there is no need to model the contact condition. So, for free inflation, equations (3.3) and equation (3.7) can be directly solved by a much faster Newton’s method instead of FTIM.

## 4. Results and discussion

The results (in non-dimensional form) presented here are for two material models *α*=0 (neo-Hookean (NH) model) and *α*=0.1 (Mooney–Rivlin (MR) model), with pre-stretch values of (*δ*_{1},*δ*_{2})=(1,1), (*δ*_{1}, *δ*_{2}) = (1.5, 0.8164) and (*δ*_{1}, *δ*_{2}) = (2, 0.7071). The value of the distributed stiffness for the substrate is taken as *k*=20, final length of membrane *l*_{f}=6, and the radial distance of the substrate from axial axis of membrane as *s*=1.3 in all considered cases.

The parameters chosen for the study do not necessarily refer to any practical problem, but are merely chosen as representative values.

### (a) Free inflation analysis

#### (i) Stretch induced softening

The effect of pre-stretch on the maximum radial displacement (measured at the axial coordinate *z*=*l*_{f}/(2*δ*_{1})) of the cylindrical membrane is observed in figure 2. It is interesting to note that, at a given pressure,

#### (ii) Limit points and cusp point

For both material models, there may exist limit points with respect to pressure, as observed in figure 2. Figure 3*a* clearly shows the occurrence of two pressure limit points for the MR membrane. The first limit point is a local maximum, and the second is a local minimum of pressure, with a snap-through zone in between. Variation of both limit points with pre-stretch *δ*_{1} is shown as a fold line [54] in figure 3*b*, which establishes a nonlinear relationship between pressure limit points and pre-stretch. At a certain value of pre-stretch, both pressure limit points converge to a cusp point (*δ*_{1}≈1.163). Above this pre-stretch value, there will not be pressure limit points for the MR membrane. Occurrence of limit points thereby depends upon a combination of material model and pre-stretch in free inflation.

#### (iii) Principal stretches

The variation of the principal stretches for an initially unstretched MR membrane with the axial coordinate *Z* for different pressures is plotted in figure 4. It is immediately observed that, meridional stretch is greater than one at the fixed boundary, as a contrast to the circumferential stretch, which is one at the fixed boundary because of the nature of the boundary condition for this study. Furthermore, figure 4 shows that, at a lower pressure, the cylindrical membrane is flatter at the centre, whereas with an increase in pressure, it starts deforming more at the centre in the radial direction.

#### (iv) Membrane profile

The comparison of profiles for an initially unstretched membrane and a pre-stretched MR membrane at the same pressure in the pre-critical and post-critical zones is shown in figure 5. Figure 5 underlines the softening behaviour of the membrane with pre-stretch in the pre-critical and post-critical zones.

### (b) Constrained inflation and deflation analysis

#### (i) Membrane deformation with pressure

The variation of maximum radial displacement *b* shows the constrained inflation of the membrane from *p*=0 to *p*=3 and a subsequent deflation from pressure *p*=3 to *p*=0 with adhesive contact. During adhesive inflation up to a certain pressure, the membrane is in free inflation, and the deformation stagnates after interaction with the substrate. As soon as deflation starts from pressure *p*=3, the deformation of membrane reduces to *p*=0. The deformation during adhesive constrained inflation and deflation match each other closely for a cylindrical membrane.

#### (ii) Principal stretches and principal stress resultants

The variation of the meridional (λ_{1}) and circumferential (λ_{2}) stretches with the axial coordinate *Z* for an initially unstretched MR membrane inflated against a frictionless soft cylindrical substrate is shown in figure 7*a*,*b*. It may be observed that the meridional and circumferential stretches exhibit continuity up to *b*. It is interesting to note that the meridional stretch λ_{1} has _{2} has *Z* for an initially unstretched MR membrane in adhesive contact with the elastic cylindrical substrate during deflation are shown in figure 7*c*. While a jump is observed in meridional stretch at the contact boundary, the circumferential stretch has *b*) and deflation (figure 7*c*).

The Cauchy principal stresses in the MR membrane for an inflation phase with adhesive contact are presented in figure 8*a*,*b*. During inflation, meridional stress shows *a*), whereas with pre-stretch meridional stretch reduces with increase in pressure (figure 8*b*) for an inflation phase, which is counterintuitive. For adhesive deflation, the meridional stress resultant increases near the contact boundary, whereas it decreases in the non-contact zone owing to relaxation. Both stress resultants *t*_{1} and *t*_{2} reduce with the reduction in pressure (figure 8*c*). The meridional stress is

#### (iii) Membrane profile

Figure 9 shows a comparison of the membrane profiles for different combinations of material parameters, pre-stretch and contact conditions at the same pressure. The membrane profiles for frictionless and adhesive contacts are very similar. Comparing the profiles in figure 9*a*,*b*, we observe that the membrane without pre-stretch has a tendency to bulge sidewards with flattening in the central zone, whereas the membrane with pre-stretch propagates in the radial direction. The indentation of the membrane into the soft cylindrical substrate is nearly similar for frictionless and adhesive contact conditions, when considering the same combination of material and pre-stretch.

During deflation without peeling, the profiles of the membrane with decreasing pressure are shown in figure 10. The paths of the material points during inflation and deflation, as marked in figure 10, exhibit different trends if pressure reduces drastically. The contact area achieved at the end of the inflation phase remains during deflation. It may be noted that the movement of the material points in the contacting region are horizontally straight, because the axial movement is arrested owing to adhesion. As the inflation pressure is reduced, the curvature of the membrane in the non-contacting region undergoes a change in sign owing to the change in the meridional curvature from concave to convex.

#### (iv) Contact geometry

The contact patch formed between the membrane and the soft substrate is of an inflated cylindrical nature and characterized by its axial length *η**. The variation of contact patch length *η** with the inflation pressure is presented in figure 11. This clearly shows the rapid initial propagation of the contact patch with pressure, and then, stagnation for initially unstretched NH and MR membranes with frictionless and adhesive contact. The growth of the contact patch with inflation is an important indicator of the surface condition. For a cylindrical membrane inflated against an elastic cylindrical substrate, the contact patch and the indentation of the membrane into the substrate closely match for frictionless and adhesive contacts, when considering the same combination of material and pre-stretch.

#### (v) Axial traction and adhesive line forces

As we assumed that vertical movements of material points are arrested during adhesive inflation and deflation, it is necessary to show the resulting traction necessary to do so. The axial traction *f*_{tη} in the contact zone during adhesive inflation and deflation is given by
*f*_{kρ} in the contact zone during adhesive inflation and deflation is given by
*a*). For adhesive deflation, adhesive traction decreases with decrease in pressure (figure 12*b*). The maximum adhesive traction is similar in magnitude to the radial spring force density *f*_{kρ}, which means that if the constraining surface is ( *f*_{tη})_{max}/( *f*_{kρ})_{max} times stiffer in the axial than in the horizontal direction, the vertical movement of points is arrested.

However, during adhesive deflation, the membrane experiences an adhesive line force at the contact boundary, which is responsible for a jump in the meridional stretch and stress. The adhesive line forces *f*_{l} at the contact boundary during deflation are given by
*c* and 8*c*). At lower pressure, the adhesive line force starts reducing for an unstretched membrane, whereas this trend is not observed for a pre-stretched membrane.

## 5. Conclusion

The effects of a soft constraining substrate on the inflation and deflation mechanics of initially unstretched and pre-stretched cylindrical membranes have been analysed in this paper. Both frictionless and adhesive contact conditions have been considered.

The order of continuity of the principal stretches (λ_{1} and λ_{2}) and stresses (*t*_{1} and *t*_{2}) is dependent on the continuity of field variables (*u* and *w*), summarized in table 1. The superscript *i* is infinite for free inflation, for constrained frictionless inflation *i*=2, for constrained adhesive inflation *i*=1 and for constrained adhesive deflation the superscript *i*=0. The variations of the contact patch with pressure have been shown here. Pre-stretch introduces softening for NH as well as for MR membranes in the pre-critical and post-critical zones. The limit point pressure depends upon combinations of material parameters and pre-stretch for free inflation, whereas it depends upon combinations of material parameters, pre-stretch and constraining properties for constrained inflation [30]. At a certain value of pre-stretch, a cusp point is observed for an MR membrane. Above this, a pre-stretched MR membrane does not exhibit a limit point behaviour.

The axial traction necessary for arresting vertical movements of material points has been shown. The relationship between adhesive line forces and pressure during deflation for different combinations of material model and pre-stretch is shown clearly.

These results are expected to be useful in balloon angioplasty where a cylindrical membrane produces stresses on the artery, and the contact conditions between membrane–artery or membrane–plaque affect the inflation process. These results are also important when studying the contact mechanics of biomembranes and cell membranes. The quasi-static and dynamic aspects of cylindrical membranes in adhesive contact with material and geometrical nonlinear constraining surfaces will be an interesting future direction to pursue.

- Received April 7, 2014.
- Accepted May 28, 2014.

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