Finite deformation of elastic membranes with application to the stability of an inflated and extended tube

A theory is formulated for the finite deformation of a thin membrane composed of homogeneous elastic material which is isotropic in its undeformed state. The theory is then extended to the case of a small deformation superposed on a known finite deformation of the membrane. As an example, small deformations of a circular cylindrical tube which has been subjected to a finite homogeneous extension and inflation are considered and the equations governing these small deformations are obtained for an incompressible material. By means of a static analysis the stability of cylindrically symmetric modes for the inflated and extended cylinder with fixed ends is determined and the results are verified by a dynamic analysis. The stability is considered in detail for a Mooney material. Methods are developed to obtain the natural frequencies for axially symmetric free vibrations of the extended and inflated cylindrical membrane. Some of the lower natural frequencies are calculated for a Mooney material and the methods are compared.     

Cylindrical membrane partially stretched on a rigid cylinder

We consider the equilibrium problem of a hyperelastic thin-walled tube. One end of the tube is placed over an immovable, rough, rigid cylinder. We assume that the deformation of the tube is finite and axisymmetric. The tube is modeled by a cylindrical membrane. The membrane is composed of an incompressible, homogeneous, isotropic elastic material. We use Bartenev–Khazanovich (Varga) strain energy function. A contact between the membrane and the rigid cylinder is with a dry friction. The membrane will not slide off the cylinder only by a friction and at a sufficient contact area. The friction is described by Coulomb's law. We study a minimum length of the membrane which is in contact with the rigid cylinder and is needed to the equilibrium of the membrane.     

Tension of a cylindrical membrane partially stretched over a rigid cylinder

We study a contact problem with friction for a hyperelastic long thin-walled tube. One end of the tube is placed over an immovable, rough, rigid cylinder and an axial force is applied to another end. We assume the deformation of the tube is finite and axisymmetric. The tube is modeled by a semi-infinity cylindrical membrane. The axial force tends to a constant value at large distances from the inclusion. The membrane is made of an incompressible, homogeneous, isotropic elastic material. A contact between the membrane and the rigid cylinder is with a dry friction. The membrane will not slide off the cylinder only by friction and at a sufficient contact area. The friction is described by Coulomb’s law. We study a minimum length of the membrane which is in contact with the rigid cylinder and is needed to hold the membrane on the rigid cylinder. We obtain an explicit solution for the Bartenev–Khazanovich (Varga) strain–energy function and numerical results for the Mooney–Rivlin and Fung models.     

Analytical models for the inflation of a polymeric tube

In the blow moulding processes, a tube-shaped parison (also called preform) made of polymer is inflated inside a mould in order to obtain the desired bottle shape. The free inflation of a cylindrical tube (a simplified kinematic of inflation for the preform before contacting the mould) has been investigated in order to develop analytical models and then provide reliable validation material for finite element software. The solutions calculated for a tube made of a Newtonian fluid and a Lodge's rubberlike liquid material using both exact volumic approach and thin shell approximation have been analysed and compared.     

On the general contact problem of an inflated nonlinear plane membrane

The nonaxisymmetric contact problem between an inflated membrane and a rigid indentor is considered. The membrane is assumed to be an initially thin plane sheet. The shape and the boundary of the contact region and the configuration of the deformed membrane under both inflation and indentation are found by employing the minimum potential energy principle subjected to an inequality constraint condition. A slack variable that converts the inequality constraint to an equality constraint condition is introduced. The coordinate functions that describe the deformed configurations of the membrane are assumed to be represented by a series of geometric admissible functions with unknown coefficients. The unknown coefficients that minimize the total potential energy are determined by Fletcher and Powell's[1] iterative descent method for finding the minimum of a function of multivariables.     

On the burst of branched polymer melts during inflation

Two molten low-density polyethylene melts, shaped as plates, have been inflated into a circular cylinder during isothermal conditions. Lowering the inflation rates allow the plates to be inflated into a larger volume of the cylinder before bursting. Numerical simulations of the inflations have been performed, using a time-strain separable constitutive K-BKZ equation based on the potential function from the Doi–Edwards theory. The material parameters in the constitutive model are based on liner viscoelastic and time dependent uniaxial elongational viscosities. The numerical calculations show quantitative agreement with the experiments, including the appearance of the burst, for a wide range of experimental conditions. This strongly suggests that the initiation of the burst in the polymer melts is a hydrodynamic phenomenon.     

Numerical and experimental dynamic characteristics of thin-film membranes

Presented is a total-Lagrangian displacement-based non-linear finite-element model of thin-film membranes for static and dynamic large-displacement analyses. The membrane theory fully accounts for geometric non-linearities. Fully non-linear static analysis followed by linear modal analysis is performed for an inflated circular cylindrical Kapton membrane tube under different pressures, and for a rectangular membrane under different tension loads at four corners. Finite-element results show that shell modes dominate the dynamics of the inflated tube when the inflation pressure is low, and that vibration modes localized along four edges dominate the dynamics of the rectangular membrane. Numerical dynamic characteristics of the two membrane structures were experimentally verified using a Polytec PI PSV-200 scanning laser vibrometer and an EAGLE-500 8-camera motion analysis system.     

On the response and stability of an inflated toroidal membrane under radial loading

Two flat annular hyperelastic membranes, stacked and bonded together at both the boundaries (equators), form a closed inflatable structure of toroidal topology. The response and stability of the inflated toroidal structure subjected to a radial line force distribution at the inner boundary are studied. The forcing is considered under constant pressure and constant amount of gas inflation conditions. Two hyperelastic models described by the corresponding relaxed strain energy density functions are considered for the membrane material. The influence of geometry, material and level of inflation on the response and stability of the structure under load has been brought out. The structure exhibits pressure limit points with increasing levels of inflation. The force–deflection (stiffness) behaviour is found to be qualitatively different below and above the pressure limit points. Below the pressure limit point, wrinkling and pull-in under loading are revealed for different inflation conditions, and the stability boundaries are determined. Under certain conditions, a counter-intuitive stretch-softening behaviour is also observed.     

Influence of Thermally Induced Chemorheological Changes on the Inflation of Spherical Elastomeric Membranes

When an elastomeric material is deformed and subjected to temperatures above some chemorheological value T _cr (near 100°C for natural rubber), its macromolecular structure undergoes time and temperature dependent chemical changes. The process continues until the temperature decreases below T _cr. Compared to the virgin material, the new material system has modified properties (often a reduced stiffness) and permanent set on removal of the applied load. A recently proposed constitutive theory is used to study the influence of chemorheological changes on the inflation of an initially isotropic spherical rubber membrane. The membrane is inflated while at a temperature below T _cr. We then look at the pressure response assuming the sphere's radius is held fixed while the temperature is increased above T _cr for a period of time and then returned to its original value. The inflation pressure during this process is expressed in terms of the temperature, representing entropic stiffening of the elastomer, and a time dependent property that represents the kinetics of the chemorheological change in the elastomer. When the membrane has been returned to its original temperature, it is shown to have a permanent set and a modified pressure-inflated radius relation. Their dependence on the initial inflated radius, material properties and kinetics of chemorheological change is studied when the underlying elastomeric networks are neo-Hookean or Mooney–Rivlin.     

Deformation of a fluid-filled compliant cylinder in a uniform flow

We investigated surface compliance effects of a fluid-filled object in flow on its shape and internal flow through numerical simulation. A two-dimensional compliant cylinder containing fluid in a flow is a simple model of a cell, e.g. an erythrocyte, leukocyte or platelet. The thin membrane of the cylinder consisted of a network of mass-spring-damper (MSD) systems, representing its mechanical characteristics. We assumed that the stiffness and damping coefficients were those of latex gum. The two-dimensional flow inside and outside the membrane was obtained by solving the two-dimensional Navier–Stokes equations by using the finite element scheme at Re=400, based on the external flow velocity and diameter of an initial circular cylinder. The deformation of the membrane was calculated by solving the equation of motion for an MSD system by using the fourth-order Runge-Kutta method. The compliant cylinder deformed more if its stiffness was smaller than that of latex gum. The initial circular section of the cylinder became oval, with a flat front and a convex rear. The aspect ratio of the lateral to streamwise axis length of the oval became larger than unity, and increased with decreasing stiffness. The drag coefficient of the oval cylinder became larger than that of the circular cylinder, and increased with decreasing stiffness. The partial vibration at the rear, caused by shedding vortices, induced oscillating internal flows between two antinodes of the vibrating membrane. Since the object with smaller stiffness had higher ductility, velocity fluctuations of the external flow influenced the internal flow of the compliant object through deformation of the membrane.     

Resonant behavior of a membrane of a dielectric elastomer

This paper analyzes a membrane of a dielectric elastomer, prestretched and mounted on a rigid circular ring, and then inflated by a combination of pressure and voltage. Equations of motion are derived from a nonlinear field theory, and used to analyze several experimental conditions. When the pressure and voltage are static, the membrane may attain a state of equilibrium, around which the membrane can oscillate. The natural frequencies can be tuned by varying the prestretch, pressure, or voltage. A sinusoidal pressure or voltage may excite superharmonic, harmonic, and subharmonic resonance. Several modes of oscillation predicted by the model have not been reported experimentally, possibly because these modes have small deflections, despite large stretches.     

On the contact of a spherical membrane enclosing a fluid with rigid parallel planes

The mechanical response of an inflated spherical membrane-fluid structure in contact with rigid parallel planes is studied. The membrane is assumed to be a two-dimensional non-linear elastic and isotropic structure, while no assumption is imposed on the fluid. A numerical procedure is employed to compute the equilibrium configurations of the membrane-fluid structure. This study provides information regarding the contact force, stress distribution and pressure in the membrane and in the enclosed fluid, respectively. It was observed that a transition between unwrinkled to partially wrinkled configurations of the membrane occurs subjected to the loading conditions. Further investigation of the wrinkled configurations is presented.     

The stability of a finitely inflated cylindrical elastic membrane under axial compression

The stability under overall axial compression of a finitely inflated cylindrical membrane composed of highly elastic material is investigated. The critical loads for inflated tubes with closed ends and with either simply-supported or fixed ends are determined in terms of the material properties of the membrane. For long tubes the results are compared with the Euler formulae for the buckling load for struts in compression. An equivalent Young's modulus is derived, and it is shown that the critical loads can be obtained from the Euler formulae by using the dimensions of the inflated state and the equivalent Young's modulus.     

Quasistatic periodic contact problem for a viscoelastic layer,a cylinder,and a space with a cylindrical cavity

This paper considers the contact problem of interaction of a rigid die, a rigid band, and a rigid insert with a viscoelastic layer, a viscoelastic cylinder, and viscoelastic space with a cylindrical cavity, respectively. It is assumed that the die, band, and insert move at a constant velocity along the boundaries of the viscoelastic bodies. In the first stage, the displacement of the boundaries of the above-mentioned bodies is determined as a function of the applied normal loads ignoring friction in the contact area. In the second stage, integral equations are derived to determine contact pressure in the contact problems. In the third stage, approximate solutions of the integral equations are constructed using a modified Multhopp-Kalandia method.     

Finite Oscillations of a Cylinder in a Coaxial Duct Subjected to Annular Compressible Flow

As a generalization considering small fluid-structural vibrations, the present paper examines the finite magnitude oscillatory motion of an elastically supported rigid cylinder in a cylindrical rigid duct conveying a compressible flow. The fluid is assumed to be inviscid and irrotational and free purely transverse vibrations of the body are dealt with. The governing equations of motion are the fully nonlinear Euler equations together with the continuity equation and a state equation (here for an ideal gas), the ordinary differential equation for the vibrating cylinder, and the kinematical transition and boundary conditions at the moving contact interface between fluid and body and the outside fluid border, respectively. A pertubation analysis is performed to calculate not only the dynamic characteristics for small coupled oscillations but also the corrections due to the inherent nonlinearities of the vibroacoustic problem. To make the calculation steps more transparent, the simpler problem of a two-dimensional channel flow between a rigid wall and an elastically supported rigid plate is also included in the present study. As an outlook, the influence of flexibility of the cylinder (or the plate) is addressed and the problem of forced vibrations is touched. This revised version was published online in July 2006 with corrections to the Cover Date.     

Propagation of Normal Axisymmetric Waves in Compliant Elastic Cylinders Filled with and Immersed in a Fluid

The paper studies the relationship between the physical characteristics of a cylinder and the properties of normal axisymmetric waves in elastic–liquid waveguides. The cylinder is made of a compliant material in which the velocity of shear waves is less than the sonic velocity in a perfect compressible liquid. The complete system of dynamic elasticity equations and the wave equation are used to describe the wave fields in the elastic cylinder and fluid, respectively. This approach allows obtaining the dispersion characteristics of coupled normal waves in compound waveguides over wide ranges of frequencies and wavelengths. The curves of real, imaginary, and complex wave numbers versus frequency are plotted for specific pairs of waveguide materials. Computations are carried out for a thick-walled cylinder filled with a fluid and immersed in either vacuum or a fluid. It is found out that compliant and rigid materials of the cylinder affect differently the wave interaction process in elastic–liquid waveguides     

Inflation of spherical rubber balloons

When a spherical rubber balloon of the sort used in meteorological applications is inflated, the onset of aspherical deformation is observed after the pressure maximum has been attained. Upon further inflation the balloon regains its spherical shape. Here, the rubber balloon is idealized as an elastic membrane and inflation is taken to be accomplished by a prescribed increase in enclosed volume. The axisymmetric equilibrium states of slightly imperfect membranes are determined numerically by means of the Ritz-Galerkin method. Several particular material models representative of the behavior of rubberlike solids are employed in order to illustrate a number of feautres associated with the aspherical deformation.     

Pressure distribution in an elastomer confined by a long thin-walled flexible hollow cylinder under the assumption of a hydrostatic stress state

The stress distribution in a pressurized elastomer confined by a hollow cylinder is of interest in various applications of material testing and manufacturing. A relatively accurate closed form solution for the pressure distribution inside an elastomer confined by a rigid hollow cylinder was presented by Yu et al. (2001). But in many practical applications the assumption of a rigid hollow cylinder is not appropriate, because the cylinder deformations have a significant influence on the stresses inside the elastomer. Thus in this paper a solution for an elastomer confined by a deformable hollow cylinder is derived. Both axial and radial deformations of the hollow cylinder are taken into account, while the bending stiffness of the cylinder wall is neglected, i.e. the cylinder wall is treated according to the membrane theory. The accuracy of the proposed closed form solution is verified by a parametric finite element simulation.     

Bifurcation of response of a nonlinear viscoelastic spherical membrane

The quasistatic inflation of a nonlinear viscoelastic spherical membrane by monotonically increasing pressure is considered. The deformation is assumed to be spherically symmetric. For the constitutive equation assumed, circumstances are shown to exist when the radius history must either have a jump discontinuity or bifurcate. A necessary condition for bifurcation and its dependence on material properties and radius history is analysed. Examples of bifurcation for various pressure histories are presented. Post-bifurcation branches are constructed and the possibility of secondary bifurcation is discussed.