On the dynamic snap-out instability of inflated non-linear spherical membranes

The dynamic behavior of a spherical membrane, made of Mooney material and subjected to a uniform inflating step-pressure, is studied. Its phase-planes and nondimensional radius versus time curves are plotted for different values of the material constants. The conditions for dynamic snapout instability are discussed. The relations between the static behavior of the membrane and the dynamic instability are pointed out.     

Equilibrium shapes of inflated inextensible membranes

This paper proposes an effective method for directly determining the final equilibrium shapes of closed inextensible membranes subjected to internal pressures. With reference to new high-performance textile materials, we assume that the mechanical response of a fabric membrane can be accurately represented by regarding it as a two-state material. In the active state, the membrane is subject to tensile stresses and is virtually inextensible; vice versa, in the passive state it is unable to sustain any compressive stress, so it contracts freely. Equilibrium of the membrane in the final configuration is enforced by recourse to the minimum total potential energy principle. The Lagrange multipliers method is used to solve the minimum problem by accounting for the aforesaid nonlinear constitutive law. The set of governing equations is solved for the unknown coordinates of the equilibrium surface points. Closed form solutions are obtained for fully wrinkled cylindrical and axisymmetric membranes under homogeneous boundary conditions, while a simple iterative procedure is used to numerically solve cases of axisymmetric membranes under various inhomogeneous boundary conditions. The soundness of the proposed method is verified by comparing the results with solutions available in the literature.     

Effects of axisymmetric loads on inflated non-linear membranes

This paper describes the effects of various external axisymmetric loads on pressurized hinged spherical membranes taking into account changes in internal pressure, volume, and temperature. “Exact” geometrical non-linearity along with generalized constitutive relations for a highly non-linearly clastic, isotropic, homogeneous, incompressible material are used in the analysis. The specialized case of a Hookean material is also treated.The non-linear equations of membrane equilibrium are derived in terms of additional finite displacements for the case of nonorthogonal curvilinear midsurface coordinates and are then specialized for the problem of an inflated hinged spherical membrane. The resulting two highly non-linear coupled second order differential equations are solved by means of a finite difference and Newton-Raphson iterative procedure. All results are presented in nondimensionalized graphical form.     

Necking of pressurized spherical membranes

The necking of spherical membranes subject to a prescribed increase in enclosed volume is investigated. Attention is restricted to axisymmetric deformations. The materials considered are incompressible, isotropic, time-independent and incrementally linear. A complete set of axisymmetric bifurcation modes is considered and a simple relation is found to govern the critical stress for bifurcation into a given mode. The limiting critical stress and the corresponding mode for short wavelengths are investigated and related to the results obtained from an independent local-necking analysis. Two perturbation methods are employed to study the growth of initial imperfections: one is valid for arbitrary modes, but restricted to small deviations from sphericity, and the other is valid only for the local-necking mode, but is not restricted to small deviations. The effect of path-dependent material behavior on the onset of local necking is explored. Path-dependent material behavior is found to encourage the preferential growth of short wavelength imperfections. Path-independent materials are shown to exhibit significant sensitivity to initial imperfections in the localized-necking mode, although this sensitivity is far less than for a path-dependent material. When account is taken of initial material-property inhomogeneities as well as initial thickness imperfections, it seems that no definite conclusion can be drawn concerning the appropriateness or inappropriateness of an explanation of the onset of localized necking based on a smooth yield-surface plasticity theory and assuming the presence of such initial inhomogeneities.     

Lateral stability of lofty air-supported spherical membranes

The lateral stability of a lofty air supported spherical membrane subjected to an unsymmetrically applied concentrated load is treated. Simple expressions for the ‘critical’ load are obtained at which the configuration of the structure changes dramatically and part of the membrane comes into contact with the support. The value of this load depends strongly on the area of the support and decreases with increasing eccentricity of the load.     

Inflation and bifurcation of compressible spherical membranes

The inflation and bifurcation of spherical membranes is considered. The membrane material is assumed to be isotropic and hyperelastic but may be arbitrarily compressible. Qualitatively the behaviour of compressible membranes is shown to be the same as that of incompressible membranes but specific forms of strain-energy functions are chosen to illustrate possible quantitative differences.     

Post-bifurcation of perfect and imperfect spherical elastic membranes

When a spherical elastic membrane is inflated it is well known that it may bifurcate into an aspherical mode after the pressure maximum is reached. Upon further inflation the spherical configuration is regained. Here we follow the developing aspherical solution path, for specific forms of strain-energy function, using a simple numerical method. For a realistic strain-energy function it is shown that the post-bifurcation solution curve connects the two bifurcation points. We also consider the inflation of imperfect spherical membranes and show that bifurcation may still occur. For the class of Ogden materials we investigate the asymptotic shape of arbitrary axisymmetric membranes.     

Ponding instability of air-supported spherical membranes with initial imperfections

The stability of a spherical, air-supported membrane subjected to a concentrated load and an accumulating ponding fluid is the subject of the present paper which consists of two parts: in the first part a spherical elastic membrane with an initial imperfection at the apex is considered. The effect of the ponding fluid accumulating in the initial depression reduces the value of the critical load significantly. Simple formulae for the critical load are obtained. In the second part, the case of a non-symmetric load applied to the membrane is considered. Results indicate that the eccentricity causes a significant increase in the value of the critical load thereby making the axisymmetric loading the governing configuration.     

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.     

Some qualitative properties of incompressible hyperelastic spherical membranes under dynamic loads

Some nonlinear dynamic properties of axisymmetric deformation are ex- amined for a spherical membrane composed of a transversely isotropic incompressible Rivlin-Saunders material. The membrane is subjected to periodic step loads at its inner and outer surfaces. A second-order nonlinear ordinary differential equation approximately describing radially symmetric motion of the membrane is obtained by setting the thick- ness of the spherical structure close to one. The qualitative properties of the solutions are discussed in detail. In particular, the conditions that control the nonlinear periodic oscillation of the spherical membrane are proposed. In certain cases, it is proved that the oscillating form of the spherical membrane would present a homoclinic orbit of type ＂∞＂, and the amplitude growth of the periodic oscillation is discontinuous. Numerical results are provided.     

Path-dependent failure of inflated aluminum tubes

Our recent investigation on the formability of Al alloy tubes under combined internal pressure and axial load is expanded by examining the effect of the loading path traced. A set of Al-6260-T4 tubes were loaded along orthogonal stress paths to failure and the results are compared to those of the corresponding radial paths. It is confirmed that failure strains are path-dependent, but also is demonstrated that failure stresses become path-dependent if the prestrain is significant. The experiments are simulated using the previously developed finite element models and the calibration of the Yld2000-2D [Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dick, R.E., Lege, D.J., Pourboghrat, F., Choi, S.-H., Chu, E., 2003. Plane stress yield function for aluminum alloy sheets-part I: theory. Int. J. Plasticity 19, 1297--1319] anisotropic yield function shown in [Korkolis, Y.P., Kyriakides, S., 2008b. Inflation and burst of anisotropic aluminum tubes. Part II: an advanced yield function including deformation-induced anisotropy. Int. J. Plasticity 24, 1625–1637] to yield accurate predictions of rupture for nine radial paths. The models are shown to reproduce the path dependence of the failure stresses and strains quite well. A group of additional radial and corner paths are subsequently examined numerically to enrich the existing data on path-dependence of failure. It is again shown that the amount of plastic prestraining in either of the two directions influences the difference of the failure stresses and strains between the radial and the corner stress paths.     

Experimental-theoretical method for determining mechanical characteristics of spherical films and membranes of complex structure

There are known methods for determining the mechanical characteristics of films and membranes with plane initial geometry. But the films and membranes can have nonplanar initial shape (shell compositions of material-structure type) and complex structure. It is impossible to determine mechanical characteristics of such objects by the standard uniaxial tension method. There are few papers dealing with mechanical characteristics of shell material-structures with defects.     

Dynamics of biological soft tissue and rubber: internally pressurized spherical membranes surrounded by a fluid

The behavior of a family of dynamical systems representing the elastodynamic response of an internally pressurized, non-linearly elastic spherical membrane lying in an incompressible external fluid is governed primarily by the strain energy function for the membrane, the specific forcing function due to the internal pressure, and the viscosity of the external fluid. It is shown that such systems with an inviscid external fluid and having a constant internal pressure are integrable but not Hamiltonian. Under periodic internal loading, and for a small spherical radius and constitutive relations typical of many biological soft tissues, a periodic orbit in phase space exists near a static equilibrium. A viscous external fluid causes the periodic orbit to be an attractor. The dynamical system is robust under small loading perturbations common in normal biological systems. Rubber models, on the other hand, may admit structural catastrophes. For small initial sphere radii, a jump from one periodic orbit to another is possible for rubber models but not for the classical soft tissue models. It is dangerous, therefore, to model soft biological tissue as a rubber either mathematically or physically in experiments because the predicted instabilities may not exist in tissue.     

Inhomogeneous deformation in metallic glasses

The present study provides a theoretical framework for the inhomogeneous deformation in metallic glasses. The free volume concentration is adopted as the order parameter, which is a function of position and time. The three processes that can change the local free volume concentration are diffusion, annihilation, and stress-driven creation. The rate functions for free volume generation and plastic flow depend on the underlying microscopic model, but the framework is generally valid for different models. A simple shear problem is solved as an example. A linear stability analysis is performed on the basis of the homogeneous solution. An inhomogeneous solution is obtained with a finite amplitude disturbance to the initial free volume distribution. Numerical simulation shows the development of the inhomogeneous deformation and strain localization.