Stability of pear-shaped configurations bifurcated from a pressurized spherical balloon

Abstract

It is well-known that for most spherical rubber balloons the pressure versus volume curve associated with uniform inflation has an N-shape (the pressure increases rapidly to a maximum, falls to a minimum, and subsequently increases monotonically), and that somewhere along the descending branch of this curve the spherical shape may bifurcate into a pear shape through localized thinning near one of the poles. The bifurcation is associated with the (uniform) surface tension reaching a maximum. It is previously known that whenever a pear-shaped configuration becomes possible, it has lower energy than the co-existing spherical configuration, but the stability of the pear-shaped configuration itself is unknown. With the use of the energy stability criterion, it is shown in this paper that the pear-shaped configuration is unstable under pressure control, but stable under mass control. Our calculations are carried out using the Ogden material model as an example, but it is expected that the qualitative stability results should also be valid for other material models that predict a similar N-shaped behaviour for uniform inflation.

Electro-mechanical coupling bifurcation and bulging propagation in a cylindrical dielectric elastomer tube

“…On inflating a purely spherical membrane refined stability analysis and experiments show that the spherical configuration in inflation is unstable. The spherical membrane may bifurcate into a pear-shaped configuration through a localized thinning near one of the poles (Needleman, 1977;Haughton and Ogden, 1978;Fu and Xie 2014). When inflating a tubular balloon, the localized bulging instability is usually observed at a critical pressure.…”

Shape Bifurcation of a Spherical Dielectric Elastomer Balloon Under the Actions of Internal Pressure and Electric Voltage

“…All the perturbations in Eqs. (27) and (28) can be expressed by power series of δr(θ) and δt(θ). To investigate the critical conditions of the bifurcation, we only keep the linear order terms of δr (θ) and δt(θ).…”

“…In this section, we will conduct stability analysis. 12 Following the energetic method adopted by different researchers [27,31,32], we first derive second variation of the free energy of the dielectric elastomer balloon system. If the second variation of the free energy of an equilibrium state is positive definite, the state is energetically stable.…”

Bifurcation of a dielectric elastomer balloon under pressurized inflation and electric actuation

“…Chen and Healey (1991) showed that the pear-shaped configuration must necessarily have lower energy than the co-existing spherical configuration, and they also derived some sufficient conditions under which the above bifurcation behavior actually occurs for a general material model. Fu and Xie (2014) analyzed the stability of the pear-shaped configuration itself with respect to further axi-symmetric perturbations, and showed that it is stable under mass or volume control but unstable under pressure control. The well-known bifurcation condition in the purely mechanical case was originally derived from the incremental theory of nonlinear elasticity, but it was shown in Fu and Xie (2014) that if attention is focused on axi-symmetric bifurcation modes then bifurcation can be detected by a simple shooting procedure based on the original governing equations.…”

“…Fu and Xie (2014) analyzed the stability of the pear-shaped configuration itself with respect to further axi-symmetric perturbations, and showed that it is stable under mass or volume control but unstable under pressure control. The well-known bifurcation condition in the purely mechanical case was originally derived from the incremental theory of nonlinear elasticity, but it was shown in Fu and Xie (2014) that if attention is focused on axi-symmetric bifurcation modes then bifurcation can be detected by a simple shooting procedure based on the original governing equations. It is this latter method that will be employed in the present paper.…”