Shape bifurcation of a pressurized ellipsoidal balloon

Abstract

It is well-known that for most spherical and cylindrical rubber balloons the pressure versus volume curve associated with uniform inflation both has an N-shape, but their shape bifurcation has different characters: whereas a spherical balloon tends to bifurcate into a pear shape through localized thinning near one of the poles, a cylindrical balloon would always bulge out locally in a symmetric manner. To understand the connection between these two different bifurcation behaviors, we study in this paper the shape bifurcation of an ellipsoidal balloon which becomes a spherical balloon when the three axes are identical, and approximates a cylindrical balloon when one axis is much larger than the other two axes. The ellipsoidal shape is obtained by rotating an ellipse about one of its axes, that gives rise to two possibilities: a rugby shape or a pumpkin shape. It is shown that for a rugby-shaped balloon, there exists a threshold axes ratio below which the slender ellipsoidal balloon behaves more like a tube and bifurcation into a pear shape becomes impossible, whereas for a pumpkin-shaped balloon bifurcation into a pear shape is always possible.

Snap-through instabilities of pressurized balloons: Pear-shaped bifurcation and localized bulging

“…where and̃are material constants. In this paper, we take the following material parameters from [32], which are widely used for the analyses of inflation of hyperelastic balloons [16,22,27,39]: 1 = 1.3, 2 = 5.0, 3 = −2.0,̃1 = 1.4919,̃2 = 0.003 and̃3 = −0.024, in which̃are dimensionless, scaled by the ground state shear modulus . Practically, these values of the parameters refer to a vulcanized natural rubber which has highly reversible hyperelastic behavior but low crystallization upon extreme stretching [40].…”

“…A rugby-shaped balloon refers to non-equal axes with ∕ < 1. In [22], the governing equation (6) is decomposed into three first-order ODEs that can be solved by numerical shooting method. Their results show that the bifurcation interval shortens with the decrease of ∕ , and then disappears at ∕ = 0.626.…”

“…Differing from rugby configuration, for a pumpkin-shaped balloon with ∕ > 1, bifurcated solutions always exist with pear-shaped deformation. In [22], it is, however, difficult to distinguish the bifurcated and primary solutions of pressure-volume curves analytically. Through finite element method, we can clearly detect bifurcated branches from pressure-volume curves.…”

“…The numerical primary and bifurcated -shaped curves are drawn in Fig. 6, in which the theoretical bifurcated thresholds [22] are compared. When the scaling volumē = ∕ rises tō1 = 6.177, the rugby-shaped balloon can initially bifurcate into pear shape (see Fig.…”

“…Lately, Geng et al [22] extended the work of Fu and Xie [16] to ellipsoidal configurations and found that it is always possible for pumpkinshaped balloon to bifurcate into pear shape, whereas there exists a threshold of aspect ratio for rugby-shaped balloon. When the aspect ratio of the major and minor axes is smaller than a critical value, the inflation behavior of the ellipsoid of revolution is approximated to the response of a tube.…”