Snap-through instabilities of pressurized balloons: Pear-shaped bifurcation and localized bulging. (January 2018) - Wang, T.; Xu, F.; Huo, Y.; Potier-Ferry, M.
Abstract: This paper investigates numerically the post-bifurcation evolution of ellipsoidal or spherical balloons subject to internal pressure, where the primary N -shaped curves of pressure vs. principal stretch and the corresponding bifurcated branch, i.e. pear-shaped deformation, are captured quantitatively. We quantify and discuss the range of pear-shaped bifurcation intervals of ellipsoids and the associated critical points. For rugby-shaped ellipsoidal balloons, we find that there exists a threshold for the aspect ratio of the major and minor axes that leads to pear-shaped bifurcation, which is detected by the finite element method. When the rugby shape becomes sufficiently slender, the nonlinear response of the ellipsoidal balloons is well approximated by the deformation of a tube with localized bulging instead of pear-shaped configuration. We obtain the nonlinear evolution of the localized bulging of rugby-like ellipsoids numerically. Furthermore, we examine the influence of various aspect ratios for rugby-shaped balloons on the localized bulging response. We find that pumpkin-shaped ellipsoids can always bifurcate into pear shape. Lastly, we provide a unified phase diagram on instability mode selection of various aspect ratios of ellipsoidal balloons, with diverse representative deformed configurations. Highlights: Finite element method was applied for post-bifurcation analyses of ellipsoidal balloons. Bifurcation curves including primary and pear-shaped branches of ellipsoids of revolution were obtained quantitatively. Phase diagram of ellipsoidal balloons including pear-shaped and localized bulging deformations was first provided. Localized bulging of slender rugby-shaped balloons was observed. Numerical results remarkably agree with analytical solutions