Some qualitative properties of incompressible hyperelastic spherical membranes under dynamic loads

Abstract

Some nonlinear dynamic properties of axisymmetric deformation are examined 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 thickness 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.

The role of constitutive relation in the stability of hyper-elastic spherical membranes subjected to dynamic inflation

“…Unlike the work of Hearn (2013a, 2013b), the constitutive sensitivity analysis developed in the present paper lies within the dynamic regime. While the dynamic inflation of elastomeric shells is less explored than the static one, it was shown by Verron, Khayat, Derdouri, and Peseux (1999), Verron, Marckmann, and Peseux (2001) and Yuan, Zhang, Ren, and Zhu (2010) that inertia has a significant influence on the mechanical stability of spherical balloons. Within this framework, we address in this paper a key issue: different constitutive models calibrated using the same set of experimental data and the same fitting procedure provide different predictions of the dynamic response of the spherical balloon.…”

Physically and geometrically non-linear vibrations of thin rectangular plates

“…Sensitivity of the dynamics to the material parameters, the structure geometry, and the applied pressure is discussed. In [34], the authors explored the dynamic symmetric response of a spherical membrane made of incompressible material with the Rivlin-Saunders constitutive law. The membrane is subjected to periodic step loads on its inner and outer surfaces.…”

Large Amplitude Vibrations of Thin Hyperelastic Plates: Neo-Hookean Model

“…Reviews of static and dynamic problems can be found in [2,3]. A majority of research studies on the dynamics of hyperelastic thin-walled structures deals with simple geometries (spherical and circular cylindrical shells, circular membranes) and assume the same simple shape of the deformed structure [4][5][6][7][8][9][10][11][12].…”