Nonlinear breathing motions and instabilities of a pressure-loaded spherical hyperelastic membrane

Abstract

This work presents the mathematical modeling, based on a variational formulation, for the analysis of the nonlinear static and dynamic breathing motions of a hyperelastic spherical membrane subjected to internal pressure. The membrane is composed of an isotropic, homogeneous, incompressible, and hyperelastic material that is modeled using the Mooney–Rivlin constitutive law. The equilibrium equations are obtained using the fully nonlinear elasticity theory. First, the static nonlinear analysis is performed, and the principal stretches and stresses are obtained as a function of the internal pressure. Additionally, a parametric analysis is conducted to study the influence of the material constants on the nonlinear equilibrium path and potential energy. Depending on the value of the two material constants, a bistable behavior may be observed, having a profound influence on the nonlinear vibrations of the membrane. Then, the equations of motion of the pressure-loaded membrane subjected to increasing pressure and an additional harmonically varying internal pressure are obtained, considering a nonlinear damping force. A detailed parametric analysis clarifies the influence of the Mooney–Rivlin material parameters and static preload on the natural frequencies and particularly on the nonlinear frequency–amplitude relation. Precise results for the backbone curves are obtained by the shooting method, using the fully nonlinear equation of motion, and compared with several approximations obtained by the harmonic balance methods, highlighting the influence of higher order nonlinear terms on the nonlinear response of hyperelastic membranes. Finally, a parametric analysis of the harmonically excited preloaded membrane is conducted to reveal the importance of the two-well potential function, damping and constitutive law on the resonance curves, bifurcation diagrams, and basins of attraction.