Dynamical modeling and analysis of hyperelastic spherical shells under dynamic loads and structural damping

Abstract

This paper investigates the effects of dynamic loads and structural damping on the nonlinear behaviors of incompressible hyperelastic spherical shells modeled by the Yeoh strain energy function. Firstly, the dynamical modeling is formulated for the nonlinear behaviors of the shells by the variational principle, and the second order nonlinear ordinary differential equation describing the radially symmetric motion is obtained. Then, the dynamic behaviors, such as periodic, quasiperiodic and chaotic motions, are discussed under different loading types. Particularly, for constant loads, the first integral of the integrable Hamiltonian system without damping is given and it is numerically proved that there exists an asymmetric "∞" homoclinic orbit for the prescribed material parameters obtained in experiments; moreover, it is found that for different prestretches, the structural parameter has a completely different role on the nonlinearity of the system, and the basins of attraction are given with the structural damping. For periodic loads, there exist some interesting dynamic phenomena, i.e., quasiperiodic oscillation in the approximately integrable Hamiltonian system, limit cycles and chaos with the damping. The criterion for chaos is discussed by the Melnikov method combined with the numerical calculation and the chaos is further analyzed with the Poincaré section and the phase plane.