Large-Amplitude Oscillations of Hyperelastic Cylindrical Membrane Under Thermal-Mechanical Fields

In this paper, the nonlinear dynamic behaviors of a hyperelastic cylindrical membrane composed of the incompressible Ogden material are examined, where the membrane is subjected to uniformly distributed radial periodic loads at the internal surface and surrounded by a thermal field. A second-order nonlinear differential equation describing the radially symmetric motion of the membrane is obtained. Then, the dynamic characteristics of the system are qualitatively analyzed in terms of different material parameter spaces and ambient temperatures. Particularly, for a given constant load, the bifurcation phenomenon of equilibrium points is examined. It is shown that there exists a critical load, and the phase orbits may be the asymmetric homoclinic orbits of the “ $$\infty $$ ” type. Moreover, for the system with two centers and one saddle point, the dynamic behaviors of the system show softening phenomena at both centers, but the temperature has opposite effects on the stiffness of the structure. For a given periodically perturbed load superposed on the constant term, some complex dynamic behaviors such as quasiperiodic and chaotic oscillations are analyzed. With the Poincare section and the maximum Lyapunov characteristic exponent, it is found that the ambient temperature could lead to the irregularity and unpredictability of the nonlinear system, and also changes the threshold of chaos.