Softening Backbone Behavior in Hyperelastic Cylindrical Shells

This study examines the frequency-amplitude relationship (backbone curves) of simply supported hyperelastic circular cylindrical shells using a modal formulation of the displacement field. The shell is modeled with the Mooney–Rivlin constitutive law, assuming an incompressible, homogeneous, and isotropic material. The strain energy density function is expanded up to fourth order via a Taylor series, while the Donnell theory is employed to establish the displacement–strain relations. The governing equations are derived using Hamilton’s principle and discretized through the Rayleigh–Ritz method with a Fourier series expansion, incorporating both axisymmetric and asymmetric modes. The proposed modal formulation accurately captures the nonlinear softening behavior in the frequency–amplitude response of the undamped and unforced shell, showing good agreement with finite element results. Furthermore, it demonstrates that second-order terms in the strain energy density function are sufficient to describe the softening behavior for vibration amplitudes up to approximately four times the shell thickness.