Nonlinear vibrations of a radially stretched circular hyperelastic membrane

This paper presents a detailed analysis of the nonlinear vibration response of a pre-stretched hyperelastic membrane subjected to finite deformations and a time-varying lateral pressure. The problem is both geometrically and materially nonlinear due to finite deformations and a hyperelastic constitutive relationship. The membrane material is assumed to be isotropic, homogeneous, and neo-Hookean. First, the exact solution of the membrane under a uniform radial stretch is obtained. The equations of motion of the pre-stretched membrane are then derived. From the linearized equations, the natural frequencies and mode shapes of the membrane are analytically obtained. The natural modes are then used to approximate the nonlinear deformation field using the Galerkin method. Several reduced order models are tested and compared with the results evaluated for the same membrane using a nonlinear finite element formulation. Excellent agreement is observed. The results show the strong influence of the stretching ratio on the linear and nonlinear oscillations of the membrane. Finally, the influence of the constitutive law on linear and nonlinear vibrations is investigated. Results show that several constitutive laws for hyperelastic rubber-like materials lead to the same frequency–amplitude relation.