Nonlinear vibrations of a rectangular hyperelastic membrane resting on a nonlinear elastic foundation

The aim of the present work is to investigate the nonlinear vibration response and stability of a pre-stretched rectangular hyperelastic membrane resting on a nonlinear elastic foundation. The membrane is composed of an isotropic, homogeneous and hyperelastic material, which is modeled with four different constitutive laws, specifically: neo-Hookean, Mooney-Rivlin, Yeoh and Ogden hyperelastic models. The elastic foundation is described by a Winkler type nonlinear model with cubic hardening or softening nonlinearity. First the exact solution of the membrane under a biaxial stretch is obtained. Then the equations of motion of the pre-stretched membrane resting on the nonlinear foundation are derived. From the linearized equations, the natural frequencies and mode shapes of the membrane are obtained analytically. Then the natural modes are used to approximate the nonlinear deformation field using the Galerkin method. A detailed parametric analysis shows the strong influence of the stretching ratios and foundation parameters on the linear and nonlinear oscillations of the membrane. Frequency-amplitude relations and resonance curves are used to illustrate its nonlinear dynamics. The present results are favorably compared with the results evaluated for the same membrane, whenever possible, using a nonlinear finite element formulation.