Nonlinear vibration and resonance analysis of a rectangular hyperelastic membrane resting on a Winkler–Pasternak elastic medium under hydrostatic pressure

Abstract

In this paper, the nonlinear vibration and resonance analyses of a rectangular hyperelastic membrane embedded within a nonlinear Winkler–Pasternak elastic medium subjected to a uniformly distributed hydrostatic pressure are investigated. The material of the membrane is incompressible, homogeneous, isotropic, and hyperelastic. The constitutive law of neo-Hookean is utilized for modeling the system. The elastic foundation includes two Winkler and Pasternak linear terms and a Winkler term with cubic nonlinearity. Using the theory of thin hyperelastic membrane, Hamilton’s principle, and assuming the finite deformations, the governing equations are obtained. Then, the motion equation in the transverse direction is discretized by applying Galerkin’s method. The transverse nonlinear oscillations of the membrane are considered in two-modes. Then, utilizing the multiple scales method, the secondary resonance for the case of [Formula: see text] is analyzed. Also, to analyze the nonlinear vibration behavior, numerical methods including the fourth-order Runge–Kutta methods are utilized. In addition, in analyzing the nonlinear vibration responses of the rectangular hyperelastic membrane, the bifurcation diagram, quasi-period motion responses, and the phase portrait are examined. Finally, the influence of the geometrical characteristics and elastic foundation parameters on the resonance analysis of a rectangular hyperelastic membrane is investigated.

Viscoelasticity in Large Deformation Analysis of Hyperelastic Structures

“…Runge-Kutta time integration and penalty methods are used in the solution. Karimi et al [ 20 ] conducted the nonlinear dynamic analysis of an embedded neo-Hookean hyperelastic membrane under a uniformly distributed hydrostatic pressure. Alibakhshi and Heidari [ 21 ] carried out the nonlinear vibration of a dielectric elastomer balloon via the Gent hyperelastic model.…”

Nonlinear dynamic and forced vibration characteristic analysis of sandwich panel with double-U auxetic core and GPLRC facing sheets embedded with piezoelectric layers

“…As observed from the previous study, the large amplitude vibration response of composite structures has been widely researched in the literature. Extensive reviews are expressed in Jamshidi and Jafari (2021), Liu et al (2018), and Karimi et al (2022). Based on classical shell theory, the nonlinear vibration of cylindrical shells is investigated based on the perturbation method by Chen and Babcock (1975).…”