Nonlinear interaction of the damped large dynamic deformations of the Mooney–Rivlin hyperelastic plates with the viscoelastic and shear reactions of the supporting substrate

Abstract

In this research, the interaction between the damped large deformation dynamic responses of the Mooney–Rivlin hyperelastic plates and the viscoelastic and shear characteristics of the supporting substrate (that constitute a visco-hyperelastic system) is investigated for various distributions and various time variations of the transverse loads and different boundary conditions. A viscoelastic Winkler–Pasternak model is chosen for the supporting substrate to account for the dissipative, shearing, and supporting features of the foundation. The governing equations of motion are obtained by using Hamilton’s principle, von Karman assumptions, left Cauchy–Green deformation tensor, and a modified classical plate theory whose accuracy is enhanced through the incorporation of an appropriate high-order incompressibility condition. The combination of the nonlinear and coupled motion equations and boundary conditions is solved by an iterative 2D differential quadrature (DQ) spatial-discretization and Newmark’s time-marching methods. The effects of the constitutive parameters of the hyperelastic material, magnitude and type of the distributed loads, the thickness and aspect ratios of the plate, parameters of the Winkler–Pasternak viscoelastic foundation, and boundary conditions on the dynamic lateral deflections of the plate are studied examined during the parametric studies. Results emphasize the role of the visco-hyperelasticity features interaction of the combined plate-substrate system on the vibration suppression and dynamic performance of the structure in different spatial and time-variation patterns of the distributed loads and various boundary conditions.