Topology optimization and internal resonances in transverse vibrations of hyperelastic plates

Nonlinear dynamics of plates synthesized using topology optimization and undergoing transverse vibrations with 1:2 internal resonances are presented. The plates are assumed to be made of hyperelastic materials; specifically two particular material models, namely, the neo-Hookean and Mooney–Rivlin material model are considered. A finite element approximation is first used in conjunction with novel topology optimization techniques to develop linearized candidate plate structures that have their lowest two natural frequencies in the ratio of 1:2. The plate structures are assumed to follow thin plate theory Kirchoff assumptions. The nonlinear dynamic response of the synthesized structures is then developed using modal superposition, and forced response to base excitations is analyzed to study the effects of material and geometric nonlinearities on nonlinear plate vibrations. The geometric nonlinearities introduced are through the assumptions of finite strains while the material nonlinearities arise due to nonlinear stress–strain or constitutive relationships for hyperelastic material models. Results are also compared with those obtained using von Karman and Novozhilov approximations for nonlinear plate vibrations. First the results are developed with the assumption that the materials are incompressible, and then this requirement is relaxed to include compressible materials as well.