Long-wave vibrations of a nearly incompressible isotropic plate with fixed faces

The three-dimensional dynamic equations of elasticity are subjected to a two-parameter asymptotic analysis for the case of long-wave vibrations of a nearly incompressible isotropic plate with fixed faces. Two non-standard types of two-dimensional approximations are revealed. The approximation of the first type is not local, as is typical for long-wave vibrations of standard linear isotropic plates and shells, occurring near zero-frequency and thickness resonance frequencies. The relevant governing equations are characterized by the appearance of trigonometric functions of the frequency parameter. The second approximation is localized near the cut-off frequencies of an incompressible plate, which do not represent thickness resonance frequencies. The analysed phenomena are specific only for symmetric motions and are not observed for other boundary conditions on the faces.