Vibrations of asymptotically and variationally based Uflyand–Mindlin plate models

Abstract

In this paper, we provide alternative Uflyand–Mindlin's plate equations taking into account rotary inertia and shear deformation, based on both asymptotic expansion and variational arguments. The aim is to derive truncated versions of Uflyand–Mindlin's equations, specifically without the fourth order derivative term with respect to time. The truncated version of Uflyand–Mindlin's plate model may be derived starting from three-dimensional elasticity equations, by using asymptotic arguments based on expansion of displacements with respect to a small geometrical parameter. This expansion method also leads to a proper identification of the shear correction factor. It is shown that suitably modified variational derivation leads to an additional term which is shown to be negligible for determination of the fundamental natural frequency of the all-round simply supported plates, but may contribute significantly in estimation of higher natural frequencies. It is argued that the proposed version of Uflyand–Mindlin's plate equations is simpler and more consistent than the original Uflyand–Mindlin equations. Likewise, it is advantageous over the equation that stems from neglecting the fourth order time derivative in original Uflyand–Mindlin equations. The two alternative truncated models serve as intermediate theories between the classical plate theory and the original Uflyand–Mindlin theory their usefulness depending on the problem at hand.