Eigen-frequencies in thin elastic 3-D domains and Reissner-Mindlin plate models

The eigen-frequencies of elastic three-dimensional thin plates are addressed and compared to the eigen-frequencies of two-dimensional Reissner-Mindlin plate models obtained by dimension reduction. The qualitative mathematical analysis is supported by quantitative numerical data obtained by the p-version finite element method. The mathematical analysis establishes an asymptotic expansion for the eigen-frequencies in power series of the thickness parameter. Such results are new for orthotropic materials and for the Reissner-Mindlin model. The 3-D and R-M asymptotics have a common first term but differ in their second terms. Numerical experiments for clamped plates show that for isotropic materials and relatively thin plates the Reissner-Mindlin eigen-frequencies provide a good approximation to the three-dimensional eigen-frequencies. However, for some anisotropic materials this is no longer the case, and relative errors of the order of 30 per cent are obtained even for relatively thin plates. Moreover, we showed that no shear correction factor is known to be optimal in the sense that it provides the best approximation of the R-M eigen-frequencies to their 3-D counterparts uniformly (for all relevant thicknesses range).