## Eigen-Frequencies in Thin Elastic 3-D Domains and Reissner-Mindlin Plate Models

*
Monique Dauge,
Zohar Yosibash
*

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 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 order of 30% 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).

Rennes, Beer Sheva, January 2001.

* Mathematical Methods in the Applied Sciences* 25, 2002, 21-48.