## Plates and shells: Asymptotic expansions and hierarchical
models

*
Monique Dauge,
Erwan Faou,
Zohar Yosibash
*

Concerning thin structures such as plates and shells, the idea of reducing the equations of elasticity to
two-dimensional models defined on the mid-surface seems relevant. Such a reduction was first performed thanks
to kinematical hypotheses about the transformation of normal lines to the mid-surface:
Let us mention, among others, the Kirchhoff-Love and Reissner-Mindlin models for plates, and the Koiter
model for shells.

As nowadays, the
asymptotic expansion of the displacement solution of the three-dimensional linear model is fully known
at least for plates and clamped elliptic shells,
we start from a description of these expansions in order to introduce the two-dimensional models
known as hierarchical models: These models extend the classical models, and pre-suppose the displacement
to be polynomial in the thickness variable, transverse to the mid-surface.

Because of the singularly
perturbed character of the elasticity problem as the thickness approaches zero, boundary- or internal layers
may appear in the displacements and stresses, and so may numerical locking effects.
We describe how hierarchical models can be
discretized by the *p*-version of finite elements, using refined meshes in the layer regions.
Thanks to the very strong
approximation properties of piecewise polynomials of high degrees in suitably designed anisotropic meshes,
such a discretization may help to overcome locking effects.

Finally, besides the classical characterization of shells as membrane and flexural,
we also present a classification through the low frequency response of shells and give illustrative examples
of modal computations on three characteristic families:

• Clamped spherical shells

• Partially clamped spherical shells *(sensitivity)*

• Partially clamped cylindric shells *(inextensional displacements)*

February 2004.

** Chapter 8**,
**Vol. I**,
pp. 199-236, in the
**
Encyclopedia for Computational Mechanics**

Edited by Erwin Stein, René de Borst, Thomas J.R. Hughes.