Plates and Shells: Asymptotic Expansions and Hierarchical Models

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. As 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. The use of hierarchical models, discretized by higher degree polynomials (p-version of finite elements) may help to overcome these severe difficulties.