Plates and Shells: Asymptotic Expansions and Hierarchic Models
Monique Dauge, Erwan Faou, Zohar Yosibash · 2004
Abstract Concerning thin structures, such as plates and shells, the idea of reducing the equations of elasticity to two‐dimensional models defined on the midsurface seems relevant. Such a reduction was first performed thanks to kinematical hypotheses about the transformation of normal lines to the midsurface. 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 presuppose the displacement to be polynomial in the thickness variable, transverse to the midsurface. 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 overcome these severe difficulties.