Classification of thin shell models deduced from the nonlinear three-dimensional elasticity. Part I: the shallow shells - Hamdouni, A.; Millet, O. (Archives of Mechanics, 2003)
The purpose of this paper is to construct a classification of asymptotic shell models (inferred from the non linear three-dimensional elasticity) with respect to the applied forces and to the geometrical data. To do this, we use a constructive approach based on a dimensional analysis of the nonlinear three-dimensional equilibrium equations, which naturally gives rise to the appearance of dimensionless numbers characterizing the applied forces and the geometry of the shell. In order to limit our study to one-scale problems, these dimensionless numbers are expressed in terms of the relative thickness ε of the shell, which is considered as the perturbation parameter. This leads, on the one hand, to distinguish shallow shells from strongly curved shells which have a different asymptotic behaviour, and on the other hand, to fix the applied force level. For each of these two classes of shells, using the usual asymptotic method, we propose a complete classification of two-dimensional shell models based on decreasing force levels, from severe to low. In the first part of this paper, we present the classification for shallow shells. We obtain successively the nonlinear membrane model, another membrane model, Koiter’s non linear shallow shell model, and the linear Novozhilov-Donnell one, respectively for severe, high, moderate and low forces.