Asymptotic Behaviors of Shell Models

Abstract

Implicit in the concept of a “shell” is the idea that the thickness is “small” compared to the other two dimensions. In practice, it is not unusual to deal with structures for which the thickness is smaller by several orders of magnitude, in which case the shell is said to be “thin” (consider, for example, the shell body of a motor car). Considering the role of the thickness parameter t in the shell models that we presented in the previous chapter (see for example Eqs. (4.36) and (4.51)), with different powers of t in the bilinear terms on the left-hand side, it is essential to determine how the properties of the models are affected when this parameter becomes small. Likewise, it is important to know whether the model converges, in some sense to be specified, towards a limit model when the thickness t “tends to zero”, so that this possibly simpler model can be used instead of the original one when t is sufficiently small, i.e. we need to study the asymptotic behavior of the shell models. Of course, our goal is also to investigate the influence of the thickness on the convergence of finite element methods, as we want to be able to identify numerical procedures for which there is no deterioration of convergence when the thickness becomes small. To that purpose, the analysis of the asymptotic behaviors of mathematical shell models clearly also represents a crucial prerequisite on which we concentrate in this chapter, whereas the issues arising in the finite element solutions themselves are addressed in the next chapters.