Limit behavior of a shell,when first the thickness approaches zero and then the shell becomes a plate

Abstract

We consider a shell, when first the thickness approaches zero and then the middle surface becomes flat. As noted by P.G. Ciarlet, the limit displacement obtained in this fashion is different from the limit displacement obtained when the shell becomes a plate first. Indeed, the ‘membrane’ and ‘bending’ effects both appear when first the shell becomes a plate, while only one of these effects appear when first the thickness approaches zero. This result is proved for shells with uniformly elliptic middle surfaces and for cylindrical shells.