ASYMPTOTIC ANALYSIS OF LINEARLY ELASTIC S HELLS .1. MEMBRANE-DOMINATED SHELLS

We consider a family of linearly elastic shells, clamp;ed along their entire lateral surface, all having the same middle surface S = phi (om egaBAR), where omega subset-of R2 is a bounded, connected, open set wi th a smooth boundary and phi : omegaBAR --> R3 is a smooth mapping. We assume that, as the thickness 2epislon approaches zero, the applied b ody force density is O(1) with respect to epsilon, and finally, we mak e an essential geometrical assumption on the middle surface S, which i s satisfied if S is uniformly elliptic. We then show that, as epsilon --> 0, the three covariant components of the displacement of the point s of the shell, once defined over the fixed open set OMEGA = omega x] - 1, 1[, converge in L2 (OMEGA) to limits u(i) that are independent of the transverse variable x3, the averages over the thickness of the ta ngential components converging in addition in H-1 (omega). Finally, we show that the averages 1/2 integral-1/-1 u(i) dx3 solve the classical bi-dimensional problem of a ''membrane-dominated'' shell, whose equat ions, posed over the space H-0(1) (omega) x H-0(1) (omega) x L2 (omega ), are therefore justified.