ASYMPTOTIC ANALYSIS OF LINEARLY ELASTIC SHELLS - GENERALIZED MEMBRANESHELLS

We consider a family of linearly elastic shells indexed by their half- thickness epsilon, all having the same middle surface S = rho(omega), with rho:omega subset of R(2) --> R(3), and clamped along a portion of their lateral face whose trace on S is rho rho(gamma(0)), where gamma (0) is a fixed portion of partial derivative omega with length gamma(0 ) > 0. Let (gamma alpha beta(eta)) be the linearized strain tensor of S. We make an essential geometric and kinematic assumption, according to which the semi-norm \.\(omega)M = {Sigma(alpha,beta)\\gamma(alpha b eta)(eta)\\(L2(omega))(2)}(1/2) is a norm over the space V(omega) = {e ta is an element of H-1(omega); eta = 0 on gamma(0)}, excluding howeve r the already analyzed 'membrane' shells, where gamma(0) = partial der ivative omega and S is elliptic. This new assumption is satisfied for instance if gamma(0) not equal partial derivative omega and S is ellip tic, or if S is a portion of a hyperboloid of revolution. We then show that, as epsilon --> 0, the averages 1/2 epsilon integral(-epsilon)(e psilon) u(i)(epsilon)dx(3)(epsilon) across the thickness of the shell of the covariant components u(i)(epsilon) of the displacement of the p oints of the shell strongly converge in the completion V-m(#)(omega) o f V(omega) with respect to the norm \.\(M)(omega), toward the solution of a 'generalized membrane' shell problem. This convergence result al so justifies the recent formal asymptotic approach of D. Caillerie and E. Sanchez-palencia. The limit problem found in this fashion is 'sens itive', according to the terminology recently introduced by J.L. Lions and E. Sanchez-Palencia, in the sense that it possesses two unusual f eatures: it is posed in a space that is not necessarily contained in a space of distributions, and its solution is 'highly sensitive' to arb itrarily small smooth perturbations of the data. Under the same assump tion, we also show that the average 1/2 epsilon integral(-epsilon)(eps ilon)udx(3)(epsilon) = (u(i)(epsilon)), and the solution xi epsilon is an element of V-K(omega) of Koiter's equations have the same principa l part as epsilon --> 0 in the same space V-M(#)(omega) as above. For such 'generalized membrane' shells, the two-dimensional shell model of W.T. Koiter is thus likewise justified.