ASYMPTOTIC ANALYSIS OF LINEARLY ELASTIC SHELLS .4. SENSITIVE MEMBRANE SHELLS

We consider a family of linearly elastic shells indexed by their half-thickness epsilon, all having the same middle surface S = phi(omega), with phi : omega subset of C R(2) --> R(3), and clamped along a portion of their lateral face whose trace on S is phi (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/./(M)(omega) defined by /(eta)/((omega))(M)= [GRAPHICS] is a norm over the space V(omega) ={(eta) is an element of H-1 (omega); eta = O on gamma(0))}, excluding however the already treated ''membrane'' case, where gamma(0) = partial derivative omega and S is elliptic; this assumption is satisfied for instance if gamma(0) not equal partial derivative omega and S is elliptic, or if S is a portion of a hyperboloid of revolution. We then show that, as epsilon --> 0, the averages across the thickness of the covariant components of the displacement of the points of the shell, strongly converge in the completion of V (omega) for /./(M)(omega), towards the solution of a ''membrane'' variational problem that is ''sensitive'' in the sense recently introduced by J.-L. Lions and E. Sanchez-Palencia.