ASYMPTOTIC ANALYSIS OF LINEARLY ELASTIC SHELLS .1. JUSTIFICATION OF MEMBRANE SHELL EQUATIONS

We consider a family of linearly elastic shells with thickness 2 epsil on, clamped along their entire lateral face, all having the same middl e surface S = phi(<(omega)over bar>) subset of R(3), where omega subse t of R(2) is abounded and connected open set with a Lipschitz-continuo us boundary gamma, and phi is an element of C-3(<(omega)over bar>R(3)) . We make an essential geometrical assumption on the middle surface S, which is satisfied if gamma and phi are smooth enough and S is ''unif ormly elliptic'', in the sense that the two principal radii of curvatu re are either both > 0 at all points of S, or both < 0 at all points o f S. We show that, if the applied body force density is 0(1) with resp ect to epsilon, the field u(epsilon) = ((u)i(epsilon)), where u(i)(eps ilon) denote the three covariant components of the displacement of the points of the shell given by the equations of three-dimensional elast icity, once ''scaled'' so as to be defined over the fixed domain Omega = omega x] - 1, 1[, converges in H-1(Omega) x H-1(Omega) x L(2)(Omega ) as epsilon --> 0 to a limit u, which is independent of the transvers e variable. Furthermore, the average zeta = 1/2 integral(-1)(1)udx(3), which belongs to the space V-M(omega) = H-0(1)(omega) x H-0(1)(omega) x L(2)(omega) satisfies the (scaled) two-dimensional equations of a ' 'membrane shell'' viz.,[GRAPHICS] for all eta = (eta(i)) is an element of V-m(omega), where a(alpha beta sigma tau) are the components of th e two-dimensional elasticity tensor of the surface S, gamma(alpha beta )(eta) = 1/2(partial derivative(alpha)eta(beta) + partial derivative(b eta)eta(alpha)) - Gamma(alpha beta)(sigma)eta(sigma) - b (alpha beta)e ta(3) are the components of the linearized change of metric tensor of S, Gamma(alpha beta)(sigma) are the Christoffel symbols of S, b(alpha beta) are the components of the curvature tensor of S, and f(i) are th e scaled components of the applied body force. Under the above assumpt ions, the two-dimensional equations of a ''membrane shell'' are theref ore justified.