Asymptotic analysis of linearly elastic shells .1. Justification of membrane shell equations

We consider a family of linearly elastic shells with thickness 2 epsilon, clamped along their entire lateral face, all having the same middle surface S = phi(<(omega)over bar>) subset of R(3), where omega subset of R(2) is abounded and connected open set with a Lipschitz-continuous 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 ''uniformly elliptic'', in the sense that the two principal radii of curvature are either both > 0 at all points of S, or both < 0 at all points of S. We show that, if the applied body force density is 0(1) with respect to epsilon, the field u(epsilon) = ((u)i(epsilon)), where u(i)(epsilon) denote the three covariant components of the displacement of the points of the shell given by the equations of three-dimensional elasticity, 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 transverse 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 the two-dimensional elasticity tensor of the surface S, gamma(alpha beta)(eta) = 1/2(partial derivative(alpha)eta(beta) + partial derivative(beta)eta(alpha)) - Gamma(alpha beta)(sigma)eta(sigma) - b (alpha beta)eta(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 the scaled components of the applied body force. Under the above assumptions, the two-dimensional equations of a ''membrane shell'' are therefore justified.