ASYMPTOTIC ANALYSIS OF LINEARLY ELASTIC SHELLS .2. JUSTIFICATION OF FLEXURAL SHELL EQUATIONS

We consider as in Part I a family of linearly elastic shells of thickn ess 2 epsilon, all having the same middle surface S = phi(<(omega)over bar>) subset of R(3), where omega subset of R(2) is a bounded and con nected open set with a Lipschitz-continuous boundary, and phi is an el ement of C-3(<(omega)over bar>;R(3)). The shells are clamped on a port ion of their lateral face, whose middle line is phi(gamma(0)), where g amma(0) is any portion of partial derivative omega with length gamma(0 ) > 0. We make an essential geometrical assumption on the middle surfa ce S and on the set gamma(0), which states that the space of inextensi onal displacements V-F(omega) = {(eta = (eta(i)) is an element of H-1( omega) x H-1(omega) x H-2(omega); eta(i) = partial derivative(nu)eta(3 ) = 0 on gamma(0), gamma(alpha beta)(eta) = 0 in omega}, where gamma(a lpha beta)(eta) are the components of the linearized change is metric tensor of S, contains non-zero functions. This assumption is satisfied in particular if S is a portion of cylinder and phi(gamma(0)) is cont ained in a generatrix of S. We show that, if the applied body force de nsity is O(epsilon(2)) with respect to epsilon, the field u(epsilon) = (u(i)(epsilon)), where u(i)(epsilon) denote the three covariant compo nents of the displacement of the points of the shell given by the equa tions of three-dimensional elasticity, once ''scaled'' so as to be def ined over the fixed domain Omega = omega x ] - 1, 1[, converges as eps ilon --> 0 in H-1(Omega) to a limit u, which is independent of the tra nsverse variable. Furthermore, the average zeta = 1/2 integral(-1)(1) udx(3), which belongs to the space V-F(omega), satisfies the (scaled) two-dimensional equations of a ''flexural shell'', viz., [GRAPHICS] fo r all eta = (eta(i)) is an element of V-F(omega), where a(alpha beta s igma tau) are the components of the two-dimensional elasticity tensor of the surface S, rho(alpha beta)(eta) = partial derivative(alpha beta eta 3) - Gamma(alpha beta)(sigma)partial derivative(sigma)eta(3) + b( beta)(sigma)(partial derivative(alpha)eta(sigma) - Gamma(alpha sigma)( tau)eta(tau)) + b(alpha)(sigma)(partial derivative(beta)eta(sigma) - G amma(beta sigma)(tau)eta(tau)) + b(alpha)(sigma)\(beta)eta(sigma) - c( alpha beta)eta(3) are the components of the linearized change of curva ture tensor of S, Gamma(alpha beta)(sigma) are the Christoffel symbols of S, b(alpha)(beta) are the mixed components of the curvature tensor of S, and f(i) are the scaled components of the applied body force. U nder the above assumptions, the two-dimensional equations of a ''flexu ral shell'' are therefore justified.