Asymptotic Analysis of Linearly Elastic Shells: 'Generalized Membrane Shells'

摘要

We consider a family of linearly elastic shells indexed by their half-thickness ε, all having the same middle surface S = φ(ω), with φ : ω is contained in R~2 →R~3, and clamped along a portion of their lateral face whose trace on S is φ(γ_0), where γ_0 is a fixed portion of partial deriv ω with length γ_0 > 0. Let (γ_(αβ)(η)) be the linearized strain tensor of S. We make an essential geometric and kinematic assumption, according to which the semi-norm ∣·∣ _ω~M defined by ∣η╚O_ω~M = {Σ_(α,β)‖γ_(αβ)(η)‖_(L~2(ω))~2}~(1/2) is a norm over the space V(ω) = {η ∈ H~1(ω);η = o on γ_0}, excluding however the already analyzed 'membrane' shells, where γ_0 = partial deriv ω and S is elliptic. This new assumption is satisfied for instance if γ_0 ≠ partial deriv ω and S is elliptic, or if S is a portion of a hyperboloid of revolution. We then show that, as ε → 0, the averages 1/(2ε) ∫_(-ε)~ε u_i~ε dx_3~ε across the thickness of the shell of the covariant components u_i~ε of the displacement of the points of the shell strongly converge in the completion V_M~# (ω) of V(ω) with respect to the norm ∣·∣_ω~M, toward the solution of a 'generalized membrane' shell problem. This convergence result also justifies the recent formal asymptotic approach of D. Caillerie and E. Sanchez-Palencia.