WEAKENED CONDITIONS OF ADMISSIBILITY OF SURFACE FORCES APPLIED TO LINEARLY ELASTIC MEMBRANE SHELLS

We consider a family of linearly elastic shells of the first kind (as defined in [2]), also known as non inhibited pure bending shells [7]. This family is indexed by the half-thickness ε. When ε approaches zero, the averages across the thickness of the shell of the covariant components of the displacement of the points of the shell converge strongly towards the solution of a "2D generalized membrane shell problem" provided the applied forces satisfy admissibility conditions [1,3]. The identification of the admissible applied forces usually requires delicate analysis. In the first part of this paper, we simplify the general admissibility conditions when applied forces h are surface forces only, and obtain conditions that no longer depend on ε [5]: find hαβ = hαβ in L2(ω) such that for all η = (ηi) in V(ω), ∫ω hi ηi dω = ∫ω hαβγαβ(η)dω where ω is a domain of ℝ2, θ is in [Formula: see text] and [Formula: see text] is the middle surface of the shells, where (γαβ (η)) is the linearized strain tensor of S and V(ω) = {η ∈ H1(ω), η = 0 on γ0}, the shells being clamped along Γ0 = θ(γ0). In the second part, since the simplified admissibility formulation does not allow to conclude directly to the existence of hαβ, we seek sufficient conditions on h for hαβ to exist in L2(ω). In order to get them, we impose more regularity to hαβ and boundary conditions. Under these assumptions, we can obtain from the weak formulation a system of PDE's with hαβ as unknowns. The existence of solutions depends both on the geometry of the shell and on the choice of h. We carry through the study of four representative geometries of shells and identify in each case a special admissibility functional space for h.