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Weakened conditions of admissibility of surface forces applied to linearly elastic membrane shells

By Robert Luce, Cécile Poutous and Jean-marie Thomas


We consider a family of linearly elastic shells of the first kind (as defined in Ciarlet [2]), also known as non inhibited pure bending shells (Sanchez-Hubert and Sanchez-Palencia [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 (Ciarlet and Lods [3], Chapelle and Bathe [1]). 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 (Luce, Poutous and Thomas [5]) : find h αβ = h βα in L 2 (ω) such that for all η = (ηi) in V(ω), ∫ ω hiηidω = ∫ ω hαβγαβ(η)dω where ω is a domain of R2, θ is in C 3 (ω, R3) and S = θ(ω) 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 h αβ, we seek sufficient conditions on h for h αβ to exist in L 2 (ω). In order to get them, we impose more regularity to h αβ and boundary conditions. Under these assumptions, we can obtain from the weak formulation a system of PDE with h αβ 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. 1 Introduction an

Year: 2014
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