Asymptotic analysis of linearly elastic shells. III. Justification of Koiter's shell equations

Abstract

We consider as in Part I a family of linearly elastic shells of thickness 2ɛ, all having the same middle surfaceS=ϕ(ϖ)⊂R3, whereω⊂R2 is a bounded and connected open set with a Lipschitz-continuous boundary, andϕ∈l3 (ϖ;R3). The shells are clamped on a portion of their lateral face, whose middle line isϕ(γ0), whereγ0 is any portion of∂ω withlength γ0>0. We make an essential geometrical assumption on the middle surfaceS and on the setγ0, which states that the space of inextensional displacements $$\begin{gathered} V_F (\omega ) = \{ \eta = (\eta _i ) \in H^1 (\omega ) \times H^1 (\omega ) \times H^2 (\omega ); \hfill \\ \eta _i = \partial _v \eta _3 = 0 on \gamma _0 ,\gamma _{\alpha \beta } (\eta ) = 0 in \omega \} , \hfill \\ \end{gathered}$$ where\(\gamma _{\alpha \beta }\)(η) are the components of the linearized change is metric tensor ofS, contains non-zero functions. This assumption is satisfied in particular ifS is a portion of cylinder andϕ(γ0) is contained in a generatrix ofS.