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

Abstract

We consider as in Parts I and II a family of linearly elastic shells of thickness 2ɛ, all having the same middle surfaceS=ϕ(ϖ)⊂R ³, whereω⊂R ² is a bounded and connected open set with a Lipschitz-continuous boundary, andϕ ∈ ℒ³ (ϖ;R ³). The shells are clamped on a portion of their lateral face, whose middle line isϕ(γ ₀), whereγ ₀ is a portion of∂ω withlength γ ₀>0. For allɛ>0, let$\zeta _i^\varepsilon$ denote the covariant components of the displacement$u_i^\varepsilon g^{i,\varepsilon }$ of the points of the shell, obtained by solving the three-dimensional problem; let$\zeta _i^\varepsilon$ denote the covariant components of the displacement$\zeta _i^\varepsilon$ a ⁱ of the points of the middle surfaceS, obtained by solving the two-dimensional model ofW.T. Koiter, which consists in finding

$$\zeta ^\varepsilon = \left( {\zeta _i^\varepsilon } \right) \in V_K (\omega ) = \left\{ {\eta = (\eta _\iota ) \in {\rm H}^1 (\omega ) \times H^1 (\omega ) \times H^2 (\omega ); \eta _i = \partial _v \eta _3 = 0 on \gamma _0 } \right\}$$

such that

$$\begin{gathered} \varepsilon \mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \gamma _{\sigma \tau } (\zeta ^\varepsilon )\gamma _{\alpha \beta } (\eta )\sqrt a dy + \frac{{\varepsilon ^3 }}{3} \mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \rho _{\sigma \tau } (\zeta ^\varepsilon )\rho _{\alpha \beta } (\eta )\sqrt a dy \hfill \\ = \mathop \smallint \limits_\omega p^{i,\varepsilon } \eta _i \sqrt a dy for all \eta = (\eta _i ) \in V_K (\omega ), \hfill \\ \end{gathered}$$

where$a^{\alpha \beta \sigma \tau }$ are the components of the two-dimensional elasticity tensor ofS,$\gamma _{\alpha \beta }$(η) and$\rho _{\alpha \beta }$(η) are the components of the linearized change of metric and change of curvature tensors ofS, and$p^{i,\varepsilon }$ are the components of the resultant of the applied forces.

Under the same assumptions as in Part I, we show that the fields$\frac{1}{{2_\varepsilon }}\smallint _{ - \varepsilon }^\varepsilon u_i^\varepsilon g^{i,\varepsilon } dx_3^\varepsilon$ and$\zeta _i^\varepsilon$ a ⁱ, both defined on the surfaceS, have the same principal part asɛ → 0, inH ¹ (ω) for the tangential components, and inL ²(ω) for the normal component; under the same assumptions as in Part II, we show that the same fields again have the same principal part asɛ → 0, inH ¹ (ω) for all their components. For “membrane” and “flexural” shells, the two-dimensional model ofW.T. Koiter is therefore justified.