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

We consider as in Parts I and II a family of linearly elastic shells of thickness 2?, all having the same middle surfaceS=?(?)?R ³, whereω?R ² is a bounded and connected open set with a Lipschitz-continuous boundary, and? ∈ ?³ (?;R ³). The shells are clamped on a portion of their lateral face, whose middle line is?(γ ₀), whereγ ₀ is a portion of?ω withlength γ ₀>0. For all?>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 as? → 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 as? → 0, inH ¹ (ω) for all their components. For “membrane” and “flexural” shells, the two-dimensional model ofW.T. Koiter is therefore justified.