Véronique Lods

Asymptotic Analysis of Linearly Elastic Shells. III. Justification of Koiter's Shell Equations

AbstractWe 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 % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaieWacG% aGC9NvamacaY1gaaWcbGaGCHqaciacaYLFgbaabKaGCbGccGaGCjik% aiadaYfHjpWDcGaGCjykaiadaYLH9aqpieaajqgaadGaa03EaGGadO% Gamaiu8D7aOHGaaiadac1E9aqpcWaGqThkaGIamaiueE7aOnacac1g% aaWcbGaGqjacacLFPbaabKaGqbGccGaGqjykaiadacLHiiIZcGaGqn% isamacacfhaaWcbKaGqfacacLaiaiuigdaaaGccGaGqjikaiadacfH% jpWDcGaGqjykaiadacLHxdaTcGaGqnisamacacfhaaWcbKaGqfacac% LaiaiuigdaaaGccGaGqjikaiadacfHjpWDcGaGqjykaiadacLHxdaT% cGaGqnisamacacfhaaWcbKaGqfacacLaiaiuikdaaaGccGaGqjikai% adacfHjpWDcGaGqjykaiacacLG7aaabaGaa8hiaiaa-bcacaWFGaGa% a8hiaiaa-bcacaWFGaGaa8hiaiaa-bcacaWFGaGaa8hiaiaa-bcaca% WFGaGaa8hiaiadaseH3oaAdGaGeTbaaSqaiaircGaGe5xAaaqajair% aOGaiair-1dacWaGezOaIy7aiairBaaaleacasKaiGgi+zhaaeqcas% eakiad0HdH3oaAdGaGeTbaaSqaiaircGaGeH4maaqajairaOGaiair% -1dacGaGe9hiaiacasuFWaGaiair-bcacGaGe13BaiacasuFUbGaia% ir+bcacWaGer4SdC2aiairBaaaleacasKaiair+bdaaeqcaseakiac% asKGSaGamaireo7aNnacas0gaaWcbGaGejadCHdHXoqycWax4qOSdi% gabKaGebGccGaxejikaiadCreF3oaAcGaxejykaiadCrKH9aqpcGax% eHimaiacCruFGaGaiWfr9LgacGaxe1NBaiacCruFGaGamWfreM8a3L% azaamacGaxakyFaiacCbOGSaaaaaa!D00E!

The Space of Inextensional Displacements for a Partially Clamped Linearly Elastic Shell with an Elliptic Middle Surface

\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}

Nonlinearly Elastic Shell Models: A Formal Asymptotic Approach II. The Flexural Model

where % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaS% baaSqaaiad4WiHXoqycWaomsOSdigabeaaaaa!3C98!

Error estimates between the linearized three-dimensional shell equations and Naghdi's model

\gamma _{\alpha \beta }

Justification asymptotique des hypothèses de Kirchhoff-Love pour une coque encastrée linéairement élastique

(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.We show that, if the applied body force density isO(ɛ2) with respect toɛ, the fieldu(ɛ)=(ui(ɛ)), whereui(ɛ) denote the three covariant components of the displacement of the points of the shell given by the equations of three-dimensional elasticity, once “scaled” so as to be defined over the fixed domain Ω=ω×]−1, 1[, converges asɛ→0 inH1(Ω) to a limitu, which is independent of the transverse variable. Furthermore, the averageζ=1/2ts−11udx3, which belongs to the spaceVF(ω), satisfies the (scaled) two-dimensional equations of a “flexural shell”, viz., % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaiaiMlaaabG% aGykacaIjIXaaabGaGykacaIjIZaaaamacWY7fqaqaialViiGajqga% GgGamaiGmaafb83kIipajeaObGaS8kadaciF-dpaeM8a3bWcbKaS8c% GccGaGacaaKdWGHbWaiaiGaaa5aWbaaSqajaiGaaa5aeacaciaaqoa% cWaGacaaKdaHXoqycWaGacaaKdaHYoGycWaGacaaKdaHdpWCcWaGac% aaKdaHepaDaaGccWaGacaaKdaHbpGCdGaGacaaKdWgaaWcbGaGacaa% KdGamaiGaaa5aq4WdmNamaiGaaa5aqiXdqhabKaGacaaKdaakiacac% iaaqoacIcaiiWacWaGacaaKdGF2oGEcGaGacaaKdGGPaGamaiGaaa5% aqyWdi3aiaiGaaa5aSbaaSqaiaiGaaa5aiadaciaaqoaeg7aHjadac% iaaqoaek7aIbqajaiGaaa5aaGccGaGacaaKdGGOaGamaiGaaa5aq4T% dGMaiaiGaaa5aiykamacacijaapakaaabGaGascaWdGaiaiGKaa8am% yyaGqaaiacacijaapa9bcaaSqajaiGKaa8aaacbiGccGaGascaWdaF% KbGaiaiGKaa8aWxEaiadaciaaGkag2da9macacipaWjaxababGaGaY% daCcqcKbaObiadaci6-dpa-TIiYdqcbaAaiaiG8aaNaiadacit-Zma% eM8a3bWcbKaGaYdaCcaakmaacmaajqgaGgqaaOWaiaiG8aaibCbmaK% azaaAabGaGaYdaGeGamaiG8aaib83kIipaleacacipaasaiiaacWaG% aIeaydWEsislcWaGaIeaydWEXaqmaeacacipaasacWaGaYmaGeWEXa% qmaaGccGaDSpOzamac0XohaaWcbKaDShac0XUaiqh7dMgaaaGccGaD% Spizaiac0X+G4bWaiqh7Baaaleac0XUaiqh7iodaaeqc0XoaaOGaay% 5Eaiaaw2haaiaa9bcacWaGut4TdG2aiai1BaaaleacasTaiai18Lga% aeqcasnakmacasTcaaqaiai1cGaGupyyaiacasDFGaaaleqcasnaki% acasnFKbGaiai18Lhaaaa!02F0!