Existence and uniqueness for the linear Koiter model for shells with little regularity

Blouza, Adel; Dret, Hervé Le

doi:10.1090/qam/1686192

Cited by 60 publications

(89 citation statements)

References 8 publications

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“…It was later given other proofs by Ciarlet & Miara [33] and Bernadou, Ciarlet & Miara [12]; then by Akian [4] and Ciarlet & S. Mardare [32], who showed that it can be directly derived from the three-dimensional Korn inequality in curvilinear coordinates (this idea goes back to Destuynder [45]); then by Blouza & Le Dret [13], who showed that it still holds under a less stringent smoothness assumption on the mapping θ. We follow here the proof of Bernadou, Ciarlet & Miara [12].…”

Section: Korn's Inequalities On a Surfacementioning

confidence: 98%

“…Together with the expression of the components γ αβ (η) in terms of η (Theorem 2.8-1), this simpler expression was efficiently put to use by Blouza & Le Dret [13], who showed that their principal merit is to afford the definition of the components γ αβ (η) and ρ αβ (η) under substantially weaker regularity assumptions on the mapping θ.…”

Section: The Linear Koiter Shell Modelmentioning

confidence: 99%

“…The culprits responsible for this regularity are the functions b τ β | α appearing in the functions ρ αβ (η). Otherwise Blouza & Le Dret [13] have shown how this regularity assumption on θ can be weakened if only the expressions of γ αβ (η) and ρ αβ (η) in terms of the field η are considered.…”

Section: The Linear Koiter Shell Modelmentioning

confidence: 99%

“…Otherwise these choices can be weakened to accommodate shells whose middle surfaces have little regularity (see Blouza & Le Dret [13]). …”

Section: Equivalently the Vector Fieldmentioning

confidence: 99%

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An Introduction to Shell Theory

Ciarlet

Mardare

2008

Series in Contemporary Applied Mathematics

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Abstract. These notes are intended to provide a thorough introduction to the mathematical theory of elastic shells.The main objective of shell theory is to predict the stress and the displacement arising in an elastic shell in response to given forces. Such a prediction is made either by solving a system of partial differential equations or by minimizing a functional, which may be defined either over a three-dimensional set or over a two-dimensional set, depending on whether the shell is viewed in its reference configuration as a three-dimensional or as a two-dimensional body (the latter being an abstract idealization of the physical shell when its thickness is "small").The first part of this article is devoted to the three-dimensional theory of elastic bodies, from which the three-dimensional theory of shells is obtained simply by replacing the reference configuration of a general body with that of a shell. The particular shape of the reference configuration of the shell does not play any rôle in this theory.The second part is devoted to the two-dimensional theory of elastic shells. In contrast to the three-dimensional theory, this theory is specific to shells, since it essentially depends on the geometry of the reference configuration of a shell.For a more comprehensive exposition of the theory of elastic shells, we refer the reader to Ciarlet [18] and the references therein for the first part of the article, and to Ciarlet [20] and the references therein for the second part. Une introduction a la théorie des coquesResumé. Ces notes sont destinéesà fournir une introduction détailléeà la théorie mathématique des coquesélastiques.L'objectif principal de la théorie des coques est de prédire les contraintes et les dépla-cements survenant dans une coqueélastique en réponseà des forces appliquées. Une telle prédiction est faite soit en résolvant un système d'équations aux dérivées partielles, soit en minimisant une fonctionnelle, qui peuventêtre définis soit dans un ensemble tridimensionnel, soit dans un ensemble bidimensionnel, selon que la coque est vue dans sa configuration de référence comme un corps tridimensionnel ou comme un corps bidimensionnel (ce dernierétant alors une idéalisation de la coque physique lorsque sonépaisseur est "petite").La première partie de cet article est consacréeà la théorie tridimensionnelle des corpś elastiques,à partir de laquelle la théorie tridimensionnelle des coques est obtenue en remplaçant simplement la configuration de référence d'un corpsélastique général par celle d'une coque. La forme particulière de la configuration de référence d'une coque ne joue aucun rôle dans cette théorie.La deuxième partie de cet article est consacréeà la théorie bidimensionnelle des coqueś elastiques. Contrairementà la théorie tridimensionnelle, cette théorie est spécifique aux coques, puisqu'elle dépend de façon essentielle de la géométrie de la configuration de référence de la coque.Pour un exposé plus complet de la théorie de coquesélastiques, nous renvoyons le lecteurà Ciarlet [18] et ses réf...

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Section: Korn's Inequalities On a Surfacementioning

confidence: 98%

Section: The Linear Koiter Shell Modelmentioning

confidence: 99%

Section: The Linear Koiter Shell Modelmentioning

confidence: 99%

“…Otherwise these choices can be weakened to accommodate shells whose middle surfaces have little regularity (see Blouza & Le Dret [13]). …”

Section: Equivalently the Vector Fieldmentioning

confidence: 99%

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An Introduction to Shell Theory

Ciarlet

Mardare

2008

Series in Contemporary Applied Mathematics

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“…It is well known that the Koiter problem (7) has a unique solution u ε ∈ U (see [4] and [10]). We recall (see [16]) that…”

Section: The Shell Problem and The Classification Basic Criteriummentioning

confidence: 99%

A Shell Classification by Interpolation

Baiocchi

Lovadina

2002

Math. Models Methods Appl. Sci.

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The shell problem and its asymptotic are investigated. A connection between the asymptotic behavior and real Interpolation Theory is established. Thus, a detailed study of the cases when neither the bending energy nor the membrane energy dominate is provided. An application to a cylindrical shell is also detailed. Although only the Koiter shells have been considered, the same procedure can be used for other models, such as Naghdi's one, for example.

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On a model of a flexural prestressed shell

Marohnić¹,

Tambača

2015

Math Methods in App Sciences

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We derive a linearized prestressed elastic shell model from a nonlinear Kirchhoff model of elastic plates. The model is given in terms of displacement and micro-rotation of the cross-sections. In addition to the standard membrane, transverse shear, and flexural terms, the model also contains a nonstandard prestress term. The prestress is of the same order as flexural effects; hence, the model is appropriate when flexural effects dominate over membrane ones. We prove the existence and uniqueness of the solutions by Lax-Milgram theorem and compare solution with the solution of the standard shell model via numerical examples.

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Existence and uniqueness for the linear Koiter model for shells with little regularity

Cited by 60 publications

References 8 publications

An Introduction to Shell Theory

An Introduction to Shell Theory

A Shell Classification by Interpolation

On a model of a flexural prestressed shell

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