The infinitesimal rigid displacement lemma in Lipschitz co‐ordinates and application to shells with minimal regularity

Anicic, Sylvia; Dret, Hervé Le; Raoult, A.

doi:10.1002/mma.501

Cited by 22 publications

(23 citation statements)

References 26 publications

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“…Both play a central rôle in the existence theory for linear two-dimensional plate equations (see, e.g., Ciarlet [1997, Theorems 1.5-1 and 1.5-2]). As shown by Blouza & Le Dret [1999], Le Dret [2004], and Anicic, Le Dret & Raoult [2005], the regularity assumptions made on the mapping θ and on the field η = (η i ) in both the infinitesimal rigid displacement lemma and the Korn inequality on a surface of Theorems 4.3-3 and 4.3-4 can be substantially weakened.…”

Section: Theorem 43-4mentioning

confidence: 96%

An Introduction to Differential Geometry with Applications to Elasticity

2005

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Section: Theorem 43-4mentioning

confidence: 96%

An Introduction to Differential Geometry with Applications to Elasticity

2005

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“…The so called 'infinitesimal rigid displacement lemma in curvilinear coordinates', a version of which can be found in [1] and which is important for linear elasticity in curvilinear coordinates (see also [2,4]) states the following: If Ω ⊂ R N is a bounded domain, Ψ ∈ W 1,∞ (Ω; R N ) satisfying det ∇Ψ ≥ c + > 0 a.e. and Φ ∈ H 1 (Ω; R N ) with sym(∇Φ ⊤ ∇Ψ) = 0 a.e., then on a dense open subset of Ω there exist locally constant mappings a : Ω → R N and A : Ω → so(N ) such that locally Φ = AΨ + a.…”

Section: Introductionmentioning

confidence: 99%

Uniqueness of Integrable Solutions ∇ζ = Gζ, ζ∣_Γ = 0 for Integrable Tensor‐Coefficients G and Applications to Elasticity

Lankeit

Neff

Pauly

2013

Proc Appl Math and Mech

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Let Ω ⊂ R N be bounded Lipschitz and ∅ = Γ ⊂ ∂Ω relatively open. We show that the solution to the linear first order system) for some p, q > 1 with 1/p + 1/q = 1 and det P ≥ c + > 0. We give a new proof for the so called 'in-finitesimal rigid displacement lemma' in curvilinear coordinates:Then there are a ∈ R 3 and a constant skew-symmetric matrix A ∈ so(3), such that Φ = AΨ + a.

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“…As shown by Blouza & Le Dret [13], Le Dret [61], and Anicic, Le Dret & Raoult [7], the regularity assumptions made on the mapping θ and on the field η in both the infinitesimal rigid displacement lemma and the Korn inequality on a surface of Theorems 2.9-2 and 2.9-3 can be substantially weakened.…”

Section: Korn's Inequalities On a Surfacementioning

confidence: 98%

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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The infinitesimal rigid displacement lemma in Lipschitz co‐ordinates and application to shells with minimal regularity

Abstract: SUMMARYWe establish a version of the inÿnitesimal rigid displacement lemma in curvilinear Lipschitz coordinates. We give an application to linearly elastic shells whose midsurface and normal vector are both Lipschitz.

Cited by 22 publications

References 26 publications

An Introduction to Differential Geometry with Applications to Elasticity

An Introduction to Differential Geometry with Applications to Elasticity

Uniqueness of Integrable Solutions ∇ζ = Gζ, ζ∣_Γ = 0 for Integrable Tensor‐Coefficients G and Applications to Elasticity

An Introduction to Shell Theory

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The infinitesimal rigid displacement lemma in Lipschitz co‐ordinates and application to shells with minimal regularity

Abstract: SUMMARYWe establish a version of the inÿnitesimal rigid displacement lemma in curvilinear Lipschitz coordinates. We give an application to linearly elastic shells whose midsurface and normal vector are both Lipschitz.

Cited by 22 publications

References 26 publications

An Introduction to Differential Geometry with Applications to Elasticity

An Introduction to Differential Geometry with Applications to Elasticity

Uniqueness of Integrable Solutions ∇ζ = Gζ, ζ∣Γ = 0 for Integrable Tensor‐Coefficients G and Applications to Elasticity

An Introduction to Shell Theory

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Uniqueness of Integrable Solutions ∇ζ = Gζ, ζ∣_Γ = 0 for Integrable Tensor‐Coefficients G and Applications to Elasticity