An estimate of the H1-norm of deformations in terms of the L1-norm of their Cauchy–Green tensors

Philippe G. Ciarlet; Cristinel Mardare

doi:10.1016/j.crma.2004.01.014

Mathematical Problems in Mechanics/Differential Geometry

An estimate of the H¹-norm of deformations in terms of the L¹-norm of their Cauchy–Green tensors
[Un majorant de la norme H¹ des déformations en fonction de la norme L¹ de leurs tenseurs de Cauchy–Green]

Présenté par : Robert Dautray

Philippe G. Ciarlet ¹ ; Cristinel Mardare ²

¹ Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
² Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France

Comptes Rendus. Mathématique, Volume 338 (2004) no. 6, pp. 505-510.

Résumé
Abstract

Soit $Ω$ un ouvert borné connexe de $ℝ^{n}$ à frontière lipschitzienne et soit $Θ \in 𝒞^{1} (\bar{Ω}; ℝ^{n})$ une déformation de l'ensemble $\bar{Ω}$ satisfaisant $dét \nabla Θ > 0$ dans $\bar{Ω}$ . On établit l'existence d'une constante $C (Θ)$ ayant la propriété suivante : quelle que soit la déformation $Φ \in H^{1} (Ω; ℝ^{n})$ satisfaisant $dét \nabla Φ > 0$ p.p. dans $Ω$ , il existe une matrice n×n de rotation $𝐑$ et un vecteur $𝐛 \in ℝ^{n}$ tels que

{∥ Φ - (𝐛 + 𝐑 Θ) ∥}_{𝐇^{1} (Ω)} ⩽ C (Θ) {∥ \nabla Φ^{T} \nabla Φ - \nabla Θ^{T} \nabla Θ ∥}_{𝐋^{1} (Ω)}^{1 / 2} .

La démonstration repose en particulier sur un « lemme de rigidité géométrique » fondamental, récemmment établi par G. Friesecke, R.D. James, et S. Müller.

Let $Ω$ be a bounded open connected subset of $ℝ^{n}$ with a Lipschitz-continuous boundary and let $Θ \in 𝒞^{1} (\bar{Ω}; ℝ^{n})$ be a deformation of the set $\bar{Ω}$ satisfying $\det \nabla Θ > 0$ in $\bar{Ω}$ . It is established that there exists a constant $C (Θ)$ with the following property: for each deformation $Φ \in H^{1} (Ω; ℝ^{n})$ satisfying $\det \nabla Φ > 0$ a.e. in $Ω$ , there exist an n×n rotation matrix $𝐑 = 𝐑 (Φ, Θ)$ and a vector $𝐛 = 𝐛 (Φ, Θ)$ in $ℝ^{n}$ such that

{∥ Φ - (𝐛 + 𝐑 Θ) ∥}_{𝐇^{1} (Ω)} ⩽ C (Θ) {∥ \nabla Φ^{T} \nabla Φ - \nabla Θ^{T} \nabla Θ ∥}_{𝐋^{1} (Ω)}^{1 / 2} .

The proof relies in particular on a fundamental ‘geometric rigidity lemma’, recently proved by G. Friesecke, R.D. James, and S. Müller.

Reçu le : 2004-01-14
Accepté le : 2004-01-14
Publié le : 2004-02-25

DOI : 10.1016/j.crma.2004.01.014

Affiliations des auteurs :

Philippe G. Ciarlet ¹ ; Cristinel Mardare ²

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     author = {Philippe G. Ciarlet and Cristinel Mardare},
     title = {An estimate of the {\protect\emph{H}\protect\textsuperscript{1}-norm} of deformations in terms of the {\protect\emph{L}\protect\textsuperscript{1}-norm} of their {Cauchy{\textendash}Green} tensors},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {505--510},
     publisher = {Elsevier},
     volume = {338},
     number = {6},
     year = {2004},
     doi = {10.1016/j.crma.2004.01.014},
     language = {en},
}

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Philippe G. Ciarlet; Cristinel Mardare. An estimate of the H¹-norm of deformations in terms of the L¹-norm of their Cauchy–Green tensors. Comptes Rendus. Mathématique, Volume 338 (2004) no. 6, pp. 505-510. doi : 10.1016/j.crma.2004.01.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.01.014/

Bibliographie
Cité par

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[6] P.G. Ciarlet; C. Mardare Recovery of a manifold with boundary and its continuity as a function of its metric tensor, J. Math. Pures Appl. (2004) (in press)

[7] P.G. Ciarlet, C. Mardare, A surface as a function of its two fundamental forms, in preparation

[8] P.G. Ciarlet, C. and Mardare, Continuity of a deformation in H¹ as a function of its Cauchy–Green tensor in L¹, in preparation

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