Rotation fields and the fundamental theorem of Riemannian geometry in $ {\mathbb{R}}^{3}$

Philippe G. Ciarlet; Liliana Gratie; Oana Iosifescu; Cristinel Mardare; Claude Vallée

doi:10.1016/j.crma.2006.08.007

Differential Geometry/Mathematical Problems in Mechanics

Rotation fields and the fundamental theorem of Riemannian geometry in

R^{3}

[Champs de rotations et le théorème fondamental de la géométrie riemannienne dans]

Présenté par : Philippe G. Ciarlet

Philippe G. Ciarlet ¹ ; Liliana Gratie ² ; Oana Iosifescu ³ ; Cristinel Mardare ⁴ ; Claude Vallée ⁵

¹ Department of Mathematics, City University of Hong Kong, 83, Tat Chee Avenue, Kowloon, Hong Kong
² Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, 83, Tat Chee Avenue, Kowloon, Hong Kong
³ Départment de mathématiques, université de Montpellier II, place Eugène-Bataillon, 34095 Montpellier cedex 5, France
⁴ Laboratoire Jacques-Louis Lions, université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France
⁵ Laboratoire de mécanique des solides, université de Poitiers, 86962 Futuroscope-Chasseneuil cedex, France

Comptes Rendus. Mathématique, Volume 343 (2006) no. 6, pp. 415-421.

Résumé
Abstract

$R^{3}$ . Soit Ω un ouvert simplement connexe de $R^{3}$ . On montre dans cette Note que, si un champ suffisamment régulier U de matrices symétriques définies positives d'ordre trois satisfait la relation de compatibilité (due à C. Vallée)

CURL Λ + COF Λ = 0 dans Ω,

où le champ Λ de matrices est défini en fonction du champ U par

Λ = \frac{1}{\det U} {U {(CURL U)}^{T} U - \frac{1}{2} (tr [U {(CURL U)}^{T}]) U},

alors il existe, typiquement dans des espaces tels que

W_{loc}^{2, \infty} (Ω; R^{3})

C^{2} (Ω; R^{3})

, une immersion

Θ : Ω \to R^{3}

telle que

U^{2} = {\nabla Θ}^{T} \nabla Θ

in Ω. Dans cette approche, on cherche à identifier directement la factorisation polaire

\nabla Θ = RU

du gradient de l'immersion inconnue Θ en une rotation R et une extension pure

U = C^{1 / 2}

Let Ω be a simply-connected open subset of $R^{3}$ . We show in this Note that, if a smooth enough field U of symmetric and positive-definite matrices of order three satisfies the compatibility relation (due to C. Vallée)

CURL Λ + COF Λ = 0 in Ω,

where the matrix field Λ is defined in terms of the field U by

Λ = \frac{1}{\det U} {U {(CURL U)}^{T} U - \frac{1}{2} (tr [U {(CURL U)}^{T}]) U},

then there exists, typically in spaces such as

W_{loc}^{2, \infty} (Ω; R^{3})

C^{2} (Ω; R^{3})

, an immersion

Θ : Ω \to R^{3}

such that

U^{2} = {\nabla Θ}^{T} \nabla Θ

in Ω. In this approach, one directly seeks the polar factorization

\nabla Θ = RU

of the gradient of the unknown immersion Θ in terms of a rotation R and a pure stretch U.

Reçu le : 2006-08-15
Publié le : 2006-09-15

DOI : 10.1016/j.crma.2006.08.007

Affiliations des auteurs :

Philippe G. Ciarlet ¹ ; Liliana Gratie ² ; Oana Iosifescu ³ ; Cristinel Mardare ⁴ ; Claude Vallée ⁵

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     author = {Philippe G. Ciarlet and Liliana Gratie and Oana Iosifescu and Cristinel Mardare and Claude Vall\'ee},
     title = {Rotation fields and the fundamental theorem of {Riemannian} geometry in $ {\mathbb{R}}^{3}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {415--421},
     publisher = {Elsevier},
     volume = {343},
     number = {6},
     year = {2006},
     doi = {10.1016/j.crma.2006.08.007},
     language = {en},
}

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Philippe G. Ciarlet; Liliana Gratie; Oana Iosifescu; Cristinel Mardare; Claude Vallée. Rotation fields and the fundamental theorem of Riemannian geometry in $ {\mathbb{R}}^{3}$. Comptes Rendus. Mathématique, Volume 343 (2006) no. 6, pp. 415-421. doi : 10.1016/j.crma.2006.08.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.08.007/

Bibliographie
Cité par

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[2] P.G. Ciarlet An Introduction to Differential Geometry with Applications to Elasticity, Springer, Dordrecht, 2005

[3] P.G. Ciarlet, L. Gratie, O. Iosifescu, C. Mardare, C. Vallée, Another approach to the fundamental theorem of Riemannian geometry in $R^{3}$ , by way of rotation fields, J. Math. Pures Appl., in press

[4] D.G.B. Edelen A new look at the compatibility problem of elasticity theory, Internat. J. Engrg. Sci., Volume 28 (1990), pp. 23-27

[5] B. Fraeijs de Veubeke A new variational principle for finite elastic displacements, Internat. J. Engrg. Sci., Volume 10 (1972), pp. 745-763

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[7] A. Hamdouni Interprétation géométrique de la décomposition des équations de compatibilité en grandes déformations, C. R. Acad. Sci. Paris, Sér. IIb, Volume 328 (2000), pp. 709-712

[8] S. Mardare The fundamental theorem of surface theory for surfaces with little regularity, J. Elasticity, Volume 73 (2003), pp. 251-290

[9] W. Pietraszkiewicz; J. Badur Finite rotations in the description of continuum deformation, Internat. J. Engrg. Sci., Volume 21 (1983), pp. 1097-1115

[10] R.T. Shield The rotation associated with large strains, SIAM J. Appl. Math., Volume 25 (1973), pp. 483-491

[11] J.C. Simo; J.E. Marsden On the rotated stress tensor and the material version of the Doyle–Ericksen formula, Arch. Rational Mech. Anal., Volume 86 (1984), pp. 213-231

[12] T.Y. Thomas Systems of total differential equations defined over simply connected domains, Ann. of Math., Volume 35 (1934), pp. 730-734

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