Recovery of a displacement ﬁeld from its linearized strain

536 P G Ciarlet et al / C R Acad Sci Paris, Ser I 344 (2007) 535–540 metric inherited from the surrounding space R3, the immersion Θ induces a Riemannian metric on Ω, deﬁned by its

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C. R. Acad. Sci. Paris, Ser. I 344 (2007) 535-540

Mathematical Problems in Mechanics

Recovery of a displacement field from its linearized strain tensor field in curvilinear coordinates

Philippe G. Ciarlet

a , Cristinel Mardare b , Ming Shen a a Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kongb

Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, Place Jussieu, 75005 Paris, France

Received and accepted 2 March 2007

Presented by Philippe G. Ciarlet

Abstract

We establish that, if a symmetric matrix field defined over a simply-connected open set satisfies the Saint Venant equations in

curvilinear coordinates, then its coefficients are the linearized strains associated with a displacement field. Our proof provides an

explicit algorithm for recovering such a displacement field, which may be viewed as the linear counterpart of the reconstruction of

an immersion from a given flat Riemannian metric.To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).?2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Résumé

Reconstruction d"un champ de déplacements à partir de son tenseur des déformations linéarisées en coordonnées curvi-

lignes.Nous montrons que, si un champ de matrices symétriques défini sur un ouvert simplement connexe vérifie les équations de

Saint Venant en coordonnées curvilignes, alors c"est le tenseur des déformations linéarisées associé à un champ de déplacements.

Notre démonstration fournit un algorithme explicite de reconstruction d"un tel champ de déplacements, qui peut être considéré

comme la version linéarisée de la reconstruction d"une immersion à partir d"une métrique riemannienne plate.Pour citer cet

1. Notations and preliminaries

Latin indices and exponents vary in the set

{1,2,3}and the summation convention with respect to repeated indices and exponents is systematically used in conjunction with this rule.

LetΩbe an open subset ofR3

and let there be given an immersionΘ?C 3 (;R 3 ). For eachx=(x i)?Ω,the three vectorsg i (x):=∂ i

Θ(x), where∂

i :=∂/∂x i , form a basis in the tangent space, identified here withR 3 ,tothe manifoldΘ(Ω)at the pointΘ(x). The vector fieldsgj , defined byg i (x)·g j (x)=δ j i for allx?Ω,form the dual basis of the basis formed by the vector fieldsg i

. The manifoldΘ(Ω)being naturally endowed with the EuclideanE-mail addresses:mapgc@cityu.edu.hk (P.G. Ciarlet), mardare@ann.jussieu.fr (C. Mardare), geoffrey.shen@student.cityu.edu.hk (M. Shen).

1631-073X/$ - see front matter?2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.crma.2007.03.012

536P.G. Ciarlet et al. / C. R. Acad. Sci. Paris, Ser. I 344 (2007) 535-540

metric inherited from the surrounding spaceR 3 , the immersionΘinduces a Riemannian metric onΩ, defined by its covariant components g ij (x)=g i (x)·g j (x)for allx?Ω. The contravariant components of this metric are defined byg k? (x)=g k (x)·g (x),or equivalently, by(g k? (x))= (g ij (x)) -1

for allx?Ω. This metric induces the Levi-Civita connection in the manifoldΩ, defined by the Christoffel

symbols kij :=1 2g k? i g j? j g i? g ij kji inΩ. Note that the regularity assumption on the immersionΘimplies thatg ij ,g k? ?C 2 ()and thatΓ kij ?C 1 ().The covariant derivatives of the covariant componentsu i ?H 1 (Ω)of a vector fieldu i g i are defined by u j?i i u j kij u k The covariant derivatives of the covariant componentsT ij ?L 2 (Ω)of a second-order tensor field are defined by T ij?k k T ij ?ki T ?j ?kj T i? and they belong to the spaceH -1 (Ω). The covariant derivatives of the covariant componentsT ijk ?H -1 (Ω)of a third-order tensor field are defined by T ijk?? T ijk t?i T tjk t?j T itk t?k T ijt and they belong to the spaceH -2 (Ω).IfT ij ?L 2 (Ω), the second-order covariant derivativesT ij?k? are defined by the relations T ij?k? T ij?k t?i T tj?k t?j T it?k t?k T ij?t =T ij??k

AdomaininR

3 is a bounded and connected open setΩwith a Lipschitz-continuous boundary, the setΩbeing locally on the same side of its boundary. Detailed proofs of the results announced in this Note are given in [4].

2. Poincaré theorem in curvilinear coordinates

Poincaré"s Theorem, which is classically proved only for continuously differentiable functions, was generalized by

Ciarlet and Ciarlet, Jr. [1] as follows:

Theorem 2.1.LetΩbe a simply connected domain ofR 3 . Leth k ?H -1 (Ω)be distributions that satisfy h k k h inH -2 (Ω).Then there exists a functionp?L 2 (Ω), unique up to an additive constant, such that h k k pinH -1 Clearly, this theorem remains valid if the functionsh k are replaced by matrix fieldsH k with componentsh ijk in H -1 (Ω), the functionpbeing then replaced by a matrix fieldPwith componentsp ij inL 2 (Ω). Using Theorem 2.1,

one can then show that a similar Poincaré theorem in curvilinear coordinates" holds as well. The mappingΘis that

introduced in Section 1. Theorem 2.2.LetΩbe a simply connected domain ofR 3 and letΘ?C 3 (;R 3 )be an immersion. LetH k be matrix fields with componentsh ijk ?H -1 (Ω)satisfying h ijk?? =h ij??k inH -2

Then there exist a matrix fieldPwith componentsp

ij ?L 2 (Ω), unique up to additive constants, such that h ijk =p ij?k inH -1

Note that Theorem 2.2 can also be established as a consequence of Theorem A.4 of [6] establishing the existence

of weak solutions to Pfaff systems, of which the equationsp ij? k =h ijk constitutes a special case. P.G. Ciarlet et al. / C. R. Acad. Sci. Paris, Ser. I 344 (2007) 535-540537

3. Saint Venant equations in curvilinear coordinates

LetΩbe a bounded open subset ofR

3 and letΘ?C 3 (;R 3 )be an immersion. The vector fieldsg i ?C 2 (;R 3 andg i ?C 2 (;R 3 )are defined as in Section 1. With every vector fieldu?H 1 (Ω;R 3 ), we associate thecovariant

components of the linearized change of metric tensor, also known as thelinearized strains in curvilinear coordinates,

defined by ij (u):=1

2(∂

i u·g j +g i j u).

Note thatε

ij (u)?L 2 (Ω)for alli,jand thatε ij (u)=ε ji (u).

The next theorem shows that the functionsε

ij (u)satisfy crucialcompatibility relations, which constitute theSaint

Venant equations in curvilinear coordinates, since they generalize the well-known Saint Venant equations in Cartesian

coordinates. The proof rests on various computations involving derivatives in the distributional sense.

Theorem 3.1.The linearized strains in curvilinear coordinatesε ij (u)?L 2 (Ω)associated with a vector field u?H 1 (Ω;R 3 )satisfy the relations ki?j? (u)+ε ?j?ik (u)-ε kj?i? (u)-ε ?i?jk (u)=0inH -2

4. Recovery of a vector field from the associated linearized change of metric tensor

We now characterize the space of all symmetric matrix fields that satisfy the Saint Venant equations in curvilinear

coordinates found in Theorem 3.1. Theorem 4.1.LetΩbe a simply-connected domain inR 3 and letΘ?C 3 (;R 3 )be an immersion. Let there be given a symmetric matrix field(e ij )?L 2 (Ω;S 3 )that satisfies the Saint Venant equations in curvilinear coordinates e ki?j? +e ?j?ik -e kj?i? -e ?i?jk =0inH -2

Then there exists a vector fieldv?H

1 (Ω;R 3 )such that e ij =1

2(∂

i v·g j +g i j v)inL 2

Sketch of proof.Since the Saint Venant equations are satisfied, Theorem 2.2 shows that there exist functions

a ij ?L 2 (Ω), unique up to additive constants, such thata ij?kquotesdbs_dbs14.pdfusesText_20

[PDF] Recovery of a displacement ﬁeld from its linearized strain