# On rigid displacements and their relation to the infinitesimal rigid displacement lemma in three-dimensional elasticity

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1631-073reserveddoi:10.10C. R. Acad. Sci. Paris, Ser. I 336 (2003) 873878

Mathematical Problems in Mechanics

n rigid displacements and their relation to the infinitesimalrigid displacement lemma in three-dimensional elasticity

Philippe G. Ciarlet a, Cristinel Mardare b

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

Received and accepted 4 April 2003

Presented by Robert Dautray

t

be an open connected subset of R3 and let be an immersion from into R3. It is established that the set formed bydisplacements of the open set () is a submanifold of dimension 6 and of class C of the space H1(). It is also

hat the infinitesimal rigid displacements of the same set () span the tangent space at the origin to this submanifold.this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 336 (2003).Acadmie des sciences. Published by ditions scientifiques et mdicales Elsevier SAS. All rights reserved.

acements rigides et leur relation au lemme du dplacement rigide infinitsimal en lasticit tri-dimensionnelle.un ouvert connexe de R3 et une immersion de dans R3. On tablit que lensemble form par les dplacementsde louvert () est une sous-varit de dimension 6 et de classe C de lespace H1(). On montre aussi que lesments rigides infinitsimaux du mme ouvert () engendrent le plan tangent lorigine cette sous-varit. Pourt article : P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 336 (2003).Acadmie des sciences. Published by ditions scientifiques et mdicales Elsevier SAS. All rights reserved.

n franaise abrge

notations sont dfinies dans la version anglaise.lemme du dplacement rigide infinitsimal en coordonnes curvilignes, qui joue un rle important ent linarise tri-dimensionnelle en coordonnes curvilignes (cf. [2, Chapitre 1]), snonce ainsi : Soit ert connexe de R3, soit une immersion suffisamment rgulire de dans un espace euclidien tri-ionnel E3, et soit v H 1() un champ de vecteurs vrifiant

ij (v)= 0 dans ,

ail addresses: mapgc@cityu.edu.hk (P.G. Ciarlet), mardare@ann.jussieu.fr (C. Mardare).

X/03/\$ see front matter 2003 Acadmie des sciences. Published by ditions scientifiques et mdicales Elsevier SAS. All rights

.

16/S1631-073X(03)00191-2

• 874 P.G. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 336 (2003) 873878

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gij (v)= 12 (i v gj + j v gi ) et gi = i .

l existe des vecteurs c E3 et d E3 tels que (cf. [2, Thorme 1.7-3])(x)= c+ d (x), x .lasticit tri-dimensionnelle en coordonnes curvilignes, lensemble () E3 est la configuration dece dun corps lastique et le champ v est un champ de dplacements de lensemble (). Les fonctionssont les composantes covariantes du tenseur linaris de changement de mtrique associ au champ v,placement de la forme ci-dessus v = c + d est appel un dplacement rigide infinitsimal deble ().jet de cette Note est dtablir que le lemme du mouvement rigide infinitsimal en coordonnes curvilignestre que la version linarise (dans un sens prcis au Thorme 4.1) du thorme de rigidit bien connu detrie diffrentielle, une fois celui-ci convenablement tendu lespace de Sobolev H 1().

e extension (Thorme 2.1) repose elle-mme sur une gnralisation du thorme de Liouville, qui est dueetnyak  (voir  pour une dmonstration particulirement courte et lgante).tablit ensuite (Thorme 3.1 et son corollaire) que lensembleMrig form des dplacements rigides (au

Thorme 2.1) de lensemble () est une sous-varit de dimension 6 et de classe C de lespace).

ontre enfin (Thorme 4.1) que lespace vectoriel engendr par les dplacements rigides infinitsimaux deble () nest autre que lespace tangent lorigine la varitMrig.

noncs des Thormes 2.1, 3.1, et 4.1 mentionns ci-dessus, accompagns desquisses de dmonstrations,vent dans la version anglaise. On trouvera les dmonstrations compltes de ces thormes dans .rsultats de cette Note sont tendus aux dplacements rigides et rigides infinitsimaux sur une surface].

liminaries

plete proofs of Theorems 2.1, 3.1, and 4.1 are found in .spaces, matrices, etc., considered are real. The notations M3, O3, O3+, and A3 respectively designate theall square matrices of order 3, of all orthogonal matrices of order 3, of all matrices Q O3 with detQ= 1,all antisymmetric matrices of order 3.n indices range over the set {1,2,3} and the summation convention with respect to repeated indices is usedunction with this rule. The identity mapping of a set X is denoted idX .notation E3 designates a three-dimensional Euclidean space and a b, a b, and |a| = a a respectivelyte the Euclidean inner product, the exterior product of a,b E3, and the Euclidean norm of a E3.

rigidity theorem in Sobolev spaces

be an open subset of R3, let xi denote the coordinates of a point x R3, and let i := /xi . C1(;E3) be an immersion. The metric tensor field (gij ) C0(;M3) of the set () (which is openince is an immersion) is then defined by means of its covariant components

ij (x) := i(x) j(x), x .

• P.G. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 336 (2003) 873878 875

The classical rigidity theorem for an open set (cf., e.g., [3, Theorem 3]) asserts that, if two immersions Cexplan

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d

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M1() := C1(;E3) and C1() have the same metric tensor fields, i.e., if gij = gij in (with self-atory notations) and is connected, then there exist a vector c E3 and a matrix Q O3 such that(x)= c+Q(x) for all x .

following result shows that a similar result holds under the assumption that H 1() :=H 1(;E3).trix (x) M3 is that whose i-th column is i(x).

m 2.1. Let be a connected open subset of R3 and let C1() be a mapping that satisfies det > 0ssume that there exists a vector field H 1() that satisfieset > 0 a.e. in and gij = gij a.e. in .ere exist a vector c E3 and a matrix Q O3+ such that(x)= c+Q(x) for almost all x .

of proof. The Euclidean space E3 is identified with the space R3 throughout the proof.o begin with, consider the special case where = id . In other words, we are given a mapping H 1()tisfies (x) O3+ for almost all x . Then, by a result of Reshetnyak  (for a modern proof, seeke, James and Mller ), there exist a vector c E3 and a matrix Q O3+ such that(x)= c+Qox for almost all x .

at this result is a generalization of the classical Liouville theorem.Consider next the general case. Let x0 be given. Since is an immersion, the local inversion theoremapplied; there thus exist bounded open neighborhoods U of x0 and U of (x0) satisfying U and(), such that the restrictionU of toU can be extended to a C1-diffeomorphism from U onto {U}.

n the composite mapping := 1U belongs to H 1(U) and satisfies (x) O3+ for almost all x U .there thus exist c R3 and Q O3+ such that (x)= c+Qox for almost all x U , hence such that(x)= c+Q(x) for almost all x U.

The assumed connectedness of the open set then implies that the last relation holds in fact for almost all

submanifold of rigid displacements

the results needed below about submanifolds in infinite-dimensional Banach spaces are found in . If Mmanifold, the tangent space toM at m M is denoted TmM.next theorem shows that the set M formed by all the mappings H 1() that satisfy the assumptionsigidity theorem for an open set (Theorem 2.1) is a finite-dimensional submanifold of the space H 1(). Thet space toM at is also identified. Another equally important characterization of the same tangent space,ng this time the linearized change of metric tensor, will be given in Theorem 4.1.

m 3.1. Let be a connected open subset of R3 and let C1() H 1() be a mapping that satisfies> 0 in . Then the set

:= { H 1(); det > 0 and gij = gij a.e. in }

• 876 P.G. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 336 (2003) 873878

is a submanifold of class C and of dimension 6 of the space H 1() and its tangent space at is given byT

Sketch

f

is injecC-dif

By t

MSince Eand ofalso a s

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T

In thdimensis irrelethe fielparticudefined

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4. The

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eM={v H 1(); c E3, A A3, v = c+A a.e. in }.

of proof. It is easily seen that the linear mapping f defined by

: (c,F ) E3 M3 f (c,F ) := c+F H 1()tive. Consequently, the image f (E3 M3) is a linear subspace of dimension 12 of H 1() and f is afeomorphism between E3 M3 and f (E3 M3).he rigidity theorem (Theorem 2.1), the setM may be equivalently defined as= f (E3 O3+

).

O3+ is a submanifold of class C and of dimension 6 of E3 M3,M is thus a submanifold of class Cdimension 6 of f (E3 M3). Besides, the closed subspace f (E3 M3) is split in H 1(). HenceM isubmanifold of class C and of dimension 6 of H 1().e f is linear and TIO3+ =A3, the tangent space toM at is given byM = Tf (0,I )f

(E

3 O3+)= f (T(0,I)

(E

3 O3+))= f (E3 A3)

= {v H 1(); c E3, A A3, v = c+A a.e. in }. ree-dimensional elasticity in curvilinear coordinates, the set () is the reference configuration of a three-ional elastic body (under the additional assumption that the immersion is injective, but this assumptionvant for our present purposes). Then, for each H 1(), the set () is a deformed configuration andd v H 1() defined by = + v is a displacement field of the reference configuration (). If inlar M, the field v defined in this fashion is called a rigid displacement, and the subsetMrig of H 1()by

= +Mrigrdingly called the manifold of rigid displacements. We now recast Theorem 3.1 in terms of theldMrig.

ary to Theorem 3.1. Let be a connected open subset of R3, and let C1()H 1() be a mappingtisfies det > 0 in . Then the manifold of rigid displacements of the set (), viz.,

rig :={v H 1(); det( +v) > 0 and gij = gij a.e. in

},

manifold of class C and of dimension 6 of the space H 1() and its tangent space at 0 is given by

0Mrig = TM= {v H 1(); c E3, A A3, v = c+A a.e. in }.

infinitesimal rigid displacement lemma in curvilinear coordinates revisited

covariant components of the linearized change of metric tensor associated with a displacement field v of() are defined by

ij (v) := 12 [gij gij ]lin,

• P.G. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 336 (2003) 873878 877

where gij and gij are the covariant components of the metric tensors of the sets () and () where := immed

e

Thise

Underis callelemmais the oof the m

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where

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The+ v, and []lin denotes the linear part with respect to v in the expression []. A formal computationiately gives

ij (v)= 12 (i v gj + j v gi ), where gi := i .expression thus shows that

ij (v) L2loc() if v H 1() and C1().this assumption on the mapping , a displacement field v H 1() that satisfies eij (v) = 0 a.e. in d an infinitesimal rigid displacement of the set (). Accordingly, the infinitesimal rigid displacementin curvilinear coordinates consists in identifying the vector space V linrig formed by such displacements. Thisbject of the next theorem, which shows that the space V linrig has a remarkably simple interpretation in termsanifoldMrig of rigid displacements introduced at the end of Section 3.

m 4.1. Let be a connected open subset of R3 and let C1() H 1() be a mapping that satisfies> 0 in . Then the space of infinitesimal rigid displacements of the set (), viz.,linrig :=

{v H 1(); [gij gij ]lin = 0 a.e. in

},

bylinrig = T0Mrig,

he tangent space T0Mrig has been identified in the Corollary to Theorem 3.1.

of proof. Let v H 1() be such that eij (v)= 0 in . An extension of the proof in [2, Theorem 1.7-3]ows that, given any x0 , there exist an open neighborhood U of x0, a vector c E3, and a matrix3 such that(x)= c+A(x) for almost all x U.e the point x0 is arbitrary and is connected, this relation holds in fact for almost all x (given any, cover any path joining x0 to x1 by a finite number of neighborhoods similar to U and repeat the previousnt). In other words,linrig =

{v H 1(); c E3, A A3, v = c+A a.e. in }.

nclusion then follows from the Corollary to Theorem 3.1. manifold of rigid displacements can be equivalently written as

rig ={v H 1(); det( +v) > 0 a.e. in , Fij (v)= 0 a.e. in

},

the mappings Fij :H 1() L1() are defined byij (v) := i( + v) j ( + v) i j, v H 1().appings are Frchet-differentiable and their Gteaux derivatives at 0 are given by DFij (0)v = 2eij (v) forH 1(), by definition of the functions eij (v). Theorem 4.1 thus shows that

0Mrig ={v H 1(); DFij (0)v = 0 a.e. in

}.

ther words, the space T0Mrig has the expression that is naturally expected, yet is often delicate to establish,angent space to a submanifold of an infinite-dimensional Banach space defined by means of equations; seerespect [1, Chapter 3].results of this Note can be extended to rigid and infinitesimal rigid displacements on a surface (cf. ).

• 878 P.G. Ciarlet, C. Mardare / C. R. Acad. Sci. Paris, Ser. I 336 (2003) 873878

References

 R. Abraham, J.E. Marsden, T. Ratiu, Manifolds, Tensor Analysis, and Applications, Second edition, Springer-Verlag, New York, 1988. P.G. Ciarlet, Mathematical Elasticity, Volume III: Theory of Shells, North-Holland, Amsterdam, 2000. P.G. Ciarlet, F. Larsonneur, On the recovery of a surface with prescribed first and second fundamental forms, J. Math. Pures Appl. 81 (2002)

167185. P.G. Ciarlet, C. Mardare, On rigid and infinitesimal rigid displacements in three-dimensional elasticity, Math. Models Methods Appl. Sci.,

to appear. P.G. Ciarlet, C. Mardare, On rigid displacements and their relation to the infinitesimal rigid displacement lemma in shell theory, C. R. Acad.

Sci. Paris, Sr. I, to appear. G. Friesecke, R.D. James, S. Mller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional

elasticity, Comm. Pure Appl. Math. 55 (2002) 14611506. Y.G. Reshetnyak, Liouvilles theory on conformal mappings under minimal regularity assumptions, Siberian Math. J. 8 (1967) 6985.