Differential Geometry: Theory and Applications (Contemporary Applied Mathematics)

0iffe re nt ia G e o met ryD h Theory and App ications Philippt G C rlet Ta-Tsien Li A

A

1

0iffe re nt ia I G e o 1e t ry : Theory and Applications

Series in Contemporary Applied Mathematics CAM Honorary Editor: Chao-Hao Gu (Fudun University) Editors: P. G. Ciarlet (City University ofHong Kong), Ta-Tsien Li (Fudan University)

1. Mathematical Finance -Theory and Practice (Eds. Yong Jiongmin, Rama Cont) 2. New Advances in Computational Fluid Dynamics -Theory, Methods and Applications (Eds. F. Dubois, Wu Huamo) Actuarial Science -Theory and Practice 3. (Eds. Hanji Shang, Alain Tosseti) 4. Mathematical Problems in Environmental Science and Engineering (Eds. Alexandre Ern, Liu Weiping) 5. Ginzburg-Landau Vortices (Eds. HaYmBrezis, Ta-Tsien Li) 6. Frontiers and Prospects of Contemporary Applied Mathmetics (Eds. Ta-Tsien Li, Pingwen Zhang) 7. Mathematical Methods for Surface and Subsurface Hydrosystems (Eds. Deguan Wang, Christian Duquennoi, Alexandre Ern) 8. Some Topics in Industrial and Applied Mathematics (Eds. Rolf Jeltsch, Ta-Tsien Li, Ian Hugh Sloan) 9. Differential Geometry: Theory and Applications (Eds. Philippe G. Ciarlet, Ta-Tsien Li)

Series in Contemporary Applied Mathematics CAM 9

0iffe Ie n t ia I G e o m e t ry : Theory and Applications editors

Philippe G Ciarlet City University of Hong Kong, China

Ta-Tsien Li Fudan University, China

Higher Education Press

xe World Scientific NEW J E R S E Y

*

LONDON

SINGAPORE

BElJlNG

*

SHANGHAI

*

HONG KONG

- TAIPEI

CHENNAI

Philippe G. Ciarlet

Ta-Tsien Li

Department of Mathematics

School of Mathematical Sciences

City University of Hong Kong

Fudan University

83 Tat Chee Avenue Kowloon, Hong Kong

220, Handan Road Shanghai, 200433

China

China

Editorial Assistants: Zhou Chun-Lian

Copyright @ 2008 by

Higher Education Press 4 Dewai Dajie, Beijing 100011, P. R. China, and World Scientific Publishing Co Pte Ltd

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any information storage and retrieval system, without permission in writing from the Publisher

ISBN 978-7-04-022283-8 Printed in P. R. China

Preface

The ISFMA-CIMPA School on “Differential Geometry: Theory and Applications” was held on 07 August 18 August 2006, in the building of the Chinese-French Institute for Applied Mathematics (ISFMA), Fudan University, Shanghai, China. This school was jointly organized by the ISFMA and the CIMPA (International Centre for Pure and Applied Mathematics), Nice, France. About sixty participants from China, Hong Kong, France, Cambodia, India, Iran, Pakistan, Philippines, Romania, Russia, Sri-Lanka, Thailand, Turkey, Uzbekistan and Vietnam attended this highly successful event. The first objective of this school was to lay down in a self-contained and accessible manner the basic notions of differential geometry, such as the metric tensor, the Riemann curvature tensor, the fundamental forms of a surface, covariant derivatives, and the fundamental theorem of surface theory etc. Although this field is with good reasons often considered as a “classical” one, it has been recently “rejuvenated”, thanks to the manifold applications where it plays an essential role. The second objective of this school was to present some of these applications, such as the theory of linearly and nonlinearly elastic shells, the implementation of numerical methods for shells, and mesh generation in finite element methods. To fulfill these objectives, four series of lectures, each series comprising ten 50min-lectures, were delivered under the following titles: “Introduction to differential geometry” , “Introduction to shell theory” , “A differential geometry approach to mesh generation”, and “Numerical methods for shells”. This volume gathers the materials covered in these lectures. As such, this volume should be very useful to graduate students and researchers in pure and applied mathematics. The organizers take pleasure in thanking the various organizations for their generous support: The ISFMA, the CIMPA, the French Embassy in Beijing, the Consulate General of France in Shanghai, the National Natural Science Foundation of China, Fudan University, Higher Education Press and World Scientific. Finally, our special thanks are due to Mrs. Zhou Chun-Lian for her patient and effective work in editing this book. ~

Philippe G. Ciarlet and Ta-Tsien Li February 2007

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Contents

Preface

Philippe G. Ciarlet: An Introduction to Differential Geometry in R3 ................................................ Philzppe G. Ciarlet, Cristinel Mardare: An Introduction to Shell Theory ...............................................

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94

Dominique Chapelle: Some New Results and Current Challenges in the Finite Element Analysis of Shells . . . . . . . . . . . 185 Pascal Frey: A Differential Geometry Approach to Mesh Generation .............................................

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An Introduction to Differential Geometry in R3 Philippe G. Ciarlet Department of Mathematics, City University of Hong Kong 83 Tat Chee Avenue, Kowloon, Hong Kong, China E-mail: [email protected]

Introduction These notes’ are intended to give a thorough introduction to the basic theorems of differential geometry in R3, with a special emphasis on those used in applications. The treatment is essentially self-contained and proofs are complete. The prerequisites essentially consist in a working knowledge of basic notions of analysis and functional analysis, such as differential calculus, integration theory and Sobolev spaces, and some familiarity with ordinary and partial differential equations. In Part 1, we review the basic notions, such as the metric tensor and covariant derivatives, arising when a three-dimensional open set is equipped with curvilinear coordinates. We then prove that the vanishing of the Riemann curvature tensor is sufficient for the existence of isometric immersions from a simply-connected open subset of E3 equipped with a Riemannian metric into a three-dimensional Euclidean space. We also prove the corresponding uniqueness theorem, also called rigidity theorem. In Part 2, we study basic notions about surfaces, such as their two fundamental forms, the Gaussian curvature, and covariant derivatives. We then prove the fundamental theorem of surface theory, which asserts that the Gauss and Codazzi-Mainardi equations constitute sufficient conditions for two matrix fields defined in a simply-connected open subset of R2 to be the two fundamental forms of a surface in a three-dimensional Euclidean space. We also prove the corresponding rigidity theorem. ‘With the kind permission of Springer-Verlag, these notes are extracted and adapted from my book “An Introduction to Differential Geometry with Applications to Elasticity”, Springer, Dordrecht, 2005, the writing of which was substantially supported by two grants from the Research Grants Council of Hong Kong Special Administrative Region, China [Project No. 9040869, CityU 100803 and Project No. 9040966, CityU 100604].

Philippe G. Ciarlet

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1 Three-dimensional differential geometry

Outline Let R be an open subset of R3,let E3 denote a three-dimensional Euclidean space, and let 0 : R 4 E3 be a smooth injective immersion. We begin by reviewing (Sections 1.1 to 1.3) basic definitions and properties arising when the three-dimensional open subset O(R) of E3 is equipped with the coordinates of the points of R as its curvilinear coordinates. Of fundamental importance is the metric tensor of the set O(R), whose covariant and contravariant components gij = gji : R + R and .. 9'3 = gja : R + R are given by (Latin indices or exponents take their values in {I, 2,3}): ..

.

.

j 923" '- gi . g j and gt3 = g' . g 3 , where g i = 8iO and gj . g i = hi.

The vector fields g i : R + R3 and gj : R + R3 respectively form the covariant, and contravariant, bases in the set O(R). It is shown in particular how volumes, areas, and lengths, in the set O(R) are computed in terms of its curvilinear coordinates, by means of the functions gij and gij (Theorem 1.3-1). We next introduce in Section 1.4 the fundamental notion of covariant derivatives vilij of a vector field wigi : R 4 R3 defined by means of its covariant components wi over the contravariant bases gi. Covariant derivatives constitute a generalization of the usual partial derivatives of vector fields defined by means of their Cartesian components. In particular, covariant derivatives naturally appear when a system of partial differential equations with a vector field as the unknown, e.g., the displacement field in elasticity, is expressed in terms of curvilinear coordinates. It is a basic fact that the symmetric and positive-definite matrix field ( g i j ) defined on R in this fashion cannot be arbitrary. More specifically (Theorem 1.5-1), its components and some of their partial derivatives must satisfy necessary conditions that take the form of the following relations (meant to hold for all i ,j , k , q E {1,2,3}): Let the functions rijq and be defined by 1 rijq= z(ajgiq+digjq-dqgij) Then, necessarily,

and l7$

= g p q r i j q , where

(gpq) = (gij)-'.

An Introduction to Differential Geometry in R3

3

The functions rij, and are the Christoffel symbols of the first, and second, kind and the functions

are the covariant components of the R i e m a n n curvature tensor of the set 0 ( R ) . We then focus our attention on the reciprocal questions: Given an open subset R of R3 and a smooth enough symmetric and positive-definite matrix field ( g i j ) defined on R, when is it the metric tensor field of an open set @(R) c E3, i.e., when does there exist a n immersion 0 : R 4 E3 such that g i j = &O ' a j 0 in R? If such an immersion exists, to what extent is it unique? As shown in Theorems 1.6-1 and 1.7-1, the answers turn out t o be remarkably simple to state (but not so simple to prove, especially the first one!): Under the assumption that R i s simply-connected, the necessary conditions R q i j k = 0 in R

are also s u f i c i e n t for the existence of such a n immersion 0 . Besides, i f R is connected, this immersion i s unique up t o isometries of E3. This means that, if 0 : R + E3 is any other smooth immersion satisfying

g $3 ..- a

-

i . ~aj0 in R,

there then exist a vector c E E3 and an orthogonal matrix Q of order three such that

~ ( x=) c +

Q ~ ( X )for

all

5

E

R.

Together, the above existence and uniqueness theorems constitute an important special case of the fundamental theorem of Riemannian geometry and as such, constitute the core of Part 1. We conclude this chapter by indicating in Section 1.8 that the equivalence class of 0 , defined in this fashion modulo isometries of E3, depends continuously o n the matrix field ( g i j ) with respect t o various topologies.

1.1

Curvilinear coordinates

To begin with, we list some notations and conventions that will be consistently used throughout. All spaces, matrices, etc., considered here are real. Latin indices and exponents range in the set {1,2,3}, save when otherwise indicated, e.g., when they are used for indexing sequences, and the summation convention with respect to repeated indices or exponents

Philippe G. Ciarlet

4

is systematically used in conjunction with this rule. For instance, the relation Si(X) = gij ( X ) d (XI

means that

3

gi(x)= Cgij(x)gi(z) for i

=

1,2,3.

j=1

Kronecker’s symbols are designated by dl, dij, or 6aj according to the context. Let E3 denote a three-dimensional Euclidean space, let a.b and a A b denote the Euclidean inner product and exterior product of a,b E E3, and let la1 = f i denote the Euclidean norm of a E E3.The space E3 is endowed with an orthonormal basis consisting of three vectors il?= Zi. Let 2i denote the Cartesian coordinates of a point 2 E E3 and let ai := a/a2ii. In addition, let there be given a three-dimensional vector space in which three vectors ea = ei form a basis. This space will be identified with R3.Let xi denote the coordinates of a point x E EX3 and let 13i := a / d x i , := d2/axidxj, and aijk:= a 3 / d ~ i d ~ j a ~ k . Let there be given an open subset 6 of E3 and assume that there exist an open subset R of R3 and an injective mapping 0 : R -+ E3 such h

aij

Figure 1.1-1: Curvilinear coordinates and covariant bases in a n open set 8 c E3. The thr? coordinates z1,x2,z3 of x E R are the curvilinear coordinates of P = O ( x ) E R. If the three vectors gi(z) = &O(z) are linearly independent, they form the covariant basis at 9= O(z) and they are tangent to the coordinate lines passing through 9.

An Introduction to Differential Geometry in lit3

5

that O(R) = 6. Then each point 2 E fi can be unambiguously written as 2 = O ( 2 ) ,2 E R, h

and the three coordinates zi of 2 are called the curvilinear coordinates of 2 (Figure 1.1-1).Naturally, there are infinitely m a n y ways of defining curvilinear coordinates in a given open set R , depending on how the open set R and the mapping 0 are chosen! Examples of curvilinear coordinates include the well-known cylindrical and spherical coordinates (Figure 1.1-2). h

b z

E3

Figure 1.1-2: Two familiar examples of curvilinear coordinates. Let the mapping 0 be defined by 0 : (cp, p, z ) E R 4( p cos 'p, p sin 9, z) E E3. Then (p, p, z ) are the cylindrical coordinates of P = 0(cp,p, z ) . Note that (p 2kn, p, z ) or (p T + 2 k ~- p, , z ) , k E Z,are also cylindrical coordinates of the same point 2 and that cp is not defined if Z is the origin of E3. Let the mapping 0 be defined by 0 : ( c p , $ , ~ ) E C l -i (rcos$cosp,rcos$sinp,rsin$) E E3. Then (9,$, T ) are the spherical coordinates of P = 0 ( p ,$, T ) . Note that ( ( p + Z k ~$+ , 2!7r, r ) or ('p 2 h , $ K + 2!~, - T ) are also spherical coordinates of the same point P and that cp and $ are not defined if 2 is the origin of E3.

+

+

+

+

In a different, but equally important, approach, an open subset R of B3 together with a mapping 0 : R 4 E3 are instead_ a priori given. If 0 E Co(R;E3) and 0 is injective, the set R := O(R) is open by the invariance of domain theorem (for a proof, see, e.g., Nirenberg [1974, Corollary 2, p. 171 or Zeidler [1986, Section 16.4]), and curvilinear coordinates inside R are unambiguously defined in this case. If 0 E C1(R; E3) and the three vectors &O(z) are linearly independent at all 2 E R, the set fi is again open (for a proof, see, e.g., Schwartz [1992] or Zeidler [1986, Section 16.4]), but curvilinear coordinates may be defined only locally in this case: Given z E R, all that can be asserted (by the local inversion theorem) is the existence of an open neighborhood h

Philippe G. Ciarlet

6

V of x in R such that the restriction of 0 to V is a C1-diffeomorphism, hence an injection, of V onto O ( V ) .

1.2 Let

Metric tensor

R be an open subset of R3 and let

be a mapping that is diflerentiable at a point x E R. If (x Sx) E R,then

+

+

O(z + Sx) = O ( x ) VO(x)bx where the 3 x 3 matrix

SX is such that

+ o(Sx),

V O ( x ) and the column vector Sx are defined by

&@I

&@I

8301

8102

8202

8302)

8103

8203

8303

(x) and Sx

=

f::) . 6x3

Let the three vectors gi(x) E R3 be defined by

i.e., gi(x) i s the i-th column vector of the matrix VO(x). Then the expansion of 0 about x may be also written as

O(x

+ Sx) = O ( x )+ 6xigi(.) + o(6x)

If in particular Sx is of the form SX = Gtei, where bt E one of the basis vectors in R3,this relation reduces to

+

O ( x + 6tei) = O ( x ) 6tgi(x)

R and

ei is

+ o(6t).

A mapping 0 : R -+ E3 is an immersion at x E R if it is differentiable at x and the matrix V O ( x ) is invertible or, equivalently, if the three vectors gi(x) = &O(x) are linearly independent. Assume that the mapping 0 i s an immersion at x. Then the three vectors gi(x) constitute the covariant basis at the point 2 = O ( x ) . In this case, the last relation thus shows that each vector gi(x) i s tangent to the a-th coordinate line passing through 2 = O ( x ) ,defined as the image by 0 of the points of R that lie on the line parallel to ei passing through x (there exist to and t l with to < 0 < t l such that the i-th coordinate line is given by t E ]to,tl[ -+ fi(t) := O ( x tei) in a

+

An Introduction to Differential Geometry in R3

7

neighborhood of 2; hence fl(0) = & O ( x ) = gi(x));see Figures 1.1-1 and 1.1-2. Returning to a general increment 6x = 6xiei, we also infer from the expansion of 0 about x that (recall that we use the summation convention) :

p(x

+ 62) - @(%)I2

+

= 6 x T V O ( x ) T V O ( x ) 6 x o(16z12) = 6xig&)

+

'gj(x)6xj o(16s12).

Note that, here and subsequentIy, we use standard notations from matrix algebra. For instance, 6xT stands for the transpose of the column vector 6x and V O ( Z )designates ~ the transpose of the matrix VO(x), the element at the i-th row and j - t h column of a matrix A is noted (A)ij, etc. In other words, the principal part with respect to 6 x of the length between the points O ( x + 6 x ) and O ( x )is {Gxigi(x).gj(x)Sxj}1/2. This observation suggests to define a matrix (gij(x))of order three, by letting

gij(x) := gz(x) . gj(x) = (vo(x)Tvo(x))ij. The elements g i j ( x ) of this symmetric matrix are called the covariant components of the metric tensor at 2 = O ( x ) . Note that the matrix VO(x) is invertible and that the matrix (gij(z)) is positive definite, since the vectors gi(x) are assumed to be linearly independent. The three vectors gi(x)being linearly independent, the nine relations = 6;

g2(x). g&)

unambiguously define three linearly independent vectors gi(x). To see this, let a priori gi(x)= Xik(z)gk(x)in the relations gi(z).gj(x)= 6;. This gives Xik(x)gkj(x)= 6;; consequently, Xik(x)= g i k ( x ) ,where (gij(x)):= (gij(2))y. Hence gi(x)= gik(x)gk(x).These relations in turn imply that

g i ( 4.

sw

= (gik(z)g&))

. (gje(4ge(4)

= gz"x)gje(x)gke(x)

= gZk(2)6% = g"(x),

and thus the vectors gi(x) are linearly independent since the matrix (gij(z))is positive definite. We would likewise establish that gi(x) = gij (4Sj(x). The three vectors gi(x)form the contravariant basis a t the point f = O ( x ) and the elements gij(x) of the symmetric positive definite

Philippe G. Ciarlet

8

matrix ( g i j ( x ) ) are the contravariant components of the metric tensor at 2 = O ( x ) . Let us record for convenience the fundamental relations that exist between the vectors of the covariant and contravariant bases and the covariant and contravariant components of the metric tensor at a point x E R where the mapping 0 is an immersion:

g i j ( x ) = g i ( x ) . g j ( x ) and gZj(x) = g i ( x ) . g j ( x ) , g i ( x ) = g i j ( x ) g j ( x ) and g i ( x ) = g i j ( x ) g j ( x ) .

A mapping 0 : R -+ E3 is an immersion if it is an immersion at each point in R, i.e., if 0 is differentiable in R and the three vectors g i ( x ) = & O ( x ) are linearly independent at each x E R. If 0 : R -+ E3 is an immersion, the vector fields gi : R + R3 and gi : R -+ R3 respectively form the covariant, and contravariant bases. To conclude this section, we briefly explain in what sense the components of the “metric tensor” may be “covariant” or “contrawariant”. - Let R and 6 be two domains in R3 and let 0 : R + E3 and 6 : R + E3 be two C’-diffeomorphisms such that O(R) = 6(6)and such that the vectors g i ( x ) := & O ( x ) and Gi(Z)= of the covariant bases at the same point O(z) = 6(Z) E E3 are linearly independent. Let g i ( x ) and $(Z) be the vectors of the corresponding contravariant bases at the same point 2. A simple computation then shows that

x

--1

= 0 o 0 E C1(R;6) (hence Z = ~ ( z ) and ) (T):= x-1 E C l ( 6 ; 0). Let g i j ( x ) and Tij ( 2 )be the covariant components, and let gzj ( x ) and ?j(Z) be the contravariant components, of the metric tensor at the same point O(z) = 6(Z) E E3. Then a simple computation shows that

where

= ( x j ) :=

These formulas explain why the components gij(x) and $ j ( z ) are respectively called “covariant” and ‘kontravariant”: Each index in g i j ( x ) “varies like” that of the corresponding vector of the covariant basis under a change of curmilinear coordinates, while each exponent in g ij( x ) (‘varies like” that of the corresponding vector of the contravariant basis.

Remark. What is exactly the “second-order tensor” hidden behind its covariant components g i j ( x ) or its contravariant exponents g i j ( z )

An Introduction to Differential Geometry in

W3

9

is beautifully explained in the gentle introduction to tensors given by Antman [1995, Chapter 11, Sections 1 to 31; it is also shown in ibid. that the same “tensor” also has “mixed” components gj(x), which turn out to be simply the Kronecker symbols 6;. 0 In fact, analogous justifications apply as well to the components of all the other “tensors” that will be introduced later on. Thus, for instance, the covariant components .{(.) and Gi(x),and the contravariant components d ( x ) and Gi(x) (both with self-explanatory notations), of a vector at the same point O ( x )= 6 ( E ) satisfy (cf. Section 1.4)

?Ji(x)gZ(z)= Gz(.)g(.)

= ? J ~ ( x ) g i ( .= ) 3(.)iji(.).

It is then easily verified that

In other words, the components w{(x) “vary like” the vectors gi(z) of the covariant basis under a change of curvilinear coordinates, while the components d ( x ) of a vector “vary like” the vectors g i ( x ) of the contravariant basis. This is why they are respectively called %ovariant” and rrcontrauariant”. A vector is an example of a “first-order” tensor. Note, however, that we shall no longer provide such commentaries in the sequel. We leave it instead to the reader to verify in each instance that any index or exponent appearing in a component of a “tensor” indeed behaves according to its nature. The reader interested by such questions will find exhaustive treatments of tensor analysis, particularly as regards its relevance to elasticity, in Boothby [1975],Marsden & Hughes [1983, Chapter 11, or Simmonds [1994].

1.3

Volumes, areas, and lengths in curvilinear coordinates

We now review fundamental formulas showing hzw volume, area, and length elements at a point 2 = O ( x ) in the set R = O(R) can be expressed either in terms of the matrix VO(x), or in terms of the matrix (Sij(X)).

These formulas thus highlight the crucial r61e played by the matrix ( g i j ( x ) ) for computing “metric” notions at the point 2 = O(z). Indeed, the “metric tensor” well deserves its name! A domain in Wd, d 2 2, is a bounded, open, and connected subset D of Wdwith a Lipschitz-continuous boundary, the set D being locally on

10

Philippe G. Ciarlet

one side of its boundary. All relevant details needed here about domains are found in NeEas [1967] or Adams [1975]. Given a domain D c R3 with boundary r, we let dx denote the volume element in D , d r denote the area element along r, and n = nii? denote the unit (In1 = 1) outer normal vector along I? ( d r is well defined and n is defined dr-almost everywhere since r is assumed to be Lipschitz-continuous) . Note also that the assumptions made on the mapping 0 in the next theorem guarantee that, if D is a domain in R3 such that D C R, then {6}-c R, { O ( D ) } - = O(D),and the boundaries 6’6 of and dD of D are related by 8 6 = O ( a D ) (see, e.g., Ciarlet [1988, Theorem 1.2-8 and Example 1.71). If A is a square matrix, C o f A denotes the cofactor matria: of A . Thus C o f A = (det A)A-T if A is invertible.

5

Theorem 1.3-1. Let R be an open subset of R3,-let 0 : R 4 E3 be an injective and smooth enough immersion, and let R = @(a). (a) The volume element d f at f = O(x) E R is given in terms of the volume element dx at x E R by

-

df = I det V O ( x ) l d z = m d x , where g(x) := det(gij(x)).

c R. The area element (b) Let D be a domain in R3 such that d f ( f ) at 3 = O(x) E 8 6 is given in terms of the area element dF(x) at x E d D by d f ( f ) = I C o f V O ( x ) n ( x ) l d r ( x )= m d n i ( x ) g i j ( x ) n j ( a : ) d I ’ ( x ) , where n ( x ) := ni(x)ei denotes the unit outer normal vector at x E d D . (c) The length element d?(f) at f = O(x) E fi is given by dT(f)

. 112

=

{ S X ~ V O ( ~ ) ~ V O ( X )= S ~{6xigij(x)6xJ} }~”

,

where Sx = 6xiei.

Proof. The relation d2 = ) d e t V O ( x ) l dx between the volume elements is well known. The second relation in (a) follows from the relation g(x) = I d e t V O ( x ) I 2 , which itself follows from the relation (gz3(x))= v o ( x ) T v o ( z ) . Indications about the proof of the relation between the area elements d?(f) and d r ( x ) given in (b) are found in Ciarlet [1988, Theorem 1.7-11 (in this formula, n ( x ) = n,(x)e2 is identified with the column vector in R3 with n,(x) as its components). Using the relations C o f ( A T ) =

An Introduction to Differential Geometry in EX3

11

(CofA)T and Cof(AB) = (CofA)(CofB),we next have:

I CofVo(x)n(x)/2= n ( x ) T Cof ( v o ( x ) T v o ( x ) ) n ( x ) = g (x)ni (x)gij( x ) n j(x).

Either expression of the length element given in (c) recalls that dF(5) is by definition the principal part with respect to 6x = Gxaei of the length l@(x 62) - O ( x ) / ,whose expression precisely led to the introduction of the matrix ( g i j ( x ) )in Section 1.2. 0

+

The relations found in Theorem 1.3-1 F e used in particular for computing volumes, areas, and lengths inside R by means of integrals insid: R , i.e., in terms of the curvilinear coordinates used in the open set R (Figure 1.3-1): Let D be a domain in R3 such that 25 c 0, let 5 := O ( D ) ,and let f ^ L1(6) ~ be given. Then

0

\t

I

R

Figure 1.3-1: Volume, area, and length elements an curvilinear coordinates. The elements dZ,dF(Z), and d@Z) at P = O(z) E are expressed in terms of dz, d f ( z ) , and 6x at z E R by means of the covariant and contravariant components of the metric tensor; cf. Theorem 1.3-1. Assume that R is a domain and that 0 is a C1diffeomorphism of R onto {a}-. Then, given a domain V such that V C R and a dr-measurable-subset A of the corresponding relatio_ns are used for computing the volume of V = O(V) c R, the area of A = @ ( A ) c 8 2 , and the length of a curve (? = O ( C )c where C = f ( I ) and I is a compact interval of R.

fi

20,

{fi}-,

Philippe G. Ciarlet

12

h

In particular, the volume of D is given by h

volD :=

d2 =

m d x .

Next, lct r := do_, let CJe a dr-measurable subset of O ( C ) c d D , and let h E L1(C) be given. Then

s,-

h(2)df;(2)=

L(x

o

r, let 5 :=

O)(x)md-dl?(x).

h

In particular, the area of C is given by h

areaC :=

s,

dF(2) =

m

d

w

d

l

?

(

x

)

.

Finally, consider a curve C = f ( I ) in R, where I is a compact interval of R and f = f iei : I + R is a smooth-enough injective mapping. Then the length of the curve 6 := O ( C )c R is given by

This relation shows in particular that the lengths of curves inside the open set O(R) are precisely those induced by the Euclidean metric of the space E3. For this reason, the set O(R) is said to be isometrically immersed in E3.

1.4 Covariant derivatives of a vector field Suppose that a vector field is defined in an open subset 6 of E3 by means of its Cartesian component_s Ci : 6 + R, i.e., this field is defined by its values Gi(2)2? at each 2 E R, where the vectors Zi constitute the orthonormal basis of E3; see Figure 1.4-1. Suppose now that the open set R is equipped with curvilinear coordinates from an open subset R of EX3, by means of an injective mapping 0 : R + E3 satisfying O(R) = 6. How does one define appropriate components of the same vector field, but this time in terms of these curvilinear coordinates? It turns out that the proper way to do so consists in defining three functions ui : R -+ R by requiring that (Figure 1.4-2) h

An Introduction to Differential Geometry in R3

13

Figure 1.41: A vector field in Cartesian coordinates. At each point 3 E 6, the vector Gi(?@' is defined by its Cartesian components Gi(.^)over an orthonormal basis of E3 formed by three vectors Zi.

Figure 1.4-2:A vector field in curvilinear coordinates. Let there be given a vector field in Cartesian coordinates defined at each 3 E 6 by its Cartesian components &(Z) over the vectors 8" (Figure 1.4-1). In curvilinear coordinates, the same vector field is defined at each z E R by its covariant components vi(z) over the contravariant basis vectors gi(z) in such a way that vi(z)gi(z) = Gi(3)ei, 3 = O(z).

where the three vectors gi(x)form the contravariant basis at 2= O ( x ) (Section 1.2). Using the relations gi(x). g j ( z ) = 6; and 2 .2j = $, we immediately find how the old and new components are related, viz.,

The three components vi(x)are called the covariant components

Philippe G. Ciarlet

14

of the vector v i ( x ) g i ( x )at 2,and the three functions vi : R + R defined in this fashion are called the covariant components of the vector field vigi : R + E3. Suppose next that we wish t o compute a partial derivative ajGi(2) at a point 2 = O ( x ) E R in terms of the partial derivatives & v k ( z ) and of the values vq(x) (which are also expected to appear by virtue of the chain rule). Such a task is required for example if we wish to write a system of partial differential equations whose unknown is a vector field (such as the equations of nonlinear or linearized elasticity) in terms of ad hoc curvilinear coordinates. As we now show, carrying out such a transformation naturally leads to a fundamental notion, that of covariant derivatives of a vector field. h

h

Theorem 1.4-1. Let R be a n open subset of R3 and let 0 : R + E3 be a n injective immersion that is also a C2-diffeomorphism of R onto 6 := O(R). Given a vector field Gigi: fi + R3 in Cartesian coordinates with components Gi E C'(fi), let vigi : R + R3 be the same field in curvilinear coordinates, i.e., that defined by h

h

-2

vi(x)e

= v i ( x ) g z ( x )for

T h e n vi E C'(R) and for all x E

all 2= O ( x ) ,x E

R.

R,

h

ajGi(2) = (vklle[gkli[gelj)(x), 2 = O(x), where 21.

.

2113

:= 8 . v . - r ? . v 3

2

23

P

and

rp. := gp . a i g j , 23

and [Si(2)]k:= g y x ) . Z k

denotes the i - t h component of g i ( x ) over the basis { Z I , Z ~ , Z ~ } . Proof. The following convention holds throughout this proof The simultaneous appearance of 2 and x in an equality means that they are related by 2 = O ( x )and that the equality in question holds for all x E R. (i) Another expression of [gi(x)]k:= g i ( x ) . Z k . Let O ( x ) = Ok(x)Zk and 6(2) = @(2)ei,where 6 : 6 + E3 denotes the inverse mapping of 0 : R + E3. Since 6 ( O ( x ) )= x for all x E R, the chain rule shows that the matrices VO(x) := ( a j O k ( x ) )(the row index is k ) and ? 6 ( 2 ) := ( a k W ( 2 ) ) (the row index is i) satisfy

-

h .

V 6 ( 2 ) V O ( x ) = I,

An Introduction to Differential Geometry in Iw3

15

or equivalently,

The components of the above column vector being precisely those of the vector g, (x),the components of the above row vector must be those of the vector gi(x)since gi(x)is uniquely defined for each exponent i by the three relations gi(x).gj(x)= 6 j , j = 1 , 2 , 3 . Hence the Ic-th component of gi(x)over the basis {21,22,23} can be also expressed in terms of the inverse mapping 0, as: h

h

A .

[gi(x)]k = ako"2). (ii) The functions

:= gq

. &g,

E Co(R).

We next compute the derivatives &gq(x) (the fields gq = gq'g' are of class C1 on R since 0 is assumed to be of class C2). These derivatives will be needed in (iii) for expressing the derivatives &Ci(2) as functions of LC (recall that Ci(2)= uk(z)[g'(x)]i). Recalling that the vectors gk(x) form a basis, we may write a priori

aesq(x)= - r & ( 4 g k ( 4 , : R + Iw. To find their thereby unambiguously defining functions expressions in terms of the mappings 0 and 0, we observe that h

r;k(x)= r;m(x)6r = r j m ( z ) g m ( x .) gk(x) = -aegq(x). gk(x). h

h

Hence, noting that &(gq(x) . gk(x))= 0 and [gq(x)lp= aP@(2),we obtain

r;k(x)= gq(x).

aegk(x)= 5 p S q ( ~ ) a e k o p =(rig(+ ~)

Since 0 E C2(R;E3) and 6% E relations show that r;kE Co(0).

C1(6;Iw3)by

assumption, the last

h

of the Cartesian components of (iii) The partial derivatives the vector field Giza E C'(6; EL3) are given at each 2 = 0 ( x ) E 6 b y

5,Gi(2) = u k l l e ( 4 [&)li

[ge(41j,

Philippe G. Ciarlet

16

and [ g k ( z ) ] iand r&(z) are defined as in (i) and (ii). h

We compute the partial derivatives ajGi(2) as functions of z by means . this end, we first note that a of the relation Zi(2) = v k ( z ) [ g k ( z ) ] iTo differentiable function w : R -+ R satisfies

& w ( 6 ( 2 ) )= aew(z)@e(2)

= aew(z)[ge(z)]j,

by the chain rule and by (i). In particular then,

5jGi (2)= 5j V k (6( 2 ) [gk ) ( z )i]+ vq (z)5j [gq(6

I)$(

i

+

= aevk(z)[ge(z)]j[Sk(")li v,(.)(ae[9q(z)li)[ge(z)lj

[gk(~)li[ge(z)lj,

= (aevk(z)- % ( Z ) % ( 4 )

since & g q ( z )

=

0

- 1 ' & ( z ) g k ( z ) by (ii).

The functions

v. . = a.v. - rP.v 43 3 2 23 P

defined in Theorem 1.4-1 are called the first-order covariant derivatives of the vector field vigi : R -+ R3. The functions = g p . zgj

a.

are called the Christoffel symbols of the second kind (the Christoffel symbols of the first kind are introduced in the next section). The following result summarizes properties of covariant derivatives and Christoffel symbols that are constantly used.

Theorem 1.4-2. Let the assumptions o n the mapping 0 : R -+ E3 be as in Theorem 1.4-1, and let there be given a vector field vigi : fl -+ R3 with covariant components v i E C'(fl). (a) The first-order cowariant derivatives villj E Co(R) of the vector field vigi : R -+ R3,which are defined b y villj := ajvi

- I?$vp, where I'P. .= gp . & g j , 7 . 2 .

can be also defined by the relations a j ( v i g i ) = vzl,jgi

*

VZllj

= {aj(wkg"} ' g i .

(b) The Christoffel symbols I?;' := g p . d i g j = I?yi E Co(R) satisfy the relations &gP = -rP.gj 23 and a j g , = I?jqgi.

An Introduction to Differential Geometry in R3

17

Proof. It remains to verify that the covariant derivatives villj, defined in Theorem 1.4-1 by Villj

=

ajvi qjVp, -

may be equivalently defined by the relations aj(Vig2)

= vzlljgi

These relations unambiguously define the functions vUillj = {aj(Vkg')} 'gi since the vectors gz are linearly independent at all points of R by assumption. To this end, we simply note that, by definition, the Christoffel symbols satisfy & g p = - r : j g j (cf. part (ii) of the proof of Theorem 1.4-1); hence aj(vigi)= (ajVi)gZ

+ Vidjgi = ( a j V i ) g i - Virjkg'

To establish the other relations a j g ,

o=

' 9 , )= -rp.gi 3% . g P

Hence ajg, =

= r i q g i , we

= 21'ZIlJSi. '

note that

+ g p . d j g , = -r;j + g p . a j g , .

m,. s P ) g p

= r:jgp.

Remark. The Christoffel symbols can be also defined solely in terms of the components of the metric tensor; see the proof of Theorem 1.5-1. If the affine space E3 is identified with R3 and O(z) := z for all R, the relation 8 j ( v i g i ) ( z )= (viiljgi)(z)reduces to aj(Gi(2)z)= (8jGi(2))Zi. In this sense, a covariant derivative of the first order constitutes a generalization of a partial derivative of the first order in Cartesian h

z E

coordinates.

1.5

Necessary conditions satisfied by the metric tensor; the Riemann curvature tensor

It is remarkable that the components gij = g j i : R 4 R of the metric tensor of a n open set O(R) c E3 (Section 1.2), defined by a smooth enough immersion 0 : R -+ E3, cannot be arbitrary functions. As shown in the next theorem, they must satisfy relations that take the form: d j r i k q - dkrijq

+ I'plCqp r;krjqp = o in 0, -

Philippe G. Ciarlet

18

where the functions rijqand 17!j have simple expressions in terms of the functions g i j and of some of their partial derivatives (as shown in the next proof, it so happens that the functions I':j as defined in Theorem 1.5-1 coincide with the Christoffel symbols introduced in the previous section; this explains why they are denoted by the same symbol). Note that, according to the rule governing Latin indices and exponents, these relations are meant to hold for all i ,j , k , q E {1,2,3}.

Theorem 1.5-1. Let R be a n open set in EX3, let 0 E C3(R;E3) be a n immersion, and let g i j :=

aio. ajo

denote the covariant components of the metric tensor of the set @(R). E C1(R)be defined by Let the functions rijqE C1(R) and 1 rijq:= -(a. . .) 2 3% . + 8. d 3 .q - a4Qz.9 rp. := g P T i j q where ( g p q ) := ( g i j ) - ' . 7

Then, necessarily,

djrikq - akrijq + rzjrkqp - r;krjqp = o in 0.

a,@.

Proof. Let gi = It is then immediately verified that the functions rij, are also given by

r t.j .q -- 8.d l j ' g q . For each

II: E

R, let the three vectors

gj(II:) be

defined by the relations

g j ( z ) . g i ( z ) = 6;. Since we also have g j = g z j g , , the last relations imply that '?:I = &gj . g p . Therefore,

aigi = r ; j g p since aigj = (&gj . gP)g,. Differentiating the same relations yields &rijq= dikgj

- g q + aigj . dkgql

so that the above relations together give

a i g j . dkgq = r : j g p . d k g ,

=

r:'rkqP.

Consequently, dikgj

'

gq = akrijq - r:jrkqp.

Since a i k g j = & j g l c , we also have aikgj

. g q = 8 j r i k q - r:krjqp,

An Introduction to Differential Geometry in R3 and thus the required necessary conditions immediately follow.

19

0

Remark. The vectors g i and g j introduced above form the covariant and contravariant bases and the functions g i j are the contravariant components of the metric tensor (Section 1.2). 0 As shown in the above proof, the necessary conditions R q i j k = 0 thus simply constitute a re-writing of the relations a i k g j = a k i g j in the form of the equivalent relations d i k g j . g q = a k i g j . g q . The functions

and

rP. = g p q r i j q 23

= digj .g p =

rP. 3%

are the Christoffel symbols of the first, and second, kinds. We saw in Section 1.4 that the Christoffel symbols of the second kind also naturally appear in a different context (that of covariant differentiation). Finally, the functions

R w . .k

:=a.r. 3 t k q -akr.. z3q f r : j r k q p

- rfkrjqp

are the covariant components of the Riemann curvature tensor of the set 0 ( R ) . The relations R q i j k = 0 found in Theorem 1.4-1 thus express that the Riemann curvature tensor of the set 0 ( R ) (equipped with the metric tensor with covariant components g i j ) vanishes.

1.6

Existence of an immersion defined on an open set in R3 with a prescribed metric tensor

Let M3,S3, and S; denote the sets of all square matrices of order three, of all symmetric matrices of order three, and of all symmetric positive definite matrices of order three. As in Section 1.2, the matrix representing the F’rkhet derivative at 2 E R of a differentiable mapping 0 = : R -+ E3 is denoted

(el)

VO(Z) := (ajo((2))E M3, where l is the row index and j the column index (equivalently, VO(z) is the matrix of order three whose j-th column vector is a j 0 ) . So far, we have considered that we are given an open set R c R3 and a smooth enough immersion 0 : R -+ E3, thus allowing us to define a matrix field

c = ( g q ) = VOTVO : R + S3>,

Philippe G. Ciarlet

20

where gz3: R --t IK are the covariant components of the metrac tensor of the open set @(R) c E3. We now turn t o the recaprocal questaons: Given an open subset R of R3 and a smooth enough matrix field C = (gt3) : R 4 S,3>,when is C the metric tensor field of an open set @(a)c E3? Equivalently, when does there exzst a n zmmerszon 0 . R 4 E3 such that c = V O ~ V Oin R,

or equavalently, such that gz3 =

a,@ . a,@

in R?

If such an immersion exists, t o what extent is it unique? The answers are remarkably simple: If R as szmply-connected, the necessary condataons

found in Theorem 1.7-1 are also suficient for the existence of such a n immersion. If R is connected, this immersion is unique u p t o isometries an E3. Whether the immersion found in this fashion is globally injective is a different issue, which accordingly should be resolved by different means. This result comprises two essentially distinct parts, a global existence result (Theorem 1.6-1) and a uniqueness result (Theorem 1.7-1). Note that these two results are established under different assumptions, on both the set R and the smoothness of the field ( g i j ) . In order t o put these results in a wider perspective, let us make a brief incursion into Riemannian Geometry. For detailed treatments, see classic texts such as Choquet-Bruhat, de Witt-Morette & Dillard-Bleick [1977], Marsden & Hughes [1983], Berger [2003], or Gallot, Hulin & Lafontaine [2004]. Considered as a three-dimensional manifold, an open set R c R3 equipped with an immersion 0 : R + E3 becomes an example of a Riemannian manifold (0;( g i j ) ) ,i.e., a manifold, viz., the set R, equipped with a Riemannian metric, viz., the symmetric positive-definite matrix field ( g i j ) : R St defined in this case by gij := &@ . d j @ in R. More generally, a Riemannian metric on a manifold is a twice covariant, symmetric, positive-definite tensor field acting on vectors in the tangent spaces t o the manifold (these tangent spaces coincide with R3 in the present instance). This particular Riemannian manifold (R; (gij )) possesses the remarkable property that its metric is the same as that of the surrounding space E3. More specifically, (a;( g i j ) ) is isometrically immersed in the Euclidean space E3,in the sense that there exists an immersion 0 : R + E3 ---f

An Introduction to Differential Geometry in R3

21

that satisfies the relations gij = &O . a j 0 . Equivalently, the length of any curve in the Riemannian manifold (R; ( g i j ) ) is the same as the length of its image by 0 in the Euclidean space E3 (see Theorem 1.3-1). The first question above can thus be rephrased as follows: Given a n open subset R of R3 and a positive-definite matrix field ( g i j ) : R -+ s,3>, when is the Riemannian manifold (R; ( g i j ) ) flat, in the sense that it can be isometrically immersed in a Euclidean space of the same dimension (three)? The answer to this question can then be rephrased as follows (compare with the statement of Theorem 1.6-1 below): Let R be a simplyconnected open subset of R3. T h e n a Riemannian manifold (0;( g i j ) ) with a Riemannian metric ( g i j ) of class C2 in R is flat i f and only if its Riemannian curvature tensor vanishes in R. Recast as such, this result becomes a special case of the fundamental theorem on flat Riemannian manifolds, which holds for a general finite-dimensional Riemannian manifold. The answer to the second question, viz., the issue of uniqueness, can be rephrased as follows (compare with the statement of Theorem 1.7-1 in the next section): Let R be a connected open subset of R3. T h e n the isometric immersions of a flat Riemannian manifold (R; ( g i j ) ) into a Euclidean space E3 are unique up to isometries of E3. Recast as such, this result likewise becomes a special case of the so-called rigidity theorem; cf. Section 1.7. Recast as such, these two theorems together constitute a special case (that where the dimensions of the manifold and of the Euclidean space are both equal to three) of the fundamental theorem of Riemannian Geometry. This theorem addresses the same existence and uniqueness questions in the more general setting where R is replaced by a p-dimensional manifold and E3 is replaced by a (p q)-dimensional Euclidean space (the “fundamental theorem of surface theory”, established in Sections 2.8 and 2.9, constitutes another important special case). When the p-dimensional manifold is an open subset of E%P+q, an outline of a self-contained proof is given in Szopos [2005]. Another fascinating question (which will not be addressed here) is the following: Given again an open subset R of EX3 equipped with a symmetric, positive-definite matrix field ( g i j ) : R -+ S,3>,assume this time that the Riemannian manifold (0;( g i j ) ) is n o longer flat, i.e., its Riemannian curvature tensor no longer vanishes in R. Can such a Riemannian manifold still be isometrically immersed, but this time in a higher-dimensional Euclidean space? Equivalently, does there exist a Euclidean space Ed with d > 3 and does there exist an immersion 0 : R + Ed such that gij = &O .ajO in R? The answer is yes, according to the following beautiful Nash theorem, so named after Nash [1954]: A n y p-dimensional Riemannian man-

+

Philippe G. Ciarlet

22

ifold equipped with a continuous metric can be isometrically immersed in a Euclidean space of dimension 2p with a n immersion of class C1; it can also be isometrically immersed in a Euclidean space of dimension (2p 1) with a globally injective immersion of class C1. Let us now humbly return to the question of existence raised at the beginning of this section, i.e., when the manifold is an open set in R3.

+

Theorem 1.6-1. Let R be a connected and simply-connected open set in R3 and let C = ( g i j ) E C2(R; S); be a matrix field that satisfies R q i j k := djrikq - a k r i j q

+ r;’rkqp - r;krjqp = 0 in 0,

where 1 r 239 . . .- -(a. . + a&,, 2 3Q2q ’-

rp. Y :=

p r i j qwith

-

($9)

an%), := ( g i j ) - l .

T h e n there exists a n immersion 0 E C3(R;E3) such that

c = V O ~ V Oin R. Pro05 The proof relies on a simple, yet crucial, observation. When a smooth enough immersion 0 = (Oe) : R + E3 is a priori given (as it was so far), its components Oe satisfy the relations &jOe = ryjapOe7 which are nothing but another way of writing the relations &gj = ryjgp (see the proof of Theorem 1.5-1). This observation thus suggests to begin by solving (see part (ii)) the system of partial differential equations &Fej = rP.Fep in R, 23 whose solutions Fej : R -+ R then constitute natural candidates for the partial derivatives ajoe of the unknown immersion 0 = (Be) : R 4 E3 (see part (iii)). To begin with, we establish in (i) relations that will in turn allow us t o re-write the sufficient conditions djrikq - a k r i j q

+ rfjrkqP - rykrjqp = o in

in a slightly different form, more appropriate for the existence result of part (ii). Note that the positive definiteness of the symmetric matrices ( g i j ) is not needed for this purpose. (i) Let R be a n open subset 0fR3 and let there be given a field ( g i j ) E C2(R; S 3 ) of symmetric invertible matrices. The functions rijq,I?;’, and gpq being defined by 1

rijq:= p j g i q + aigjq - aqgij), rTj := g p q r 23q7 .

( S P 4 ) := (gijl-l,

An Introduction to Differential Geometry in IR3

23

define the functions

ajrikq- akrijq + r:jrkqp- rTkrjqP, := a j q k akr$+ rfkrye rfjqe.

R q i j k :=

R&

-

-

Then

Rp.. '23k = gPqRqijk and

R p i j k = gpqR:jk.

Using the relations rjqet rejq = ajgqe

and

r i k q = gqtrfk,

which themselves follow from the definitions of the functions and noting that

rijqand

(gPqajgqe+ gqeajgPq)= aj (gPqgqe)= 0 , we obtain

gPq(ajrikq - rLkrjqT) = ajrYk- rikqajgpq - retkgPq(ajg,e

-

rejq)

a j q k t F f k r y Q- rfk(gPqajgqet gqeajgPq) = a j q k + &rye. =

Likewise,

gpq(dkrijq - rLjrkqT) = akrYj- rfjrie, and thus the relations R:jk = gPqRqijk are established. The relations Rq23k . . - gpqRpijk are clearly equivalent to these ones. We next establish the existence of solutions to the system ai Fej

=':?I

Fep in R.

(ii) Let R be a connected and simply-connected open subset of IR3 and = E C'(R) satisfying the relations let there be given functions

ajqk akrfj+ rfkr;e- rfjrge= o in R, -

which are equivalent t o the relations

ajrikq - akrijq + r$rkqp- rTkrjqP = o in R, by part (i). Let a point xo E R and a matrix (F&) E M3 be given. T h e n there exists one, and only one, field (Fej) E C2(R; M3) that satisfies &FQ~(= x ) I'Fj(x)Fep(x),x E 0, Fej(x0)= Fej. 0

Philippe G. Ciarlet

24

Let x1 be an arbitrary point in the set R , distinct f r o m xo. Since R is connected, there exists a path y = (ri)E C1([O, l];R3) joining xo to x1 in R; this means that

y(0) = xo,$1) = xl, and y ( t )E R for all 0

< t 6 1.

Assume that a matrix field (Fej)E C1(R;M3) satisfies &Fej(x) = E 0. Then, f o r each integer C E {1,2,3}, the three functions c j E C1([O, 11) defined by (for simplicity, the dependence on C is dropped) C j ( t ) := Fej(r(t)), 0 6 t 6 1,

ryj(x)Fep(x),x

satisfy the following Cauchy problem for a Linear system of three ordinary differential equations with respect to three unknowns:

em c;, =

where the initial values

are given by

co3 := F&. Note in passing that the three Cauchy problems obtained by letting

C = 1 , 2 , or 3 only differ by their initial values . The regularity assumption on the components g i j of the symmetric positive definite matrix field C = (gij) made in Theorem 1.6-1, viz., that g i j E C2(R), can be significantly weakened. More specifically, C. Mardare [2003] has shown that the existence theorem still holds if gij E C’(fl), with a resulting mapping 0 in the space C2(R;Ed). Then S. Mardare [2004] has shown that the existence theorem still holds if gij E WLCm(R), with a resulting mapping 0 in the space W$‘T((R; E d ) . As expected, the sufficient conditions R q i j k = 0 in of Theorem 1.6-1 are then assumed to hold only in the sense of distributions, viz., as

for all p E D(R). The existence result has also been extended “up to the boundary of the set R” by Ciarlet & C. Mardare [2004b]. More specifically, assume that the set R satisfies the “geodesic property” (in effect, a mild smoothness assumption on the boundary dR, satisfied in particular if dR is Lipschitzcontinuous) and that the functions gij and their partial derivatives of the symmetric order 6 2 can be extended by continuity to the closure matrix field extended in this fashion remaining positive-definite over the set Then the immersion 0 and its partial derivatives of order 6 3 can be also extended by continuity to 2. Ciarlet & C. Mardare [2004a]have also shown that, if in addition the geodesic distance is equivalent to the Euclidean distance on R (a property stronger than the “geodesic property”, but again satisfied if the boundary dR is Lipschitz-continuous), then a matrix field (gij) E C 2 ( a ; S y )

a,

a.

An Introduction to Differential Geometry in W3

31

with a Riemann curvatu_re tensor vanishing in R can be extended to a matrix field (&) E C2(R;S,n>)defined on a connected open set 6 containing and whose Riemann curvature tensor still vanishes in 6. This result relies on the existence of continuous extensions to of the immersion 0 and its partial derivatives of order 6 3 and on a deep extension theorem of Whitney [1934].

a

1.7 Uniqueness up to isometries of immersions with the same metric tensor In Section 1.6, we have established the existence of an immersion 0 : 52 c W3 -+ E3 giving rise to a set @(a)with a prescribed metric tensor, provided the given metric tensor field satisfies ad hoc sufficient conditions. We now turn to the question of uniqueness of such immersions. This uniqueness result is the object of the next theorem, aptly called a rigidity theorem in view of its geometrical interpretation: It asserts that, if two immersions 6 E C1(R;E3) and 0 E C1(R;E3) share the same metric tensor field, then the set O(52) is obtained by subjecting the set 6 ( R ) either to a rotation (represented by an orthogonal matrix Q with det Q = l),or to a symmetry with respect t o a plane followed by a rotation (together represented by an orthogonal matrix Q with det Q = -l), then by subjecting the rotated set to a translation (represented by a vector c ) . The terminology “rigidity theorem” reflects that such a geometric transformation indeed corresponds to the idea of a “rigid transformation” of the set 0(52)(provided a symmetry is included in this definition). Let O3 denote the set of all orthogonal matrices of order three.

Theorem 1.7-1. Let 52 be a connected open subset of R3 and let 0 E C1(R; E3) and 6 E C1(R;E3) be two immersions such that their associated metric tensors satisfy

VOTVO

=

~

6 in R. ~

~

6

T h e n there exist a vector c E E3 and a n orthogonal matrix Q 6 O3 such that o(IL:) = c + Q ~ ( xfor ) all IL: E R. Proof. Fhe three-dimensional vector space R3 is identified throughout this proof with the Euclidean space E3. In particular then, R3 inherits the inner product and norm of E3. The spectral n o r m of a matrix A E M3 is denoted ( A (:= sup{lAb(; b E R3,( b (= 1).

32

Philippe G. Ciarlet

To begin with, we consider the special case where 6 : R -+ E3 = R3 is the identity mapping. The issue of uniqueness reduces in this case to finding 0 E C1(R; E3) such that

V O ( ~ ) ~ V O=(I~for ) all x E R. Parts (i) to (iii) are devoted to solving these equations. (i) We first establish that a mapping 0 E C1(R; E3) that satisfies

V O ( ~ ) ~ V O=( I~for ) all x E R i s locally a n isometry: Given a n y point xo E 0, there exists a n open neighborhood V of xo contained in R such that IO(y) - O(x)I = Iy - xi for all z, y E V Let B be an open ball centered at xo and contained in R. Since the set B is convex, the mean-value theorem (for a proof, see, e.g., Schwartz [1992]) can be applied. It shows that

I O ( y ) - O(x)I < sup IVO(z)l/y- z / for all z,y E B. zElx>Yl

Since the spectral norm of an orthogonal matrix is one, we thus have lO(y) - O(x)1

< ly - xi for all z,y E B.

Since the matrix VO(zo) is invertible, the local inversion theorem (for a proof, see, e.g., Schwartz [1992]) shows that there exist an open neighborhood V of xo contained in R and an open neighborhood p of O ( x o )in E3 such that the restriction of 0 to V is a C1-diffeomorphism from V onto p. Besides, thereis no loss of generality in assuming that V is contained in B and that V is convex (to see this, apply the local inversion theorem first to the restriction of 0 to B , thus producing a "first" neighborhood V' of zo contained in B , then to the restriction of the inverse mapping obtained in this fashion to an open ball V centered at O ( x o )and contained in O ( V ' ) ) . Let 0-l : p + V denote the inverse mapping of 0 : V + p. The chain rule applied to the relation O-'(O(z)) = x for all z E V then shows that

VO-'(Z) = V O ( ~ ) - 'for all

z=

x

E

V.

The matrix VO-'(Z) being thus orthogonal for all 2 E p,the meanvalue theorem applied in the convex set shows that

le-'(Q) - W'(Z)I < 1 ~ -21for all Z , Q E V, h

An Introduction to Differential Geometry in R3

33

or equivalently, that

( y - z ( < ( O ( y )- O(z)(for all z, y E V. The restriction of the mapping 0 to the open neighborhood V of zo is thus an isometry. (ii) We next establish that, if a mapping 0 E C1(R;E3) is locally a n isometry, in the sense that, given any xo E R, there exists an open neighborhood V of xo contained in R such that I O ( y ) - O(z)(= Iy - ZI for all x , y E V , then its derivative i s locally constant, in the sense that

VO(Z)= VO(Z') for all z

E

K

The set V being that found in (i), let the differentiable function F : V x V + R be defined for all x = ( z p )E V and all y = ( y p )E V by

F ( X , Y ) :=

(WY)

@ e ( x ) ) ( @ e ( y) @e(z))- (Ye - Q)(Y!

-

ze)

for all z, y E V . For a fixed y E V, each function Gi(.,y ) : V differentiable and its derivative vanishes. Consequently,

+

Then F ( z , y )

dGa dXi

-(z,

=0

-

for all z, y E V by (i). Hence

d@e dOe y ) = --(y)-(z) dya dzj

JR is

+ 6 i j = 0 for all z, y E V ,

or equivalently, in matrix form,

Letting y = xo in this relation shows that

VO(z) = V O ( X ' )for all z E V. (iii) By (ii), the mapping V O : R + M3 is differentiable and its derivative vanishes in R. Therefore the mapping 0 : R + E3 is twice differentiable and its second Fre'chet derivative vanishes an a. The open set R being connected, a classical result from differential calculus (see, e.g., Schwartz [1992, Theorem 3.7.101) shows that the mapping 0 is a f i n e in R, i.e., that there exists a vector c E E3 and a matrix Q E M3 such that (the notation oz designates the column vector with components zi) O(z) = c + Qoz for all x E 0.

34

Philippe G. Ciarlet

= I by assumption, the Since Q = VO(zo) and VO(zO)TVO(zO) matrix Q is orthogonal.

-

(iv) We now consider the general equations gij = g23 in R, noting that they also read

vo(z)Tvo(z) = v G ( ~ ) ~ v G (for ~ )all z E R.

c

Given any point zo E R, let ;he neighborhoods V of zo and of O(zo) and the mapping 0-1 : V -+ V be defined as in part (i) (by assumption, the mapping 0 is an immersion; hence the matrix V@(zo) '"is invertible). Consider the composite mapping

& := 6 o 0-l : 9 -+ E3. Clearly, 6 E C1@; E3) and

v q q = VG(z)V@-yq =

v G ( ~ ) v E I ( ~ ) - 'for all 2= ~ ( z )z,E V.

Hence the assumed relations

V O ( ~ ) ' V O ( ~= )v

G ( ~ ) ~ v G (for~ )all z E R

imply that

e65(2)T6&(2) = I for all z

E V.

By parts (i) to (iii), there thus exist a vector c Q E O3 such that

E

R3 and a matrix

6(2)= G(z) = c + QO(Z) for all 2= ~ ( z )z, E V, hence such that =(z) :=

v G ( ~ ) v o ( ~ ) -=~ Q for all z E V.

The continuous mapping E : V -+M3 defined in this fashion is thus locally constant in R. As in part (iii), we conclude from the assumed connectedness of C2 that the mapping E is constant in R. Thus the proof is complete. 13 An isometry of E3 is a mapping J : E3 --+ E3 of the form J(z) = c+Q OX for all z E E3,with c E E3 and Q E O3 (an analogous definition holds verbatim in any Euclidean space of dimension d 2 2). Clearly, a n isometry preserves distances in the sense that

IJ(y) - J(z)I = Iy - $1 for all z, y E R.

An Introduction to Differential Geometry in R3

35

Remarkably, the converse is also true, according to the classical Mazur-Ulam theorem, which asserts the following: Let R be a connected subset in Rd, and let 0 : R -+ Rd be a mapping that satisfies

I@(Y)

-

@(%)I

= ly -

XI

for all x , y E R.

T h e n 0 is a n isometry of Rd. Parts (ii) and (iii) of the above proof thus provide a proof of this theorem under the additional assumption that the mapping 0 is of class C1 (the extension from R3 to Rd is trivial). In Theorem 1.7-1, the special case where 0 is the identity mapping of R3 identified with E3 is the classical Liouville theorem. This theorem thus asserts that if a mapping 0 E C1(R; E3) is such that V O ( x )E O3 for all x E R, where R is a n open connected subset of R3, then 0 is a n isometry. Two mappings 0 E C1(R;E3) and 6 E C1(R;E3) are said to be isometrically equivalent if there exist c E E3 and Q E O3 such that 0 = c f QO in R, i.e., such that 0 = J o 6 ,where J is an isometry of E3. Theorem 1.7-1 thus asserts that two immersions 0 E C1(R; E3) and 6 E C1(R; E3) share the same metric tensor field over a n open connected subset R of R3 i f and only i f they are isometrically equivalent. I

Remark. In terms of covariant components gij of metric tensors, parts (i) to (iii) of the above proof provide the solution to the equations gij = &i in R, while part (iv) provides the solution to the equations 9 23. . - &0 . aj6 in R , where 6 E C1(R; E3) is a given immersion. 0 While the immersions 0 found in Theorem 1.6-1 are thus only defined up to isometries in E3, they become uniquely determined if they are required t o satisfy ad hoc additional conditions, according t o the following corollary to Theorems 1.6-1 and 1.7-1.

Theorem 1.7-2. Let the assumptions o n the set R and o n the matrix field C be as in Theorem 1.6-1, let a point xo E R be given, and let Fo E M3 be any matrix that satisfies

FTF,, = c ( x o ) . T h e n there exists one and only one immersion 0 E C3(R;E3) that satisfies v @ ( x ) ~ ~ @ ( x=) C ( X )for all x E R, O ( x o )= 0 and V O ( x 0 )= Fo.

Proof. Given any immersion

a E C3(R; E3) that satisfies

Va(z)TVa(x) = C(x)

Philippe G. Ciarlet

36

for all x E R (such immersions exist by Theorem 1.6-1), let the mapping 0 : R 3 R3 be defined by

O ( x ) := FoV+(xO)-l(+(x) - +(zo)) for all x

E R.

Then it is immediately verified that this mapping 0 satisfies the announced properties. Besides, it is uniquely determined. To see this, let 0 E C3(R;E3) and E C3(R; E3) be two immersions that satisfy

+

v @ ( ~ ) ~ v=oV( +x()S ) ~ V + (for ~ )all x E R. Hence there exist (by Theorem 1.7-1) c E R3 and Q E O3 such that a(x)= c QO(x) for all 5 E $2, so that V@(x) = Q V O ( x ) for all x E R. The relation VO(x0) = V+(xo) then implies that Q = I and 0 the relation O(z0) = ~ ( Z O in ) turn implies that c = 0.

+

Remark. One possible choice for the matrix Fo is the square root of 0 the symmetric positive-definite matrix C (x 0 ) . Theorem 1.7-1 constitutes the “classical” rigidity theorem, in that both mappings 0 and 6 are assumed to be in the space C1(R; E3). The next theorem is an extension, due to Ciarlet & C. Mardare [2003], that covers the case where one of the mappings belongs t o the Sobolev space H1(R; E3). The way the result in part (i) of the next proof is derived is due to F’riesecke, James & Muller [2002]; the result of part (i) itself goes back to Reshetnyak [1967]. Let 0 : denote the set of all proper orthogonal matrices of order three, i.e., of all orthogonal matrices Q E O3 with det Q = 1. Theorem 1.7-3. Let R be a connected open subset of R3, let 0 C1(R; E3) be a mapping that satisfies

E

d e t V O > 0 in R,

and let

6 E H1(R; E3) be a mapping

that satisfies

det 06 > 0 a.e. in R and VOTVO = 06’06 a.e. in R.

T h e n there exist a vector c E E3 and a matrix Q

6(z) = c + Q@(Z)

E

0; such that

for almost all x E R.

Proof. The Euclidean space E3 is identified with the space R3 throughout the proof.

An Introduction to Differential Geometry in R3

37

(i) To begin with, we consider the special case where Q ( x ) = x for all x E 0. In other words, we arc given a mapping 6 E H1(R) that satisfies vO(z) E 0: for almost all x E R. Hence C o f V G ( z ) = (det V 6 ( x ) ) V 6 ( ~ ) = - ~V ~ ( X ) - ~ for almost all

II:

E R, on the one hand. Since, on the other hand,

div C o f V 6 = 0 in ( D ' ( B ) ) 3 in any open ball B such that B c R (to see this, combine the density of C2(B)in H 1 ( B )with the classical Piola identity in the space C2(B); for a proof of this identity, see, e.g., Ciarlet [1988, Theorem 1.7.1]),we conclude that

A6 Hence

= divCofV6 = 0

6 = (6j) E (C"(R))3. a(a&aiGj)

in ( D ' ( B ) ) 3 .

For such mappings, the identity

+ 2t4k6jaik6j,

= aai6jai(~6j)

-

-

together with the relations AC3j = 0 and &Oj&6j = 3 in R, shows that aikC3j = 0 in R. The assumed connectedness of R then implies that there exist a vector c E E3 and a matrix Q E 0; (by assumption, V ~ ( ZE )0;for almost all II: E 0) such that

-

6 ( x )= c + Q ox for almost all x E R. (ii) Consider next the general case. Let xo E R be given. Since 0 is a n immersion, the local inversion theorem can b_e applied; there thus exist bounded open neighborhoods U of zo and U of Q(Q) satisfying U c R and {6}c Q(R), such that the restriction Q U of 0 to U can be extended to a C1-diffeomorphism from onto {6}-. Let 0 ,' : 64 U denote the inverse mapping of O U ,which therefore satisfies VQG'(2) = VO(x)-l for all 2 = Q ( x ) E (the notation 6 indicates that differentiation is carried out with respect to the variable 2E Define the composite mapping

u

6

G).

Since from

6

E H1(U) and 0;' can be extended t o a C1-diffeornorphism {6}-onto g , it follows that 6 E H1(6;R3) and that VG(2) = V6(2)60,1(2) = v o ( x ) v o ( x ) - '

Philippe G. Ciarlet

38

6

for almost all P = O(z) E (see, e.g., Adams [1975, Chapter 31). Hence the assumptions det VO > 0 in R, det V 6 > 0 a.e. in R, and

VOTVO = V G T V 6 a.e. in R, together imply that 6 & ( P ) E 0 : for almost all 2 E 6.By (i), there thus exist c E E3 and Q E 0 : such that h

&(2)= 6(z) = c + Q 02 for almost all P = ~ ( z E) U , or equivalently, such that ~ ( z:= )

v O ( z ) v ~ ( z ) -=l Q for almost all z E U.

Since the point zo E R is arbitrary, this relation shows that E E Ltoc(R). By a classical result from distribution theory (cf. Schwartz [1966, Section 2.6]), we conclude from the assumed connectedness of R that E(z) = Q for almost all z E R, and consequently that

6(z) = c + Q O ( ~for ) almost all z E Q.

0

Remarks. (1) The existence of 6 E H1(R; E3) satisfying the assumptions of Theorem 1.7-3 thus implies that 0 € H1(R;E3) and 6 E C1(R; E3). (2) If 6 E C1(R;E3), the assumptions d e t V O > 0 in R and d e t V 6 > 0 in R are no longer necessary; but then it can only be concluded that Q E 03:This is the classical rigidity theorem (Theorem 1.7-1), of which Liouville's theorem is the special case corresponding to O(z) = z for all z E R. (3) The result established in part (i) of the above proof asserts that, given a connected open subset R of R3, if a mapping 0 E H1(R; E3) is such that VO(z) E 0%for almost all z E R, then there exist c E E3 and Q E 0 : such that O(z) = c &ox for almost all z E R. This result thus constitutes a generalizatio_n of Liouville 's theorem. (4)By contrast, if the mapping 0 is assumed to be instead in the space H1(R;E3)(as in Theorem 1.7-3), an assumption about the sign of det 06 becomes necessary. To see this, let for instance R be an open ball centered-at the origin in R3, let O(z) = z, and let 6(z) = z if z 1 2 0 and O(z) = ( - z 1 , 2 2 , 2 3 ) if z 1 < 0. Then 6 E H1(R;E3) and V 6 E O3 a.e. in R; yet there does not exist any orthogonal matrix such that 6(z) = Qo.: for all z E R, since 6 ( R ) c {z E R3;z1 3 0) (this counter-example was kindly communicated to the author by Sorin Mardare). (5) Surprisingly, the assumption d e t V O > 0 in R cannot be replaced by the weaker assumption det VO > 0 a.e. in R. To see this, let for instance R be an open ball centered at the origin in R3, let

+

An Introduction to Differential Geometry in R3

O ( x ) = O(z) if 22

39

O ( x ) = ( x 1 ~ ~ , ~ 2 , and 2 3 ) let 2 0 and 6 ( x ) = ( - q x i , - 2 2 , ~ ) if 22 < 0 (this counter-example was kindly communicated to the author by HervB Le Dret). (6) If a mapping 0 E C1(R;E3) satisfies d e t V O > 0 in 0, then 0 is an immersion. Conversely, if R is a connected open set and O E C1(R; E3) is an immersion, then either det VO > 0 in R or det VO < 0 in R. The assumption that det VO > 0 in R made in Theorem 1.7-3 is simply intended to fix ideas (a similar result clearly holds under the other assumption). (7) A little further ado shows that the conclusion of Theorem 1.73 is still valid if 6 E H1(R;E3)is replaced by the weaker assumption

6 E Hk,(R;E3).

0

Like the existence results of Section 1.6, the uniqueness theorems of this section hold verbatim in any dimension d 2 2, with R3 replaced by Rd and Ed by a d-dimensional Euclidean space.

1.8

Continuity of an immersion as a function of its metric tensor

Let R be a connected and simply-connected open subset of R3. Together, Theorems 1.6-1 and 1.7-1 establish the existence of a mapping 3 that associates with any matrix field C = ( g i j ) E C2(R;S); satisfying R q i j k := ajrikq-

+ r;$'kqp

-

I'ykrjqp = 0 in 0,

where the functions rij, and l7$ are defined in terms of the functions g i j as in Theorem 1.6-1, a well-defined element 3 ( C ) in the quotient set C3(R; E3)/R,where ( O , 6 )E R means that there exist a vector a E E3 and a matrix Q E CD3 such that O(z) = a + Q6(z) for all 2 E R. A natural question thus arises as to whether the mapping 3 defined in this fashion is continuous. Equivalently, is a n immersion a continuous function of its metric tensor? When both spaces C2(R; S3) and C3(R; E3) are equipped with their natural Frkchet topologies, a positive answer to this question has been provided by Ciarlet & Laurent [2003]. More specifically, let R be an open subset of R3. The notation K R 0, and means that K is a compact subset of R. If g E C'(R;W),t K cz R, we define the semi-norms

aa

where stands for the standard multi-index notation for partial derivatives. If 0 E Ce(R; E3) or A E Ce(R; M3), 1 2 0, and K @ R, we likewise

Philippe G. Ciarlet

40

set

where 1.1 denotes either the Euclidean vector norm or the matrix spectral norm. Then, for any integers C 3 0 and d 3 1, the space Ce(L?;Rd)becomes a locally convex topological space when it is equipped with the Fre'chet topology defined by the family of semi-norms /l./le,K, K R, defined above. Then a sequence (On),20 converges t o 0 with respect t o this topology if and only if lim

n-cc

/I@"

- Olle,~ = 0 for all

K G R.

Furthermore, this topology is metrizable: Let (Ki)+o be any sequence of subsets of R that satisfy

u 00

Ki CE L? and Ki

i 3 0, and R

C int Ki+l for all

=

Ki.

i=O

Then

~ 0 for all K G R lim 110" - O l l e , = ?L--tco

+ lim

n-+m

&(On,0)= 0,

where

For details about F'rkchet topologies, see, e.g., Yosida [1966, Chapter 11. Let C3(R; E3) := C3(R; E3)/R denote the quotient set of C3(R; E3) by the equivalence relation R,where E R means that 0 and 6 are isometrically equivalent (Section 1.7), i.e., that there exist a vector c E E3 and a matrix Q E O3 such that 0 ( x ) = c + Q G ( x ) for all x E R. Then it is easily verified that the set C3(R; E3) becomes a metric space when it is equipped with the distance d 3 defined by

(0,G)

&(6,+) =

{ where

inf.

dg()i,

;z7

x)=

inf

&(GIc

{ z::

+ Q+),

6 denotes the equivalence class of 0 modulo R.

An Introduction to Differential Geometry in R3

41

The continuity of an immersion as a function of its metric tensor has then been established by Ciarlet & Laurent [2003], according to the following result (if d is a metric defined on a set X , the associated metric space is denoted { X ;d } ) .

Theorem 1.8-1. Let R be a connected and simply-connected open subset ofR3. Let

Cz(R; S;)

:= {(gij) E

C2(R; s”,; R q i j k

=0

in

a},

and, given any matrix field C = ( g i j ) E C,2(R;S : ) , let F(C)E C3(R; E3) denote the equivalence class modulo R of any 0 E C3(R2;E3) that satisfies VOTVO = C in R. T h e n the mapping

F : {C,”(R; S;); d2)

-

defined in this fashion is continuous.

{C3(R; E3);d 3 )

0

As shown by Ciarlet & C. Mardare [2004b], the above continuity result can be extended “up to the boundary of the set R”, as follows. If R is bounded and has a Lipschitz-continuous boundary, the mapping F can be extended to a mapping that is locally Lipschitz-continuous with respect to the topologies of the Banach spaces C2(a;S3)for the continuous extensions of the symmetric matrix fields C , and C3@; E3) for the continuous extensions of the immersions 0. Another extension, essentially motivated by nonlinear three-dimensional elasticity, is the following: Let R be a bounded and connected subset of W3, and let B be an elastic body with R as its reference configuration. Thanks mostly to the landmark existence theory of Ball [1977], it is now customary in nonlinear three-dimensional elasticity to view any mapping 0 E H1(R;E3)that is almost-everywhere injective and satisfies det V 0 > 0 a.e. in R as a possible deformation of B when B is subjected to ad hoc applied forces and boundary conditions. The almosteverywhere injectivity of 0 (understood in the sense of Ciarlet & NeEas [1987])and the restriction on the sign of det V O mathematically express (in an arguably weak way) the non-interpenetrability and orientationpreserving conditions that any physically realistic deformation should satisfy. It then turns out that the metric tensor field VOTVO E L1(R; S 3 ) , also known as the Cauchy-Green tensor field in elasticity, associated with a deformation 0 E H1(R;E3)pervades the mathematical modeling of three-dimensional nonlinear elasticity (extensive treatments of this subject are found in Marsden & Hughes [1983] and Ciarlet [1988]). Conceivably, an alternative approach to the existence theory in threedimensional elasticity could thus regard the Cauchy-Green tensor as the primary unknown, instead of the deformation itself as is usually the case.

Philippe G. Ciarlet

42

Clearly, the Cauchy-Green tensors depend continuously o n the deformations, since the Cauchy-Schwarz inequality immediately shows that the mapping

0 E H1(O; E3)

---f

VOTVO E L1(O; S3)

is continuous (irrespectively of whether the mappings 0 are almosteverywhere injective and orientation-preserving). Then Ciarlet & C. Mardare [2004c] have shown that, under appropriate smoothness and orientation-preserving assumptions, the converse holds, i. e., the deformations depend continuously o n their Cauchy-Green tensors, the topologies being those of the same spaces H1(R;E3)and L1(0;S3) (by contrast with the orientation-preserving condition, the issue of non-interpenetrability turns out to be irrelevant to this issue). This continuity result is itself a simple consequence of the following nonlinear Korn inequality, which constitutes the main result of ibid.: Let R be a bounded and connected open subset of R3 with a Lipschitzcontinuous boundary and let 0 E C1(n;E3)be a mapping satisfying det V 0 > 0 in Then there exists a constant C(0) with the following property: For each orientation-preserving mapping 9 E H1(QE3), there exist a proper orthogonal matrix R = R ( 9 ,0 )of order three (i.e., an'orthogonal matrix of order three with a determinant equal to one) and a vector b = b ( 9 ,0 )in E3 such that

n.

1 1 9- (b

+ R O ) I ( H ~ ( ~6; C(@)IIV@V@ E~) - V@TV@/I$n;s3).

That a vector b and an orthogonal matrix R should appear in the lefthand side of such an inequality is of course reminiscent of the classical rigidity theorem (Theorem 1.7-1), which shows that, if two mappings 6 E C1(R; E3) and 0 E C1(R; E3) satisfying det 06 > 0 and det V 0 > 0 in an open connected subset R of R3 have the same metric tensor field, then the two mappings are isometrically equivalent, i.e., there exist a vector-b in E3 and a proper orthogonal matrix R of order three such that 0 ( z ) = b RO(z) for all z E 0. More generally, we shall say that two orientation-preserving mappings 6 € H1(R; E3) and 0 € H1(R; E3) are isometrically equivalent if there exist a vector b in E3 and an orthogonal matrix R of order three (a proper one in this case) such that

+

6(z) = b + R e ( % for ) almost all z E R. One application of the above key inequality is the following sequential continuity property: Let O k E H1(R;E3),k 3 1, and 0 E C1(2;E3) be orientation-preserving mappings. Then there exist a constant C ( 0 )

An Introduction to Differential Geometry in R3

43

-k

and orientation-preserving mappings 0 E H1(R; E3),k 2 1, that are isometrically equivalent to O k such that

-k

)El

Hence the sequence (0 converges to 0 in H1(R; E3) as k + 00 if the sequence ( ( v o ~ ) ~ v oconverges ~ ) ~ =to. _ V ,O ~ V Oin L ~ ( R~; 3 as )

k+m. Should the Cauchy-Green strain tensor be viewed as the primary unknown (as suggested above), such a sequential continuity could thus prove to be useful when considering injimizing sequences of the total energy, in particular for handling the part of the energy that takes into account the applied forces and the boundary conditions, which are both naturally expressed in terms of the deformation itself. They key inequality is first established in the special case where 0 is the identity mapping of the set R, by making use in particular of a fundamental “geometric rigidity lemma” recently proved by F’riesecke, James & Muller [2002]. It is then extended to an arbitrary mapping 0 E C1@ EXn) satisfying det V 0 > 0 in 2,thanks in particular to a methodology that bears some similarity with that used in Ciarlet & Laurent [2003]. Such results are to be compared with the earlier, pioneering estimates of John [1961], John [1972] and Kohn [1982], which implied continuity at rigid body deformations, i.e., at a mapping 0 that is isometrically equivalent to the identity mapping of R. The recent and noteworthy continuity result of Reshetnyak [2003]for quasi-isometric mappings is in a sense complementary to the above one (it also deals with Sobolev type norms).

2

Differential geometry of surfaces

Outline We saw in Part 1 that an open set 0 ( R ) in E3,where R is an open set in R3 and 0 : R + E3 is a smooth injective immersion, is unambiguously defined (up to isometries of E3) by a single tensor field, the metric tensor field, whose covariant components g i j = gji : R -+ R are given by 923” ’- ai0 . a j 0 . Consider instead a surface G = O(w) in E3, where w is a twodimensional open set in R2 and 8 : w + E3 is a smooth injective immersion. Then by contrast, such a “two-dimensional manifold” equipped with the coordinates of the points of w as its curvilinear coordinates,

44

Philippe G. Ciarlet

requires two tensor fields for its definition (this time up to proper isometries of E3),the first and second fundamental forms of G.Their covariant components a,p = ap, : w -+ R and bag = bp, : w R are respectively given by (Greek indices or exponents take their values in {1,2}): -+

a,p

= a, . a g

and b,p

= a3 . & u p ,

The vector fields ai : w ’-+ R3 defined in this fashion constitute the covariant bases along the surface G I while the vector fields ai : w R3 defined by the relations ai . aj = 6; constitute the contravariant bases along 2. These two fundamental forms are introduced and studied in Sections 2.1 to 2.5. In particular, it is shown how areas and lengths, i.e., “metric notions”, o n the surface G are computed in terms of its curvilinear coordinates by means of the components a,p of the first fundamental form (Theorem 2.3-1). It is also shown how the curvature of a curve on G can be similarly computed, this time by means of the components of both fundamental forms (Theorem 2.4-1). Other classical notions about “curvature”, such as the principal curvatures and the Gaussian curvature, are introduced and briefly discussed in Section 2.5. We next introduce in Section 2.6 the fundamental notion of covariant derivatives qil, of a vector field qiai : w -+R3 o n G I thus defined here by means of its covariant components qi over the contravariant bases az. We establish in this process the formulas of Gauss and Weingarten (Theorem 2.6-1). Covariant derivatives of vector fields on a surface (typically, the unknown displacement vector field of the middle surface of a shell) pervade the equations of shell theory; see, e.g., Ciarlet [2000], Ciarlet [2005],or Ciarlet & C. Mardare [2007]in this Volume. It is a basic fact that the symmetric and positive definite matrix field (a,g) and the symmetric matrix field (hap) defined on w in this fashion cannot be arbitrary. More specifically, their components and some of their partial derivatives must satisfy necessary conditions taking the form of the following relations (meant to hold for all a , p, (T,T E { 1,2}), which respectively constitute the Gauss, and Codazzi-Mainardi, equations (Theorem 2.7-1): Let the functions rapT and be defined bY 1 r a p T = a(dpa,, &upT - &-a,,) and I’Ep = a U T a p T , --f

+

where (auT):= (a,p)-’.

8.d’,,,

-

apb,,

Then, necessarily,

+ r&ruTp r:,,rPTp= b,,bpT - bapbU, in - aUb,g + rgubfip - I ’ ~ o b u p= O in w.

W,p,

-

W,

An Introduction to Differential Geometry in R3

45

The functions r a p 7 and rgp are the Christoffel symbols of the first, and second, kind. We also establish in passing (Theorem 2.7-2) the celebrated Theorema Egregium of Gauss: At each point of a surface, the Gaussian curvature is a given function (the same for any surface) of the components of the first fundamental form and their partial derivatives of order 2 at the same point. We then turn to the reciprocal questions: Given an open subset w of R2 and a smooth enough symmetric and positive definite matrix field (a,p) together with a smooth enough symmetric matrix field (b,p) defined over w, when are they the first and second fundamental forms of a surface 8 ( w ) C E3, i.e., when does there exist an immersion 8 ; w + E3 such that


0 such that the symmetric matrices ( g i j ) are positive definite at all points in Ge, where

Re

:= we x

Finally, the open set

R

:=

I - E ~ ,E [ [ .

U Re e>o

is connected and simply-connected. Let we, t 2 0, be open subsets of w with compact closures We C w such that w = Ue20~e, For each C, a set Re := we x I - E ~ , E ~ [ can then be constructed in the same way that the set Ro was constructed in part (9. It is clear that the set R := Ue20 Re is connected. To show that R is simply-connected, let y be a loop in R , i.e., a mapping y E Co([O, 1];R3) that satisfies

y(0) = y(1) and y ( t ) E R for all 0 6 t 6 1. Let the projection operator 7r : R + w be defined by ~ ( y2 ,3 ) = y for all (y, x3) E R, and let the mapping po: [0,1] x [0,1] --t R3 be defined by

po(t,X):= (1- X)y(t)+ X7r(y(t))for all 0 Q t 6 I, O Q X 6 1. Then pois a continuous mapping such that p o ( [ O , 11 x [0,1]) c R, by definition of the set R. Furthermore, po(t,O)= y ( t ) and p o ( t , l )= 7r(y(t))for all t E [0, I]. The mapping y : = 7 r o y ECO([O,l];R2) is a loop in w since T ( 0 ) = rr(y(0)) = x(y(1)) = y(1) and y(t) E w for all 0 Q t 6 1. Since w is simply connected, there exist a mapping p1E Co([O, 11 x [0,1];R2) and a point yo E w such that

p l ( t ,1) = 7 and p l ( t ,2) = yo for all 0 6 t 6 1 and

p l ( t , X )E w for all 0 6 t

< 1, 1 Q X 6 2.

Then the mapping p E Co([O, 11 x [0, 2];R3) defined by

p ( t , X ) = po(t,X) for all 0 6 t 6 1, 0 < X 6 1, p ( t ,A) = pl(t,A) for all 0 Q t 6 1, 1 6 X 6 2, is a homotopy in R that reduces the loop y to the point ( y o l o ) E R. Hence the set R is simply-connected.

An Introduction t o Differential Geometry in R3 (ix) B y parts (iv) to (viii), the functions

rij,

E

81

C ' ( R ) and I;''

E

C1(R) constructed as in part (iii) satisfy

in the connected and simply-connected open set R . B y Theorem 1.6-1, there thus exists a n immersion 0 E C3(R; E3) such that gij

= 8iO. 8 j 0

in R,

where the matrix field ( g i j ) E C2(R;S3>) is defined by gap = a,p

-

2~3b,p

+ x3cap and gi3 = 2

6i3

in R.

T h e n the mapping 8 E C3(w;E 3 ) defined by 8 ( y ) = 0 ( y , O ) for all y

E w:

satisfies

Let gi := 8i0. Then 8 3 3 0 = 83g3 = I'g3gp = 0; cf. part (iii). Hence there exists a mapping 8l E C3(w;E 3 ) such that

+ X ~ @ ( Y ) for all ( y ,2 3 ) E 0, = 8,8 + ~ 3 8 ~ 8and ' g 3 = 8 l . The relations

@ ( Y ,2 3 ) = O ( Y )

and consequently, g, gi3 = g i . g 3 = &3 (cf. part (i)) then show that

(da8

+ ~ 3 8 ~ 8 ' ) = 0 and 8l . 8 l = 1.

These relations imply that 8,8 . 8l 8 1 = -a3 in w , where a3 :=

=

0. Hence either 8l

dl8 A 828 ld18 A 8281 '

But 8' = -a3 is ruled out since

( 8 1 8 A 828). 8l

= det(gij)lz3,0

> 0.

Noting that

8,8.

a3

= 0 implies 8,B. 8 p a 3 = - 8 4 3 . a3,

we obtain, on the one hand, gap

=

( a d+ x d a a 3 ) . ( 8 ~+8 x38pa3)

= 8,8.

8p8 - 2x3dap8. a3 + xidaa3 . 8pa3 in R.

= a3

or

82

Philippe G. Ciarlet

Since, on the other hand,

by part (i), we conclude that a,p

= d,8.

ape and b,p

= dap8.a 3 in

as desired. This completes the proof.

w, 0

Remarks. (1) The functions cap = bzbp., = d,a3 . spa, introduced in part (i) are the covariant components of the third fundamental form of the surface O(w). (2) The series expansion gap = l ) x ~ a a f f ( B "found )~ in part (i) is known; cf., e.g., Naghdi [1972c (3) The functions b:lp and b,plg introduced in part (ii) are the covariant derivatives of the second fundamental form of the surface 8(w); for details, see, e.g., [Ciarlet, 2005, Section 4.21. (4) The Gauss equations are used only once in the above proof, for showing that R l 2 1 2 = 0 in part (vii). 0

c,,o(n+

The regularity assumptions made in Theorem 2.8-1 on the matrix fields (a,p) and (b,p) can be significantly relaxed in several ways. First, Cristinel Mardare has shown by means of an ad hoe, but not trivial, modification of the proof given here, that the existence of an immersion 8 E C3(w;E3) still holds under the weaker (but certainly more natural, in view of the regularity of the resulting immersion 8) assumption that (b,p) E C1(w;S2), all other assumptions of Theorem 2.8-1 holding verbatim. In fact, Hartman & Wintner [1950] had already shown the stronger result that the existence theorem still holds if (a,p) E C'(w;S;) and (baa) E C o ( w ; S 2 ) ,with a resulting mapping 8 in the space C2(u;E3). Their result has been itself superseded by that of S. Mardare [2003b],who established that if (asp) E Wll,',"(w; S;) and (hap) E Lgc(w;S2) are two matrix fields that satisfy the Gauss and Codazzi-Mainardi equations in the sense of distributions, then there exists a mapping 8 E W;"F(u;E3) such that (a,p) and (b,p) are the fundamental forms of the surface 8 ( w ) . The last word in this direction seems to belong to S. Mardare [2005], who was able to further reduce these regularities, to those of the spaces W:d,p(w;S2>)and LFoc(w;S2)for any p > 2, with a resulting mapping 8 in the space W;":(w; E3).

An Introduction to Differential Geometry in JR3

2.9

83

Uniqueness up to proper isometries of surfaces with the same fundamental forms

In Section 2.8, we have established the existence of an immersion 8 : w c IR2 -+ E3 giving rise t o a surface 8 ( w ) with prescribed first and second fundamental forms, provided these forms satisfy ad hoc sufficient conditions. We now turn t o the question of uniqueness of such immersions. This is the object of the next theorem, which constitutes another rigidity theorem, called the rigidity theorem for surfaces. It asserts that, if two immersions 2 E C 2 ( q E 3 ) and 8 E C2(u;E3)share the same fundamental forms, then the surface 8(w) is obtained by subjecting the surface g(w) to a rotation (represented by an orthogonal matrix Q with det Q = l),then by subjecting the rotated surface to a translation (represented by a vector c ) . Such a geometric transformation of the surface G(w) is sometimes called a “rigid transformation” , t o remind that it corresponds to the idea of a “rigid” one in E3. This observation motivates the terminology “rigidity theorem” . As shown by Ciarlet & Larsonneur [2001] (whose proof is adapted here), the issue of uniqueness can be resolved as a corollary to its “threedimensional counterpart” , like the issue of existence. We recall that O3 denotes the set of all orthogonal matrices of order three. In addition, we let 0; := {Q E 0 3 ; d e t Q = 1) denote the set of all proper orthogonal matrices of order three.

Theorem 2.9-1. Let w be a connected open subset of R2 and let 8 E C2(w; E3) and 2 E C2(w; E3) be two immersions such that their associated first and second fundamental forms satisfy (with self-explanatory notations)

a,p

-

= a,p

-

and b,p

= b,p

in w.

T h e n there exist a vector c E E3 and a matrix Q E 0 : such that 8(y) = c

+ QZ(y) for all y E w.

Proof. Arguments similar to those used in parts (i) and (viii) of the proof of Theorem 2.8-1 show that there exist open subsets we of w and real numbers ~g > 0, C 0, such that the symmetric matrices ( g i j ) defined by

.>

gap := a,p - 2x3bap

where cap := a“‘b,,bp,,

+ x3c,p 2

and

gi3 = bi3,

are positive definite in the set

Q := U W e)o

e

x

I-E~,E~[.

Philippe G. Ciarlet

84

The two immersions 0 E C1(R;E3)and (with self-explanatory notations) O(y, 2 3 ) := 8 ( y )

6 E C1(R;E3)defined by

+ 2303(y) and 6 ( y , 2 3 ) := 8(y) + 23&(y)

for all (y, q )E R therefore satisfy gij

= & in

0.

By Theorem 1.7-1, there exist a vector c E E3 and an orthogonal matrix Q such that ~ ( yx 3, ) = c

+ Q G ( ~x3) , for all (y, x 3 ) E R.

Hence, on the one hand, det VO(y, 2 3 )

= det Q det V 6 ( y ,2 3 )

for all (y, 2 3 ) E R.

On the other hand, a simple computation shows that

) 0, wliere for all ( y , ~ E := aP“(Y)b*o(Y), 3 E w ,

so that

det VO(y,z3) = det V 6 ( y , 2 3 ) for all

(y,23)

E

0.

Therefore det Q = 1, which shows that the orthogonal matrix Q is in fact proper. The conclusion then follows by letting 2 3 = 0 in the relation @(y, 2 3 ) = c

+ Q&,

23)

for all (y, 2 3 ) E 0.

A proper isometry of E3 is a mapping J+ : E3 + E3 of the form J+(z) = c Qoz for all 2 E E3, with c E E3 and Q E 0;.Theorem 2.9-1 thus asserts that two immersions 8 E C1(w;E3) and is E C1 (w; E3) share the same fundamentaJ forms over a n open connected subset w of R3 i f and only if 8 = J+ o 8 , where J+ is a proper isometry of E3.

+

Remark. By contrast, the “three-dimensional” rigidity theorem (Theorem 1.7-1) involves isometries of E3 that may not be proper. 0 Theorem 2.9-1 constitutes the “classical” rigjdity theorem for surfaces, in the sense that both immersions 8 and 8 are assumed to be in the space C2(w;E3).

An Introduction t o Differential Geometry in R3

85

As a preparation to our next result, we note that the second fundamental form of the surface 8 ( w ) can still be defined under the weaker assumptions that 8 E C 1 ( w ; E 3 )and a3 = a1 A a 2 E C 1 ( w ; E 3 )by , la1 A a21

means of the definition

b,p := -acu.dpa3,

which evidently coincides with the usual one when 8 E C2(w;E 3 ) . Following Ciarlet & C. Mardare [2004a], we now show that a similar result holds under the assumptions that G E H 1 ( w ; E 3 )and G3 :=

61 A 6 2 161 A 621

E

H1( w ;E 3 ) (with self-explanatory notations). Naturally, our

first task will be to verify that the vector field 6 3 , which is not necessarily well defined a.e. in w for an arbitrary mapping 5 E H1(w;E 3 ) , is nevertheless well defined a.e. in w for those mappings G that satisfy the assumptions of the next theorem. This fact will in turn imply that the functions b,p := -a, . 8 p 6 3 are likewise well defined a.e. in w.

-

I

Theorem 2.9-2. Let w be a connected open subset of R2 and let 8 E a 3 E C1(w;E 3 ) . Assume that there exists a vector field 5 E H 1 ( w ;E 3 ) that satisfies

C1(w;E 3 ) be an immersion that satisfies

-a,p

= a,p

a.e. in w,

53 E

H 1 ( w ; E 3 ) , and

-

b,p = b,p

a.e. in w.

Then there exist a vector c E E 3 and a matrix Q E 0 : such that

-

8(y) = c + Q8(y) for almost all y E w .

Proof. The proof essentially relies on the extension t o a Sobolev space setting of the “three-dimensional” rigidity theorem established in Theorem 1.7-3. (i) To begin with, we record several technical preliminaries. First, we observe that the relations Zap = a,p a.e. in w and the assumption that 8 E C1(w;E 3 ) is a n immersion together imply that I

la1 A 6 2 1 =

d m Jdcto> o =

a.e. in w.

,., Consequently, the vector field defined a.e. an w. Second, we establish that

63,

and thus the functions b,p,

-

b,p = bp, in w and b,p = bp, a.e. in w ,

are well

86

Philippe G. Ciarlet

i.e., that a, . 8pa3 = a p . &a3 in w and Zi, . 8pZi3 = Zip ' daZ3 a.e. in w . To this end, we note that either the assumptions 8 E C 1 ( w ; E 3 ) and a 3 E C 1 ( w ; E 3 ) together, or the assumptions 8 E H 1 ( w ; E 3 ) and a 3 E H 1 ( w ; E 3 ) together, imply that a, . 8pa3 = 8,0 .8pa3 E LiOc(w), hence that 8,e. 8pa3 E D ' ( w ) . Given any cp E D ( w ) , let U denote an open subset of R2 such that supp cp c U and TI is a compact subset of w . Denoting by X ' (., .)xthe duality pairing between a topological vector space X and its dual X ' , we have V'(W)

'

8pa3i

=

p ) v ( W )

Observing that a,8.

a3

cpaCXO

'

dy

= 0 a.e. in w and that

=

H - 1 ( u ; ~) 3 (8,

(a, 0)

1

pa3 ) H A ( ;~3 ) 7

u

we reach the conclusion that the expression (8,8 . @a,, c p ) ~ ( ~is) symmetric with respect to Q and p since 8,pO = 8p,0 in ;D'(U). Hence 8,8 . dpa3 = ap0 . &a, in LtOc(w),and the announced symmetries are established. Third, let

-cap := d,Zi3,

aPZ3and cap := & a 3 . 8pa3.

Then we claim that .Zap = cap a.e. in w . To see this, we note that the matrix fields (Zap) := (Zap)-' and ( a m p ) := (a,p)-l are well defined and equal a.e. in w since 8 is an immersion and Zap = a,p a.e. in w . The formula of Weingarten (Section 2.6) can thus be applied a.e. in w , showing that Cap = Zu7bu,b7p a.e. in w . The assertion then follows from the assumptions b,p = b,p a.e. in w .

- -

-

(ii) Starting from the set w and the mapping 0 (as given in the statement of Theorem 2.9-2), we next construct a set R and a mapping 0 that satisfy the assumptions of Theorem 1.7-2. More precisely, let O(y, 2 3 ) := 8 ( g )

+ 23a3(y) for all ( y ,

23)

Ew x

R.

Then the mapping 0 := w x IR + IE3 defined in this fashion is clearly continuously differentiable on w x R and

An Introduction to Differential Geometry in W3

87

for all (y,x3) E w x R, where

Let wnr n 2 0 , be open subsets of R2 such that Zn is a compact subset of w and w = Un20wn. Then the continuity of the functions a,p, asp, b,p and the assumption that 8 is an immersion together imply that, for each n 3 0, there exists E, > 0 such that det VO(y, 5 3 ) > 0 for all (y, 2 3 ) E zn x

[- E ~,E ~ ] .

Besides, there is no loss of generality in assuming that property will be used in part (iii)). Let then R := (J(wn x ]-En,&,[).

E,


0 in R. Finally, note that the covariant components gij E Co(R) of the metric tensor field associated with the mapping 0 are given by (the symmetries b,p = bp, established in (i) are used here) gap = a,p

-

223b,p

+ x$c,p,

ga3 = 0,

g33 = 1.

(iii) Starting with the mapping 8 (as given in the statement of Theorem 2.9-2), we construct a mapping 0 that satisfies the assumptions of Theorem 1.7-2. To this end, we define a mapping 6 : R -+ IE3 by letting O(y, 2 3 ) := 8(y)

+ 23?i3(y) for all (y,x3) E R,

where the set R is defined as in (ii). Hence 6 E H 1 ( R ; E 3 ) since , R c w x 1-1, l[. Besides, det V 6 = det V O a.e. in R since the functions bg := Zp"b,,, which are well defined a.e. in w , are equal, again a.e. in w , to the functions bg. Likewise, the components - & E L1(R) of the metric tensor field associated with the mapping 0 satisfy & = gij a.e. in R since Zap = a,@ and b,p = b,p a.e. in w by assumption and E,p = cap a.e. in w by part (i).

-

-

(iv) By Theorem 1.7-2, there exist a vector c E E3 and a matrix Q E 0; such that I

8(y)

+ x363(y) = c + Q(B(y) + 23a3(y)) for almost all (y, 23) E R.

88

Philippe G. Ciarlet

Differentiating with respect to 2 3 in this equality between functions in H1(!22;E3)shows that &(y) = Qu3(y) for almost all y E w . Hence 0 8(y) = c Q8(y) for almost all y E w as announced.

+

Remarks. (1) The existence of i? E H 1 ( w ;E3) satisfying the assumptions of Theorem 2.9-2 implies that i? E C1(w; E3) and G3 E C1(w; E3), and that 8 E H 1 ( w ;E3) and u3 E H 1 ( w ;E3). (2) It is easily seen that the conclusion of Theorem 2.9-2 is still valid if the assumptions i? E H 1 ( w ;E3) and 6 3 E H 1 ( w ;E3) are replaced by the weaker assumptions i? E H:,,(W; E3) and Z3 E H;,,(w; E3).

Continuity of a surface as a function of its fundamental forms

2.10

Let w be a connected and simply-connected open subset of R2. Together, Theorems 2.8-1 and 2.9-1 establish the existence of a mapping F that associates with any pair of matrix fields (a,p) E C2(w;S;) and (b,p) E C2(w;S2) satisfying the Gauss and Codazzi-Mainardi equations in w a well-defined element F((a,p), (hap)) in the quotient set C3(w;E3)/R, where (0,s)E R means that there exists a vector c E E3 and a matrix QE0 : such that 8(y) = c Q&) for all y E w . A natural question thus arises as to whether the mapping F defined in this fashion is continuous. Equivalently, is a surface a continuous function of its fundamental forms? When both spaces C2(w;S2) and C3(w;E3) are equipped with their natural F'rkchet topologies, a positive answer to this question has been provided by Ciarlet [2003], along the following lines. To begin with, we introduce the following two-dimensional analogs to the notations used in Section 1.8. Let w be an open subset of W3. The notation K G w means that K is a compact subset of w . If f E Ce(w;R) or 8 E Ce(w;E3),C 2 0, and K G w , we let

+

\\f\\t,K :=

SUP

{ r$<e

IW(Y)I

I I ~ I I:= ~ , SUP ~

iayy)i,

&.: : I{

where d" stands for the standard multi-index notation for partial derivatives and 1. 1 denotes the Euclidean norm in the latter definition. If A E Ce(w;€MI3)),C 2 0, and K G w , we likewise let llAlle,K = SUP la"A(Y)I, e :{: I :

where

1. 1

denotes the matrix spectral norm

An Introduction to Differential Geometry in R3

89

We recall (see Section 1.8) that, for any integers k 2 0 and d 2 1, the space Ce(w;Rd) becomes a locally convex topological space when it is equipped with the &.e'chet topology defined by the family of semi-norms l l . l l e , K , IF, G w . Then a sequence (8".),,0 converges to 8 with respect to this topology if and only if

Furthermore, this topology is metrizable: Let of subsets of w that satisfy

(IF,~)+o

be any sequence

u 00

IF,^

cw

and

IF,^

c int I F , ~ +for I all i 2 0, and w =

IF,^

i=O

Then lim I18n-8jle,K

=0

for all

IF,

Gw

e lim de(8", 8)= 0 , n-+m

71-00

where

-

Let C3(w;E3) := C3(w;E 3 ) / R denote the quotient set of C3(w;E3) by the equivalence relation R, where ( 8 , O ) E R means that there exist a vector c E E3 and a matrix Q E 0 : such that 8 ( y ) = c + Q G ( y ) for all y E w. Then the set C 3 ( w ; E 3 )becomes a metric space when it is equipped with the distance d 3 defined by

where 6 denotes the equivalence class of 8 modulo R. The continuity of a surface as a function of its fundamental forms has then been established by Ciarlet [2003], according to the following result (if d is a metric defined on a set X , the associated metric space is denoted { X ;d } ) : Theorem 2.10-1. Let w be connected and simply connected open subset ofR2. Let

C,"(w;: S x S2):= {((a,p), (b,p)) E C2(w; ):52 x C2(w;S2); apcaur - aucapr C&Curp - C&rCprp = baubpT - bapbur in apb,, - &b,p C:,bp, - C$$,, = 0 in w }

+

+

W,

Philippe G. Ciarlet

90

Given any element ((a,p), (b,p)) E C ; ( u ; S ; x S’), let F(((a,p), (hap))) E C3(w;E3) denote the equivalence class modulo R of any immersion 8 E C 3 ( u ;E3) that satisfies

T h e n the mapping

F : { C ~ ( W: S; x S2);dz}

+ {C3(w;E

defined in this fashion is continuous.

3 ;d3

)

}

0

The above continuity results have been extended “up t o the boundary

of the set w” by Ciarlet & C. Mardare [2005].

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CIARLET, P .G. [1988]: Mathematical Elasticity, Volume I: Three-Dimensional Elasticity, North-Holland, Amsterdam.

CIARLET,P.G. [2000]: Mathematical Elasticity, Volume 111: Theory of Shells, North-Holland, Amsterdam.

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F. [2001]: On the recovery of a surface with CIARLET,P.G.; LARSONNEUR, prescribed first and second fundamental forms, J. Math. Pures Appl. 81, 167-185.

CIARLET,P.G.; LAURENT, F. [2003]: Continuity of a deformation as a function of its Cauchy-Green tensor, Arch. Rational Mech. Anal. 167, 255-269. CIARLET,P.G.; MARDARE, C. [2003]: On rigid and infinitesimal rigid displacements in three-dimensional elasticity, Math. Models Methods Appl. Sci. 13, 1589-1598.

CIARLET,P.G.; MARDARE, C. [2004a]: On rigid and infinitesimal rigid displacements in shell theory, J. Math. Pures Appl. 83, 1-15. CIARLET, P.G.; MARDARE, C. [2004b]: Recovery of a manifold with boundary and its continuity as a function of its metric tensor, J . Math. Pures Appl. 83, 811-843. CIARLET, P.G.; MARDARE, C. [2004c]: Continuity of a deformation in H 1 as a function of its Cauchy-Green tensor in L1, J. Nonlinear Sci. 14, 415-427. CIARLET,P.G.; MARDARE, C. [2005]: Recovery of a surface with boundary and its continuity as a function of its two fundamental forms, Analysis and Applications, 3, 99-117. CIARLET, P.G.; MARDARE, C . [2007]: An introduction to shell theory. CIARLET,P.G.; NECAS, J. [1987]: Injectivity and self-contact in nonlinear elasticity, Arch. Rational Mech. Anal. 19, 171-188. FLANDERS, H. [1989]: Differential Forms with Applications to the Physical Sciences, Dover, New York. FRIESECKE, G.; JAMES,R.D.; MULLER,S. [2002]: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity, Comm. Pure Appl. Math. 5 5 , 1461-1506. J. [2004]: Riemannian Geometry, GALLOT,S.; HULIN,D.; LAFONTAINE, Third Edition, Springer, Berlin. GAUSS, C.F. [1828]: Disquisitiones generales circas superficies curvas, Commentationes societatis regiae scientiarum Gottingensis recentiores 6, Gottingen. HARTMAN, P.; WINTNER,A. [1950]: On the embedding problem in differential geometry, Amer. J. Math. 72,553-564. JACOBOWITZ, H. [1982]: The Gauss-Codazzi equations, Tensor, N.S. 39, 15-22.

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JANET, M. [1926]: Sur la possibilitb de plonger un espace riemannien donne dans un espace euclidien, Annales de la Socie'te' Polonaise de Mathimatiques 5 , 38-43. JOHN, F. [1961]: Rotation and strain, Comm. Pure Appl Math. 14,391-413. JOHN,F. [1972]: Bounds for deformations in terms of average strains, in Inequalities 111 (0. Shisha, Editor), pp. 129-144, Academic Press, New York. KLINGENBERG, W. [1973]: Eine Vorlesung uber Differentialgeometrze, Springer-Verlag, Berlin (English translation: A Course in Differential Geometry, Springer-Verlag, Berlin, 1978). KOHN,R.V. [1982]: New integral estimates for deformations in terms of their nonlinear strains, Arch. Rational Mech. Anal. 78,131-172. KUHNEL,W. [2002]: Differentialgeometrie, Fried. Vieweg & Sohn, Wiesbaden (English translation: Differential Geometry: Curves-Surfaces-Manifolds, American Mathematical Society, Providence, 2002). MARDARE,C. [2003]: On the recovery of a manifold with prescribed metric tensor, Analysis and Applications 1,433-453. MARDARE, S. [2003b]: The fundamental theorem of surface theory for surfaces with little regularity, J . Elasticity 73,251-290. MARDARE,S. [2004]: On isometric immersions of a Riemannian space with little regularity, Analysis and Applications 2, 193-226. MARDARE, S. [2005]: On Pfaff systems with L p coefficients and their applications in differential geometry, J. Math. Pures Appl., to appear. MARSDEN, J.E.; HUGHES,T.J.R. [1983]: Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs (Second Edition: 1999). NAGHDI,P.M. [1972]: The theory of shells and plates, in Handbuch der & C. TRUESDELL, Editors), pp. 425-640, Physik, Vol. VIa/2 ( S . FLUGGE Springer-Verlag, Berlin. NASH,J. [1954]: C1 isometric imbeddings, Annals of Mathematics, 6 0 , 383396. NECAS,J. [1967]: Les Me'thodes Directes e n The'orie des Equations Elliptiques, Masson, Paris. NIRENBERG, L. [1974]: Topics in Nonlinear Functional Analysis, Lecture Notes, Courant Institute, New York University (Second Edition: American Mathematical Society, Providence, 1994). RESHETNYAK, Y .G. [1967]: Liouville's theory on conformal mappings under minimal regularity assumptions, Sibirskii Math. J. 8,69-85. RESHETNYAK, Y.G.[2003]: Mappings of domains in R" and their metric tensors, Siberian Math. J. 44, 332-345. SCHWARTZ, L. [1966]: The'orie des Distributions, Hermann, Paris. SCHWARTZ,L. [1992]: Analyse 11: Calcul Diffe'rentiel et Equations Diffe'rentielles, Hermann, Paris. SIMMONDS,J.G. [1994]: A Brief o n Tensor Analysis, Second Edition, Springer-Verlag, Berlin.

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SPIVAK,M. [1999]: A Comprehensive Introduction to Differential Geometry, Volumes I to V, Third Edition, Publish or Perish, Berkeley. STOKER,J.J. [1969]: Differential Geometry, John Wiley, New York. SZCZARBA, R.H. [1970]: On isometric immersions of Riemannian manifolds in Euclidean space, Boletim da Sociedade Brasileira de Matemcitica 1,31-45. SZOPOS,M. [2005]: On the recovery and continuity of a submanifold with boundary, Analysis and Applications 3, 119-143. TENENBLAT, K. [1971]: On isometric immersions of Riemannian manifolds, Boletim da Sociedade Brasileira de Matemcitica 2, 23-36. WHITNEY,H. [1934]: Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. SOC.36,63-89. YOSIDA,K. [1966]: Functional Analysis, Springer-Verlag, Berlin. ZEIDLER, E. [1986]: Nonlinear Functional Analysis and its Applications, Vol. I: Fixed-Point Theorems,

94

An Introduction to Shell Theory Philippe G. Ciarlet Department of Mathematics, City University of Hong Kong 8 3 Tat Chee Avenue, Kowloon, Hong Kong, China E-mail: mapgcQcityu. edu.hlc

Cristinel Mardare Universite‘ Pierre et Marie Curie-Paris6 UMR 7598 Laboratoire Jacques-Louis Lions, Paris, F- 75005 France E-mail: [email protected] r

Introduction 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 threedimensional 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 rBle 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. ‘With the kind permission of Springer-Verlag, some portions of these notes are extracted and adapted from the book by the first author “An introduction t o Differential Geometry with Applications t o Elasticity”, Springer, Dordrecht, 2005, the writing of which was substantially supported by two grants from the Research Grants Council of Hong Kong Special Administrative Region, China [Project No. 9040869, CityU 100803 and Project No. 9040966, CityU 1006041.

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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.

1 Three-dimensional theory Outline In this first part of the article, the displacement and the stress arising in an elastic shell, or for that matter in any three-dimensional elastic body, in response to given loads are predicted by means of a system of partial differential equations in three variables (the coordinates of the physical space). This system is formed either by the equations of nonlinear threedimensional elasticity or by the equations of linearized three-dimensional elasticity. Sections 1.2-1.4 are devoted to the derivation of the equations of three-dimensional elasticity in the form of two basic sets of equations, the equations of equilibrium and the constitutive equations. The equations of nonlinear three-dimensional elasticity are then obtained by adjoining appropriate boundary conditions to these equations. The equations of linearized three-dimensional elasticity are obtained from the nonlinear ones by linearization with respect to the displacement field. Sections 1.5-1.6 study the existence and uniqueness of solutions to the equations of linearized three-dimensional elasticity. Using a fundamental lemma, due to J.L. Lions, about distributions with derivatives in “negative” Sobolev spaces (Section 1.5), we establish in Section 1.6 the fundamental Korn inequality, which in turn implies that the equations of linearized three-dimensional elasticity have a unique solution. In sections 1.7-1.8, we study the existence of solutions t o the equations of nonlinear three-dimensional elasticity, which fall into two distinct categories: If the data are regular, the applied forces are “small”, and the boundary condition does not change its nature along connected portions of the boundary, the equations of nonlinear three-dimensional elasticity have a solution by the implicit function theorem (Section 1.7). If the constituting material is hyperelastic and the associated stored energy function satisfies certain conditions of polyconvexity, coerciveness, and growth, the minimization problem associated with the equations of nonlinear three-dimensional elasticity has a solution by a fundamental theorem of John Ball (Section 1.8).

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1.1

Notation, definitions, and some basic formulas

All spaces, matrices, etc., are real. The Kronecker symbol is denoted dl. The physical space is identified with the three-dimensional vector space R3 by fixing an origin and a Cartesian basis (e1,e2,e3).In this way, a point x in space is defined by its Cartesian coordinates xi,x2,23 or by the vector x := xiei.The space R3 is equipped with the Euclidean inner product u . w and with the Euclidean norm 1u1,where u, w denote vectors in R3.The exterior product of two vectors u, w E R3 is denoted

xi

UAW.

For any integer n 2 2, we define the following spaces or sets of real square matrices of order n:

M": the space of all square matrices,

A": the space of all anti-symmetric matrices, S": the space of all symmetric matrices, My: the set of all matrices A E Mn with det A > 0,

Sy : the set of all positive-definite symmetric matrices, 0": the set of all orthogonal matrices,

07:the set of all orthogonal matrices R E 0" with det R = 1. The notation ( a i j ) designates the matrix in M" with aij as its element at the i-th row and j-th column. The identity matrix in M" is denoted I := (6;). The space M", and its subspaces A" and S" are equipped with the inner product A : B and with the spectral norm (A1defined by

i>j

1A) := sup{lAwl; w E R",IwI

< I},

where A = ( a i j ) and B = ( b i j ) denote matrices in M". The determinant and the trace of a matrix A = ( a i j ) are denoted det A and tr A. The cofactor matrix associated with an invertible matrix A E M" is defined by CofA := (det A)A-T. Let R be an open subset of R3.Partial derivative operators of order m 2 1 acting on functions or distributions defined over R are denoted

+ +

where Ic = (ki)E N3 is a multi-index satisfying llcl := kl k2 k3 = m. Partial derivative operators of the first, second, and third order are also denoted di := d / d x i , dij := d2/dxidxj, and dijk := d 3 / 8 ~ l a ~ 2 a ~ 3 .

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f :R

4 IR is the vector field gradf := The gradient of a vector field v = (vi) : R 4 R” is the matrix field Vv := (djwi), where i is the row index, and the divergence of the same vector field is the function div v := Ci &vi. Finally, the divergence of a matrix field T = ( t i j ) : R + M” is the vector field divT with components 8jtij)i. The space of all continuous functions from a topological space X into a normed space Y is denoted C o ( X ;Y ) ,or simply C o ( X )if Y = R. For any integer m 2 1 and any open set R c It3,the space of all realvalued functions that are m times continuously differentiable over R is m 2 1, is defined as that consisting denoted Cm(R). The space of all vector-valued functions f E P ( R ) that, together with all their partial derivatives of order 6 m, possess continuous extentions to the closure of R. If R is bounded, the space C ” ( 2 ) equipped with the norm

The gradient of a function

(aif),where i is the row index.

(Cj”=,

em@),

is a Banach space. The space of all indefinitely derivable functions cp : R 4 R with compact support contained in R is denoted D(R) and the space of all distributions over R is denoted D’(R). The duality bracket between a distribution T and a test function cp E D(R) is denoted ( T ,cp). The usual Lebesgue and Sobolev spaces are respectively denoted Lp(R), and WmJ’(R) for any integer m 3 1 and any p 2 1. If p = 2, we use the notation Hm(R) = Wm>’(R). The space W,oT’p(R) is the space of all mesurable functions such that flu E WmJ’(U) for all U G R, where the notation flu designates the restriction to the set U of a function f and the notation U G R means that is a compact set that satisfies -

u

u c R.

The space WT1p(R)is the closure of D(R) in WmJ’(R) and the dual of the space Wrlp(R) is denoted WPm+”(R),where p’ = If the boundary of R is Lipschitz-continuous and if ro c 80 is a relatively open subset of the boundary of R, we let

5.

w::(R)

:= {f E

w’J’(R);f = o on ro},

w:;P(~) := {f E w~J’(R); f = 8,f

= o on

r,,},

where 8, denote the outer normal derivative operator along 8 R (since is Lipschitz-continuous, a unit outer normal vector ( u i ) exists 80-almost everywhere along 80, and thus 8, = ui8i). If Y is a finite dimensional vectorial space (such as R”, Mn, etc.), the notation Cm(R; Y ) , Y ) ,LP(R; Y ) and W”iP(R; Y ) designates the spaces of all mappings from R into Y whose components in Y are respectively in Cm(R), LP(R) and Wm>P(n). If Y is equipped with

em@; em@),

Philippe G. Ciarlet, Cristinel Mardare

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the norm I ' 1 , then the spaces LP(R2;Y )and W">P(R; Y )are respectively equipped with the norms

and

Throughout this article, a domain in R" is a bounded and connected open set with a Lipschitz-continuous boundary, the set R being locally on the same side of its boundary. See, e.g., Adams [2], Grisvard [54], or NeEas [73]. If R c R" is a domain, then the following formula of integration by parts is satisfied

s,

div F . v d x

=

-

.I

F : Vv d x

+ l a ( F n ). v da

for all smooth enough matrix field F : R 4 M k and vector field v : R -+ Rk,k 1 (smooth enough means that the regularity of the fields F and v is such that the above integrals are well defined; for such instances, see, e.g., Evans & Gariepy [47]). The notation da designates the area element induced on the surface dR by the volume element d x . We also record the Stokes formula:

1.2

Equations of equilibrium

In this section, we begin our study of the deformation arising in an elastic body in response to given forces. We consider that the body occupies the closure of a domain R c R3 in the absence of applied forces, henceforth called the reference configuration of the body. Any other configuration that the body might occupy when subjected to applied forces will be defined by means of a deformation, that is, a mapping

that is orientation preserving (i.e., det V + ( x ) > 0 for all z E 2)and injective on the open set R (i.e., no interpenetration of matter occurs). The image is called the deformed configuration of the body defined

+(a)

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by the deformation CP. The “difference”between a deformed configuration and the reference configuration is given by the displacement, which is the vector field defined by u := CP - id,

a a

where id : + is the identity map. It is sometimes more convenient to describe the deformed configuration of a body in terms of the displacement u instead of the deformation CP, notably when the body is expected to undergo small deformations (as typically occurs in linearized elasticity). Our objective in this section is to determine, among all possible deformed configurations of the body, the ones that are in “static equilibrium” in the presence of applied forces. More specifically, let the applied forces acting on a specific deformed configuration fi := CP(Cl) be represented by the densities

f : fi + R3 and g

: f’1 + R3,

where f’,c i3fi is a relatively open subset of the boundary of fi. If the body is subjected for instance to the gravity and to a uniform pressure on f ‘ ~then , the densities and g are given by f(5) = -g5(5)e3 and g ( 5 ) = - ~ f i ( 5 ) ,where g is the gravitational constant, 5, : fi + JR is the mass density in the deformed configuration, 5 denotes a generic point in n(5)is the unit outer normal to afi, and T is a constant, called pressure. These examples illustrate that an applied force density may, or may not, depend on the unknown deformation. Our aim is thus to determine equations that a deformation CP corresponding to the static equilibrium of the loaded body should satisfy. To this end, we first derive the “equations of equilibrium” from a fundamental axiom due to Euler and Cauchy. The three-dimensional equations of elasticity will then be obtained by combining these equations with a “constitutive equation” (Section 1.3). Let

{a}-,

s 2 := {v

E R3; 1v1 = l}

denote the set of all unit vectors in R3.Then, according to the stress principle of Euler and Cauchy, a body fi c W3subjected to applied forces of densities f : fi --t R3 and g : f’1 -+ R3 is in equilibrium if there exists a vector field

z : {a}- x s2

-+

R3

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A C 6,

such that, for all domains

/A

f d2 +

/-

i ( 2 ,i i ( 2 ) )dii = 0,

LA

LJA

2 A f d2 +

2 A i ( 2 ,n ( 2 ) )dii

= 0,

where i i ( 2 ) denotes the exterior unit normal vector at Z to the surface aA (because is a domain, i i ( 2 ) exists for d2-almost all 2 E d A ) . This axiom postulates in effect that the “equilibrium” of the body to the applied forces is reflected by the existence of a vector field i that depends only on the two variables 2 and i i ( 2 ) . The following theorem, which is due to Cauchy, shows that the dependence of on the second variable is necessarily linear:

z

Theorem 1.2-1. I f Z ( . , i i ) : {6}- + R3 is of class C1 f o r all ii E S’J, -+ R3 is continuous f o r all 2 E {6}-, and f : {6}- + R3 is continuous, then i : (62)-x S’J+ R3 is linear with respect to the second variable.

i(Z,.) : Sz

Proof. The proof consists in applying the stress principle to particular subdomains in For details, see, e.g., Ciarlet [18] or Gurtin & Martins [55]. 0

{a}-.

In other words, there exists a matrix field ?’ : {6}- + M3 of class

C1 such that i ( 2 , i i )= T(2)ii for all 2 E

(62)-and all ii E S’J.

Combining Cauchy’s theorem with the stress principle of Euler and Cauchy yields, by means of Stokes’ formula (see Section 1.1),the following equations of equilibrium in the deformed configuration:

Theorem 1.2-2. The matrix field T : (62)-+ M3 satisfies -divp(2) = f(2) for all 2 E

6,

? ‘ ( ~ ) i i=( g(2) ~ ) for all 2 E Fl,

?(z)

E

s3 for all

2E

(1.2-1)

6.

a,

The system (1.2-1) is defined over the deformed configuration which is unknown. Fortunately, it can be conveniently reformulated in terms of functions defined over the reference configurence R of the body, which is known. To this end, we use the change of variables

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a

5 = @(x) defined by the unknown deformation @ : + {fi}-, assumed to be injective, and the following formulas between the volume and area elements in {fi}- and (with self-explanatory notations)

a

d5 = I det V@(x)/dx, f i ( 5 ) dSi = CofV@(x)n(x) da.

We also define the vector fields f : R + R3 and g : Fl

+

R3 by

f(5)d5 = f (x)dx, g(5) dSi = g(x) da. Note that, like the fields f and g, the fields f and g may, or may not, depend on the unknown deformation First of all, assuming that is smooth enough and using the change of variables 4j : 2 + (6)-in the first equation of (1.2-l), we deduce that, for all domains A c R,

+

f ( x ) dx The matrix field T : defined by

+

+.

1,

T(@(x))CofV+(x)n(x) da

= 0.

a + M3 appearing in the second integral, viz., that

~ ( x:=) T(+(x))CofV@(x)for all x E

a,

is called the first Piola-Kirchhoff stress tensor field. In terms of this tensor, the above relation read

which implies that the matrix field T satisfies the following partial differential equation:

-divT(x)

= f (x)

for all x E R.

Using the identity

'

V @(x) T (x) = V@(x) [det V@(x)?'( 9(x))]V+ (x)--T, which follows from the definition of T ( x ) and from the expression of the inverse of a matrix in terms of its cofactor matrix, we furthermore deduce from the symmetry of the matrix T(5) that the matrix ( V @ ( x ) - ' T ( x ) ) is also symmetric.

Philippe G. Ciarlet, Cristinel Mardare

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It is then clear that the equations of equilibrium in the deformed configuration (see eqns. (1.2-1)) are equivalent with the following equations of equilibrium in the reference configuration: -divT(z) = f(z) T(z)n(z= ) g(z) V+(Z)-~T(Z)

E s3

for all z E R , for all z E

rl,

(1.2-2)

for all z E R,

where the subset rl of a R is defined by = +(ri). Finally, let the second Piola-Kirchhoff stress tensor field E : R -+ S3 be defined by ~ ( z:= ) V + ( Z ) - ~ T ( ~ ) for all z E

R.

Then the equations of equilibrium defined in the reference configuration take the equivalent form

-div (V+(z)E(z))= f ( z ) for all z E R,

(V+(z)E(z))n(z)= g(z) for all z E

r1,

(1.2-3)

in terms of the symmetric tensor field E. The unknowns in either system of equations of equilibrium are the deformation of the body defined by the vector field : 2 + R3,and the stress field inside the body defined by the fields T : -+ MI3 or X : S3.In order to determine these unknowns, either system (1.2-2) or (1.2-3) has to be supplemented with an equation relating these fields. This is the object of the next section.

+

n

1.3

-+

Constitutive equations of elastic materials

It is clear that the stress tensor field should depend on the deformation induced by the applied forces. This dependence is reflected by the constitutive equation of the material, by means of a response function, specific to the material considered. In this article, we will consider one class of such materials, according to the following definition: A material is elastic if there exists a function Tn: x M t 4 M3 such that

n

~ ( z=) @(z, v+(z)) for all z E

a.

Equivalently, a material is elastic if there exists a function Xfl : a x M $ S3 such that ~ ( z=) @(z, v+(z))for all z E

4

n.

Either function Tnor Ed is called the response function of the material.

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A response function cannot be arbitrary, because a general axiom in physics asserts that any “observable quantity” must be independent of the particular orthogonal basis in which it is computed. For an elastic material, the “observable quantity” computed through a constitutive equation is the stress vector field i. Therefore this vector field must be independent of the particular orthogonal basis in which it is computed. This property, which must be satisfied by all elastic materials, is called the axiom of material frame-indifference. The following theorem translates this axiom in terms of the response function of the material. Theorem 1.3-1. An elastic material satisfies the axiom of material frame-indifference i f and only if Tn(z,Q F ) = QTn(z, F ) for all x E

a and Q E 0: and F E Mi$,

or equivalently, if and only i f

Xn(x,Q F ) = Xn(x,F ) for all x E

and Q E 0 : and F E M.:

The second equivalence implies that the response function En depend on F only via the symmetric positive definite matrix U := ( F T F ) ’ l 2 , the square root of the symmetric positive definite matrix ( F T F )E S;. To see this, one uses the polar factorisation F = RU, where R := FU-’ E 0:,in the second equivalence of Theorem 1.3-1to deduce that

En(,, F ) = Xfl(x,U ) for all z E

n and F = RU E M.:

This implies that the second Piola-Kirchhoff stress tensor field E : -+ S3 depends on the deformation field @ : 32 + R3 only via the associated metric tensor field C := V@TV@, i.e., ~ ( z =) %(z,~ ( z ) )for all

where the function

% : fi x S;

%(x, C ) := Xn(x,

+ S3

z E a,

is defined by

for all z E

a and C E Sj3>.

We just saw how the axiom of material frame-indifference restricts the form of the response function. We now examine how its form can be further restricted by other properties that a given material m a y possess. An elastic material is isotropic at a point z of the reference configuration if the response of the material “is the same in all directions”, i.e., if the Cauchy stress tensor $(Z) is the same if the reference configuration is rotated by an arbitrary matrix of 0 : around the point z.An elastic material occupying a reference configuration fi is isotropic if it is isotropic at all points of The following theorem gives a characterisation of the response function of an isotropic elastic material:

a.

Philippe G. Ciarlet, Cristinel Mardare

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Theorem 1.3-2. An elastic material occupying a reference configuration R is isotropic i f and only if

-

@(z, F Q ) = Tfl(z, F ) Q for all z E 2 and Q E 0 : and F E M,:

or equivalently, i f and only if Efl(z,F Q ) = QTXfl(z,F ) Q for all z E

and Q E 0 : and F E M;.

Another property that an elastic material may satisfy is the property of homogeneity: An elastic material occupying a reference configuration is homogeneous if its response function is independent of the particular point z E 2 considered. This means that the response function Tfl : 2 x M: 4 M3, or equivalently the response function Xfl : a x M : 4 S3,does not depend on the first variable. In other words, there exist mappings (still denoted) T n: MLn"+4 M3 and Xfl : M: 4 S3 such that

a

T n ( z , F= ) T n ( F )for all x E

n and F E M,:

and

X n ( x l F )= d ( F ) for all z E s2 and F

E

M.:

The response function of an elastic material can be further restricted if its reference configuration is a natural state, according to the following definition: A reference configuration is called a natural state, or equivalently is said to be stress-free, if

a

T # ( ~ ,= I )o for all z E 2, or equivalently, if ~ f l ( zI ,) =

o for all z E 2.

- We have seen that the second Piola-Kirchhoff stress tensor field X : R 4 S3 is expressed in terms of the deformation field CP : 4 B3 as

X(z)

= %(z,C(z)), where

C(z) = VCP'(z)VCP(z) for all z E ;2.

If the elastic material is isotropic, then the dependence of X(z) in terms of C(z) can be further reduced in a remarkable way, according to the following Rivlin-Ericksen theorem: Theorem 1.3-3. If a n elastic material is isotropic and satisfies the principle of material frame-indifference, t h e n there exists functions : R x B3 -+ EX, i E {1,2,3}, such that

-,!

+

+

X(z) = +yo(z)I yi(z)C(z) yZ(z)c2(z) for all x where ri(z)= $(z, tr C ,tr(CofC), det C ) .

E

2,

An Introduction to Shell Theory

Proof. See Rivlin & Ericksen [74] or Ciarlet [18].

105

0

Note that the numbers tr C(z), tr(CofC(z)), and det C(z) appearing in the above theorem constitute the three principal invariants of the matrix C(z). Although the Rivlin-Ericksen theorem substantially reduces the range of possible response functions of elastic materials that are isotropic and satisfy the principle of frame-indifference, the expression of X is still far too general in view of an effective resolution of the equilibrium equations. To further simplify this expression, we now restrict ourselves to deformations that are “close to the identity”. In terms of the displacement filed u : R + R3, which is related to the deformation @ : R + R3 by the formula @(z)= z

+ ~ ( z for ) all z E 1,

the metric tensor field C has the expression C ( X ) =I

+ 2E(z),

where

1 E ( z ) := - ( V u T ( z ) V u ( z ) V u T ( z ) V u ( z ) ) 2 denotes the Green-St Venant strain tensor at z. Thanks to the above assumption on the deformation, the matrices E ( z ) are “small” for all z E and therefore one can use Taylor expansions to further simplify the expression of the response function given by the Rivlin-Ericksen theorem. Specifically, using the Taylor expansions

+

+

a,

+ 2 tr E ( z ) , + 4 t r E ( z ) + o((E(z)I), + 2 t r E ( z ) + o(lE(z)I), C2(z)= 1+ 4E(z) + o(IE(z)I),

tr C(z) = 3 tr(CofC(z)) = 3 det C(z) = 1

and assuming that the functions yitl are smooth enough, we deduce from the Rivlin-Ericksen theorem that

where the real-valued functions X(z) and p ( z ) are independent of the displacement field u. If in addition the material is homogeneous, then X and p are constants. To sum up, the constitutive equation of a homogeneous and isotropic elastic material that satisfies the axiom of frame-indifference must be such that X(X) = ~ t f ( zI ,)

+ X(tr E ( ~ ) )+I2 p ~ ( z+) o z ( l ~ ( z ) for ~ ) all z E 1.

Philippe G. Ciarlet, Cristinel Mardare

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If in addition R is a natural state, a natural candidate for a constitutive equation is thus ~ ( z=) X(tr E

( ~ ) )+I 2 p ~ ( z for ) all z E

a,

and in this case X and p are then called the Lam6 constants of the material. A material whose constitutive equation has the above expression is called a St Venant-Kirchhoff material. Note that the constitutive equation of a St Venant-KirchhoE material is invertible, in the sense that the field E can be also expressed as a function of the field E as 1

~ ( z=) -E(x) 2P

-

v -(tr E(Z))I for all z E 2.

E

Remark. The Lami: constants are determined experimentally for each elastic material and experimental evidence shows that they are both strictly positive (for instance, X = 106kg/cm2 and p = 820000kg/cm2 for steel; X = 40000kg/cm2 and p = 1200kg/cm2 for rubber). Their explicit values do not play any r81e in our subsequent analysis; only their positivity will be used. The Lami: coefficients are sometimes expressed in terms of the Poisson coeficient v and Young modulus E through the expressions X P(3X + 2cL) v= 2 ( X + p ) and = X+P '

1.4 The equations of nonlinear and linearized three-dimensional elasticity It remains to combine the equations of equilibrium (equations (1.2-3) in Section 1.2) with the constitutive equation of the material considered (Section 1.3) and with boundary conditions on ro := d R \ I ' l . Assuming that the constituting material has a known response function given by T uor by Xu and that the body is held fked on ro, we conclude in this fashion that the deformation arising in the body in response to the applied forces of densities f and g satisfies the nonlinear boundary value problem: -divT(s) = f(z), z E R, a(.)= 2, 2 E r,,, (1.4-1) T ( z ) n ( z= ) g ( z ) , 5 r1,

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where

T ( z )= Tn(z,V@(z))= V+(z)Xn(z,v@(z)) for all z

E

D.

(1.4-2)

The equations (1.4-1) constitute the equations of nonlinear threedimensional elasticity. We will give in Sections 1.7 and 1.8 various sets of assumptions guaranteeing that this problem has solutions. Consider a body made of an isotropic and homogeneous elastic material such that its reference configuration is a natural state, so that its constitutive equation is (see Section 1.3):

where X > 0 and p > 0 are the Lam6 constants of the material. The equations of linearized three-dimensional elasticity are obtained from the above nonlinear equations under the assumption that the body will undergo a "small" displacement, in the sense that +(z) = z

+ u ( z )with IVu(z)l 0 and that the densities of the applied forces satisfy f E L6//5(R;R3) and g E L4/3(l?l;E%3). If areal70 > 0, the variational problem (1.6-1) has a unique solution in the space

p

Proof. It suffices to apply the Lax-Milgram lemma to the variational equation (1.6-1), since all its assumptions are clearly satisfied. In particular, the coerciveness of the bilinear form appearing in the left-hand side of the equation (1.6-1) is a consequence of Korn's inequality established in the previous theorem combined with the positiveness of the Lam6 constants, which together imply that, for all w E H;JR; R3),

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Theorem 1.6-3. Assume that the Lame' constants satisfy X 3 0 and p > 0 and that the densities of the applied forces satisfy f E L6/5(R;R3) and g E L4/3(dR;R3). Ifarearo = O a n d J n f.wdJ:+Jang.wda=O f o r a l l w e H 1 ( R ; R 3 ) satisfying e(w) = 0 , then the variational problem (1.6-1) has a solution in H1(R;R3), unique u p to a n infinitesimal rigid displacement field. Sketch of proof. It is again based on the Lax-Milgram lemma applied to the variational equations (1.6-1), this time defined over the quotient space H1(R;R3)/I&, where I& is the subspace of H1(R;R3) consisting of all the infinitesimal rigid displacements fields. By the infinitesimal rigid displacement lemma (see Part (ii) of the proof of Theorem 1.6-1), Ro is the finite-dimensional space

{w : R

+ R3;

w(x) = a

+ b A 2, a,b E R3}.

The compatibility relations satisfied by the applied forces imply that the variational equation (1.6-1) is well defined over the quotient space H1(R;R3)/I&, which is a Hilbert space with respect to the norm

The coerciveness of the bilinear form appearing in the left-hand side of the equation (1.6-1) is then established as a consequence of another Korn's inequality:

lle(qllLyn;s3) 3 Cll4lH1(n2;W3)/Rg for all

+ E H1(QR 3 ) / I & .

The proof of this inequality follows that of Theorem 1.6-1, with Part (iii) adapted as follows: The sequence ( w n ) n E ~ is now defined in H1(R; R3)/& and satisfies

/l+nIIH1(R;R3)/Ro = 1 for all 12, lle(+n)llLyn;s3) 0 as n m. +

+

Hence there exists an increasing function a : N 4 N such that the subsequence ( w ~ ( ~is) )a Cauchy sequence in H1(R;R3). This space being complete, there exists w E H1(R; R3) such that

wa(") + w in H1(R;R3), and its limit satisfies

e ( w ) = lim e(wa(n)) = 0 . 71-00

Therefore w E This implies that

by Part (iii), hence ( w ~ ( -~ w) )

G

I I W ~ ( ~-)

Il~a(n)IIH~(n;R3)~~

wIIH1(n2;R3)

-+

-+ 0

in H1(R;R3).

0 as n

00,

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= 1 for all n. which contradicts the relation ((i~,(,)llHi(n;p)/~~ The variational problem (1.6-1) is called a pure displacement problem when ro = do, a pure traction problem when rl = dR, and a displacement-traction problem when arearo > 0 and areal71 > 0.

0 Since the system of partial differential equations associated with the linear three-dimensional variational model is elliptic, we expect the solution of the latter to be regular if the data f,g , and dR arc regular and if there is no change of boundary condition along a connected portion of dR. More specifically, the following regularity results hold (indications about the proof arc given in Ciarlet [18, Theorem 6.3-61). Theorem 1.6-4 (pure displacement problem). Assume that ro = 80. I f f E W"J'(R; R3) and 80 is of class C m f 2 for some integer m 2 0 and real number 1 < p < co satisfying p 3 &, then the solution u to the variational equation (1.6-1) is in the space Wmf2J'(C12;W3)and there exists a constant C such that ll~IIWm+2.P(n;W3)

G

CllfllW~+2~P(n;R3).

Furthermore, u satisfies the boundary value problem:

-divu(x) U ( . )

x E R, = 0, x E 80.

= f,

Theorem 1.6-5 (pure traction problem). Assume that rl = dR and f . w dx g . w da = 0 for all vector fields v E H1(R; R3) satisfying e(w) = 0 . Iff E W">p(R;R3),g E W"+1-1/PJ'(I'~;R3), and dR is of class for some integer m 2 0 and real number 1 < p < co satisfying p 2 &, then any solution u to the variational equation (1.6-1) is in the space W"f2~p(0;W3)and there exist a constant C such that

sn

+,s,

ern+'

Furthermore, u satisfies the boundary value problem: -divu(z) = f(x), x E 0,

fT(x)n(x) = g(x), x E dR.

1.7

Existence theory in nonlinear three-dimensional elasticity by the implicit function theorem

The question of whether the equations of nonlinear three-dimensional elasticity have solutions has been answered in the affirmative when the

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data satisfy some specific assumptions, but remains open in the other cases. To this day, there are two theories of existence, one based on the implicit function theorem, and one, due to John Ball, based on the minimization of functionals. We state here the existence theorems provided by both theories but we will provide the proof only for the existence theorem based on the implicit function theorem. For the existence theorem based on the minimization of functionals we will only sketch of the proof of John Ball (Section 1.8). The existence theory based on the implicit function theorem asserts that the equations of nonlinear three-dimensional elasticity have solutions if the solutions to the associated equations of linearized threedimensional elasticity are smooth enough, and the applied forces are small enough. The first requirement essentially means that the bodies are either held fixed along their entire boundary (i.e., = dR), or nowhere along their boundary (i.e., rl = 80). We restrict our presentation to the case of elastic bodies made of a St Venant-Kirchhoff material. In other words, we assume throughout this section that

Y

+

X = X(tr E ) I + 2pE and E = - VuT Vu 2 where X IJ

:R

--f

+ VuTVu) ,

(1.7-1)

> 0 and p > 0 are the Lam6 constants of the material and R3 is the unknown displacement field. We assume that ro = dR

(the case where rl = dR requires some extra care because the space of infinitesimal rigid displacements fields does not reduce to (0)).Hence the equations of nonlinear three-dimensional elasticity assert that the displacement field u : R 4 R3 inside the body is the solution to the boundary value problem

-div ( ( I+ V U ) = ~ f) in R, u = 0 on dR,

(1.7-2)

where the field X is given in terms of the unknown field u by means of (1.7-1). The existence result is then the following Theorem 1.7-1. T h e nonlinear boundary value problem (1.7-1)-(1.7-2) has a solution u E W2>p(R; R3)i f R is a domain with a boundary dR of class C2, and f o r some p > 3, f E Lp(R;R3) and IlfllLp(n;R3) is small

enough. Proof. Define the spaces

x

w2>p(R; R~); w = o on an}, Y := U ( R ; R3). := (w E

116

Philippe G. Ciarlet, Cristinel Mardare

Define the nonlinear mapping

F :X

4

Y by

F(w) := -div ( ( I + V W ) ~for) any w E X , where

x = X(tr E ) I + 2

p and ~ E =

Y v w T + v w+

-

2

VW~VW

It suffices t o prove that the equation

3‘(u) =f has solutions in X provided that the norm of f in the space Y is small enough. The idea for solving the above equation is as follows. If the norm of f is small, we expect the norm of u to be small too, so that the above equation can be written as

Since F ( 0 ) = 0, we expect the above equation to have solution if the linear equation F”(0)U =f has solutions in X . But this equation is exactly the system of equations of linearized three-dimensional elasticity. Hence, as we shall see, this equation has solutions in X thanks to Theorem 1.6-4. In order to solve the nonlinear equation F(u) = f , it is thus natural to apply the inverse function theorem (see, e.g., Taylor [85]). According to this theorem, if F : X 4 Y is of class C1 and the F’rkchet derivative F’(0): X 4 Y is a n isomorphism (i.e., an operator that is linear, bijective, and continuous with a continuous inverse), then there exist two open sets U c X and V c Y with 0 E U and 0 = F ( 0 ) E V such that, for all f E V ,there exists a unique element u E U satisfying the equation F(U)= f . Furthermore, the mapping

f EVHUEU is of class C1. It remains to prove that the assumptions of the inverse function theorem are indeed satisfied. First, the function F is well defined (i.e., F(u)E Y for all u E X)since the space W’J’(R) is an algebra (thanks t o the assumption p > 3). Second, the function .F : X 4 Y is of class

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C1 since it is multilinear (in fact, F is even of class P ) .Third, the Frkchet derivative of .F is given by F’(0)u= -diva, where 1

c := A ( t r e ) l + 2 p e and e := -(VuT 2

+ Vu),

from which we infer that the equation F’(0)u= f is exactly the equations of linearized three-dimensional elasticity (see (1.4-3)-(1.4-4) with To = an). Therefore, Theorem 1.6-4 shows that the function F’(0): X 4 Y is an isomorphism Since all the hypotheses of the inverse function theorem are satisfied, the equations of nonlinear three-dimensional elasticity (1.7-1)-(1.7-2) have a unique solution in the neighborhood U of the origin in W2iP(Q; R3) if f belongs t o the neighborhood V of the origin in L*(Cl;R3). In particular, if S > 0 is the radius of a ball B(0,S) contained in V , then the problem (1.7-1)-(1.7-2) has solutions for all Ilfll~~cn, < 6. 0 The unique solution u in the neighborhood U of the origin in W2+’(Cl; R3) of the equations of nonlinear three-dimensional elasticity (1.7-1)(1.7-2) depends continuously on f ,i.e., with self-explanatory notation f, + f in L * ( R ; I w ~j ) u,

--f

u in W ~ ~ ~ ( R ; R ’ ) .

This shows that, under the assumptions of Theorem 1.7-1, the system of equations of nonlinear three-dimensional elasticity is well-posed. Existence results such as Theorem 1.7-1 can be found in Valent [86], Marsden & Hughes [SS], Ciarlet & Destuynder [25], who simultaneously and independently established the existence of solutions to the equations of nonlinear three-dimensional elasticity via the implicit function theorem.

1.8

Existence theory in nonlinear three-dimensional elasticity by the minimization of energy (John Ball’s approach)

We begin with the definition of hyperelastic materials. Recall that (see Section 1.3) an elastic material has a constitutive equation of the form ~ ( z:= ) ~ f l ( zv+(z)) , for all z E K,

where Til : x M$ -+ M3 is the response function of the material and T ( z )is the first Piola-Kirchhoff stress tensor at 2.

Philippe G. Ciarlet, Cristinel Mardare

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Then an elastic material is hyperelastic if there exists a function + R, called the stored energy function, such that its response function Tncan be fully reconstructed from W by means of the relation

W : s2 x M$

where denotes the F'rkchet derivative of W with respect to the variable F . In other words, a t each x E D, g ( x , F )is the unique matrix in M3 that satisfies

for all F E M$ and H E M3 (a detailed study of hyperelastic materials can be found in, e.g., Ciarlet [18, Chap. 41). John Ball [9] has shown that the minimization problem formally associated with the equations of nonlinear three-dimensional elasticity (see (1.4-1)) when the material constituting the body is hyperelastic has solutions if the function W satisfies certain physically realistic conditions of polyconvexity, coerciveness, and growth. A typical example of such a function W , which is called the stored energy function of the material, is given by

W ( x ,F ) = allFllP + bllCofFllq

+ cI det 3'1'

-

dlog(det F )

forallFEM:,wherep>2,q> ~,r>l,a>O,,b>O,c>O,d>O, and 11 . 11 is the norm defined by IlFll := {tr(FTF)}'12 for all F E M3. The major interest of hyperelastic materials is that, for such materials, the equations of nonlinear three-dimensional elasticity are, at least formally, the Euler equation associated with a minimization problem (this property only holds formally because, in general, the solution to the minimization problem does not have the regularity needed to properly establish the Euler equation associated with the minimization problem). To see this, consider first the equations of nonlinear threedimensional elasticity (see Section 1.4):

where, for simplicity, we have assumed that the applied forces do not depend on the unknown deformation a.

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A weak solution @ to the boundary value problem (1.8-1) is then the solution to the following variational problem, also known as the principle of virtual works:

a

for all smooth enough vector fields v : 4 R3 such that = 0 on ro. If the material is hyperelastic, then T#(x, V@(s))= s ( x ,V Q ( x ) ) , and the above equation can be written as

J’(+)w = 0, where J’ is the Friichet derivative of the functional J defined by

J ( 9 ) :=

1

W(x,V*(x))dx

-

f .9 d x -

1,

g.9du,

for all smooth enough vector fields 9 : 4 R3 such that 9 = id on ro. The functional J is called the total energy. Therefore the variational equations associated with the equations of nonlinear three-dimensional elasticity are, at least formally, the Euler equations associated with the minimization problem

J(+)

=

min J ( 9 ) ,

*‘EM

where M is an appropriate set of all admissible deformations @ : R + R3 (an example is given in the next theorem). John Ball’s theory provides an existence theorem for this minimization problem when the function W satisfies the following fundamental definition (see [9]): A stored energy function W : x Mr”+4 R is said to be polyconvex if, for each x f there exists a convex function W ( x ,.) : M3 x M3 x (0, m) 4 R such that

a

a,

W ( x ,F ) = W ( x ,F , CofF, det F ) for all F E M.: Theorem 1.8-1 (John Ball). Let R be a domain in R3 and let W be a polyconvex function that satisfies the following properties: The function W ( . ,F , H , 6 ) : R 3 R is measurable f o r all ( F ,H , 6 ) E M3 x M3 x (0, X I ) . There exist numbers p 3 2, q 3 r > 1, a > 0 , and ,f3 E R such that W ( x ,F ) 3 a( llFllP IICofFIIq I det FIT)- P

3,

+

for almost all x

E

R and f o r all F E M.:

+

Philippe G. Ciarlet, Cristinel Mardare

1.20

For almost all x E R, W ( x , F )+ +m i f F E M: i s such that det F + 0. Let r1 be a relatively open subset of dR, let I'o := dR \ rl, and let there be given fields f E L6l5(R;EX3) and g E L4/3(171;EX3). Define the functional f (z) . @(z)d x -

W ( x ,V@(z))d z and the set

M

:= {@ E

W17p(R;R3);C o f ( V @ ) E L4(R;M3), det(V@) E L'((R), d e t ( V 9 ) > 0 a.e. in R , @ = id o n ro}.

Finally, assume that arearo > 0 and that infgiiM J ( @ ) < 00. T h e n there exists @ E M such that

J ( @ )= inf J ( 9 ) . *EM

Sketch of proof (see Ball 191 or Ciarlet [18],for a detailed proof). Let @, be a infimizing sequence of the functional J , i.e., a sequence of vector fields G n E M such that

J(@,)

+ infgiEM J

( 9 ) < m.

The coerciveness assumption on W implies that the sequences (a,), (Cof(V@,)), and (det(V@,)) are bounded respectively in the spaces W1>p(R;EX3), L4(R; M3), and L'(R). Since these spaces are reflexive, there exist subsequences (@,(,)), (Cof (V@,(n))), and (det(V@,(,))) such that (3denotes weak convergence) +u(n)

H,(n) := Cof(V9,(,)) 6m(n) := det(V9,(,))

in w'J'(R;R ~ ) ,

3

3

H

2

6

in Lq(R;M3), in L'(R).

For all a E W1J'(R;EX3), H E LQ(R;M3),and 6 E L'(R) with S > 0 almost everywhere in R, define the functional

J(9,H , 6) :=

J,

W ( ZV , @ ( x ) ,H ( x ) S(x)) , dx

where, for each x E R, W ( x ,.) : M3 x M3 x (0, m) + IR is the function given by the polyconvexity assumption on W . Since W ( x ,.) is convex, the above weak convergences imply that

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But J ( @ u ( n ) ,H u ( n ) , 6 u ( n ) ) = J ( @ c ( n ) )and J ( @ n ) inf*EM J(*). Therefore J ( 9 H , , 6) = inf*,,M J(*). A compactness by compensation argument applied to the weak convergences above then shows that +

H = C o f ( V 9 ) and 6 = d e t ( V 9 ) . Hence J ( + ) = J ( 9 H , , 6). It remains to prove that E M . The property that W ( F )4 +m if F E M+ is such that det F + 0, then implies that d e t ( V 9 ) > 0 a.e. in R. Finally, since 9, 9 in W1>P(!2;R3)and since the trace operator is linear, it follows that 9 = id on ro. Hence 9 E M . Since J ( 9 ) = J ( 9 H , , 6 ) = inf,p,,M J(*), the weak limit 9 of the sequence aU(,) satisfies the conditions of the theorem.

+

2

0

A St Venant-Kirchhoff material with Lam6 constants X > 0 and p > 0 is hyperelastic, but not polyconvex. However, Ciarlet & Geymonat [26] have shown that the stored energy function of a S t Venant-Kirchhoff material, which is given by

can be “approximated” with polyconvex stored energy functions in the following sense: There exists polyconvex stored energy functions of the form

W b ( F )= allF112 + bllCofFI12 + cI det FI2 - dlog(det F ) + e with a

> 0, b > 0, c > 0, d > 0, e E R,that satisfy

+

w~(F =W ) ( F ) C ? ( I I F~ 1F1 1 3 ) . A stored energy function of this form possesses all the properties required for applying Theorem 1.8-1. In particular, it satisfies the coerciveness inequality: W b ( F )3 a(llF112+ /ICofFII’

2

+ ( d e t F ) 2 )+ p, with a > 0 and p E R.

Two-dimensional theory

Outline In the first part of the article, we have seen how an elastic body subjected t o applied forces and appropriate boundary conditions can be modeled

122

Philippe G. Ciarlet, Cristinel Mardare

by the equations of nonlinear or linearized three-dimensional elasticity. Clearly, these equations can be used in particular to model an elastic shell, which is nothing but an elastic body whose reference configuration has a particular shape. In the second part of the article, we will show how an elastic shell can be modeled by equations defined on a two-dimensional domain. These new equations may be viewed as a simplification of the equations of three-dimensional elasticity, obtained by eliminating some of the terms of lesser order of magnitude with respect to the thickness of the shell. This simplification i s done by exploiting the special geometry of the reference configuration of the shell, and especially, the assumed “smallness” of the thickness of the shell. In the next section, we begin our study with a brief review of the geometry of surfaces in R3 defined by curvilinear coordinates. Of special importance are their first and second fundamental forms. In Section 2.2, we define the reference configuration of a shell as the set in R3 formed by all points within a distance 6 E from a given surface in R3.This surface is the middle surface of the shell and E is its half-thichness. We then define a system of three-dimensional curvilinear coordinates inside the reference configuration of a shell. In Section 2.3, the equations of nonlinear or linearized threedimensional elasticity, which were written in Cartesian coordinates in the first part of the article, are recast in terms of these natural curvilinear coordinates, as a preliminary step toward the derivation of twodimensional shell theories. In Section 2.5, we give a brief account of the derivation of nonlinear membrane and flexural shell models by letting the thickness E approach zero in the equations of nonlinear three-dimensional elasticity in curvilinear coordinates. The same program is applied in Section 2.6 to the equations of linearized three-dimensional elasticity in curvilinear coordinates to derive the linearized membrane and flexural shell models. In Sections 2.7-2.10, we study the nonlinear and linear Koiter shell models. The energy of the nonlinear Koiter shell model is defined in terms of the covariant components of the change of metric and change of curvature tensor fields associated with a displacement field of the middle surface of the reference configuration of the shell. The linear Koiter shell model is then defined by linearizing the above tensor fields. Finally, the existence and uniqueness of solutions to the linear Koiter shell equations are established, thanks to a fundamental K o r n inequality o n a surface and to an infinitesimal rigid displacement lemma o n a surface.

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123

A quick review of the differential geometry of surfaces in R3

To begin with, we briefly recapitulate some important notions of differential geometry of surfaces (for detailed expositions, see, e.g., Ciarlet ~ 2 231). , Greek indices and exponents (except v in the notation 8,) range in the set {1,2}, Latin indices and exponents range in the set { 1 , 2 , 3 } (save when they are used for indexing sequences), and the summation convention with respect to repeated indices and exponents is systematically used. Let w be a domain in EX2. Let y = (y,) denote a generic point in the set 9 and let d, := d/dy,. Let there be given an immersion 0 E C3(9;R3),i.e., a mapping such that the two vectors

%(Y)

:= 8 , q Y )

are linearly independent at all points y E G. These two vectors thus span the tangent plane to the surface

s := e(w) a t the point O(y), and the unit vector

is normal to S at the point O(y). The three vectors %(y) constitute the covariant basis at the point 0(y), while the three vectors ai(y)defined by the relations a2(y) . .j(Y)

=

q,

where dj is the Kronecker symbol, constitute the contravariant basis at the point 0(y) E S. Note that a3(y) = a3(y) and that the vectors aa(y) are also in the tangent plane to S at O(y). As a consequence, any vector field rl : w + R3 can be decomposed over either of these bases as r] = via2= $ai,

where the coefficients vi and vi are respectively the covariant and the contravariant components of q. The covariant and contravariant components a,p and affp of the first and the covariant fundamental f o r m of S , the Christoffel symbols and mixed components b,p and bf of the second fundamental f o r m of S are then defined by letting:

a,p := a, . ap,

aap := aa ' ap,

b,p := a3 . dpa,,

rzp:= a'.

b t := ap'bua.

dpaa,

Philippe G. Ciarlet, Cristinel Mardare

124

The area element along S is &dy, where

a

:= det(a,a).

Note that one also has & = la1 A a2l. The derivatives of the vector fields ai can be expressed in terms of the Christoffel symbols and of the second fundamental form by means of the equations of Gauss and Weingarten: a,ap

=

r ; p , + b,pa3,

&a3

=

-bLa,.

Likewise, the derivatives of the vector fields aj satisfy

+ bza3,

&ar

=

-r;,,aY

3

=

-bauau.

&a

These equations, combined with the symmetry of the second derivatives of the vector field a, (i.e., a,(auaa)= 6',(&a,)), imply that

These relations are equivalent to the Gauss and Codazzi-Mainardi equations, namely,

where

R:,,,

:=

a,q,

-

arr;, + rrar;@- rgarYp

are the mixed components of the Riemann curvature tensor associated with the metric (a,@). If R?, := U"~R?,,~, then one can see that all these functions vanish, save for R12,,. This function is the Gaussian curvature of the surface S , given by

We will see that the sign of the Gaussian curvature plays an important r6le in the two-dimensional theory of shells.

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2.2

125

Geometry of a shell

Let the set w c R2 and the mapping 8 : CJ 4 R3 be as in Section 2.1. In what follows, the surface S = 8(G) will be identified with the middle surface of a shell before deformation occurs, i.e., S is the middle surface of the reference configuration of the shell. The coordinates y1,y2, of the points y E W constitute a system of “two-dimensional” curvilinear coordinates for describing the middle surface of the reference configuration of the shell. More specifically, consider an elastic shell with middle surface S = O(Z) and (constant) thickness 2~ > 0 , i.e., an elastic body whose reference configuration is the set {fi‘}- := @@‘), where (cf. Figure 2.2-1)

RE := w x

(-E,

E)

and 0 ( y ,x;) := 8(y) f xga3(y) for all ( y ,xj) E

a‘.

The more general case of shells with variable thickness or with a middle surface described by several charts (such as an ellipsoid or a torus) can also be dealt with; see, e.g., Busse [16] and S. Mardare [67]. Naturally, this definition makes sense physically only if the mapping 0 is globally injective on the set Fortunately, this is indeed the case if the immersion 8 is itself globally injective on the set Z and E is small enough, according to the following result (due to Ciarlet [20, Theorem 3. I-11).

a‘.

Theorem 2.2-1. Let w be a domain in R2, let 8 E C3(G;R3) be a n injective immersion, and let 0 : W x R 4 R3 be defined by 0 ( y , 23) := B ( y )

+ 23a3(y) for all ( y ,x3) E w x R.

T h e n there exists E > 0 such that the mapping 0 is a C2-diffeomorphism f r o m W x [ - E , E ] onto @(a x [ - E , E ] ) and det(g,,g2,g3) > 0 in w x [-&,&I, where gi := &0.

Proof. The assumed regularity on 8 implies that 0 E C2(W x E > 0. The relations

[-E,

E]; R3)

for any

g, = &0 = a,

+ xsd,a3

and g3 = a30 = a 3

imply that det(gl,g2,g3)lZ3=o= det(al,az,as) > 0 in W. Hence det(g,, g2,g3) > 0 on W x [-E, E ] if E > 0 is small enough. Therefore, the implicit function theorem can be applied if E is small enough: It shows that, locally, the mapping 0 is a C2-diffeomorphism: Given any y E W,there exist a neighborhood U ( y ) of y in G and ~ ( y>) 0

Philippe G. Ciarlet, Cristinel Mardare

126

Figure 2.2-1: T h e reference Configuration of a n elastic shell. Let w be a domain in R2,let RE = w x ( - E , E ) , let 8 E C3((w;R3) be an immersion, and let the mapping 0 : @ + R3 be defined by 0(y, zg) = B(y) +zga3(y) for all (y,zg) E -

if the immersion 8 is RE. Then the mapping 0 is globally injective on globally injective on W and E > 0 is small enough (Theorem 2.2-1). In this case, the set may be viewed as the reference configuration of an elastic shell with thickness 2~ and middle surface S = 8 ( W ) . The coordinates (yl,y2,zg) are then viewed as curvilinear coordinates of the of an arbitrary point z E E point 2" = O ( x E ) of the reference configuration of the shell.

@(a")

such that 0 is a C2-diffeomorphism from the set U(y) x [--E(Y),E(Y)] onto O ( U ( y ) x [ - - ~ ( y ) , ~ ( y ) ] )See, . e.g., Schwartz [81, Chapter 31 (the proof of the implicit function theorem, which is almost invariably given for functions defined over open sets, can be easily extended to functions defined over closures of domains, such as the sets W x [-&,&I; see, e.g., Stein [SZ]). To establish that the mapping 0 : W x [ - - E , E ] + R3 is injective provided E > 0 is small enough, we proceed by contradiction: If this property is false, there exist E" > 0, (yyn,z;), and (p",;), n 3 0, such that E,

-+0

as n

-+

(y",z;)

00,

y" E W,

# (y",2;)

f7 E W,

and O ( y n , z);

6 E,, =

IZFI

< E",

0(p, E?).

Since the set W is compact, there exist y E W and

Q E 9,and there

An Introduction to Shell Theory

exists an increasing function : N

127

N such that

Hence

by the continuity of the mapping 0 and thus y = g since the mapping 8 is injective by assumption. But these properties contradict the local injectivity (noted above) of the mapping 0.Hence there exists E > 0 such that 0 is injective on the set iZ x [ - E , E ] . 0 In what follows, we assume that E > 0 is small enough so that the conclusions of Theorem 2.2-1 hold. The reference configuration of the considered shell is then defined by

{fi"}-

:=

0(2'),

where R' := w x ( - E , E ) and 6' := @(RE). Let x" = (xg)denote a generic point in the set G" (hence xz = y a ) and let 2" = ( 2 ; ) denote a generic point in the reference configuration {&}-. The reference configuration of the shell can thus be described either in terms of the "three-dimensional" curvilinear coordinates y1, y2, xj, or in terms of the Cartesian coordinates 2:, 25,2j, of the same point 2' = 0 ( x ' ) E {&}-. To distinguish functions and vector fields defined in Cartesian coordinates from the corresponding functions and vector fields defined in curvilinear coordinates, we henceforth adopt the following convention of notation: Any function or vector field defined on flEis denoted by letters surmounted by a hat (e.g., ijEis a function defined on &, is a vector field defined on !?F,etc.). The corresponding functions and vector fields defined in curvilinear coordinates are then denoted by the same letters, but without the hat (e.g., g" is the function defined on RE by gE(xE)= t j & ( P for ) all xE E RE, f " is the vector field defined on 0' by f"(zCE) = f (2.) for all xE E RE,etc., the points 2" and zE being related by PE = @(x')). Let 8; := (hence = d / d y a ) and let 8; := ala2i-f.For each xE E D', the three linearly independent vectors gz(x') := afO(x") constitute the cowarian? basis at the point O ( x E )and , the three (likewise linearly independent) vectors gj?"(x")defined by the relations gj"(x') . gg(xE) = d i constitute the contravariant basis at the same point. As a consequence, any vector field uE: RE + R3 can be decomposed over either basis as

i"

A &

a/ax:

a/axz

u"= UZgi?" z Q l E g ; ,

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Philippe G. Ciarlet, Cristinel Mardare

where the coefficients uf and uz+are respectively the covariant and the contravariant components of U ' . The functions gtj(zE):= gf(zE). g;(z') and g i j > " ( z E ):= gi>"(x') . gjiE(z') are respectively the covariant and contravariant components of the metric tensor induced by the immersion 0. The volume element in O ( n E is ) then f l d z ' , where

gE := det(gzj). For details about these notions of three-dimensional differential geometry, see Ciarlet [23, Sections 1.1-1.31

2.3

The three-dimensional shell equations

In this section, we consider an elastic shell whose reference configuration is (6')- := O ( n E )(see Section 2.2), and we make the following assumptions. The shell is subjected to applied body forces given by their densities : & -+ R3 (this means that d P is the body force applied to the volume d2' at each 2' E 6'). For ease of exposition, we assume that there are no applied surface forces. The shell is subjected to a homogeneous boundary condition of place along the portion @(yox [-E, €1) of its lateral face 0 ( d w x [ - E , E]), where yo is a measurable subset of the boundary aw that satisfies lengthy0 > 0. This means that the displacement field of the shell vanishes on the set @(yo x [-&,El). The shell is made of a homogeneous hyperelastic material, thus characterized by a stored energy function (see Section 1.8)

1"

1"

: M3 -+

R.

Such a shell problem can thus be modeled by means of a minimization problem (Section 1.8), which is expressed in Cartesian coordinates, in the sense that all functions appearing in the integrands depend on three variables, the Cartesian coordinates 2' = ( 2 ; ) of a point in the reference configuration {&}- of the shell. We now recast this problem in terms of the curvilinear coordinates zE = (zz) describing the reference configuration {&}- = of the same shell. This will be the natural point of departure for the two-dimensional approch to shell theory described in the next sections. More specifically, the minimization problem consists in finding a minimizer 8' : {h'}- -+ R3 of the functional (see Section 1.8) defined bv

@(a')

s'

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129

over a set of smooth enough vector fields 9"= {A"}R3 satisfying &"(2') = 2" for all 2" E @(yo x [-&,&I). Recall that the functional k is the total energy of the shell. This minimization problem can be transformed into a minimization problem posed over the set i.e., expressed in terms of the "natural" curvilinear coordinates of the shell, the unknown : --f R3 of this new problem being defined by

a",

+ & ( x E )= 2 & ( i E for) all

+' a"

2" =

xE E

a'. a"

If E > 0 is small enough, the mapping 0 is a C'-diffeomorphism of onto its image {AE}- = and det(VEO)> 0 in (Theorem 2.21). The formula for changing variables in multiple integrals then shows that (P" is a minimizer of the functional J" defined by

@(a")

a'

1,.

~ " ( 9:=" ) %(V9"(x")(VEO(xE))-') det V'O dz"

-

In.

f"(xE) . 9 " ( x E )det V"O dx",

a'

where the matrix field V"9" : + M3 is defined by V"W = (cf. Section 1.1)and the vector field f" : + R3 is defined by f E ( x E ):=

]"(P) for all 2'

n"

=

(aj$;)

~ ( x ' )x~ , E RE.

Note that the function det V"O is equal to the function f l ,where gE = det(gtj); cf. Section 2.2. Consider next a linearly elastic shell with Lam6 constants X > 0 and p > 0. In this case, the minimization problem associated with the equations of linearized three-dimensional elasticity (Section 1.4) consists in finding a minimizer & : {A"}- + R3 over a set of smooth enough vector fields 9 ' = {&}- -+ R3 satisfying G"(2")= 2" for all 2" E @(yo x [-&,&I) of the functional k defined by

where X i@(= ~ -(tr(FT ) 8

+ F - 21))' + ~4

+

I I F -~21112 for all F E

~3

(this stored energy function for a linearly elsatic material easily follows from the equations of linearized three-dimensional elasticity given in Section 1.4). Its expression shows that the functional J' is well defined if 9' E Hl(A"; R3).

130

Philippe G. Ciarlet, Cristinel Mardare

As in the nonlinear case, this minimization problem can be recast in curvilinear coordinates. As such, it consists in finding a minimizer W : D" + R3 over the set of all vector fields * I E E H 1 ( f i ER3) ; satisfying W = 0 on yo x [ - E , E ] of the functional J" defined by

As usual in linearized elasticity, it is more convenient to express this energy in terms of the displacement field U" : 2" R3,defined by ---f

w(x")

=

~ ( x "+)u E ( x Efor ) all x E E 52".

nE

Likewise, let v E : 4 R3 be such that forward calculation shows that

W

=0

+ v'.

Then a straight-

r/ir( V E Q(V EE O ) - ' )= AijkelE efj (v')eie (v"),

1

e:j(vE):= - ( d f v E . g ; 2

+ d;v'

.g:).

The functions AijkeiEand e:j (u")denote respectively the contravariant components of the three-dimensional elasticity tensor in curvilinear coordinates, and the covariant components of the linearized strain tensor associated with the displacement field v".It is then easy to see that U" is a minimizer over the vector space

v ( V ):= { u " = ufgi+; uf E H'(R"), u: = o on 7 0 x

(-E,E)},

of the functional J" defined by

This minimization problem will be used in Section 2.6 as a point of departure for deriving two-dimensional linear shell models.

2.4

The two-dimensional approach to shell theory

In a two-dimensional approach, the above minimization problems of Section 2.3 are "replaced" by a, presumably much simpler, two-dimensional problem, this time "posed over the middle surface S of the shell". This means that the new unknown should be now the deformation cp : -+ R3 of the points of the middle surface S = t9(U), or, equivalently, the displacement field : U + R3 of the points of the same surface S (the

F

c

An Introduction t o Shell Theory

131

deformation and the displacement fields are related by the equation ‘p = 8 6); cf. Figure 2.4-1. The two-dimensional approach to shell theory yield a variety of twodimensional shell models, which can be classified in two categories (the same classification applies for both nonlinear and linearized shell models) : A first category of two-dimensional models are those that are obtained from the three-dimesional equations of shells “by letting E go to zero”. Depending on the data (geometry of the middle surface of the shell, boundary conditions imposed on the displacement fields, applied forces) one obtains either a membrane shell model, or a flexural shell model, also called a bending shell model. A brief description of these models and of their derivation is given in Sections 2.5 and 2.6.

+

h

Y

E > 0 “small enough” and data of ad hoc orders of magnitude, the three-dimensional shell problem is “replaced” by a “two-dimensional shell problem”. This means that the new unknowns are the three covariant components Ci : GJ + E% of the displacement field Ciai : GJ + R3 of the points of the middle surface S = O@). In this process, the “three-dimensional” boundary conditions on !?o need to be replaced by ad hoc L‘twO-dimensional’’boundary conditions on yo. For = 0 on 70 (used instance, the “boundary conditions of clamping” Ci = in Koiter’s linear equations; cf. Section 2.8) mean that the points of, and the tangent spaces to, the deformed and undeformed middle surfaces coincide along the set O(y0).

Figure 2.4-1: An elastic shell modeled as a two-dimensional problem. For

132

Philippe G. Ciarlet, Cristinel Mardare

A second category of two-dimensional models are those that are obtained from the three-dimensional model by restricting the range of admissible deformations and stresses by means of specific a priori assumptions that are supposed to take into account the “smallness” of the thickness (e.g., the Cosserat assumptions, the Kirchhoff-Love assumptions, etc.). A variety of two-dimensional models of shells are obtained in this fashion, as, e.g., those of Koiter, Naghdi, etc. A detailed description of Koiter’s model is given in Sections 2.7 and 2.8.

2.5

Nonlinear shell models obtained by r-convergence

Remarkable achievements in the asymptotic analysis of nonlinearly elastic shells are due to Le Dret & Raoult [64] and to F’riesecke, James, Mora & Miiller [49], who gave the first (and only ones to this date) figorous proofs of convergence as the thickness approaches zero. In so doing, they extended to shells the analysis that they had successfully applied to nonlinearly elastic plates in Le Dret & Raoult [62] and F’riesecke, James & Miiller [48]. We begin with the asymptotic analysis of nonlinearly elastic membrane shells. H. Le Dret and A. Raoult showed that a subsequence of deformations that minimize (or rather “almost minimize” in a sense explained below) the scaled three-dimensional energies weakly converges in W1>p(R;R3)as E 4 0 (the number p > 1 is governed by the growth properties of the stored energy function). They showed in addition that the weak limit minimizes a “membrane” energy that is the r-limit of the (appropriately scaled) energies. We now give an abridged account of their analysis. Let w be a domain in R2 with boundary y and let 8 E C2(G; R3) be an injective mapping such that the two vectors a,(y) = d,8(y) are linearly independent at all points y = (ye) E SLi. Consider a family of elastic shells with the same middle surface S = 8(G) and whose thickness 2~ > 0 approaches zero. The reference configuration of each shell is thus the image @@&) c R3 of the set -& s1 c R3 through a mapping 0 : + R3 defined in Section 2.2. By Theorem 2.2-1, if the injective mapping 8 : W 4 R3 is smooth enough, the mapping 0 : 4 R3 is also injective for E > 0 small enough and y1, y2, x$ then constitute the “natural” curvilinear coordinates for describing each reference configuration O ( @ ) . Assume that all the shells in the family are made,of the same hyperelastic homogeneous material (see Section 1.8), satisfying the following properties : The stored energy function I%’ : M3 4 R of the hyperelastic material

An Introduction to Shell Theory satisfies the following assumptions: There exist constants C 0, p E EX,and 1 < p < ca such that

133

> 0, a >

It can be verified that the stored energy function of a St VenantKirchhoff material, which is given by

satisfies such inequalities with p

= 4.

Remark. By contrast, the stored energy function of a linearly elastic material, which is given by P @ ( F ) = -1IF 4

+ FT

-

where JJFJJ := (tr FTF}'12,satisfies the first inequality with p not the second one.

= 2,

but

0

It is further assumed that, for each E > 0, the shells are subjected in their interior t o applied body forces of density f" = ffgZ+ : fi" 4 R3 per unit volume, where ff E Lq(fi") and $ = 1, and that these densities do not depend on the unknown deformation. Applied surface forces on the "upper" and "lower" faces of the shells could be likewise considered, but are omitted for simplicity; see in this respect Le Dret & Raoult [64] who consider a pressure load, an example of applied surface force that depends on the unknown deformation. Finally, it is assumed that each shell is subjected to a boundary condition of place along its entire lateral face O(y x [-&, &]), where y := dw, i.e., that the displacement vanishes there. The three-dimensional problem is then posed as a minimization probl e m in terms of the unknown deformation field

+i

a'(.")

+

:= O ( x " )

of the reference configuration, where

U 8 ( Z E ) ,Z E

U"

:

E

a",

fiE 4 R3 is its displacement

Philippe G. Ciarlet, Cristinel Mardare

134

field (Section 2.3): It consists in finding

a"E M(R") and J"(a")= M(R')

:=

a" such that J E ( S E )where ,

inf

W€M(W)

{a"E W11p(R";IW3);9"= 0 on y x -

[-E,

E]},

In f" 9" .V"0dx". .

det

This minimization problem may have no solution; however, this is not a shortcoming as only the existence of a "diagonal infimizing family", whose existence is always guaranteed, is required in the ensuing analysis (Theorem 2.5-1). The above minimization problem is then transformed into an analogous one, but now posed over the fixed domain R := W X ] - 1, 1[. Let x = ( 2 1 , 2 2 , 2 3 ) denote a generic point in and let & := 8/8xi. With each point x E we associate the point xE E through the bijection

a

a,

7r"

: 2 = (21, 2 2 , 2 3 ) E

R

a"

-+ 2" = (2" i ) = (21, 2 2 , E Z g ) E

n".

We then define the unknown scaled deformation a(&): by letting @ ( E ) ( Z ) := @ " ( x E ) for all 2" = T ' Z , z E

a + R3

a.

Finally, we assume that the applied body forces are of order O(1) with respect to E , in the sense that there exists a vector field f E L2(R;R3) independent of E such that

fE(x") = f ( 2 ) for all zE = T"Z,IL: E

a.

Remark. Should applied surface forces act on the upper and lower faces of the shells, we would then assume that they are of order O(E) with respect to E . 0 In what follows, the notation (bl;b2; b3) stands for the matrix in M3 whose three column vectors are bl, b2, b3 (in this order). These scalings and assumptions then imply that the scaled deformation % ( E ) satisfies the following minimization problem:

a(&)E M ( & ;0) and J ( E ) ( @ ( E ) ) M ( E ;R)

=

J ( E ) ( ~where ),

inf

*'EM("; n) := {9E W1>p(R;W3); 9 = + O ( E ) on

J ( E ) ( * ) :=

1 R

y x [-I, I]},

1 I@( (819;8 2 s ; -&*)(G(E))-') & det G ( Edz ) r

- /n

f . +det G(E)dz,

An Introduction to Shell Theory where the vector field

+o(E)

:

135

fi + R3 is defined for each E > 0 by

+ o ( ~ ) ( z:= ) @(xE) for all x E = T ' Z , x E '32, and the matrix field G(E):

--+

M3 is defined by

G ( E ) ( z:= ) V e O ( x Efor ) all xE = T ' X , x E

n.

The scaled displacement U(E)

:= +(&)

- a+)(&)

therefore satisfies the following minimization problem: U(E)

E

W(R;IR3) and J(E)(u(E)) =

w(R;R~)

:=

J ( E ) ( w:= )

{w

h

E

inf

J ( E ) ( w )where ,

v€w(n;R3)

w ' > P ( s ~w;= IR o on ~ )y; x [-I, 11)

%(I

1 + (&w; d2w; -&w)(G(&))-') det G ( Edx )

5

f . ('Po(&)

+ w)det G(E)dx.

Central to the ensuing result of convergence is the notion of quasiconvexity, due to Morrey [71, 721 (an account of its importance in the calculus of variations is provided in Dacorogna [38, Chap 51): Let M m x n denote the space of all real matrices with m rows and n columns; a function I@ : M m x n 4 R is quasiconvex if, for all bounded open subsets U c R",all F E M m x n , and all 6 = (&)gl E W,'l"(U;R"),

where V6 denotes the matrix (aj 0. The family ( J ( E ) ) ~ > is osaid to r-converge as E -+ 0 if there exists a functional J : V -+ R U {+co}, called the F-limit of the functionals J ( E ) ,such that V(E)

-+

v as E

-+

0 + J ( v ) < liminf J ( E ) ( v ( E ) ) , E+O

on the one hand and, given any v E V, there exist V ( E ) 6 V, E > 0, such that V ( E ) -+ v as E -+ 0 and J ( v ) = lim J ( E ) ( v ( E ) ) , E+O

on the other. As a preparation to the application of I?-convergence theory, the scaled energies J ( E ): M(R) -+ R found above are first extended to the larger space LP(R; EX3) by letting J(E)(W) =

i

J ( E ) ( w if ) w E M(Q), f c o if w E LP(R;R3) but w $ M(R).

Such an extension, customary in F-convergence theory, has inter alia the advantage of “incorporating” the boundary condition into the extended functional. Le Dret & Raoult [64] then establish that the family ( J ( E ) ) ~ of > ~ extended energies F-converges as E -+ 0 in LP(R; EX3) and that its r-limit can be computed by means of quasiconvex envelopes. More precisely, their analysis leads to the following remarkable convergence theorem, where the limit minimization problems are directly posed as two-dimensional problems (part c)); this is licit since the solutions of these limit problems do not depend on the transverse variable (part (b)). Note thaq, while minimizers of J ( E )over M ( Q ) need not exist, the existence of a “diagonal infimizing family” in the sense understood below is always guaranteed because inf,,M(n) J ( E ) ( w > ) -co. In what follows, the notation ( b l ; b2) stands for the matrix in M3x2 with bl, b2 (in this order) as its column vectors and &dy denotes as usual the area element along the surface S .

An Introduction to Shell Theory

137

Theorem 2.5-1. A s s u m e that the applied body forces are of order O(1) with respect to E , and that there exist C > 0 , a > 0, p E R, and 1 < p < 00 such that the stored energy function W : M3 3 R satis$es the following growth conditions: for all F E M3, IW(F)I < C ( I + IF\”) for all F E M ~ W(F) b a\FI” + p (W(F) - W/(G)I6 C(I + l ~ l p - l + I G \ ~ - ~ -) IGFI for all F , G E ~

, 3

Let the space M(R) be defined by

M(R) := {v E W11p(R;R3);v = 0 on y x [-I, I]}, and let (U(E)),>O be a “diagonal infimizing family” of the scaled energies, i.e., a family that satisfies U(E)

E M(C2)

and J(E)(u(E)) 6

inf

J ( E ) ( v )+ h(&)for all

VEM(n)

E

> 0,

where h is any positive function that satisfies h(&)+ 0 as E -+ 0 . Then: (a) The family ( U ( E ) ) , > ~ lies in a weakly compact subset of the space Wl~”(R;R3). (b) T h e limit u E M(R) as E -+ 0 of any weakly convergent subsequence of ( U ( E ) ) ~ > O satisfies 8321 = 0 in R and is thus independent of the transverse variable. 1 (c) The vector field 6 := S-, u dx3 satisfies the following minimization problem:

C E W ~ 1 p ( w ; Rand 3 ) j ~ ( 6=)

inf

j~(rl),

17EW;’P(W;P3)

where

G ( Y ):= (ai(~); a 2 ( ~ )as(^)), ; the vectors a i ( y ) forming for each y E 2 the covariant basis at the point O(y) E S , and Q W o ( y , .) denotes for each y E G the quasiconvex envelope of W o ( Y , .). 0

.

Philippe G. Ciarlet, Cristinel Mardare

138

It remains to de-scale the vector field C. In view of the scalings performed on the deformations, we are naturally led to defidng for each E > 0 the limit displacement field C" : Z -+EX3 of the middle surface by

s

C"

:= 6.

It is then immediately verified that C" satisfies the following minimization problem (the notations are those of Theorem 2.5-1):

C"

E W~'*(w;EX3) and j$(C") =

jG(r]),where

inf 1)EW,1'P(W;R3)

The unknown r] in the above minimization problem appears only by means of its first-order partial derivatives a,r] in the stored energy function r] E

W1Jyw;R3)

--f

€&*&

+

(a1 8177;a2

+

a2r]>>

found in the integrand of the energy jh. Assume that the original stored energy function is frame-indifferent, in the sense that

I@(RF)= I@(F) for all F E M3 and R 6 0;. This relation is stronger than the usual one, which holds only for F E M3 with det F > 0 (see Ciarlet [18, Theorem 4.2-11); it is, however, verified by the kinds of stored energy functions to which the present analysis applies, e.g., that of a St Venant-Kirchhoff material. Under this stronger assumption, Le Dret & Raoult [64, Theorem 101 establish the crucial properties that the stored energy function found in jh, once expressed as a function of the points of S, is frame-indifferent and that it depends only on the metric of the deformed middle surface. For this reason, this theory is a frame-indifferent, nonlinear "membrane" shell theory. It is remarkable that the stored energy function found in jh can be explicitly computed when the original three-dimensional stored energy function is that of a St Venant-Kirchhoff material; see Le Dret & Raoult [64, Section 61. Again for a St Venant-Kirchhoff material, Genevey [50] has furthermore shown that, when the singular values of the 3 x 2 matrix fields (8,vi) associated with a field r] = viai belong to an appropriate compact subset of R2 (which can be explicitely identified), the expression

An Introduction to Shell Theory

139

j & ( q ) takes the simpler form

where aapuT .- %a@aUT

a,O(q)

+ 2p(aa‘-7aP7 + aOrTaPU),

x + 2p := a,(e + viai) . a,(e

+ qjaj).

This is precisely the expression of j & ( q ) that was found by Miara [69] to hold “for all fields q” (i.e., without any restriction on the fields q such as that found by Genevey [50]),by means of a formal asymptotic analysis. This observation thus provides a striking example where the limit equations found by a formal asymptotic analysis “do not always coincide” with those found by means of a rigorous convergence theorem. Le Dret & Raoult [64, Section 61 have further shown that, if the stored energy function is frame-indifferent and satisfies l@(F)3 $ ( I ) for all F E M3 (as does the stored energy function of a St Venant-Kirchhoff material), then the corresponding shell energy i s constant under compression. This result has the striking consequence that “nonlinear membrane shells o#er n o resistance t o crumpling. This is an empirical fact7 witnessed by anyone who ever played with a deflated balloon” (to quote H. Le Dret and A. Raoult). We now turn our attention to the asymptotic analysis, by means of I?convergence theory, of nonlinearly elastic flexural shells. In its principle, the approach is essentially the same (although more delicate) as ‘that used for deriving the nonlinear membrane shell equations. There are, however, two major differences regarding the assumptions that are made at the onset of the asymptotic analysis. A first difference is that the applied body forces are now assumed to be of order C ~ ( E with ~ ) respect to E (instead of 0(1)), in the sense that there exists a vector field f E L2(a; Pi3) independent of E such that

fg(xE)= E ~ ~ (forx all ) x E = T‘X

E s2.

A second difference (without any counterpart for membrane shells) is that the set denoted M F ( win ) the next theorem contains other fields than 8 (the interpretation of this key assumption is briefly commented upon after the theorem). Under these assumptions, F’riesecke, James, Mora & Muller [49] have proved the following result. The notations as($), asp($), and hap($)

Philippe G. Ciarlet, Cristinel Mardare

140

used in the next statement are self-explanatory: Given an arbitrary VFCtor field E H2(u;JR3),

+

Theorem 2.5-2. A s s u m e that the applied body forces are of order C ~ ( E ~ ) with respect t o E . A s s u m e in addition that the stored energy function W : M3 4 R satisfies the following properties: It is measurable and of class C2 in a neighborhood of Ot,it satisfies

* ( I ) = 0 and i @ ( R F )= $ ( F ) for all F E M3 and R E O:, and, finally, it satisfies the following growth condition: There exists a constant C > 0 such that

IL+(F)I

c REO? inf

I F- ~1~

for all F E M ~ .

Finally, assume that the set

M F ( w ):= {+

E

H2(w;JR3);a,p(+) = aap in w ; = 8 and as(+) = a3 on YO},

+

contains other vector fields than 8 . T h e n the scaled energies J ( E ) , E > 0 , constitute a family that I?converges as E + 0 in the following sense: A n y "diagonal infimizing family" (defined as in Theorem 2.5-1) contains a subsequence that strongly converges in H1(R; R3). Besides, the limit of any such subsequence is independent of the transverse variable, and the vector field cp :=

where

An introduction to Shell Theory

141

The assumption that the set M F ( u )contains other vector fields than 8 means that there exist nonzero displacement fields viai of the middle surface 6 ( 9 ) that are both inextensionaZ, in the sense that the surfaces 8(Z) and +(G), where := 8 viai, have the same metric (as reflected by the assumption a,p(+) = a,p in w ) , and admissible, in the sense that the points of, and the tangent spaces to, the surfaces 8(Z) and +(Z) coincide along the set 6(yo) (as reflected by the boundary conditions = 8 and a3(+) = a3 on 70). it follows that the “de-scaled” unknown deformation pE: Z + R3 of the middle surface of the shell is a minimizer over the set M F ( w )of the functional j g defined by

+

+

+

When the original three-dimensional stored energy function is that of a St Venant Kirchhof material, the expression j$(+) takes the simpler form

where

interestingly, exactly the same expression jg(+) was found for all

+ E M F ( w by ) means of a formal asymptotic analysis by Lods & Miara

[65], as the outcome of sometimes exceedingly delicate computations. This observation is thus in sharp contrast with that made for a membrane shell, whose limit equations cannot always be recovered by a formal approach, as noted earlier.

Remark. Although r-convergence automatically provides the existence of a minimizer of the r-limit functional, the existence of a minimizer of the functional j~ over the set M F ( u )can be also established by means of a direct method of calculus of variations; cf. Ciarlet & Coutand [24].c3

142

2.6

Philippe G. Ciarlet, Cristinel Mardare

Linear shell models obtained by asymptotic analysis

In this section, we briefly review the genesis of those two-dimensional linear shell theories that can be found, and rigorously justified, as the outcome of an asymptotic analysis of the equations of three-dimensional linearized elasticity as E -+ 0 . The asymptotic analysis of elastic shells has been a subject of considerable attention during the past decades. After the landmark attempt of Goldenveizer [53],a major step for linearly elastic shells was achieved by Destuynder [44] in his Doctoral Dissertation, where a convergence theorem for “membrane shells” was “almost proved”. Another major step was achieved by Sanchez-Palencia [77],who clearly delineated the kinds of geometries of the middle surface and boundary conditions that yield either two-dimensional membrane, or two-dimensional flexural, equations when the method of formal asymptotic expansions is applied to the variational equations of three-dimensional linearized elasticity (see also Caillerie & Sanchez-Palencia [17] and Miara & Sanchez-Palencia [70]). Then Ciarlet & Lods [27, 281 and Ciarlet, Lods & Miara [31] carried out an asymptotic analysis of linearly elastic shells that covers all possible cases: Under three distinct sets of assumptions on the geometry of the middle surface, on the boundary conditions, and on the order of magnitude of the applied forces, they established convergence theorems in H I , in L 2 , or in ad hoc completion spaces, that justify either the linear two-dimensional equations of a “membrane shell”, or those of a “generalized membrane shell”, or those of a ‘‘j?exural shell”. More specifically, consider a family of linearly elastic shells of thiclcness 2~ that satisfy the following assumptions: All the shells have the same middle surface S = 8(C)c R3, where w is a domain in R2 with boundary y,and 8 E C 3 ( Z ; R 3 ) .Their reference configurations are thus of the form O(fi“),E > 0, where

RE := w x (-&,&) and the mapping 0 is defined by

@ ( y , x ~:= ) 6(y)

+ x ; a 3 ( y ) for all ( y , ~ : ) .

All the shells in the family are made with the same homogeneous isotropic elastic material and that their reference Configurations are natural states. Their elastic material is thus characterized by two Lam6 constants X > 0 and p > 0. The shells are subjected to body forces and that the corresponding applied body force density is O(EP) with respect t o E , for some ad hoc power p (which will be specified later). This means that, for each E > 0,

An Introduction to Shell Theory the contravariant components f' = fi)'gS are of the form

fz?"

E

143

L2(RE)of the body force density

filE(yE , Z ~=) & p f i ( y ,z3) for all (y, z3) E R := w x 1-1,

I[,

and the functions f i E L2(R) are independent of E (surface forces acting on the "upper" and "lower" faces of the shell could be as well taken into account but will not be considered here, for simplicity of exposition). Let then the functions pi>' E L2(w)be defined for each E > 0 by

J -€

Finally, each shell is subjected to a boundary condition of place on the portion @(yo x [-E, €1) of its lateral face, where yo is a fixed portion of y,with lengthy0 > 0. Then the displacement field of the shell satisfies the following minimization problem associated with the equations of linearized threedimensional elasticity in curvilinear coordinates (see Section 2.3): uEE V ( V )and J E ( u E=)

min

JE(v'), where

V"EV(W)

V ( R E ):= { v E= v;gi+; w;

E

H ~ ( R ) ,u:

=

o on TO

x

(-E,E)}.

For each E > 0, this problem has one and only one solution u' E V ( R ) . For any displacement field q = V i a i : w R3, let ---f

1 y d v ) = ~ ( d p vacu . + aav.ap) and pOlg(rl)= ( % g q - r&a,q) . a3

denote as usual the covariant components of the linearized change of metric, and linearized change of curvature, tensors. In Ciarlet, Lods & Miara [31] it is first assumed that the space of linearized inextensional displacements (introduced by Sanchez-Palencia

PI) v,(w)

:= { q = viai;

qa E ~ ' ( w ) ,773 E H ~ ( w ) ; vi = dyv3 =

0 on yo, ycyp(q)= 0 in w}

contains non-zero functions. This assumption is in fact one in disguise about the geometry of the surface S and on the set 7 0 . For instance, it is satisfied if S is a portion of a cylinder and O(70) is contained in one or two generatrices of S , or if S is contained in a plane, in which case the shells are plates.

Philippe G. Ciarlet, Cristinel Mardare

144

Under this assumption Ciarlet, Lods & Miara [31]showed that, i f the applied body force density is O ( e 2 ) with respect to E , then

1'

!2E

uEdz: -+

C in H 1(w;R3) as E + 0,

-&

:= &ai belongs t o the space V F ( Wand ) where the limit vector field satisfies the equations of a linearly elastic "flexural shell" , viz.,

E"3 S, a a p c T p u T ( < ) p f f p ( v ) h d =y b p i l ~ v i h d y for all 77 = viai E V F ( WObserve ). in passing that the limit 6 is indeed independent of E , since both sides of these variational equations are of the same order (viz., E ~ ) because , of the assumptions made on the applied forces. Equivalently, the vector field 6 satisfies the following constrained minimization problem:

5 E V F ( Wand ) G(5) = inf&(v), where

for all

v = viai E V F ( Wwhere ) , the functions

are precisely the familiar contravariant components of the shell elasticity tensor. If V F ( W#){ 0 } , the two-dimensional equations of a linearly elastic 'Vexural shell" are therefore justified. If V F ( W=)( 0 } ,the above convergence result still applies. However,

SE

the only information it provides is that u' d & J-+ 0 in H1(w; R3) 2E -& as E 0. Hence a more refined asymptotic analysis is needed in this case. A first instance of such a refinement was given by Ciarlet & Lods [27], where it was assumed that yo = y and that the surface S is elliptic, in the sense that its Gaussian curvature is > 0 everywhere. As shown in Ciarlet & Lods [27] and Ciarlet & Sanchez-Palencia [35], these two conditions, together with ad hoc regularity assumptions, indeed imply that V p ( w ) = (0). -+

An Introduction to Shell Theory

145

In this case, Ciarlet & Lods [28] showed that, if the applied body force density is 0(1)with respect to E , then 2E

/'

u: dxg

+ 0, let 8, denote the outer normal derivative operator along dw, and let the space V(w) be defined by

va E ~ ' ( w ) , q E3 ~ ~ ( w ) ,=q dvr/3 i = o on yo}. Then the displacement field C" = 0 such that

B :B

> 0.

< ,BC : C for all B = A - l C

the announced inequality also holds if - p

Since there clearly

E MI2,

<x 0,

with

y = (2pp)-l in this case. (ii) We next show that, for any y E W and any nonzero symmetric matrix ( t a p ) ,

a"fluT(Y)tuTtap 3 yaau(Y)aP7(y)to,t,p

> 0,

where y = y(x, p ) > 0 is the constant found in (i). Given any y E W and any symmetric matrix ( t a p ) , let

A(!/)= (a%))

and T

= (tap),

let K ( y ) E S2 be the unique square root of A ( y ) (i.e., the unique positivedefinite symmetric matrix that satisfies ( K ( Y )=) A ~ ( y ) ) , and let B ( y ) := K ( y ) T K ( y )E S2.

An Introduction to Shell Theory

173

Then

By the inequality established in part '(i), there thus exists a constant a ( X , p ) > 0 such that 1

-2a a O n ~

if x

(y)tU,tap 2 a t r (B(Y)TB(Y))

+ p > 0 and p > 0, or equivalently, if 3X + 2p > 0 and p > 0 (iii) Conclusion: Since the mapping

is continuous and its domain of definition is compact, we infer that

Hence ltapI2 6 aau(Y)aflT(Y)t,Ttap a,P

and thus

c

ItapI2 6 ceaapUT(y)t,,tap

%B

for all y E i J and all symmetric matrices ( t a p ) , with ce := (yb)-l.

0

Combined with Korn's inequality "with boundary conditions" (Theorem 2.9-3), the positive definiteness of the elasticity tensor leads to the existence of a weak solution, i.e., a solution to the variational equations of the linear Koiter shell model.

Theorem 2.10-2. Let w be a domain in R2, let yo be a subset of y = dw with length yo > 0, and let 8 E C3(w;R3) be an injective immersion. Finally, let there be given constants X and p that satisfy 3X 2p > 0 and p > 0, and functions pa>€E L T ( w ) f o r some r > 1 and p3+ E L 1 ( w ) . Then there is one and only one solution = 0 for all y E i;s. Finally, the Korn inequality "with boundary conditions" (Theorem 2.9-3) and the uniform positive definiteness of the elasticity tensor of the shell (Theorem 2.10-1) together imply that 2

min {e, y j c j V

2

^ ^ \\va\\m {uj)

for all 17 = v i a i E V(w). Hence the bilinear form B is V(w)-elliptic.

An Introduction to Shell Theory

175

The Lax-Milgram lemma then shows that the variational equations have one and only one solution. Since the bilinear form is symmetric, this solution is also the unique solution of the minimization problem stated in the theorem. 0 The above existence and uniqueness result applies to linearized pure displacement and displacement-traction problems, i.e., those that correspond to lengthyo > 0. We next derive the boundary value problem that is, at least formally, equivalent to the variational equations of Theorem 2.10-2. In what follows, 7 1 := 7\70, (ua)is the unit outer normal vector along 7, 7 1 := - 2 4 , 7-2 := u1, and arx := rJax denotes the tangential derivative of x in the direction of the vector (ra).

Theorem 2.10-3.Let w be a domain in R2 and let 0 E C3(W;R3)be a n injective immersion. Assume that the boundary y of w and the functions pi>€are smooth enough. If the solution C" = 1. Assume finally that, for some integer m 3 1 and some real number q > 1, y is of class Cm+4 and 9 E C"+4(sj;R3). Then a regularity result of Agmon, Douglis & Nirenberg [3] implies that

C"a E W"+3i4(w) and 1, then N' = N .

Figure 3 Example of metric intersection in two and three dimensions using the simultaneous reduction of two quadratic forms.

1.2.5

Metric interpolation

Let us consider a parametrization of the segment p q as c : [0,1] + R2, c(t) = (1 - t ) p + t q (a mesh edge, for instance) and suppose that Mp and M, are two symmetric positive definite matrices representing two metric tensors associated with the endpoints. We are looking for the metric tensor at t , hence for a matrix Mt defined along the segment c ( t ) for any value of the parameter t E [0,1]. The definition of this matrice Mt involves the interpolation of the eigenvalues of the matrices M p and

240

Pascal F’rey

M,. This procedure allows to define a continuous metric field along the segment. To this end, we suggest a linear interpolation scheme, for which the metric at any point c ( t ) is given by: Mt

=

OO

at po E U if detII,, is

=0

< 0.

Regarding the intrinsic properties of a surface, we can state:

Definition 1.26. Let c ( t ) = f ou(t) be a curve o n a surface f : U c ( t ) is a geodesic if V C ( t ) / d t = 0 .

-+

R3;

Theorem 1.2. Let X E Tpof be a tangent vector to a surface f . T h e n for suficiently small E > 0 there exists a unique geodesic c ( t ) = f o u(t), It1 < E satisfying the initial conditions u(0)= po and C(0) = X .

246

Pascal Frey

As a consequence, all the nonconstant geodesics on a sphere are great circles.

To give compatibility equations for the theory of surfaces, we introduce the six Christoffel symbols:

where gij,k = d g i j / d u k . This will allow us t o write the following result: Proposition 1.6 (Integrability conditions). The Gauss formula and Codazzi-Mainardi equations provide relations between g i k , h i k , g i k , l , hik,l and l?Fj,z: rF,k

-

r2,j+ C(rfjr2 rfkr;) = C(hijhk1 - h i k h j z ) g z , -

I

(3)

I

Finally, we can give the fundamental result, known as Bonnet’s theorem: Theorem 1.3 (Fundamental theorem of surface theory). Let U be a n open, simply connected subset of W2. Suppose I, and II, are quadratic forms o n TpW2, p E U , whose coeficients ( g i k ( p ) ) and ( h i k ( p ) ) are differentiable functions of p . If I, is positive definite and the Gauss and Codazzi-Mainardi equations are satisfied, then: i) there exists a surface f : U -+ W3 whose first and second fundamental forms are I, and IIp. ii) any two surfaces f and f” defined o n U which have the same first and second fundamental forms differ b y a n isometry:

f =B of

,

B an isometry of W3.

The theorem asserts that the knowledge of the first and second fundamental forms determines a surface locally. This result concludes our short introduction to the differential geometry of curves and surfaces. We will use these concepts in Section 3, when dealing with the tedious problem of meshing surfaces.

2

A geometric error estimate

Anisotropic error estimation has been studied extensively in the last two decades. In this section, we briefly review the concept of error estimation

A Differential Geometry Approach to Mesh Generation

247

and we propose an approach to obtain an anisotropic bound for the interpolation error suitable for linear simplicia1elements in any dimension. In addition] we will explain how this bound will provide a characterization of the optimal mesh elements] both in terms of shape and of size. We assume the reader to be familiar with the concept of Hilbert spaces and with variational (weak) formulation for elliptic problems.

2.1

A priori error analysis

We will primarily focuss on error estimation in the context of finite element methods (or FEM in short). The concepts presented in this section can however be transposed to other classical discretization methods such as finite differences or finite volumes, for instance.

2.1.1

Interpolation error in L2 norm and H 1 seminorm

Let a be a bounded domain of Rd and let u be the solution of an elliptic differential problem on under Dirichlet boundary conditions on dol the domain boundary. For an elliptic problem of order 2, for which we can write a variational formulation: ‘fJind u E v such that a(u,v) = l(v), vw E V ” where V is a Hilbert space, a is a bilinear continuous elliptic form, 1 is a linear continuous form on V , we will recall the classical error bound given by Cea’s lemma [9] and already mentioned in the introduction of this chapter. A bilinear form a(., .) is continous means that:

3A4

> 0,

such that

(u(w,w)(< A4 ( ( w (( (( ~ 1 1 ,vw,wE V .

and a(., ,) is a-elliptic or coercive if

3a

> 0,

such that

la(wlw)I