Ciarlet_A Brief Introduction to Mathematical Shell Theory

8/12/2019 Ciarlet_A Brief Introduction to Mathematical Shell Theory

1/60

A Brief Introduction to Mathematical Shell Theory

Philippe G. Ciarlet

City University of Hong Kong

AbstractIn the first chapter, we study basic notions about surfaces, such as their

two fundamental forms, the Gaussian curvature and covariant derivatives. We also

state the fundamental theorem of surface theory, which asserts that the Gau and

Codazzi-Mainardi equations constitute sufficient conditions for two matrix fields

defined in a simply-connected open subset ofR2 to be the two fundamental forms

of a surface in a three-dimensional Euclidean space. We also state the corresponding

rigidity theorem.

The second chapter, which heavily relies on Chapter 1, begins by a detailed

description of the nonlinear and linear equations proposed by W.T. Koiter for mod-

eling thin elastic shells. These equations are two-dimensional, in the sense that

they are expressed in terms of two curvilinear coordinates used for defining the

middle surface of the shell. The existence, uniqueness, and regularity of solutions

to the linear Koiter equations is then established, thanks this time to a fundamental

Korn inequality on a surface and to an infinitesimal rigid displacement lemma

on a surface.

Lecture Notes for the Advanced School Classical and Advanced Theories of Thin Structures:

Mechanical and Mathematical Aspects, International Center for Mechanical Sciences, Udine,

June 0509, 2006. Theses notes are adapted from Chapters 2 and 4 of my book An In-

troduction to Differential Geometry with Applications to Elasticity, Springer, Dordrecht,

2005.

1


2/60

Contents

Contents 2

1 Differential geometry of surfaces 3

1.1 Surfaces defined by means of curvilinear coordinates . . . . . . . . . . . . 31.2 First fundamental form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Areas and lengths on a surface . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Second fundamental form; curvature on a surface . . . . . . . . . . . . . . 101.5 Principal curvature; Gaussian curvature . . . . . . . . . . . . . . . . . . . 161.6 Covariant derivatives of a vector field defined on a surface; the Gau and

Weingarten formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.7 The Gau and Codazzi-Mainardi equations . . . . . . . . . . . . . . . . . 231.8 The fundamental theorem of surface theory . . . . . . . . . . . . . . . . . 26

2 An introduction to shell theory 29

2.1 What is a two-dimensional shell problem? . . . . . . . . . . . . . . . . . . 292.2 The nonlinear Koiter shell equations . . . . . . . . . . . . . . . . . . . . . 322.3 The linear Koiter shell equations . . . . . . . . . . . . . . . . . . . . . . . 362.4 A fundamental lemma of J.L. Lions . . . . . . . . . . . . . . . . . . . . . . 432.5 Korns inequality on a surface . . . . . . . . . . . . . . . . . . . . . . . . . 452.6 Existence and uniqueness theorems for the linear Koiter shell equations . 50

References 58

2


3/60

1 Differential geometry of surfaces

1.1 Surfaces defined by means of curvilinear coordinates

Latin indices and exponents range in the set{1, 2, 3}, Greekindices and exponentsrange in the set{1, 2}, and the summation convention be systematically used in conjunc-tion with these rules. For instance, the relation (see Theorem 1.6.1)

(iai) = (| b3)a + (3|+b)a3

means that

3

i=1

iai

=

2=1

(| b3)a +

3|+

2=1

ba3 for = 1, 2.

Kroneckers symbolsare designated by , , or according to the context.

Let there be given a three-dimensional Euclidean space E3, equipped with an or-thonormal basis consisting of three vectors

ei =

ei, and let a b, |a|, and a b denote

the Euclidean inner product, the Euclidean norm, and the vector product of vectors a, bin the space E3.

y1 y

y2

R2

(y)

a1(y)a1(y)

a2(y)

=()

E3

a2(y)

Figure 1.1.1.Curvilinear coordinates on a surface and covariant and contravariant bases

of the tangent plane. Let = () be a surface in E3. The two coordinates y1, y2 ofy are the curvilinear coordinates ofy = (y). If the two vectorsa(y) = (y)are linearly independent, they are tangent to the coordinate lines passing throughy andthey form the covariant basis of the tangent plane to

at

y = (y). The two vectors

a(y) from this tangent plane defined by a(y)

a(y) = form its contravariant basis.

3


4/60

In addition, let there be given a two-dimensional vector space, in which two vectorse =eform a basis. This space will be identified withR

2. Letydenote the coordinatesof a point y R2 and let := /y and :=2/yy.

Finally, let there be given an open subset of R2 and a smooth enough mapping : E3 (specific smoothness assumptions on will be made later, according to eachcontext). The set := ()is called a surface in E3.

If the mapping : E3 is injective, each pointy can be unambiguouslywritten as y= (y), y,and the two coordinates y ofy are called the curvilinear coordinates ofy (Figure1.1.1). Well-knownexamplesof surfaces and of curvilinear coordinates and their corre-sponding coordinate lines (defined in Section 1.2) are given in Figures 1.1.2 and 1.1.3.

Naturally, once a surface is defined as = (), there are infinitely many otherways of defining curvilinear coordinates on, depending on how the domain andthe mapping are chosen. For instance, a portion

of a sphere may be represented

by means of Cartesian coordinates, spherical coordinates, or stereographic coordinates

(Figure 1.1.3). Incidentally, this example illustrates the variety of restrictions that haveto be imposed on according to which kind of curvilinear coordinates it is equippedwith!

1.2 First fundamental form

Let be an open subset ofR2 and let

= iei : R2 () =E3be a mapping that is differentiable at a point y . If y is such that (y+ y) ,then

(y+ y) = (y) + (y)y +o(y),

where the 3

2 matrix (y) and the column vector y are defined by

(y) :=

11 2112 2213 23

(y) and y = y1y2

.

Let the two vectors a(y)R3 be defined by

a(y) := (y) =

123

(y),i.e., a(y)is the-thcolumn vector of the matrix(y). Then the expansion of abouty may be also written as

(y+ y) = (y) +y

a(y) +o(y).

4


5/60

x

y

x

y

u

v

u

v

Figure 1.1.2. Several systems of curvilinear coordinates on a sphere. Let E3 bea sphere of radius R. A portion of contained in the northern hemisphere can berepresented by means of Cartesian coordinates, with a mapping of the form:

: (x, y)(x,y, {R2 (x2 +y2)}1/2)E3.A portion of that excludes a neighborhood of both poles and of a meridian (to

fix ideas) can be represented by means of spherical coordinates, with a mapping of theform:

: (, )(R cos cos , R cos sin , R sin )E3.A portion of that excludes a neighborhood of the North pole can be represented

by means of stereographic coordinates, with a mapping of the form:: (u, v)

2R2uu2 +v2 +R2

, 2R2v

u2 +v2 +R2, R

u2 +v2 R2u2 +v2 +R2

E3.

The corresponding coordinate lines are represented in each case, with self-explanatorygraphical conventions.

5


6/60

z

z

Figure 1.1.3. Two familiar examples of surfaces and curvilinear coordinates. A portion of a circular cylinder of radius R can be represented by a mapping of the form: (, z)(R cos , R sin , z)E3.

A portion of a torus can be represented by a mapping of the form: (, )((R+r cos )cos , (R+r cos )sin , r sin )E3,

with R > r.The corresponding coordinate lines are represented in each case, with self-explanatory

graphical conventions.

6


7/60

If in particular y is of the form y = te, where t Rand e is one of the basisvectors in R2, this relation reduces to

(y+te) = (y) +ta(y) +o(t).

A mapping : E3 is an immersion at y if it is differentiable at y and the3

2 matrix (y) is of rank two, or equivalently if the two vectors a(y) = (y) are

linearly independent.Assume from now on in this section thatthe mapping is an immersion aty . In this

case, the last relation shows that each vectora(y) is tangent to the -th coordinateline passing throughy = (y), defined as the image by of the points of that lie ona line parallel to e passing through y (there exist t0 andt1 with t0 < 0 < t1 such thatthe-thcoordinate line is given by t]t0, t1[f(t) :=(y+te) in a neighborhoodofy; hence f(0) =(y) = a(y)); see Figures 1.1.1, 1.1.2, and 1.1.3.

The vectors a(y), which thus span the tangent planeto the surface aty = (y),form the covariant basis of the tangent plane to aty; see Figure 1.1.1.

Returning to a general increment y = ye, we also infer from the expansion of about y that (yT and (y)T respectively designate the transpose of the columnvector y and the transpose of the matrix (y))

|(y+ y) (y)|2 =yT(y)T(y)y +o(|y |2)=ya(y) a(y)y +o(|y |2).

In other words, the principal part with respect to y of the length between the points(y+ y) and (y) is{ya(y) a(y)y}1/2. This observation suggests to define amatrix (a(y)) of order two by letting

a(y) :=a(y) a(y) =(y)T(y)

.

The elementsa(y) of this symmetric matrix are called the covariant componentsof the first fundamental form, also called the metric tensor, of the surface at

y = (y).Note thatthe matrix(a(y)) is positive definitesince the vectors a(y) are assumed

to be linearly independent.The two vectors a(y) being thus defined, the four relations

a(y) a(y) = unambiguously define two linearly independent vectors a(y) in the tangent plane. Tosee this, let a prioria(y) = Y(y)a(y) in the relationsa(y) a(y) = . This givesY(y)a(y) = ; hence Y

(y) =a(y), where

(a(y)) := (a(y))1.

Hence a(y) = a(y)a(y). These relations in turn imply that

a(y) a(y) =a(y)a(y)a(y) a(y)

=a

(y)a

(y)a(y) = a

(y)

= a

(y),

7


8/60

and thus the vectors a(y) are linearly independent since the matrix (a(y)) is positivedefinite. We would likewise establish that a(y) = a(y)a

(y).The two vectors a(y) form the contravariant basis of the tangent plane to the

surface aty = (y) (Figure 1.1.1) and the elements a(y) of the symmetric matrix(a(y)) are called the contravariant components of the first fundamental form,or metric tensor, of the surface aty= (y).Let us record for convenience the fundamental relations that exist between the vectorsof the covariant and contravariant bases of the tangent plane and the covariant andcontravariant components of the first fundamental tensor:

a(y) = a(y) a(y) and a(y) = a(y) a(y),a(y) = a(y)a

(y) and a(y) =a(y)a(y).

A mapping : E3 is animmersionif it is an immersion at each point in , i.e.,if is differentiable in and the two vectors (y) are linearly independent at eachy.

If : E3 is an immersion, the vector fields a : R3 and a : R3respectively form the covariant, and contravariant, bases of the tangent planes.

1.3 Areas and lengths on a surface

We now review fundamental formulas expressing areaand length elementsat a pointy = (y) of the surface= () in terms of the matrix (a(y)); see Figure 1.3.1.These formulas highlight in particular the crucial role played by the matrix (a(y))

for computing metric notions aty= (y).Theorem 1.3.1. Let be an open subset of R2, let : E3 be an injective andsmooth enough immersion, and let= ().

(a) The area elementda(y) aty= (y) is given in terms of the area elementdyaty by

d

a(

y) =

a(y) dy, where a(y) := det(a(y)).

(b) The length elementd(y) aty= (y) is given byd(y) = ya(y)y1/2 .

Proof. The relation (a) between the area elements is well known.

The expression of the length element in (b) recalls that d(y) is by definition theprincipal part with respect to y = ye of the length|(y + y)(y)|, whoseexpression precisely led to the introduction of the matrix (a(y)).

The relations found in Theorem 1.3.1 are used for computing surface integrals andlengthson the surface

by means of integrals inside , i.e., in terms of the curvilinear

coordinates used for defining the surface (see again Figure 1.3.1). In what follows, : E3 is again an injective and smooth enough immersion.

8


9/60

(y+y)

y+y

A

dy

dl(y)

(y

) = y

CE

3

I f

t

R2

R

y

C

da(y)

A

Figure 1.3.1. Area and length elements on a surface. The elements da(y) and d(y)aty = (y) are related to dy and y by means of the covariant components ofthe metric tensor of the surface; cf. Theorem 1.3.1. The corresponding relations areused for computing the area of a surfaceA = (A) and the length of a curveC= (C)

, where C=f(I) andI is a compact interval ofR.

Let A be a bounded, open, and connected subset ofR2 with a Lipschitz-continuousboundary, that satisfies A. LetA:= (A), and letfL1(A) be given. Then

bA

f(y) da(y) = A

(f )(y)a(y) dy.In particular, the area ofA is given by

area

A:=

bA

d

a(

y) =

A

a(y) dy.

9


10/60

Consider next a curve C = f(I) in , where I is a compact interval ofR and f =fe : I is a smooth enough injective mapping. Then the length of the curveC:= (C) is given by

lengthC:= I d

dt

(

f)(t) dt= Ia(f(t))

df

dt

(t)df

dt

(t) dt.

The last relation shows in particular that the lengths of curves inside the surface ()are precisely those induced by the Euclidean metric of the space E3. For this reason, thesurface () is said to be isometrically imbedded in E3.

1.4 Second fundamental form; curvature on a surface

Consider a surface given as the image () E3 of a two-dimensional open set R2 by a smooth enough immersion : R2 E3. Then it is intuitively clearthat this surfacecannotbe defined by its metric alone, i.e., by its sole first fundamentalform.

As suggested by Figure 1.4.1, the missing information is provided by the curvatureof a surface. A natural way to give substance to this otherwise vague notion consists in

specifying how the curvature of a curve on a surfacecan be computed. As shown in thissection, solving this question relies on the knowledge of the second fundamental formof a surface, which naturally appears for this purpose through its covariant components(Theorem 1.4.3).

Consider as in Section 1.1 asurface= () in E3, where is an open subset ofR2and : R2 E3 is a smooth enough immersion. For each y, the vector

a3(y) := a1(y) a2(y)|a1(y) a2(y)|

is thus well defined, has Euclidean norm one, and is normal to the surface at the point

y = (y).

Remark 1.4.1. The denominator in the definition ofa3(y) may be also written as

|a1(y) a2(y)|=

a(y),

where a(y) := det(a(y)). This relation, which holds in fact even ifa(y) = 0, will beestablished in the course of the proof of Theorem 2.3.2.

Fixy and consider a planeP normal to aty= (y), i.e., a plane that containsthe vector a3(y). The intersectionC= P is thus a planar curve on.

As shown in Theorem 1.4.3, it is remarkable that the curvature ofC aty can becomputed by means of the covariant components a(y) of the first fundamental form ofthe surface

= () introduced in Section 1.2, together with the covariant components

b(y) of the second fundamental form of. The definition of the curvature of aplanar curveis recalled in Figure 1.4.2.

10


11/60

0

1

2

Figure 1.4.1. A metric alone does not define a surface inE3. A flat surface0 maybe deformed into a portion1 of a cylinder or a portion2 of a cone without alteringthe length of any curve drawn on it (cylinders and cones are instances of developablesurfaces, i.e., that can be, at least locally, continuously rolled out, or developed,onto a plane, without changing the metric of the intermediary surfaces in the process;see Stoker [1969, Chapter 5, Sections 2 to 6] or Klingenberg [1973, Section 3.7]). Yet itshould be clear that in general0 and1, or0 and2, or1 and2, are not identicalsurfacesmodulo an isometry ofE3, i.e., up to a translation and a rotation in E3.

11


12/60

(s)

p(s) + R(s)

(s)

(s+ s)

p(s)

p(s+ s)

(s+ s)

(s)

Figure 1.4.2. Curvature of a planar curve. Let be a smooth enough planar curve,parametrized by its curvilinear abscissa s. Consider two points p(s) andp(s + s) withcurvilinear abscissaes and s + sand let (s) be the algebraic angle between the twonormals (s) and (s+ s) (oriented in the usual way) to at those points. When

s0, the ratio (s)

s has a limit, called the curvature ofat p(s). If this limit isnon-zero, its inverseR is called the algebraic radius of curvature ofat p(s) (the signofR depends on the orientation chosen on ).

The point p(s) +R(s), which is intrinsically defined, is called the center of curva-ture of at p(s): It is the center of the osculating circle at p(s), i.e., the limit ass0 of the circle tangent to at p(s) that passes through the point p(s+ s). Thecenter of curvature is also the limit as s 0 of the intersection of the normals (s)and(s + s). Consequently, the centers of curvature oflie on a curve (dashed on thefigure), called la developpee in French, that is tangent to the normals to .

12


13/60

If the algebraic curvature ofCatyis= 0, it can be written as 1R

, andR is then called

thealgebraic radius of curvature of the curveCaty. This means that the center ofcurvatureof the curveCatyis the point (y + Ra3(y)); see Figure 1.4.3. WhileR is notintrinsically defined, as its sign changes in any system of curvilinear coordinates wherethe normal vectora3(y) is replaced by its opposite, the center of curvature is intrinsically

defined.If the curvature ofC aty is 0, the radius of curvature of the curveC aty is said to

be infinite; for this reason, it is customary to still write the curvature as 1

Rin this case.

Note that the real number 1

R is always well defined by the formula given in the

next theorem, since the symmetric matrix (a(y)) is positive definite. This implies inparticular that the radius of curvature never vanishes along a curve on a surface()defined by a mapping satisfying the assumptions of the next theorem, hence in particular

of classC2 on.

Remark 1.4.2. It is intuitively clear that if R = 0, the mapping cannot be toosmooth. Think of a surface made of two portions of planes intersecting along a segment,

which thus constitutes a foldon the surface. Or think of a surface () with 0 and(y1, y2) =|y1|1+ for some 0 < < 1 and (y1, y2) in a neighborhood of 0, so that C1(;E3) but / C2(;E3): The radius of curvature of a curve on such a surfacecorresponding to a constant y2 vanishes at y1= 0.

Theorem 1.4.3. Let be an open subset of R2, let C2(;E3) be an injectiveimmersion, and lety be fixed.

Consider a planeP normal to= () at the pointy= (y). The intersectionPis a curveC on, which is the imageC= (C) of a curveCin the set. Assume that,in a sufficiently small neighborhood ofy , the restriction ofC to this neighborhood is theimage f(I) of an open interval I R, where f = fe : I R is a smooth enoughinjective mapping that satisfies

df

dt

(t)e=0, wheretI is such thaty= f(t) (Figure1.4.3).

Then the curvature 1

Rof the planar curveC aty is given by the ratio

1

R =

b(f(t))df

dt

(t)df

dt

(t)

a(f(t))df

dt

(t)df

dt

(t)

,

wherea(y) are the covariant components of the first fundamental form of aty (Sec-tion 1.1) and

b(y) := a3(y) a(y) =a3(y) a(y) = b(y).

13


14/60

y=(y)

C

P

y+Ra3(y)

a3(y)

E3

If=f e

t

e2

e1 R2

R

df

dt(t)e

y= f

(t)e

C

Figure 1.4.3. Curvature on a surface. Let P be a plane containing the vector

a3(y) =

a1(y)

a2(y)

|a1(y) a2(y)| , which is normal to the surface = (). The algebraiccurvature

1

Rof the planar curveC= P= (C) aty= (y) is given by the ratio

1

R=

b(f(t))df

dt

(t)df

dt

(t)

a(f(t))df

dt

(t)df

dt

(t)

,

where a(y) and b(y) are the covariant components of the first and second

fundamental forms of the surface aty and dfdt

(t) are the components of the vector

tangent to the curveC= f(I) aty = f(t) = f(t)e. If 1

R= 0, the center of curvature

of the curveC aty is the point (y + Ra3(y)), which is intrinsically defined in theEuclidean space E3.

14


15/60

Proof. (i) We first establish a well-known formula giving the curvature 1

R of a planar

curve. Using the notations of Figure 1.4.2, we note that

sin(s) =(s) (s+ s) ={(s+ s) (s)} (s+ s),so that

1

R := lims0

(s)

s = lims0

sin(s)

s =d

ds (s) (s).

(ii) The curve ( f)(I), which is a priori parametrized by t I, can be alsoparametrized by its curvilinear abscissa s in a neighborhood of the pointy. There thusexist an interval IIand a mapping p : JP, whereJ Ris an interval, such that

( f)(t) =p(s) and (a3f)(t) =(s) for all tI , sJ.

By (i), the curvature 1

R ofCis given by

1

R=d

ds(s) (s),

where

d

ds(s) =

d(a3f)dt

(t)dt

ds =a3(f(t))

df

dt

(t)dt

ds,

(s) =dp

ds(s) =

d( f)dt

(t)dt

ds

=(f(t))df

dt

(t)dt

ds=a(f(t))

df

dt

(t)dt

ds.

Hence1

R =a3(f(t)) a(f(t)) df

dt

(t)df

dt

(t) dt

ds

2.

To obtain the announced expression for 1

R, it suffices to note that

a3(f(t)) a(f(t)) = b(f(t)),by definition of the functions b and that (Theorem 1.3.1 (b))

ds=

ya(y)y1/2

=

a(f(t))df

dt

(t)df

dt

(t)1/2

dt.

Remark 1.4.4. The knowledge of the curvatures of curves contained in planes normaltosuffices for computing the curvature ofanycurve on. More specifically, the radiusof curvatureR aty of any smooth enough curveC (planar or not) on the surface isgiven by

cos

R

= 1

R, where is the angle between the principal normal to

Cat

y and

a3(y) and

1

R is given in Theorem 1.4.3; see, e.g., Stoker [1969, Chapter 4, Section 12].

15


16/60

The elements b(y) of the symmetric matrix (b(y)) defined in Theorem 1.4.3 arecalled the covariant components of the second fundamental form of the surface= () aty = (y).1.5 Principal curvature; Gaussian curvature

The analysis of the previous section suggests that precise information about the shapeof a surface= () in a neighborhood of one of its pointsy= (y) can be gathered byletting the planePturn around the normal vector a3(y) and by following in this processthe variations of the curvatures aty of the corresponding planar curves P, as givenin Theorem 1.4.3.

As a first step in this direction, we show that these curvatures span a compact intervalofR. In particular then, they stay away from infinity.

Note that this compact interval contains 0 if, and only if, the radius of curvature ofthe curve P is infinite for at least one such plane P.Theorem 1.5.1. (a) Let the assumptions and notations be as in Theorem 1.4.3. Fora fixed y , consider the setP of all planes P normal to the surface = () aty = (y). Then the set of curvatures of the associated planar curvesP, P P, is acompact interval ofR, denoted

1R1(y)

, 1

R2(y)

.

(b) Let the matrix(b(y)), being the row index, be defined by

b(y) := a (y)b(y),

where (a(y) ) = (a(y))1 (Section 1.2) and the matrix (b(y)) is defined as inTheorem 1.4.3. Then

1

R1(y)+

1

R2(y)=b11(y) +b

22(y),

1R1(y)R2(y)

=b11(y)b22(y) b21(y)b12(y) = det(b(y))det(a(y)) .

(c)If 1

R1(y)= 1

R2(y), there is a unique pair of orthogonal planesP1 P andP2 P

such that the curvatures of the associated planar curvesP1 andP2 are precisely1

R1(y) and

1

R2(y).

Proof. (i) Let (P) denote the intersection ofP Pwith the tangent plane T to thesurfaceaty, and letC(P) denote the intersection ofP with. Hence (P) is tangenttoC(P) aty.

16


17/60

In a sufficiently small neighborhood ofy the restriction of the curveC(P) to thisneighborhood is given byC(P) = ( f(P))(I(P)), whereI(P)Ris an open intervaland f(P) = f(P)e : I(P) R2 is a smooth enough injective mapping that satisfiesdf(P)

dt (t)e= 0, where t I(P) is such that y = f(P)(t). Hence the line (P) is

given by

(P) =y+ d( f(P))

dt (t); R

={y+a(y); R} ,

where :=df(P)

dt (t) ande=0 by assumption.

Since the line{y+ e; R} is tangent to the curve C(P) := 1(C(P)) aty (the mapping : R3 is injective by assumption) for each such parametrizingfunction f(P) : I(P) R2 and since the vectors a(y) are linearly independent, thereexists a bijection between the set of all lines (P)T, P P, and the set of all linessupporting the nonzero tangent vectors to the curve C(P).

Hence Theorem 1.4.3 shows that when P varies inP, the curvature of the corre-sponding curvesC =C(P) aty takes the same values as does the ratio

b(y)

a(y)

when :=1

2

varies in R2 {0}.

(ii) Let the symmetric matricesA and B of order two be defined by

A:= (a(y)) and B:= (b(y)).

Since A is positive definite, it has a (unique) square root C, i.e., a symmetric positivedefinite matrix Csuch that A = C2. Hence the ratio

b(y)

a(y) =TB

TA=TC1BC1

T , where = C,

is nothing but the Rayleigh quotientassociated with the symmetric matrix C1BC1.When varies in R2 {0}, this Rayleigh quotient thus spans the compact interval ofRwhose end-points are the smallest and largest eigenvalue, respectively denoted

1

R1(y)and

1

R2(y), of the matrix C1BC1 (for a proof, see, e.g., Ciarlet [1982, Theorem 1.3-1]).

This proves (a).Furthermore, the relation

C(C1BC1)C1 =BC2 =BA1

shows that the eigenvalues of the symmetric matrix C1BC1 coincide with those ofthe (in general non-symmetric) matrix BA1. Note that BA1 = (b(y)) withb

(y) :=

a(y)b(y), being the row index, since A1

= (a(y)).

17


18/60

Hence the relations in (b) simply express that the sum and the product of the eigen-values of the matrix BA1 are respectively equal to its trace and to its determinant,

which may be also written as det(b(y))

det(a(y)) since BA1 = (b(y)). This proves (b).

(iii) Let 1 = 11

21 = C1 and 2 = 12

22 = C2, with 1 = 11

21 and 2 =1222

, be two orthogonal (T1 2 = 0) eigenvectors of the symmetric matrix C

1BC1,

corresponding to the eigenvalues 1

R1(y) and

1

R2(y), respectively. Hence

0 =T1 2 = T1C

TC2 = T1A2 = 0,

since CT = C. By (i), the corresponding lines (P1) and (P2) of the tangent plane

are parallel to the vectors 1 a(y) and 2a(y), which are orthogonal since

1 a(y) 2a(y)= a(y)12 =T1A2.If

1

R1(y)= 1

R2(y), the directions of the vectors 1 and 2 are uniquely determined

and the lines (P1) and (P2) are likewise uniquely determined. This proves (c).

We are now in a position to state several fundamental definitions:

The elements b(y) of the (in general non-symmetric) matrix (b(y)) defined in The-

orem 1.5.1 are called the mixed components of the second fundamental form ofthe surface= () aty= (y).

The real numbers 1

R1(y) and

1

R2(y)(one or both possibly equal to 0) found in The-

orem 1.5.1 are called the principal curvatures of aty.If 1R1(y)

= 0 and 1R2(y)

= 0, the real numbers R1(y) and R2(y) are called the

principal radii of curvature of aty. If, e.g., 1R1(y)

= 0, the corresponding radius

of curvatureR1(y) is said to be infinite, according to the convention made in Section 1.4.While the principal radii of curvature may simultaneously change their signs in anothersystem of curvilinear coordinates, the associated centers of curvature are intrinsicallydefined.

The numbers 1

2

1R1(y)

+ 1

R2(y)

and

1

R1(y)R2(y), which are the principal invariants

of the matrix (b(y)) (Theorem 1.5.1), are respectively called the mean curvature andthe Gaussian, or total, curvature of the surface

at

y.

A point on a surface is an elliptic, parabolic, or hyperbolic, point according as

its Gaussian curvature is > 0, = 0 but it is not a planar point, or < 0; see Figure 1.5.1.

18


19/60

Figure 1.5.1. Different kinds of points on a surface. A point is elliptic if the Gaussiancurvature is > 0 or equivalently, if the two principal radii of curvature are of the samesign; the surface is then locally on one side of its tangent plane. A point is parabolic ifexactly one of the two principal radii of curvature is infinite; the surface is again locallyon one side of its tangent plane. A point is hyperbolic if the Gaussian curvature is


20/60


21/60

y

y2

y1

R2

y

3(y)

i(y)ai(y)

a3(y)a2(y)

2(y) a1(y)

1(y)=()

E3

Figure 1.6.1. Contravariant bases and vector fields along a surface. At each pointy = (y)= (), the three vectors a i(y), where a(y) form the contravariant basisof the tangent plane to aty (Figure 1.1.1) and a3(y) = a1(y) a2(y)|a1(y) a2(y)| , form thecontravariant basis aty. An arbitrary vector field defined on may then be defined byits covariant components i : R. This means that i(y)ai(y) is the vector at thepointy.

21


22/60

Proof. Since any vector c in the tangent plane can be expanded as c = (ca)a =(c a)a, since a3 is in the tangent plane (a3 a3 = 12(a3 a3) = 0), and sincea

3 a =b (Theorem 1.4.3), it follows that

a3 = (a

3 a)a =ba.

This formula, together with the definition of the functionsb(Theorem 1.5.1), impliesin turn that

a3 = (a3a)a =b(a a)a =baa =ba.

Any vectorc can be expanded as c = (c ai)ai = (c aj)aj . In particular,

a = (aa)a+ (a a3)a3 = a+ba3,

by definition of andb. Finally,

a = (a

a)a + (a a3)a3 =a +ba3,

sincea

a3 =a a3 = baa =b.Thatia

i C1(;R3) ifi C1() is clear sinceai C1(;R3) if C2(;E3). Theformulas establishedsupraimmediately lead to the announced expression of(iai).

The relations (found in Theorem 1.6.1)

a = a+ba3 anda

=a +ba3

and

a3 = a3 =ba =ba,

respectively constitute the formulas of Gau and Weingarten. The functions (alsofound in Theorem 1.6.1)

|= and3| = 3

are the first-order covariant derivatives of the vector field iai : R3, and the

functions

:=a a =a a

are the Christoffel symbols of the second kind (the Christoffel symbols of the firstkind are introduced in the next section).

Remark 1.6.2. The Christoffel symbols can be also defined solely in terms of the

covariant components of the first fundamental form; see the proof of Theorem 1.7.1.

22


23/60

1.7 The Gau and Codazzi-Mainardi equations

It is remarkable that the componentsa =a : R and b =b : R ofthe first and second fundamental forms of a surface() cannot be arbitrary functions.

As shown in the next theorem, they must satisfy relations that take the form:

+

= bb

bb in ,

b b+ b b = 0 in ,

where the functions and have simple expressions in terms of the functions

a and of some of their partial derivatives (as shown in the next proof, it so happensthat the functions as defined in Theorem 1.7.1 coincide with the Christoffel symbolsintroduced in the previous section; this explains why they are denoted by the samesymbol).

These relations, which are meant to hold for all ,,, {1, 2}, respectively con-stitute the Gau, and Codazzi-Mainardi, equations.

Theorem 1.7.1. Let be an open subset of R2, let C3(;E3) be an immersion,and let

a := and b := 1 2|1 2|denote the covariant components of the first and second fundamental forms of the surface

(). Let the functions C1() and C1() be defined by

:= 1

2(a+a a),

:=awhere (a

) := (a)1.

Then, necessarily,

+ = bb bb in ,b

b+

b

b = 0 in .

Proof. Let a = . It is then immediately verified that the functions are alsogiven by

=aa.

Let a3 = a1a2|a1a2| and, for each y , let the three vectors a

j(y) be defined by

the relations aj(y) ai(y) = ji . Since we also havea = aa and a3 = a3, the lastrelations imply that the functions are also given by

=a a.

Consequently,

a = a+ba3,

23


24/60

since a = (aa)a+ (aa3)a3. Differentiating the same relations yields

=a a+ a a,

so that the above relations together give

a a=

a a+ba3 a =

+bb.

Consequently,aa = bb.

Sincea = a, we also have

aa = bb.

Hence the Gau equations immediately follow.Sincea3 = (a3 a)a + (a3 a3)a3 anda3 a =b=aa3, we

havea3 =ba.

Differentiating the relationsb=aa3, we obtainb= aa3+a a3.

This relation and the relations a = a +ba3 and a3 =ba togetherimply that

a a3 =b.Consequently,

aa3 = b+ b.Sincea = a, we also have

aa3 = b+ b.

Hence the Codazzi-Mainardi equations immediately follow.

Remark 1.7.2. The vectors aand a introduced above respectively form the covariant

and contravariant bases of the tangent plane to the surface (), the unit vectora3 = a3

is normal to the surface, and the functions a are the contravariant components of thefirst fundamental form (Sections 1.2 and 1.3).

As shown in the above proof, the Gau and Codazzi-Mainardi equations thus simplyconstitute a re-writing of the relations a = a in the form of the equivalentrelations a a=aa andaa3 = aa3.

The functions

=

1

2 (a+a a) = a a=

24


25/60

and=a

=aa = are the Christoffel symbols of the first, and second, kind. We recall that theChristoffel symbols of the second kind also naturally appeared in a different context(that of covariant differentiation; cf. Section 1.6).

Finally, the functions

S:= + are the covariant components of the Riemann curvature tensor of the surface().

The definitions of the functions and imply that the sixteen Gau equationsare satisfied if and only if they are satisfied for = 1, = 2, = 1, = 2 and that the Codazzi-Mainardi equations are satisfied if and only if theyare satisfied for = 1, = 2, = 1 and = 1, = 2, = 2 (other choices of indiceswith the same properties are clearly possible).

In other words, the Gau equations and the Codazzi-Mainardi equations in fact re-spectively reduce to oneand two equations.

Letting = 2, = 1, = 2, = 1 in the Gau equations gives in particular

S1212 = det(b).

Consequently, the Gaussian curvature at each point (y) of the surface () can bewritten as

1

R1(y)R2(y)=

S1212(y)

det(a(y)), y,

since 1

R1(y)R2(y) =

det(b(y)

det(a(y))(Theorem 1.5.1). By inspection of the functionS1212,

we thus reach the astonishing conclusion that, at each point of the surface, a notioninvolving the curvature of the surface, viz., the Gaussian curvature, is entirely de-

termined by the knowledge of the metric of the surface at the same point, viz., thecomponents of the first fundamental forms and their partial derivatives of order 2 atthe same point! This startling conclusion naturally deserves a theorem:Theorem 1.7.3. Let be an open subset ofR2, let C3(;E3) be an immersion, leta = denote the covariant components of the first fundamental form of thesurface(), and let the functions andS1212 be defined by

:=1

2(a+a a),

S1212 :=1

2(212a12 11a22 22a11) +a(1212 1122).

Then, at each point(y) of the surface(), the principal curvatures 1R1(y) and 1R2(y)

satisfy1

R1(y)R2(y)

= S1212(y)

det(a(y))

, y

.

25


26/60

Theorem 1.7.3 constitutes the famedTheorema Egregiumof Gau [1828], so namedby Gau who had been himself astounded by his discovery.

1.8 The fundamental theorem of surface theory

Let M2, S2, and S2>denote the sets of all square matrices of order two, of all symmetric

matrices of order two, and of all symmetric, positive definite matrices of order two.So far, we have considered that we are given an open set R2 and a smoothenough immersion : E3, thus allowing us to define the fields (a) : S2> and(b) : S2, where a : R and b : R are the covariant components ofthe first and second fundamental formsof the surface ()E3.Remark 1.8.1. The immersion need not be injective in order that these matrix fieldsbe well defined.

We now turn to the reciprocal questions:Given an open subset ofR2 and two smooth enough matrix fields (a) : S2>

and (b) : S2, when are they the first and second fundamental forms of a surface()E3, i.e., when does there exist an immersion : E3 such that

a=

andb=

1 2|1 2| in ?

If such an immersion exists, to what extent is it unique?The answers to these questions turn out to be remarkably simple: If is simply-

connected, the necessary conditions ofTheorem 1.7.1, i.e.,the Gau and Codazzi-Mainardiequations, are also sufficient for the existence of such an immersion. If is connected,this immersion is unique up to isometries inE3.

Remark 1.8.2. Whether an immersion found in this fashion is injective is a differentissue, which accordingly should be resolved by different means.

These existence and uniqueness results together constitute the fundamental theo-rem of surface theory. This theorem comprises two essentially distinct parts, a globalexistence result(Theorem 1.8.3) and auniqueness result(Theorem 1.8.4), the latter beingalso called rigidity theorem. Note that these two results are established under different

assumptionson the set and on the smoothness of the fields (a) and (b). We beginwith the issue of existence.

A direct proof of the fundamental theorem of surface theory is given in Klingenberg[1973, Theorem 3.8.8]. See also Ciarlet & Larsonneur [2001] or Ciarlet [2005] for aself-contained, and essentially elementary, proof. A proof of the local version of thistheorem, which constitutesBonnets theorem, is found in, e.g., do Carmo [1976] or Kuhnel[2002].

Theorem 1.8.3. Let be a connected and simply-connected open subset ofR2 and let(a) C2(;S2>) and (b) C1(; S2) be two matrix fields that satisfy the Gau andCodazzi-Mainardi equations, viz.,

+ = bb bb in ,

b b+

b

b = 0 in ,

26


27/60

where

:= 1

2(a+a a),

:=awhere (a

) := (a)1.

Then there exists an immersion

C3(;E3) such that

a= and b= 1 2

|1 2|

in .

The regularity assumptionsmade in Theorem 1.8.3 on the matrix fields (a) and(b) can be significantly relaxed in several ways.

In this direction, Hartman & Wintner [1950] first showed that the existence theoremstill holds if (a) C1(; S2>) and (b) C0(; S2), with a resulting mapping in thespaceC2(;E3). Then S. Mardare [2003b] established that if (a)W1,loc (; S2>) and(b) Lloc(; S2) are two matrix fields that satisfy the Gau and Codazzi-Mainardiequations in the sense of distributions, there exists a mapping W2,loc (;E3) suchthat (a) and (b) are the fundamental forms of the surface (). The last word seems

to belong to S. Mardare [2005], who was able to further reduce these regularities, to thoseof the spaces W1,ploc(; S

2>) for the field (a) and L

ploc(;S

2) for the field (b) for any

p >2, with a resulting mapping in the space W2,ploc(;E3).

Theorem 1.8.3 establishes the existenceof an immersion : R2 E3 giving riseto a surface () with prescribed first and second fundamental forms, provided theseforms satisfy ad hocsufficient conditions. We now turn to the question ofuniquenessofsuch immersions.

This is the object of the next theorem, which is called the rigidity theorem forsurfaces. This result asserts that, if two immersions C2(;E3) and C2(;E3)share the same fundamental forms, then the surface () is obtained by subjecting the

surface

() to arotation(represented by an orthogonal matrixQ with detQ= 1), then

by subjecting the rotated surface to a translation (represented by a vector c). Note inpassing that the converse clearly holds.

Such a geometric transformation of the surface() is sometimes called a rigidtransformation, to remind that it corresponds to the idea of a rigid one in E3. Thisobservation motivates the terminology rigidity theorem.

Theorem 1.8.4. Let be a connected open subset of R2 and let C2(;E3) and C2(;E3) be two immersions such that their associated first and second fundamentalforms satisfy (with self-explanatory notations)

a = a andb = b in .Then there exist a vectorcE3 and a matrixQO3+ such that

(y) =c + Q

(y) for all y.

27


28/60

More details about the various notions of classical Differential Geometry considered inChapter 1 are found in classic texts such as Stoker [1969], Klingenberg [1973], do Carmo[1976], Berger & Gostiaux [1987], Spivak [1999], or Kuhnel [2002].

28


29/60

2 An introduction to shell theory

2.1 What is a two-dimensional shell problem?

A domain in R2 is an open, bounded, connected subset ofR2, whose boundary isLipschitz-continuous, the set being locally on one side of its boundary. To begin with,we briefly recapitulate some important notions introduced and studied in Chapter 1.

Note in this respect that we shall extend without further noticeall the definitions given,or properties studied, on arbitrary open subsets ofR2 in Chapter 1 to their analogs ondomains inR2.

Greek indices and exponents (exceptin the notation) range in the set {1, 2}, Latinindices and exponents range in the set{1, 2, 3}(save when they are used for indexing se-quences), and the summation convention with respect to repeated indices and exponentsis systematically used. The notation E3 denotes a three-dimensional Euclidean space,the Euclidean scalar product and the exterior product ofa, bE3 are noted a b anda b, and the Euclidean norm ofaE3 is noted|a|.

Let be a domain in R2. Let y = (y) denote a generic point in the set , and let := /y. Let there be given an immersion C3(;E3), i.e., a mapping such thatthe two vectors

a(y) := (y)

are linearly independent at all points y . These two vectors thus span the tangentplane to the surface

S:= ()

at the point (y), and the unit vector

a3(y) := a1(y) a2(y)|a1(y) a2(y)|

is normal to Sat the point (y). The three vectors a i(y) constitute the covariant basisat the point (y), while the three vectors ai(y) defined by the relations

ai(y) aj(y) = ij,

whereij is the Kronecker symbol, constitute thecontravariant basisat the point (y)S(recall that a3(y) = a3(y) and that the vectors a(y) are also in the tangent plane toSat (y)).

The covariant and contravariant components a and a of the first fundamentalform ofS, the Christoffel symbols , and the covariant and mixed components bandb of the second fundamental form ofSare then defined by letting:

a :=aa, a :=a a , :=a a,b :=a

3 a, b:= ab.

The areaelement alongS is

a dy, where

a:= det(a).

29


30/60

Note that one also has

a=|a1a2|.Let := ], [, letx = (xi) denote a generic point in the set (hence x = y),

and leti := /xi. Consider anelastic shellwithmiddle surfaceS= () andthickness2 >0, i.e., an elastic body whose reference configurationis the set( [, ]), where(cf. Figure 2.1.1)

(y, x3) :=(y) +x3a3(y) for all (y, x3) [, ] .Naturally, this definition makes sense physically only if the mapping is globally injec-tive on the set . As shown by Ciarlet [2000a, Theorem 3.1-1], this is indeed the case ifthe immersion is itself globally injective on the set and is small enough, accordingto the following result.

x3

x

= ] , [ R3

2

S=(

)

x

= () E3

a3(y)

2

y

Figure 2.1.1. The reference configuration of an elastic shell. Let be a domain inR2, let = ], [ > 0, let C3(;E3) be an immersion, and let the mapping

: E3 be defined by (y, x3) = (y) +x3a3(y) for all (y, x3) . Then themapping is globally injective on if the immersion is globally injective on and >0 is small enough (Theorem 2.1.1). In this case, the set () may be viewed as thereference configuration of an elastic shell with thickness 2and middle surface S= ().The coordinates (y1, y2, x3) of an arbitrary point x are then viewed as curvilinearcoordinates of the point

x= (x) of the reference configuration of the shell.

30


31/60

Theorem 2.1.1. Let be a domain inR2 and let C3(;E3) be an injective immer-sion. Then there exists > 0 such that the mapping: E3 defined by

(y, x3) :=(y) +x3a3(y) for all (y, x3),where := ], [, is aC2-diffeomorphism from onto ()anddet(g1, g2, g3)> 0in, wheregi := i.

In what follows, we assume that > 0 is small enough so that the conclusions ofTheorem 2.1.1 hold. In particular then, the coordinatesy1, y2, x3 of the points x constitute a system of three-dimensional curvilinear coordinates for describing thereference configuration () of the shell.

Then for each x, the three linearly independent vectors g i(x) := i(x) consti-tute the covariant basisat the point (x), and the three (likewise linearly independent)vectorsgj(x) defined by the relationsgj(x) gi(x) = ji constitute thecontravariant basisat the same point.

Let 0 be a measurable subset of the boundary := that satisfies length0 >0.It is assumed that the shell is subjected to a homogeneous boundary condition of placealong the portion (0 [, ]) of its lateral face ( [, ]). This means that thedisplacement field of the shell vanishes on the set (0 [, ]).

The shell is subjected to applied body forcesin its interior() and toapplied surfaceforces on its upper and lower faces (+) and (), where := {}.Such applied forces are given by the contravariant components fi L2() and hi L2(+) of their densities per unit volume and per unit area, respectively (this meansthatfi(x)gi(x)

g(x) dxis the body force applied to the volume

g(x) dxat eachx;

a similar interpretation holds for the surface forces).Finally, the elastic material constituting the shell is assumed to be homogeneous and

isotropicand the reference configuration () of the shell is assumed to be a naturalstate. Hence the material is characterized by two Lame constants and satisfying3+ 2 > 0 and >0.

Such a shell, endowed with its curvilinear coordinates, (y1, y2, x3) , can thus bemodeled by means of theequations of three-dimensional, linear or nonlinear, elasticity incurvilinear coordinates (for a detailed description of these equations, see Ciarlet [2005]).

In such equations, the natural unknowns are the three covariant components ui :R of the displacement fielduigi : R of the points of the reference configuration(), where the vector fields gi denote the contravariant bases; cf. Figure 2.1.2.

In a two-dimensional approach, the above three-dimensional problem is replacedby a, presumably much simpler, two-dimensional problem, this time posed over themiddle surfaceS of the shell. This means that the new unknowns should be now thethree covariant components i : R of the displacement field iai : E3 of thepoints of the middle surfaceS= (); cf. Figure 2.1.3.

During the past decades, considerable progress has been made towards a rigorousjustification of such a replacement. The central idea is that of asymptotic analysis:It consists in showing that, if the data are of ad hoc orders of magnitude, the three-dimensional displacement vector field(once properly scaled) converges in an appropri-ate function space as 0 to a limit vector field that can be entirely computed bysolving a two-dimensional problem.

31


32/60

2

x3

x

y

0 0

S

2

(0)

g2(x)

ui(x)gi(x)

g1(x)

g3(x)

Figure 2.1.2. An elastic shell modeled as a three-dimensional problem. Let = ], [. The set(), where (y, x3) = (y) +x3a3(y) for all x = (y, x3) , is thereference configuration of a shell, with thickness 2and middle surfaceS= () (Figure2.1.1), which is subjected to a boundary condition of place along the portion (0) ofits lateral face (i.e., the displacement vanishes on (0)), where 0 = 0 [, ] and0 = . The shell is subjected to applied body forces in its interior () andto applied surface forces on its upper and lower faces (+) and () where = {}. Under the influence of these forces, a point (x) undergoes a displacementui(x)gi(x), where the three vectorsgi(x) form the contravariant basis at the point (x).The unknowns of the problem are the three covariant components ui : R of thedisplacement fielduig

i :

R3 of the points of(), which thus satisfy the boundary

conditionsui= 0 on 0. The objective consists in findingad hocconditions affording thereplacement of this three-dimensional problem by a two-dimensional problem posedover the middle surface S if is small enough; see Figure 2.1.3.

Note that, for the sake of visual clarity, the thickness is overly exaggerated.

In this direction, see Ciarlet [2000a, Part A] for a thorough overview in the linearcase and the key contributions of Le Dret & Raoult [1996] and Friesecke, James, Mora& Muller [2003] in the nonlinear case.

2.2 The nonlinear Koiter shell equations

We now describe the nonlinear Koiter shell equations, so named after Koiter

[1966], and since then a two-dimensional nonlinear model of choice in computational

32


33/60

y

0

(0)

a3(y)

i(y)ai(y)

S

a2(y)

a1(y)

Figure 2.1.3. An elastic shell modeled as a two-dimensional problem. For > 0 small enoughand data ofad hocorders of magnitude, the three-dimensional shell problem (Figure 2.1.2) is replacedby a two-dimensional shell problem. This means that the new unknowns are the three covariantcomponents i : R of the displacement field ia

i : R3 of the points of the middle surfaceS= (). In this process, the three-dimensional boundary conditions on 0need to be replaced by adhoctwo-dimensional boundary conditions on 0. For instance, the boundary conditions of clampingi = 3 = 0 on 0 (used in Koiters linear equations; cf. Section 2.3) mean that the points of, and thetangent spaces to, the deformed and undeformed middle surfaces coincide along the set (0).

mechanics (its relation to an asymptotic analysis as 0 is briefly discussed at the endof this section).

To begin with, we need to define two important tensor fields. Given an arbitrarydisplacement field ia

i : R3 of the surface S with smooth enough componentsi: R, define the vector field := (i) : R3 and let

a() := a() a(), where a() := (+iai),denote the covariant components of the first fundamental form of the deformed surface(+iai)(). Then the functions

G() :=

1

2 (a() a)

33


34/60

denote the covariant components of the change of metric tensor associated withthe displacement fieldiai ofS.

If the two vectors a() are linearly independent at all points of , let

b() := 1

a()

(+iai) {a1() a2()},

wherea() := det(a()),

denote the covariant components of the second fundamental form of the deformed surface(+ia

i)(). Then the functions

R() := b() bdenote the covariant components of the change of curvature tensor field associ-ated with the displacement field ia

i ofS. Note that

a() =|a1() a2()|.Note thatbothsurfaces() and (+iai)() are equipped with the samecurvilinear

coordinates y1, y2.As a point of departure, consider an elastic shell modeled as a three-dimensional prob-

lem (Section 2.1). The nonlinear two-dimensional equations proposed by Koiter [1966]for modeling an elastic shell are then derived from those of nonlinear three-dimensionalelasticity on the basis of two a priori assumptions: One assumption, of a geometricalnature, is the Kirchhoff-Love assumption. It asserts that any point situated on a normalto the middle surface remains on the normal to the deformed middle surface after thedeformation has taken place and that, in addition, the distance between such a pointand the middle surface remains constant. The other assumption, of a mechanicalnature,asserts that the state of stress inside the shell is planar and parallel to the middle surface(this second assumption is itself based on delicate a prioriestimates due to John [1965,1971]).

Taking these a prioriassumptions into account, W.T. Koiter then reached the con-clusion that the unknown vector field = (i) should be a stationary point, in particulara minimizer, over a set of smooth enough vector fields = (i) : R3 satisfying adhocboundary conditions on0, of the functionalj defined by (cf. Koiter [1966, eqs. (4.2),

(8.1), and (8.3)]):

j() =1

2

aG()G() +

3

3aR()R()

a dy

pii

a dy,

where the functions

a := 4

+ 2aa + 2(aa +aa)

denote the contravariant components of the shell elasticity tensorand the func-tions pi L2() are defined by

pi :=

fi dx3

+hi(, +) +hi(

,

).

34


35/60

The above functional j is called Koiters energy for a nonlinear elastic shell.The stored energy function wKfound in Koiters energy j is thus defined by

wK() =

2aG()G() +

3

6aR()R()

forad hocvector fields . This expression is the sum of the membrane part

wM() = 2

aG()G()

and of the flexural part

wF() = 3

6aR()R().

The long-standing question of how to rigorously identify and justify the nonlineartwo-dimensional equations of elastic shells from three-dimensional elasticity was finallysettled in two key contributions, one by Le Dret & Raoult [1996] and one by Friesecke,James, Mora & Muller [2003], who respectively justified the equations of a nonlinearlyelastic membrane shelland those of a nonlinearly elastic flexural shellby means of -convergence theory(a nonlinearly elastic shell is a membrane shell if there are no nonzeroadmissible displacements of its middle surface S that preserve the metric ofS; it is aflexural shell otherwise).

The stored energy functionwMof a nonlinearly elastic membrane shell is an ad hocquasiconvex envelope, which turns out to be only a function of the covariant componentsa() of the first fundamental form of the unknown deformed middle surface (the notionof quasiconvexity, which plays a central role in the calculus of variations, is due to Morrey[1952]; an excellent introduction to this notion is provided in Dacorogna [1989, Chapter

5]). The function wM reduces to the above membrane part wM in Koiters storedenergy function wKonly for a restricted class of displacement fields iai of the middlesurface. By contrast, the stored energy function of a nonlinearly elastic flexural shell isalways equal to the above flexural partwF in Koiters stored energy function wK.

Remark 2.2.1. Interestingly, a formal asymptotic analysis of the three-dimensionalequations is only capable of delivering the above restricted expression wM(), but

otherwise fails to provide the general expression, i.e., valid for all types of displacements,found by Le Dret & Raoult [1996]. By contrast, the same formal approach yields thecorrect expressionwF(). For details, see Miara [1998], Lods & Miara [1998], and Ciarlet[2000a, Part B].

Another closely related set of nonlinear shell equations of Koiters type has beenproposed by Ciarlet [2000b]. In these equations, the denominator

a() that appears in

the functionsR() =b() bis simply replaced bya, thereby avoiding the pos-sibility of a vanishing denominator in the expressionwK(). Then Ciarlet & Roquefort[2001] have shown that the leading term of a formal asymptotic expansion of a solutionto this two-dimensional model, with the thickness 2as the small parameter, coincideswith that found by a formal asymptotic analysis of the three-dimensional equations. Thisresult thus raises hopes that a rigorous justification, again by means of -convergence

theory, of either types of nonlinear Koiters models might be possible.

35


36/60

2.3 The linear Koiter shell equations

Consider the Koiter energy j for a nonlinearly elastic shell, defined by (cf. Section2.1)

j() =1

2 aG()G() +

3

3aR()R()

a dy

pii

a dy,

for smooth enough vector fields = (i) : R3. One of its virtues is that theintegrands of the first two integrals are quadraticexpressions in terms of the covariantcomponentsG() andR() of the change of metric, and change of curvature, tensorsassociated with a displacement field ia

i of the middle surfaceS= () of the shell. Inorder to obtain the energy corresponding to the linear equations of Koiter [1970], whichwe are about to describe, it suffices, by definition, to replace the covariant components

G() =1

2(a() a) and R() = b() b,

of these tensors by their linear parts with respect to = (i), respectively denoted()and () below. Accordingly, our first task consists in finding explicit expressions ofsuch linearized tensors. To begin with, we compute the components ().

A word of caution. The vector fields

= (i) and := iai,which are both defined on , must be carefully distinguished! While the latter has anintrinsic character, the former has not; it only provides a means of recovering the field viaits covariant components i. Theorem 2.3.1. Let be a domain inR2 and let C2(;E3)be an immersion. Givena displacement field

:= iai of the surface S = () with smooth enough covariant

componentsi:

R, let the function() :

R be defined by

() :=1

2[a() a]lin ,

where a and a() are the covariant components of the first fundamental form ofthe surfaces() and (+ iai)(), and [ ]lin denotes the linear part with respect to = (i) in the expression [ ]. Then

() =1

2( a+a) = ()

=1

2(|+ |) b3

=1

2(+)

b3,

36


37/60

where the covariant derivatives| are defined by| = (Theorem 1.6.1).In particular then,

H1() and 3L2()()L2().

Proof. The covariant componentsa() of the metric tensor of the surface (+ iai)()

are by definition given bya() = (+) (+).

The relations(+ ) = a+

then show that

a() = (a+) (a+ )=a+ a+ a+ ,

hence that

() =1

2[a() a]lin =1

2(

a+

a).

The other expressions of() immediately follow from the expression of =(iai) given in Theorem 1.6.1.

The functions () are called the covariant components of the linearizedchange of metric tensor associated with a displacement iai of the surfaceS.

We next compute the components ().

Theorem 2.3.2. Let be a domain in R2 and let C3(;E3) be an immersion.Given a displacement field := iai of the surfaceS = () with smooth enough andsmall enough covariant componentsi : R, let the functions() : R bedefined by

() := [b() b]lin,where b and b() are the covariant components of the second fundamental form ofthe surfaces() and (+ iai)(), and [ ]lin denotes the linear part with respect to = (i) in the expression [ ]. Then

() = ( ) a3 = ()= 3| bb3+b|+b|+b|= 3 3 bb3

+b( ) +b( )+(b

+

b

b),

where the covariant derivatives| are defined by| = (Theorem 1.6.1)and

3| :=3 3 and b

| := b

+

b

b

.

37


38/60

In particular then,

H1() and 3H2()()L2().The functionsb| satisfy the symmetry relations

b| = b

| .

Proof. For convenience, the proof is divided into five parts. In parts (i) and (ii), weestablish elementary relations satisfied by the vectors ai and a

i of the covariant andcontravariant bases along S.

(i) The two vectorsa = satisfy|a1a2|=

a, wherea= det(a).Let A denote the matrix of order three with a1,a2,a3 as its column vectors. Conse-

quently,

detA= (a1a2) a3 = (a1a2) a1a2|a1a2| =|a1a2|.

Besides,(detA)2 = det(ATA) = det(a)= a,

since aa = a and aa3= 3. Hence|a1 a2|=a.(ii) The vectorsai anda

are related bya1a3=aa2 anda3a2 =aa1.To prove that two vectors c and d coincide, it suffices to prove that c ai = d ai for

i {1, 2, 3}. In the present case,(a1a3) a1 = 0 and (a1a3) a3 = 0,(a1a3) a2 =(a1 a2) a3 =

a,

since

aa3 = a1a2 by (i), on the one hand; on the other hand,aa2 a1=

aa2 a3 = 0 and

aa2 a2 =

a,

since ai

aj =ij . Hence a1

a3 =

aa2. The other relation is similarly established.

(iii) The covariant componentsb() satisfy

b() = b+ ( ) a3+ h.o.t.,where h.o.t. stands for higher-order terms, i.e., terms of order higher than linearwith respect to = (i). Consequently,

() := [b() b]lin =

a3 = ().Since the vectors a = are linearly independent in and the fields = (i) are

smooth enough by assumption, the vectors( +iai) are also linearly independent in provided the fields are small enough, e.g., with respect to the norm of the spaceC1(;R3). The following computations are therefore licit as they apply to a linearizationaround = 0.

38


39/60

Let

a() :=(+ ) = a+ anda3() := a1() a2()a()

,

wherea() := det(a()) and a() := a() a().

Then

b() =a() a3()

= 1

a()(a+) (a1a2+ a1 2+1 a2+ h.o.t.)

= 1

a()

a(b+ a3)

+ 1

a()

(a+ ba3) (a1 2+1 a2) + h.o.t. ,

since b = a a3 and a = a + ba3 by the formula of Gau (Theorem1.6.1 (a)). Next,

(a+ba3) (a1 2)= 2a2 (a12) b2 (a1a3)=

a22 a3+b2 a2 ,

since, by (ii), a2 (a1 2) =2 (a1a2) =a2a3 and a1a3 =aa2;likewise,

(a+ba3) (1 a2) = a(11 a3+b1 a1).Consequently,

b() = aa() b(1 + a) + ( ) a3+ h.o.t. .There remains to find the linear term with respect to = (i) in the expansion1

a()=

1a

(1 + ). To this end, we note that

det(A +H) = (detA)(1 + trA1H+o(H)),

with A := (a) and A + H := (a()). Hence

H= ( a+ a + h.o.t.) ,since [a() a]lin =

a+

a (Theorem 2.3.1). Therefore,

a() = det(a()) = det(a)(1 + 2a + h.o.t.),

39


40/60

since A1 = (a); consequently,

1a()

= 1

a(1 a + h.o.t.).

Noting that there are no linear terms with respect to = (i) in the product (1 a)(1 + a), we find the announced expansion, viz.,

b() = b+ ( ) a3+ h.o.t.(iv) The components() can be also written as

() =3| bb3+b|+ b|+b|,where the functions3| andb

| are defined as in the statement of the theorem.

By Theorem 1.6.1 (b),

= ( b3)a + (3+b)a3.Hence

a3 =(3+b),since ai a3 = i3. Again by Theorem 1.6.1 (b),

a3 = ( b3)a + (3+b)a3 a3= ( b3)a a3

+(3+ (b)+b

)a

3 a3+ (3+b)a3 a3= b( ) bb3+3+ (b)+b,

sincea a3 = (a +ba3) a3 = b,

a3 a3=ba a3 = 0,

by the formulas of Gau and Weingarten (Theorem 1.6.1 (a)). We thus obtain() = ( ) a3

= b( ) bb3+3+ (b)+ b(3+b).

While this relation seemingly involves only the covariant derivatives 3| and | ,it may be easily rewritten so as to involve in addition the functions | and b

|. The

stratagem simply consists in using the relation b b = 0! This gives

() = (3 3) bb3+b( ) +b( )

+(b+

b

b

b

).

40


41/60

(v) The functionsb| are symmetric with respect to the indices and.Again, because of the formulas of Gau and Weingarten, we can write

0 = a a =

a +ba3 a +ba3= ()a + a ba3 + (b)a3 bba

+()a a + ba3 (b)a3 +bba.Consequently,

0 = (a a) a3 =b b+ b b,

on the one hand. On the other hand, we immediately infer from the definition of thefunctionsb| that we also have

b| b| =b b+ b b,

and thus the proof is complete.

The functions () are called the covariant components of the linearizedchange of curvature tensorassociated with a displacementiai of the surfaceS. Thefunctions

3|=3 3 and b| = b+ b brespectively represent a second-order covariant derivative of the vector field ia

i

and a first-order covariant derivative of the second fundamental form of S,defined here by means of its mixedcomponents b .

Remarks 2.3.3. (1) Covariant derivatives b| can be likewise defined. More specifi-cally, each function

b| := b b brepresents a first-order covariant derivatives of the second fundamental form, defined

here by means of its covariant components b. By a proof analogous to that given inTheorem 2.3.2 for establishing the symmetry relations b| = b| , one can then showthat these covariant derivatives likewise satisfy the symmetry relations

b| = b| ,

which are themselves equivalent to the relations

b b+ b b = 0,

i.e., the familiar Codazzi-Mainardi equations (Theorem 1.7.1)!(2) The functions c := b

b =c appearing in the expression of() are the

covariant components of the third fundamental form ofS. For details, see, e.g., Stoker

[1969, p. 98] or Klingenberg [1973, p. 48].

41


42/60

(3) The functions b() are not always well defined (in order that they be, thevectorsa() must be linearly independent in ), but the functions () are alwayswell defined.

(4) The symmetry () = () follows immediately by inspection of the ex-pression () = (

)a3 found there. By contrast, deriving the same

symmetry from the other expression of() requires proving first that the covariant

derivatives b| are themselves symmetric with respect to the indices and (cf. part(v) of the proof of Theorem 2.3.1).

While the expression of the components () in terms of the covariant componentsi of the displacement field is fairly complicated but well known (see, e.g., Koiter [1970]),that in terms of = iai is remarkably simplebut seems to have been mostly ignored,although it already appeared in Bamberger [1981]. Together with the expression of thecomponents() in terms of(Theorem 2.3.1), this simpler expression was efficientlyput to use by Blouza & Le Dret [1999], who showed that their principal merit is toafford the definition of the components () and () under substantially weakerregularity assumptions on the mapping .

More specifically, we were led to assume that C3(;E3) in Theorem 2.3.1 in orderto insure that ()L2() ifH1() H1() H2(). The culprits responsiblefor this regularity are the functions b

| appearing in the functions (). OtherwiseBlouza & Le Dret [1999] have shown how this regularity assumption on can be weakenedif only the expressions of() and() in terms of the field are considered.

We are now in a position to describe the linear Koiter shell equations. Let0be a measurable subset of = that satisfies length0 > 0, let denote the outernormal derivative operator along (since is a domain, a unit outer normal vector() exists d-almost everywhere along, and thus =), and let the space V()be defined by

V() :== (i)H1() H1() H2(); i= 3 = 0 on 0

.

Then the unknown vector field = (i) : R3, where the functions i are thecovariant components of the displacement field iai of the middle surface S = () ofthe shell, should be a stationary point over the spaceV() of the functional j defined

by

j() =1

2

a()() +

3

3a()()

a dy

pii

a dy

for all V(). This functional j is called Koiters energy for a linearly elasticshell.

Equivalently, the vector field V() should satisfy the variational equations

a()() +

3

3a()()

a dy

= piia dy for all = (i)V().

42


43/60

We recall that the functions

a := 4

+ 2aa + 2(aa +aa)

denote the contravariant components of the shell elasticity tensor (and are the Lame

constants of the elastic material constituting the shell), () and () denote thecovariant components of thelinearized change of metric, andchange of curvature, tensorsassociated with a displacement field ia

i ofS, and the given functions pi L2() accountfor the applied forces. Finally, the boundary conditions i = 3 = 0 on0 express thatthe shell is clampedalong the portion (0) of its middle surface (see Figure 2.1.3).

The choice of the function spaces H1() and H2() for the tangential components and normal components 3 of the displacement fields ia

i is guided by the naturalrequirement that the functions () and() be both in L

2(), so that the energyis in turn well defined for = (i)V(). Otherwise these choices can be weakened toaccommodate shells whose middle surfaces have little regularity (see Blouza & Le Dret[1999]).

Remark 2.3.4. Koiters linear equations can be fully justified by means of an asymptoticanalysis of the three-dimensional equations of linearized elasticity as 0; see Ciarlet[2000a, Chapter 7] and the references therein.

2.4 A fundamental lemma of J.L. Lions

We first review some essential definitions and notations, together with a fundamentallemma of J.L. Lions (Theorem 2.4.1). This lemma will play a key role in the proofs ofthe Korn inequalities on a surface (Section 2.5).

A domain in Rd is an open, bounded, connected subset of Rd with a Lipschitz-continuous boundary , the set being locally on one side of . As is Lipschitz-continuous, a measured can be defined along and a unit outer normal vector =(i)di=1 (unit means that its Euclidean norm is one) exists d-almost everywhere along

.Let be a domain in Rd. For each integer m 1, Hm() and Hm0 () denote the

usual Sobolev spaces. In particular,

H1() :={vL2(); ivL2(), 1id},H2() :={vH1();ijvL2(), 1i, jd},

whereiv andijv denote partial derivatives in the sense of distributions, and

H10 () :={vH1();v = 0 on },

where the relation v = 0 on is to be understood in the sense of trace. The norm in

L2() is noted0, and the norm in Hm(), m 1, is notedm,. In particular

43


44/60

then,

v0,:=

|v|2 dx1/2

ifvL2(),

v

1,:= v

20,+

d

i=1 iv20,

1/2

ifv

H1().

We also consider the Sobolev space

H1() := dual space ofH10 ().

Another possible definition of the space H10 () being

H10 () = closure ofD() with respect to 1,,

whereD() denotes the space of infinitely differentiable real-valued functions definedover whose support is a compact subset of , it is clear that

v

L2() =

v

H1() and iv

H1(), 1

i

n,

since (the duality between the spacesD() andD() is denoted by, ):

|v, |=

v dx v0,1,,

|iv, |=| v, i|=

vi dx v0,1,

for all D(). It is remarkable (but also remarkably difficult to prove!) that theconverse implication holds:

Theorem 2.4.1. Let be a domain inRd and letv be a distribution on. Then

{v

H1() and iv

H1(), 1

i

d}

=

v

L2().

This implication was first proved by J.L. Lions, as stated in Magenes & Stampacchia[1958, p. 320, Note (27)]; for this reason, it will be henceforth referred to as the lemmaof J.L. Lions. Its first published proof for domains with smooth boundaries appearedin Duvaut & Lions [1972, p. 111]; another proof was also given by Tartar [1978]. Variousextensions to genuine domains, i.e., with Lipschitz-continuous boundaries, are given inBolley & Camus [1976], Geymonat & Suquet [1986], and Borchers & Sohr [1990]; Am-rouche & Girault [1994, Proposition 2.10] even proved that the more general implication

{v D() and ivHm(), 1in} = vHm+1()

holds for arbitraryintegersm Z.

44


45/60

Our objective in the next sections is to study the existence and uniqueness of thesolution to the above variational equations. To this end, we shall see (Theorem 2.6.1)that, under the assumptions 3+ 2 > 0 and >0, there exists a constant ce> 0 suchthat

,

|t|2 cea(y)tt

for all y and all symmetric matrices (t). When length0 > 0, the existence anduniqueness of a solution to this variational problem by means of the Lax-Milgram lemmawill then be a consequence of the existence of a constant c such that

21,+ 322,1/2

c

,

()20,+,

()20,1/2

for all V().

The objective of the next section precisely consists in showing that such a fundamentalKorns inequality on a surface indeed holds (Theorem 2.5.3).

2.5 Korns inequality on a surface

To begin with, we establish a Korns inequality on a surface, without bound-ary conditions, as a consequence of the lemma of J.L. Lions(cf. Theorem 2.4.1). Wefollow here Ciarlet & Miara [1992] (see also Bernadou, Ciarlet & Miara [1994]).

Theorem 2.5.1. Let be a domain inR2 and let C3(;E3) be an injective immer-sion. Given = (i)H1() H1() H2(), let

() :=1

2( a+ a)L2(),

() :=

(

) a3

L2()

denote the covariant components of the linearized change of metric, and linearized change

of curvature, tensors associated with the displacement field := iai of the surfaceS= (). Then there exists a constantc0 = c0(, ) such that

21,+ 322,1/2

c0

20,+ 321,+,

()20,+,

()20,1/2

for all= (i)H1() H1() H2().

Proof. The fully explicit expressions of the functions () and (), as found inTheorems 2.3.1 and 2.3.2, are used in this proof, simply because they are more convenient

for its purposes.

45


46/60

(i) Define the space

W() :== (i)L2() L2() H1();

()L2(), ()L2()

.

Then, equipped with the normW() defined by

W() :=

20,+ 321,+ ,

()20,+ ,

()20,1/2,the spaceW() is a Hilbert space.

The relations () L2() and () L2() appearing in the definitionof the space W() are to be understood in the sense of distributions. They mean thata vector field L2() L2() H1() belongs to W() if there exist functions inL2(), denoted () and(), such that for all D(),

() dy =

12

(+) + +b3

dy,

() dy = 3+ 3+bb3+ (b

) +b

+ (b) +b

b+ b b dy.Let there be given a Cauchy sequence (k)k=1 with elements

k = (ki)W(). Thedefinition of the normW() shows that there exist L2(), 3 H1(), L2(), and L2() such that

k in L2(), k3 3 in H1(),(k) in L2(), (k) in L2()

ask . Given a function D(), lettingk in the relations (k) d=. . . and

(k) d= . . . then shows that =() and=().

(ii) The spacesW() andH1() H1() H2() coincide.Clearly, H1()H1()H2() W(). To prove the other inclusion, let =

(i)W(). The relations

s() :=1

2(+ ) = () +

+b3

then imply that e() L2() since the functions and b are continuous on .Therefore,

H1(),

() ={s() +s() s()} H1

(),

46


47/60

sinceL2() impliesH1(). HenceL2() by the lemma of J.L. Lions(Theorem 2.4.1) and thus H1().

The definition of the functions(), the continuity overof the functions , b , b

,

and b, and the relations ()L2() then imply that 3L2(), hence that

3H2().

(iii) Korns inequality without boundary conditions.The identity mapping from the space H1()H1() H2() equipped with its

product norm = (i) {

21, +322,}1/2 into the space W() equippedwith W() is injective, continuous, and surjective by (ii). Since both spaces arecomplete (cf. (i)), the closed graph theorem then shows that the inverse mapping 1

is also continuous or equivalently, that the inequality of Korns type without boundaryconditions holds.

In order to establish a Korns inequality with boundary conditions, we have toidentify classes of boundary conditions to be imposed on the fields = (i)H1() H1() H2() in order that we can get rid of the norms0, and31, in theright-hand side of the above inequality, i.e., situations where the semi-norm

= (i) ,

()20,+ ,

()20,1/2becomes a norm, which should be in addition equivalentto the product norm.

To this end, we need an infinitesimal rigid displacement lemma on a surface (theadjective infinitesimal reminds that only thelinearizedparts() and() of thefull change of metric and curvature tensors 12(a() a) and (b() b) arerequired to vanish in ), which is due to Bernadou & Ciarlet [1976, Theorems 5.1-1 and5.2-1]; see also Bernadou, Ciarlet & Miara [1994, Lemmas 2.5 and 2.6], Blouza & Le Dret[1999, Theorem 6], or Ciarlet [2005, Theorem 4.3-3].

Part (a) in the next theorem is an infinitesimal rigid displacement lemma on asurface, without boundary conditions, while part (b) is an infinitesimal rigiddisplacement lemma on a surface, with boundary conditions.

Theorem 2.5.2. Let there be given a domain in R2 and an injective immersion C3(;E3).

(a) Let= (i)H1() H1() H2() be such that() = () = 0 in .

Then there exist two vectorsa, bR3 such thati(y)a

i(y) = a + b (y) for all y.(b) Let0 be ad-measurable subset of= that satisfieslength 0 > 0 and let a

vector field= (i)H1() H1() H2() be such that() = () = 0 in and i = 3 = 0 on 0.

Then= 0 in .

47


48/60

We are now in a position to prove the announced Korns inequality on a surface,with boundary conditions.

This inequality was first proved by Bernadou & Ciarlet [1976]. It was later given otherproofs by Ciarlet & Miara [1992] and Bernadou, Ciarlet & Miara [1994]; then by Akian[2003] and Ciarlet & S. Mardare [2001], who showed that it can be directly derived fromthe three-dimensional Korn inequality in curvilinear coordinates (this idea goes back to

Destuynder [1985]); then by Blouza & Le Dret [1999], who showed that it still holdsunder a less stringent smoothness assumption on the mapping . We follow here theproof of Bernadou, Ciarlet & Miara [1994].

Theorem 2.5.3. Let be a domain inR2, let C3(;E3) be an injective immersion,let0 be a d-measurable subset of= that satisfies length0 > 0, and let the spaceV() be defined as:

V() :={= (i)H1() H1() H2(); i = 3 = 0 on 0}.

Given = (i)H1() H1() H2(), let

() := 1

2( a+aL

2(),

() :=

( ) a3L2()denote the covariant components of the linearized change of metric and linearized change

of curvature tensors associated with the displacement field := iai of the surfaceS=(). Then there exists a constantc= c(, 0, ) such that

21,+ 322,1/2

c

,

()20,+,

()20,1/2

for all= (i)V().

Proof. Let

H1()H1()H2() :=

21,+ 322,1/2.If the announced inequality is false, there exists a sequence (k)k=1 of vector fields

k V() such that

kH1()H1()H2()= 1 for all k ,

limk

,

(k)20,+,

(k)20,1/2

= 0.

Since the sequence (k)k=1 is bounded in H1() H1() H2(), a subsequence

()=1 converges inL2() L2()H1() by the Rellich-Kondrasov theorem. Further-

more, each sequence (())=1 and (())=1 also converges in L2() (to 0, but

48


49/60

this information is not used at this stage) since

lim

,

()20,+,

()20,1/2

= 0.

The subsequence ()=1 is thus a Cauchy sequence with respect to the norm

20,+ 321,+,

()20,+,

|()|20,1/2

,

hence with respect to the norm H1()H1()H2()byKorns inequality without bound-ary conditions (Theorem 2.5.1).

The space V() being complete as a closed subspace of the space H1() H1() H2(), there exists V() such that

in H1() H1() H2(),

and the limit satisfies

()0, = lim

()0, = 0,()0, = lim

()0, = 0.

Hence = 0 by Theorem 2.5.2. But this last relation contradicts the relations H1()H1()H2() =1 for all 1, and the proof is complete.

If the mapping is of the form(y1, y2) = (y1, y2, 0) for all (y1, y2), the inequalityof Theorem 2.5.3 reduces to two distinct inequalities (obtained by letting first = 0,then3 = 0):

32,c,

320,1/2

for all 3H2() satisfying 3 = 3 = 0 on 0, and

21,1/2

c

12

(+)20,

1/2for allH1() satisfying = 0 on 0. The first inequality is a well-known propertyof Sobolev spaces. The second inequality is the two-dimensional Korn inequality inCartesian coordinates. Both play a central role in the existence theory for linear two-dimensional plate equations(see, e.g., Ciarlet [1997, Theorems 1.5-1 and 1.5-2]).

As shown by Blouza & Le Dret [1999], Le Dret [2004], and Anicic, Le Dret & Raoult[2005], the regularity assumptions made on the mapping and on the field = (i) inboth the infinitesimal rigid displacement lemma and the Korn inequality on a surface of

Theorems 2.5.2 and 2.5.3 can be substantially weakened.

49


50/60

It is remarkable that,for specific geometries and boundary conditions, a Korn inequal-ity can be established that only involves the linearized change of metric tensors. Morespecifically, Ciarlet & Lods [1996] and Ciarlet & Sanchez-Palencia [1996] have establishedthe following Korn inequality on an elliptic surface:

Let be a domain in R2 and let C2,1(;E3) be an injective immersion with theproperty that the surfaceS= () is elliptic, in the sense that all its points are elliptic

(this means that the Gaussian curvature is > 0 everywhere on S; cf. Section 1.5). Thenthere exists a constant cM=cM(, )> 0 such that

21,+ 320,1/2

cM

,

()20,1/2

for all = (i)H10 () H10 () L2().Remarks 2.5.4. (1) The norm32, appearing in the left-hand side of the Korninequality on a general surface (Theorem 2.5.3) is now replaced by the norm30,.This replacement reflects that it is enough that = (i) H1() H1() L2() inorder that()L2(). As a result, no boundary condition can be imposed on 3.

(2) The Korn inequality on an elliptic surface was first established by Destuynder[1985, Theorem 6.1 and 6.5], under the additional assumption that theC0()-norms ofthe Christoffel symbols are small enough.

Only compact surfaces defined by a single injective immersion C3() have beenconsidered so far. By contrast, a compact surface S without boundary (such as anellipsoid or a torus) is defined by means of a finite number I2 of injective immersionsi C3(i), 1 i I, where the sets i are domains in R2, in such a way that S =

iIi(i). As shown by S. Mardare [2003a], the Korn inequality without boundaryconditions (Theorem 2.5.1) can be both extended to such surfaces without boundary.

2.6 Existence and uniqueness theorems for the linear Koiter shell equations

Let the space V() be defined by

V() :={ = (i)H1() H1() H2(); i= 3 = 0 on 0},

where 0 is a d-measurable subset of:= that satisfies length0> 0. Our primaryobjective consists in showing that the bilinear form B : V() V()Rdefined by

B(,) :=

a()() +

3

3a()()

a dy

for all (,)V() V() is V()-elliptic.As a preliminary, we state the uniform positive-definiteness of the elasticity tensor

of the shell, given here by means of its contravariant components a (note that theassumptions on the Lame constants, viz., 3 +

Documents

Ciarlet_A Brief Introduction to Mathematical Shell Theory