....

"

THE CLASSICAL VARIATIONAL PRINCIPLES OF MECHANICS

J. T. ODEN

March, 1977

0159

..

1.

THE CLASSICAL VARIATIONAL PRINCIPLES OF MECHANICS

J. T. Oden

Contents

INTRODUCTIONPage

2

2. MATHEMATICAL PRELIMINARIES 3

2.12.22.3

Transposes and Adjoints of Linear OperatorsPivot SpacesSobo1ev Spaces . . . .

388

3. GREEN'S FORMULAE FOR OPERATORS ON HILBERT SPACES 13

3.13.23.33.43.5

A General Comment . . . .Abstract Trace Property .Bilinear Forms and AssociatedGreen's Formulas ....Mixed Boundary Conditions . .

Operators

1313152024

4. ABSTRACT VARIATIONAL BOUNDARY-VALUE PROBLEMS

4.1 Some Linear Boundary-Value Problems4.2 Compatibility of the Data.4.3 Existence Theory .

5. CONSTRUCTION OF CLASSICAL VARIATIONAL PRINCIPLES .

6. THE CLASSICAL VARIATIONAL PRINCIPLES .

7. APPLICATIONS IN ELASTICITY

8. DUAL PRINCIPLES

8.1 The Dual Problem8.2 Application to Elastostatics

REFERENCES . . . . .

1

26

263238

43

45

52

60

6062

65

2

1. INTRODUCTION

The last twenty years have been marked by some of the most

significant advances in variational mechanics of this century. These

advances have been made in two independent camps. First and foremost,

the entire theory of partial differential equations has been recast in

"variational" framework that has made it possible to significantly ex-

pand the theory of existence, uniqueness, and regularity of solutions

of both linear and nonlinear boundary-value problems. In this regard,

the treatise of LIONS and MAGENES [l] on linear partial differential

equations, the works of LIONS [2], BREZIS [3], and BROWDER [4], on non-

linear equations, DUVAUT and LIONS [5] and LIONS and STAMPPACHIA [6]

on variational inequalities should be mentioned. Secondly, the under-

lying structure of classical variational principles of mechanics are

better understood. The work of VAINBERG [7] led to important general-

izations of classical variational notions related to the minimization

of functionals defined on Banach spaces, and an excellent memoir on

variational theory and differentiation of operators on Banach spaces

was contributed by NASHED [8]. Generalizations of Hamiltonian theory

were described by NOBEL [9] and NOBEL and SEWELL [10] and this led to

generalization of the concept of complementary variational principles

by AUTHURS [ll] and others. TONTI [l2,l3] showed that many of the

linear equations of mathematical physics share a common structure that

leads naturally to several dual and complementary variational princi-

ples, and these notions were further developed by ODEN and REDDY [l4].

A quite general theory of complementary and dual variational principles

•

3

for linear problems in mathematical physics was given in the monograph

of ODEN and REDDY [15], and a more complete historical account of the

subject, together with additional references, can be found in that work.

This essay deals with a general theory of variational methods

for linear problems in mechanics and mathematical physics. This work

is partly expository in nature, since one of its principal missions is

to develop, in a rather tutorial way, complete with examples, a rather

general mathematical framework for both the modern theory of variation-

al boundary-value problems and the "classical" primal, dual, and com-

plementary variational principles of mechanics. However, many of the

ideas appear to be new, and generalized Green's formulas are given

which make it possible to further generalize the recent theories of

ODEN and REDDY [l5].

2. MATHEMATICAL PRELIMINARIES

2.1 Transposes and Adjoints of Linear Operators. It is frequently im-

portant to distinguish between the transpose of a linear operator on

Hilbert spaces, and its adjoint. We set the stage for this discussion

by introducing some notation: Let

U, V be Hilbert spaces with inner products (u1,u2)

and (vl,v2), repsectively.

U' , V' be the topological duals of U and V; i. e., the

spaces of continuous linear functionals on U and V,

respectively.

4

("')u' (·,·)V denote duality pariings on U' x U and

V' x V, repsectively; Le., if e E U' and q E V' ,

we write

e (u) = (e, u )U and q (v) = (q,v)V

We shall denote by A a continuous linear operator from U

into V.

Now, following the standard arguments, we note that since

Au E V, the linear functional q(Au) = (q,Au)V also identifies a con-

tinuous linear functional on U; i.e., a correspondence exists between

•

q E V' and the elements of the dual space U' We describe this cor-

respondence by introudcing an operator A': V' ~ U' such that

or, equivalently,

A'q(u) q(Au) ,

( A' q, u)U = (q ,Au)V (Z.l)

The operator A' is called the transpose of A; it is clearly

linear and continuous from V' into U'.

Now the fact that U and V are Hilbert spaces allows us to

enter into a considerable amount of additional detail in describing A'

and other relationships between U and V. Since U is a Hilbert

space, we know from the Riesz representation theorem that for every

linear functional e E U, there exists a unique element ue E U such

that

( t, u)U 't:J u E U

5

Indeed, this relationship defines an isometric isomorphism KU: U ~ U'

such that

(2.2)

and Ku is called the Riesz map corresponding to the space U. Sim-

ilar1y, if KV is the Riesz map corresponding to the space V, we

have

In view of the definitions of the Riesz maps, we see that

!

't:JvEV, uEU

.. tlvEV, uEU

Thus, we discover in a very natural way that for each continuous linear

operator A: U ~ V satisfying the above relations there corresponds an

operator A*: V ~ U given by the composition

A*

such that

(2.3)

(2.4)

..

•

The operator A* is linear and continuous and is called the adjoint of

A. The relationship between A,A' , and A* is depicted symbolically

in Fig. 1.

A

IA

6

"

Figure 1 Relationship between the transpose AI

and the adjoint A* of a linear operator A.

...

7

The following theorem establishes some important porperties of

the transpose•

Theorem 2.1. Let A' E £(V',U') (here £(V',U') is the space

of continuous linear operators mapping V' into U') denote the

transpose of a continuous linear operator A mapping Hilbert spaces U

into V. Then the following hold:

(i) N (A' )l.

R(A)

-..

~where N(A') is the null space of A' and R(A) is the orthogonal

complement of the range R(A) of A in V'

(ii) A' is injective if and only if R(A) is dense in V

Proof: This is a well-known theorem; see, for example, DUNFORD

and SCHWARTZ [l6]. Since its proof is informative and brief, we shall

outline it below.

(i) Obviously, if q E N(A'), then

( A' q , u )U = (q, Au ) V o 'if u E U

~which means that q E R(A)

~Thus N(A') c R(A)

~If q E R (A)

then, by the same token, (A'q,u)U = 0~

R(A) c N(A')

VuE U. Hence

(ii) To prove (ii), we use the fact that a subspace M of a..

Hilbert space V is dense in V if and only if any continuous linear

functional on V that vanishes identically on M also vanishes identi-.-cally on V • If this is true, then (ii) follows immediately from (i) •

8

2.2 Pivot Spaces. The Riesz map KU: U ~ U' is an isometric isomorph-

ism form U onto its dual U'. Consequently, it is possible to iden-

tify U with its dual. In many instances, particularly in the theory

of linear boundary value problems, we encounter collections of Hilbert

spaces satisfying U c U 1 c . . . c Uz C Ul C U , the inclusionsm m- 0

being dense and continuous. When one member of this set is identified

with its dual, say U , we call it the pivot space, and writeo

U = U'o 0

The term "pivotlt arises from the fact that

1. e., Uo provides a "pivotlt between the spaces Ui t i::::0, and

their duals.

2.3 Sobolev Spaces. The notion of a Sobolev space is fundamental to

the modern theory of boundary-value problems, and most of the

applications of our theory can be developed within the framework of

Sobo1ev spaces.

Let SG 1be a smooth, open, bounded domain in :R.n. We shall

mdenote by H (SG) the Sobolev space of order m t which is a linear

space of functions (or equivalence classes of functions) defined by

1We shall assume throughout that Q is simply-connected with a~C -boundary aSG. However, most of the results and examples we citesubsequently hold if aSG is only Lipschitzian. See, for examplet

NECAS [l7] or ADAMS [18].

9

HID(Q)= {u : u and all of its distributional derivatives

...D!2-u of order ~ m are in LZ (Q), m ~ O}

Here we employ standard multi-index notations (see, e.g., ODEN and

REDDY [19]):

(2.5)

a 'ax nn

'~I

. .. , a )n integer ~ 0 (2.6)

x E Q C Rn .,

mNow the Sobolev spaces (2.5) are Hilbert spaces. Indeed, H (Q)

is complete with respect to the norm associated with the inner product

(u,v) m )H (Sil

mand the norm on H (Q) is

I L ngu n£vdx

52 1~I~m(2.7)

Ilull m )H (52

= (2.8)

•For example, if 52C :R2 ,

or if 52

(u,v) H2(Q)

(a,b) C :R ,

I [uv + au • av + au • av + a2

u •a2

vax ax ay ay a 2 a 2

52 x x

2 2 2 2]+~.~ au avaxay axay + 2 .2 dxdy

ay ay

10

•

(u,v) m( b)H a, f

b m

a{;m mThe subspace of H (Q) consisting of those functions in H (52)

whose derivatives of order ~m - 1 vanish on the boundary aQ of Q

is denoted ~(52)o

o , lal ~ m - I} (2.9)

Equivalently, can be defined as the completion of the space

00C (52) of infinitely differentiable functions with compact support ino

with respect to the Hm-Sobolev norm defined in (2.8)

The so-called negative Sobolev spaces are defined as the duals

mof the H (Q) spaces:o

m ::: 0 (2.10)

II

The Sobolev spaces are important in making precise the "degree

of smoothness" of functions. The following list summarizes some of

their most important properties:

(i)

(2.11)

Indeed,

(2.l2)

The remarkable fact is that these inclusions are continuous and dense.

m kIndeed, H (52) is dense in H (52), m ~ k ~ 0, and

/Iu/l k )H (52

~ Ilull Hm(52)(2.13)

(ii) Obviously, if m is sufficiently large, the elements of

Hm(52) can be identified with continuous functions. Just how large m

must be in order that each u E Hm(Q) be continuous is determined by

one of the so-called Sobolev embedding theorems: If 52 is smooth

(e.g., if Q C ~n satisfies the cone condition) and if m> n/2,

then Hm

(52) is continuously and compactly embedded in the space CO(Q)

of continuous functions; consequently, there exists a constant c > 0

..

..

such thatmVuE H (Q),

supxEQ

lu(x)1 ~ (2.14)

l2

s(iii) Sobolev spaces H (aQ) can be defined for classes of

functions whose domain is the boundary aQ. There is an important

relation between these boundary spaces and those containing functions

•

defined on Q. We define the trace operators Yj as

o ~ j ~ m - 1 (2.l5)

L e., they are the normal derivatives at aQ of order ~ m - 1. Let

1and define the norm

1I\p1I j 1 == inf {!lullm- -~ mH (aQ) uEHm(Q) H (Q)

(2.16)

•The completion of L2(aQ) with respect to this norm is a Hilbert space

of boundary functions denoted

o ~ j ~ m - 1 (2.17)

Two important properties of these spaces are called the trace

properties of Sobolev spaces:

can be extended to continuous linear(iii.l) The operators

operators mapping ~(Q) onto

Yj

Hm-j-\aQ); Le., there exist constants

such that

o ~ j ~m - 1 (2.18)

lThere are other more constructive ways of defining thesespaces. See, for example, aDEN and REDDY [19].

;

w

Y

13

(iii.2) The kernel of the collection of trace operators

(YO,Yl, .•. , Ym-l) is H~(Q) and H~(Q) is dense in gm(Q)

o , j O,l, ... , m - 1 (2.l9)

,

•

3. GREEN'S FORMULAE FOR OPERATORS ON HILBERT SPACES

3.l A General Comment. One of the most important applications of the

notion of adjoints of linear operators involves cases in which it makes

sense to distinguish between spaces of functions defined on the inte-

rior of some domain and spaces of functions defined on the boundary of

a domain. The introduction of boundary values is obviously essential

in the study of boundary value problems in Hilbert spaces, and it leads

us to the idea of an abstract Green's formula for linear operators.

In this section, we develop a general and abstract Green's for-

mula which extends those given previously in the literature. Our for-

mat resembles that of AUBIN [20], who developed Green's formulae for

elliptic operators of even order. Our results involve formal opera-

tors associated with bilinear forms B: H x G, Hand G being Hil-

bert spaces (see Sect. 3.3 below) and reduce to those of AUBIN when

collapsed to the very special case, G ~ H •

3.2 Abstract Trace Property. An abstraction of the idea of boundary

values of elements in Hilbert spaces is embodied in the concept of

spaces with a trace property. A Hilbert space H is said to have the

trace property if and only if the following conditions hold:

l4

(i) H is contained in a larger Hilbert space U which has a

weaker topology than H.

(ii) H is dense in U and U is a pivot space; i.e.,

H c U U' c H' (3.l)

(iii) There exists a linear operator y that maps H onto

another Hilbert space aH such that the kernel

y is dense in U; Le.,

Ho of

ker y = H c H •o '

H c Uo U' c H'

o (3.2)

The space aH corresponds to a space of boundary values, and

the operator y extends the elements of H, which can be thought of

as functions defined on the interior of some domain, onto the space of

boundary values aH. The operator y is sometimes called the trace

operator.

Example 3.lmThe Sobolev spaces H (Q) described in Section 2.3 pro-

vide the best examples of spaces with the trace property. Note that

(i) Hm(Q) is dense in LZ (Q) , m ~ 0

(ii) L2(Q) can be identified with its dual

(iii) extensions of the trace operators yj of (2.l5) map

Hm(Q) onto the boundary spaces Hm-j-~(aSG),

O~j~m-l, of (2.l7)

(iv) "

:

l5

In particular, suppose that Q is a smooth simply-connected

2open bounded domain in]R. with a smooth boundary aQ, the outward

normal to which is denoted n, and let us introduce the Sobolev

spaces,

o and au = 0an on

closure of L2

(aQ) with respect to the norm

•In this case,

II<pIlH~(aQ) inf {liullHl(Q); <p = u on aQ}

..H u aH

ker y and yu

property -

2are dense in L2

(Q), H (Q) has the trace

•

3.3 Bilinear Forms and Associated Operators. Let Hand G denote

two real Hilbert spaces (the extension of our results to complex spaces

is trivial), and let both Hand G have the trace property; i.e.,

H c U U' c H' G c V v' c G'

l6

y : H -+ aH y* : G -+ aG(3.3)

ker y H o ker y* Go

H c Uo

U' c H'o '

G c V = V' c G'o 0

The inclusions H c U ,

continuous.

G c V , H c U ,o G c V, are dense ando

Next, we introduce an operator B which maps pairs (u,v),

u E H and ~ E G, linearly and continuously into real numbers:

B:HxG~]R (3.4)

Wedenote the values of B in :R by B(u, v), and we refer to B as

a continuous bilinear form on H x G. That B is bilinear means that

(3.5)

tJ u,ul'uZ E H, '~>~1'~2 E G, a,S E B. That B is continuous means

that it is bounded; i.e., there exists a positive constant M such

that

tJuEH, tJ v E G (3.6)•

Nowlet u be a fixed element of H and let v E G. Theno

B(u,v) describes a continuous linear functional euon the space Go

!

functioal e depends linearly and continuously on u, and we de-u

,for each choice of u E H : B(u,v) = e (v), v E G .

UNO

describe this dependence in terms of a linear operator

The linear

Au == eu

l7

Thus

B(u,v) = (Au,v)G ' v E Go (3.7)

The operator A is called the formal operator associated with the bi-

linear form B. Clearly

A E £(H,G')o (3.8)

•

In a similar manner, if we fix v E G, B(u,v) defines a con-

tinuous linear functional on H , and we define a continuous linearo

operator A* by

u E Ho (3.9)

The operator A* is known as the formal adjoint of A, and

(3.l0)

The fact that H = ker y and G = ker y* enables us to establisho 0

the following fundamental lemma.

Theorem 3.1. Let Hand G denote the Hilbert spaces with the trace

properties described above, and let B denote a continuous bilinear

form from H x G ~ :m. with formal association operators A E £ (H ,G')o

..and A* E £ (G,W) .

oMoreover, let H

Aand GA* denote subspaces

{v E G :

Au E V}

A*v E U}

(v

(u

v' c G')o

u' c H')o

18

(3.11)

Then there exist uniquely defined operators 6 E £(HA,aG') and

6* E £(GA*,aH') such that the following formulas hold:

B(u,v) = (~,Au)V + (6u'Y*~)aG ' u E HA, v E G

)~

(3.l2)

B(u,v) = (A*~,u)U + (6*~'YU)aH ' u E H , v E G *~ A

Here (',' )aH and (',' )aG denote duality pairings on aH' x aH and

aG' x aG respectively.

Proof: We first consider the operator A.

V' c G~ for u E HA, we may write

Since Au E V =

tJ v E Go

Now pick v E G and introduce the operator CA defined by

v E G (3.13)

plement

into the orthogonal com-Since (CAu'~)G = 0

~G c G' •o 0

Next, we recall that y* maps G onto aG, and, therefore,

its transpose y*' is a bijective map of aG' onto

in G'. Hence y*' is invertible and the range of

its closed rangel.

Y*' is G •o '

19

..

1.e., (y*') -l maps G onto aG' . Set0

-1o = (y*') CA

Then 6 maps HA onto aG' , and we can write

-l = y*' QU , u E HAC u = y*' (y*' ) C uA a

(3.l4)

(3.l5)

Consequently, for CAu E G' we have

Thus, for u E HA

we have

B(u,v)

as asserted.

(3.l6)

That 0 is unique follows by contradiction: assume otherwise,

i.e., let there exist another operator 60 E £(HA,aG') such that the

above formula holds. Then, for any u E HA '

6u-l

(y*') y*'oOu

Hence 6 = 50 .

The derivation of the second formula follows identical argu-l.

ments. We introduce an operator CA* :GA* ~ HO such that

(CA*'!,u)H = B(u,~) - (A*'!,u)U' Then

..5* (3.l7)

20

is a uniquely defined linear operator such that the second formula

holds •

Let B: H x G +]R. be a continuous bilinear form on Hilbert

spaces Hand G having the trace property. Let in addition, A be

the formal operator associated with B(.,.) and let A* be its formal

adjoint. The operators y E £(H,aH) and 6 E £(HA,aG') described

above are called the Dirichlet operator and the Neumann operator, re-

spectively, corresponding to the operator A. Likewise, the operators

y* E £(G,aG) and 5* E £(GA*,aH') described above are called the

Dirichlet and Newmann operators, respectively, corresponding to the

operator A*.

3.4 Green's Formulas. The relationships derived in Theorem 3.l are

called Green's formulas for the bilinear form B(·,·) :

B(u,v) = (~,Au)V + (6u'Y*~)aG ' u E HA, v E G

I~

(3.l8)

B(u,v) = (A*~,u)U + (6*~,.yu )aH ' u E H , ~ E GA*

If we take u E HA and ~ E GA*, we obtain the abstract Green's for-

mula for the operator A E £(H,G~) n £(HA,V)

(3.19)

The collection of boundary terms,

r(u,v) (3.20)

2l

is called the bilinear concommitant of A; r: HA x GA* ~ ]R •

Example 3.2 Consider the case in which Q is a smooth open bounded

subset of Jt2 with a smooth boundary aQ and

1H = G = H (Q)

Let a = a(x,y), b = b(x,y), and c = c(x,y) be sufficiently smooth

1functions of x and y (e.g., a, b, c E C (Q», and define the bi-

1 1linear form B: H (Q) x H (Q) -+:R by

B(u,v) f (alJu. 'i/v+ bvu + cvu )dxdyQ ~ ~ x y

where 'Vu = grad u = (u ,u ). We observed in Example 3.l that Hl(Q)~ x y

has the trace property; it is dense in LZ(Q) and the trace operator

1yu = ul

aQcan be extended to continuous operator from H (Q) onto the

space H~(aQ) of functions defined as the closure of LZ(aQ) in the

norm 11·11 1 • ThusH~(aQ)

aH ker y u = 0 on aQ}

In this case,

'II

..

= J v (-V,Q ~

(a'iJu)+ bu + cu )dxdy +x y fau

a an vds

aQ

Thus, the formal operator corresponding to B(.,.) is

if we take

Au -11• (al1u)+ b au + c auax ay

22

Also, we now have

{u E H1(Q):

auleu = a an aS2 ' y*v

Similarly, if v E GA* = H A' we have

where n, n are the components of the unit outward normal n tox y

aSG • Thus,

A*v -11• (al1v)_ abv _ ocvax ay

Yu = ul 'aS2

e*v ~ av + (bn + cn )) Ir an x y ~ aS2

The bilinear concomitant is then

r(u,v) = f tv ~~ -av _

au an (cn + bn )u)dSy x:J •

..

Example 3.3 Let Q be as in Example 3.2 and define

B(u,v) = f grad u • v dxdyQ

23

as a bilinear form on H x G, where

{u: u, ux' uy' E L2(Q)}

G

1 1H (Q) is dense in U = L2(Q) and ~ (Q) is dense in V = ~2(Q)

L2(Q) x L2(Q). In this case, we may take y*~ = ~'~laQ' but

5 = O. Indeed, if the formal operator associated with the given bi-

1 linear form is

Au = grad u

Then

.1CA : HA ~ Go is identically zero. We next note that

B(u,~) = f u(-div ~)dxdy + f ~ . ~ udsQ aQ

Thus, GA* 1~ (Q) and

A*v = -div v ; yu u I aQ ; 6*v

The Green's formula is the classical relation,

f (grad U • v + u div v)dxdy

Q

f u v • n ds

aQ•

24

3.5 Mixed Boundary Condition. An abstract Green's formula appropri-

ate for operators with mixed boundary conditions can be obtained by

introducing some additional operators and ,spaces.

Let

n! = a continuous linear projection defined on aG

into itself

rr* = I - rr* I being the identity map from aG2 1 '

onto itself

y* = TT*Y* y* = TT*Y* and y* = y* + y*1 1 ' 2 2 1 2

J = ker y* = {v E G : TT*y*v = o}1 ~ 1 ~ ~

(3.21)

The space J is a closed linear subspace of G with the property

G cJcGo

(3.22)

The operators y*1 and effectively decompose aG into two sub-

spaces

y~(G) I(3.23)

25

A similar collection of operators can be introduced on aH

ITl = a continuous linear projection of aH into aH

IT 2 = I' - IT 1 (I': aH ~ aH)

(3.24)F = ker y 2 = {u E H : IT 2yu = a}

H c F c Ho

The Green formulae (3.l2) now yield

v E J

(3.25)

We observe that for v E J ,

u E F

where

(3.26)

5 = IT*'J::2 2 u ,(3.27)

26

Finally, collecting (3.25) and (3.26), we arrive at the abstract

Green's formula,

tJ u E HA n F , v E GA* n J (3.28)

4. ABSTRACT VARIATIONAL BOUNDARY-VALUE PROBLEl1S

4.1 Some Linear Boundary-Value Problems. The Green's formulae and

various properties of the bilinear forms B(·,·) described in the pre-

vious section provide the basis for a theory of boundary-value prob-

1ems involving linear operators on Hilbert spaces. We shall continue

to use the notations and conventions of the previous section: Hand

G are Hilbert spaces with the trace property, densely embedded in

pivot spaces u and V, respectively, and y* : G ~ aG, ker y* = G ,o

y : H ~ aH, ker y = H ,o etc.

Let B: H x G -+ JR be a continuous bilinear form and let A be

the formal operator associated with B(u,v). We consider three types

of boundary-value problems associated with A

1. The Dirichlet Problem for A. Given data f E V and

g E aH, the problem of finding u E HA such that

Au = f

yu = g} (4.1)

is called the Dirichlet problem for the operator A.

27

2. The Neumann Problem for A. Given data f E V and

s E aG', the problem of finding u E HA such that

Au = f

l5u = s} (4.2)

3. The Mixed Problem for A.

is called the Neumann problem for the operator A.

Given data f E V, g E aHl '

and s E aGZ' the problem of finding u E HA such that

Au = f

is u = s2

is called the mixed problem for the operator A.

(4.3)

Now the bilinear form B: H x G -+:R described in the previous

section can be used to contruct variational boundary-value problems

analogous to those for A. We shall consider the following varia-

tiona1 problems:

1. The Variational Dirichlet Problem for A. Given data

f E V and g E aH, find w E H = ker y such that0

-l 'if v E G (4.4)B(w,v) = (f,~)V - B(y g,~) ~ 0

where -1 is a right inverse ofy y.

28

2. The Variational Neumann Problem for A. Given data f E V

and s E aG', find u E H such that

B(u,v) (f,~)V + (s'Y*~)aG v v E G (4.5)

3. The Variational Mixed Problem for A. Given data f E V ,

g E aHl, and s E aG2, find w E ker Y1 = ker 1ilY such that

B(w,v) = (f,~)V - B(y~lg,~) + (S'Y*~)aG2 tJ v E ] (4.6)

where -1 is a right inverse of and ] = ker Y~ = ker TT 1y* •Yl Yl

Remark. There are, of course, several other abstract boundary-value

problems for the operator A that could be mentioned. For example, a

second bilinear form b: aH x aG ~ JR could be introduced at this point

which would permit the construction of oblique boundary conditions and

which would lead to a formulation more general than (4.3). However,

such technical generalizations obscure the simple structure of the

ideas we wish to clarify here, and so they are omitted. -

Theorem 4.1 Problems (4.l), (4.2), and (4.3) are equivalent

to problems (4.4), (4.5), and (4.6) respectively; i.e., if u is a so-

lution to (4.l) (or (4.2) or (4.3) respectively), then u is a solu-

tion to (4.4) (or (4.5) or (4.6) respectively) and, conversely, if u

is a solution of (4.4) (or (4.5) or (4.6) respectively) then u is a

solution of (4.1) (or (4.2) or (4.3) respectively).

Proof: It suffices to prove the equivalence of the mixed problems.

Then, for v E ker yt 'First, let w be a solution of (4.6) and let w •

o

29

B(w,v) (Aw,~)V + (ow'Y*~)aG

(f,v)V - (Aw ,v)v - (ow ,y*v)aG + (s,y*v)aG~ 0 ~ 0 ~ ~ 2

Set u w + w •o Then g and

Hence

o 'if v E ker *1

Au = f and

i.e., u is a solution of (4.3)

Next, let u be a solution of (4.3) and denote w=u-w o

where wo is any element of H such that Then

w E ker Yl . If v E J, we have

B(w,v) B(u,v)

Thus, w is a solution of (4.6).

The proof that (4.l) and (4.4) are equivalent and (4.2) and

(4.5) are equivalent follow similar lines and are omitted -

Example 4.1 Let

30

and aH1

aG = H~(aQ), where Q is a smooth open bounded domain in

:R2. The bilinear form

B(u,v) = J (~u. ~v + auv)dxdyQ

where a is a non-negative constant, is a continuous bilinear form

from H1(Sl) x H1(Q) into :JR. The formal operator associated with

B(·,·) is A = -6 + a , where 222226 = Il• Il= Il = a lax + a lay is

the Laplacian operator. We denote

The Dirichlet problem for A is to find u E H1(A,Sl) such tnat

u

f

g

in Q,

on aSil,

In view of Theorem 4.l, this problem is equivalent to the problem of

finding such that

J (Ilw'Ilv+ awv)dxdySl N N J fvdxdy - f (Ilw • Ilv+ aw v)dxdy

NON 0Q Q

where wo1is any function in H (Q) such that on aQ . Then

u = w + w is the solution of the Dirichlet problem for A.o

The Neumann problem for A is to find 1u E H (A,SG)

31

such that

-flu + au f in SG,

auan = s on aS2,

and it is equivalent to seeking u E Hl(S2) such that

J (lJu·IJv+ auv)dxdySG ~ ~

J fvdxdy +S2

f s v ds

aSG

where the contour integral denotes duality pairing on

•

We observe that boundary conditions enter the statement of a

variational boundary-value problem in two distinct ways. The essential

or stable boundary conditions enter by simply defining the spaces Ho

and G on which the problem is posed. The natural or unstable bound-o

ary conditions are introduced in the definition of the bilinear form

B(utv) and are defined on the spaces HA

and GA*.

Example 4.2 Returning to Example 4.l, take a = 0 with s = 0 and

aSG. Then the variational Dirichlet problem is to find

such that

I yu· yvdxdy52

ISG fvdxdy

1whereas the variational Neumann problem is to find u E H (Q) such that

J V'u• V'vdxdys;( ~

J fvdxdyQ

1\f v E H (Q)

32

While these problems look very similar, they are quite different. The

essential boundary conditions for the variational Dirichlet problem are

tied up in the statement that the equality holds for u and for every

1v E H (Q). The natural boundary conditions enter the Neumann varia-o

tional problem in the definition of the bilinear form. Here

B(u,v) = J ~u • yvdxdy = J fvdxdy + f svds 'rJ v E Hl(Q)Q Q aQ

1 1and B(-,-) is defined on H (Q) x H (Q). The contour integral hap-

pens to vanish because of our choice s = O. •

4.2 Compatibility of the Data. We shall discuss briefly here on the

issue of the compatibility of the data f,g,s with various boundary-

value problems. The ideas are derived from the classical theorem (re-

call Theorem 2.l):

Theorem 4.2. Let A be a bounded linear operator from a Hil-

bert space H into a Hilbert space G. Let A* be its adjoint. Let

N(A) , N(A*) denote the null spaces of A and A* respectively and

R(A) and R(A*) denote the ranges of A and A* respectively. Then

~R(A) N(A*) ,

N(A) , R(A*)

.1N(A*)

l.= N(A)

} (4.7)

33

where 1. denotes the orthogonal complement of the spaces indicated and

an overbar indicates the closure.

Proof: For a proof, see, for example, TAYLOR [21].•

We shall first address the compatibility question in connec-

tion with the Dirichlet problem for the operator A: find u E H A

such that

Au=f, f E V

yu g , g E aH

Let A* E £(GA*,H~) denote the formal adjoint of A and let

y* E £(G,aG), ker y* = G. Then the adjoint Dirichlet problem cor-o

responding to A is the problem of finding v E GA* such that

A*v f* , f* E U

y*v = g*,

We also introduce the null spaces,

g* E aG

N (A,y) = {u E HA :Au = 0 , yu = o}

N(A*,y*) = {~ E GA* : A*~ = 0 , y*~ = o}

We shall assume that these spaces are finite-dimensional.

Now if N(A,y) is finite dimensional, it is closed in U and

we have

where

u .1N(A,y) ffi N(A,y)

34

1N (A, y) = {u E U : (u,v)u = 0 V v E N(A,y)}

Then A can be regarded as a continuous linear operator from1N(A,y)

onto its range R(A) C V. By the Banach theorem, a continuous inverse

-lA

Thus

exists from R(A)

V = R(A) $ R(A)l

onto

where

1N(A,y) • and R(A) is closed in V.

1R(A) = {~ E V: (~,~)V = 0 V ~ E R(A)}

The Dirichlet problem for A clearly has at least one solution when-

ever the data fER (A) and g E R (y). Data satisfying these require-

menta are said to be compatible with the operators (A,y).

A conventient test for compatibility of the data is given in

the following theorem.

Theorem 4.3. Let f E V and g E aH be data in a Dirichlet

problem for the operator A. Then a necessary and sufficient condi-

tion that there exists a solution of the Dirichlet problem is that

(f,~)V - (o*'!:,g)aH = 0 tJ v E N(A*,y*) (4.8)

Proof: The necessity of this condition is obvious and follows

immediately from Green's formula. Sufficiency follows from the fact

1that (4.8) implies that the data (f,g) E N(A*,y*) = R(A,y), and

R(A,y) is closed. For additional details, see, for example

aDEN [22] •

35

f'For the Neumann problems

Au = f , A*v = f*~

au = s , a*v = s*

We define

N(A,a) = {u E HA

: Au = 0, au = o}

N(A*,a*) = {~ E GA* : A*v = 0 , a*v = o}

and, analogously, have

Theorem 4.4. A necessary and sufficient condition for the ex-

istence of at least one solution of the Neumann problem for the operator

A is that the data (f,s) satisfy

(f ,~)V + ( s, y*~) aG o tJ v E N(A*,6*) (4.9)

Proof: The proof is similar to that of Theorem 4.3 and is

omitted. -

A similar compatibility condition can be developed for mixed

boundary-value problems:

Theorem 4.5. A necessary and sufficient condition for the ex-

istence of a solution to the mixed boundary-value problem for A is

that the data (f,g,s) satisfy

36

(4.10)•

where ,.

{v E GA* : A*'!. o , o , o*vL a} _ (4.11)

Whenever the compatibility conditions hold, a solution to the

Dirichlet, Neumann, or mixed problems may exist, but it will not neces-

sarily be unique. The solution u is unique, of course, whenever

N(A,y) = {O}, for the Dirichlet problem, whenever N(A,o) = {O} for

the Neumann problem, and when N(A,y,B) = {OJ for the mixed problem.

However, these conditions often do not hold. We can, however, force

the solutions to either class of problems to be unique by imposing an

additional condition.

Theorem 4.6. Let u E HA be a solution to the Dirichlet prob-

lem for the operator A. Then u is the only solution to this prob-

lem if

(u,w)u = 0 tj w E N(A,y) (4.12)

Likewise, a solution u of the Neumann (mixed) Problem for A is

unique if (u,w)U = 0 't:J w E N(A,o) (~ w E N(A,y,o», respectively.

Proof: If (u,w)u = 0 't:J w E N(A,y), then 1u E N(A,y) • Let

so thatThen AUl = f = AU2

Hence ul - u2 E N(A,y), a contradiction. A similar

be two such solutions.and

argument applies to the Neumann and mixed problems. -

37

Example 4.3. Consider the Neumann problem for the Laplacian operator,

2in Q C:R :

-6u

au =an

f ,

s ,

This is a self-adjoint problem (recall Example 3.2) so that the ele-

ments of the space N(A*,6*) are those functions in1

H (l1.Q) (recall

,

Example 4.l) such that

-!Jv = 0

av = 0an

The solution of these homogeneous equations are simply constants:

v E N (l1 ~) => v'anconstant

Take v = 1 . Then the Neumann problem has a solution if and only if

the data f and s are such that

t f dxdy

The solution u is unique if

o

J u dxdy = 0

{constand. _

38

4.3 Existence Theory. The theory presented thusfar suggests the

following general setting for linear variational boundary-value

problems.

• Let Hand G be arbitrary Hilbert spaces (now Hand G

are not necessarily the spaces appearing earlier in this section--they

do not necessarily have the trace property)

• B: H x G ~ JR is a bilinear form. Then find an element

u E H such that

,

B(u,v) f(v) 'tJ v E G

where f E G' .

The essential question here is what conditions can be imposed

so that we are guaranteed that a unique solution exists which depends

continuously on the choice of data f? This question was originally

resolved for certain choices of B by Lax and Milgrim. We shall prove

a more general form of their classic theorem made popular by BABUSKA

[23] (see also NE~AS [17] and ODEN and REDDY [l9]).

Theorem 4.7. Let B: H x G -+ B be a bilinear functional on

H x G, Hand G being Hilbert spaces, which has the following three

properties.

(i) There exists a constant M > 0 such that

•

'tJ u E H , v E G

where II·IIH and II·IIG denote the norms on Hand G, respectively,

39

(ii) There exists a constant y > 0 such that

(iii)

inf sup IB (u,v) I~y>OuEH vEG

lIullH=l IlvllG~l ~ 1

sup IB (u,v) I > 0 ,uEH

v "t 0

Then there exists a unique solution to the problem of finding u E H

such that

B(u,v) f(v) VvEG, f E G'

Moreover, the solution u depends continuously on the data; in fact,o

lIu II ~ lllfilo H Y G'

Proof: Proofs of this theorem can be found in BABUSKA and AZIZ

[24] and ODEN and REDDY [19]. We shall outline the essential ideas for

the sake of completeness.

For each fixed u E H, B(u,v) defines a linear functional Fu

on G, and this functional is continuous by virtue of property (i):

F (v)u

B(u,v)

IIF II = supu G' vEG

40

Thus. by the Riesz representation theorem, there esixts a unique vF

such that Fu(v) = (vF,v)G. (·.·)G being the inner product on G.

This vF depends linearly on the choice of u, and we write this de-

pendence as vF = Au. where A is a linear operator mapping H into

G • Thus

and

B(u,v) F (v)u

II Au II = IIFII s: MilullG u G' H

which means that A is continuous.

We also have

JB(u,v)L II II I (u JIsup IIvl/ = sup u H B M'vvEG G IlvllGS:l 'H

i.e., A is bounded below:

Now we recall that every linear operator that is bounded below

has an inverse which is continuous on its range. The range of A,R(A)

..

is closed in G; indeed, if A(u ) is Cauchy. then so isn {un} in H

by virtue of the continuity of A. We now show that its complement

41

R(Al is empty. Assume otherwise. Then there exists a v ~ 0 E R(A)l0

such that

(Au,v) = 0o G

but

(Au,v) = B (u ,v ) , sup !B(u,v)1 > 0, v ~ ao G 0 uEH 0 0

by virtue of (iii).

and R(A) = G .

Hence, we arrive at a contradiction. Thus v = 0o

Let g be the element in G corresponding to the linear func-

tional f E G' via the Riesz theorem. Then (g,v)G = f(v) 't:J v E G ,

~d

Hence, a solution

Moreover,

uo

B(u,v) = (Au,v)G

exists and

-lu = A go

as asserted.

It remains to be shown that u is unique. Assume the con-o

trary; i.e., let ul ~ Uo also be a solution. Then

42

o

which means that !luI- u2!1 = 0, a contradiction. -H

Property (i) B(.,·) is, of course, a continuity requirement;

B(.,-) is assumed to be a bounded linear functional on H and on G.

Properties (ii) and (iii) serve to establish the existence of a contin-

uous inverse of the associated operator A.

Corollary 4.8. Let B: H x H + B be a bilinear form on a

real Hilbert space H. Let there exist positive constants M and y

such that

tj u,v E H

B (u, u)

Then there exists a unique u E H such that

and

B(u,v) = f(v) VvEH, f E H'

Proof: This follows immediately from Theorem 4.7 and details

are left to the reader. -

43

5. CONSTRUCTION OF VARIATIONAL PRINCIPLES

Let W be a real Banach space and W' its topological dual.

If P is an operator from W into W', not necessarily linear~ we

may consider the abstract problem of finding u E W such that

P(u) o , o E W' (5.l)

Now it is well known that in many cases an alternative problem

can be formulated, equivalent to (5.1), which involves seeking a u E W

such that

K' (u) = 0

where K is an appropriate functional defined on Wand K'(u) is

the Gateaux derivative of K at u; Le. ~

lim 1. (K(u + £'l'") ) - K (u») = (K' (u),'1 )e~ £

where (.,.) denotes duality pairing on W' x W . Thus, if there ex-

ists a Gateaux differentiable functional K: W -+ lR such that

P K' (5.2)

then (5.1) is equivalent to the classical variational problem of find-

ing elements u E W which are critical points of the functional K.

We say that P is the gradient of K and we sometimes write P = grad K .

Any operator P: W -+ W' for which there exists a functional

K :W -+ lR such that P = grad K is called a potential operator. It

44

is well known (see, e.g., VAINBERG [7] or NASHED [8]) that if a con-

tinuous Gateaux differential 6P(u,~) of P exists, then a necessary

and sufficient condition that P be potential is that it be symmetric

in the sense that

( 6P (u ,z;:> , ~ ) 't:J u,~,z.: E W

Given a potential operator P: W ~ W', the problem of deter-

mining a functional K such that (5.2) holds is called the inverse

problem of the calculus of variations. Its solution is provided by

the following theorem, the proof of which can be found in the mono-

graph of VAINBERG [7].

Theorem 5.1. Let P: W ~ W' be a potential operator on the

Banach space W. Then P is the gradient of the functional K: W -+ R

given by

K(u) =11

(P(u + s(u - u »),u - u >ds + Ko 0 0 0 0(5.3)

where uo is a fixed point in W, Ko

K(u ), ando s E [0,1] . •

By an appropriate identification of the space Wand the

duality (.,.), all of the classical variational principles of math-

ematical physics can be constructed using (5.3). A lengthy list of

applications of (5.3) to this end can be found in Chapter 5 of ODEN and

REDDY [l7].

..

45

6. THE CLASSICAL VARIATIONAL PRINCIPLES

6.1 A General Class of Boundary-Value Problems. We shall now describe

a general class of abstract boundary-value problems that is encountered

with remarkable frequency in linear problems of mathematical physics

and mechanics. Continuing to use the notation of the previous section,

we introduce a linear, symmetric, operator

E : V' ~ V (6.1)

which effects a continuous, isometric isomorphism of the dual of the

pivot space V' onto V, which has a continuous inverse, E-l: V ~ V' •

If v E V', we shall denote the elements in E(V') by

Ev

The Green's formula (3.19) can now be written

(6.2)

(6.3)

Now let us consider the following abstract problem: Given

fEU', g E aH1

, and s E aGZ' find u E H A such that

46

A*EAu = f

g (6.4)

where we have used the notations of (3.2l) and (3.24). We shall ex-

ploit the fact that this problem can be rewritten in the following

canonical form: fino

(u.v,a) E HA x V' x GA*

such that

Au = v y u = g1

Ev = aN

A*a = f y*a = s2N

(6.5)

(6.6)

We would now like to construct a variational principle (i.e., a

potential functional) corresponding to (6.6). Toward this end, we in-

troduce the matrix operator

0 -1 A 0 0 a I 0N

-1 E 0 0 0 v 0N

A* 0 0 0 0 u = f (6.7)

0 0 0 0 1Tllly*a g

0 0 0 1T* o IIYU I I s2

or

P(\) = f

where P is the coefficient matrix of operators in (6.7),

T\ = (a,v,u,y*a,yu) E GA* x V' x HA x aG x aH

Tf = (O,O,f,g,s) E V' x V x U' x aH x aG

1 2

If

w = V x V' x U x aH' x aG'

and

( • , • )w == ( . , . ) V + ( • , • )V ' + < • , • ) U + ( • , .) H' + ( • , .) G'a 1 a 2

We compute easily the functional,

47

(6.8)

L(u,v,a)

1 A A

= - (P (\) , \) - (f, \ )2 ~ ~ W ~ W

48

wherein

TX = (a,v,u,6*a,6u) E V x V' x U x aH' x aG'

1Applying Green's formula to the term -2(A*a, u ) gives~ U

1.1

L (u,v ,a) = "2 ( E~ '~ )V ' + (Au - ~,~ )V - ( f,u )U

(6.9)

The Euler equations are (6.6). Indeed,

6L(u,v,a;u,v,o) (Ev,v)V' + (Au - v,o)V - (f,u)U + (Ylu - g,5*;:) ,~ ~ ~ ~ N aH

1

(Ev - ~,~)V' + (Au - ~,~)V + (A*~'- f,u)U

where we have applied Green's formula into (Au,~)V'

Next, we list a number of additional principles that can be

derived directly from the functional L:

II. Constraint: -1v = E a (6.10)

R (u,a)a ~ L(u,v(a),a)

49

(6.11)

Euler equations: Au -1E a g

(6.12)

A*a = f

Constraint: a = Ev

y*a = s2~

(6.13)

R (u,v)v ~ L(u,v,a(v»)

(6.14)

Euler equations:

III. Constraints:

Au = v

A*Ev = f

v = Au

a = EAu

Ylu = g

y*Ev = S2 ~

(6.l5)

(6.l6)

I(u) 1L(u,v(u),a(u» = I (Au,EAu}V - (f,u)U - (ou,s}aG2

(6.17)

Euler equations: A*EAu f

50

(6.18)

OI(u,u) (A~,EAu)V - ( f,~)U - (O~,s )aG2

- (f,u)U - (Ou,S)aG2

= (A*EAu - f,u)U + (Bu,y~EAu - s)aG ;2

u E ker "(1

IV. Constraints:-1

v = E 0

(6.19)

A*o f y*o = s2~

J(o) = L(u(o),v(o),o)

(6.20)

Euler equations: Au -1E 0 g (6.21)

BJ(o,cr)

g)aH1

; ~ E ker A* n ker y~

..

,. V. Constraints: Au = v

A*a = f y*a = s2~

51

(6.22)

0- E {o: A*o = f , y*o = s}2~

(6.23)

Euler equation:

VI. Constraints:

Ev = 0

Au = v

(6.24)

(6.25)

1 -1M(u,~) = (Au,EAu)V +Z(E ~'~)v - (f,u)U

- (Au,a)V - (Bu,s)aG~ 2

(6.26)

Euler equations: Au -1E 0

(6.27)

VII. Constraints:

A*(2EAu - 0)

A*o = f

f y*(2EAu - a) = s2

y*o = s2~(6.28)

1N(v,o) ="2 (~,E~)V - (~,~)V + (B*~,g)aH1 (6.29)

Euler equations: v = Au

o = Ev

(6.30)

52

6N(v,a;v,cr)

= (v,Ev - ~)V + (~ - Au,~)V + (6*cr,g

a E ker A* n ker y~

We summarize these results in Table 6.1.

7. APPLICATIONS IN ELASTICITY

We now apply the theory in the previous section to the con-

struction of seven variational principles of linear e1astostatics.

The Lame-Navier equations of linear elasticity are

(A*EAu)j = _(Eijrsu ) = fj in S'2r,s ,i

(Yl~)i = ui = ui on as'2l ~ (7.1)

" i"rs A"

(A*EAu)J = n (E J u ) = TJ on as'222 N i r,s

which correspond to equations (6.4). The canonical equations (6.6) are

the strain-displacement equations, kinematical boundary conditions,

constitutive equations, and equilibrium equations:

TABLE 6.1

S3

Functional Definition Constraints Euler Equations

Au =~; Yl u = g

I L(u, v ,0) (6.9) --- Ev = 0N N ~ ~

A*o= f; y*o= s~ 2N

-1

R (u,o) (6.11) -1 Au :::E ~; Y1u = gv=E 0o N N

A*o= f; y*o = s2NII

Au = v; Y u = gR (u,v) (6.14) 0= Ev 1v N ~ N A*Ev = f; y*Ev = s~ 2 ~

I(u) (6.17)~ = Au; Y1u = g

A*EAu = f; y~EAu = sIII0= EAu~

-1v=E 0 -1IV J(o) (6.20) N N

Au = E ~; Y1u = g~A*o = f; y*o = s~ 2~

Au = v; Y u = gK(v) (6.23) N 1 Ev = 0V N A*o = f; y*o = s

~ N~ 2N

-1Au = E 0~

VI M(u,o) (6.26) Au =~; Y1u = g A*(2EAu - 0) = fN N

y*(2EAu - 0) = s

(6.29) A*o = f; y*o = s~ = Au; Y1u = g

VII N(v,o)N N ~ 2~ 0= EvN N

(Au)i' = t (ui . + u. i) =N J ,J J,

in

S4

ija in st

(A*a)j = _aij = fjf:j ,i

in st (Y*2a)j = n aij = 'idIII i

(7.2)

In this case,

U = ~2 (S'2) = {~ : ui E L2 (2) , i ~ 1,2,3} I(7.3)

V = b(Sn = {~: aij E L2(S'2) , i,j = 1,2,3}

and

1 { 1i ~ 1,2.3} IH = H (st) = ~: Ui E H (st) ,

(7.4)1 { 1 i,j = 1,2,3}G = H (S'2) = a: a .. E H (S'2) ,

f:$ ::J 1J

Hence, the canonical problem of the elasticity can be expressed

as follows: given

(7.5)

find

such that equations (7.2) are satisfied. Here

..HA = ~1 (A,S'2)= {~ E ~1 (S'2): A~ E ~2 (S'2) } )

GA* = ~l(A*,S'2)= {g E ~1(S'2): A*~ E ~2(S'2)}

The Green's formula (6.3) becomes, for this problem,

ss

(7.7)

iju.n.a dsJ 1

1VuE H (A,S'2), (7.8)

from which we can identify the operators 6 E £(H1 (A,S'2),G-~(aS'2») and

6* E £(Hl(A*,S2) ,H-~(aS'2»):III

(6u). = u, on aS22N J J

I(7.9)

ij i'(6*a) = -n a J on as'21III i

Now we are able to construct the variational theory for linear

elastostatics. According to Table 6.1, we obtain:

I. The Hu-Washizu Principle.

L(u, e:,a)~ Rl III

I ij A f Aj- nia (u. - u,)ds - u.T dsaS2 J J as'2 J

1 2

(7,10)

Euler Equations:

in Q;

S6

ij rs e:E rs

ij-0 ,i

ijo in S2

in S2;

(7.11)

II. The He1linger-Reissner Principle.

(i). Constraints:

e;rs (7.12)

Euler equations:

(7.13)

(7.14)

ij-0 ,i

(ii). Constraints:

i'o J

in Q ;

in Q (7.15 )

57

Euler equations:

(7.l6)

1-2 (ui . + u. i) =, J J,

in S'2;

(7.17)

in S'2; Eijrsni e:rs

III. Potential Energy Principle.

Constraints:

in S'2;

(7.18)

I(u) (7.19)

Euler equations:

Eijrsn ui r,sTJ on aS22 (7.20)

IV. Complementary Energy Principle.

Constraints:

58

e:rs

ij-0 , i

C ijijrsO in S2

in S'2;

(7.21)

Euler equations:

(7.22)

1- (u + u )2 r,s s,r

i'Ci. 0 JJrs in S2; on (7.23)

V. A Constitutive Vari.ationa1Principle.

Constraints:

e:ij in Q;

(7.24 )Ii' , i' A'

-0 J = fJ in !it ; n J = TJ on aS22,i i

K(~) = f [1 ijrs Aij) (7.25)"2 e:i .E e: - e: •. 0 dxJ rs 1J

Euler equations:

(7.26).J

VI. A Constitutive-Potential Energy Principle.

Constraints:

59

(7.27)

M(u,o)N III f r: Eijrs

S2 Li,j ur,s1 ij rs+ -2 Ci· 0 0Jrs

Euler equations:

1- (u + u )2 r,s s,r

C i'ijrsO J in Q

(7.28)

_{Eijrs(u + u ) - oij} = fj in Qr,s s,r ,i

ni{Eijrs(u + u ) - oij} = Tj on as'22r,s s,r

VII. A Compatibility-Constitutive Variational Principle.

Constraints:

(7.29)

(7.30)

N(e:,o)III '"

(7.31)

60

Euler equations:

(7.32)

ijo in S'2

8. DUAL PRINCIPLES

8.l The Dual Prob1eTl!.'It was pointed out by ODEN and REDDY [l5] (see

also [l4]) that a parallel collection of so~called dual variational

principles can be constructed in one-to-one correspondence with the

variational principles described in the previous section, While we

shall not elaborate on the detailed features of these dual functionals,

we shall outline briefly the essential concepts for the sake of

completeness.

To construct the dual principles, we consider a new Hilbert

space S and a linear operator

C:S-+G

such that

(8.1)

R (C)

N(c*)

:J N(A*) }

c R(A)

(8.2)

Next, we consider the dual problem of finding ~ E S such that

where

(C)Yl ~ = p

(C)* -1y E C~ = q2

6l

(8.3)

T) E S', P E aGi ' q E as'2 (8.4)

Owing to the similarity of (8.3) and (6.4), we can immediately

construct the following dual functionals:

I' .

II' .

or

(C)* (C) (C) )+(6 u'Yl ~-p)aG,-(q,6 ~aS'

1 2

(C)*-1 (C) (C)+ ( 6 Eo, Y 1 <P - p) aG' - (q, 6 <P ) as1 2

(8.6)

(8.7)

62

III' .1 -l. (C) (8.8)1 (~) = "2 (E C<p,c~)V' - (''l'~)S - (q,6 ~ )as ...

2

:..

IV' .1 (C)* (8.9)T (u) = -"2 ( u, Eu )V' - < 6 u ,P )aG'

1

1 -1(B.lO)V' . K(a) = "2 (E a,a)V' - (u ,a)V'

VI' ..

VII' .

(C)- (u, C<p)V ' - ( q, 5 ~ )as

2

1 -l (C)*N(a,u) = "'2 (E a,a)V' - (u,a)V' - (6 U,P )aG'

1

(8.11)

(8.12)

Clearly, a table similar to Table 6.l ca~ also be constructed

directly from (8.5) - (B.l2) by using the following correspondences:

G' rv G , SrvH, <prvµ, C"'A,

C* '" A* , E-l rv E a"'u, u"'a'r (8.13),

y(C) "'y* y(C)* '" y q"'g, P rv s1 2 ' 2 1 '

8.2 Application to Elastostatics. In the case of ~inear elasticity,

we set

<p = <Pij tensor of stress functions

63

stress tensor

= strain tensor

TJ'"TJij = dislocation tensor

Then (8.5) - (8.12) assume the following specific forms:

ij£ (<p,. a , E:i .)1J, J

ims jkr )e e <Prs,km

- TJij<PiJdX + (o*e:ij 'Y(~ij) - p)oS'2l

where, for compactness in notation, we have denoted

(8.l4)

wherein

Fimjnptqu _ imr jnsc tpt kqu- e e rstke e

i,j,k,t,m,n,p,q,r,s,t,u, = 1,2,3

64

Fimjnptqun _ ijqp<Ptu,mn - ql

Ion aS22,

Fimjnptqun _ ijq<Ptu,mnp - q2

_ f (l ijrs ims jkr ij)Re:(<Pij,Eij) - S'2 -ZEijE Era + Eije e lPrs,km - TJ <Pij dx

+ (5*e:ij,y(<Pij) - P)aS'2l

- (q,O(<Pij»aQ2

ij _ f t 1 ij rs ij rmp skqR (<Pij,o ) - - -20 Ci. 0 + Cij 0 e e ~ ko S'2 Jrs rs pq,m

- TJij<PiJdX + (O*(CijrSors) 'Y(~ij) - p )aQl

- (q,o(<Pij) )aS22.

(8.15)

(8.l6)

f a C imp jqu rta sbk ij Jd- e e e e - xQ 2 ijrs <Pmn,pqYak,tb <Pij

(8.l7)

..

65

_ f (.! rs ij ij )- S'2 2 Cijrsa a - a E:ij dx

(8.l8)

(8.l9)

ijrs irs jmk ij l+ E e:ije:rs- 2E:ije e <Psk,rm - T) ~ijJdx

(8.20)

(8.2l)

Acknowledgement: The support of this work by Grant 74-2660 from theU.S. Air Force Office of Scientific Research is gratefullyacknowledged. I also wish to make a special note of thanks to GonzaloAlduncin, who read the entire manuscript and made many helpful suggestions.

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66

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v ~ ; ~

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"

..

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