Chapter 1 The Classical Variational Principles of …This chapter deals with a general theory of variational methods for linear problems in mechanics and mathematical physics. This

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Chapter 1

The Classical Variational Principles ofMechanicsJ. T. Oden

1.1 INTRODUCTION

The last twenty years have been marked by some of the most significantadvances in variational mechanics of this century. These advances have beenmade in two independent camps. First and foremost, the entire theory ofpartial differential equations has been recast in a 'variational" frameworkthat has made it possible to significantly expand the theory of existence.uniqueness. and regularity of solutions of both linear and nonlinearboundary-value problems. In this regard, the treatise of Lions and Magenes I

on linear partial differential equations, the works of Lions.2 Brezis.3 andBrowder4 on nonlinear equations. and of Duvaut and Lions5 and Lions andStampacchiatJ on variational inequalities should be mentioned. Secondly, theunderlying structure of classical variational principles of mechanics arebetter understood. The work of Vainberg' led to important generalizationsof classical variational notions related to the minimization of functionalsdefined on Banach spaces, and an excellent memoir on variational theoryand differentiation of operators on Banach spaces was contributed byNashed.8 Generalizations of Hamiltonian theory were described by Noble9

and Noble and Sewell 10 and this led to generalization of the concept ofcomplementary variational principles by Arthurs 1 1 and others. TontiI2

•13

showed that many of the linear equations of mathematical physics share acommon structure that leads naturally to several dual and complementaryvariational principles. and these notions were further developed by Odenand Reddy.14 A quite general theory of complementary and dual variationalprinciples for linear problems in mathematical physics was given in themonograph of Oden and Reddy,15 and a more complete historical accountof the subject, together with additional references. can be found in thatwork.

This chapter deals with a general theory of variational methods for linearproblems in mechanics and mathematical physics. This work is partlyexpository in nature, since one of its principal missions is to develop, in arather tutorial way, complete with examples, a rather general mathematical

2 Energy Methods in Finite Element Analysis

· }.

framework for both the modern theory of variational boundary-value prob-lems and the 'classical" primal, dual, and complementary variational princi-ples of mechanics. However, many of the ideas appear to be new, andgeneralized Green's formulas are given which make it possible to furthergeneralize the recent theories of Oden and Reddy.I5

1.2 MATHEMATICAL PRELIMINARIES

1.2.1 Transposes and adjoints of linear operators

It is frequently important to distinguish between the transpose of a linearoperator on Hilbert spaces, and it~ adjoint. We set the stage for thisdiscussion by introducing some notation.

Let 011, 'Y be Hilbert spaces with inner products (Ub U2) and (VI. V2),respectively.

Let q/, 'Y' be the topological duals of 011 and 'Y, i.e. the spaces ofcontinuous linear functionals on 011 and 'Y. respectively.

Let ('. ')<11, (' •• )". denote duality pairings on Oft x 011 and 'JI" x 'Y, respectively,i.e. if I E Oft and q E 'JI" we write

I(u) = (I, u)", and q(v) = (q, v)".

We shall denote by A a continuous linear operator from 011 into 'Y.Now, following the standard arguments. we note that since Au E 'Y, the

linear functional q(Au) = (q. Au)". also identifies a continuous linear func-tional on au. i.e. a correspondence exists between q E'JI" and the element<; ofthe dual space Oft. We describe this correspondence by introducing anoperator A': 'JI" - ifl.' such that

(A'q)(u)=q(Au) or (A'q. u)",=(q, Au)". (L1)

The operator A' is called the transpose of A; it is clearly linear andcontinuous from 'JI" into au'.

Now the fact that OU and 'Yare Hilbert spaces allows us to enter into aconsiderable amount of additional detail in describing A' and other relation-ships between OU and 'Y. Since au is a Hilbert space, we know from the Rieszrepresentation theorem that for every linear functional I E Oft there exists aunique element UI E au such that

VUE6i1

Indeed. this relationship defines an isometric isomorphism K",: au - Oft suchthat

( 1.2)

....!

The Classical Variational Principles of Mechanics 3

and K", is called the Riesz map corresponding to the space 011.Similarly, ifK". is the Riesz map corresponding to the space 'Y, we have (K,...vu. v)". =(vo. v),... VVn. V E 'Y. In view of the definitions of the Riesz maps. we see that

VVE'Y,UEOUand

VVE'Y. UEOU

Thus, we discover in a very natural way that for each continuous linearoperator A : au -+ 'V satisfying the above relations there corresponds anoperator A * :'Y - au given by the composition

(1.3)

such that

( \.4)

The operator A * is linear and continuous and is called the adjoint of A.The following theorem establishes some important properties of the

transpose.

Theorem 1.2.1 Let A' E !t(r'. CU)(here :t('JI", au') is the space of contilluouslinear operators mapping r' into au') denote the transpose of a cO/ltilllwusli/lear operator A mapping I-Iilbert spaces au into 'Y. Theil tlte following hold:

(i)X(A')=~(AV

where X(A') is the /lull space of A' and ~(A).L is the orthogonal cOlllpleme,zrof the range ~(A) of A i/l 'JI".(ii) A' is injective if and only if ~(A) is dense in 'Y.

Proof: This is a well-known theorem; see. for example, Dunford andSchwartz. 16 •

t .2.2 Pi\'ot Spaces

The Riesz map K",: au - Oft is an isometric isomorphism from au Onto its dualOft. Consequently, it is possible to identify au with its dual. In many instances.particularly in the theory of linear boundary-value problems, we encountercollections of Hilbert spaces satisfying ou,.. c t51J....-l C ... c ifl.z c au( c 0/.4" theinclusions being dense and continuous. When one member of this set isidentified with its dual, say lll4" we call it the pivot space, and write

lll4,=%


·'

( 1.6)

( I.7)

The term 'pivot' arises from the fact that

... C OU2 C OU1 C % == OU{) c OIl! c ~ c ...

i.e. 1114) provides a 'pivot' between the spaces OUj, i ~O, and their duals.

1.2.3 Sobolev spaces

The notion of a Sobolev space is fundamental to the modern theory ofboundary-value problems. and most of the applications of our theory can bedeveloped within the framework of Sobolev spaces.

Let 0 be a smooth,t open. bounded domain in R". We shall denote byH"'(n) the Sobolev space of order m, which is a linear space of functions (orequivalence classes of functions) defined by

HIlI(n) = {ll : Il and all of its distributional derivatives

D"'u of order ~m are in L2(O). m ~ O} (1.5)

Here we employ standard multi-index notations (see, Oden and ReddyI9).Clearly, HU(n) = L 2(0).

Now the Sobolev spaces (l.5) are Hilbert spaces. Indeed, HIlI(O) IS

complcte with respect to the norm associated with the inner product

(ll, V )11"'(!ll == 1 L D"'IlDav dxn 1a1"1lI

and the norm on HIlI(!}) is

IlllllH'''(n)== J(u. U)H~(o) = { L IIDaIlIlL(n)}~lalc",

The subspace of HIlI(n) consisting of those functions in HIlI(n) whosedcrivatives of order ~m - 1 vanish on the boundary ao of 0 is denotedH;;'(O):

H~t(O)={ll E H"'(n) :D"'llian =0, lal~ m -I} (1.8)

Equivalently. H~'(n) can be defined as the completion of the space Lo(O) ofinfinitely differentiable functions with compact support in 0 with respect tothe HIlI-Sobolev norm defined in (1.8)

The so-called negative Sobolev spaces are defined as the duals of theH~\(O) spaces:

H-"'(O) = (H~\(m)', m~O ( 1.9)

t We shall assume throughout that n is simply-connected with a C"'-boundary an. However.most of the results and examples we cite subsequently hold if an is only Lipschitzian. See, forexample. Nceas 17 or Adams.ls


The Sobolev spaces are important in making precise the 'degree ofsmoothness' of functions. The following list summarizes some of their mostimportant properties:(i)

Indeed.m;:a:k;:a:O (LlO)

(1.11)

The remarkable fact is that these inclusions are continuous and dense.Indeed, H"' (n) is dense in Hk(n), m;:a:k;:a: 0, and IluIIHk(O)"'; IluIlH'"(O)(ii) Obviously, if m is sufficiently large, the elements of H"' (0) can beidentified with continuous functions. Just how large m must be in order thateach U E H"'(O) be continuous is determined by one of the so-called Sobolevembedding theorems: If 0 is smooth (e.g. if 0 c IR" satisfies the conecondition) and if m > n/2, then H"' (0) is continuously and compactlyembedded in the space CO(fi) of continuous functions.

(iii) Sobolev spaces H' (an) can be defined for classes of functions whosedomain is the boundary ao. There is an important relation between theseboundary spaces and those containing functions defined on O. We define thetrace operators 'Y; as

( 1.12)

i.e. they are the nomlal derivatives at ao of order ",;m -1. Let cp E L2(ilO)and define the norm1"

IIcpIIHm-H12(ao) == inf {lluIIHm({}); cP = 'Yju}lleHm(O)

(1.13)

The completion of L2(aO) with respect to this norm is a Hilbert space ofboundary functions denoted

H",-;-1/2(an), 0",;j",;m-1 (1.14)

Two important properties of these spaces are called the trace properties ofSobolev spaces:

(iii. 1) The operators 1'; can be extended to continuous linear operatorsmapping H'" (0) onto H",-i-I12(aO). i.e. there exist constants C;>0 suchthat

( 1.15)

t There are other more constructive ways of defining these spaces. See. for example. Oden andReddy. 19


,(iii.2) The kernel of the collection of trace operators 'Y = ('Yo.'YI' .... 'Ym-I) isH;;'(H) and H~'(H) is dense in HUl(n);

'Y;(Ho"(n)) == 0, j == O. 1, .... m - 1 (1.16)

1.3 GREEN'S FORMULAE FOR OPERATORS ON HILBERTSPACES

1.3.1 A general comment

One of the most important applications of the notion of adjoint<; of linearoperators involves cases in which it makes sense to distinguish betweenspaces of functions defined on the interior of some domain and spaces offunctions defined on the boundary of a domain. The introduction of bound-ary values is obviously essential in the study of boundary value problems inHilbert spaces. and it leads us to the idea of an abstract Green's formula forlinear operators.

In this section, we develop a general and abstract Green's formula whichextends those given previously in the literature. Our format resembles thatof Aubin,21l who developed Green's formulae for elliptic operators of evenorder. Our results involve formal operators associated with bilinear formsB :'Je x f§, 'Je and f§ being Hilbert spaces (see Section 1.3.3) and reduce tothose of Aubin when collapsed to the very special case, f§ = 'Je.

1.3.2 Abstract trace property

An abstraction of the idea of boundary values of elements in Hilbert spacesis embodied in the concept of spaces with a trace property. A Hilbert space'Je is said to have the trace property if and only if the following conditionshold:

(i) 'Je is contained in a larger Hilbert space au which has a weaker topologythan 'Je.(ij) 'Je is dense in au and au is a pivot space. i.e.

( 1.17)

(iii) There exists a linear operator y that maps 'Je onto another Hilbert space(J'Je such that the kernel 'Jeo of 'Y is dense in au, i.e.

( 1.18)

The space (J<Je corresponds to a space of boundary values, and theoperator 'Y extend., the elements of <Je, which can be thought of as functions

The Classical Variational Principles of Mechanisms 7

defined on the interior of some domain, onto the space of boundary valuesiJ;Je. The operator 'Y is sometimes called the trace operator.

The spaces H'" (D.) a.nd the operators 'Yj in (1.12)-( 1.16) are examples:H"'(D.) is dense in ~(n), m ~O. L2(n) can be identified with its dual,extensions of the trace operators 'Yj of (l.12) map H'" (n) onto the boundaryspaces H'"-I-1/2(iJD.). O~ j ~ m -1, and ker 'Y == ker( 'Yo, Yt ....• 'Ym-l) ==H;;'(D.) is dense in L2(!1).

1.3.3 Bilinear forms and associated operators

Let 'J{ and ~ denote two real Hilbert spaces (the extension of our results tocomplex spaces is trivial), and let both 'J{ and ~ have the trace property. i.e.

'J{cau==q(c'J{',

'Y : 'J{ - iJ'J{,

ker 'Y = ;Jeo,

'J{oc au = q( c ;Jeo.( 1.19)

The inclusions 'J{c au, ~c 'Y. ;Jeoc au, ~IlC 'Y, are dense and continuous.Next, we introduce an operator B which maps pairs (u, v). U E'J{ and

VE~. linearly and continuously into real numbers:

(1.20)

We denote the values of B in IR by B(u. v). and we refer to B as acontinuous bilinear form on ;Jex~. That B is bilinear mcan that

B(aUt+{3u2,v)=aB(uI,v)+{3B(u2.v) } (1.21)B(u, aVI + (3v:J = aB(u. VI) + (3B(u. V2)

Vu. ul, U2E'J{. v, VI' V2E~, a, {3 EIR. That B is continuous means that it isbounded. i.e. there exists a positive constant M such that

B(u, v) ~ M111l11J1u1k. Vu E'J{. VVE~ ( 1.22)

Now let Il be fixed element of 'J{ and let v E~o. Then B(u. v) describes acontinuous lincar functional l.. on the space ~ll for each choice of U E

'J{: B(u. v) = l,.(v). V E~o' The linear functional l.. depends linearly andcontinuously on u. and we describe this dependence in terms of a linearoperator All == I... Thus

B (u. v) = (Au. v )~. v E ~o (1.23)

The operator A is called the formal operator associated with the bilinear fonnB. Clearly

A E !t('J{, ~o) ( 1.24)


In a similar manner, if we fix VElO, B(u, v) defines a continuous linearfunctional on <Jeo, and we define a continuous linear operator A * by

B(u, v)=(A*v. U);K, u E <Jeo (1.25)

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

A * E !t(lO. <Je[I) (1.26)

The fact that <Jeo = ker -y and lOo= ker -y* enables us to establish the follow-ing fundamental lemma.

Theorem 1.3.1 Let <Je and lO denote the Hilbert spaces with the traceproperties described above, and let B denote a continuous bilinear form from<Jex lO-IR with formal association operators A E !£(<Je. lO~) alld A * E !£(lO, <Je~).Moreover, let <JeA and lOA- denote subspaces

<JeA = {u E<Je: Au E 'Y} ('Y = 'Y' c: lO~) } (1.27)'&A. = {v E lO: A *v E 6U} (OU = OU'c: ile~)

Then there exists uniquely defined operators 8 E !t(<JeA, iYd) alld 8* Ef£(lOAo• ilile') such that tlte following formulae hold:

B(u,v) = (v,:U)y+(811.*-y*V)iI'§' UE<JeA, VElO } (1.28)B(u.v)==(A v.u)",+(8 V.-YU)il:K' UE<Je. VElOAo

Here ('. ');1:K and ('. ')iI'§ denote duality pairings on aile'xiI<Je and a<dxiIlOrespectivel y.

For a proof of the theorem. see Oden.22

Let B :<Jex lO-IR be a continuous bilinear form on Hilbert spaces <Je andlO having the trace property. Let in addition, A be the formal operatorassociated with B('.·) and let A* be its formal adjoint. The operators'Y E !£(<Je, aile) and 8 E !£(<JeA, a<d) described above are called the Dirichletoperator and the Neumann operator, respectively, corresponding to theoperator A. Likewise, tbe operators -y*E!£(lO,alO) and 8*E!t(lOA-.aile') de-scribed above are called the Dirichlet and Neumann operators. respectively.corresponding to the operator A *.

1.3.4 Green's formulae

VElO }VElOA* (1.29)U E <Je,

The relationships derived in Theorem 1.3.1. are called Green's formulae forthe bilinear form B(', .):

B((~, v) == (v, AU)y+(8u, -Y*V)"6'

B(u, v) == (A *v, u)",+(8*v, -yu);J{,


(1.31)

If we take UE:/{A and VE<§A*' we obtain the abstract Green's formula forthe operator A E fe(:/{, <§h) n fe(:/{ A' 'V):

(A *v, u)",= (v, Au)y+(Su, )'*V)iIIf-(S*,', )'u)ax, U E'lIeA, VE<§A*

( 1.30)The collection of boundary terms,

r(u, v) = (Su, )'*v)aw - (S*v, )'u)ax

is called the bil inear concomitant of A; r: 'lieA X <§A. - R

Example 1.3.1 Consider the case in which n is a smooth open boundedsubset of lW with a smooth boundary an and

OU='Y=~(n)

Let a=a(x,y), b=b(x,y), and c=c(x,y) be sufficiently smooth functionsof x and y (e.g. a, b, CE Clai)), and define the bilinear form B: H1(n) xH1(n)_1R by

B(u,v)= 1(aVu·Vv+bvux+cvUy)dxdy

where Vu = grad u = (u", fly). Thus.

a'lle=(j<§= Hl/2(an); ker)'= HMn}={u EHI(n); u =0 on an}

In this case,

1 1 auB(u,v)= v(-V'(aVu)+bux+cUy)dxdy+ a-vds

00 a1lThus, the formal operator corresponding to B (., .) is

au auAu =-V· (aVu)+b-+c-ax ay

if we take

Also, we now have

aulSu=a- .an an

Similarly. if v E '!)A * = 'lieA' we have

-y*v = vlan

abv acv) dx dy + )-1 u(-V'(aVv)-ax-- ay 1 ( aV+bvunx+cvUtly dsB(n, u)- + au an

<In

10 Energy Methods in Finite Element Analysis\

where n", fly arc the components of the unit outward normal n to ao. Thus,

* abv acvA v=-V· (aVv)----ax iJy

-yu = ulan, S*v = [a iJv + (bn" + C1ly)v] Ian an

The bilinear concomitant is then

f [au av ]r(u,v)= av--au--(cfly+bn,,)uv ds •an anExample 1.3.2 Let 0 be as in Example 1.3. I and define

B(u.v)= 1gradu 'vdxdy

as a bilinear form on ':f{ x t:§. where

':f{ = Hl(O) = {u: u, u.:, lly, E L2(0)}

t:§=Hl(O)={v=(vl' vz): VI, Vlx' Vty, V2, V2x' V2y EL2(0)}

H 1(0) is dense in au = L2(0) and W (0) is dense in 'Y = L:z(0) =L2(0) x L2(0). In this case. we may take -y*v= v . nlan' but S = O. Indeed, ifthe formal operator associated with the given bilinear form is

Au = grad u

then '!leA = '!Ie = Hl(O) and CA : '!leA - '!l* is identically zero. We next notethat

B(u. v)= 1u(-divv) dx dy +" V· nu dsInThus. t:§A. = Hl(O) and

A *v= -divv; S*v=v' nlan

The Green's formula is the classical relation,

"(gradu'v+udivv)dxdy='c' 14v'nds •tl til1.3.5 Mixed boundary condition

An abstract Green's formula appropriate for operators with mixed boundaryconditions can be obtained by introducing some additional operators andspaces.

Let


(1.32)

71'f = a continuous linear projection defined on if§into itself

7T~= 1- 7rf. I being the identity map from ac§

onto itself

1'f = 7rf-y*, 1'~ = 7rh* and 1'* = 1'f + 1'f

.1= ker 1'f == {v E C§: 7Tf1'*v =O}

The space .1 is a closed linear subspace of C§ with the property

C§() c .1c 'fJ (1.33)

The operators 1'f and 1'~ effectively decompose ifO into two subspaces

aC§1== 1'f(C§), if§2 = 1'f(~)} (1.34)a~ = ifO. +iYB2

VEj } (1.36)U E[¥

A similar collection of operators can be introduced on a~:7T1= a continuous linear projection of a~ into a~7T2= l'-71'1(1': a~ - a:Jl)

1'1 = 71'11', 1'2 = 71'21', l'= 1'1 + 1'2

[¥ = ker 1'2 = {u E ~: 7r21'U = O}~oc ~ C 'l/{

a~t't=1'I(~)' a~2=1'2(:Jl), a:Jl=a:Jll+a~2

The Green formulae (1.28) now yield

B(u. v) = (v, AU)y+(8u, 1'!V>ifN,.B(u. v) = (A *v, u)",+(8*v, 1'IU>a:N'"

We observe for v E .1,

(8u, 1'*V>OIf= (5u, 7rf1'*V)ifif. +(5u, 7Tf1'*V)OIf,

= (5u. 7r!1'*V)ifif,

= (7T*' 5u 'V*v) = (5 u 'V*V>2 ' r if§, 2. r 0If,

( 1.35)

(1.37)where

82 = 7r~'8, 52 E ft(:Jl A' if§~) (1.38)

Finally, collecting (1.36) and (1.37), we arrive at the abstract Green'sformula,(A *v, u)<1I= (v, AU)y-(5*v, 1'IU)a:N',+(52u, 1'*v)a:N',

Vu E:JlA n~, VE 'OA' n.1 (1.39)


1.4 ABSTRACT VARIATIONAL BOUNDARY -VALUE PROBLEMS

1.4.1 Some linear boundary-value problems

The Green's formulae and various properties of the bilinear forms B(', .)described in the previous section provide the basis for a theory of boundary-value problems involving linear operators on Hilbert spaces. We shallcontinue to use the notations and conventions of the previous section: 'll and<IJare Hilbert spaces with the trace property, densely embedded in pivotspaces au and 'Y, respectively, and y*:<IJ-a<IJ. kery*=<lJo, y:'ll-a'll,ker y = 'llo, etc.

Let B : 'll x <IJ-IR be a continuous bilinear form and let A be the formaloperator associated with B(u, v). We consider three types of boundary-valueproblems associated with A.

(1) The Dirichlet problem for A. Given data f E 'Y and g E a'll, the problemof finding u E'll A such that

Au = fl (1.40)yu = g1

is called the Dirichlet problem for the operator A.

(2) The Neuman problem for A. Given data f E 'Y and s E Cfd, the problemof finding U E 'JeA such that

Au = fl (1.41)8u =s 1

is called the Neuman/J problem for the operator A.

(3) TIle mixed problem for A. Given data f E 'Y, g E a'Jel' and s E a<IJ2, theproblem of finding U E 'JeA such that

AU=f}YIU = g (1.42)

82u == s

is called the mixed problem for the operator A.

Now the bilinear form B :'llx<IJ-1R described in the previous section can beused to construct variational boundary-value problems analogous to thosefor A. We shall consider the following variational problems:

(1) The variational Dirichlet problem for A. Given data f E 'Y and g E a'll,find W E 'Jeo = ker y such that

B(w, v) == (f, ")".- B(y-1g, v) 'v'VE<lJo (1.43)where y-1 is a right inverse of 1'-

Tlte Classical Variational Principles of Mechanics 13

(2) TIle variational Neumann problem for A. Given data IE'Y and S E ifd,find u E'Jt such that

B(u, v) = (f, v),..+(s, y*v)a<.§VVE C§ (1.44)

(3) The variational mixed problem for A. Given data f E 'Y, g E iJ'Jt1• ands E~, find WE ker 'Yl = ker 7T1Y such that

B( w, v) = (f. v),..- B( Yll g, v)+(s, y*v)a<.§~VVE,j (1.45)

where Ylt is a right inverse of Y1 and j!i=keryf=ker7T1Y*.

Remark. There are, of course. several other abstract boundary-value prob-lems for the operator A that could be mentioned. For example, a secondbilinear form b :iJ'Jtx iJC§ -IR could be introduced at this point which wouldpermit the construction of oblique boundary conditions and which wouldlead to a formulation more general thall (1.42). However, such technicalgeneralizations obscure the simple structure of the ideas we wish to clarifyhere, and so they are omitted. •

Theorem 1.4.1 The Dirichlet problem (1.40) for the operator A and tltevariational Dirichlet problem (1.43) are equivalent in the following sense. Lety-1 be the inverse map of Y; if u is a solution of (1.40) then w == u - y-lg is asolution of (1.43). Conversely, if w is a solution of (1,43), then u == w +y-1g isa solution of (1.40). Moreover. the Neumann problem (1.41) for the operatorA is equivalent to the variational Neumann problem (1.44) in the sense t/tatany solution of (1.41) is also a solution of (1.44) and, conversely, any solutionof (1.44) is a solution of (1.41). Likewise, problems (1.42) and (1,45) areequivalent in a similar sense. •

For a proof see Reference 22.

Example 1.4.1 Let 'Jt=<§=HI(o.), OU=='Y=L2(o.). 'Jto=c§o=HMO}, andiJ'Jt=iJ<tJ=HI12(ilO), where 0. is a smooth open bounded domain in~. Thebilinear form

B(u,v)== 1(Vu ·Vr;+alw)dxdy

where a is a non-negative constant, is a continuous bilinear form fromH1(o.) x H1(!l) into IR. The formal operator associated with B(' .. ) is A ==-Ll+a. where Ll=V . V = V2 =a2/iJx2+a2/iJy2 is the Laplacian operator. Wedenote


The Dirichlet problem for A is to find U E HI(A, n) such that

-all +au == f in n.u = g on an,

fE ~(n)g E H1/2(an)

In view of Theorem 1.4.1, this problem is equivalent to the problem offinding W E HMn) such that

fa (Vw ·Vv+awv)dxdy= ifvdxdy

-i(Vwo·Vv+awov)dxdy Vv E H~(n)

-all +au = f in n,au an,-=s onan

where Wo is any function in H len) such that Wo = g on an. Then II = W + Wo isthe solution of the Dirichlet problem for A.

The Neumann problem for A is to find U E HI(A, n) such that

fEL2(n)

and it is equivalent to seeking U EHI(n) such that

In (VU . Vv +auv) dx dy ==itv dx dy +tn sv ds Vv E Hl(fl)

where the contour integral tan denotes duality pairing on (H 1/2(an))' xH1/2(an). •

We observe that boundary conditions enter the statement of a variationalboundary-value problem in two distinct ways. The essential or sta/JIe bound-ary conditions enter by simply defining the spaces 'Ito and '00 on which theproblem is posed. The natural or u11Stable boundary conditions are intro-duced in the definition of the bilinear form B(u. v) and are defined on thespaces 'itA and 'OA*'

1.4.2 CompatIbility of the data

We shall discuss briefly here the issue of the compatibility of the data f, g, swith various boundary-value problems. The ideas are derived from theclassical theorem (recall Theorem 1.2.1).

Theorem 1.4.2 Let A be a bounded linear operator from a Hilbert space 'J{

infO a Hilbert space '0. Let A * be its adjoint. LeI .N'(A), .N'(A *) denote the null


spaces of A and A * respectively and ~(A) and ~(A *) denote the ranges of Aand A * respectively. Then

~(A).1. = .N'(A *),

~(A *).1.= .N'(A),

~(A) = .N'(A *).1. }

~(A *) = .N'(A).1.(1.46)

where 1. denotes the orthogonal complement of the spaces indicated and anoverbar indicates the closure.

Proof: For a proof see, for example, Taylor.2l•

We shall first address the compatibility question in connection with theDirichlet problem for the operator A: find U E ';}{A such that

Au = f, fE'Y

yu = g, g E a';}{

Let A * E ~(/§A" ';}{ij) denote the formal adjoint of A and let 1'* E !£(/§, fM).ker 1'* = /§o. Then the adjoint Dirichlet problem corresponding to A is theproblem of finding v E/§A* such that

A*v==f*,

y*v=g*,

We also introduce the null spaces,

.N'(A, 1') = {u E ';}{A: Au = 0, yu = O}

.N'(A*, y*)={VE/§A*:A*v==O. y*v=O}

We shall assume that these spaces are finite-dimensional.Now if .N'(A, 1') is finite-dimensional, it is closed in au and we have

whereau = .N'(A, y)Etl.N'(A, 1').1.

.N'(A, 1').1.= {u E au: (u. v)", = 0 Vv E .N'(A, y)}

Then A can be regarded as continuous linear operator from .N'(A, 1').1.ontoits range !17i(A) c 'Y. By the Banach theorem, a continuous inverse A-I existsfrom gil(A) onto .N'(A, 1').1.,and gil(A) is closed in 'Y. Thus 'Y=gil(A)E9~(A).1.where !fl(A).i.={VE'Y:(V, w)y==OVwE~(A)}. The Dirichlet problem for A.clearly has at least one solution whenever the data fEf¥t(A) and gE~(Y).Data satisfying these requirements are said to be compatible with theoperators (A, 1').

A convenient test for compatibility of the data is given in the followingtheorem.


Theorem 1.4.3 Let f E 'Y and g E a:'le be data in a Dirichlet problem for theoperator A. Then a necessary and sufficient condition that there exists asolWion of the Dirie/llet problem is that

(f, v)y-(S*v, g)ax= 0

For the Neumann problems

Au = f,Su = s.

we define

'Iv E J((A *, ,,*)

A*v=rS*v = s*

(1.47) •

J(A, S) ={u E:'IeA: Au = O. Su == O}

J((A*, S*)={VE§Ao: A*v=O, S*v=O}

and, analogously, have

Theorem 1.4.4 A necessary and sufficient condition for the existence of atleast one solution of the Neumann problem for the operator A is rltat tlte data(f. s) satisfy

(f, v)y+(s, ,,*v)il'§=O 'Iv E J((A *, s*) (1.48) •

A similar compatibility condition can be developed for mixed boundary-value problems:

Theorem 1.4.5 A necessary and sufficient condition for the existence of asolution (0 the mixed boundary-value problem for A is that the data (f, g. s)sutisfy

where

(f, v)y- (S*v, g)a:I(, + (s, y*v)il'82

== 0 'Iv E J((A *, "f. Sf) (1.49)

J(A *, "t,Sn = {v E<§Ao: A *v = 0, "tv == 0, sfv = O} (1.50) •

Whenever the compatibility conditions hold. a solution to the Dirichlet,Neumann, or mixed problems may exist but it will not necessarily beunique. The solution u is unique. of course, whenever J((A. ,,) = {a}, for theDirichlet problem, whenever J(A, S) = {O}for the Neumann problem. andwhen J((A,,,, S) ={O} for the mixed problem. However, these conditionsoften do not hold. We can, however, force the solutions to either class ofproblems to be unique by imposing an additional condition.

Theorem 1.4.6 Let u E:'IeA be a solution to the Dirichlet problem for theoperator A. Then u is the only solwion to this problem if

Vw E J((A. ,,) ( 1.51)


Likewise, a solution u of the Neumann (mixed) problem for A is unique if(u, w)",=O\fwE.N'(A, 8)(yWE.N'(A, 1', 8», respectively .•

1.4.3 Existence theory

The theory presented thus far suggests the following general setting forlinear variational boundary-value problems.

-Let iJe and f§ be arbitrary Hilbert spaces (now iJe and f§ are not necessarilythe spaces appearing earlier in this section-they do not necessarily have thetrace property).-B :iJexf§-1R is a bilinear form. Then find an element u EiJe such that

B(u, v)=f(v)

where f E f§'.

The essential question here is: What conditions can be imposed so that weare guaranteed that a unique solution exists which depends continuously onthe choice of data f? This question was originally resolved for certainchoices of B by Lax and Milgram. A morc general form of their classictheorem made popular by BabuSka23 (see also Necas17 and Oden andReddyl9) is given as follows.

Theorem 1.4.7 Let B :iJe x ~ -IR be a bilinear functional on iJe x~, iJe and ~being Hilbert spaces, which has the following three properties.(i) There exists a constant M> 0 such that

B(u, v).,.;M1lullJlvllw \fu EiJe. v Ef§

where 1\·lbc and 11·11'6 denote the nomts on iJe and f§, respectively,(ii) There exists a c011Stant l'>0 such that

inf sup IB(II, v)I~'Y>OUE~ UEVJ

lIu11.r= 1 Uull.... 1

(iii) We have

sup IB(u, v)I>O. v;cOUEX

Then there exists a unique solution to the problem of finding u E iJe such that

B(u, v) == f(v) \fv E<§, fE~'

Moreover, tlte solution Uo depends continuously on the data; itt fact,

•


Property (i) B(',') is, of course, a continuity requirement; B(·,·) is assumedto be a bounded linear functional on 'if{ and on CfJ. Properties (ii) and (iii)serve to establish the existence of a continuous inverse of the associatedoperator A.

Corollary 1.4.8 Let B ::1L'X'if{-1R be a bilinear form on a real Hilbert space'leo Let there exist positive conSlants M and 'Y such that

B(u, v)~MlluIlJlvl~B(u, u);;;''Yllulli-

Then there exists a unique u E'le such that

'tIu, v E'le

1Ilull:rt~-llfll1(''Y

andB(u, v) == f(v) 'tIvE'le,

•1.5 CONSTRUCTION OF VARIATIONAL PRINCIPLES

Let W be a real Banach space and W' its topological dual. If (jJ is anoperator from W into W', not necessarily linear, we may consider theabstract problem of finding u E W such that

(jJ(u)=O, DE W' (1.52)

Now it is well known that in many cases an alternative problem can beformulated, equivalent to (1.52), which involves seeking a liE W such that

K'(u)=()

where K is an appropriate functional defined on Wand K'(u) is theGateaux derivative of K at u, i.e.

lim~(K(u+€'Y/)-K(u»=(K'(u), 'Y/)€~()E

where (".) denotes duality pamng on W' x W. Thus, if there exists aGateaux differentiable functional K: W -IR such that

(jJ= K' (1.53)

then (1.52) is equivalent to the classical variational problem of findingelements U E W which are critical points of the functional K. We say that (jJis the gradient of K and we sometimes write (jJ= grad K.

Any operator (jJ: W - W' for which there exists a functional K: W -IRsuch that ~ = grad K is called a potential operator. It is well known (see, forexample, Vainberg' or Nashed8) that if a continuous Gateaux differential


8~u, 11) of fj> exists, then a necessary and sufficient condition that fj> bepotential is that it be symmetric in the sense that

(8~(u, 11), () = (8~(u, (l, 11) VU, TJ, ~E W

(1.54)

Given a potential operator (i}: W - W', the problem of determining afunctional K such that (1.53) holds is called the inverse problem of tltecalculus of variations. Its solution is provided by the following theorem, theproof of which can be found in the monograph of Vainberg'.

Theorem 1.5.1 Let fj>: W - W' be a potential operator on the Banach spaceW. Then ~ is the gradient of the functional K: W -IR given by

K(u)= r (~(uO+S(U-II()). uo)ds+Ko

where Uo is a fixed point in W, Ko=K(uo). and SE[O, 1]. •

By an appropriate identification of the space Wand the duality (', .), all ofthe classical variational principles of mathematical physics can be con-structed using (1.54). A lengthy list of applications of (1.54) to this end canbe found in Chapter 5 of Oden and Reddy 17.

1.6 THE CLASSICAL VARIATIONAL PRINCIPLES

1.6.1 A general class of boundary-value problems

We shall now describe a general class of abstract boundary-value problemsthat is encountered with remarkable frequency in linear problems ofmathematical physics and mechanics. Continuing to use the notation of theprevious section, we introduce a linear, symmetric, operator

(1.55)

which effects a continuous, isometric isomorphism of the dual of the pivotspace Y onto 'Y, which has a continuous inverse, E-1

: 'Y - 'JI". If v E 'JI", weshall denote the elements in E('JI") by

(T = Ev (1.56)

The Green's formula (1.30) can now be written

(A *(T, u)",= (Au, 0")y+(8*0", 'Y1Il)<UI':,-(Suo Y~(0")~2 Vu E'f/{A,

(1.58)

(1.60)


Now let us consider the following abstract problem. Given f E q(. g E iJ:1l't,and s E ;m2, find U E'ltA such that

A*EAu =f }Y1U = g

'YfEAu =swhere we have used the notations of (1.32) and (1.35). We shall exploit thefact that this problem can be rewritten in the following canonical form: find

(U,V,O")E'ltAXYX<OA. (1.59)such that

Au=v Y1U=g}Ev = 0"

A *0" = f yfO" = s

We would now like to construct a variational principle (i.e. a potentialfunctional) corresponding to (1.60). Towards this end, we introduce thematrix operator

o -1 A 0 0 0" 0-1 E 0 0 0 v 0

A* 0 0 0 0 u = f0 0 0 0 7Tl *0" g

0 0 0 7Tf 0 yu S-or~A)=f

(1.61)

(1.62)

where r/' is the coefficient matrix of operators in (1.61),

AT = (0", v, u, Y*O", Yll) E <OA. X Y X 'itA X iJ<O X iJ'J{

fT = (0,0, f, g, s) E Y X'Y xq( xa'ltl X if92

If'W = 'Y x'JI" X OUx a'Jt X if§'

and(', .),....=(', ')y+(', ')".'+(', ')<11+(', ').w;+(', ')a'§;

We compute easily the functionaL

L(u, v, 0") = r (rJ'(SA) - f, A.),.... ds1 A A= 2(r/'(A), A),.... - (t. A),....

= ~-v+ Au, (T),...+~-o"+ Ev, v)y+!(A *0'. u)",

+~7TIYU, S*O")il:ll!";+!(7T~Y*O", SU)ilW;

-(f, u)",-(g, S*O")il:ll!";-(s, SU)a'§2


wherein

~T = (a, v. u, cS*a, cSU)E'YxYxouxa1t"xa<d

Applying Green's formula to the term 1(A *a, u)", gives

I. L(u, v, a) =!(Ev, v)"..+(Au -v, a)y-(f, u)",

+(Y1l4 - g, cS*a)ilK) -(s, cSu)il't9; (1.63)

The Euler equations are (1.60). Indeed,

cSL(u, v, a; ii, V, a) = (Ev, v)y.+(Au - v, o')y- (f, u)<fl+ (YI u - g, cS*O').wi

- (s, cSu)~ +(Au -v, a)y+(YIU, cS*a)aK.

= (Ev-a. v)y·+(All-v, o-)y+(A *a- t, in",

+ (Y1 u - g, cS*O')ar. + (y!a - s, cSu)il't92

where we have applied Green's formula into (Au, a)"..Next, we list a number of additonal principles that can be derived directly

from the functional L:n. Constraint:

R,.(u, a) = L(u, v(a), a)

= -~E-1a. a)". +Au, a)". -(f, It)<fl

+(cS*o', y,u -g)a"", -(cSu, s)iI'§,

Euler equations:

Au = E-1O' Y1U = g

A*O'=f y!a==sConstraint:

(t= Ev

Rv(u, v) = L(u, v. a(v»

= -~v, EV)y+(Au. Ev)".-(f, u)'ll

+(cS*O', Y1U -g)o'lK, -(cSu, s)il'62

Euler equations:

(1.64)

(1.65)

( 1.66)

(1.67)

(1.68)

Au=vA*Ev==f

( 1.69)

III. Constraints:

v==Au Y1U=g O'=EAl4 (1.70)

leu) = L(u. v(u), a(u» =~Au. EAu)".-(f, u)",-(cSu, S)iI'§2 (1.71)

22

Euler equations:

Energy Methods in Finite Element Analysis

A *EAu = f "Y~EAu = S (1.72)

8I(u, l-:t) = (Au, EAu)"..-(f. u)",-(8a, S)il'62

= (A *EAu, u)",-(o*Eau, "Y1U)il:N'1+(8u, "Y~EAu)a\f2

-(f. u)",-(8u, S)<l'62

== (A *EAu - f, u)",+(8u, "Y~EAu -S)a\f2; UE ker Y1

IV. Constraints:

Euler equations:

v == E-1a A *0' = f "Y~a = S

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

= _~E-1(J", 0')"..-(8*0', g)a:N',

(1.73)

(1.74)

(1.75)

8J(a, a) = -(E-1a, O')y-(8*O', g)a:K,+(Au, 0').".+(8*0', "Y1U)il:N'1 -(8u, "Y~a)a\f2-(A *0', u)",

= (Au - E-1a, O')y+(8*o-, "Y1U- g)a:K,; a E ker A * nker "Y~

V. Constraints:

Au=v

A *0' = fK(v) ==~(v,Ev)"..-(v, a),.,

Euler equation:

"Y1U=gyfa = s

a E {a: A *0' = f, "Yfa = s}

( 1.76)

(1.77)

Bv = a

VI. Constraints:

(1.78)

Au==v (1.79)

M(u, a) = (Au, EAu)y+!<E-1a, a).,...-(f, u)",

-(Au, 0)y-(5u, s)n<a,

Euler equations:

(1.80)

A *(2EAu -a) = f "Y~(2EAu -a) = s (1.81)

•

The Classical Variational Principles of Mechanics

VII. Constraints:

A *cr= f

N(v, cr)=!<v, Ev)y-(v, cr)y+(8*cr, g)iJ.:ll'1

Euler equations:

23

(1.82)

(1.83)

v = Au Y111 = g cr=Ev (1.84)

8N(v.u; v. a) =(v, Ev-cr)y+(v, O')y+(8*O', g)«<,

+(A *0', u)",-(Au, O')y+(8u, yfO')iJ'lP2

-(8*0', Y1U)«<,= (v, Ev-cr)y+(v- Au, O')y+(8*O'. g - Y1U)«<,

a E ker A * nker Y~

We summarize these results in Table 1.1.

Tabte 1.1

II Ra(u.a) (1.65) t' = E-IO'R,,(II.v) (1.68) 0'= Ev

11/ 1(11) (1.71)v=Au; 'Ylu=g

a=EAII

( 1.74)v=E-la

IV 1(0')A·O'=,; 'Y!O'=s

V K(v) (1.77)Au=v; 'Y.u=g

A·a='; 'Y!O'=s

VI M(II.O') ( 1.80) Au =v; 'YIII= g

vn N(v,lr) (1.83) A-O'=,; 'YrO'=s

FunClional

L(u. v. a)

Definition

(1.63)

Constraints Euler equations

Au=,,; 'Ylu=gEv=a

A ·Ir =,; 'Y!O'= s

Au = E-1a; 'YIU = gA-O'='; 'Y!lr=sAu=v·'Ylu=g

A-Ev='; 'Y!Ev=s

A·EAll =,; 'Y!EAu =s

Ev=a

All=E-IO'

A·(2EAu-a)=''Y·(2EAu -a) = s

v=Au: )'Iu=gO'=Ev

1.7 APPLICATIONS IN ELASTICITY

We now apply the theory in the previous section to the construction of sevenvariational principles of linear elastostatics.


·'

The Lame-Navier equations of linear elasticity are

(A *EAu)i = - (Eiirsu.,s).i == r in G }

('Y1U)i=U;=r1, on an](A~EAu)j = ni(Ei;rsur,s) = ti on a~

(1.85)

which correspond to equations (1.58). The canonical equations (1.60) arethe strain-displacement equations. kinematical boundary conditions, con-stitutive equations, and equilibrium equations:

(Au)'j =-21(JI .. + u··) = e·· in n·• -'.J J.' I' ,

(BE yi =Eiirse .. = (Tij in .0

(A *U)i =_(T.:i = fi in .0:

In this case,

('Y1U); == Uj = uj on aGI 1(1.86)

('Y~U)i = n·(Tii = tl on 00..,I ~

and

au=~(n)=={u:u1E~(G),i=1,2,3} }'Y = L2(n) = {u: (Tij EL2(n), i. j = 1,2, 3}

<Je = H 1(.0) = {u : 14iE H I(G), i= I, 2, 3} }

~= HI(n) = {u: (Til EH1(O), i, j = 1, 2, 3}

(1.87)

( 1.88)

Hence, the canonical problem of the elasticity can be expressed asfollows: given

(f, ii, t) E q( X a<Jel x a~2 = (L2(0)' x H 112(0.01) X H-I/2(a~) (1.89)

find

such that equations (1.86) are satisfied. Here

<JeA=H1(A,n)={UEW(n):AUELin)} (1.91)'DA- = H1(A*, O)={UE Hl(n): A *u EL2(On

The Green's formula (1.57) becomes, for this problem,

\fUE W(A. .0), VUEH1(A*,n) (1.92)

from which we can identify the operators l) E~(HI(A. 0),w-1/2(aG» and

",

•

The Classical Variational Principles of MecMnics 25

8* E ~(W(A *, !l), H-1/2(aO»:

(80~~=uI .. on a~} (1.93)(5*0')'J=_rljU" on aOl

Now we are able to construct the variational theory for linear elastosta-tics. According to Table 1.1, we obtain:

I. The Hu- Washizu principle

L( )- r 11 Ell" +[1( + ) ] il fi }dU, £, 0' - .10 UEil E" 2 ~I ~.i - Eij U - ~ x

Euler equations:

1(ui.1+ uI.;) = Eil

Eii"E" = uii

_(Ti!= fl.1

in 0;

\l1 0111 0.;

Uj = uj on an, }

njuij = 1'1 on a~(1.95)

II. The Hellinger-Reissner principle

(i) Constraints:E = C· (Tii(e. a (Eii"')-I)

rs 1'F3 .'n (1.96)

Ru(o, 0') ==i[-!Cii",uiiU'" +!(~.i + Uj.i)uii - f~] dx

-1 rljUii(~-Ui)ds-1 ujti ds (1.97)<ifl, <10;,

Euler equations:

!(~.i+lIl.i)=C"'iP"'" in 0:-u~:= f' in 0.:

(ii) Constraints:

Ui = U.rljuij = 1'/

( 1.98)

R. (0, £) = r [-!eiIEji"'E,..+!(~.I + ~)Eii"'e", - fu;) dxIn

-1 njuil(Ui - 14/) ds -1 uiT; dsdn. ~

(1.99)

(1.100)

26

Euler equations:

Energy Methods ill Finite Element Analysis

~(U;.I +~J= Eii in 0;-(Ei/"E,,) = r in fl;

III. Potential energy principle

Constraints:

U; ==ilin;Ei/"E" = yi

on aol}

on il~(1.101)

Eii =!(Uj.1 + ui.i) in 0; Uj =~(Tli =!Eij"(u,. ...+ us) in 0

(1.102)

Euler equations:

( 1.103)

-(Ei/"u )·=f· in fl'r,s ,I ,

IV. Complementary energy principle

Constraints:

(1.104)

E,. = CijrsUil in 0 }-u[! = fl in O· n·crij = 1'i on aO~.1 , I ..

J(u) = -llc. (TIIUn dx + i nuiiil· ds2: IJn I J'.JO,

Euler equations:

!(ur.s + us.,) == Cj/rsuj/ in n;

V. A constitutive variational principle

Constrain ts:

(1. 105)

(1.106)

(1.107)

!(Uj,/ + ul•l) = Ell in 0;-(T~I= r in n;

U; == ui

n;aii = yl (1.108)

K(E) == f GEijEiirse,. - Ei,.uii) dx

cTli E{aii : -u~!== f in n, Il;uii = 1'/ on a02lEuler equations:

(1.109)

(1.110)


VI. A c011Stilutive-potential energy principle

Constraints:

27

~(Lli.;+ Ui.l) == eli in !l; (1.111)

M(n, 0') =i['1 .. Eii"'u,. +lC. criicr'"-'.1 ,s 2 IJF'S

-!(tt;.i + u;.Jcrii - ftt;]dx- I uiTi ds (1.112)ilU

Euler equations:

~(u,. .• + u",) = Ci;",crij in !l-{Eil"'(ur•s + us)-cri;L = fni{Eii"'(u,. .. +Us,,)-cri;}=Ti on

VII. A compatibility-co11Stitutive variational principle

Constraints:

(1.113)

(1.114)

(1.115)

Euler equations:

gii =!(4.;+uiJ in !l;cril == E;;'se,. in !l

on on, 1 (1.116)

1:8 DUAL PRINCIPLES

1.8.1 The dual problem

It was pointed out by Oden and Reddyl5 (see also Reference 14) that aparallel collection of so-called dlwl variational principles can be constructedin one-to-one correspondence with the variational principles described inthe previous section. While we shall not elaborate on the detailed features ofthese dual functionals, we shall outline briefly the essential concepts for thesake of completeness.

To construct the dual principles, we consider a new Hilbert space Yanda linear operator

(1.117)


such that

91i( C) => X(A *)}X( C*) c 91i(A)

(1.118)

Next, we consider the dual problem of finding cp E g such that

(1.119)

where

TJEY', (1.120)

Owing to the similarity of (1.119) and (1.58), we can immediately con-struct the following dual functionals:

I'

II'.

.!£(cp, a, v) == !<E-1a, u)Y' + (v. Ccp - a)Y'-(TJ, cp)9'+ (S(C)"0, yr) cp- P )iY6;- (q, SiC)cp)a9',

92.,(cp, v)=-!<v, Ev)".,+(v. Ccp)y·-(TJ, cp)9'+(S<Cl*u, y\Clcp - p)il'II;-(q, SCC)cp)a9',

(1.121)

(1.122)

or

91i" (cp, a) = -~(E-Ia, a),. + (E-la, Ccp)Y' - (TJ, cp)9'

+(SCC)"E-ta, y\Clcp - p)iY6;-(q. 8CC)CP)il9',(1. 123)

( 1.124)

(1.125)

(1.126)IV'.

V'.VI'.

g(cp) == ~(E-ICcp, Ccp)"'-(TJ, cp)9'-(q, 8(C)CP),1..'I'2

~(v) = --!<v, Ev )y,-(S(C·)· v, p )iY6;

X(a) = !(E-1a, a)y.-(u, a)y.

Al(cp. v) = !(E-1 Ccp, Ccp)y. +~(v. Eu)Y' - (TJ, cp)9'

-(v, Ccp)y.-(q, 8(C)CP),w2 (1.127)

VII'. X(a, v) = !(E-ta, a)'Y' -(v, u)y.-(S(Cl· v, P)wDi (1.128)

Clearly, a table similar to Table 1.1 can also be constructed directly from(1.121)-(1.128) by using the following correspondences:

III'.

CfJ'-CfJ, g-'J(, cp-µ, C-A, }C*-A*, E-1_E, a-v, v-a. (1.129)

yr)-yt yf)*-y., q-g, p-s


1.8.2 Application to elastostati£s

In the case of linear elasticity. we set

cP = CPi; = tensor of stress functionsCT - CTi;= stress tensorv - Ei; = strain tensor

rr - rri; = dislocation tensor

Then (1.121 H1.128) assume the following specific forms:

W( i;) -1 [1 IiC rs _ (i; _ in.- ;kT ).L CPjj.CT • Eij - 2CT i;",CT Ei; CT e e CP""km

- rri;CPi;]dx + (ll*Ei;' Y(CPi;)- r>ao,-(q, ll(CPjj»ao,

where. for compactness in notation, we have denoted

("* () -) -1 [(fimi"l'''lu - imjP<l)U Eij, Y CPij P ao, - n,.cP", PI Ejm,jp'lani

_ (p'minP'<lu _ ij"'l) ] d""CPr,,_m P2 Ei;''''1 S

wherein

29

(1.130)

i, j. k, I, m. 11, P, q, r, s. t, U = 1,2. 3

on a02

"" ( )_ r ( 1 Eijrs + i'"'' jkr ij ) d:no CPij'Eij -.lo -ZEij Ers Eije e CPrs,km- rr qJij x

-rrijqJij] dx +(8*( Cij",CTrs),Y(qJij)- P)ao,

-(q,8(CPij»ao,

(1.131)

(1.132)


6.( ) -1 rlC imp jqu Ita .bk - i/ ] doT CPij- L2 ij,.e e e e CPmn.pqYak,lbT/ CPi; X

-(q, S(CPij»all2

.1(Ei/) = L (-~Eii"'EiIErs) dx -(S*Eil' P)ao,

:K(Uil) == 1(!Cijrsunuij -uijEij) dx

U( ) _ r [IC ipm /qu Ita .bk.NL CPij'Eij - .In .2 ijrse e e e CPmn.pqYak.lb

+EijnE ..E _2E ..eirseimkcp - ....ijcp.] dxII n II sk.",,', Ij

-(q, S(CPi/)}aG,

J{( ii ) -1- ij d +I ij d -("'* )u , Eij - U Eij X EjjCP S 0 Eij,Pan,an,

(1.133)

(1.134)

(1.135)

(1.136)

(1.137)

ACKNOWLEDGEMENTThe support of this work by Grant 74-2660 from the U.S. Air Force Officeof Scientific Research is gratefully acknowledged. I also wish to make aspecial note of thanks to Gonzalo Alduncin, who read the entire manuscriptand made many helpful suggestions.

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Chapter 1 The Classical Variational Principles of …This chapter deals with a general theory of variational methods for linear problems in mechanics and mathematical physics. This