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11',\ International Conference on

Variational methodsin engineeringSEPTEMBER 25-29 1972.

SESSION IT

OPENING LECTURE

Monday - September 25th

Department of Civil Engineering. Southampton University. England

2.1

SESSION II

VARIATIONAL PRINCIPLES IN

NONLINEAR CONTINUUM MECHANICSby

J. T. Oden

2.1 IntrodlctiooThe utility of variational statements of various problems in contimmmmechanics has been recognized for nearly three hundred years.Variational formulations of physical laws describe global properties ofthe whole system rather than its component parts; they providepowerful tools for studying the existence of solutions to problems inmathematical physics; they can encompass in concise statementsentire collections of boondary-, initial-, and jump-conditions anddifferential and integral equations governing the problem under con-sideration; they may naturally include the effects of constraints in thesystem; and they may lead to information 00 bounds to solutions.pfdifferential and integral equations. Finally, and of paramountimportance, they provide a natural means for approximation.The use of variational principles in continuum mechanics began in thelate seventeenth and early eighteenth centuries in the writings ofNewton, Euler, Lagrange, and J. Bernoolli; the theory was nourishedby Kirchhoff, Green, Kelvin, Hamilton, and Castigliano in thenineteenth century and Hellinger in the early part of this century.A historical account of the subject as it stood in the early 1950's canbe found in the treatise of Truesdell and Toopin1• Among contri-butions to variational principles for purely mechanical laws the well-knownprinciples of Reissner2 must be mentioned; variousgeneralizations and additional references can be found in the book 01Washizu3

• In 1955and 1956, Biot· 5 formulated variational principlesfor linear irreversible thermodynamics, with emphasis on linearlyvisCOielasticand thermoelastic solids and the linear theory of heatconmction. Similar variational theorems were presented by PrigogiDe'in connection with linear thermodynamics of irreversible processes,and followinga different approach, GurtinT • developed variationalprinciples for linear elastodynamics and linear isothermal visco-ela~ticity. Nickell and Sackmanll presented a variational principle forthe linear theory of thermo-elasticity and, more recently, Sandhu and

2.2

Pister10 11 presented a variety of variational principles correspondingto linear theories of anisotropic isothermal viscoeleasticity, thermo-elastici~, and mixtures using generalizations of theorems ofMikhlin1

• The opportunity for broad generalizations of the theory ofvariational methods for nonlinear continua came from the pioneeringwork of Vainberg13 who developeda general theory in the context ofBanach spaces. In recent work, Oden1•15and Oden and Bhandari1617

have presented general variational principles for nonlinear continua,together with extensions of certain results of Vainberg, which includeprinciples for nonlinear viscoelasticity, thermoviscoplasticity, andheat conduction. We examine some of these results in the presentpaper.

The present lecture deals with general variational principles for anextremely wide class of nonlinear problems in continuum mechanicsand thermodynamics. Bya variational principle, we mean here atheorem concerning stationary values of a well-defined functional, asdistinct from Galerkin integrals or "virtual work" principles associ-ated with a given physical law. Much of this lecture is a summary andcondensation of the results presented by the author in earlier contri-mt(.ms1• 151617. Followingthis introduction we review the essentialfeatures of the general theory of potential operators on Banachspaces13 14 15and the field equations governing the thermodynamics ofcontinuau. We then follow the plan outlined in14 151617to obtain aflblctional K(~) which assumes a stationary value when the entirecollection of field equations governing the behavior of the continuumis satisfied. This study is followedby the identification of a variety ofspecial cases. The theory is based on the properties of nonlinearpotential operators. If a given operator is not potential, the questionof the existence of an associated variational principle and of the inter-pretatiOn of Galerkin approximations of the operator arises. Thisquestion is pursued in the final section of the paper wherein therelationship between Galerkin' s integrals (virtual work), least squares,invariant operator imbeddings, and direct variational methods arediscussed.

2.3

of position Z and time t will be the S""-space defined by the cartesianproduct of the closure .ff1 of an open, bounded, connected region .!If inthe three-dimensional Eu.clidean s~ce (or a body ~ isomorphic toJi')and time interval (-00, 00), denoted.!lfx (-00, oo)~ €.!If, t € (-00, 00)]. Toclarify tqese ideas, suppose tha~ 'the operator .!i' carries elementsfrom a linear vector, space 'lJ' int(j another linear space r. Theelements of ~ and rare functidnS of some class defined on Ji' + aSPand the usual properties of linear vector spaces hold (1. e. ml1(x) +l3u2(x) €~ , where a,13 are scalars and Ul, U2€~ , etc.). Supposealso that it is possible to define on i" and r a norm; that is, a realnumber associated with each element u(v), denoted Dull(IIvll),whichobeys the norm axioms; Ilull~ 0, = 0 iff u = 0, IIcrull= lallluU, andlIu1+u211:SlIullI+ llu211.If Z' and rare complete, (i.e., if Cauchysequences {Un}of elements converge in the norms to limits 110in ~ inthe sense that lim llu n- UoII ~ 0, '110 € &'), then they are referred to as

n-oo

Banach spaces. Moreover, if li~l.!i'(u n) - .!i'(uo>II = 0, then.!i' is saidn-OO

to be continuous at Uo. It is understood that (2.1) is to be satisfied inthe interior of ~ and, on the boundary a.!lf , we must impose certainboondary conditions, ..2"(u)= 0, where 2' is also a nonlinear operator.Our aim here is to construct a variational statement of (2.1) for thecase in which it corresponds to the equations governing the nonltneartheory of thermodynamical behavioor of simple materials. To make'the terms "variational statement" more precise, we must introcl1ceinto this abstract setting one of two genera.li7~tions of the ideas of ..differentiation and of differentials. The Gatf!llUXdifferential of .!i' (u),denoted G.!i'(u, h) and the hechet differential of .!i'(u), denoted(\9(u, h) are defined by

LimII!..[.!i'(u+ ah) - .!i'(u)] - G.!i'(u,h)II = 0' (2.2)a-O a

where .9 is a nonlinear operator from a dense set il C Z'into r, z.-and ,r being real Banach spaces, u = u(z, t) is an element of il, ande € r. Here and subsequently the domain of definition for all functions

2.2 Variational methods for nonlinear operatorsFollowingVainberg13, we outline here certain basic properties ofvariational methods for nonlinear operators on Banach spaces. In anabstract sense, we wish to consider a special class of nonlinearoperator equations of the form

(2.3)

g(u) = e (2.1)

Lim1I.!i'(u+ h) -.!i' (u) - 6.!i'(u,h)II = 0IIhll-owhere h is an arbitrary: element in i". The linear operators .?(u), (h)and.9' (u)(h), where .9(u)(h) = G.9(u, h) and.!i" (u)(h) = 6.9(u, h)are called Gateaux and Frechet derivatives of.!Jl'at U, respectively.Since it can be shown that if.?(~) exists and is contlnuoos in theneighborhood of u, then G.!JI'(u,h) exists and is identically the same as69(u, h), we shall use the definitions and notations of (2.2) and (2.3)interchangeably, 1. e., we henceforth assume the existence andcontinuity of9(u).A particular type of operator K:'V -R which associates with elementsof 'V real (or complex) numbers a € R is called a functional. IfK(cru + I3v)= aK(u) + I3K(v), K( ) is called a linear functional.

2.4

is the gradient of K(u), and we write 9(u) = g;ad K(u).

The familiar notions of maxima, minima, and critical points ofordinary functions from elementary calculus also apply to functionalson Banac~ spaces. For example, a point Uo€ 9 is a critical point ofa functional K(u) defined on 9 if grad K(uo) = e, e being the zeroelement in 9 C z.'. Moreover, i'f K(u) ~ K(uo) for all u € 90 c 9,then Uois called a global minimizer of K on 90 (or, if K(u) ~ K(uo) forall u in 9 n 9 , .9' being an open neighborhood of Uo, then Uo is alocal minimizer of K). As expected, if Uois a local minimizer of aGateaux-differentiable functional, then grad K(uo) = 0; that is, Uois acritical point of K(u). If the second Gateaux differential GZK(u,h, h)exists and is positive definite (1. e. GZK(u,h, h) > 0 for all h € 90) at acritical point Uoof K(u), then Uois a proper local minimizer of K(u)(I.e. K(u) > K(uo) for all u € 90), If GZK(uo,h, h) is positive-semidefinite at Uo, Uois a local minimizer of K(u). Finally, if Uois acritical point of K(u) and GZK(uo,h,h) is negative definte or negativesemidefinite, then Uois a proper maximizer or a local maximizer ofK(u), respectively. Second and higher-order differentials offunctionals (and other operators) can be obtained by repeated applica-tion of (2.3). For example, the second Frechet (Gateaux) differential

is GZK(u, h, k) = c5ZK(u,h, k) = lim f [OK(u + edt, h)] which is bilineara--O a

in h and k; and by continued differentiations it is sometimes possibleto construct Taylor-type expansions of functionals.An operator 9: ~ - ~ * is said to be a potential operator on someset 9 c z.- if there exists a functional K(u) such that grad K(u) = 9(u)for every u € 9 c ~. It is shown by Vainberg13 that a necessary andsufficient condition for an operator 9: ~ -- r* to be potential on9 c ~ is that the bilinear functional <09(u, hI), hz> be symmetric inthe sense that <09(u, hI), h2> = <o!i' (u, hz), hI> for every hI, hz €~

and u €9, where <09 (u, hI)' h2> is assumed to be continuous on g.Thus, if!i' in (2.1) is potential, solutians of (2. 1) will be critical pointsof some functional K(u). This, in fact, is the essence of variationalmethod in mathematical physics: instead of seeking solutions directlyto (2.1), determine the critical points of a functional K(u) which hasthe property grad K(u) = !i' (u). The equation oK(u, h) = <!i'(u), h> = 0(or !i' (u) = e) is then the Euler-Lagrange equation for K(u) and at ·thecritical points u such that oK(u, h) = 0, we say K(u) assumes astatianary value.The converse problem; given a potential operator 9 (u), find afunctional K(u) for which grad K(u) = !i'(u), is an important question inthe theory of variational methods. It is resolved by the followingfundamental theorem by Vainberg13:

Theorem 1. Let!i': ~ -- Z' * be a potential operator on 9 c Z', notnecessarily linear. Then there exists a unique functional K(u), whosegradient is!i'(u), which is given by

2.5

(2.6)

(2.5)

(2.7)

9(u) = grad K (u)

The gradient of a given functional can be readily identified if we pointout ~ome adcUtional structure of the spaces ~ and ~* = r. If ~and ~* are vector spaces, suppose that we introduce a mapping of~ ® ~* into R which associa,tes with pairs of elements u € ~ ,v l' ~ * real numbers· <u, v> with the properties

<11,v> is linear in u and v

<u,.> = 0 for fixed !€S'* and all ~ ....y: = 0

<U, v> = 0 for fixed u: € S' and all v -u = 0

a<'?(u),11> = lim a- K(u + ah) = OK(u,h)a--o a

Since K, whether linear or nonlinear, is also an operator on ~, wecan also speak of derivatives of functionals in the same spirit as (2. 2)and (2.3). In fact, for a given Gateaux differentiable functional K(u)~we compute

1 a· IOK(u, h) = lim - [K(u+ah) - K(u)] = -a K\u+ah) (2.4)a-oa a

a=OThe Gateaux differential OK(u~'h)is also called the first vanatianof K(u); it is, in general, linear in h so that for fixed U, llK(u, h)describes a linear functional on ~ and thereby identifies a mapping ofsorts of·~ into its conjugate space ~*. This mapping, denoted 9(u),is called·the gradient of K(u) and we write

Then' Z' and ~* are dual and <u, v> is the scalar product of u and v.If ~ and ~* are coincident (or related by a one-to-one, onto,positive definite mapping), <u, v> can be associated with the innerprodlct on j'. Alternately, from the Reisz:-Fischer theorem19

, if f(h)is a linear functional on Z'(h € j'), there is a unique v such thatf(h) =.<h, v>; then j'* can be regarded as the conjugate space of ~(i.e.~ the space of linear functionals of ~).

Returning now to (2.5), we recall that oK(u, h) is a linear functionalon ~ which defines an element !i'(u) in S'*. It is now clear that wecan associate llK(u,h) directly with the bilinear mapping <!i'(u), 11>,where .9'(u) is the gradient of K(u). In other words, llK(u, h) is linearin h and grad K(u) can be defined so tha.t llK(u, h) is also linear in !i'{u).If we assume further that, for this biltftear operator, conditions, (2. 5)(and possibly (2.6» hold, then we can ~ssociate oK(u, h) directly with<9(u),I1>, where .9'(u) is the gradient of K(u). In summary,ifK(u) possesses a linear Gateaux differential an some subset9 c ~, then the operator.9(u) defined by

2.7

where € is the internal energy density, q1 is the heat flux and h is theheat supplied per unit mass from internal sources. Alternately, if weintroduce the Helmholtz free energy density cp(~ t) and the internaldissipation a, by the definitions

where the superpoosed dot indicates partial differentiation withrespect to time, p(x) is the mass density of the solid in the referenceconfiguration, a1J are the. contravariant components of the secondPiola-Kirchhoff stress tensor, measuring stress per unit area of thematerial surface in the reference configuration. That is, if£.(e, t) is the stress tensor and Q1(~ t) = ZJ(e, t), 11,1are natural tangentvectors to the deformed material coordinate lines, then£. = (a'J/v'G)Q1 ® QJ where G = det Q1' QJ'

Eq. (3.1), (3.3), (3.4) and (3.5) describe purely mechanical lawsgoverning the motion of a continuum. To accommodate thermodynamicphenomena, we must, of course, add to the mechanical laws theprinciple of conservation of energy. If mass is conserved and (3.4)is satisfied, the local form of the principle of conservation of energycan be written

2.6

1

K(u) = f <9'(Uo + s (u - Uo», (u - uo»ds + Ko (2.8)

owherein Ko = K(Uo)is a constant, and s a real parameter.

In the following section, we identify nonlinear operators appearing inthe governing equations of nonlinear continuum mechanics and applythe above theory for special definitions of the bilinear form <u, v>.

2.3 The equations of nonlinear contiDIum mechanics

Consider a continuoos material body .!II with surface 0.!8 in motion inthree-dimensional space E3

• Material points in the body are denotedx = (Xl, X2, X3), and we trace the motion of the body relative to a fixedspatial frame of reference Z i. For simplicity, the materialcoordinates xi, are assumed to be rectangular cartesian while thebody occupies a referen~ configuration Co C E3

• If.!8 =.!8 + a.!8 isthe closure of .!II, then.!ll x (-00, 00) is the domain of aU functions ofposition ~ and time t to be considered. Let!!.~ t), ~ t), 2~ t),a(!o t), f<1s. t), q(~ t) and ~~ t) denote, respectively, the displacementvector, the strain tensor, the stress tensor, the internal dissipation,the body force vector, the heat flux vector and the thermal gradientvector and let 6 (x, t), 71(~t), and hex, t), respectively, denote the"absolutetemperature, the specific entropy per unit mass, and therate of internal heat generated per unit mass within the body. Thenthe strain-displacement relations are

pl. = a1J'hJ + q~l + ph

cp = E - 716

(3.6)

, (3.7)

Alternately, we can use as a measure of the deformation, the Greendeformation tensor

where the comma denotes partial differentiation with respect to thematerial coordinates, and

wherein z.~ = U.,l + olll.are the components of the deformation gradient.

Since we adopt a material descrip,tion 'of the motion here, ·the localforms of the laws of balance of linear momentum, angular momentumare, respectively, To the system of field equations just cited, we must, of course, adjoin

the initial and boundary conditions. Let a.!8~ and a.!lla denote thedisjoint sets whose union is a.!8. Here a.!ll~ is taken as the portion ofthe boundary on which displacements are prescribed and 0.!8. representsthe portion over which tractions are prescribed. Also let 0!1J. and~ q denote disjoint sets whose union is a.!8, a.!ll. being the portion ofthe boundary on which temperatures are specified and a.!8q is the

1YiJ = '2 (Ui'J + UJ,l + U.,i u.,J)

gi = 6,1

G1J = 2Y1J+ OiJ = Z.,l z.,J

[aiJ (o"J + u.,J)J ,i + pF., = pu.a1J = aJ1

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

and

a = a1JY1J - p(eP + 716)

then the local form of the balance of energy can be written

p6r, = q~1 + ph + aor

PeP = a1JY'J - P718 - (j

where, again 71is the specific entropy.

(3.8)

(3.9)

(3.10)

2.8 2.9

portion over which normal heat flux is prescribed. Further, let theoutward unit normal vector to agbe denoted n. Then the boondaryconditions considered here are ~

_ A A

where Um, Til, (), and (/ are prescribed displacements, tractions,temperatures, and normal heat fluxes on the boundary. Without loss ingenerality, we can also assume that the body is at rest and in a naturalunstressed state at a uniform temperature To prior to some referencetime t = O. Then~ =.Q, !!. =.Q, X = ~ and () = To on!6 x (-co, 0).

Now up to this point, all equations presented in this section pertain toany continuous body. To proceed further, we must identify, if only ina general sense, the constitutive equations for the specific material ofwhich the body is composed. We retain a higher degree of generality inour results by considering so-called simple materials for which weneed assert no more about the forms of the constitutive equations thanthe following: the free energy, entropy, stress, and heat flux at aparticle 15at time t are determined by the history of the deformationand absolute temperature at x and the current value of the temperaturegradient. That is, we consider a class of materials described by acollection of constitutive equations of the form

u. = UII on a!6u x (-co, 00)

T· = n101J(0a,) + u.,.l) = I'll on ag. x (_co, co)

8 = () on ag. x (_00, co)

(/ = qin1 = il on cgq x (-00, co)

(3.11)

(3.12)

(3.13)

(3.14)

Here:l (s) = z<.!, t-8) and 81 (s) = 8<.!, t-s) are, res,pectively, thetotalmstories of the strain and the absolute temperature (0:$ s < co)and the dependence of the constitutive variables on ~ is understood.i and .r.are the scalar-valued functionals and .!"and ~ are,respectively, the tensor -valued and vector ",valued functional8 of theindicated arguments.

Coleman18 has shown that certain simple materials can be characterizedby only two constitutive functionals, one describing the free energywhich is independent of g and the other the heat flux; e.g.

'"'00 00

cp = ~ lrr, 8~; 'Y, 8] ql = (/1 lY.-, 0~;'Y, 0, ~J (3.16)s:::O '"' "" s:::O "" "" ;r:

Here we have decomposed the total histories -r (8) = 'Y (x, t-s) and'"' ..., ~

(}t (s) = 0 (x, t-s), 8 £ [0, co), into the "past histories" Yr (s) = r (s)"" "" ""

and ()~(s) = 0\ (8) ls £ (0, oo)J and the current values of strain andco

temperature 'Y(t)and o(t) (s = 0) respectively. Then, if ~ [rr, o~;'Y, 0]"" s:::O "" ""

possesses sufficient smoothness jYYoperties, the stress and the enkopyare determined by the functionals

co co

all =.!r' I [rr, ()~;'Y, (}J = pa'7 ~ [Yr, ()~; 'Y, ()]s:::O - "" - s:::O ..., ...

(3.17)co co

1] = .r ['Y~, ()~; 'Y, ()] = -a. ~ ['Y~, 6~; 'Y, 8]s:::O - - s=O "" ""

while the internal dissipation is given by

00

cp = ~ [yl (5), 81 (s); g(t)]O - -s=

co co co

f1 = 9 [rr, ()~;'Y, ()] = - p {oor ~ [-11~]+ o. ~ H8~]}s:::O .... ~ - s=O "" s=O

(3.18)

co co

1'/

00

./ [yl (s), (}I(s); get)]8=0 - -

where 0'7 ~ [; I.] and 0.' 4' [; I.] denote the Frechet differentials-s:::O s:::O

which are linear in their arguments to the right of the vertical stroke;

o = .r [yl (8), (}.I (s); get)]- 5";;0 - -

co

co

q = (/ [yt (s), 01 (s); get)]5;;'0 - -

(3.15)

co co

Cor ~ [;] and a. ~ [;] denote ordinary partial differentiation of ~-s=o s=O

with respect to 'Yand e respectively. This completes the review of thebasic equations governing the behavior of nonlinear continua. amother generalizations could be added (e. g., electromagnetic effects,microstructure, etc.), but the basic procedure discussed below remainsessentially the same.

2.11

g*[O'II (6"J + u.., J)]' I + 11.. - PI .. = 0

which, together with the boundary conditions (3.11) - (3.14) com-pletely determine the thermomechanical response of the continuum.,To proceed to construct general variational principles, we first put(4.4) in a concise operator form. Let ~ denote an element in a

24-dimensional "thermodynamic state" space given by the orderedoctuple

2.10

2.4 General variational principles

We wish to develop here a general variational principle involving afunctional which assumes a stationary value when all of the fundamentalequations describing the thermomechanical behavior of nonlinearcontinua are satisfied. We recall from Section 2 that such a functionalexists whenever the operators appearing in proposed Euler-Lagrangeequations are potential operators, Unfortunately, the nonlinearoperators appearing in the kinematical, mechanical, and thermo-dynamical equations for continua are not, in general, potential, so thatit is necessary to first develop alternate but equivalent forms whichpossess the desired symmetry properties. This involves a trans-formation, of sorts, together with an appropriate definition of thescalar product <u, v> in l2. 6). We emphasis that such transformationsare not unique', and that several variational principles of completelydifferent form may exist for a single problem.

The method followed here is similar to that used for linear theories(e. g.7 8 II 10 11), and is based on the observation that time derivativesof various terms in the field equations can be eliminated by taking theLaplace transform of appropriate equations. Here we introduce theconvolution u*v of two-scalar valued functions on g x (_00, oo) as thefunction on .. x [0, co} defined by

t

u*v=!U(X, t-r)v(x, r)dr (4.1)"" ""o

It is easily verified that u*v = v*u and u*(v~) = (u*v)*w. Applying(4.1) to both sides of (3.4) and (3.9) we obtain

g*[O'll (6"j + urn, j) J, I + Ii..= pu ..

00

0'11 _.r1j [Yr, 8~; 'Y, 8] =0s~ "" ""

1'YII -2(u'q +Uj,l +U .. ,I +u",j) = 0

co

a - 9 [~,8~; 'Y, 8J =0s~ "" ""

gl - e, I = 0

coq I _ (J I [Yr, 9~; 'Y, 9, g] = 0

s~ "" "" ""

g'*q: I + ph + E - pT 0 ~ = 0

co

1'/ - .r [~,8~; 'Y, 9j = 0s~ "" ""

~"t (~, t) = {UI, 'Ylj, (Tlj, a, ql, gl, 8, 7]}

(4.4)

(4.5)

co co 00 co

rT<!,t) = {+pfl' -g* ylj [ ],0, -g* 9 [;],0, -g* q\ g*(pi1+i;), -g*.r [ ]}8=0 s=O s=O saO

(4.6)

g'*q: I + ph + i; = pT 0 ~

wherein get) = t; g' (t) = 1

f .. = [g*F ..] (x,t) + p(x)[tv", (x) +dm (x)J,.., - L - ,..,

h + i; = g'*[h + &] (x, t) + p(x)T 0 ~ (x, 0)"" '" ""

(4.2)

(4.3)

Similarly, let

Then (4.4) can be written concisely as

'?(A) = .?' {A) - r = e,.., roJ _ ,.., (4.7)

Here Urn <!, 0) = dm ~), Urn ~,O) = Vm ~) are the initial displacementsand velocities an~, for simplicity in performing the transformation,we have defined fj = 9iJ/T o. Thus, collecting the entire set of equationsdescribing the behavior of nonlinear continua, we have

where.§' (.l is obviously the nonlinear operator defined by (4.4) whichmaps I! into r (with the constitutive functional appropriately prescribed)and ~ is the 24-dimensional zero vector ~"t = {Q, ,Q. Q, 0, Q, .Q, 0, O}.

2.12 2.13

Now, returning to (4.1), we introduce as the scalar product

= f {SUm*JAlm- um*g*[saIJ (Oml+ sum, I)]' 1 - um*{lm - sg*aIJ*'YIJ

§

<u, v> =f u*vdJ (4.8)g

Then, if S is a scalar parameter, and temporarily assuming homo-geneous boundary conditions (i. e.~ Om= 0, tm = 0, e = 0, g = 0) wemay compute < .!J'{s~), /);> = < .!i' (s~) - r,~>

+g* ;'Ij [-r:., 6~; 'Y, 6l*'Ylj - sg*'Ylj*all +-21g*[SUI, I + SUI' I

8=0 "" ~

002... J* Ij *~~ *~ if1t [".t 6t• 6] * *+ s-Urn, I Urn,I a - sg u'a +g U' .:7 Tr, r, 'Y, - sg TI T/

s=O '" ""

00

+g*'l* f l'Y~, 8~; 'Y, 6] - sg*{gl - 6,1)*ql _'sg*ql*gls=O ~ ""

00

+g* gl lr.., 6~; 'Y, 6, gJ*gl - sg*g'*ql, 1*6 - g* (ph + i:)*68=0 '" "" ~

+ pT "g*T1*6}dsd§

(4.9)

where we have set Ko = 0, ~o = ,2.Thus, we have derived a variational principle associated with the non-linear operator equations (4. 7); that is, K(~) of (4;.10) assumes astationary value when ~ satisfies (4.7) provided .!J' is potential. That.!J' is indeed potential, is verified by the following theorem17•

Theorem 2, Let u I (~, t), 'YIJ(~, t), ~IJ (~t), (1(~, t), q I (x, t), ~I (~, t), 8 (~, t)and TI (x, t), all defined for (x, t) (~X (-00,00), satisfy the field equations(4.7) WIth homogeneous boundary conditions. Then the functionalK(~) of (4.10) assumes a stationary value at ~ = {~, ;t, £, 0-, ~, ~, 8, TI}.

Proof: Let ~ = {~, 2, £, ~, g, i, e, m denote an arbitrary elementin the domain of the operator .9-. Then, by (2.4),

1[ - J 0 -ILim - K(A + aA) - K{A) =;- K(A + aA)a-O a "" "" "" va "" '" a=O

= f {(PUm- g*[all (Oml+ urn, I)]' I - pfm)*Um + g*(~ [UI, 1+ uI' I + Um,! um'l]~

00

- 'YII)*;j'lJ + g* {yll l'Y~, 6~; 'Y, 6] - all)*Ylj -g*{gl - 6, \)*ijl-10 "" ""S-v

00 ~ 00

+g*(g ['Y~, e~;'Y, 6]-fr)*O:+g* (f[Yr, 8~;'Y, e]-T/)*Tj_g*{qls=O ""! s=O "" ""

00

- QI ['Y~, e~; 'Y, e, l!])*gl - g*{g'*ql,! +ph +i: - PTof1)*8}dg=s=O "" "" ~

Now introducing (4. 9) into (2.7), integrating by parts, and making useof the homogeneous boundary conditions and the commutativity andassociativity of the convolution operator [u*v = v*u, u*(v*w) = (u*v)*w j,we obtain the functional

1/ 00K{ A)= -2 Wum*um - 2g*um *a~":- 2g*'Ylm*alm +2g*'Ylm* ylm ('Y~ 6~; 'Y, e].... s=O.... ""

g00

-g*fr*(1 +2g*fr* 9 (Y.-, e~; 'Y, e] - g*um*{a\Jum, J)' \ - 2pum*fm

8=0 '" '"

<g(A), X>~ -That is, .9 (A) = grad K (A) which proves that (4.7) is indeed the Euler-Lagrange equation of K (~).

Aside from a great deal of additional algebra, the case of non-homogeneous boundary conditions can also be accommodated. Considerthe functional

11 ~. fK a~{~) = 2' {g*[nlaIJ (2 OmJ+ Urn,J) - 2TmJ*uJd.9' + g* um*Tmd.9'

ag. agu

00 00

-g*'l*T/ + 2g*T1* f ('Y~, 6~; 'Y, e] -2g*ql*gl + 2g* (/1 b?, e~; 'Y, 8, g]*g!s=O ~ ~ s:::O .... .... '"

-29*g*g'*ql, \ - 2g*{pb + L)*e + 2pT .,g*17*e}d§ (4.10)

+f {g*g'*(Q - i- )*e}d.9' +! g*g'*9* (/d.9'

a~ q o.!JI,

(4.11)

2.15

2.14

where K(A) is given by (4.10) and KaJ (A) by (4.11), assumes astationary value whenever A satisfies (4.7) and the nonhomogeneousboundary conditions (3.11)-13.14).

Unless noted otherwise, we shall henceforth assume homogeneousboundary conditions. The functional (4. 11) was obtained by expandingthe domain of 9' (~)to include a.!JJ .

then, by following a procedure identical to that used in deriving (4.10),we may show

Theorem 3. The functional

(5.4)

(5.3)

11 coK{u,9} =-2 {ul*PU1+2g*Ylj*ri[Yr, 9~;Y, 9]-2pul*fl- 'gJ s=O - -

Nanlinear Viscoelasticity. The variational theorem given in18 for non-linear viscoelasticity is obtained from (4.10) by simply setting to zero9" 71, and gl in (4.10):

A 11 coK (A2) = -2 {PUm*Um- 2g*um*0-:~ + 2g*Ylm*..rIm [r (s)] - g*&*u~ .!JJ s~ ~

co ~- g*ql*gl - 9*g*g'*q: I +2g* (71 [; ]*gl - 2pg*h*6 +2pTo g*ij*9}d~

s~

co

- 2Y,,]*0-" - 2pul*fl - g*TJ*71+ 2g*TJ*.r [;]-g*(gl - 8,,)*q' (5.2)8=0

co

- 2g*Ytm*a1m + 2g*u* 9 [yt (s)] - g*um* (al,um, ,), 1- 2PUm*fm}d~s~ ~

Now consider the case when the displacement field ~(!,t) and thetemperature field e(x, t) are kinematically adm1ssible (1. e. the dis-placements u (x, t) and the temperatures 9(x, t) satisfy the strain dis-placement relation and the temperature-tnermal gradient relation.Then the functional (5.2) takes the form

co co

For special case in which y1i [y;, 9~; y, 9]q1 = (71 (y;, 9~;y, 9, g], and8=0 - - s=O - ~ -

K(~l) =! 1{Ul*PUI + 2g*Ylj* f~'[;j + g*[UI, j + uj' I + u"" I Um,j

~

+ 2g*g'*ql*9" - 2pg*h*9 + 2pTo g*~*9}d!8(5.1)

(4.12)K(~) = K(~) + Ka.. (~)

co

!{um*pum- 2g*um*0-:i - 2g*Ylm*0-Im+ 2g*Y,m..r,m [ ; ]s=o

.!JJco

- g*um*(al'um, ,), 1- 2pum*fm-g*71*71+ 2g*7)* f [ ; j -2g*ql*gls=O

1K(~l) = 2

2.5 Some special cases

To emphasize the generality of the variational principle given inTheorem 2, we shall now point out a number of special variationalprinciples that derive from (4.10) by placing restrictions on the classof admissible functions.

Equivalent Functians. Among a large number of functionals equivalentto K(A) of (4.10), we illustrate two obtained by deleting the internaldissipation equation from (4.7) and, following a similar procedure asbefore, obtain the new functional

00

+ 2g* (71 [ ; ] *gl-2 9*g*g'*q: I - 2pg*h* 9 + 2pT 0 g*~*9} d..ifs=O

co

~ = f [y;, 9;; Y, 9] are linear in strain history Yt and the tempera-s=o - - -

where, for simplicity in writing, we have used the abbreviated notation ture history e t, (5.4) reduces so that oeo for homogeneous boundaryconditions .

co co

.rIm [ ; ] =..rlm [y~, e~; Y, 9], etc. The functional K (AI) of (5.1) iss~ s=O - - ~

again written into an alternate form by making use of the divergencetheorem and homogeneous boundary conditions; 1.e.

Nanlinear Heat Conduction. By ignoring all mechanical variables andretaining only 6, 71, q>, g i. and 9 we obtain a general variationalprinciple for nonlinear heat conduction:

Linear Coupled Thermoelasticity. For linearized coupled thermo-elasticity, the system of field equations in terms of displacementvector ~ and temperature 8 as given by Nickell and Sackman

2.16

C(A,) = ! f {-g*u*u + 2g*u* .; [at (s)] - g*TJ*TJ + 2g*TJ* ;. [at (s)]- 2 jJ s=O s=O

00

_ 2g*ql*gi + 2g* (/i [at (s); ~(t)] *gl - 2a*g*g'*q~ is=O

- 2g* (ph + !;)*a + 2pT 0 g*~*a}d~

g*(CiJkt Uk,' - (3ij 8), j - pU1 + fi = 0

~o *(K1J 8'J)'i - .fo Co8 - (31jUI,j +~ = 0

(5.5)

(5.6a)

(5.6b)

• J

2.1'1

Other Principles - Complementary Energy, and Variants. A varietyof other variation principles, too numerous to mention, can be obtained..For example, we observe that the variational principles of GurUn'.Sandhu and Pisteru for the ~ -static motion of linear viscoelasticand linear elastic solids become special cases of (5.8) and (5.9).Moveover, by ignoring all thermal effects, assuming each of theremaining mechanical constitutive e~ations are satisfied, ignoringrate terms in (3.4), (3.6) and (3.10) (and, hence avoiding thenecessity of the operation g*), we set

2YiJ (yiJ - aij) = C(Yij) - 2(aij Ylj - W)

The quantity C(YiJ) depends ()DIyon the strains and can be omittedsince we plan to vary only a1J. The quantity in parenthesis is ~(aiJ>.the complementary energy density. Incorporating all of theserestrictions, and applying the Green -Gauss theorem, the functional(4.12) reduces to

K~) = f [-fZl<£) + iaiju., i U. j] dg +1 U. T·d.9' (5.10)

g a!I.

Application of divergence theorem to (5.8) yields

We follow the procedure adopted in section 4, and set

Now making use of the definition (3.14) for the bilinear mapping<u, v>, we obtain

8 a a- O*g*{3ijUi,J -To *g*CoO +To *g*g'*(K1jO'J),1 +2To *g*h}d.!Jf

2.6 Noupotential operators

The convolution operation employed in Section 4 is only one of severaltechniques that can be used to transform a given problem character-ized by nonpotential operators into an equivalent problem character-ized by potential operators. In fact, the general idea of transformingoperators so as to identify potential operators provides a convenientavenue for showing the relationship between strict Ritz methods andGalerkin -methods of approximation.

Suppose for eXfUIlple, that.!i' is a nonlinear operator from 8' into 1'*,not necessarily potential, and consider the problem of solving theequation !?(u) = 8, where 0 is the zero element of 8'* (we use e insteadof 0 temporarily to distinguish elements of 1'* from the real number0). Let g: Z'* .....R be a functional with the following properties:

(1) g(e) < g(v) for every v € Z' *, V ~ e(ti) g(O) = 0

It is easily verified that if variations in K <.v are considered due to. variations in stress which satisfy the equilibrium e~ations and thetraction boundary conditions (3.12), then 6~ <.v vanishes if, and onlyif, (3.1) holds.

""

(5.8)

(5.7)

r = 1--;: IlUi

~=

8

J{~}= ~1{Ui *pU1 - U1*g*(CiJkl Uk,! )tJ + Ui *g*({31J8)q - 2Ui*CjJ

.J{~}=i1{Ui*PU. + g*Ui,j*C1JIr.f Ult,t - 2ui*fi - 2g*a*{31JU1,J

'.!I

- Jt *g'*a iKiJ8 - JL *o*c. e + 2 JL *8*h}d~To ' 'J To To

which is identical to Eq. (45) of10•

(5.9) Setting

j(U) = g(!?(u» (6.1)

2.19

2.18

We find that the solution u* of .9(u) = 8 is a (unique) global minimizerof j(u). Furthermore, if

(iii) g(u) is Gateaux differentiable and has a unique critical point at8 € 'Jr, then

Since h can assumedly be taken from outside the null space of thelinear operator ~(u), vanishing of (6.5) implies that £1..9(u*» = O.But since £I' is positive definite, the unique solution of~(.§'(u*» = 0 is .§'(u*) = 8, which is the solution of the originalproblem.We remark that, in general, the operation (6.1) may lead to derivativesof higher-order in j(u) that would ordinarily be encoontered if ~(u)Were, at the onset, potential. However, in some cases differentiabilityof the argument u can actually be decreased by imbeddings such as(6.5), particularly if £l'is an integral operator 111 •

then

j(u + ah) - j(u) = <.9(u + ah), .9(u + ah» - <.9(u), .9(u»

= Illl.9(u, all)1l2 + 2<ll.9(u, ah), .9(u» + 2<w(u, ah), .9(u»

+ <.9(u + all), w(u, ah»

and1 . .'

lim -[j(u + ah) - j(u)] = 2<9'(u)(h), .9(u» = g(.9(u» ..9(u)(h) (6.4)a-O a

Hence, if grad j(u*) = 0, g(8) = 0, which implies that v = 8 is thecritical point of g. Likewise. if .9 (u*) = 8, then grad j(u*) = O.

Obviously, the choice (6.3) leads to least-squares approximations.Still other choices are possible. For example, if !£' is a positive-definite, symmetric, linear operator on z.-*, the procedure leading to(6.4) with

References

1 Truesdell, C. A. and R. A. Toupin, "The Classical Field Theories, "Encyclopedia of Physics, Edited by S. Flugge, Springer-Verlag,Berlin, 1960.

2 Reissner, E., "On a Variational Theorem for Finite ElasticDeformation, " Journal of Mathematical Physics, Vol. 32,pp. 129-135, 1953.

3 Washizu, K., Variational Methods in Elasticity and Plasticity,Pergamon Press, New York, 1967.

" Biot, M. A., "Variational Principles in Irreversible Thermo-dynamics with Application to Viscoelasticity, " Physics Reviews,Vol. 97, No.6, pp. 1463-1469, 1955.

5 Biot, M. A., "Variational and Lagrangian Methods in Visco-elasticity", Deformation and Flow of Solids. Edited by R. Gramme!,IUTAM Colloquium, Springer Verlag, Berlin, 1955.

6 Prigogine, I., Introduction to Thermodynamics of IrreversibleProcesses. 2nd ed., Interscience Publication, John Wiley andSons, New York, 1961.

1 Gurtin, M. E ... "Variational Principles for Linear Elastodynamics,"Archive for Rational Mechanics and Analysis, Vol. 16, pp. 234-250,1964.

8 Gurtin, M. E., "Variational Principles in the Linear Theory ofViscoelasticity," Archive for Rational Mechanics and Analysis,Vol. 3, pp. 179-191, 1963.

9 Nickell, R. E. and J. L. Sackman, "Variational Principles forLinear Coupled Thermoelasticity," Quarterly Applied Mathematics,Vol. 26, pp. 11-26, 1968.

10 Sandhu, R. S. and K. S. Pister, "A Variational Principle forLinear Coupled Problems in Continuum Mechanics," InternationalJournal of Engineering Science. Vol. 8, pp. 989-999, 1971.

11 Sandhu, R. S. and K. S. Pister, "Variational Principles forBoundary-Value and Initial Boundary-Value Problems in ContinuumMechanics," International Journal of Solids and Structures, Vol. 7,pp. 639-654, 1971.

12 Mikhlin, S. G., Variational Methods in Mathematical Physics(Trans. from the 1957 Russian edition by T. Boddington) theMacmillan Company, New York, 1964.

13 Vainberg, M. M., Variational Methods for the Study of Nonlinearoperators (Trans. from the 1956 Russian edition by A. Feinstein),Holden-Day, San Francisco, 1964.

Acknowledgement. The support of this research by the U. S. AirForce Office of Scientific Research under Contract F44620-69-C-0124is gratefully acknowledged.

:

(6.5)

(6.6)

(6.3)

(6.2)

and v = 9(u) the residual, leads to

<grad j(u), Ii> = 2<£I'(.9(u», 9(u) (h»

j(u) = <2v, v>

For example, suppose that Ilv1/ denotes the norm in ~ *. If we set

g(v) = Ilvll2 = <v, v>

grad j{u*) = 0 .9(u*) = 8

2.20

14 Oden, J. T., "Finite Element Formulation of Nonlinear Boundary-Value Problems," NATO Advanced Study Institute on FiniteElement Methods in Continuum Mechanics, Lisbon, 1971.

15 Oden, J. T., "Finite Element Models of Nonlinear OperatorEquations, "Proceedings. Third Conference on Matrix Methods inStructural Mechanics, Wright-Patterson AFB, Dayton, Ohio,October 1971 (to appear).

11 Oden, J. T. and D. R. Bhandari, "Variational Principles forNonlinear Viscoelasticity," International Journal oj Solids andStructures (to appear).

1T Bhandari, D. R. and Od~n, J. T., "Variational Principles forNonlinear Thermoviscoplasticity, " International Journal of NonlinearMechanics, (to appear).

18 Coleman, B. D" "Thermodynamics of Materials with Memory,"Archive for Rational Mechanics and Analysis. Vol. 17, pp. 1-46,1964.

111 Oden, J. T" "An ImbeddingTechnique for the Generation ofWeak-Weak Finite Element Approximations of Linear and Non-linear Operators". Journal of the Engineering Mechanics Division,ASCEt (to appear).

30 Nickell, R. E" "stress Wave Analysis in Ifa.yered Thermovisco-elastic Materials by Extended Ritz Method," Teclmical Report S-175.Vol, D, Room and Haas Co., Huntsville, Alabama, 1968.