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A nonlinear onsager theory of irreversibility
hr. J. Engng Sci.. 1972. Vol. 10, pp. 48 I-490.
Pergamon Press.
Printed in Great Britain
A NONLINEAR ONSAGER THEORY IRREVERSIBILITY DOMINIC G. B. EDELEN Center for the Application of Mathematics, Lehigh University, Ahstraet-
OF
Bethlehem, Pennsylvania
Let V denote a material body and let
denote the internal production of entropy of V, where J. are the thermodynamic fluxes and X, are the thermodynamic forces. A constitutive theory is obtained which determines the thermodynamic fluxes in terms of variational derivatives of a convex functional O[W;Xj which assumes its absolute minimum at X, = 0. It is shown that for all X, and for all additional parameters W,, and hence we have identical satisfaction of the Second Law of Thermodynamics. If Ct,= Jr &W;X)du, then J,, = &j&X, and we obtain the reciprocity relations &I&X, = a&k&,. The constitutive theory is not restricted to linear relations since the only requirements on $(W;X) are convexity in X, and (a/aX,)+J r._,, = J&,_,, = 0. Specific constitutive relations are obtained for the heat flux vector and the dissipative stress tensor. Since $ can depend on the parameters W, as well as on the X0’s, the dissipative stress and the heat flux can depend on the strain and temperature as well as on the rate of strain and the temperature gradient. 1. INTRODUCTION
THE RECIPROCITYrelations of Onsager[l] and the minimum entropy production principle of Prigogine[2] have provided access to both the understanding and the analysis of a wide range of physical phenomena. The principle drawback of the theory is that it is restricted to linear thermodynamic constitutive relations, so that phenomena have to be either linear or else the derivatives from equilibrium have to be sufficiently small that a linear approximation is valid. For phenomena which represent large nonlinear deviations from equilibrium states, such as large displacements of thermo-viscoelastic bodies, the Onsager theory can not be applied. This paper gives an extension of the Onsager theory wherein the restriction to linear constitutive relations is eliminated. After an examination of a number of equivalent formulations of the Onsager theory, it was finally found that the theory could be cast in a form which was not dependent on the assumption that the constitutive relations were linear. The essential properties of this formulation obtained from two conditions on functionals which are incorporated in the hypothesis H given in section 3. The theory that derives from these two conditions allows for quite general kinds of nonlinearity in the constitutive relations and the theory can be applied all the way down to the class of ideal bodies that are completely reversible. In addition, the constitutive relations that we obtain allow for inclusion of all of the thermodynamic variables rather than just those which appear as ‘thermodynamic forces’ in the reduced entropy production equation. One interesting aspect of the results is that the theory gives both the lower bound of zero and upper bounds for the reduced entropy production and the upper bound is, in fact, a least upper bound for a nontrivial class of functionals which satisfy hypothesis H and a nonvacuous class of states. 481
482 2. ENTROPY
DOMINIC PRODUCTION
AND
G. B. EDELEN
THE
SECOND
LAW OF THERMODYNAMICS
The notation used in this section follows, primarily, that of Truesdell and Toupin [3]. For a nonpolar body, the entropy production equation is given by (see [3], equations (256.12) and (257.1-257.3))
(2.1) In the general case, (2.1) is replaced by
where P, = ma. If q is viewed as the supply of energy per unit volume that comes from agencies external to the body, and if h is viewed as the supply of energy per unit area of the boundary of the body (h’ da* = h da) from agencies external to the body, then the right-hand-sides of (2.1) and (2.2) represent the total supply of entropy from agencies internal to the body. We accordingly refer to the quantity P, defined by P=
5 $+hf+),,}
dv,
(2.3)
l-1
as the total reduced entropy production of the body. Since there is not universal agreement on the form of the second law of thermodynamics under all circumstances, we confine our attention in this paper to bodies whose internal thermodynamic processes satisfy the following postulate of irreversibility: the total reduced entropy production is nonnegative at each instant of time, that is P L 0.
(2.4)
Because (2.4) is required to hold for each instant of time, there is no loss of generality in suppressing the explicit time dependence of functions and functionals in what follows. After the analysis is complete, explicit time dependences are easily accounted for by simply inserting t as an explicit argument in addition to those considered. The purpose of this paper is to obtain a general class of constitutive relations for the quantities {I’,, hi} whereby the postulate of irreversibility is ‘satisfied identically’. Since our results are based upon functional analysis, it is unnecessary to require satisfaction of (2.4) for any proper subset of the body. Of course, appropriate additional assumptions easily lead to particularizations of the general theory for which the integrand in (2.3) will be nonnegative at each point of the body. 3. FUNCTIONAL
CONSIDERATIONS
AND
CONVEXITY
The constitutive relations we obtain in the next section rely on certain specific results dealing with convex functionals. Although these results are easily obtained, they do not appear to be available in the literature. Let V be an arc-wise connected, compact point set in Es with a piece-wise smooth boundary and a nonzero Euclidean volume measure. Let CI(Y, M) denote the normed
Nonlinear Onsager theory of irreversibility
483
linear space of ordered M-tuples of functions {IV,} which are continuous on V, where the norm is that of uniform convergence, and let C,(V, N) denote the normed linear space of ordered N-tuples of functions (X,} which are continuous on V and where the norm is again that of uniform convergence. (One obviously can use different normed linear spaces, the only proviso being that the space be complete with respect to the norm selected.) Let @[W,x] be a continuous functional on C,(V, M) x C,( V, N) which was a continuous first functional derivative with respect to (X,} for each ( IV,} E C,(V, M). We use the notation L[ W, X; Y] = lim r-0
~[W,X+EYl-~[W,Xl [
E
1
for functional differentiation, so that L[W,X;aY+bZ]=aL[W,X;Y]+bL[W,X;Z]
(3.2)
holds for all real numbers (a, b), for all {W,} l C1(V, M) and for all {X,, Y,, 2,) E G(~,~)xC,(Y,~) XG(V,N). A functional @[W, X] with the continuity and digerentiability above is said to satisfy hypothesis H if and only if @[W,aX+(l-a)Y]
properties
Sa@[W,X]+(l-a)Q[W,Y], L(W, 0; Y] = 0
stated
(3.2) (3.3)
holds fir each ae[O, I], for each {W,) E C1(V, M) and for each {X,, Y,) E C2(V, N) X C,(V, N), where 0 stands for the zero element, X, = 0, of C,(V, N). The first condition simply states that a[ W, x] is a convex functional with respect to
its second set of arguments {X,}, while the second condition states that X, = 0 renders stationary with respect to the second set of arguments. It should be noted that we have made no assumptions concerning the existence of second functional derivatives of a,[ W, x] , nor do we need to make any.
@[W,Xj
1 IfQ,[W, X] satisjes hypothesis H, then
Lemma
L[W,X;Y]
G @[W,X+Y]-@,[W,X]
(3.4)
for each {W,} l C,(V, M) andfor each {X,, Y,} E C2(V, N) x C&V, IV). Proof. By convexity, we have @[W,X+aY]=Q)[W,a(X+Y)+(l-a)X]
sa@[W,X+Y]+(l-a)@[W,X],
and hence Q,[W,X+aY]
-Q[W,X]
d a(@[W,X+Y]--@[W,X]).
The result then follows on allowing a to approach zero and using the definition of L[W, x; Y].
484
DOMINIC
G. B. EDELEN
Lemma 2 If @[W, X] satisfies hypothesis H, then
@W,xl 3 @[W,Ol for
(3.5)
each { Wa} E Cl@‘, M) and for each {X,} E C2(V, N). Hence
g.1.b. @,[W, Xj = {X,1 E C,
@[W,XJ = @[W, 01. Proof. When we combine (3.3) and (3.4), we have
qw,
0 = L[W, 0; Y] s
Y] -@[W,
O]
since 0 is the identity of addition in C,(V, N). Theorem 1 If @ [ W, X] satisfies hypothesis H, then L[W,X;x]
2
0
(3.6)
for each { W,} E C,( V, M) andfor each (Xa} E C,( V, N). Proof. We have -L[W, X; Xj = L[W, X; -xJ by (3.2), and hence Lemma 1 gives -L[W, X; Xj s @[W, 0] -@[W, Xl. Since the right-hand-side of this inequality is nonpositive by Lemma 2, the result is established. Corollary 1 If @ [ W, X] satisfies hypothesis H, then
0 =sL[W,X;X]
6 @[w,2x]-@(w,x]
(3.7)
for each {W,} E C,(V, M) andfor each {X,} E C,(V, N). Proof. The result is a direct combination of Theorem 1 and Lemma 1. Corollary 2 If @[ W, X] satisfies hypothesis H, then
@I?, (1 +a)Xl
2
+[W,X],O
< a 6 1,
(3.8)
for each { W,} E C,(V, M) andfor each {X,) E C, (V, N). Proof. Using (3.2), (3.6) and (3.7), we have 0 G aL[W,X;X]
=L[W,X;aX]
s +[W,
(l+a)X]-@,CW,X].
Corollary 3 If @[ W, Xl satisfies hypothesis H and if @[W,X]=@[W,bX]
(3.9)
485
Nonlinear Onsager theory of irreversibility
forany
one value ofb, 0 <
b < 1, andfor each {IV,} EC#, @W, xl =
M), then
qw, 01
(3.10)
for each {W,} EC,(V, M). Proof We have @[W,aX] =@[w,ax+(l--)O]
G a@[w,x]+(l--a)@[JV,O],
0 < a < 1 by (3.2). If (3.9) holds for given b with 0 < b < 1, then setting a = b we have 0 c (l-b)(@[FV,OJ--Q[W,X]). Since 0 < b < 1, we obtain 0 c @[IV, O]Cp[ W, XJ.On the other hand, Lemma 2 gives 0 G @[W, X] -a [ W, 0) and the result follows.
If Q,[ W, X] satisfies hypothesis N and if @[W,2X] =ct)[W,X]
(3.11)
L[W, ax; ax] = 0
(3.12)
for all (W,} e C#‘, M), then
for al6 {W,) ECl (V, 44) and all values ofa in the interval [0, I]. Proof. The result follows by direct combination of Theorem 1 and Corollary 3. Corollary 5 if @[W, XJ satisfies hypothesis H and the ~-t~ple of quantities { J,( W, X)) is dejinedonC1(V,M)XC2(F”,N) by L[ W, X; Y] = j+, i J,Y, do a-1
(3.13)
for each ( Wn} e C, (If, M) andfor each {X,, Y,) EC, (V, N) X C, (V, N) then
0s
j-, ii1JaXadu G @[IV, 2X] -@[W,X]
(3.14)
foreach {W,)
e Cl (If, M) indoor each (X,) E C2(T/, IV). ProojI The existence of the quantities Ja follows from the standard theorem on the representation of a continuous linear functional (recall that Q,[ W, X] is continuous with a continuous first functional derivative). Since (3. I 1) holds for each {Ya}ECz (V, N) , it holds for Y, = X,, and hence we have L[W,X;X]
=sv
3 J,X,& a=1
(3.15)
for each {Wa} ECl (V, M) and for each {X,} E Cz (V, N) . The result then follows from Corollary 1.
486
DOMINIC
4. NONLINEAR
CONSTITUTIVE
G.
B. EDELEN
RELATIONS
FOR
IRREVERSIBLE
PROCESSES
The results established in the last section provide a simple and direct method of obtaining constitutive relations for irreversible processes which identically satisfy the postulate of i~eversibility. We start by making the following identifications. Set N = K+ 3 and identify the Ntuple {Xa} with {z&,(I/@),i}. We identify the M-tuple {W,} with all other thermodynamic state variables, such as temperature, displacement gradients, polarization, and so forth. Theorem 2 Let Cp[ W, X] satisfy hypothesis H and be invariant under superimposed rigid body motions, then the (nonlinear) constitutive relations
rr = r,(
W, X) = f?J,, 1 G r G K,
h’= h’(W,X) satisty
the postulate
=-JK+i,
1s i d 3
(4.1) (4.2)
of irreversibility for each { Wcr} (FC,(V, M) and for each {P,,
(1l(3),t} ECz (V, N) and satisfy the principle of material indi&rence. Proof. If the constitutive relations (4.1) and (4.2) hold, then (4.3) and hence (2.3) gives (4.4)
Corollary 5 then gives P 3 0 and hence (4.1) and (4.2) satisfy the postulate of irreversibility for each {Wa} E C,(V, M) and for each (X,} EC,(V, N). Since @[W,x] is invariant under superimposed rigid body motions, it follows from (3.1) and (3.13) that (I’,, h”} s&tisfythe principle of material indifference. Theorem 3 If the hypotheses of Theorem 2 are satisfied, then we have the bounds (4.5) for each { W,) E C1 (V, M) andfor each {Xa) e Cp ( V, N) .
Proof. The result is an immediate consequence of Theorem 2 and Corollary 5. The above result shows that not only do the constitutive relations (4.1) and (4.2) satisfy the postdate of irreversibility identically for each choice of the functional Q,[ W, X] which satisfies hypothesis H, but afso we have the upper bound for the total reduced entropy production. This property of providing an upper bound for the total reduced entropy production appears to be absent in the current literature. (Of course, if
487
Nonlinear Onsager theory of irreversibility
bound in the class of functionals @[W, X] that satisfy hypothesis N. Consider the functional Q [ W, X] that is defined by @[W, X-J= f, # du
(4.6)
where for 1) for
+(x) = {$$X,l-
lX,Ial IX,1 > 1.
The functional (4.6) satisfies hypothesis H, as is easily demonstmted, and L[W, X; x] = 1Oa2Jy dv, Q[W, 2Xj = 19a2 .JYdu and ip[W,XI = 9u2 Jy do for XI = 5. The upper bound is thus achieved by this @[W,x] for all {IV,} c C,( V, N) and for all {X,} E C1(V, N) with X, = 5. In fact, it is clear that whenever @[W, X] has a ‘linear portion’, we attain the upper bound for any state X which lies within the ‘linear portion’. If we substitute the constitutive relations (4.1’)and (4.2) into (3.13), we obtain the ‘vacations principle’ L[W, X; Y] = I, (2
?Y,.-hiY,,,}
dv.
r=1
We can thus summarize the whole theory by the following statement: @(W,uX+ (l-a)Y]
d
#~[W,X]+(l-ff)~[W,Y],
L[W, 0; Y] = 0, L[W,X;
Y] = I, (5
$Y,-KY,,,}
(4.7) (4.8)
du
(4.9)
l-1
shall hold for each a E IO, 11,for each { W,}e C, Uf, M) and for each {X,, Y,) ECZ (V, N) X C,( V, N) with (X,} = {t,, (l/e), i}, N = K + 3, in which case we have
OsP=
5 $-hi(j),
*} dv
7-l
=L[W,X;X]
c 4j[WJx]--a,[W,X].
(4.10)
It now follows immediately from (4.8) and (4.9) or (4.10) that {T’,,h*} vanishes with {t,, ( l/e),} . Thus, although we have not assumed that @[ W, X] possesses second functional derivatives, the total reduced entropy production vanishes at least to second order in {it,., (l/e), $1at the ‘equilib~um’ state 6,. = 0, (l/&Q,t = 0. We also note that the set of constant functionais satisfies (4.7) and (4.8) and hence the theory also includes bodies which are completely thermodynamically
reversible.
For general functionals which satisfy (4.7) and (4.8), we do not obtain the pointwise inequality (4.11) However, we will now show that there are particular classes of functionals for which the point-wise inequality (4.11) does obtain. Let &(W, ;u) be a function of M + N
488
DOMINIC
G. B. EDELEN
arguments which (I) is a convex function in the arguments {X,} for each { W,} E C, (I’, M), (2) has continuous first partial derivatives with respect to the N arguments {X,} for each {IV,} E C1(V, M), (3) 134/8X,, = 0 at X, = 0 for each {IV,} E C,(V, M), and (4) is such that (4.12) @W,Xl =I,. $du is invariant under superimposed rigid body motions. (Although condition (2) is actually implied by condition (l), it is included in the above list in the interests of ease for the reader.) It is easily demonstrated that the functional defined by (4.12) satisfies (4.7) and (4.8), and hence we have the constitutive relations (4.13) (4.14)
If we go back to the particular stress as
case represented
by (2.1), (4.13) gives the dissipative (4.15)
Thus $J may be viewed as a potential for the dissipative stress in the same way that the internal energy may be viewed as a potential for the recoverable stress. This analogy also carries over in the general case, for Edelen and Laws [4] among others have shown that a variational principle similar to (4.9) can be used to derive the potential representation for the recoverable stress and the temperature in terms of derivatives of the integral of the internal energy over the body V. In fact, it is now clear that specification of the internal energy function and the function 4 determine the energetic and mechanical responses of a material body. In the general case, we see that specification of the integral of the internal energy over the body as a functional of the thermodynamic state and the functional a[ W, X] determine the energetic and mechanical responses of material bodies. If 4 has continuous second derivatives, then (4.13) and (4.14) give the ‘reciprocity relations’ (4.16) ar, _ -_--
ahi
(4.17)
Suppose that we choose 4 to be independent of { W,} and homogeneous of degree two in {Xa}. It is then an easy matter to show that the quadratic form must be nonnega-
489
Nonlinear Onsager theory of irreversibility
tive definite in order to satisfy the convexity property and that (4.16) through (4.18) are just the Onsager reciprocity relations[l]. Further, the variational principle expressed by (4.9) is exactly what Prigogine[2] refers to as the principle of ‘minimum entropy production’ and which is used as a means of deriving Onsager’s reciprocity relations. It thus seems appropriate to refer to the theory presented here as a nonlinear Onsager theory of irreversibility. Acknowledgmenr-The author is indebted to R. S. Rivlin and N. Laws for the many informative discussions of the practices of thermodynamics. These discussions illuminated the subject and provided the motivation for the-work reported here. _ REFERENCES [l] L. ONSAGER, Phys. Rev. 37,40.5 (193 1); 38,2265 (193 1). 121 I. PRIGOGINE. Bull. Acad. r. Bela. Cl. Sci. 31.600 (1945). i3j C. TRUESDELL and R. A. TOUPIN, Hand&h dhrPhy;ik III/l. Springer-Verlag 141 D. G. B. EDELEN and N. LAWS, Arch. ration. Mech. Analysis 43,24. (Receioed
14 November
(1960).
197 1)
I&sun&- Soit V un corps matCrie1 et soit:
la production inteme d’entropie de V, ou J, sont les flux thermodynamiques et X. sont les forces thermodynamiques. Une theorie constitutive est obtenue qui determine les flux thermodynamiques en termes de derivees variationnelles d’un fonctionnel convexe @ [ W; X ] lequel prend son minimum absolu a X, = 0. On montre que 0 c P @[W;2X]-@[Iv;X] pour tout X, et pour tout param&.re. additionnel W, et par consequent nous pouvons satisfaire dune facon identique a la deuxibme loi de la thermodynamique. Si @ = Jr r/~(IV; X) du, alors J, = &$/ax. et nous obtenons les relations de r&iprocite ti./aX, = aJ,/aX,. La thCorie constitutive n’est pas limitee a des relations lineaires puisque les seules conditions requises pour +I( W; X) sont la convexite en X. et (a/ax,) 4 \xa_O= J, I.Ya=Ll=0. Des relations constitutives spicifiques sont obtenues pour le vecteur de flux de chaleur et pour le tenseur de tension de dissipation. Puisque $J peut dependre des parametres IV, de mEme que des parambtres X,,, la tension de dissipation et le flux de chaleur peuvent dependre de la contrainte et de la temperature, de m&me que du taux de contrainte et du gradient de la temp&ature. Zusammenfassung-
I’s011 einen materiellen Kiirper bezeichnen und
die inteme Entropieproduktion von V, worin J, die thermodynamischen Striimungen und X, die thermodynamischen Kriifte sind. Eine Materialtheorie wird erhalten, die die thennodynamischen Striimungen in Termen von variationellen Ableitungen eines konvexen Funktionals @ [ W; X] bestimmt, das sein absolutes minimum bei X, = 0 einnimmt. Es wird gezeigt dass 0 6 P s @[W;2X]-@[W;X] fir alle X,, und fir alle zusatzlichen Parameter W, ist, und wir deshalb identische Befriedigung des zweiten Grundgesetzes der Thermodynamik haben. Wenn cp= _fv4 (IV; X) do, dann J. = a4/aXa und wir erhalten die Reziprozitatsbeziehungen ti,,/aX, = aJ,/aX,,. Die Materialtheorie ist nicht auf lineare Beziehungen beschrknkt weil die einziien Anforderungen an d(W; X) Konvexitiit in X. und @/aXa +&,,_O= J&,=,, = 0 sind. Spezifische Materialbeziehungen werden fir den Warmeflussvektor und den Zerstreuungsspannungstensor erhaben. Weil Q, von den Parametem IV, als such von den X.‘s abh%ngen kann, kormen die Zerstreuungsspanmmg und der Warmefluss von der Spannung und Temperatur aIs such von der Spannungsrate und dem Temperaturgefalle abh%ngen.
DOMINIC
490 Sommarjo-
G. B. EDELEN
Che V denoti un corpo materiale e the
denoti la produzione in~madientropiadi V, dove J, SOROi flussi ~e~~in~~i e X, le forze te~~in~che. Si ottiene una teoria costitutiva the determina i flussi termodinamici in termki di Lrivative variazionali di unafunzionale convessacP(W;X] the assumed propriominimoassolutoa X, = 0; Sidimostrache 0 c Ps
@[W;2X]-@[W;X]
per tutti gii X, e per tutti i parametri supplementari u/,, e quindi abbiamo una soddisfazione identica della seconda legge delia termodinamica. Se Qt = I,, 4 f W; X) du, allora J, = a#jaX= ed otteniamo i rapporti di reciprocita &IJc?X~= CRUX=. La teoria costitutiva non t ristretta ai rapporti lineari in quanta le uniche impasizionisuQ,(W;X)sonolacoavessit~inX,e(~/~X,)(p~,=,=J,Jx,=o=0. Si ottengono rapporti costitutive specifici per il vettore de1 flusso termico e il tensore della sollecitazione dissipativa. Dal moment0 the b, pub dipendere dai parametri W, quanta dall’ X., la sollecitazione dissipativa e il flusso termico possono dipendere dalIa deformazione e dalla temperatura come pure dal tasso di deformazione e dalia curva di tempemtura.
e4xb eny~pexi~ee npo~3~OAc~~o 3~~pomiki Tena V, rAe J.Tep~o~awiwc~e noTo~~, X.---TepMoAHHaME¶ecXxie C&fAbL ~OJQWleTCfl KOHCTkiTyTKBHlUi TeOpKK, KOTOpWl OlTpeAWeT Te~MOAHHaMHUeCKHe i?OTOKH uepe3 BapwarproHHbIe npoH3BoAHbIe sbmyKnor0 @ymcwfofm.na @,[W,X], KOTO~& nprmblbfaer M?fHEMaJn8Hoe 3Ha9eme npx x.=o.&ISl Bcex x* H AJiirBceX AOnOJIHUTeJIbHblX napac~oea6conxxxoe MeIJ?OB ~,~~O,¶TO
o~P~#[w,2x]-@[w, CJleAOBaTWibHO
Q=lv(P(W;
Mb1
X)dv,
EMeM
HqeiiTHWOe
aa?
TOU'.=,~,
BbmOxHeHlie
HhSbInOJIyYEMcOoTHomeHHJl
Xl, BTOpOrO
3aXOHa
TepMOAliIfaMHKH.
Ewxi
BYUU~HO~T~~~=~~KOH~T~~~TIIBH~
CII K JDiHe&ibIM COOTHOmeHHRM, TaK KaK OTHOCHTe~HO ‘&w,x) Tpe6yeTCX reopw neorpm TOAbKO BbRIyXJIoCTb II0 x,, H (ajax3~l,,,,=J,I,=,=o. nOJWIeHb1 CReIWIbHbZe XOHCTHWTWibIe COOTHOnreHIfS #IS BeKTOpa TermOBnrO nOTOKa M AJIK TeH3Opa pacceHBarourer0 HallpJEi@HHX. T~K KaK @MOXeT!SiBiiWrbOTBceX &EiOTrEipZ&S?T$OB W,,TO~~m~H~pffPreHae~TennoBoftnOTOK MOl-yT 38BSiCeTb OT CKOpoCrrr L@U3pMWiH H T@MnfQiirypHOrO rpaSSi@HTa, a T8glge OT Ae@pbi~ H TehmepaTypbr.
Pergamon Press.
Printed in Great Britain
A NONLINEAR ONSAGER THEORY IRREVERSIBILITY DOMINIC G. B. EDELEN Center for the Application of Mathematics, Lehigh University, Ahstraet-
OF
Bethlehem, Pennsylvania
Let V denote a material body and let
denote the internal production of entropy of V, where J. are the thermodynamic fluxes and X, are the thermodynamic forces. A constitutive theory is obtained which determines the thermodynamic fluxes in terms of variational derivatives of a convex functional O[W;Xj which assumes its absolute minimum at X, = 0. It is shown that for all X, and for all additional parameters W,, and hence we have identical satisfaction of the Second Law of Thermodynamics. If Ct,= Jr &W;X)du, then J,, = &j&X, and we obtain the reciprocity relations &I&X, = a&k&,. The constitutive theory is not restricted to linear relations since the only requirements on $(W;X) are convexity in X, and (a/aX,)+J r._,, = J&,_,, = 0. Specific constitutive relations are obtained for the heat flux vector and the dissipative stress tensor. Since $ can depend on the parameters W, as well as on the X0’s, the dissipative stress and the heat flux can depend on the strain and temperature as well as on the rate of strain and the temperature gradient. 1. INTRODUCTION
THE RECIPROCITYrelations of Onsager[l] and the minimum entropy production principle of Prigogine[2] have provided access to both the understanding and the analysis of a wide range of physical phenomena. The principle drawback of the theory is that it is restricted to linear thermodynamic constitutive relations, so that phenomena have to be either linear or else the derivatives from equilibrium have to be sufficiently small that a linear approximation is valid. For phenomena which represent large nonlinear deviations from equilibrium states, such as large displacements of thermo-viscoelastic bodies, the Onsager theory can not be applied. This paper gives an extension of the Onsager theory wherein the restriction to linear constitutive relations is eliminated. After an examination of a number of equivalent formulations of the Onsager theory, it was finally found that the theory could be cast in a form which was not dependent on the assumption that the constitutive relations were linear. The essential properties of this formulation obtained from two conditions on functionals which are incorporated in the hypothesis H given in section 3. The theory that derives from these two conditions allows for quite general kinds of nonlinearity in the constitutive relations and the theory can be applied all the way down to the class of ideal bodies that are completely reversible. In addition, the constitutive relations that we obtain allow for inclusion of all of the thermodynamic variables rather than just those which appear as ‘thermodynamic forces’ in the reduced entropy production equation. One interesting aspect of the results is that the theory gives both the lower bound of zero and upper bounds for the reduced entropy production and the upper bound is, in fact, a least upper bound for a nontrivial class of functionals which satisfy hypothesis H and a nonvacuous class of states. 481
482 2. ENTROPY
DOMINIC PRODUCTION
AND
G. B. EDELEN
THE
SECOND
LAW OF THERMODYNAMICS
The notation used in this section follows, primarily, that of Truesdell and Toupin [3]. For a nonpolar body, the entropy production equation is given by (see [3], equations (256.12) and (257.1-257.3))
(2.1) In the general case, (2.1) is replaced by
where P, = ma. If q is viewed as the supply of energy per unit volume that comes from agencies external to the body, and if h is viewed as the supply of energy per unit area of the boundary of the body (h’ da* = h da) from agencies external to the body, then the right-hand-sides of (2.1) and (2.2) represent the total supply of entropy from agencies internal to the body. We accordingly refer to the quantity P, defined by P=
5 $+hf+),,}
dv,
(2.3)
l-1
as the total reduced entropy production of the body. Since there is not universal agreement on the form of the second law of thermodynamics under all circumstances, we confine our attention in this paper to bodies whose internal thermodynamic processes satisfy the following postulate of irreversibility: the total reduced entropy production is nonnegative at each instant of time, that is P L 0.
(2.4)
Because (2.4) is required to hold for each instant of time, there is no loss of generality in suppressing the explicit time dependence of functions and functionals in what follows. After the analysis is complete, explicit time dependences are easily accounted for by simply inserting t as an explicit argument in addition to those considered. The purpose of this paper is to obtain a general class of constitutive relations for the quantities {I’,, hi} whereby the postulate of irreversibility is ‘satisfied identically’. Since our results are based upon functional analysis, it is unnecessary to require satisfaction of (2.4) for any proper subset of the body. Of course, appropriate additional assumptions easily lead to particularizations of the general theory for which the integrand in (2.3) will be nonnegative at each point of the body. 3. FUNCTIONAL
CONSIDERATIONS
AND
CONVEXITY
The constitutive relations we obtain in the next section rely on certain specific results dealing with convex functionals. Although these results are easily obtained, they do not appear to be available in the literature. Let V be an arc-wise connected, compact point set in Es with a piece-wise smooth boundary and a nonzero Euclidean volume measure. Let CI(Y, M) denote the normed
Nonlinear Onsager theory of irreversibility
483
linear space of ordered M-tuples of functions {IV,} which are continuous on V, where the norm is that of uniform convergence, and let C,(V, N) denote the normed linear space of ordered N-tuples of functions (X,} which are continuous on V and where the norm is again that of uniform convergence. (One obviously can use different normed linear spaces, the only proviso being that the space be complete with respect to the norm selected.) Let @[W,x] be a continuous functional on C,(V, M) x C,( V, N) which was a continuous first functional derivative with respect to (X,} for each ( IV,} E C,(V, M). We use the notation L[ W, X; Y] = lim r-0
~[W,X+EYl-~[W,Xl [
E
1
for functional differentiation, so that L[W,X;aY+bZ]=aL[W,X;Y]+bL[W,X;Z]
(3.2)
holds for all real numbers (a, b), for all {W,} l C1(V, M) and for all {X,, Y,, 2,) E G(~,~)xC,(Y,~) XG(V,N). A functional @[W, X] with the continuity and digerentiability above is said to satisfy hypothesis H if and only if @[W,aX+(l-a)Y]
properties
Sa@[W,X]+(l-a)Q[W,Y], L(W, 0; Y] = 0
stated
(3.2) (3.3)
holds fir each ae[O, I], for each {W,) E C1(V, M) and for each {X,, Y,) E C2(V, N) X C,(V, N), where 0 stands for the zero element, X, = 0, of C,(V, N). The first condition simply states that a[ W, x] is a convex functional with respect to
its second set of arguments {X,}, while the second condition states that X, = 0 renders stationary with respect to the second set of arguments. It should be noted that we have made no assumptions concerning the existence of second functional derivatives of a,[ W, x] , nor do we need to make any.
@[W,Xj
1 IfQ,[W, X] satisjes hypothesis H, then
Lemma
L[W,X;Y]
G @[W,X+Y]-@,[W,X]
(3.4)
for each {W,} l C,(V, M) andfor each {X,, Y,} E C2(V, N) x C&V, IV). Proof. By convexity, we have @[W,X+aY]=Q)[W,a(X+Y)+(l-a)X]
sa@[W,X+Y]+(l-a)@[W,X],
and hence Q,[W,X+aY]
-Q[W,X]
d a(@[W,X+Y]--@[W,X]).
The result then follows on allowing a to approach zero and using the definition of L[W, x; Y].
484
DOMINIC
G. B. EDELEN
Lemma 2 If @[W, X] satisfies hypothesis H, then
@W,xl 3 @[W,Ol for
(3.5)
each { Wa} E Cl@‘, M) and for each {X,} E C2(V, N). Hence
g.1.b. @,[W, Xj = {X,1 E C,
@[W,XJ = @[W, 01. Proof. When we combine (3.3) and (3.4), we have
qw,
0 = L[W, 0; Y] s
Y] -@[W,
O]
since 0 is the identity of addition in C,(V, N). Theorem 1 If @ [ W, X] satisfies hypothesis H, then L[W,X;x]
2
0
(3.6)
for each { W,} E C,( V, M) andfor each (Xa} E C,( V, N). Proof. We have -L[W, X; Xj = L[W, X; -xJ by (3.2), and hence Lemma 1 gives -L[W, X; Xj s @[W, 0] -@[W, Xl. Since the right-hand-side of this inequality is nonpositive by Lemma 2, the result is established. Corollary 1 If @ [ W, X] satisfies hypothesis H, then
0 =sL[W,X;X]
6 @[w,2x]-@(w,x]
(3.7)
for each {W,} E C,(V, M) andfor each {X,} E C,(V, N). Proof. The result is a direct combination of Theorem 1 and Lemma 1. Corollary 2 If @[ W, X] satisfies hypothesis H, then
@I?, (1 +a)Xl
2
+[W,X],O
< a 6 1,
(3.8)
for each { W,} E C,(V, M) andfor each {X,) E C, (V, N). Proof. Using (3.2), (3.6) and (3.7), we have 0 G aL[W,X;X]
=L[W,X;aX]
s +[W,
(l+a)X]-@,CW,X].
Corollary 3 If @[ W, Xl satisfies hypothesis H and if @[W,X]=@[W,bX]
(3.9)
485
Nonlinear Onsager theory of irreversibility
forany
one value ofb, 0 <
b < 1, andfor each {IV,} EC#, @W, xl =
M), then
qw, 01
(3.10)
for each {W,} EC,(V, M). Proof We have @[W,aX] =@[w,ax+(l--)O]
G a@[w,x]+(l--a)@[JV,O],
0 < a < 1 by (3.2). If (3.9) holds for given b with 0 < b < 1, then setting a = b we have 0 c (l-b)(@[FV,OJ--Q[W,X]). Since 0 < b < 1, we obtain 0 c @[IV, O]Cp[ W, XJ.On the other hand, Lemma 2 gives 0 G @[W, X] -a [ W, 0) and the result follows.
If Q,[ W, X] satisfies hypothesis N and if @[W,2X] =ct)[W,X]
(3.11)
L[W, ax; ax] = 0
(3.12)
for all (W,} e C#‘, M), then
for al6 {W,) ECl (V, 44) and all values ofa in the interval [0, I]. Proof. The result follows by direct combination of Theorem 1 and Corollary 3. Corollary 5 if @[W, XJ satisfies hypothesis H and the ~-t~ple of quantities { J,( W, X)) is dejinedonC1(V,M)XC2(F”,N) by L[ W, X; Y] = j+, i J,Y, do a-1
(3.13)
for each ( Wn} e C, (If, M) andfor each {X,, Y,) EC, (V, N) X C, (V, N) then
0s
j-, ii1JaXadu G @[IV, 2X] -@[W,X]
(3.14)
foreach {W,)
e Cl (If, M) indoor each (X,) E C2(T/, IV). ProojI The existence of the quantities Ja follows from the standard theorem on the representation of a continuous linear functional (recall that Q,[ W, X] is continuous with a continuous first functional derivative). Since (3. I 1) holds for each {Ya}ECz (V, N) , it holds for Y, = X,, and hence we have L[W,X;X]
=sv
3 J,X,& a=1
(3.15)
for each {Wa} ECl (V, M) and for each {X,} E Cz (V, N) . The result then follows from Corollary 1.
486
DOMINIC
4. NONLINEAR
CONSTITUTIVE
G.
B. EDELEN
RELATIONS
FOR
IRREVERSIBLE
PROCESSES
The results established in the last section provide a simple and direct method of obtaining constitutive relations for irreversible processes which identically satisfy the postulate of i~eversibility. We start by making the following identifications. Set N = K+ 3 and identify the Ntuple {Xa} with {z&,(I/@),i}. We identify the M-tuple {W,} with all other thermodynamic state variables, such as temperature, displacement gradients, polarization, and so forth. Theorem 2 Let Cp[ W, X] satisfy hypothesis H and be invariant under superimposed rigid body motions, then the (nonlinear) constitutive relations
rr = r,(
W, X) = f?J,, 1 G r G K,
h’= h’(W,X) satisty
the postulate
=-JK+i,
1s i d 3
(4.1) (4.2)
of irreversibility for each { Wcr} (FC,(V, M) and for each {P,,
(1l(3),t} ECz (V, N) and satisfy the principle of material indi&rence. Proof. If the constitutive relations (4.1) and (4.2) hold, then (4.3) and hence (2.3) gives (4.4)
Corollary 5 then gives P 3 0 and hence (4.1) and (4.2) satisfy the postulate of irreversibility for each {Wa} E C,(V, M) and for each (X,} EC,(V, N). Since @[W,x] is invariant under superimposed rigid body motions, it follows from (3.1) and (3.13) that (I’,, h”} s&tisfythe principle of material indifference. Theorem 3 If the hypotheses of Theorem 2 are satisfied, then we have the bounds (4.5) for each { W,) E C1 (V, M) andfor each {Xa) e Cp ( V, N) .
Proof. The result is an immediate consequence of Theorem 2 and Corollary 5. The above result shows that not only do the constitutive relations (4.1) and (4.2) satisfy the postdate of irreversibility identically for each choice of the functional Q,[ W, X] which satisfies hypothesis H, but afso we have the upper bound for the total reduced entropy production. This property of providing an upper bound for the total reduced entropy production appears to be absent in the current literature. (Of course, if
487
Nonlinear Onsager theory of irreversibility
bound in the class of functionals @[W, X] that satisfy hypothesis N. Consider the functional Q [ W, X] that is defined by @[W, X-J= f, # du
(4.6)
where for 1) for
+(x) = {$$X,l-
lX,Ial IX,1 > 1.
The functional (4.6) satisfies hypothesis H, as is easily demonstmted, and L[W, X; x] = 1Oa2Jy dv, Q[W, 2Xj = 19a2 .JYdu and ip[W,XI = 9u2 Jy do for XI = 5. The upper bound is thus achieved by this @[W,x] for all {IV,} c C,( V, N) and for all {X,} E C1(V, N) with X, = 5. In fact, it is clear that whenever @[W, X] has a ‘linear portion’, we attain the upper bound for any state X which lies within the ‘linear portion’. If we substitute the constitutive relations (4.1’)and (4.2) into (3.13), we obtain the ‘vacations principle’ L[W, X; Y] = I, (2
?Y,.-hiY,,,}
dv.
r=1
We can thus summarize the whole theory by the following statement: @(W,uX+ (l-a)Y]
d
#~[W,X]+(l-ff)~[W,Y],
L[W, 0; Y] = 0, L[W,X;
Y] = I, (5
$Y,-KY,,,}
(4.7) (4.8)
du
(4.9)
l-1
shall hold for each a E IO, 11,for each { W,}e C, Uf, M) and for each {X,, Y,) ECZ (V, N) X C,( V, N) with (X,} = {t,, (l/e), i}, N = K + 3, in which case we have
OsP=
5 $-hi(j),
*} dv
7-l
=L[W,X;X]
c 4j[WJx]--a,[W,X].
(4.10)
It now follows immediately from (4.8) and (4.9) or (4.10) that {T’,,h*} vanishes with {t,, ( l/e),} . Thus, although we have not assumed that @[ W, X] possesses second functional derivatives, the total reduced entropy production vanishes at least to second order in {it,., (l/e), $1at the ‘equilib~um’ state 6,. = 0, (l/&Q,t = 0. We also note that the set of constant functionais satisfies (4.7) and (4.8) and hence the theory also includes bodies which are completely thermodynamically
reversible.
For general functionals which satisfy (4.7) and (4.8), we do not obtain the pointwise inequality (4.11) However, we will now show that there are particular classes of functionals for which the point-wise inequality (4.11) does obtain. Let &(W, ;u) be a function of M + N
488
DOMINIC
G. B. EDELEN
arguments which (I) is a convex function in the arguments {X,} for each { W,} E C, (I’, M), (2) has continuous first partial derivatives with respect to the N arguments {X,} for each {IV,} E C1(V, M), (3) 134/8X,, = 0 at X, = 0 for each {IV,} E C,(V, M), and (4) is such that (4.12) @W,Xl =I,. $du is invariant under superimposed rigid body motions. (Although condition (2) is actually implied by condition (l), it is included in the above list in the interests of ease for the reader.) It is easily demonstrated that the functional defined by (4.12) satisfies (4.7) and (4.8), and hence we have the constitutive relations (4.13) (4.14)
If we go back to the particular stress as
case represented
by (2.1), (4.13) gives the dissipative (4.15)
Thus $J may be viewed as a potential for the dissipative stress in the same way that the internal energy may be viewed as a potential for the recoverable stress. This analogy also carries over in the general case, for Edelen and Laws [4] among others have shown that a variational principle similar to (4.9) can be used to derive the potential representation for the recoverable stress and the temperature in terms of derivatives of the integral of the internal energy over the body V. In fact, it is now clear that specification of the internal energy function and the function 4 determine the energetic and mechanical responses of a material body. In the general case, we see that specification of the integral of the internal energy over the body as a functional of the thermodynamic state and the functional a[ W, X] determine the energetic and mechanical responses of material bodies. If 4 has continuous second derivatives, then (4.13) and (4.14) give the ‘reciprocity relations’ (4.16) ar, _ -_--
ahi
(4.17)
Suppose that we choose 4 to be independent of { W,} and homogeneous of degree two in {Xa}. It is then an easy matter to show that the quadratic form must be nonnega-
489
Nonlinear Onsager theory of irreversibility
tive definite in order to satisfy the convexity property and that (4.16) through (4.18) are just the Onsager reciprocity relations[l]. Further, the variational principle expressed by (4.9) is exactly what Prigogine[2] refers to as the principle of ‘minimum entropy production’ and which is used as a means of deriving Onsager’s reciprocity relations. It thus seems appropriate to refer to the theory presented here as a nonlinear Onsager theory of irreversibility. Acknowledgmenr-The author is indebted to R. S. Rivlin and N. Laws for the many informative discussions of the practices of thermodynamics. These discussions illuminated the subject and provided the motivation for the-work reported here. _ REFERENCES [l] L. ONSAGER, Phys. Rev. 37,40.5 (193 1); 38,2265 (193 1). 121 I. PRIGOGINE. Bull. Acad. r. Bela. Cl. Sci. 31.600 (1945). i3j C. TRUESDELL and R. A. TOUPIN, Hand&h dhrPhy;ik III/l. Springer-Verlag 141 D. G. B. EDELEN and N. LAWS, Arch. ration. Mech. Analysis 43,24. (Receioed
14 November
(1960).
197 1)
I&sun&- Soit V un corps matCrie1 et soit:
la production inteme d’entropie de V, ou J, sont les flux thermodynamiques et X. sont les forces thermodynamiques. Une theorie constitutive est obtenue qui determine les flux thermodynamiques en termes de derivees variationnelles d’un fonctionnel convexe @ [ W; X ] lequel prend son minimum absolu a X, = 0. On montre que 0 c P @[W;2X]-@[Iv;X] pour tout X, et pour tout param&.re. additionnel W, et par consequent nous pouvons satisfaire dune facon identique a la deuxibme loi de la thermodynamique. Si @ = Jr r/~(IV; X) du, alors J, = &$/ax. et nous obtenons les relations de r&iprocite ti./aX, = aJ,/aX,. La thCorie constitutive n’est pas limitee a des relations lineaires puisque les seules conditions requises pour +I( W; X) sont la convexite en X. et (a/ax,) 4 \xa_O= J, I.Ya=Ll=0. Des relations constitutives spicifiques sont obtenues pour le vecteur de flux de chaleur et pour le tenseur de tension de dissipation. Puisque $J peut dependre des parametres IV, de mEme que des parambtres X,,, la tension de dissipation et le flux de chaleur peuvent dependre de la contrainte et de la temperature, de m&me que du taux de contrainte et du gradient de la temp&ature. Zusammenfassung-
I’s011 einen materiellen Kiirper bezeichnen und
die inteme Entropieproduktion von V, worin J, die thermodynamischen Striimungen und X, die thermodynamischen Kriifte sind. Eine Materialtheorie wird erhalten, die die thennodynamischen Striimungen in Termen von variationellen Ableitungen eines konvexen Funktionals @ [ W; X] bestimmt, das sein absolutes minimum bei X, = 0 einnimmt. Es wird gezeigt dass 0 6 P s @[W;2X]-@[W;X] fir alle X,, und fir alle zusatzlichen Parameter W, ist, und wir deshalb identische Befriedigung des zweiten Grundgesetzes der Thermodynamik haben. Wenn cp= _fv4 (IV; X) do, dann J. = a4/aXa und wir erhalten die Reziprozitatsbeziehungen ti,,/aX, = aJ,/aX,,. Die Materialtheorie ist nicht auf lineare Beziehungen beschrknkt weil die einziien Anforderungen an d(W; X) Konvexitiit in X. und @/aXa +&,,_O= J&,=,, = 0 sind. Spezifische Materialbeziehungen werden fir den Warmeflussvektor und den Zerstreuungsspannungstensor erhaben. Weil Q, von den Parametem IV, als such von den X.‘s abh%ngen kann, kormen die Zerstreuungsspanmmg und der Warmefluss von der Spannung und Temperatur aIs such von der Spannungsrate und dem Temperaturgefalle abh%ngen.
DOMINIC
490 Sommarjo-
G. B. EDELEN
Che V denoti un corpo materiale e the
denoti la produzione in~madientropiadi V, dove J, SOROi flussi ~e~~in~~i e X, le forze te~~in~che. Si ottiene una teoria costitutiva the determina i flussi termodinamici in termki di Lrivative variazionali di unafunzionale convessacP(W;X] the assumed propriominimoassolutoa X, = 0; Sidimostrache 0 c Ps
@[W;2X]-@[W;X]
per tutti gii X, e per tutti i parametri supplementari u/,, e quindi abbiamo una soddisfazione identica della seconda legge delia termodinamica. Se Qt = I,, 4 f W; X) du, allora J, = a#jaX= ed otteniamo i rapporti di reciprocita &IJc?X~= CRUX=. La teoria costitutiva non t ristretta ai rapporti lineari in quanta le uniche impasizionisuQ,(W;X)sonolacoavessit~inX,e(~/~X,)(p~,=,=J,Jx,=o=0. Si ottengono rapporti costitutive specifici per il vettore de1 flusso termico e il tensore della sollecitazione dissipativa. Dal moment0 the b, pub dipendere dai parametri W, quanta dall’ X., la sollecitazione dissipativa e il flusso termico possono dipendere dalIa deformazione e dalla temperatura come pure dal tasso di deformazione e dalia curva di tempemtura.
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o~P~#[w,2x]-@[w, CJleAOBaTWibHO
Q=lv(P(W;
Mb1
X)dv,
EMeM
HqeiiTHWOe
aa?
TOU'.=,~,
BbmOxHeHlie
HhSbInOJIyYEMcOoTHomeHHJl
Xl, BTOpOrO
3aXOHa
TepMOAliIfaMHKH.
Ewxi
BYUU~HO~T~~~=~~KOH~T~~~TIIBH~
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