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A thermodynamic analysis of rigid heat conductors
hf. I Engng Sci. Vol. 18. pp. 121-139 @ Pergamon Press Ltd.. 1990. Printed in Great Britain
A THERMODYNAMIC ANALYSIS OF RIGID HEAT CONDUCTORS G. LEBON Thermodynamique des Phtnomknes IrrCversibles, UniversitC de Litge, Sart Tilman, Building B6, B-4000 Litge, Belgium (Communicated by D. G. B. EDELEN) (Receioed IO July 1979)
Abstract-A thermodynamic approach to rigid heat conductors is proposed: it introduces the heat flux vector as indeoendent variable while its temporal evolution is governed by a first order differential equation. The for; of the second law is that proposed by Miiller wherein the eniropy flux and the entropy sourceare not given a priori but determined through constitutive equations. Restrictions on the constitutive equations are placed by the second law. Some properties, valid in the vicinity of equilibrium are established. In particular, it is shown that the present theory leads to a hyperbolic heat conduction equation, allowing for the propagation of heat as a thermal wave with a finite velocity. The concept of thermodynamic forces and fluxes is also introduced. The latter are seen to derive from a potential function plus an additional term. Finally, it is established under which conditions symmetry relations are satisfied.
1. INTRODUCTION THIS PAPER contains
the ingredients of a new thermodynamic formalism. In order to make the ideas clear and explicit, the simple problem of the rigid heat conductor will be developed. A first class of thermodynamic theories rests on the local equilibrium hypothesis [ l-33. They are gathered under the vocable of classical non-equilibrium thermodynamics. Accordingly, every point in space is characterized by a locally stable equilibrium state. As a consequence, all the quantities defined in equilibrium, like absolute temperature and entropy, remain univocally defined in non-equilibrium. Such a hypothesis is certainly justified in the very vicinity of equilibrium&71 but cannot pretend to cover situations far from equilibrium. Moreover, other basic results of these theories, like the Curie principle and the Onsager reciprocity relations, have been acridly contested [8]. This has motivated Coleman [9] and No11[ lo] to propose a new theory based on completely different grounds, and nowadays termed as rational thermodynamics. Entropy is now introduced through the Clausius-Duhem inequality written as
The above inequality will be used as a restrictive condition on the possible forms of the constitutive equations. In expression (l.l), 17denotes the specific entropy, p is the density, T is the absolute temperature, q is the heat flux vector and r is the rate of heat supply; an upper dot represents the material time derivative. The entropy and the temperature are introduced as, quoting Truesdell[ll], “primitive, undefined variables described only by such properties as are laid down for them in mathematical terms”. This has been severely criticized by many authors, especially Rivlin[ 121 and Meixner[l3] who raised the acute problem of the measurement of the non-equilibrium temperature. Moreover, the entropy appears through an inequality and, therefore, cannot pretend to be a unique quantity, as explicitely shown by Day [ 141.Clausius-Duhem inequality also implies that heat pulses propagate with an infinite velocity[l5,16]. However, the paradox of an infinite speed of propagation can be removed by assuming that the heat flux is a functional of the entire history of the temperature gradient [ 171. 121
728
G.LEBON
A further drawback has been pointed out by Green and Naghdi[l4]. They consider a homogeneous isotropic rigid heat conductor admitti~ some particular dependence of the entropy and the internal energy on T and p and obeying the Fourier law. Inequality (1. I) requires then curiously that a continuous supply of heat produces a decrease of the temperature of the medium. Miiller[l%20] strongly objected against the forms (l/jr)q of the entropy flux and (l/T)r of the rate of entropy supply. These expressions were suggested by thermostatics but were not justified far from equilibrium. On the contrary, the kinetic theory of gases [21] indicates clearly that beyond the so-called Navier-Stokes approximation, the entropy flux contains supplementary terms in addition to (l/T)q. All these reasons incited Miiller to present a new formalism wherein the entropy flux and the rate of entropy supply are not guessed Q priori, but are considered as constitutive quantities. An extra assumption is also introduced: the normal component of the entropy flux is continuous at a wall where the temperature is continuous. This allows Miilier to define a new concept, the coldness, which for a certain class of materials is an universal function of the empirical temperature 8 and its time derivative 6 The introduction of 4 as a genuine variable does not receive a definite physical interpretation, nor do the concepts of temperature and coldness[l3]. All that is learned is that in equilibrium, the coldness is the reciprocal of the absolute temperature. Although Mtiller’s approach yields a heat conduction equation of hyperbolic nature, this is obtained by introducing an ad hoc assumption, which is discussed in Section 4. It should also be mentioned that the extension of Miiller’s theory to anisotropic systems is far from being a trivial matter [ IS]. Up to now, only heat conduction in anisotropic rigid bodies has been studied~201. In the same way as Mullet, Green and Laws[22] give up the Clausius~~hem inequality (1.1) in favor of
In equilibrium, 4 is the absolute temperature while outside equilibrium, 4 requires a constitutive equation. The essential difference with Mtiller’s is that the form of the rate of entropy supply is imposed from the start to be equal to r/#. Green and Laws recover most of Mtiiler’s results. This brief survey shoutd not be complete by not ~ntioning the cont~butions of Meixnerf 13,231 and Day[24]. These authors refuse to handle with a non~u~ib~urn entropy. Although their ideas are undoubtedly valuable, they will not be discussed here because they are not directly related with the content of the present work. Our purpose is to develop a new approach to continuum thermodynamics by discussing a typical example, the rigid heat conductor. The next section is devoted to the formulation of the constitutive equations and to their justification. The temperature, its gradient and the heat flux vector are selected as independent variables. The dependent variables are taken to be the internal energy (or Gibbs’ free energy function), the entropy, the entropy flux, the entropy source supply and the time variation of the heat flux. The restrictions imposed by the second law are laid down in Section 3. Some properties, valid in the vicinity of equ~ib~um, are derived in Section 4. In particular, it is shown that the heat equation is of hyperbolic character. In Section 5, the notion of therm~ynamic fluxes and forces is introduced. The fluxes are shown to take the form of the gradient of a potential plus a so-called powerless term. The potential function bears some resemblance to JXdelen’s dissipation function[25-281. Some symmetry relations analogous to Onsager’s reciprocal relations are also derived. 2.THECONSTITUTIVEEQUATIONS
Consider a rigid continuous conductor of constant density p submitted to non-mechanical external constraints. To each point x of the continuum, one assigns empirical temperature 6; it will be seen that fl reduces to the absolute Kelvin temperature T in equilibrium.
A thermodynamic
analysis of rigid heat conductors
129
~tho~h the assumption that there exists a non~qu~ib~um temperature has been repeatedly criticized in the past[12,13,29], this hypothesis is widely used in contem~~y thermodynamics [9,18,22,23,24,30]. The two problems open to discussion remain the physical reability of the concept of non-equilibrium temperature and the means of its measurement. The objective of thermodynamics is the determination of the scalar field f3(x,t) at each instant of time. This can be achieved by means of the balance of energy pi = -qj,j + pr
(2.1)
where E is the specific internal energy while r, the rate of heat supply, is arbitrary chosen to satisfy (2.1). A comma stands for the derivation with respect to the spatial variables. To solve the problem, two ways are open. Either formulate l and qi in terms of e or, and this is the attitude adopted in the present work, write E in terms of e and qi
and add a supplementary equation expressing that the rate of change of qi is governed by a function Q of B and Q
Since it is wanted to recover a Maxwell-Cattaneo type constitutive equation, E and Qi are in addition assumed to be functions of 8,i
4
=
Qi(& e.i, Sib
(2.4)
It is a well known fact that the energy balance equation does not impose enough ~s~ctions. Supplement~y constraints are placed on the process by the second law of therm~ynamics, which in its most general form[20] is written as
This fun~men~ in~u~ity introduces three supple~nt~ variables: the specific entropy n, the entropy flux vector 4 and the rate of supply of entropy i. This inequality is more general than Clausius-Duhem’s inequality in that the entropy flux is not identified with (l/e)q and that $ is not a p&n’ equal to (l/e)r. To be complete, one must add supplementary constitutive equations expressing the new variables in terms of 8, B,i and qi, namely
7) = cbi =
V(& e.i, 4i) Me,
e,i, Sill
i = ft e, @,i* @I-
(2.6) (2.7) (2.8)
By no means does the inequality (2.5) define the entropy. It is only assumed that there exists a non-equilibrium entropy r) obeying the second law of thermodynamics expressed by inequality (2.5). Instead of e, it is convenient to work with @(= E - &1) as dependent variable and to replace the constitutive relation (2.3) by
730
GLEBON
Finally, one does not alter appreciabIy the generality of the theory by replacing (2.7) and (2.8) by
A=
+a f
Me,
B= g(e,
(2.9)
fb, d
(2.10)
B,i,qi)r.
In short, our form~ism applies to the class of systems described by the set of constitutive equations
$ = +(e,e,h 4i)
(2.11)
de9 e,i, 4)
(2.12)
t =
j = g(e, 4
=
Qi(4
Bi,
(2.14)
4i)r
(2.15)
e,i,4i)-
Limitations on the possible forms of these constitutive equations are provided by the inequality (2.5) together with the energy balance (2.1). in classical irreversible thermodynamics, a distinction is made between state equations (involving only quantities which do not vanish at equilibrium) and phenomenological equations (which vanish identically in equilibrium, like Fourier’s law), This distinction is abandoned here. 3.CONSEQUENCESOFTHEENTROPYINEQIJALITY Using (2.1) and the definition of tj, the entropy inequality (2.5) becomes -
ptJI+ 7)tj) + e#i.i -
C&i -
p(ei -
r) k 0.
(3.1)
With the constitutive eqns (2.11)-(2.15), the above condition writes as
(3.2) which is linear in the derivatives 8, d,i, 8,ij, qij and r. The above inequality must hold for arbitrary values of fl, 0.i and qi and in particular for arbitrary values of their derivatives and r. This is realized by setting all the coefficients of 4, ii, B,ii, qi~ and r equal to zero, which gives (3.3) (3.4)
ge-i=o
(3.5)
ak,=O
a@
(3.6)
-$+$=o
(3.7)
J
,I
A thermodynamic analysis of rigid heat conductors
731
13.8) From (3.3), it is seen that $ does not depend on the temperature gradient; this is also true for the entropy accordingly to (3.4) which states that, like in equilib~um, the entropy is derivable from a potential. Expression (3.5) indicates that the entropy supply reduces to its classical vahre, namely * 1 S=-r. e
Equation (3.6) expresses that k?is independent of the heat flux components while the solution of (3.7) is &i = flij(B)B,j
f
$(e)
Szij= -0,
(3.10)
where 0, and fii can depend on 8 only. Moreover, Qi must be taken equal to zero because ki is imposed to vanish in equilibrium i.e. for 8,i. After substitution of (3.10) in (3.8) one obtains (3.11) This inequality simplifies further; indeed Qi must be taken equal to zero because ki is imposed to vanish in equ~ib~um i.e. for 8.i = 0. The final form of the fundamental inequality is then given by (3.12) The left hand side represents the energy dissipated per unit time and unit volume inside the medium due to thermal effects. In the classical theory[l, 2], only the second term should be present; the same classical expression is obtained in Coleman-Noll’s formalism by assuming that 0, 4 and B,i are the only independent variables[l5]. Muller’s theory[l5,18] yields, instead of (3.12) (3.13) where A denotes the “coldness” while Green and Laws1221 propose (3.14) #J is a function requiring a constitutive equation and is equal to the temperature equilibrium.
4. EQUILIBRIUM
~uilib~um
0 in
PROPERTIES
is defined by f3.i= qi = 0.
(4.1)
The property (4.1) remains valid in presence of mechanical deformations, i.e. for thermoelastic bodies. This means that whatever the values of the temperature and the strain fields, the heat flux vanishes at equilibrium, ruling out the piezo-electric effect in equilibrium. Denoting U&S. IS/S-F
732
G.LEBON
by a(& B,i,qi) the 1.h.s. of (3.12), it is clear that outside equilibrium (4.2) while at equilibrium, S is minimum and equal to zero. Denoting by lE the value at equilibrium, one has
SIE= S(& 0,O) = 0.
(4.3)
As a consequence of (4.2) and (4.3)
as
I
as
o
T&T=
I =o
(4.4)
a4iE
and
a26
-I
aqiaqj E
(4.5)
From (4.4), it follows that
a*
I
4ilE =
aqiE=O
O
(4.6)
while from (4.5), it is checked that -z
a2s dqi
I E
= 2p- aV SC0 W%j
(4.7)
3%
(4.8) (i, j = 1,2,3 but no summation on P$&lEgIE+$(E=O (i,j, k = 1,2,3 but j# k).
i)
(4.9)
(4.10)
From now on, the developments in this section will be limited to isotropic systems. In this case, 4 depends on qiqi only so that (4.7) and (4.9) reduce to (i = 1,2,3 but no summation on i)
P$%lE~lE =-iI, (i = 1,2,3 but no summation on i)
(4.11)
(4.12)
while (4.10) vanishes identically. Expand $ around its equilibrium value. By virtue of (4.6) and (3.4), expressing that JI is independent of B,i,l(i is given by (4.13) where
A thermodynamic
analysis of rigid heat conductors
733
Since $ assumes its minimum value in equilibrium, y(0) is a positive quantity. With this result in mind, (4.11) and (4.12) give 3
aqi E
~0
and
(no summation on i).
(4.14)
In the linear approximation, the constitutive eqn (2.15) takes the following form (ji =
gilE
qi (no summation on i) ai,i(Ee.i +zIE
(4.15)
I
(4.16)
+dqi
or more simply, by setting adi
a=-z
(no summation on i)
E’
and recalling that in equilibrium, (ji]E vanishes 4i = -Cre,i- @i.
(4.17)
According to (4.14), a and p are positive quantities and from (4.12), it is seen that (Yand y are related by (4.18) The above results may be directly generalized to anisotropic bodies. The quantities a, p, y are then tensors which, as a consequence of (4.5). are non-negative definite. In the literature[31-331, the constitutive eqn (4.17) is usually written in the form (4.19) where r is a positive relaxation time and K the heat conductivity. This relation has been justified by experiments and theory; for metals, K is generally taken as a constant whereas T is inversely proportional to the temperature. By comparison of (4.17) and (4.19), it is seen that K (y=-
(4.20)
p=f. 7
This indicates that the coefficients a and /3 can receive a physical interpretation and confirms the positive definite character of both coefficients. It will also be shown that a relation like (4.17) or (4.19) produces a hyperbolic heat conduction equation. By substitution of the constitutive equation
l(e,
qi) =
E(e)JE +
$$IEq:+ ***t I
in the energy balance (2.1), one obtains after that second order term in b2, diqi, diqi and 42 have been dropped (4.22) tThe absence of B., as variable and the vanishing of the first order term in (4.21) are a direct consequence of eqns (3.3). (3.4) and (4.6).
734
G.LEBON
Since &/c%& is positive in order to obey the principie of stable local equilibrium, the above equation is clearly of hyperbolic nature and therefore allows for propagation of temperature waves with a finite velocity. At the same order of approximation, Miiller[lS, 191derives the following heat conduction equation
In contrast to expression (4.221, the c~~~ient of 4 is no longer (a~~~~)~but (a~~~~)~and this coefficient may be either positive or negative. Therefore, if it is wanted to obtain an hyperbolic equation, a supplementary ad hoc assumption must be required, namely that (&.#a& be positive. Such an assumption is not needed in our approach. A last observation concerns the establishment of the Gibbs equation. From the definition of (I, and eqns (3.3) and (3.4), it is directly derived that (4.23) which in equilib~um writes as
This expression is nothing but the classical Gibbs equation of thermostatics at the condition to identify 01, with the absolute Kelvin temperature. Observe however, that outside equilibrium, and contrary to the classical irreversible thermodynamics[l, 21, the Gibbs equation is replaced by (4.23).
5.ONTHERMODYNAMICFLUXESANDFORCES
It may be asked if, in analogy with the classical irreversible thermodyn~ics, thermodynami~ thrxes and forces can be introduced in the frame of the present approach. Moreover, by ad~tting that the fluxes and forces are related by phenomenoIo~c~ relations, to what extent do they satisfy reciprocity relations “a la Onsager”? It is well known that the Onsager reciprocical relations are only justified in the linear domain. Outside this range, Edeien[26-281 has demonstrated that the reciprocity relations 8J” -=-
dXB
aJp ax”
61)
are not generally valid; J” and X* denote, respectively, the fluxes and the forces. In this section, a particular set of fluxes and forces is defined. It is shown that the fluxes are derivable from a potential function, which bears some resemblance to E?delen’s dissipation potenti~[2~28]. Stying from this result, it is determined under which conditions reciprocal relations exist. The fluxes and forces are imposed to meet the following requirements. I. The forces vanish in equilibrium X”IE = 0.
(5.2)
2. The fluxes can be expressed in terms of the forces and a set of extra variables ak, like the temperature and density, which do not vanish in equilibrium J” = J”(X@,M’).
(5.3)
3. The energy ~ssipated per unit time and unit volume takes the form of a qua~atic
A thermodynamic analysis of rigid heat conductors
135
expression in the fluxes and the forces, respectively (5.4) For a rigid heat conducting problem, one selects as forces the independent variables 0.i and - (l/e)qi, which clearly vanish at equilibrium
An inspection of expression (3.12) of 6 suggests to identify the fluxes, respectively, with J = (-/$y
(5.6)
4).
It is clear from (4.1) that in equilibrium, the fluxes vanish. It must be noted that the choice (5.5) and (5.6) differs from that proposed by Edelen[26] who takes for the rigid heat conductor
and
(5.7)
5.1 A potential function Our next purpose is to formulate the above defined fluxes (5.6) in terms of a potential function, Therefore, introduce the notion of power P associated with the forces X”; by definition
P =C (I
0
J=(x@, e)x== -p$ e,j- 2 J
21 #I&
(5.8)
The factor (Q/K~)’has been introduced for dimensional reasons, TOand ~~ denote, respectively, a reference relaxation time and a reference heat conductivity. Expression (5.8) is not to be confused with the entropy production. Although it vanishes in equilibrium, it is not necessarily positive definite outside equilibrium. Moreover, one can always express the vector-valued flux J” in the general form[26,28]
~a
_
a@txB, gk)+ ua(x.8 axa
,Ok)
(5.9)
with
T X”U”(X@, wk) = 0.
(5.10)
The quantities @ and U” are, respectively, given by I @(XS, wk) =
I0
X”J”(AX~, wk) dh
(5.11)
(5.12)
(5.13) where one has put hi = +
(5.14)
and where use has been made of the property that # is independent of 4.i.Derivation of (5.13) with respect to 8,; yields
(5.IS)
Substituting (5.16)in (5.15)and identifying the last integral in (5.15)with Ui+expression (5.15) becomes simply :=*I-& ,i
Xn the
or
-p$=$+& $
,i
same way, the derivation of CDwith respect to the force iri leads to
wherein the arguments of +i and ci;have been omitted.
(5.17)
A thermodynamic analysis of rigid heat conductors
137
After integrating the second integral by parts and calling - W the remaining integrals, eqn (5.18) can be written as (5.19) It is verified that the vector-valued functions Ui and Wi obey the orthogonality relation
U#,i t W,hj = 0
(5.20)
WE= Wils = 0.
(5.21)
and that in equilibrium
The fluxes -p(&$@) and Qj are clearly expressible as the sum of two terms: the first consists of the gradient of the potential Cpwith respect to the conjugated force while the second is of powerless essence. It remains to demonstrate that the above decomposition (5.17) and (5.19) is unique. Suppose that two set of functions @, Ui, Wi and a*, UT, W? can be found. It follows then from eqns (5.17) and (5.19) that
-j+-@*)=-(GUT) .I
(5.22)
-&@-lp*)=-(wWT). I
(5.23)
After multiplication, respectively, by 0,i and hi, one obtains in virtue of the orthogonality condition (5.20)
a
fi,i*Q,
a - @*)+ hi$@ - @*)= 0.
(5.24)
From Euler’s theorem on homogeneous functions, it can be shown that the general solution of (5.24) depends only on 8[28] and the decomposition is unique. 5.3. Symmetry reiati~~ As a preliminary remark, it must be noted that in contrast to other theories[!& 16,17,19], one cannot define a heat conductivity tensor by aqi
I
Kii = be, E
because one has selected qi and @,Ias inde~ndent variabIes. In most theories, the sym~try property of Kij does not arise from the second principle of thermodynamics but is rather a consequence of a supplementary statement, like the microscopic reversibility[34], the invariance of the infinitesimal entropy production with respect to time reversal[35] or stability concepts [36]. Other authors, however[l6,19,20], were able to prove that the symmetry of the heat conductivity tensor is a direct consequence of the second law. In our approach, the existence of symmetry relations of the form
aJa aJ@ --_axfl axa
(5.25)
is subordinated to the observance of restrictive conditions. To establish them, de&ate (5.17)
738
GLEBON
and (5.15%respectively, with respect to @~, and hi. This teads to the ~o~~owi~relations (5.26)
15.27)
(5.28)
By assuming that the r.h.s, vanish, the above relations take the form (525) of &sager’s reciprocity relations, namely (5.29)
Of course, these relations are automatically satisfied when ul:= Wi= 0. It is interesting to discuss in further detail the linear situation, for which 4
=
-rU{j6,j+
(5.30)
&hj
(5.31)
Equation (5.30) is the simplest non-stationary extension of Fourier’s law obtained by setting Qi= 0 and identifying & with the Kronecker symbol. From the results of Section 4, all the * coefficientsin (5.30)and (5.31)are positive and besides yjj is symmetric. At the same Iinear order of approximation, V;:and U$ are of the form lI&=
lijej
f
(5.32)
@lijhj
w =pij@J“t
(5.33)
nijhj
with & = 1~as can directly be seen by deriving (X17) with respect to @,j.The ~~~go~l~~ condition (5.20)places some restrictions on the values of the coefficients 1ij,mii, pij, nij. It is easily checked that
As a consequence, (5.32)and (5.33)reduce to
Wi = -ds;
where m stands for REP. With these m&s,
+ nijhj
Q.26j+.B) give the syrup pij - @jjI- 2&j
“ij = ryii
cle/arly,& is the onlycoefficient which is not symmetric,
refatiuns
A thermodynamic analysis of rigid heat conductors
739
6. CONCLUSIONS
It has been shown that by introducing the heat-flux vector as independent variable and a supplementary constitutive equation involving its time derivative, one obtains some interesting results about rigid heat conductors. Among other things, the Maxwell-Cattanco equation comes out in a natural way and the coefficients in it are seen to have the correct sign; the heat conduction equation allows now for finite thermal wave speed without making appeal to ad hoc conditions. Thermodynamic fluxes and forces are also defined. In contrast to the classical point of view, the entropy production does no longer appear as a bilinear form in the fluxes and the forces, but rather as the sum of two quadratic expressions in the fluxes and the forces, respectively. The fluxes are seen to derive from a potential function plus an additional powerless term, the latter obeying an orthogonality condition. The existence of symmetry relations, extending the well-known Onsager’s reciprocity laws, is not guaranteed but linked to the particular form of the powerless term. When the latter vanish, symmetry laws are established. Although the problem developed here is rather simple, there should be no fundamental difficulty in extending the above results to more complicated systems like deformable bodies or viscous thermo-fluids. These topics are intended to be discussed in subsequent papers. REFERENCES [I] I. PRIGOGINE, Introduction to Thermodynamicsof Irreversible Processes. Interscience, New York (l%l). [2] S. R. De GROOT and P. MAZUR, Non-Equilibtirm Thermodynamics.North-Holland, Amsterdam (1962). [3] I. GYARMATI, Non-Equilibrium Thermodynamics. Springer Verlag, Berlin (1970). [4] I. PRIGOGINE, Bull. Acad. R. Sci. Belgique 31,600 (1945). [5] J. MEIXNER and H. G. REIK, Thermodynamicder Irreversible Prozesses, Hd der Phys., III/2. Springer Verlag, Berlin (1959). [6] G. LEBON, Bull. Acad. R. Sci. Relgique 64,456 (1978). [71G. LEBON, D. JOU and J. CASAS, J. Physics A. (1979). [E] C. TRUESDELL, Rational Thermodynamics.Mc-Graw Hill, New York (1%9). [9] B. COLEMAN, Arch. Rat. Mech. Anal. 17, 1 (1964). [IO] W. NOLL, The Foundations of Mechanics and Thermodynamics.Springer Verlag, Berlin (1974). [ll] C. TRUESDELL, In I. U.T.A.M.Symp. Vienna. (Edited by Parkus and Sedov). p. 373. Springer Verlag, Berlin (1966). 1121R. S. RIVLIN, Recent Adv. in Engng Sci. 8, 1 (1977). [13] J. MEIXNER, Arch. Rat. Mech. Anal. 57,281 (1974). [14] W. DAY, Acta Mechanica 27,251 (1977). [15] K. HU’ITER, Acta Mechanica, 27, 1 (1977). [16] A. E. GREEN and P. M. NAGHDI, Proc. Roy. Sot. London, .4357,253(1977). [17] M. G&JRTINand A. PIPKIN, Arch. Rat. Mech. Anal. 31, 113(1968). [18] I. MIjLLER, Arch. Rat. Mech. Anal. 26, 118(1%7). [19] I. MULLER, Arch. Rat. Mech. Anal. 40, 1 (1971). [20] I. MULLER, Arch. Rat. Mech. Anal. 41,319 (1971). [21] S. CH,2PMAN and T. COWLING, The Mathematical Theory of Non-Uniform Gases, 3rd Edn. Cambridge University Press, Cambridge (1970). [22] A. E. GREEN and N. LAWS, Arch. Rat. Mech. Anal. 45,47 (1972). [23] J. MEIXNER, Arch. Rat. Mech. Anal. 33,33 (1%9). [24] W. DAY, The Thermodynamics of Simple Materials with Fading Memory, Tracts in Natural Philosophy, Vol. 22. Springer Verlag, Berlin (1972). [25] D. EDELEN, Arch. Rat. Mech. Anal. 51,218 (1973). [26] D. EDELEN, Znt.J. Engng Sci. 12, 121(1974). [27] D. EDELEN, J. Non-Equil. Thermodyn.2,205 (1977). [28] J. BATAILLE, D. EDELEN and J. KESTIN, Znt.J. Engng Sci. 17,563 (1979). [29] J. DOMINGOS, M. NINA and J. WHITELAW (editors), Foundations of Continuous Thermodynamics.Macmillan, London (1974). [30] S. NEMAT-NASSER, In Mechanics Today. (Edited by Nemat-Nasser), Vol. 2, p. 94. Pergamon, New York (1975). 1311P. VERNOlTE, C. R. Acad. Sci. Paris 247,3154 (1958). [32] H. WILLEM and S. CHOI, J. them. Phys. 63,2119 (1975). 1331G. LEBON and J. LAMBERMONT, I; Mkanique 15,579 (1976). 1341L. ONSAGER, Phys. Reu. 38,2256 (1931). [35] M. GURTIN, Arch. Rat. Mech. Anal. 44, 387 (1972). (Received 10 July 1979)
A THERMODYNAMIC ANALYSIS OF RIGID HEAT CONDUCTORS G. LEBON Thermodynamique des Phtnomknes IrrCversibles, UniversitC de Litge, Sart Tilman, Building B6, B-4000 Litge, Belgium (Communicated by D. G. B. EDELEN) (Receioed IO July 1979)
Abstract-A thermodynamic approach to rigid heat conductors is proposed: it introduces the heat flux vector as indeoendent variable while its temporal evolution is governed by a first order differential equation. The for; of the second law is that proposed by Miiller wherein the eniropy flux and the entropy sourceare not given a priori but determined through constitutive equations. Restrictions on the constitutive equations are placed by the second law. Some properties, valid in the vicinity of equilibrium are established. In particular, it is shown that the present theory leads to a hyperbolic heat conduction equation, allowing for the propagation of heat as a thermal wave with a finite velocity. The concept of thermodynamic forces and fluxes is also introduced. The latter are seen to derive from a potential function plus an additional term. Finally, it is established under which conditions symmetry relations are satisfied.
1. INTRODUCTION THIS PAPER contains
the ingredients of a new thermodynamic formalism. In order to make the ideas clear and explicit, the simple problem of the rigid heat conductor will be developed. A first class of thermodynamic theories rests on the local equilibrium hypothesis [ l-33. They are gathered under the vocable of classical non-equilibrium thermodynamics. Accordingly, every point in space is characterized by a locally stable equilibrium state. As a consequence, all the quantities defined in equilibrium, like absolute temperature and entropy, remain univocally defined in non-equilibrium. Such a hypothesis is certainly justified in the very vicinity of equilibrium&71 but cannot pretend to cover situations far from equilibrium. Moreover, other basic results of these theories, like the Curie principle and the Onsager reciprocity relations, have been acridly contested [8]. This has motivated Coleman [9] and No11[ lo] to propose a new theory based on completely different grounds, and nowadays termed as rational thermodynamics. Entropy is now introduced through the Clausius-Duhem inequality written as
The above inequality will be used as a restrictive condition on the possible forms of the constitutive equations. In expression (l.l), 17denotes the specific entropy, p is the density, T is the absolute temperature, q is the heat flux vector and r is the rate of heat supply; an upper dot represents the material time derivative. The entropy and the temperature are introduced as, quoting Truesdell[ll], “primitive, undefined variables described only by such properties as are laid down for them in mathematical terms”. This has been severely criticized by many authors, especially Rivlin[ 121 and Meixner[l3] who raised the acute problem of the measurement of the non-equilibrium temperature. Moreover, the entropy appears through an inequality and, therefore, cannot pretend to be a unique quantity, as explicitely shown by Day [ 141.Clausius-Duhem inequality also implies that heat pulses propagate with an infinite velocity[l5,16]. However, the paradox of an infinite speed of propagation can be removed by assuming that the heat flux is a functional of the entire history of the temperature gradient [ 171. 121
728
G.LEBON
A further drawback has been pointed out by Green and Naghdi[l4]. They consider a homogeneous isotropic rigid heat conductor admitti~ some particular dependence of the entropy and the internal energy on T and p and obeying the Fourier law. Inequality (1. I) requires then curiously that a continuous supply of heat produces a decrease of the temperature of the medium. Miiller[l%20] strongly objected against the forms (l/jr)q of the entropy flux and (l/T)r of the rate of entropy supply. These expressions were suggested by thermostatics but were not justified far from equilibrium. On the contrary, the kinetic theory of gases [21] indicates clearly that beyond the so-called Navier-Stokes approximation, the entropy flux contains supplementary terms in addition to (l/T)q. All these reasons incited Miiller to present a new formalism wherein the entropy flux and the rate of entropy supply are not guessed Q priori, but are considered as constitutive quantities. An extra assumption is also introduced: the normal component of the entropy flux is continuous at a wall where the temperature is continuous. This allows Miilier to define a new concept, the coldness, which for a certain class of materials is an universal function of the empirical temperature 8 and its time derivative 6 The introduction of 4 as a genuine variable does not receive a definite physical interpretation, nor do the concepts of temperature and coldness[l3]. All that is learned is that in equilibrium, the coldness is the reciprocal of the absolute temperature. Although Mtiller’s approach yields a heat conduction equation of hyperbolic nature, this is obtained by introducing an ad hoc assumption, which is discussed in Section 4. It should also be mentioned that the extension of Miiller’s theory to anisotropic systems is far from being a trivial matter [ IS]. Up to now, only heat conduction in anisotropic rigid bodies has been studied~201. In the same way as Mullet, Green and Laws[22] give up the Clausius~~hem inequality (1.1) in favor of
In equilibrium, 4 is the absolute temperature while outside equilibrium, 4 requires a constitutive equation. The essential difference with Mtiller’s is that the form of the rate of entropy supply is imposed from the start to be equal to r/#. Green and Laws recover most of Mtiiler’s results. This brief survey shoutd not be complete by not ~ntioning the cont~butions of Meixnerf 13,231 and Day[24]. These authors refuse to handle with a non~u~ib~urn entropy. Although their ideas are undoubtedly valuable, they will not be discussed here because they are not directly related with the content of the present work. Our purpose is to develop a new approach to continuum thermodynamics by discussing a typical example, the rigid heat conductor. The next section is devoted to the formulation of the constitutive equations and to their justification. The temperature, its gradient and the heat flux vector are selected as independent variables. The dependent variables are taken to be the internal energy (or Gibbs’ free energy function), the entropy, the entropy flux, the entropy source supply and the time variation of the heat flux. The restrictions imposed by the second law are laid down in Section 3. Some properties, valid in the vicinity of equ~ib~um, are derived in Section 4. In particular, it is shown that the heat equation is of hyperbolic character. In Section 5, the notion of therm~ynamic fluxes and forces is introduced. The fluxes are shown to take the form of the gradient of a potential plus a so-called powerless term. The potential function bears some resemblance to JXdelen’s dissipation function[25-281. Some symmetry relations analogous to Onsager’s reciprocal relations are also derived. 2.THECONSTITUTIVEEQUATIONS
Consider a rigid continuous conductor of constant density p submitted to non-mechanical external constraints. To each point x of the continuum, one assigns empirical temperature 6; it will be seen that fl reduces to the absolute Kelvin temperature T in equilibrium.
A thermodynamic
analysis of rigid heat conductors
129
~tho~h the assumption that there exists a non~qu~ib~um temperature has been repeatedly criticized in the past[12,13,29], this hypothesis is widely used in contem~~y thermodynamics [9,18,22,23,24,30]. The two problems open to discussion remain the physical reability of the concept of non-equilibrium temperature and the means of its measurement. The objective of thermodynamics is the determination of the scalar field f3(x,t) at each instant of time. This can be achieved by means of the balance of energy pi = -qj,j + pr
(2.1)
where E is the specific internal energy while r, the rate of heat supply, is arbitrary chosen to satisfy (2.1). A comma stands for the derivation with respect to the spatial variables. To solve the problem, two ways are open. Either formulate l and qi in terms of e or, and this is the attitude adopted in the present work, write E in terms of e and qi
and add a supplementary equation expressing that the rate of change of qi is governed by a function Q of B and Q
Since it is wanted to recover a Maxwell-Cattaneo type constitutive equation, E and Qi are in addition assumed to be functions of 8,i
4
=
Qi(& e.i, Sib
(2.4)
It is a well known fact that the energy balance equation does not impose enough ~s~ctions. Supplement~y constraints are placed on the process by the second law of therm~ynamics, which in its most general form[20] is written as
This fun~men~ in~u~ity introduces three supple~nt~ variables: the specific entropy n, the entropy flux vector 4 and the rate of supply of entropy i. This inequality is more general than Clausius-Duhem’s inequality in that the entropy flux is not identified with (l/e)q and that $ is not a p&n’ equal to (l/e)r. To be complete, one must add supplementary constitutive equations expressing the new variables in terms of 8, B,i and qi, namely
7) = cbi =
V(& e.i, 4i) Me,
e,i, Sill
i = ft e, @,i* @I-
(2.6) (2.7) (2.8)
By no means does the inequality (2.5) define the entropy. It is only assumed that there exists a non-equilibrium entropy r) obeying the second law of thermodynamics expressed by inequality (2.5). Instead of e, it is convenient to work with @(= E - &1) as dependent variable and to replace the constitutive relation (2.3) by
730
GLEBON
Finally, one does not alter appreciabIy the generality of the theory by replacing (2.7) and (2.8) by
A=
+a f
Me,
B= g(e,
(2.9)
fb, d
(2.10)
B,i,qi)r.
In short, our form~ism applies to the class of systems described by the set of constitutive equations
$ = +(e,e,h 4i)
(2.11)
de9 e,i, 4)
(2.12)
t =
j = g(e, 4
=
Qi(4
Bi,
(2.14)
4i)r
(2.15)
e,i,4i)-
Limitations on the possible forms of these constitutive equations are provided by the inequality (2.5) together with the energy balance (2.1). in classical irreversible thermodynamics, a distinction is made between state equations (involving only quantities which do not vanish at equilibrium) and phenomenological equations (which vanish identically in equilibrium, like Fourier’s law), This distinction is abandoned here. 3.CONSEQUENCESOFTHEENTROPYINEQIJALITY Using (2.1) and the definition of tj, the entropy inequality (2.5) becomes -
ptJI+ 7)tj) + e#i.i -
C&i -
p(ei -
r) k 0.
(3.1)
With the constitutive eqns (2.11)-(2.15), the above condition writes as
(3.2) which is linear in the derivatives 8, d,i, 8,ij, qij and r. The above inequality must hold for arbitrary values of fl, 0.i and qi and in particular for arbitrary values of their derivatives and r. This is realized by setting all the coefficients of 4, ii, B,ii, qi~ and r equal to zero, which gives (3.3) (3.4)
ge-i=o
(3.5)
ak,=O
a@
(3.6)
-$+$=o
(3.7)
J
,I
A thermodynamic analysis of rigid heat conductors
731
13.8) From (3.3), it is seen that $ does not depend on the temperature gradient; this is also true for the entropy accordingly to (3.4) which states that, like in equilib~um, the entropy is derivable from a potential. Expression (3.5) indicates that the entropy supply reduces to its classical vahre, namely * 1 S=-r. e
Equation (3.6) expresses that k?is independent of the heat flux components while the solution of (3.7) is &i = flij(B)B,j
f
$(e)
Szij= -0,
(3.10)
where 0, and fii can depend on 8 only. Moreover, Qi must be taken equal to zero because ki is imposed to vanish in equilibrium i.e. for 8,i. After substitution of (3.10) in (3.8) one obtains (3.11) This inequality simplifies further; indeed Qi must be taken equal to zero because ki is imposed to vanish in equ~ib~um i.e. for 8.i = 0. The final form of the fundamental inequality is then given by (3.12) The left hand side represents the energy dissipated per unit time and unit volume inside the medium due to thermal effects. In the classical theory[l, 2], only the second term should be present; the same classical expression is obtained in Coleman-Noll’s formalism by assuming that 0, 4 and B,i are the only independent variables[l5]. Muller’s theory[l5,18] yields, instead of (3.12) (3.13) where A denotes the “coldness” while Green and Laws1221 propose (3.14) #J is a function requiring a constitutive equation and is equal to the temperature equilibrium.
4. EQUILIBRIUM
~uilib~um
0 in
PROPERTIES
is defined by f3.i= qi = 0.
(4.1)
The property (4.1) remains valid in presence of mechanical deformations, i.e. for thermoelastic bodies. This means that whatever the values of the temperature and the strain fields, the heat flux vanishes at equilibrium, ruling out the piezo-electric effect in equilibrium. Denoting U&S. IS/S-F
732
G.LEBON
by a(& B,i,qi) the 1.h.s. of (3.12), it is clear that outside equilibrium (4.2) while at equilibrium, S is minimum and equal to zero. Denoting by lE the value at equilibrium, one has
SIE= S(& 0,O) = 0.
(4.3)
As a consequence of (4.2) and (4.3)
as
I
as
o
T&T=
I =o
(4.4)
a4iE
and
a26
-I
aqiaqj E
(4.5)
From (4.4), it follows that
a*
I
4ilE =
aqiE=O
O
(4.6)
while from (4.5), it is checked that -z
a2s dqi
I E
= 2p- aV SC0 W%j
(4.7)
3%
(4.8) (i, j = 1,2,3 but no summation on P$&lEgIE+$(E=O (i,j, k = 1,2,3 but j# k).
i)
(4.9)
(4.10)
From now on, the developments in this section will be limited to isotropic systems. In this case, 4 depends on qiqi only so that (4.7) and (4.9) reduce to (i = 1,2,3 but no summation on i)
P$%lE~lE =-iI, (i = 1,2,3 but no summation on i)
(4.11)
(4.12)
while (4.10) vanishes identically. Expand $ around its equilibrium value. By virtue of (4.6) and (3.4), expressing that JI is independent of B,i,l(i is given by (4.13) where
A thermodynamic
analysis of rigid heat conductors
733
Since $ assumes its minimum value in equilibrium, y(0) is a positive quantity. With this result in mind, (4.11) and (4.12) give 3
aqi E
~0
and
(no summation on i).
(4.14)
In the linear approximation, the constitutive eqn (2.15) takes the following form (ji =
gilE
qi (no summation on i) ai,i(Ee.i +zIE
(4.15)
I
(4.16)
+dqi
or more simply, by setting adi
a=-z
(no summation on i)
E’
and recalling that in equilibrium, (ji]E vanishes 4i = -Cre,i- @i.
(4.17)
According to (4.14), a and p are positive quantities and from (4.12), it is seen that (Yand y are related by (4.18) The above results may be directly generalized to anisotropic bodies. The quantities a, p, y are then tensors which, as a consequence of (4.5). are non-negative definite. In the literature[31-331, the constitutive eqn (4.17) is usually written in the form (4.19) where r is a positive relaxation time and K the heat conductivity. This relation has been justified by experiments and theory; for metals, K is generally taken as a constant whereas T is inversely proportional to the temperature. By comparison of (4.17) and (4.19), it is seen that K (y=-
(4.20)
p=f. 7
This indicates that the coefficients a and /3 can receive a physical interpretation and confirms the positive definite character of both coefficients. It will also be shown that a relation like (4.17) or (4.19) produces a hyperbolic heat conduction equation. By substitution of the constitutive equation
l(e,
qi) =
E(e)JE +
$$IEq:+ ***t I
in the energy balance (2.1), one obtains after that second order term in b2, diqi, diqi and 42 have been dropped (4.22) tThe absence of B., as variable and the vanishing of the first order term in (4.21) are a direct consequence of eqns (3.3). (3.4) and (4.6).
734
G.LEBON
Since &/c%& is positive in order to obey the principie of stable local equilibrium, the above equation is clearly of hyperbolic nature and therefore allows for propagation of temperature waves with a finite velocity. At the same order of approximation, Miiller[lS, 191derives the following heat conduction equation
In contrast to expression (4.221, the c~~~ient of 4 is no longer (a~~~~)~but (a~~~~)~and this coefficient may be either positive or negative. Therefore, if it is wanted to obtain an hyperbolic equation, a supplementary ad hoc assumption must be required, namely that (&.#a& be positive. Such an assumption is not needed in our approach. A last observation concerns the establishment of the Gibbs equation. From the definition of (I, and eqns (3.3) and (3.4), it is directly derived that (4.23) which in equilib~um writes as
This expression is nothing but the classical Gibbs equation of thermostatics at the condition to identify 01, with the absolute Kelvin temperature. Observe however, that outside equilibrium, and contrary to the classical irreversible thermodynamics[l, 21, the Gibbs equation is replaced by (4.23).
5.ONTHERMODYNAMICFLUXESANDFORCES
It may be asked if, in analogy with the classical irreversible thermodyn~ics, thermodynami~ thrxes and forces can be introduced in the frame of the present approach. Moreover, by ad~tting that the fluxes and forces are related by phenomenoIo~c~ relations, to what extent do they satisfy reciprocity relations “a la Onsager”? It is well known that the Onsager reciprocical relations are only justified in the linear domain. Outside this range, Edeien[26-281 has demonstrated that the reciprocity relations 8J” -=-
dXB
aJp ax”
61)
are not generally valid; J” and X* denote, respectively, the fluxes and the forces. In this section, a particular set of fluxes and forces is defined. It is shown that the fluxes are derivable from a potential function, which bears some resemblance to E?delen’s dissipation potenti~[2~28]. Stying from this result, it is determined under which conditions reciprocal relations exist. The fluxes and forces are imposed to meet the following requirements. I. The forces vanish in equilibrium X”IE = 0.
(5.2)
2. The fluxes can be expressed in terms of the forces and a set of extra variables ak, like the temperature and density, which do not vanish in equilibrium J” = J”(X@,M’).
(5.3)
3. The energy ~ssipated per unit time and unit volume takes the form of a qua~atic
A thermodynamic analysis of rigid heat conductors
135
expression in the fluxes and the forces, respectively (5.4) For a rigid heat conducting problem, one selects as forces the independent variables 0.i and - (l/e)qi, which clearly vanish at equilibrium
An inspection of expression (3.12) of 6 suggests to identify the fluxes, respectively, with J = (-/$y
(5.6)
4).
It is clear from (4.1) that in equilibrium, the fluxes vanish. It must be noted that the choice (5.5) and (5.6) differs from that proposed by Edelen[26] who takes for the rigid heat conductor
and
(5.7)
5.1 A potential function Our next purpose is to formulate the above defined fluxes (5.6) in terms of a potential function, Therefore, introduce the notion of power P associated with the forces X”; by definition
P =C (I
0
J=(x@, e)x== -p$ e,j- 2 J
21 #I&
(5.8)
The factor (Q/K~)’has been introduced for dimensional reasons, TOand ~~ denote, respectively, a reference relaxation time and a reference heat conductivity. Expression (5.8) is not to be confused with the entropy production. Although it vanishes in equilibrium, it is not necessarily positive definite outside equilibrium. Moreover, one can always express the vector-valued flux J” in the general form[26,28]
~a
_
a@txB, gk)+ ua(x.8 axa
,Ok)
(5.9)
with
T X”U”(X@, wk) = 0.
(5.10)
The quantities @ and U” are, respectively, given by I @(XS, wk) =
I0
X”J”(AX~, wk) dh
(5.11)
(5.12)
(5.13) where one has put hi = +
(5.14)
and where use has been made of the property that # is independent of 4.i.Derivation of (5.13) with respect to 8,; yields
(5.IS)
Substituting (5.16)in (5.15)and identifying the last integral in (5.15)with Ui+expression (5.15) becomes simply :=*I-& ,i
Xn the
or
-p$=$+& $
,i
same way, the derivation of CDwith respect to the force iri leads to
wherein the arguments of +i and ci;have been omitted.
(5.17)
A thermodynamic analysis of rigid heat conductors
137
After integrating the second integral by parts and calling - W the remaining integrals, eqn (5.18) can be written as (5.19) It is verified that the vector-valued functions Ui and Wi obey the orthogonality relation
U#,i t W,hj = 0
(5.20)
WE= Wils = 0.
(5.21)
and that in equilibrium
The fluxes -p(&$@) and Qj are clearly expressible as the sum of two terms: the first consists of the gradient of the potential Cpwith respect to the conjugated force while the second is of powerless essence. It remains to demonstrate that the above decomposition (5.17) and (5.19) is unique. Suppose that two set of functions @, Ui, Wi and a*, UT, W? can be found. It follows then from eqns (5.17) and (5.19) that
-j+-@*)=-(GUT) .I
(5.22)
-&@-lp*)=-(wWT). I
(5.23)
After multiplication, respectively, by 0,i and hi, one obtains in virtue of the orthogonality condition (5.20)
a
fi,i*Q,
a - @*)+ hi$@ - @*)= 0.
(5.24)
From Euler’s theorem on homogeneous functions, it can be shown that the general solution of (5.24) depends only on 8[28] and the decomposition is unique. 5.3. Symmetry reiati~~ As a preliminary remark, it must be noted that in contrast to other theories[!& 16,17,19], one cannot define a heat conductivity tensor by aqi
I
Kii = be, E
because one has selected qi and @,Ias inde~ndent variabIes. In most theories, the sym~try property of Kij does not arise from the second principle of thermodynamics but is rather a consequence of a supplementary statement, like the microscopic reversibility[34], the invariance of the infinitesimal entropy production with respect to time reversal[35] or stability concepts [36]. Other authors, however[l6,19,20], were able to prove that the symmetry of the heat conductivity tensor is a direct consequence of the second law. In our approach, the existence of symmetry relations of the form
aJa aJ@ --_axfl axa
(5.25)
is subordinated to the observance of restrictive conditions. To establish them, de&ate (5.17)
738
GLEBON
and (5.15%respectively, with respect to @~, and hi. This teads to the ~o~~owi~relations (5.26)
15.27)
(5.28)
By assuming that the r.h.s, vanish, the above relations take the form (525) of &sager’s reciprocity relations, namely (5.29)
Of course, these relations are automatically satisfied when ul:= Wi= 0. It is interesting to discuss in further detail the linear situation, for which 4
=
-rU{j6,j+
(5.30)
&hj
(5.31)
Equation (5.30) is the simplest non-stationary extension of Fourier’s law obtained by setting Qi= 0 and identifying & with the Kronecker symbol. From the results of Section 4, all the * coefficientsin (5.30)and (5.31)are positive and besides yjj is symmetric. At the same Iinear order of approximation, V;:and U$ are of the form lI&=
lijej
f
(5.32)
@lijhj
w =pij@J“t
(5.33)
nijhj
with & = 1~as can directly be seen by deriving (X17) with respect to @,j.The ~~~go~l~~ condition (5.20)places some restrictions on the values of the coefficients 1ij,mii, pij, nij. It is easily checked that
As a consequence, (5.32)and (5.33)reduce to
Wi = -ds;
where m stands for REP. With these m&s,
+ nijhj
Q.26j+.B) give the syrup pij - @jjI- 2&j
“ij = ryii
cle/arly,& is the onlycoefficient which is not symmetric,
refatiuns
A thermodynamic analysis of rigid heat conductors
739
6. CONCLUSIONS
It has been shown that by introducing the heat-flux vector as independent variable and a supplementary constitutive equation involving its time derivative, one obtains some interesting results about rigid heat conductors. Among other things, the Maxwell-Cattanco equation comes out in a natural way and the coefficients in it are seen to have the correct sign; the heat conduction equation allows now for finite thermal wave speed without making appeal to ad hoc conditions. Thermodynamic fluxes and forces are also defined. In contrast to the classical point of view, the entropy production does no longer appear as a bilinear form in the fluxes and the forces, but rather as the sum of two quadratic expressions in the fluxes and the forces, respectively. The fluxes are seen to derive from a potential function plus an additional powerless term, the latter obeying an orthogonality condition. The existence of symmetry relations, extending the well-known Onsager’s reciprocity laws, is not guaranteed but linked to the particular form of the powerless term. When the latter vanish, symmetry laws are established. Although the problem developed here is rather simple, there should be no fundamental difficulty in extending the above results to more complicated systems like deformable bodies or viscous thermo-fluids. These topics are intended to be discussed in subsequent papers. REFERENCES [I] I. PRIGOGINE, Introduction to Thermodynamicsof Irreversible Processes. Interscience, New York (l%l). [2] S. R. De GROOT and P. MAZUR, Non-Equilibtirm Thermodynamics.North-Holland, Amsterdam (1962). [3] I. GYARMATI, Non-Equilibrium Thermodynamics. Springer Verlag, Berlin (1970). [4] I. PRIGOGINE, Bull. Acad. R. Sci. Belgique 31,600 (1945). [5] J. MEIXNER and H. G. REIK, Thermodynamicder Irreversible Prozesses, Hd der Phys., III/2. Springer Verlag, Berlin (1959). [6] G. LEBON, Bull. Acad. R. Sci. Relgique 64,456 (1978). [71G. LEBON, D. JOU and J. CASAS, J. Physics A. (1979). [E] C. TRUESDELL, Rational Thermodynamics.Mc-Graw Hill, New York (1%9). [9] B. COLEMAN, Arch. Rat. Mech. Anal. 17, 1 (1964). [IO] W. NOLL, The Foundations of Mechanics and Thermodynamics.Springer Verlag, Berlin (1974). [ll] C. TRUESDELL, In I. U.T.A.M.Symp. Vienna. (Edited by Parkus and Sedov). p. 373. Springer Verlag, Berlin (1966). 1121R. S. RIVLIN, Recent Adv. in Engng Sci. 8, 1 (1977). [13] J. MEIXNER, Arch. Rat. Mech. Anal. 57,281 (1974). [14] W. DAY, Acta Mechanica 27,251 (1977). [15] K. HU’ITER, Acta Mechanica, 27, 1 (1977). [16] A. E. GREEN and P. M. NAGHDI, Proc. Roy. Sot. London, .4357,253(1977). [17] M. G&JRTINand A. PIPKIN, Arch. Rat. Mech. Anal. 31, 113(1968). [18] I. MIjLLER, Arch. Rat. Mech. Anal. 26, 118(1%7). [19] I. MULLER, Arch. Rat. Mech. Anal. 40, 1 (1971). [20] I. MULLER, Arch. Rat. Mech. Anal. 41,319 (1971). [21] S. CH,2PMAN and T. COWLING, The Mathematical Theory of Non-Uniform Gases, 3rd Edn. Cambridge University Press, Cambridge (1970). [22] A. E. GREEN and N. LAWS, Arch. Rat. Mech. Anal. 45,47 (1972). [23] J. MEIXNER, Arch. Rat. Mech. Anal. 33,33 (1%9). [24] W. DAY, The Thermodynamics of Simple Materials with Fading Memory, Tracts in Natural Philosophy, Vol. 22. Springer Verlag, Berlin (1972). [25] D. EDELEN, Arch. Rat. Mech. Anal. 51,218 (1973). [26] D. EDELEN, Znt.J. Engng Sci. 12, 121(1974). [27] D. EDELEN, J. Non-Equil. Thermodyn.2,205 (1977). [28] J. BATAILLE, D. EDELEN and J. KESTIN, Znt.J. Engng Sci. 17,563 (1979). [29] J. DOMINGOS, M. NINA and J. WHITELAW (editors), Foundations of Continuous Thermodynamics.Macmillan, London (1974). [30] S. NEMAT-NASSER, In Mechanics Today. (Edited by Nemat-Nasser), Vol. 2, p. 94. Pergamon, New York (1975). 1311P. VERNOlTE, C. R. Acad. Sci. Paris 247,3154 (1958). [32] H. WILLEM and S. CHOI, J. them. Phys. 63,2119 (1975). 1331G. LEBON and J. LAMBERMONT, I; Mkanique 15,579 (1976). 1341L. ONSAGER, Phys. Reu. 38,2256 (1931). [35] M. GURTIN, Arch. Rat. Mech. Anal. 44, 387 (1972). (Received 10 July 1979)