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Thermodynamics of anisotropie solids near absolute zero
Comput. Modelling Vol. 28, No. 3, pp. 7Q-39, 1998 @ 1998 Eleevier Science Ltd. All rights reserved Printed in Great Britain 0895-7177/98 $19.00 + 0.00 PII: s0895-7177(98)00100-9 Mathl.
Pergamon
Thermodynamics of Anisotropic Solids Near Absolute Zero Department
V. A. CIMMELLI of Mathematics, University I-85100, Potenza, Italy [email protected]
(Received and accepted February
of Basilicata
1998)
Abstract-A thermodynamic theory of thermoelastic bodies at cryogenic temperatures ia developed in the Mework of a gradient generalization of thermodynamics with internal state variables. Compatibility of model equations with second law of thermodynamics is investigated. Finally, the corresponding theory for rigid bodies, developed in a series of papers by Koshlski and coworkers is obtained ea a particular cese of the present one. @ 1998 Elsevier Science Ltd. All rights nserved. Keywords-Internd
state variable,
Semiempiricaltemperature scale, Hyperbolic heat conduc-
tion.
1. INTRODUCTION Second sound, i.e., thermal wave propagation, is a typical low temperature phenomenon, which is observed in solid He3 and He4 and in dielectric crystals like sodium fluoride, bismuth, sodium iodide, and lithium fluoride [l-5]. Speed of propagation of heat waves has been measured with a sufficient precision. Coleman and Newman [S] and recently, Cimmelli and Frimuth [7] obtained a curve fitting experimental data for sodium fluoride (NaF) and bismuth (Bi). In the literature, one can find many proposals modifying the classical Fourier’s theory of heat conduction, which fails in describing heat waves. On that subject, let us refer the reader to the review articles [&ll]. In a series of papers, Coleman, Fabrizio and Owen (12,131 proposed a continuum theory, modeling second-sound phenomenon in anisotropic rigid conductors, based on a generalization of the classical Cattaneo’s rate type evolution equation for heat flux [14]. A thermodynamically consistent theory of thermal pulse propagation was developed by Morro and Ruggeri [15,16] in the framework of thermodynamics with internal state variables. Recently, Cimmelli and Kosifiski [17] proposed a phenomenological theory of heat conduction which rests upon a gradient generalization of thermodynamics with internal state variables [18].’ The authors introduced a nonequilibrium scalar variable, the senziemp+ical tempemttq whose gradient is related to the heat flux by a Fourier’s type heat conduction law. The theory has been tested numerically by Wihmuth and Cimmelli [23,24] and their results seem to be in accordance with the experiments. In spite of the efforts by experimental%& in order to minimize crystal deformations, the observations clearly indicate a propagation of elastic waves together with the thermal ones. In particular, a fastest longitudinal elastic wave, a slower transverse elastic perturbation, and finally, a temperature wave The author would lii to thank W. Kositiki, from Warsaw, for many useful diiussiona and for the precious help in revising the paper. lThe application of a gradient theory with internal state varlablee in studying complicated phenomena has been pointed out recently by some authora in different frameworks (see, for instance, [lQ-221).
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V. A. CIMMELLI
80
have been detected. Hence, in order to achieve a satisfactory comparison with experiments, it is necessary to construct an appropriate theory of thermoelssticity. To encompass the behaviour of thermoelastic solids, the Coleman, Fabrizio and Owen model has been generalized by &ii and Moodie [25]. On the other hand, Caviglia, Morro and Straughan [26] extended the Morro-Ruggeri results to thermoelasticity at cryogenic temperatures. As far as semiempirical model is concerned, let us quote Kosir’iski’s paper [27], where only thermally isotropic bodies are considered. Indeed, since dielectric crystals are highly anisotropic, one can expect that thermal wave propagation depends strongly on its direction. Then, it would seem to us to be of some interest to extend the semiempirical model to anisotropic thermoelastic bodies. The present paper is addressed to this goal. We start by postulating an evolution equation for a vector internal state variable A, such that (i) vector A is irrotational; (ii) it is related to the heat flux vector by a linear heat conduction law. Due to (i), if the body is simply connected, there exists a scalar field /?, such that A=Grad,&
(1.1)
Then, because of (1.1) and (ii), a Fourier’s type heat conduction law holds for the additional scalar field ,0. We call fi nonequilibrium semiempirical temperature scale. Our analysis will be pursued by postulating that constitutive quantities depend on the absolute temperature 8, the deformation tensor C, and the vector state variable A. In Section 3, we discuss the main properties of model equations. Section 4 is devoted to determining the restrictions due to second law of thermodynamics on constitutive equations. These are summarized by the Dissipation Theorem, which constitutes our main result. In Section 5, we shall analyze some meaningful consequences of this theorem, in particular, we prove that the principle of maximum entropy at the equilibrium is fulfilled. Compatibility of the present theory with the principle of material frame-indifference is examined in Section 6. Finally, in Section 7, we consider an anisotropic rigid heat conductor and prove that the isotropic theory developed in [17] and [7] is contained in the present one ss a particular case. 2.
NOTATION
AND
MAIN
DEFINITIONS
Let B denote a continuum body and let B occupy a compact and simply connected iixed region C,, reference configuration, of a three-dimensional Euclidean space Es. Moreover, let the position of the points of C, be denoted by a vector X of the associated vector space Es. A motion x(X, 7) of B is a continuous and almost everywhere invertible function of C, x [0, oo[ into Es. For a fixed instant t, function x(X,t) maps C, into a compact and simply connected region Ct c Es called actual configuration. The position of points of C, will be identified by the vector position x = x(X, t). A positive mass measure is assigned on C, by setting
m(c*>=
s
P*
dc,
(2.1)
C.
where m(~) is the mass of the subpart of B which occupies a subset c,, of C, and p. : C, -+ [0, co[ is the referential mass density. The deformation gradient F at X, at time t, is given by F = Grad x(X, t),
(2.2)
where Grad denotes the gradient operator made with respect to X.2 The condition J = det F > 0 will ensure the motion is invertible and orientation preserving. The velocity v of X at time t is given by v(X, t> = k(X, Q,
(2.3)
21n what follows, the symbols Grad, Div, and Curl will indicate the standard differential operators calculated with respect to referential coordinates.
Anisotropic Solids
81
where a superposed dot denotes the time derivative when the position X of the material point X E C, is kept f&d. After the localization procedure, under suitable smoothness assumptions, balance laws of mass, linear momentum, angular momentum, and energy yield
cw
P* = JP,
p.+ = Div S + p+b, FST = SFT, p,e=S:@-DivQ+p,r,
(2.5) (2.6) (2.7)
where p is the actual mass density; S the first Piola-Kirchhoff stress tensor; b the specific body force; E the specific internal energy; r the radiative heat supply per unit of mass, and Q the referential heat flux vector. Besides the previous equations, we postulate the following phenomenological evolution equation for the internalvector state variable A: TA+TAC=G-EA,
(2.3)
where T and E are nonsingular, positive definite, second-order tensors, G = Grade is the absolute temperature, and C = FTF is the right Cauchy-Green deformation tensor. We require that vector A satisfiesthe additional kinematicconstraint Curl A = 0.
(2.9)
Since B is simply connected, by (2.9), there exists a scalar field /3(X, t) such that A=Grad&
(2.10)
We shall analyze the meaning of (2.8)-(2.10) in the next section. Finally, the second law of thermodynamics, i.e., Clausius-Duheminequality,takes the form (2.11) with function 77representingthe specific entropy. Legendre’s transformation $=c-811,
(2.12)
which defines the specific free energy $, together with (2.7), allows us to rewrite C-D inequality in the form p.(~+ple)-~Y:c+~Q.G~O, (2.13) where Y = F-‘S
is the second PiolaKirchhoff stress tensor.
3. DISCUSSION
OF MODEL EQUATIONS
Let us assume the following constitutive equations: Y = Y*(0, A, C),
(3.1)
Q = Q*(6’, A, C) = -K(B, C)A,
(3.2)
e = c*(k),A, C),
(3.3)
$ = $*(@, A, C),
(3.4)
where the heat conductivity tensor K(8, C) is nonsingular. As far as the material functions T and E are concerned, we suppose that T = T’(8, C), x =
c*(e, c),
(3.5) (3.6)
82
V. A. CIMMELLI
Constitutive equations (3.1)-(3.4) represent a very special case of a gradient theory with internal state variable, because function p does not enter constitutive equations and only its gradient appears as an independent thermodynamic variable. The same point of view has been applied by Kosixiskiand Cimmelli [28], in modeling the behaviour of liquid helium below 2.7K. Since K(B, C) representsheat conductivity, then equation (2.9) and dimensional analysis allow us to conclude that p plays the role of a temperaturefield. Indeed, the main idea underlyingthe present approach assumesthat absolute temperatureis well defined only at the thermal equilibrium, and hence, it fails in describing highly nonequilibriumphenomena like second sound propagation. To overcome thii discrepancy,we introduce a nonequilibriumtemperature,but conserve the classical proportionality law between heat flux and gradient of temperature. If we let T --t 0, then A = E-‘G
(3.7)
Q = -KY’G.
(3.8)
and
We define the Fourier’s heat conductivity,
KF = K!E-‘.
(3.9)
Q = -KFG
(3.10)
Then, the classical Fourier’s law,
is recovered. In such a case, A is nothing but a resealedgradient of absolute temperature. The above considerationsallow us to call tensor K = KFE,
(3.11)
the dynamicalheat conductivity. Let us remarkthat the requirementCurl A = 0 is in accordance with the experimentalresults,since there is not evidence of rotation of the gradientof temperature in second sound propagation. As far as the isotropic ont+dimensionalcase is concerned, it may be easily proved that the speed of propagation of a wave travellingin a rigid conductor at the thermal equilibrium (Q = 0) is given by (3.12) where k(8) means the heat conductivity, c(e) the specific heat, and T(8) the relaxation time. Equation (3.12) led some authors to admit that T cannot physically tend to zero. From the experimental point of view, second sound propagates in a very narrow range of temperatures, outside of which the diffusiveregimeis recovered. There is a given criticul tempemtwe, depending on the material at hand, at which thermal waves appear. If the temperature increases, then a progressive broadening of the pulses is noted and, lastly, the signal is no longer a wave (see, for instance, [24]). Commonly, thii fact is interpreted by admitting that outside the critical range of temperature,thermal perturbationinstantaneouslyreachesany point of the body, i.e., it propagates with inilnite speed. Heat conductivity, which is very high at the critical temperature, now decreases toward standard values, while relaxation time tends to zero. Our point of view is that a general theory of second sound should include that phenomenology, and hence, it is better to avoid any constraint limiting T(B) from below.
Anisotropic Solids
a3
4. DISSIPATION PRINCIPLE Our aim here is to develop a thermodynamic theory of thermoelastic materials near absolute zero. To this end, let us characterizethe thermodynamic states and processes. A thermodynamic state is defined by the following functions of X and t:
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)
a motion x = x(X, t); the referentialmass density p*; the absolute temperature 8; the internal vector variable A; the first Piol&Kirchhoff stress tensor S; the specific body force b; the referentialheat flux vector Q; the radiative heat supply r; the specific internal energy c; the specific entropy q; the relaxation tensor T; the resealingtensor EC.
Such a set of fields, defined for all X in C, and all t in some interval of amplitude 7, will be called a thermodynamicprocess of duration T if it is compatible with local balance laws of mass, linear momentum, and energy, together with the kinetic equation (2.8), the additional kinematic constraint (2.9), and the constitutive equations (3.1)-(3.6). Further,a thermodynamic process is said to be admissible if it satisfiesthe Clausius-Duheminequality (2.13) for all X in C, and all t in some interval [to, to + T]. Let us suppose for a while, we are given some fields r and b, an initial time to, and some initial conditions x,JX, t), v,(X,t), po.(X, t), A,(X, t), and 0,(X, t), which are smooth enough to ensure the existence of a unique solution of (2.4)-(2.9) for all X in C, and all t in some interval [to, to +T]. Then, from the constitutiveequations, we can compute the fields S, Q, q, e, T, and El. We conclude, that to any sufficientlysmooth initial condition and exterior fields b and T, there corresponds a unique thermodynamic process in [to, to + T]. Moreover, if the Clausius-Duheminequality is satisfied,such a process is admissible. Let us now investigate the conditions under which our constitutive equations are compatible with the second law of thermodynamics. To this end, we follow a classicalprocedure introduced by Koshiski and Perzyna [29] and by Coleman and Gurtin [30]. In order to get a clearer symbology, in what follows, we will use the indicial notation with summation over repeated indices. Hence, we rewrite (2.8) and (2.13) as follows: (4-i)
TijAj + TijAlCij = Gi - CuAj, -p*($ + 74) + aYj,C,, - ~Q~G~10.
(4.2)
Taking into account constitutive relations, together with (4.1), the previous inequality yields
Piially, by grouping the terms in front of 8, Aj, C;j, we rearrange (4.3) as follows:
-
p*$ + fQiTij 3
(4.4) Aj - $QiCijAj >
> 0.
V. A. ClMMELLl
84
Now, let so = (00, Ao, CO) be an arbitrary point of the domain of the response functions $,, q,Y, Q. Moreover, for a given point X, and initial time to, let x(X,t), p,(X, t), e(X,t), A(X, t) be a solution of field equationssuch that @(Xc, to) = t90,C(X,,, to) =Cc, A@,,, to) = Ao. Taylor’s expansion of this solution around the initial point (Xc, to) yields p*(X, t) = P*c + crO@- to) + r” * (X - X0) + * * + ,
(4.5)
e(x, t) = e. + +p(t - to) + Go *(X - Xc) + *** )
(4.6)
C(X, t) = Co + Lo@ - to) + MO . (X - Xc,) + . . . ,
(4.7)
A(X,t)=&+@(t-to)+No~(X-X,,)+~~~,
(4.8)
where the meaning of symbols is obvious. Note that, at the point X, and time to, we get 4 (x0, to) = r”,
C(Xo,to)
= LO,
A (X0, to) = Ho,
(4.9)
hence, for the process under consideration, the Clausius-Duheminequality at the point Xs and time to leads to
In (4.10), the downscript so means that the response functions are evaluated at the point SOof their domain of definition. Owing to the arbitrarynessin the choice of the points (Xc, to) and SO, the relation above must hold for any value of TO,Lo, and Ho. In turn, this is true if and only if rl =--
a+ ae’
(4.11) (4.12) (4.13) (4.14)
QiCijAj 5 0. Furthermore,due to constitutive equation &i = -KijAj,
(4.15)
we rewrite (4.13) as follows: (4.16) where By integrating (4.16), we obtain
1/,= A(4 C) +
&zjlAjA*
(4.18)
Then, from (4.11), it follows: 7)= -w
Be
+ IMj,AjA,,
P+
(4.19)
AnisotropicSolids
85
(4.20) with
(4.21) (4.22) Finally, from (4.12), we get W 2p* w
Yjl=
- iZj&mA
-
(4.23)
So, we have proved the following theorem. DISSIPATION THEOREM. Constitutive equations (3.1)-(3.6)
are compatible with the Clausius-
Duhem inequality (2.13) if and only if functions 111,17,E, and the second Piola-Kirchhoffstress tensor Y satisfy the thermodynamic restrictions (4.18)-(4.23). Moreover, the reduced entropy inequality (4.24) KmlLjAlAj > 0 must be fulfilled. Let us notice that Debye’s theory of specific heat of crystals at low temperature yields for E, the following expression: E(e,C) = eo(C)04 + e(C). (4.25) Of course, equation (4.25) has some very restrictive consequences in our theory. from (4.20) and (4.25), it follows that Wl
$1 - Bae = foe4 + e(C),
a
z
Owing to (4.26), we conclude that h(e,
c>
=
( >=o. Zjl
-
(4.26) (4.27)
82
el(c)e
First of all,
Q(C) - 3
8
4
+ e(C),
(4.28)
while (4.27) allows us to write z(e, c) = z”(c)e2.
(4.29)
Due to (4.29), the specific entropy 1) takes the form
q=-ae
ah --
$Z,4A,Al.
*
(4.30)
In the next section, we will prove that Z”(e, C) is positive semidefinite, and hence, by equ& tion (4.30), the principle of maximum entropy at the equilibrium is fulfilled. As a consequence, the states of thermal equilibrium of the system are asymptotically stable (see [30]). The question of accepting constitutive equation (4.25) or to generalize it by including a nonequilibrium term is still open. Some authors consider (4.25) to be convenient and not severely restrictive (see [26]), while some others consider its consequences too restrictive, so that they sssume P different from zero (see [25]). In the present paper, we accept the first point of view, because of the following consideration. For nonvanishing P, function c(e, c, A) = g,
(4.31)
representing the specific heat of the body, could assume negative values in some range of temperature, against the physical evidence. Then, the theory should be restricted to the interval of 8 where E is increasing. Fkischmuth and Cimmelli [31] tested numerically a model of a rigid heat conductor with internal energy depending on the heat flux. Their results show that the condition c > 0 is too severe and reduces drastically the interval of applicability of the theory.
V.A. CIMMELLI
86
5. SOME CONSEQUENCES
OF DISSIPATION THEOREM
We showed in Section 4 that our constitutiveequationsobey the second law of thermodynamics if and only if some restrictions are fulfilled. On the other hand, those relations imply some meaningful consequences on the material functions K, T, and E. Now we point out these consequences.
THEOREM 1. Tensor K is positive semidefinite. PROOF. F’romthe reduced entropy inequality(4.24), it follows that KTE is positive semidefinite. Our thesis is a consequence of the positive definitenessof EC.
THEOREM 2. Tensor Z is symmetric and positive semidefinite. PROOF. From (4.13), it fbllows that (5.1) (5.2) Inverting the position of i and j, we get
a=+ aAjaAi
1
z
(5.3)
= p+B ji*
Symmetry of Z is now a trivial consequence of (5.2), (5.3), and Schwa&s theorem on the symmetry of second partial derivative of +. Fmally, the positive semidefinitenessof Z follows from the positive definitenessof T and Theorem 1.
THEOREM 3. Tensor Z”(C) is positive semidefinite. PROOF. It is a consequence of Theorem 2 and of (4.29). We say that B is in a state of then& equilibtium at the instant t* if Q(X, t*) = OVX E C,. Since K is positive definite, B is in a thermal equilibrium state if and only if A(X, t*) = OVX E C.. Finally, B is said to be in a Fourier’s stute atthe instantt* if T(X, t’) = OVX E C,.
COROLLARY 1. PRINCIPLEOF MAXIMUM ENTROPY. The specific entropy q 8thhS maximum when the body is in a state of thermal equilibriumor in a Fourier’s state.
8
local
PROOF. It is a consequence of (4.30) and Theorem 3. COROLLARY 2. GIBBS PRINCIPLE.The specific f&e energy $J attains a local minimum when the body is in a state of thermal equilibriumor in a Fourier’s state. PROOF. It is a consequence of (4.18) and Theorem 3. REMARK. Let us notice that our responsefunctionscontain a classicalterm plus a nonequilibrium one, which vanishes in some particular conditions. Fist of all, it vanishes when A = 0, i.e., in the absence of heat flux. However, it is zero also when T vanishes. In such a case, although A is diierent from zero, our theory reduces to the Fourier’s one. Finally, vector A is in a stationary state if EJA = 0. (5.4) at Let us notice that now A and G are still related by an evolution equation. Neither T nor A is zero and still a nonequilibrium term appears in the response functions. This is a typical example in which equilibriumof internalstate variable and equilibriumof the system have to be distinguished(see [22]).
87
Anisotropic Solids 6.
MATERzIAL FRAMEINDIFFERENCE
The principle of material frameindifference states that the constitutive equations characterizing the response of a material must be invariantunder a change of frame or observer [32]. The
same is true for the phenomenologicalevolution equations of internal state variables. Indeed, if these reflect some inelastic properties of the material at hand, their form cannot depend on the frame of reference. Hence, the frame-indifferenceof the kinetic equation (2.8) and the additional kinematic constraint (2.9) seem to be a necessary property of our theory. In order to prove it, let us consider a rigid transformationof coordinates of the form
x’ = c(t) + O@)x,
(6.1)
where x representsthe vector position of a point in the actual configuration C, and 0 belongs to the proper orthogonal group C(3). It is straightforwardto prove that, from (6.1), we get
A' = A,
G’ = G,
Q’=Q,
C’=C,
Y’=Y.
(6.2)
The invariance of the kinematic constraint is a direct consequence of equation (6.2)i. Furthermore, (6.2)i and (6.2)4, together with the constitutive relation (3.2) imply K’ (e’, C') = K(8, C). Let us observe that when the diisive to G. Hence, (6.2)4 and (6.3) lead to
(6.3)
(Fourier’s) situation is recovered, then vector A reduces
KIF (O’,C') = KF(O, C).
(6.4)
As a consequence, from (6.4) and (3.11), we get
12 (et, c’) = qe, c),
(6.5)
while (4.12), (6.2) 4, and (6.2)s yield T’
(e’, C’) =
T(8, C).
(6.6)
We have now all the elementsto conclude that, under (6.1), kinetic equation (2.8) becomes T’A’ + T/A’@
= G’ - E’A’,
(6.7)
i.e., it is invariant under a change of frame of reference.
7. REDUCTION
TO RIGID BODY THEORY
A semiempiricalheat conduction theory for isotropic rigid bodies has been developed in a series of papers by Koshiski and coworkers[7,17]. Of course, presenttheory should include the previous one as a particular case. To prove that this is true, let us introduce the Green-Saint Venant deformation tensor &= +-I), (7.1) where I is the identity tensor. Due to the one to one correspondence between e and C, we can rewrite free energy function $.Jas follows: (7.2)
V. A. CIMMELLI
88
Lagrange’s mean value theorem assuresthat there exist two scalarser and ~2,in the interval]O,l[, such that (7.3)
(7.4 Let us define ql(e) = &(e, O),
(7.5)
Zjl(t9) = 2jl(0,0).
(7.6)
Since & vanishesfor rigid motions, we conclude that, in such a case, free energy takes the form
t,b(e, A) = +@) +
.
&,~j~Ce)Apk *
(7-V
Analogously, for the internalenergy e, we get
(7.8) e,,(c)
= &,
(&) = &J(O)+
V-9)
and hence, for a rigid motion, equation (4.25) yields
e(e) = toe4 +
e>.
(7.10)
Finally, for an isotropic rigid conductor, Kij = k(O)Sij,
(7.11)
Tij = T(e)&,,
(7.12)
and, as a consequence,
t,b(e,A) = $l(e) + Qi
=
‘~~~‘Gr~P2, *
(7.13) (7.14)
-k(e)$ t
Equations (7.10), (7.13), and (7.14) represent the semiempiricalmodel of isotropic rigid heat conductor proposed by Cimmelli and Kosifiski [17] and by Cimmelli and Friichmuth [7].
8. CONCLUSIONS In this paper, we proposed a model of a thermoelsstic heat conductor which describes the thermomechanicalbehaviour of dielectric crystals at low temperatures. It generalizesthe model proposed by Kosiliski and coworkersfor isotropic rigid conductors, which seems to be in accordance with experimental results. The model lies on a given evolution equation for the internal vector variable A, together with the hypothesisthat heat flux is a linear function of A (Fourier’s heritage). The evolution equation is invariant under a change of frame of reference. The response functions depend on vector A, beside the classicalstate variables. Compatibility of given constitutive equations with the principle of material frame indifferenceand second law of thermodynamics has been investigated. Finally, the theory of anisotropic rigid heat conductors has been obtained as a particular case of the general one. A numericaltest of the model, for linear governing equations, has been performed by Frllchmuth and Cimmelli [33]. Their results are in accordance with the experiments. A numericalcomputation for the fully nonlinear system is under preparation and will be presentedin a forthcoming paper.
Anisotropic Solids
89
REFERENCES 1. CC. Ackerman and W.C. Overton, Second sound in helium-3, Phys. Rev. Lett. 25, 764 (1969). 2. HE. Jackzon and C.T. Walker, Thermal conductivity, second sound, and phonon-phonon interactions in NaF, P/q/s. Rev. Lett. 3 (4), 1428 (1971). 3. T.F. McNelly, et al., Heat pulses in NaF: Onset of second sound, Phys. Rev. Lett. 24, 109 (1970). 4. R.J. Hardy and S.S. Jaswal, Velocity of second sound in NaF, Phys. Rev. 3 3, 4385 (1971). 5. V. Narayanamurti and R.C. Dynes, Observation of second sound in ~~rnuth, Phys. Rev. Lett., 28, 1401 (1972). 6. B.D. Coleman and D.C. Neumann, Implication of a nonlinearity in the theory of second sound in solids, P&o. Rev. B 82, 1492 (1988). 7. V.A. Cimmelli and K. Frischmuth, Determination of material functions through second sound meesurements in a hyperbolic heat conduction theory, Matht. Comput. M~e~~~ng 24 (12), 19-28 (1996). 8. V.A. Cimmelli and W. Kosifiski, Heat waves in continuous media at low temperatures, Quaderni de1 Dipartimento di Matematica, Potenza, Report n. 6, (1992). 9. D.D. Joseph and L. Preziosi, Heat wave8, Rev. Mod. Phys. 61 (41), 1 (1989). 10. D.D. Joseph and L. Preziosi, Addendum to the paper “Heat Waves”, Rev. Mod. Phys. 61 (41), 1 (1989); Rev. Mod. Phys. 62 (2), 375 (1990). 11. W. Dreyer and H. Struchtrup, Heat pulse experiments revisited, Continuum Mech. Thermodyn. 5, 3 (1993). 12. B.D.Coleman, M. Fabrizio and DR. Owen, On the thermodynamics of second sound in dielectric crystals, Arch. Rat. Me&. Anal. 80, 135 (1982). 13. B.D. Coleman, M. Fabrizio and D.R. Owen, If second0 suono nei crlstalli: Termodinamica ed equazioni costitutive, Rend. Sem. Mat. Univ. Padova 68, 208 (1982). 14. C. Cattaneo, Sulla conduzione de1 calore, Atti Sem. Mat. Fis. Univ. Malena 3, 83 (1948). 15. A, Morro and T. Ruggeri, Second sound and internal energy in solids, Znt. J. Non Linew Mech. 22, 27 (1987). 16. A. Morro and T. Ruggeri, Non-equilibrium properties of solids, obtained from second sound meaeurements, J. Phys. C21, 1743 (1988). 17. V.A. Cimmelli and W. Kosiriski,, Nonequilibrium semi-empirical temperature in materials with thermal relaxation, Arch. Mech., 43 (6), 753 (1991). 18. W. Kosiidski and W. Wojno, Gradient generalization to internal state variable approach, Ati. Me& 47 (3), 523 (1995). 19. G.A. Maugin, Vectorial internal variables in magnetoelezticity, J. Mecanique 18, 541 (1979). 26. K.C. Valanis, A gradient theory of internal variables, Acta Mechanica 116, 1 (1996). Acta Mechanica 113, 169 (1995). 21. K.C .Valanis, The concept of physical metric in therm~yn~i~, 22. G.A. Maugin and W. Muschll, Thermodynamics with internal variables. Part I. General concepts, J. Non-Equilib. Thermodyn. 19, 217 (1994). 23. K. Friachmuth and V.A. Cimmelli, Numerical reconstruction of heat pulse experiments, ht. J. Engng. SC. 33 (2), 209 (1995). 24. K. ~hmuth and V.A. Cimmelli, Hyperbolic heat conduction with variable relaxation time, J. Theor. Appi. Medwtics 34, 67 (1996). 25. T.S. &ii and T.B. Moodie, On the constitutive relations for second sound in elastic solids, Arch. Rat. Mech. Anal. 121, 87 (1992). 26. G. Caviglia, A. Morro and B. Straughan, Thermoelasticity at cryogenic temperatures, Znt. J. Non Linear Mech. 27,251 (1992). 27. W. Kosbiski, Thermodynamics of continua with heat waves, In Proc. ASME Joint Applied Mechanica and Materials Summer Meeting, UCLA 1995, AMD, Volume 18, p. 19 (1995). 28. W. Kosi&ki and V.A. Cimmelli, Gradient generalization to internal state variablea and a theory of superfluidity, J. Theor. Appl. Mechanics 35, 763 (1992) 29. W. Kosh’iski and P. Perzyna, Analysis of acceleration waves in materials with internal parameters, Arch. Mech. 24 (4), 629 (1972). 30. B.D. Coleman and M.E. Gurtin, Thermodynamics with internal state variables, J. Chem. Phys. 47, 597 (1967). and V.A. Cimmelli, Identifi~tion of constitutive functions and verification of model as31. K. ~hmuth sumptions in semi-empirical heat conduction theory, Quademi de1 Dipartimento di Matematica, Potenza, Beport n. 4, (1994). 32. W. Noll, A mathematical theory of the mechanical behavior of continuous media, Arch. Rat. Mech. Anal. 2, 197 (1958/59). 33. K. Wihmuth and V.A. Cimmelli, Coupling in thermo-mechanical wave propagation in NaF at low temperature, Arch. Mech. (to appear).
Pergamon
Thermodynamics of Anisotropic Solids Near Absolute Zero Department
V. A. CIMMELLI of Mathematics, University I-85100, Potenza, Italy [email protected]
(Received and accepted February
of Basilicata
1998)
Abstract-A thermodynamic theory of thermoelastic bodies at cryogenic temperatures ia developed in the Mework of a gradient generalization of thermodynamics with internal state variables. Compatibility of model equations with second law of thermodynamics is investigated. Finally, the corresponding theory for rigid bodies, developed in a series of papers by Koshlski and coworkers is obtained ea a particular cese of the present one. @ 1998 Elsevier Science Ltd. All rights nserved. Keywords-Internd
state variable,
Semiempiricaltemperature scale, Hyperbolic heat conduc-
tion.
1. INTRODUCTION Second sound, i.e., thermal wave propagation, is a typical low temperature phenomenon, which is observed in solid He3 and He4 and in dielectric crystals like sodium fluoride, bismuth, sodium iodide, and lithium fluoride [l-5]. Speed of propagation of heat waves has been measured with a sufficient precision. Coleman and Newman [S] and recently, Cimmelli and Frimuth [7] obtained a curve fitting experimental data for sodium fluoride (NaF) and bismuth (Bi). In the literature, one can find many proposals modifying the classical Fourier’s theory of heat conduction, which fails in describing heat waves. On that subject, let us refer the reader to the review articles [&ll]. In a series of papers, Coleman, Fabrizio and Owen (12,131 proposed a continuum theory, modeling second-sound phenomenon in anisotropic rigid conductors, based on a generalization of the classical Cattaneo’s rate type evolution equation for heat flux [14]. A thermodynamically consistent theory of thermal pulse propagation was developed by Morro and Ruggeri [15,16] in the framework of thermodynamics with internal state variables. Recently, Cimmelli and Kosifiski [17] proposed a phenomenological theory of heat conduction which rests upon a gradient generalization of thermodynamics with internal state variables [18].’ The authors introduced a nonequilibrium scalar variable, the senziemp+ical tempemttq whose gradient is related to the heat flux by a Fourier’s type heat conduction law. The theory has been tested numerically by Wihmuth and Cimmelli [23,24] and their results seem to be in accordance with the experiments. In spite of the efforts by experimental%& in order to minimize crystal deformations, the observations clearly indicate a propagation of elastic waves together with the thermal ones. In particular, a fastest longitudinal elastic wave, a slower transverse elastic perturbation, and finally, a temperature wave The author would lii to thank W. Kositiki, from Warsaw, for many useful diiussiona and for the precious help in revising the paper. lThe application of a gradient theory with internal state varlablee in studying complicated phenomena has been pointed out recently by some authora in different frameworks (see, for instance, [lQ-221).
79
V. A. CIMMELLI
80
have been detected. Hence, in order to achieve a satisfactory comparison with experiments, it is necessary to construct an appropriate theory of thermoelssticity. To encompass the behaviour of thermoelastic solids, the Coleman, Fabrizio and Owen model has been generalized by &ii and Moodie [25]. On the other hand, Caviglia, Morro and Straughan [26] extended the Morro-Ruggeri results to thermoelasticity at cryogenic temperatures. As far as semiempirical model is concerned, let us quote Kosir’iski’s paper [27], where only thermally isotropic bodies are considered. Indeed, since dielectric crystals are highly anisotropic, one can expect that thermal wave propagation depends strongly on its direction. Then, it would seem to us to be of some interest to extend the semiempirical model to anisotropic thermoelastic bodies. The present paper is addressed to this goal. We start by postulating an evolution equation for a vector internal state variable A, such that (i) vector A is irrotational; (ii) it is related to the heat flux vector by a linear heat conduction law. Due to (i), if the body is simply connected, there exists a scalar field /?, such that A=Grad,&
(1.1)
Then, because of (1.1) and (ii), a Fourier’s type heat conduction law holds for the additional scalar field ,0. We call fi nonequilibrium semiempirical temperature scale. Our analysis will be pursued by postulating that constitutive quantities depend on the absolute temperature 8, the deformation tensor C, and the vector state variable A. In Section 3, we discuss the main properties of model equations. Section 4 is devoted to determining the restrictions due to second law of thermodynamics on constitutive equations. These are summarized by the Dissipation Theorem, which constitutes our main result. In Section 5, we shall analyze some meaningful consequences of this theorem, in particular, we prove that the principle of maximum entropy at the equilibrium is fulfilled. Compatibility of the present theory with the principle of material frame-indifference is examined in Section 6. Finally, in Section 7, we consider an anisotropic rigid heat conductor and prove that the isotropic theory developed in [17] and [7] is contained in the present one ss a particular case. 2.
NOTATION
AND
MAIN
DEFINITIONS
Let B denote a continuum body and let B occupy a compact and simply connected iixed region C,, reference configuration, of a three-dimensional Euclidean space Es. Moreover, let the position of the points of C, be denoted by a vector X of the associated vector space Es. A motion x(X, 7) of B is a continuous and almost everywhere invertible function of C, x [0, oo[ into Es. For a fixed instant t, function x(X,t) maps C, into a compact and simply connected region Ct c Es called actual configuration. The position of points of C, will be identified by the vector position x = x(X, t). A positive mass measure is assigned on C, by setting
m(c*>=
s
P*
dc,
(2.1)
C.
where m(~) is the mass of the subpart of B which occupies a subset c,, of C, and p. : C, -+ [0, co[ is the referential mass density. The deformation gradient F at X, at time t, is given by F = Grad x(X, t),
(2.2)
where Grad denotes the gradient operator made with respect to X.2 The condition J = det F > 0 will ensure the motion is invertible and orientation preserving. The velocity v of X at time t is given by v(X, t> = k(X, Q,
(2.3)
21n what follows, the symbols Grad, Div, and Curl will indicate the standard differential operators calculated with respect to referential coordinates.
Anisotropic Solids
81
where a superposed dot denotes the time derivative when the position X of the material point X E C, is kept f&d. After the localization procedure, under suitable smoothness assumptions, balance laws of mass, linear momentum, angular momentum, and energy yield
cw
P* = JP,
p.+ = Div S + p+b, FST = SFT, p,e=S:@-DivQ+p,r,
(2.5) (2.6) (2.7)
where p is the actual mass density; S the first Piola-Kirchhoff stress tensor; b the specific body force; E the specific internal energy; r the radiative heat supply per unit of mass, and Q the referential heat flux vector. Besides the previous equations, we postulate the following phenomenological evolution equation for the internalvector state variable A: TA+TAC=G-EA,
(2.3)
where T and E are nonsingular, positive definite, second-order tensors, G = Grade is the absolute temperature, and C = FTF is the right Cauchy-Green deformation tensor. We require that vector A satisfiesthe additional kinematicconstraint Curl A = 0.
(2.9)
Since B is simply connected, by (2.9), there exists a scalar field /3(X, t) such that A=Grad&
(2.10)
We shall analyze the meaning of (2.8)-(2.10) in the next section. Finally, the second law of thermodynamics, i.e., Clausius-Duheminequality,takes the form (2.11) with function 77representingthe specific entropy. Legendre’s transformation $=c-811,
(2.12)
which defines the specific free energy $, together with (2.7), allows us to rewrite C-D inequality in the form p.(~+ple)-~Y:c+~Q.G~O, (2.13) where Y = F-‘S
is the second PiolaKirchhoff stress tensor.
3. DISCUSSION
OF MODEL EQUATIONS
Let us assume the following constitutive equations: Y = Y*(0, A, C),
(3.1)
Q = Q*(6’, A, C) = -K(B, C)A,
(3.2)
e = c*(k),A, C),
(3.3)
$ = $*(@, A, C),
(3.4)
where the heat conductivity tensor K(8, C) is nonsingular. As far as the material functions T and E are concerned, we suppose that T = T’(8, C), x =
c*(e, c),
(3.5) (3.6)
82
V. A. CIMMELLI
Constitutive equations (3.1)-(3.4) represent a very special case of a gradient theory with internal state variable, because function p does not enter constitutive equations and only its gradient appears as an independent thermodynamic variable. The same point of view has been applied by Kosixiskiand Cimmelli [28], in modeling the behaviour of liquid helium below 2.7K. Since K(B, C) representsheat conductivity, then equation (2.9) and dimensional analysis allow us to conclude that p plays the role of a temperaturefield. Indeed, the main idea underlyingthe present approach assumesthat absolute temperatureis well defined only at the thermal equilibrium, and hence, it fails in describing highly nonequilibriumphenomena like second sound propagation. To overcome thii discrepancy,we introduce a nonequilibriumtemperature,but conserve the classical proportionality law between heat flux and gradient of temperature. If we let T --t 0, then A = E-‘G
(3.7)
Q = -KY’G.
(3.8)
and
We define the Fourier’s heat conductivity,
KF = K!E-‘.
(3.9)
Q = -KFG
(3.10)
Then, the classical Fourier’s law,
is recovered. In such a case, A is nothing but a resealedgradient of absolute temperature. The above considerationsallow us to call tensor K = KFE,
(3.11)
the dynamicalheat conductivity. Let us remarkthat the requirementCurl A = 0 is in accordance with the experimentalresults,since there is not evidence of rotation of the gradientof temperature in second sound propagation. As far as the isotropic ont+dimensionalcase is concerned, it may be easily proved that the speed of propagation of a wave travellingin a rigid conductor at the thermal equilibrium (Q = 0) is given by (3.12) where k(8) means the heat conductivity, c(e) the specific heat, and T(8) the relaxation time. Equation (3.12) led some authors to admit that T cannot physically tend to zero. From the experimental point of view, second sound propagates in a very narrow range of temperatures, outside of which the diffusiveregimeis recovered. There is a given criticul tempemtwe, depending on the material at hand, at which thermal waves appear. If the temperature increases, then a progressive broadening of the pulses is noted and, lastly, the signal is no longer a wave (see, for instance, [24]). Commonly, thii fact is interpreted by admitting that outside the critical range of temperature,thermal perturbationinstantaneouslyreachesany point of the body, i.e., it propagates with inilnite speed. Heat conductivity, which is very high at the critical temperature, now decreases toward standard values, while relaxation time tends to zero. Our point of view is that a general theory of second sound should include that phenomenology, and hence, it is better to avoid any constraint limiting T(B) from below.
Anisotropic Solids
a3
4. DISSIPATION PRINCIPLE Our aim here is to develop a thermodynamic theory of thermoelastic materials near absolute zero. To this end, let us characterizethe thermodynamic states and processes. A thermodynamic state is defined by the following functions of X and t:
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)
a motion x = x(X, t); the referentialmass density p*; the absolute temperature 8; the internal vector variable A; the first Piol&Kirchhoff stress tensor S; the specific body force b; the referentialheat flux vector Q; the radiative heat supply r; the specific internal energy c; the specific entropy q; the relaxation tensor T; the resealingtensor EC.
Such a set of fields, defined for all X in C, and all t in some interval of amplitude 7, will be called a thermodynamicprocess of duration T if it is compatible with local balance laws of mass, linear momentum, and energy, together with the kinetic equation (2.8), the additional kinematic constraint (2.9), and the constitutive equations (3.1)-(3.6). Further,a thermodynamic process is said to be admissible if it satisfiesthe Clausius-Duheminequality (2.13) for all X in C, and all t in some interval [to, to + T]. Let us suppose for a while, we are given some fields r and b, an initial time to, and some initial conditions x,JX, t), v,(X,t), po.(X, t), A,(X, t), and 0,(X, t), which are smooth enough to ensure the existence of a unique solution of (2.4)-(2.9) for all X in C, and all t in some interval [to, to +T]. Then, from the constitutiveequations, we can compute the fields S, Q, q, e, T, and El. We conclude, that to any sufficientlysmooth initial condition and exterior fields b and T, there corresponds a unique thermodynamic process in [to, to + T]. Moreover, if the Clausius-Duheminequality is satisfied,such a process is admissible. Let us now investigate the conditions under which our constitutive equations are compatible with the second law of thermodynamics. To this end, we follow a classicalprocedure introduced by Koshiski and Perzyna [29] and by Coleman and Gurtin [30]. In order to get a clearer symbology, in what follows, we will use the indicial notation with summation over repeated indices. Hence, we rewrite (2.8) and (2.13) as follows: (4-i)
TijAj + TijAlCij = Gi - CuAj, -p*($ + 74) + aYj,C,, - ~Q~G~10.
(4.2)
Taking into account constitutive relations, together with (4.1), the previous inequality yields
Piially, by grouping the terms in front of 8, Aj, C;j, we rearrange (4.3) as follows:
-
p*$ + fQiTij 3
(4.4) Aj - $QiCijAj >
> 0.
V. A. ClMMELLl
84
Now, let so = (00, Ao, CO) be an arbitrary point of the domain of the response functions $,, q,Y, Q. Moreover, for a given point X, and initial time to, let x(X,t), p,(X, t), e(X,t), A(X, t) be a solution of field equationssuch that @(Xc, to) = t90,C(X,,, to) =Cc, A@,,, to) = Ao. Taylor’s expansion of this solution around the initial point (Xc, to) yields p*(X, t) = P*c + crO@- to) + r” * (X - X0) + * * + ,
(4.5)
e(x, t) = e. + +p(t - to) + Go *(X - Xc) + *** )
(4.6)
C(X, t) = Co + Lo@ - to) + MO . (X - Xc,) + . . . ,
(4.7)
A(X,t)=&+@(t-to)+No~(X-X,,)+~~~,
(4.8)
where the meaning of symbols is obvious. Note that, at the point X, and time to, we get 4 (x0, to) = r”,
C(Xo,to)
= LO,
A (X0, to) = Ho,
(4.9)
hence, for the process under consideration, the Clausius-Duheminequality at the point Xs and time to leads to
In (4.10), the downscript so means that the response functions are evaluated at the point SOof their domain of definition. Owing to the arbitrarynessin the choice of the points (Xc, to) and SO, the relation above must hold for any value of TO,Lo, and Ho. In turn, this is true if and only if rl =--
a+ ae’
(4.11) (4.12) (4.13) (4.14)
QiCijAj 5 0. Furthermore,due to constitutive equation &i = -KijAj,
(4.15)
we rewrite (4.13) as follows: (4.16) where By integrating (4.16), we obtain
1/,= A(4 C) +
&zjlAjA*
(4.18)
Then, from (4.11), it follows: 7)= -w
Be
+ IMj,AjA,,
P+
(4.19)
AnisotropicSolids
85
(4.20) with
(4.21) (4.22) Finally, from (4.12), we get W 2p* w
Yjl=
- iZj&mA
-
(4.23)
So, we have proved the following theorem. DISSIPATION THEOREM. Constitutive equations (3.1)-(3.6)
are compatible with the Clausius-
Duhem inequality (2.13) if and only if functions 111,17,E, and the second Piola-Kirchhoffstress tensor Y satisfy the thermodynamic restrictions (4.18)-(4.23). Moreover, the reduced entropy inequality (4.24) KmlLjAlAj > 0 must be fulfilled. Let us notice that Debye’s theory of specific heat of crystals at low temperature yields for E, the following expression: E(e,C) = eo(C)04 + e(C). (4.25) Of course, equation (4.25) has some very restrictive consequences in our theory. from (4.20) and (4.25), it follows that Wl
$1 - Bae = foe4 + e(C),
a
z
Owing to (4.26), we conclude that h(e,
c>
=
( >=o. Zjl
-
(4.26) (4.27)
82
el(c)e
First of all,
Q(C) - 3
8
4
+ e(C),
(4.28)
while (4.27) allows us to write z(e, c) = z”(c)e2.
(4.29)
Due to (4.29), the specific entropy 1) takes the form
q=-ae
ah --
$Z,4A,Al.
*
(4.30)
In the next section, we will prove that Z”(e, C) is positive semidefinite, and hence, by equ& tion (4.30), the principle of maximum entropy at the equilibrium is fulfilled. As a consequence, the states of thermal equilibrium of the system are asymptotically stable (see [30]). The question of accepting constitutive equation (4.25) or to generalize it by including a nonequilibrium term is still open. Some authors consider (4.25) to be convenient and not severely restrictive (see [26]), while some others consider its consequences too restrictive, so that they sssume P different from zero (see [25]). In the present paper, we accept the first point of view, because of the following consideration. For nonvanishing P, function c(e, c, A) = g,
(4.31)
representing the specific heat of the body, could assume negative values in some range of temperature, against the physical evidence. Then, the theory should be restricted to the interval of 8 where E is increasing. Fkischmuth and Cimmelli [31] tested numerically a model of a rigid heat conductor with internal energy depending on the heat flux. Their results show that the condition c > 0 is too severe and reduces drastically the interval of applicability of the theory.
V.A. CIMMELLI
86
5. SOME CONSEQUENCES
OF DISSIPATION THEOREM
We showed in Section 4 that our constitutiveequationsobey the second law of thermodynamics if and only if some restrictions are fulfilled. On the other hand, those relations imply some meaningful consequences on the material functions K, T, and E. Now we point out these consequences.
THEOREM 1. Tensor K is positive semidefinite. PROOF. F’romthe reduced entropy inequality(4.24), it follows that KTE is positive semidefinite. Our thesis is a consequence of the positive definitenessof EC.
THEOREM 2. Tensor Z is symmetric and positive semidefinite. PROOF. From (4.13), it fbllows that (5.1) (5.2) Inverting the position of i and j, we get
a=+ aAjaAi
1
z
(5.3)
= p+B ji*
Symmetry of Z is now a trivial consequence of (5.2), (5.3), and Schwa&s theorem on the symmetry of second partial derivative of +. Fmally, the positive semidefinitenessof Z follows from the positive definitenessof T and Theorem 1.
THEOREM 3. Tensor Z”(C) is positive semidefinite. PROOF. It is a consequence of Theorem 2 and of (4.29). We say that B is in a state of then& equilibtium at the instant t* if Q(X, t*) = OVX E C,. Since K is positive definite, B is in a thermal equilibrium state if and only if A(X, t*) = OVX E C.. Finally, B is said to be in a Fourier’s stute atthe instantt* if T(X, t’) = OVX E C,.
COROLLARY 1. PRINCIPLEOF MAXIMUM ENTROPY. The specific entropy q 8thhS maximum when the body is in a state of thermal equilibriumor in a Fourier’s state.
8
local
PROOF. It is a consequence of (4.30) and Theorem 3. COROLLARY 2. GIBBS PRINCIPLE.The specific f&e energy $J attains a local minimum when the body is in a state of thermal equilibriumor in a Fourier’s state. PROOF. It is a consequence of (4.18) and Theorem 3. REMARK. Let us notice that our responsefunctionscontain a classicalterm plus a nonequilibrium one, which vanishes in some particular conditions. Fist of all, it vanishes when A = 0, i.e., in the absence of heat flux. However, it is zero also when T vanishes. In such a case, although A is diierent from zero, our theory reduces to the Fourier’s one. Finally, vector A is in a stationary state if EJA = 0. (5.4) at Let us notice that now A and G are still related by an evolution equation. Neither T nor A is zero and still a nonequilibrium term appears in the response functions. This is a typical example in which equilibriumof internalstate variable and equilibriumof the system have to be distinguished(see [22]).
87
Anisotropic Solids 6.
MATERzIAL FRAMEINDIFFERENCE
The principle of material frameindifference states that the constitutive equations characterizing the response of a material must be invariantunder a change of frame or observer [32]. The
same is true for the phenomenologicalevolution equations of internal state variables. Indeed, if these reflect some inelastic properties of the material at hand, their form cannot depend on the frame of reference. Hence, the frame-indifferenceof the kinetic equation (2.8) and the additional kinematic constraint (2.9) seem to be a necessary property of our theory. In order to prove it, let us consider a rigid transformationof coordinates of the form
x’ = c(t) + O@)x,
(6.1)
where x representsthe vector position of a point in the actual configuration C, and 0 belongs to the proper orthogonal group C(3). It is straightforwardto prove that, from (6.1), we get
A' = A,
G’ = G,
Q’=Q,
C’=C,
Y’=Y.
(6.2)
The invariance of the kinematic constraint is a direct consequence of equation (6.2)i. Furthermore, (6.2)i and (6.2)4, together with the constitutive relation (3.2) imply K’ (e’, C') = K(8, C). Let us observe that when the diisive to G. Hence, (6.2)4 and (6.3) lead to
(6.3)
(Fourier’s) situation is recovered, then vector A reduces
KIF (O’,C') = KF(O, C).
(6.4)
As a consequence, from (6.4) and (3.11), we get
12 (et, c’) = qe, c),
(6.5)
while (4.12), (6.2) 4, and (6.2)s yield T’
(e’, C’) =
T(8, C).
(6.6)
We have now all the elementsto conclude that, under (6.1), kinetic equation (2.8) becomes T’A’ + T/A’@
= G’ - E’A’,
(6.7)
i.e., it is invariant under a change of frame of reference.
7. REDUCTION
TO RIGID BODY THEORY
A semiempiricalheat conduction theory for isotropic rigid bodies has been developed in a series of papers by Koshiski and coworkers[7,17]. Of course, presenttheory should include the previous one as a particular case. To prove that this is true, let us introduce the Green-Saint Venant deformation tensor &= +-I), (7.1) where I is the identity tensor. Due to the one to one correspondence between e and C, we can rewrite free energy function $.Jas follows: (7.2)
V. A. CIMMELLI
88
Lagrange’s mean value theorem assuresthat there exist two scalarser and ~2,in the interval]O,l[, such that (7.3)
(7.4 Let us define ql(e) = &(e, O),
(7.5)
Zjl(t9) = 2jl(0,0).
(7.6)
Since & vanishesfor rigid motions, we conclude that, in such a case, free energy takes the form
t,b(e, A) = +@) +
.
&,~j~Ce)Apk *
(7-V
Analogously, for the internalenergy e, we get
(7.8) e,,(c)
= &,
(&) = &J(O)+
V-9)
and hence, for a rigid motion, equation (4.25) yields
e(e) = toe4 +
e>.
(7.10)
Finally, for an isotropic rigid conductor, Kij = k(O)Sij,
(7.11)
Tij = T(e)&,,
(7.12)
and, as a consequence,
t,b(e,A) = $l(e) + Qi
=
‘~~~‘Gr~P2, *
(7.13) (7.14)
-k(e)$ t
Equations (7.10), (7.13), and (7.14) represent the semiempiricalmodel of isotropic rigid heat conductor proposed by Cimmelli and Kosifiski [17] and by Cimmelli and Friichmuth [7].
8. CONCLUSIONS In this paper, we proposed a model of a thermoelsstic heat conductor which describes the thermomechanicalbehaviour of dielectric crystals at low temperatures. It generalizesthe model proposed by Kosiliski and coworkersfor isotropic rigid conductors, which seems to be in accordance with experimental results. The model lies on a given evolution equation for the internal vector variable A, together with the hypothesisthat heat flux is a linear function of A (Fourier’s heritage). The evolution equation is invariant under a change of frame of reference. The response functions depend on vector A, beside the classicalstate variables. Compatibility of given constitutive equations with the principle of material frame indifferenceand second law of thermodynamics has been investigated. Finally, the theory of anisotropic rigid heat conductors has been obtained as a particular case of the general one. A numericaltest of the model, for linear governing equations, has been performed by Frllchmuth and Cimmelli [33]. Their results are in accordance with the experiments. A numericalcomputation for the fully nonlinear system is under preparation and will be presentedin a forthcoming paper.
Anisotropic Solids
89
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