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Canonical formulation and conservation laws of thermoelasticity without dissipation
Vol. 53 (2004)
REPORTS ON MATHEMATICAL PHYSICS
No. 3
CANONICAL FORMULATION AND CONSERVATION LAWS OF THERMOELASTICITY WITHOUT DISSIPATION V. K. K A L P A K I D E S Department of Mathematics, Division of Applied Mathematics and Mechanics, University of Ioannina, GR-45110, Ioannina, Greece (e-mail: [email protected])
and G. A . M A U G I N Laboratoire de Mod61isation en M6canlque, Universit6 Pierre et Made Curie 75252 Pads Cedex 05, France (e-mail: [email protected]) (Received August 26, 2003 - Revised February 16, 2004)
This work is concerned with the derivation of conservation laws for the Green-Naghdi theory of nonlinear thermoelasficity without dissipation. The lack of dissipation allows for a variational formulation which is used for the application of Noether's theorem. The balance laws on the material manifold are derived and the exact conditions under which they hold are rigorously studied. Also, the relationship with the "classical" theory is examined. Keywords: symmetries, Noether's theorem, conservation laws, configurational mechanics, Hamilton's principle, thermoelasticity. AMS-2000 Mathematics Subject Classification: 70H25, 70H33, 74A15, 74F05,
1.
Introduction
This work is devoted to the Lagrangian formulation and the derivation of conservation laws of nonlinear thermoelasticity provided by Green and Naghdi hereafter called GN theory. Green and Naghdi [9] formulated an alternative version of what is called hyperbolic thermoelasticity in which thermal disturbances propagate with finite wave speeds. The uniqueness theorem for an initial boundary value problem for the linearized and isotropic version of GN theory was proved by Chandrasekaraiala [4]. In the same framework, the continuous dependence of the solution upon initial data and body loads was given by Iesan [12] and a Saint Venant's principle was presented by Nappa [28]. The main feature of GN theory is that it does not admit energy dissipation allowing us to ask if it is possible to look for a variational formulation of the Hamilton type. Indeed, below we present such a variational formulation for nonlinear dissipationless thermoelasticity. [3711
372
v . K . KALPAKIDES and G. A. MAUGIN
Our main interest lies in conservation laws related to the dissipationless theory of thermoelasticity, and to deduce them we use the celebrated theorem of Noether exploiting in this manner the obtained variational principle. It is well known in continuum mechanics that the Noether theorem allows one to obtain a conservation law for every given variational symmetry. But it seems that what Lovelock and Rund [17] call "invariance identity", that is, a necessary a n d sufficient condition for the action integral to be invariant under a given infinitesimal group of transformations, is not often used. We used this condition rather systematically not only to explore necessary conditions for invariance of the Lagrangian but rather to obtain the nonhomogeneous terms of the material balance laws, i.e. the material forces or some kind of moment of such forces. In this manner we obtain all equations of interest, that is, the balance of linear momentum, the equation of entropy, the balance of canonical momentum, the balance of scalar moment of canonical momentum and the energy equation, all in the apparently "dissipationless form". But these equations can be transformed to those of the classical theory of (obviously thermally dissipative) nonlinear elastic conductors [26]. Therefore, we have a good starting point for a true canonical formulation of dissipative continuum mechanics, that could be developed in the future. Conceming conservation laws in thermoelasticity, it is worthwhile to refer to the work of Honein et al. [11]. They provided a method to obtain conservation laws even for dissipative phenomena. This method was applied by Chien and Hemnann [2] for the derivation of conservation laws for classical dissipative thermoelasticity (for other applications in dissipative systems see [13]). Dascalu and Maugin [6] used the GN theory to formulate the corresponding canonical balance laws of momentum and energy--of interest in the design of fracture criteria--which, contrary to the expressions of the classical theory, indeed present no source of dissipation and canonical momentum, e.g. no thermal source of quasi-inhomogeneities [7]. In recent works [22, 25] the consistency between the expressions of intrinsic dissipation and source of canonical momentum in dissipative continua has been established. This is developed within the framework of the socalled material or configurational forces, that world of forces which, for instance, drive structural rearrangements and material defects of different types on the material manifold. From another viewpoint, the present work can be laid in what is called "'Eshelbian mechanics" [21], that is, an approach of continuum mechanics focused on the material configuration, providing new insights and a unified view of disparate topics like fracture, phase transitions, dislocations etc. The interested reader can find an extended exposition of this view in the book of Maugin [20] and in the recently published books of Gurtin [10] and of Kienzler and Heuu,ann [13]. We use throughout the paper the vectorial as well as the index notation to represent Cartesian vectors and tensors, thus rectangular coordinate systems are adopted in all cases. The motion of a thermoelasfic body is described by the smooth mapping X~ = X f l ( X A ) , A, fl = 1, 2, 3, 4, where X4 = t, x4 = or, a = or(X, t) = a(XA) is the thermal displacement field. Also,
CANONICAL FORMULATION AND CONSERVATION LAWS
373
we use the notation X to denote the material space variable, and x for the spatial position of the particle X at time t. In a coordinate system these variables will be written as XL, L = 1, 2, 3 and xi, i = 1, 2, 3, respectively. Thus, the motion is alternatively written as Xi = xi(XL,
t),
a =
ot(XL, t),
i, L = 1, 2, 3.
Generally, if not otherwise denoted, Greek indices will range from 1 to 4, while the lower-case Latin ones will range from 1 to 3. Also, the capital letters K, L, M . . . . will range from 1 to 3 and A, B . . . . from 1 to 4. We use two distinct differential operators O/OXA and D/DXA. The former is the usual partial derivative operator while the latter denotes the partial derivative which accounts for the underlying function composition. For instance,
D --F(XB,x~,(XB))
OF OF Ox× = ~ + ----.
DXA
OXA
Oxy OXA
Also, the usual notations GradF = VRF = D F / D X L , D i v F = D F L / D X L and b" = D F / D t for the gradient, divergence and material time derivative, respectively are used. .
Preliminaries
According to the GN theory the field equations of thermoelasticity of type II [9], i.e. the momentum and energy equations are given respectively as
0p - - - D i v T = 0, 0t
-
~
o0)
+ -~-0
+ tr(T~') - S. VR0 = 0,
(1) (2)
0x where p -= pRv is the physical momentum, v = 57--the velocity field, 0 - - t h e free energy function per unit volume, T the first Piola-Kirchhoff stress, F - - t h e deformation gradient, S--the entropy flux vector, ~--the entropy density per unit volume, and 0--the temperature field. In the conventional thermoelasticity, the temperature gradient VR0 is chosen as an independent constitutive variable along with the temperature 0 and the deformation gradient F. In the dissipationless thermoelasticity, Green and Naghdi [9] choose Vo~ as a constitutive variable in place of V R O , where ot = a(X, t) is the so-called thermal displacement field defined in terms of the temperature as o/(X, t) := f
0(X, t)dt.
We note that this is a rather old notion introduced by Van Dantzig (cf. Von Laue, 1921 [15]) as thermacy. It was used by Maugin [18, 19]; it was proved that thermacy is nothing but the Lagrange multiplier introduced to account for isentropy in a variational formulation.
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v . K . KALPAKIDES and G. A. MAUGIN
The main constitutive assumption tunas out to be ~: = ~ ( F , Vot, &). Following the procedure of Green and Naghdi, one can prove that the constitutive equations take the form T = O~ OF'
O~r ~ -- - 0&'
O~ S = - 0-fl'
(3)
where fl = VRa. Eqs. (3a) and (3b) look like the corresponding ones of conventional thermoelasticity. To reveal the physical sense of Eq. (3c) let us confine ourselves to a lineafized version of the GN theory. Thus, assuming a quadratic free energy function one can obtain [5] q = --kVROt, where q = S / 0 is the heat flux vector and k is a constant tensor related to the thermal constants of the material. Differentiating the above equation with respect to time we obtain aq - - = --kVRO, (4) Ot which is a law for the heat conduction instead of the classical Fourier law of heat conduction. According to Eq. (4), a temperature gradient established at a point results in an instantaneous heat flux rate at that point while the classical Fourier law results in an instantaneous heat flux. Notice that Eq. (4) corresponds to the limiting case 7: ---> oo of the Cattaneo-Vernotte equation, where z- is the thermal relaxation time [3, 5]. Using Eq. (4) one obtains a hyperbolic heat conduction equation which characterizes a wave-like behaviour of heat transport. In the sequel, for the needs of the present work we give some fundamental elements related to variational symmetries and Noether's theorem. Let
L = L ( X A , Xcl,Xcl,A),
A=l,2
. . . . . n,
ot = 1,2 . . . . m,
be a C 2 function, where XA ~ G, G is a smooth domain of ~n and x~(XA) is a sufficiently smooth function. Consider the functional I : C2(G) -+ ]R given as follows: o I (xu) = I
L(XA, x~, X~,A)dV.
(5)
J ~G
Hereafter we shall refer to functional (5) as the action integral. The necessary condition for the functional I to attain an extremum is given by the well-known Euler-Lagrange equations OL
Ox~,
D
(OL~=o,
VXAEG,
(6)
DXA \ OXy,A/
where the summation convention is used over repeated indices. Consider now the (n + m)-dimensional Euclidean space E m+n made up by the dependent and independent variables and the continuous group (actually it is a Lie
CANONICAL FORMULATION AND CONSERVATION LAWS
375
group) of point transformations in this space,
2 a = XA(XB, Xfl; •w), YC~=Yc~(XB,Xtj;Ew),
W = 1,2, . . . , #,
(7)
with
f~a(XB, Xfl, O) = XA,
xa(XB, Xfl; O) = Xu,
w h e r e ~I~a and xa are C °O with respect to XB, x• and analytic with respect to Ew
in the domain of their definition. Ew denotes the /z-dimensional parameter of the group. The corresponding group of infinitesimal transformations will be given by the relations
where
XA = Xa + EwZ~,
(8)
~ = x~ + EweS,
(9)
Of~A . Z~(XB, XB) = O-ff~-tEw = 0),
OXot ¢ff(XB, xa) = O-~w(Ew = 0).
(10)
The vector field over the space ]Em+n defined by the relation Vw = Z w + ~'~ A 8X A c~ 8xcl
(11)
will be called the infinitesimal generator of the group (7). We say that the vector field (11) or equivalently the group (7) is a variational symmetry [1, 29] of the action integral (5) if the latter is invariant under any member of the transformation group (7), that is
fG L ( X A, Xa, Xu,A)dV =
L(f~ A, Xa, Xa,A)d~",
(12)
where a tilde over a quantity denotes the transformation of this quantity under Eqs. (7). The following theorem [29] will provide the so-called infinitesimal criterion in order that a functional be invariant under a continuous group of point transformations. THEOREM. The group of transformations (7) is a variational symmetry of the functional (5) if and only if
DZ~ --0, w = l , 2 . . . . . /z, (13) DXA where V(w1) denotes the first prolongation [29] of the infinitesimal generator (11). V(I~T. w _+L
After that we can easily prove that Eq. (13) is equivalent to the equation
8L OL w OL ( D ( w DZ~ DZ~ --0, OX----~Z~ "~ ~X~ ~et "]- O-ff~,A~ DX"AA xct'BO X A ,J ~ Z o x ~
(14)
376
v . K . KALPAKIDES and G. A. MAUGIN
referred to by [17] as invariance identity. We proceed now to Noether's theorem of which we give a convenient version [17]. THEOREM OF NOETHER. If the functional (5) is invariant under the I~-parameter group of transformations given by Eqs. (8-9), then there exist I~ conservation laws of Euler-Lagrange equations (6) given by
DO~ where
3.
[OL
D [ OL \q w DXa \ ox~,,a / .l
OL OL w\ O; = - LZ~ - Ox~,---~x~,sZ~+ O--~,A(: ).
(16)
The variational principle
DEFINITION 3.1. The Lagrangian function of a thermoelastic body without dissipation is defined to be of the form
L(XL,.~i, or, -Oxi -, 8XL
O~XL)= 21 . O_~XL) . :pg(XL)JciJci - ~ ( XL, -OXi - , or, OXL
(17)
The above definition indirectly provides the independent constitutive variables which should be
3xi , OXL
0=~,
fl-
3ot OXL'
in complete accordance with the corresponding ones that Green and Naghdi introduced in what they call thermoelasticity theory of type II [9]. Then, the functional I for the case under discussion will take the form
I (xi, cO =
f?fo
L(XL, Jci, &, xi,L, a r)dVdt,
(18)
where f2 is a smooth domain of •3 and [tl, t2] an interval of N. Notice that L is not an explicit function of xi by virtue of Galilean invariance (translations in physical space of placements). Neither is it an explicit function of ot itself, this implying a sort of gauge invariance very similar to that of electrostatic for the electric potential. To proceed to the variational principle we have to add initial and boundary conditions. Let us suppose that the functions x~ = (xi, oO satisfy the following restrictions
x~(XL, t) = g~(XL, t), x=(XL, tl) = ha(XL), xa(XL, t2) = ma(XL),
XL c 0~2, XL E ~2, XL 6~2,
t a [tl, t2], (19)
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CANONICAL FORMULATION AND CONSERVATION LAWS
where g~, ha, m~ are C 2 functions on the domain of their definition and moreover fulfil the compatibility relations
g~(XL, q) = h~(XL),
g~(XL, t2) = m~(XL).
The variational statement for the dissipationless thermoelasticity can be written as follows. PROPOSITION 3.1. Let the constitutive relations (3) hold. Among all admissible
functions of motion and thermal displacement for a thermoelastic body without dissipation which satisfy the initial boundary conditions (19), those yielding extreme value to the action integral defined by (17-18), will satisfy the field equations (1-2). Proof: We can write in a more elegant and compact form the argument of the Lagrangian as L = L
(
XA, O X A , ] ,
(20)
where now x4 := or. Hence, the corresponding Euler-Lagrange equations, i.e. Eqs. (6), take the form
D
( 8_L ~ = 0 ,
Y X A ~ f 2 × [ q , t2].
(21)
D X A ~kOXc~,A/] What remains is to analyse Eq. (21). For our problem, they can be written as DXA
DXL =
-~ + ~
,~
+ ~-~
= 0.
(22)
Taking into account the form of Lagrangian (17), Eqs. (22) can be written as
DXL
-- - ~ (pRJci) = O,
XL 6 f2,
t C [q, t2],
DXL
-I- - ~
X L E ~d ,
t E [tl,t2].
-~
:0,
Inserting now the constitutive relations (3) into the above equations they transform
to DTLi
D
DXL
Dt DSL --+ DXL
(pRJCi) = O,
DO =0. Dt
(23) (24)
Thus the variational statement provides two equations. The first of them, i.e. Eq. (23) is the equation of motion and coincides with the corresponding one of Green and Naghdi, that is Eq. (1). The second one, i.e. Eq. (24), is an equation for the balance
378
V . K . KALPAKIDES and G. A. M A U G I N
of entropy which is also included in the treatment of Green and Nahdhi [9]. Thus, the Proposition 3.1 is partly proved; it remains to prove that Eq. (2) holds too. REMARK 3.1. As far as the variational principle is considered, the field equations for thermoelasticity without dissipation are Eqs. (23)-(24) instead of (1)-(2) of the GN theory. The other required equation, i.e. Eq. (2), is an energy equation and cannot directly arise from a variational principle. What can be expected is that it appears as a consequence of Noether's theorem considering invariance in time translations. 4.
Variational symmetries and conservation laws
Having obtained the variational principle, we can proceed to explore particular cases of variational symmetries related to it. 4.1.
Invariance under translations
First, we shall consider invariance in material space and time translations. LEMMA 4.1. If the action integral of a thermoelastic body without dissipation is invariant under the group of space and time translations, then it is a homogeneous body. Proof: The group of translations in material space and time is given by the following relations: X a "~- XA "+"~ w A ,
if) = 1, 2, 3, 4,
~ = x~,
(25)
which means that Z~ = t~wa and ~w = 0. Then the proof is a simple consequence of the invariance identity (14), which for the group (25) results in OL -- 0, OXL
OL = 0. Ot
--
(26)
The second of (26) is satisfied by the definition of the Lagrangian and the first one is what we want to prove. [] Next, we give the main result of this subsection. PROPOSITION 4.1. Let the motion and the temperature functions xi and 0 satisfy the field equations (23-24)for a homogeneous thermoelastic body without dissipation through constitutive relations (3), on the domain f2 × [q, t2]. Then the following conservation laws also hold on f2 × [tl, t2].
D --(L~KL DXL
D q- TLiXi,K -- SLflK) -- --~(PRJCiXi,K -~- l"]flK) = O,
(27)
D D LDX-(TLiJci -- SLO) + ot-:-:(L - pRJCi~Ci-- r/0) = 0.
(28)
379
CANONICAL FORMULATION AND CONSERVATION LAWS
Proof: The assumptions of the proposition imply that the symmetry (25) holds, thus we can use the general form of the conservation laws (15). With the aid of Eq. (16), the quantity 0~ for the case under discussion takes the form OL
O~ = - - L S w a -~ -y-------Xa,w, OXet,A
W = 1, 2, 3, 4,
(29)
0,
(30)
and the corresponding conservation laws will be D
OL
--(LSwA
DXA
--
DXA
OX~,a
Xc~,w)=
W = 1, 2, 3, 4.
Eq. (30) can be expanded in
D DXA
and
(LSLA
\ -- - - X a , L )
: O,
OXo:,A
L=1,2,3,
D ( DXA
- L ~ 4 A - - 3 L xot,4~ = O. ] Oxot,A
Furthermore, developing the last equations we obtain
ox,
+
oLKxiLDXK
- - Xi q~Xi, K
Ol
D (L_ -OL. "Jr" " ~ -Xi
=0,
OL.)
--
(31)
Finally, inserting the constitutive relations in (31) we obtain the required relations (27-28), hence Proposition 4.1 has been proved. The second of the just obtained conservation laws, Eq. (28), corresponds to time translations, thus it is related to the conservation of energy. In the sequel, we shall prove that it could provide the second of the equations of Green and Naghdi, i.e. Eq. (2). Indeed, inserting the Lagrangian (17) into (28) we obtain D ( D'-XL
) TLilli -- SLO
D(1 -- ~
) "~PRViVi "[- lp "[- 170
:
O.
(32)
After some simple calculation and taking into account equation of motion (23), we can write Eq. (32) in the form - - ( ~ "[- 017) q'- TLiVi,L -- SLO, L = O.
(33)
Equation (33) coincides with Eq. (2), hence Proposition 3.1 has been completely proved. REMARK 4.1. It is noted that to obtain Eq. (33), the homogeneity of the Lagrangian with respect to the material space variables is not required. What is really necessary is the homogeneity with respect to time which is assumed by
380
V.K. KALPAKIDES and G. A. MAUGIN
Definition 3.1 of the Lagrangian. That is why there is no requirement of homogeneity in the statement of Proposition 3.1. REMARK 4.2 It is easy to confirm that the invariance of the action integral (18) under the group of translations in physical space will provide the Euler-Lagrange equations, i.e. Eqs. (23) and (24). 4.2.
Invariance under the scaling group
In this case we will use the following one-parameter group of scalings in material and physical space XA : XA -~ EXA,
~ = x,~ - Ex~.
(34)
Thus, invoking the relations (10) we obtain ZA = XA,
(¢t = --X~,
(35)
which in turn are substituted into the invariance identity (14) to yield the following result. LEMMA 4.2 The action integral of a thermoelastic body without dissipation is invariant under the transformation group (34) /f and only if its Lagrangian fulfils the identity OL XA -- 2 OXA
Xa,A + 4L = 0.
(36)
We remark that for a homogeneous thermoelastic body, namely when its Lagrangian does not depend explicitly on XA, Eq. (36) becomes 0L OXot,A
x~,a = 2L,
(37)
which is nothing but Euler's identity (equation) of degree 2. Consequently, for a homogeneous thermoelastic body all homogeneous Lagrangians of degree 2 meet the invariance identity for the scaling group. Among them of practical importance are the quadratic Lagrangians because they provide linear constitutive relations. Hence, the following results hold. LEMMA 4.3 Let us assume that a homogeneous thermoelastic body without dissipation admits linear constitutive relations deduced from (3). Then its Lagrangian satisfies the invariance identity for the scaling group. PROPOSITION 4.2 Let the motion and the temperature functions xi and 0 satisfy the field equations (23-24)for a homogeneous thermoelastic body without dissipation through linear constitutive relations, in the domain f2 x [tl, t2]. Then the following
CANONICAL FORMULATION AND CONSERVATION LAWS
381
conservation law also holds on f2 x [tl, t2].: D --[(L~KL -t- TLiXi,K -- SLflK)XK -t- (rLiXi -- SLO)t -t- TLiXi -- SLOt] DXL -t- - ~ [ - -
where
e = e(X,
(~PRfCi2i + e ) t -- (PR2iXi, K -l- rlflK)XK -- PR2iXi -- rlOt] = o,
(38)
t) is the internal density function per unit volume.
Proof: The linearity and the homogeneity of the thermoelastic body imply that Lagrangian meets the invariance identity (36). Thus, for the case under discussion, the action integral is invariant under the group (34). That means we can invoke the general form for the conservation law given by Eq. (15). Due to the fact that the group (34) has only one parameter the quantity (16) will take the form OA "-- .
(
. LXA
.
OLx~,BxB .
OXot,A.
OL
OXot,Ax~
)
(39)
and the conservation law corresponding to the symmetry (34) will be
DOA DXA
-
D (
DXA
- 0 = ,
LXA-
OL X~,BXB
OXot,A
OL x~)=O.
OXot,-----A
(40)
Eq. (40) can be analyzed as follows,
D
DXL
OL LXL -- --(X~,MXM + 2~t) -- OL x.') OXa,L
OXc~,L /
Ox~OL(X~,MXM +2fl)-- ~OLx~,I = 0 + --~O ( L t - 77-. OL OL OL OL ) D LXL -- - - ( X i , M X M -t- )git) -- -~'-"-(OtMXM -~- dtt) xi - - - o t DXL OXi,L OOtL OXi,L OOt,L -~D ( __OL OL OL OL ) + Lt -- ~Xi(Xi. ,MXM + ]tit) -- -~-~(OtMXM -~- &t) -- --xi02i - -~7daot = O. (41) Taking now into account that the internal energy is related with the free energy function by e=~p+O0 and recalling the relations (3) and (17), we can easily obtain the required conservation law (38) from (41). []
382
V.K. KALPAKIDES and G. A. MAUGIN
4.3.
Invariance under the group of rotations In this section we will examine the invariance of action integral (18) under rotations of spatial variables. This is the group SO(3) in physical space which is given by the equations:
Y~A = X A ,
A = 1,2,3,4, i, j = 1, 2, 3,
Xi -~ Q i j x j ,
(42)
:74 = x4, where Q is an orthogonal matrix with detQ = 1. The corresponding infinitesimal group is given by XK =XK, iCi = Xi
~I~4.~-X4,
if- eiwj~wXj,
-~4 = x4,
i, j, K, w = 1, 2, 3.
(43)
So, we take for the quantities given by Eqs. (10) Z ~ --~ Z ~ -~- O,
~o = eiwjXj,
~
= O.
(44)
Inserting Eqs. (44) into the invariance identity (14) we obtain the following result. LEMMA 4.4. The action integral of a thermoelastic body without dissipation is invariant under the transformation group (42) if and only if its Lagrangian fulfils the identity OL eiwj OXi, K X j, K = O.
(45)
Invoking the constitutive relations (4) we easily obtain
ewij TKiXj,K = 0.
(46)
Consequently, one can easily prove the following proposition. P R O P O S I T I O N 4.2. Let the motion and the temperature functions xi and 0 satisfy the field equations ( 2 3 - 2 4 ) f o r a thermoelastic body without dissipation through constitutive relations (3) on the domain ~ × [tl, t2]. Moreover, let the Lagrangian fulfil the identity (45). Then the following conservation law also holds on f2 × [tl, t2], D D DXL (eiwjxjTxi) - - ~ (pReiwj/CiXj) = 0. (47)
REMARK 4.3. Eq. (47) is the balance of angular momentum for dissipationless thermoelasticity. Certainly, this equation together with equation of momentum (23) can provide us, as usual, the symmetry of the tensor TLixj,L. But this is erroneous because it is an assumption for the Proposition 4.2 (see Eq. 46) and not a consequence of it. 5.
Material balance laws So far, we have presented conservation laws for the field equations of dissipationless thermoelasticity. From the point of view of material mechanics, it is interesting
383
CANONICAL FORMULATION AND CONSERVATION LAWS
to focus on what can be called material balance laws, that is these corresponding to conservation laws which are nonhomogeneous equations. To obtain such equations we must allow for the presence of sources in the already derived equations. This in turn can be done by relaxing the assumptions we have imposed in order to obtain them. In this way, for every conservation law, having found the conserved quantity, we can obtain a balance law. Depending on the particular equation, we expect these nonhomogeneous terms to be the so-called material forces or moments of such forces. We will apply this procedure to conservation laws (27) and (38). 5.1.
The canonical m o m e n t u m balance law
We proceed now to the above mentioned procedure for Eq. (27) by removing the homogeneity of the Lagrangian from the assumptions of Proposition 4.1. Let us assume that the Lagrangian (17) depends explicitly on material variables, that is
OL OXL
--50,
L = 1,2,3.
In this case no invariance with respect to space translations can be secured. In spite of this, the Euler-Lagrange equations still hold and it is possible to produce a balance law by calculating the expression
DO~
w = 1,2,3, A = 1,2,3,4, DXA' which certainly does not vanish identically any more. Indeed, calculating this term we obtain
DO~ DXA
--
--
D DXA
OL OX w
L~wA
OL
.,--Xa OXot,A
w_ ]
OL D Oxot,aXot,aw -+" ~ + b- A
( OL x OL O ~k-OXa,A - } ,~,w "~- -~Xa,a D X A (Xot,w)
xo,w.
(48)
The last term in the right-hand side of (48) vanishes due to the Euler-Lagrange equations, thus we conclude aL D X ~ - - aXw" (49) But the left-hand side of (49) has already been calculated in Proposition 4.1, hence equating the right-hand side of Eqs. (30) and (49) and taking into account Eq. (31a), we can write
D D OL , (L~KL -}- TLiXi,K - SLflK) -- "-~t (lOR-~iXi,K "~- OflK) -DXL OXK
--
K , L = 1,2,3,
(50)
which is the expected material balance law referred to by some authors [20] as canonical momentum equation or pseudomomentum equation.
384 5.2.
V. K. KALPAKIDES and G. A. MAUGIN
The scalar moment of canonical momentum balance law
In the same way, we can develop the conservation law (38) related to the scaling symmetry. Hence, removing the assumptions of linearity of constitutive relations as well as the homogeneity of the Lagrangian with respect to material variables we obtain directly the balance law
OL ~____~A(LXA_OLx~,,XS_oX~,Ax~)
DOA DXA
OXot,A
@L XL OXL
_
--
(51)
X~,A + 4L,
ct, A = 1, 2, 3,4,
L = 1,2,3.
DOA The term --~-~a has been calculated in the proof of Proposition 4.2 (see Eq. (38)), thus Eq. (51) takes the form
D D X L [(L~KL -1- TLiXi,K -- S L f l K ) X K -]- (TLi-~i -- SLO)t -I- ZLiXi -- SLCt]
°[
q- - - ~
--
pRJCiJQ+ e
)
t - (pRfQXi,K + 17flK)X K -- pRXiXi -- rlct _
@L XL OXL
-- 2
(&)
]
X~,A +4L.
(52)
We remark that Eq. (52) is valid for nonlinear, nonhomogeneous thermoelasticity in the framework of GN theory. Hence, for linear thermoelasticity, implementing Lemma 4.3, the balance law originated by the scaling symmetry will become D
DXL [(Lt~KL "~ ZLiXi,K --SLflK)XK "~ (TLiJQ -- SLO)t Jr ZLiXi -- SLOt] D
1
"~ N [ - - ( ~ p R J C i J Q q - e ) t - (pRxiXi,K 21- l~flK)X K -- pRJQXi -- Oct] :
3L
OXL X L. (53)
Eq. (53)holds for linear, nonhomogeneous thermoelasticity and represents a balance law for scalar moment of pseudomomentum or canonical momentum. The corresponding balance equation in physical space is not often used because it does not play any role in the description of the equilibrium or the motion of a body as does, for instance, the momentum or angular momentum equation. In the case of physical space, the factors that balance the rate of scalar moment of momentum are referred to as scalar moments or virials. So the right-hand side term of (53) is a sort of material scalar moment or material virial. 5.3.
Comparisons with previous results and some remarks
The obtained above results can be compared to previously published work of other researchers. We must especially refer to the work of Dascalu and Maugin
CANONICAL FORMULATION AND CONSERVATION LAWS
385
[6] for GN thermoelasticity and Maugin [20] and Fletcher [8] for elasticity. Let us return to Eq. (27) which represents the canonical momentum conservation law. It can be written in the form D ((1 ) ) D DXL -~PR2i2i - - ~r S K L ~ - TLiXi, K - - a L l K - - "-~(PR2iXi,K q- Of K) : O, or Div
PR-~- - 0
I + TF - S ® f
- ~-~ (pRFrV + Off) = 0.
(54)
Eq. (41) coincides with the corresponding one derived through a vectorial approach in [6]. The conservation laws (27), (28) and (38) restricted to the case of elasticity are in full agreement with these given by [20] and [8]. To obtain a clearer comparision we introduce the following definitions (compare to the general definitions (A.16), (A.17), (A.14) and (A.15) in [23] ' ~ L~ : = --(pR:~f,F -I- rift L) :
b th := {bLK :
pnh
--(Lt~LK nt- TLiXi, K
OL
1
0X
2
:. . . .
- - p " F - ~fl = emech _ t/E, -- SLflK)},
O0 i . i V R p R -- - -
0X'
1
(55)
(56) (57)
7-/:= ~ p e i " i + e,
(58)
Q := T~ - S0,
(59)
for canonical momentum of the present approach, canonical material stress tensor (Eshelby stress tensor), material force of true inhomogeneities, Hamiltonian density (total energy density) and material energy flux (Umov-Poynting vector), respectively. Also, we have defined the mechanical canonical momentum by pmech :__ _ p . F. The presence of fl in 79L ~ may appear strange on a first look. However, it happens that the additional contribution 0fl in the definition (55) is supported by other theories. In a variational approach to the modelling of liquid helium it was shown by Lhuillier et al. [16] that the quantity -Off is nothing more than the internal momentum (accounting for the relative motion of normal and superfluid components) of the superfluid. But in a Lagrangian variational formulation the canonical momentum is nothing more than the pseudomomenturn and thus this internal momentum is truly a part of the pseudomomentum. After definitions (55)-(59), the balance equations (50) and (53) and the energy equation (28) can be written in vectorial form as follows -
DT:,th Div b th + - - finh, Dt
D i v ( - b th • X
+ Qt + T. x -
(60) Sot) + -~O (-7-/t + T ~th • X - p . x - t/ot) = finh. X, L) t
(61)
386
v . K . KALPAKIDES and G. A. MAUGIN
and
DH Dt
-
-
Q = 0,
VR.
(62)
respectively. We recall that Eq. (61) holds for nonhomogeneous but linear thermoelasticity. Under this restriction and in the framework of elasticity it can be compared with Eq. (4.89) of [20]. If we assume, as in standard continuum thermodynamics, that entropy and heat flux are related by the usual relation q
S = ~, we have
(63)
a~
y q = - 0 m0/~
(64)
and Eq. (62) takes the classical form of the energy conservation equation [26]
DH Dt
- -
-
VR • (T. v -
q) = 0.
(65)
REMARK 5.1. After some rearrangements we can write the canonical momentum equation (60) (note that Curlfl = 0; T =transposed) in the form Dpmech D i v b mech =
Dt
rlVRO + S" (VR~) T +
firth,
(66)
where ~meeh = bth _ S ® ft. Also, the energy equation (28), by virtue of constitutive relations (3) and the momentum equation (23), can be written as
___DSL 0 DXL
-
SrOL +
SLt~L + 170 - _O- - - ( T O ) Dt
= O,
or, taking into account (63), we can write
D(Orl_____~)+ VR "q = 7/0 + S. ~. (67) Dt This equation is interesting by itself because of its structure---especially the right hand side--which is similar to that of (66), time derivatives replacing material space derivatives. REMARK 5.2. Under some conditions, a remarkable identity can be produced by scrutinizing Eq. (61) (or equivalently Eq. (53)). We remind that Eq. (37) is a necessary condition for Eq. (61) (or Eq. (53)) to hold. Assuming moreover material homogeneity and developing the terms including stress, entropy flux, momentum and entropy density we can write Eq. (61) in the form
387
CANONICAL FORMULATION AND CONSERVATION LAWS
Div(-b th.X+Qt)+
(~th.x_~t)+ +
(
DivT-
o0)
-DivS--D-~-
Dt
.x
oe-p./~-rl&-T'F-S.~=O
Inserting into the above equation the momentum and entropy balance equations and using the constitutive relations as well as the definition of the Lagrangian, we obtain D i v ( - b th. X + Qt) +
_~
(,pth. X - ~ t )
3L
3L.
3L
3L
= -~--f~ + ~--~o~+ 3---F : F + ~-~./~.
To conclude we have just to recall that the right-hand side of the above equation is equal to 2L due to identity (37), hence one can write D i v ( - b th . X + Qt) 4- ~ t (T~th "X-7-/t) = 2L,
(68)
consequently, we have proved that /f the Lagrangian fulfils Eqs. (26a) and (37), then (68) holds. Summing up, from the Lagrangian density (17) we have deduced all field equations, balance laws and constitutive relations for the theory of material inhomogeneous, finitely deformable, thermoelastic conductors of heat. As a matter of fact, Eqs. (23), (47) and (65) are the local balance equations of linear momentum, angular momentum (in physical space) and energy, respectively. This is completed by the balance equation of mass which here trivially reads
3pR 3t
- - 0.
(69)
These are all formally identical to those of the classical thermoelasticity of conductors (e.g. as recalled in [26]). Apart from the functional dependence of 7t the equation of angular momentum, i.e. Eq. (47), it is also formally identical to the classical counterpart. Only the equation of canonical momentum (60) differs from the one originally obtained by Epstein and Maugin [7] in material thermoelasticity. But, abstraction being made of material inhomogeneities, it is the same as the one obtained by direct algebraic manipulations by Dascalu and Maugin [6] in the "dissipationless" formulation of thermoelasticity. Indeed, canonical momentum (55) is made of two parts, a strictly mechanical part--which is nothing more than the pull back, changed of sign, of the physical momentum--and a purely thermal part given by the constitutive behaviour. In addition, the canonical stress tensor (56)--also called Eshelby stress tensor---contains a contribution of 1~, because from its very definition, it captures material gradients of all fields. One should note that the source term in Eq. (60) has no dissipation contents. Furthermore, contrary to common use, even the entropy equation (24) is source free. This means that in the absence of material inhomogeneities, all equations obtained are strict conservations laws, hence the qualification of "dissipationless theory". In this rather strange--we
388
v.K. KALPAKIDES and G. A. MAUGIN
admit it--approach, the entropy flux and heat flux are derived from the free energy, on the same footing as the entropy density, and stress (Eqs. (3) and (64)).
6.
Relationship to the classical theory
Although the equation of energy looks like the corresponding classical one (compare Eq. (65) with Eq. (4.67) of Maugin [20] or with Eq. (5.3) of [27]), there are essential differences stemming from the functional dependencies of the free energy function 7t in the two cases. Moreover, the equation of canonical momentum (60) deviates from the classical one even more due to the explicit presence of fl in it. Due to the utmost importance of these equations in fracture and phase-transitions front studies [26], we will try to explore their relation to the classical theory; actually we will examine under which conditions a passage to the classical equations is possible. We start from Eqs. (50) and (28) which can be written in another form by the insertion of the entropy and momentum equation (Eqs. (23)-(24)) into them,
DL Dflg OL a tTL,iXi,KL -PRJCiVi,K : __OX-~ SL atrIO'K qOX---~' D X-----~ DL TL'iVi'L Dt PR.~iJfi=O, LSL + 00,
(70)
(71)
where the fact that V × fl = 0 holds is taken into account. Next, we need to isolate the contribution of fl in order to get some "classical" limit: to this end we expand the free energy function in a Taylor series with respect to fl at f l = 0 ,
7~(x, F, 0, #) = ~(X, F, 0, # = 0) + ~-~M(X, F, 0, # = 0)/~M + 0(# 2) and retaining the first-order terms we can write 7t(X, F, O, fl) = ~(X, F, O) + ~M(X, F, O)flM,
(72)
where ~ denots the free energy at the limit and ~PM = O~/OflM[~=O. AlSO, we define the entropy flux at the limit as SM(X, t) = --TiM(X, F, 0),
(73)
following the general definition of entropy flux, i.e. Eq. (4b). With the aid of (72) we compute all the relevant terms in Eqs. (70) and (71) as follows:
Dap DX~-
D~t DaPMo D X~L + "-D-~LpM +
DflM ~M-DXL
D~ DTtM n -- DXL -+ -D-~LpM --
DflM S M -DXL -
D~ (01]?M OI~M OF O;M 0 0 ) DflM = DX----~+ \ OXL + O---F-OZ---~+ - - OXL tiM -- ~SMDXL '
(74)
389
CANONICAL FORMULATION AND CONSERVATION LAWS
T=0~ z
~-F 6
OF o, _
O0
SL--
a• abe o4/
~
]~M
a6
o,M
80
O0
(75)
"~-~ - ~ ~M,
(76)
OU
(77)
- - ~ L = sL, o~M
(78)
3XL
ax~ + 5-~ ~ '
D~p Dt
D6 DTtM ~ D~M + --bT-pM + ~v D---7Dt
-- D---7-+
Dflm
F+
(79)
0 tiM + ~PM Dt
where hereafter the hat over any quantity denotes this quantity at the limit. Inserting Eqs. (74)-(79) into Eqs. (70) and (71) we obtain
oL
^
aL
DX--~ + TL,iXi,KL -- pRXiVi,K = OO,K + OX---K' fL'il)i'L
DL
(80)
PR-~iXi 00,
Dt
(81)
=
respectively. We demand now that the momentum and entropy balance equations as well as the relation between the heat and entropy flux at the classical limit hold, thus DTLi DXK
D Dt (pRJci) = 0,
DSL DO ~SLe,L --DXL + - ~ + --O -- 0,
qL SL = --.O
(82)
Inserting now (82a) and (82) into (80) and (81), respectively, we obtain D
D
DXK
oL
(L + ~Lix~,KL) -- -~(;R~iXi, K) = OO K + OX----K'
O (f,~,~ _ 4L) - O DXL ~-~ (K + g) = 0,
(83) (84)
where ~ is the internal energy at the limit and its standard relation with the free energy function (thus 6 = ~ - 00) was used. Eqs. (80) and (81) are the ones of pseudomomentum and energy for the classical thermoelasticity as given by Epstein and Maugin [7]. Hence, we have proved the following proposition. PROPOSITION 6.1. Let the field equations, the canonical momentum equation and the energy equation for dissipationless thermoelasticity and the balance equations for momentum and entropy of classical thermoelasticity hold, then we can pass from
390
v . K . KALPAKIDES and G. A. MAUGIN
the canonical momentum and energy equations of dissipationless thermoelasticity to the corresponding ones of the classical thermoelasticity by assuming the form for the free energy function given by Eq. (72). REMARK 6.1. We must underline that the described above passage from the dissipationless thermoelasticity to the classical one is accompanied by a drawback; this is the relation (73) providing the entropy flux at the limit. The functional dependence of the entropy flux is unacceptable from the viewpoint of thermodynamics. What is really missed is a dependence of S on the temperature gradient. Maybe one could assume a less smooth function for the free energy function 7z with respect to /], especially for fl = 0. This could provide the necessary freedom for the free energy function at the classical limit, a perspective that, in any case, needs more investigation in the framework of convex analysis. REMARK 6.2. Eq. (83) can be written in the form Dpmech
Div bmech
=
fina + fth,
(85)
fth := ~VR0.
(86)
Dt where b mech :
- ( [ , I g + T . F),
The last introduced quantity is the material thermal force of quasi-inhomogeneity clearly defined by Epstein and Mangin [7] in their general theory of material uniformity and inhomogeneity. Thus (85) recovered its "classical" form the quotation marks here emphasize that, in fact, although this equation is "classical" from our viewpoint, practically unknown to most people, it is the one on which thermoelastic generalizations of J-integral of fracture could be based (cf. the review by Maugin and Berezovski [26]; Maugin and Trimarco [27]). 7.
Conclusions
We have effectively formulated a canonical theory of the thermoelasticity of conductors. All field equations, balance laws and constitutive equations follow from it. The relationship with the "classical" formulation was examined. To proceed further, one must envisage the case where nonthermal dissipative processes (e.g. anelasticity) are present. Considering the theory of internal variables of state in order to describe these phenomena is a sufficiently general approach as demonstrated in a recent book [24]. The only a priori change would be accounting for the dependency of the free energy 7z on a new set of variables collectively represented by the symbol ~b. The corresponding equations of state reads ep + (0ap/0~b)= 0. The main problem, however, is to built the evolution equation of q~, normally a relationship between ~b and the thermodynamical force qb constrained by the second law of thermodynamics. Thus the very presence of q~ is related to dissipative processes and a priori not amenable by means of a canonical variational formulation (~p(X, t) is not a classical field; it has neither inertia, nor is its gradient introduced). But it was recently shown how the variables ~b and 0 could play parallel roles in a certain reformulation of the
CANONICAL FORMULATION AND CONSERVATION LAWS
391
anelasticity of thermoconductors [22, 25]. This is the trend to follow for it is shown that Eqs. (66) and (67) are simply modified by the addition of terms ~ . (VR~b)~ and q~.~, respectively, respecting thus the already noticed space-time symmetry. This hints at considering pmech and 07 as the dual space-time canonical momenta conjugated to space-time coordinates X and t (compare to the nondissipative case in [14]). Acknowledgement The authors are grateful to the anonymous reviewers, whose constructive remarks led to some improvements upon an earlier version of the paper. Part of this work was completed during a stay of VKK at LMM-UPMC with financial support of the European TMR 98-0229 <
REPORTS ON MATHEMATICAL PHYSICS
No. 3
CANONICAL FORMULATION AND CONSERVATION LAWS OF THERMOELASTICITY WITHOUT DISSIPATION V. K. K A L P A K I D E S Department of Mathematics, Division of Applied Mathematics and Mechanics, University of Ioannina, GR-45110, Ioannina, Greece (e-mail: [email protected])
and G. A . M A U G I N Laboratoire de Mod61isation en M6canlque, Universit6 Pierre et Made Curie 75252 Pads Cedex 05, France (e-mail: [email protected]) (Received August 26, 2003 - Revised February 16, 2004)
This work is concerned with the derivation of conservation laws for the Green-Naghdi theory of nonlinear thermoelasficity without dissipation. The lack of dissipation allows for a variational formulation which is used for the application of Noether's theorem. The balance laws on the material manifold are derived and the exact conditions under which they hold are rigorously studied. Also, the relationship with the "classical" theory is examined. Keywords: symmetries, Noether's theorem, conservation laws, configurational mechanics, Hamilton's principle, thermoelasticity. AMS-2000 Mathematics Subject Classification: 70H25, 70H33, 74A15, 74F05,
1.
Introduction
This work is devoted to the Lagrangian formulation and the derivation of conservation laws of nonlinear thermoelasticity provided by Green and Naghdi hereafter called GN theory. Green and Naghdi [9] formulated an alternative version of what is called hyperbolic thermoelasticity in which thermal disturbances propagate with finite wave speeds. The uniqueness theorem for an initial boundary value problem for the linearized and isotropic version of GN theory was proved by Chandrasekaraiala [4]. In the same framework, the continuous dependence of the solution upon initial data and body loads was given by Iesan [12] and a Saint Venant's principle was presented by Nappa [28]. The main feature of GN theory is that it does not admit energy dissipation allowing us to ask if it is possible to look for a variational formulation of the Hamilton type. Indeed, below we present such a variational formulation for nonlinear dissipationless thermoelasticity. [3711
372
v . K . KALPAKIDES and G. A. MAUGIN
Our main interest lies in conservation laws related to the dissipationless theory of thermoelasticity, and to deduce them we use the celebrated theorem of Noether exploiting in this manner the obtained variational principle. It is well known in continuum mechanics that the Noether theorem allows one to obtain a conservation law for every given variational symmetry. But it seems that what Lovelock and Rund [17] call "invariance identity", that is, a necessary a n d sufficient condition for the action integral to be invariant under a given infinitesimal group of transformations, is not often used. We used this condition rather systematically not only to explore necessary conditions for invariance of the Lagrangian but rather to obtain the nonhomogeneous terms of the material balance laws, i.e. the material forces or some kind of moment of such forces. In this manner we obtain all equations of interest, that is, the balance of linear momentum, the equation of entropy, the balance of canonical momentum, the balance of scalar moment of canonical momentum and the energy equation, all in the apparently "dissipationless form". But these equations can be transformed to those of the classical theory of (obviously thermally dissipative) nonlinear elastic conductors [26]. Therefore, we have a good starting point for a true canonical formulation of dissipative continuum mechanics, that could be developed in the future. Conceming conservation laws in thermoelasticity, it is worthwhile to refer to the work of Honein et al. [11]. They provided a method to obtain conservation laws even for dissipative phenomena. This method was applied by Chien and Hemnann [2] for the derivation of conservation laws for classical dissipative thermoelasticity (for other applications in dissipative systems see [13]). Dascalu and Maugin [6] used the GN theory to formulate the corresponding canonical balance laws of momentum and energy--of interest in the design of fracture criteria--which, contrary to the expressions of the classical theory, indeed present no source of dissipation and canonical momentum, e.g. no thermal source of quasi-inhomogeneities [7]. In recent works [22, 25] the consistency between the expressions of intrinsic dissipation and source of canonical momentum in dissipative continua has been established. This is developed within the framework of the socalled material or configurational forces, that world of forces which, for instance, drive structural rearrangements and material defects of different types on the material manifold. From another viewpoint, the present work can be laid in what is called "'Eshelbian mechanics" [21], that is, an approach of continuum mechanics focused on the material configuration, providing new insights and a unified view of disparate topics like fracture, phase transitions, dislocations etc. The interested reader can find an extended exposition of this view in the book of Maugin [20] and in the recently published books of Gurtin [10] and of Kienzler and Heuu,ann [13]. We use throughout the paper the vectorial as well as the index notation to represent Cartesian vectors and tensors, thus rectangular coordinate systems are adopted in all cases. The motion of a thermoelasfic body is described by the smooth mapping X~ = X f l ( X A ) , A, fl = 1, 2, 3, 4, where X4 = t, x4 = or, a = or(X, t) = a(XA) is the thermal displacement field. Also,
CANONICAL FORMULATION AND CONSERVATION LAWS
373
we use the notation X to denote the material space variable, and x for the spatial position of the particle X at time t. In a coordinate system these variables will be written as XL, L = 1, 2, 3 and xi, i = 1, 2, 3, respectively. Thus, the motion is alternatively written as Xi = xi(XL,
t),
a =
ot(XL, t),
i, L = 1, 2, 3.
Generally, if not otherwise denoted, Greek indices will range from 1 to 4, while the lower-case Latin ones will range from 1 to 3. Also, the capital letters K, L, M . . . . will range from 1 to 3 and A, B . . . . from 1 to 4. We use two distinct differential operators O/OXA and D/DXA. The former is the usual partial derivative operator while the latter denotes the partial derivative which accounts for the underlying function composition. For instance,
D --F(XB,x~,(XB))
OF OF Ox× = ~ + ----.
DXA
OXA
Oxy OXA
Also, the usual notations GradF = VRF = D F / D X L , D i v F = D F L / D X L and b" = D F / D t for the gradient, divergence and material time derivative, respectively are used. .
Preliminaries
According to the GN theory the field equations of thermoelasticity of type II [9], i.e. the momentum and energy equations are given respectively as
0p - - - D i v T = 0, 0t
-
~
o0)
+ -~-0
+ tr(T~') - S. VR0 = 0,
(1) (2)
0x where p -= pRv is the physical momentum, v = 57--the velocity field, 0 - - t h e free energy function per unit volume, T the first Piola-Kirchhoff stress, F - - t h e deformation gradient, S--the entropy flux vector, ~--the entropy density per unit volume, and 0--the temperature field. In the conventional thermoelasticity, the temperature gradient VR0 is chosen as an independent constitutive variable along with the temperature 0 and the deformation gradient F. In the dissipationless thermoelasticity, Green and Naghdi [9] choose Vo~ as a constitutive variable in place of V R O , where ot = a(X, t) is the so-called thermal displacement field defined in terms of the temperature as o/(X, t) := f
0(X, t)dt.
We note that this is a rather old notion introduced by Van Dantzig (cf. Von Laue, 1921 [15]) as thermacy. It was used by Maugin [18, 19]; it was proved that thermacy is nothing but the Lagrange multiplier introduced to account for isentropy in a variational formulation.
374
v . K . KALPAKIDES and G. A. MAUGIN
The main constitutive assumption tunas out to be ~: = ~ ( F , Vot, &). Following the procedure of Green and Naghdi, one can prove that the constitutive equations take the form T = O~ OF'
O~r ~ -- - 0&'
O~ S = - 0-fl'
(3)
where fl = VRa. Eqs. (3a) and (3b) look like the corresponding ones of conventional thermoelasticity. To reveal the physical sense of Eq. (3c) let us confine ourselves to a lineafized version of the GN theory. Thus, assuming a quadratic free energy function one can obtain [5] q = --kVROt, where q = S / 0 is the heat flux vector and k is a constant tensor related to the thermal constants of the material. Differentiating the above equation with respect to time we obtain aq - - = --kVRO, (4) Ot which is a law for the heat conduction instead of the classical Fourier law of heat conduction. According to Eq. (4), a temperature gradient established at a point results in an instantaneous heat flux rate at that point while the classical Fourier law results in an instantaneous heat flux. Notice that Eq. (4) corresponds to the limiting case 7: ---> oo of the Cattaneo-Vernotte equation, where z- is the thermal relaxation time [3, 5]. Using Eq. (4) one obtains a hyperbolic heat conduction equation which characterizes a wave-like behaviour of heat transport. In the sequel, for the needs of the present work we give some fundamental elements related to variational symmetries and Noether's theorem. Let
L = L ( X A , Xcl,Xcl,A),
A=l,2
. . . . . n,
ot = 1,2 . . . . m,
be a C 2 function, where XA ~ G, G is a smooth domain of ~n and x~(XA) is a sufficiently smooth function. Consider the functional I : C2(G) -+ ]R given as follows: o I (xu) = I
L(XA, x~, X~,A)dV.
(5)
J ~G
Hereafter we shall refer to functional (5) as the action integral. The necessary condition for the functional I to attain an extremum is given by the well-known Euler-Lagrange equations OL
Ox~,
D
(OL~=o,
VXAEG,
(6)
DXA \ OXy,A/
where the summation convention is used over repeated indices. Consider now the (n + m)-dimensional Euclidean space E m+n made up by the dependent and independent variables and the continuous group (actually it is a Lie
CANONICAL FORMULATION AND CONSERVATION LAWS
375
group) of point transformations in this space,
2 a = XA(XB, Xfl; •w), YC~=Yc~(XB,Xtj;Ew),
W = 1,2, . . . , #,
(7)
with
f~a(XB, Xfl, O) = XA,
xa(XB, Xfl; O) = Xu,
w h e r e ~I~a and xa are C °O with respect to XB, x• and analytic with respect to Ew
in the domain of their definition. Ew denotes the /z-dimensional parameter of the group. The corresponding group of infinitesimal transformations will be given by the relations
where
XA = Xa + EwZ~,
(8)
~ = x~ + EweS,
(9)
Of~A . Z~(XB, XB) = O-ff~-tEw = 0),
OXot ¢ff(XB, xa) = O-~w(Ew = 0).
(10)
The vector field over the space ]Em+n defined by the relation Vw = Z w + ~'~ A 8X A c~ 8xcl
(11)
will be called the infinitesimal generator of the group (7). We say that the vector field (11) or equivalently the group (7) is a variational symmetry [1, 29] of the action integral (5) if the latter is invariant under any member of the transformation group (7), that is
fG L ( X A, Xa, Xu,A)dV =
L(f~ A, Xa, Xa,A)d~",
(12)
where a tilde over a quantity denotes the transformation of this quantity under Eqs. (7). The following theorem [29] will provide the so-called infinitesimal criterion in order that a functional be invariant under a continuous group of point transformations. THEOREM. The group of transformations (7) is a variational symmetry of the functional (5) if and only if
DZ~ --0, w = l , 2 . . . . . /z, (13) DXA where V(w1) denotes the first prolongation [29] of the infinitesimal generator (11). V(I~T. w _+L
After that we can easily prove that Eq. (13) is equivalent to the equation
8L OL w OL ( D ( w DZ~ DZ~ --0, OX----~Z~ "~ ~X~ ~et "]- O-ff~,A~ DX"AA xct'BO X A ,J ~ Z o x ~
(14)
376
v . K . KALPAKIDES and G. A. MAUGIN
referred to by [17] as invariance identity. We proceed now to Noether's theorem of which we give a convenient version [17]. THEOREM OF NOETHER. If the functional (5) is invariant under the I~-parameter group of transformations given by Eqs. (8-9), then there exist I~ conservation laws of Euler-Lagrange equations (6) given by
DO~ where
3.
[OL
D [ OL \q w DXa \ ox~,,a / .l
OL OL w\ O; = - LZ~ - Ox~,---~x~,sZ~+ O--~,A(: ).
(16)
The variational principle
DEFINITION 3.1. The Lagrangian function of a thermoelastic body without dissipation is defined to be of the form
L(XL,.~i, or, -Oxi -, 8XL
O~XL)= 21 . O_~XL) . :pg(XL)JciJci - ~ ( XL, -OXi - , or, OXL
(17)
The above definition indirectly provides the independent constitutive variables which should be
3xi , OXL
0=~,
fl-
3ot OXL'
in complete accordance with the corresponding ones that Green and Naghdi introduced in what they call thermoelasticity theory of type II [9]. Then, the functional I for the case under discussion will take the form
I (xi, cO =
f?fo
L(XL, Jci, &, xi,L, a r)dVdt,
(18)
where f2 is a smooth domain of •3 and [tl, t2] an interval of N. Notice that L is not an explicit function of xi by virtue of Galilean invariance (translations in physical space of placements). Neither is it an explicit function of ot itself, this implying a sort of gauge invariance very similar to that of electrostatic for the electric potential. To proceed to the variational principle we have to add initial and boundary conditions. Let us suppose that the functions x~ = (xi, oO satisfy the following restrictions
x~(XL, t) = g~(XL, t), x=(XL, tl) = ha(XL), xa(XL, t2) = ma(XL),
XL c 0~2, XL E ~2, XL 6~2,
t a [tl, t2], (19)
377
CANONICAL FORMULATION AND CONSERVATION LAWS
where g~, ha, m~ are C 2 functions on the domain of their definition and moreover fulfil the compatibility relations
g~(XL, q) = h~(XL),
g~(XL, t2) = m~(XL).
The variational statement for the dissipationless thermoelasticity can be written as follows. PROPOSITION 3.1. Let the constitutive relations (3) hold. Among all admissible
functions of motion and thermal displacement for a thermoelastic body without dissipation which satisfy the initial boundary conditions (19), those yielding extreme value to the action integral defined by (17-18), will satisfy the field equations (1-2). Proof: We can write in a more elegant and compact form the argument of the Lagrangian as L = L
(
XA, O X A , ] ,
(20)
where now x4 := or. Hence, the corresponding Euler-Lagrange equations, i.e. Eqs. (6), take the form
D
( 8_L ~ = 0 ,
Y X A ~ f 2 × [ q , t2].
(21)
D X A ~kOXc~,A/] What remains is to analyse Eq. (21). For our problem, they can be written as DXA
DXL =
-~ + ~
,~
+ ~-~
= 0.
(22)
Taking into account the form of Lagrangian (17), Eqs. (22) can be written as
DXL
-- - ~ (pRJci) = O,
XL 6 f2,
t C [q, t2],
DXL
-I- - ~
X L E ~d ,
t E [tl,t2].
-~
:0,
Inserting now the constitutive relations (3) into the above equations they transform
to DTLi
D
DXL
Dt DSL --+ DXL
(pRJCi) = O,
DO =0. Dt
(23) (24)
Thus the variational statement provides two equations. The first of them, i.e. Eq. (23) is the equation of motion and coincides with the corresponding one of Green and Naghdi, that is Eq. (1). The second one, i.e. Eq. (24), is an equation for the balance
378
V . K . KALPAKIDES and G. A. M A U G I N
of entropy which is also included in the treatment of Green and Nahdhi [9]. Thus, the Proposition 3.1 is partly proved; it remains to prove that Eq. (2) holds too. REMARK 3.1. As far as the variational principle is considered, the field equations for thermoelasticity without dissipation are Eqs. (23)-(24) instead of (1)-(2) of the GN theory. The other required equation, i.e. Eq. (2), is an energy equation and cannot directly arise from a variational principle. What can be expected is that it appears as a consequence of Noether's theorem considering invariance in time translations. 4.
Variational symmetries and conservation laws
Having obtained the variational principle, we can proceed to explore particular cases of variational symmetries related to it. 4.1.
Invariance under translations
First, we shall consider invariance in material space and time translations. LEMMA 4.1. If the action integral of a thermoelastic body without dissipation is invariant under the group of space and time translations, then it is a homogeneous body. Proof: The group of translations in material space and time is given by the following relations: X a "~- XA "+"~ w A ,
if) = 1, 2, 3, 4,
~ = x~,
(25)
which means that Z~ = t~wa and ~w = 0. Then the proof is a simple consequence of the invariance identity (14), which for the group (25) results in OL -- 0, OXL
OL = 0. Ot
--
(26)
The second of (26) is satisfied by the definition of the Lagrangian and the first one is what we want to prove. [] Next, we give the main result of this subsection. PROPOSITION 4.1. Let the motion and the temperature functions xi and 0 satisfy the field equations (23-24)for a homogeneous thermoelastic body without dissipation through constitutive relations (3), on the domain f2 × [q, t2]. Then the following conservation laws also hold on f2 × [tl, t2].
D --(L~KL DXL
D q- TLiXi,K -- SLflK) -- --~(PRJCiXi,K -~- l"]flK) = O,
(27)
D D LDX-(TLiJci -- SLO) + ot-:-:(L - pRJCi~Ci-- r/0) = 0.
(28)
379
CANONICAL FORMULATION AND CONSERVATION LAWS
Proof: The assumptions of the proposition imply that the symmetry (25) holds, thus we can use the general form of the conservation laws (15). With the aid of Eq. (16), the quantity 0~ for the case under discussion takes the form OL
O~ = - - L S w a -~ -y-------Xa,w, OXet,A
W = 1, 2, 3, 4,
(29)
0,
(30)
and the corresponding conservation laws will be D
OL
--(LSwA
DXA
--
DXA
OX~,a
Xc~,w)=
W = 1, 2, 3, 4.
Eq. (30) can be expanded in
D DXA
and
(LSLA
\ -- - - X a , L )
: O,
OXo:,A
L=1,2,3,
D ( DXA
- L ~ 4 A - - 3 L xot,4~ = O. ] Oxot,A
Furthermore, developing the last equations we obtain
ox,
+
oLKxiLDXK
- - Xi q~Xi, K
Ol
D (L_ -OL. "Jr" " ~ -Xi
=0,
OL.)
--
(31)
Finally, inserting the constitutive relations in (31) we obtain the required relations (27-28), hence Proposition 4.1 has been proved. The second of the just obtained conservation laws, Eq. (28), corresponds to time translations, thus it is related to the conservation of energy. In the sequel, we shall prove that it could provide the second of the equations of Green and Naghdi, i.e. Eq. (2). Indeed, inserting the Lagrangian (17) into (28) we obtain D ( D'-XL
) TLilli -- SLO
D(1 -- ~
) "~PRViVi "[- lp "[- 170
:
O.
(32)
After some simple calculation and taking into account equation of motion (23), we can write Eq. (32) in the form - - ( ~ "[- 017) q'- TLiVi,L -- SLO, L = O.
(33)
Equation (33) coincides with Eq. (2), hence Proposition 3.1 has been completely proved. REMARK 4.1. It is noted that to obtain Eq. (33), the homogeneity of the Lagrangian with respect to the material space variables is not required. What is really necessary is the homogeneity with respect to time which is assumed by
380
V.K. KALPAKIDES and G. A. MAUGIN
Definition 3.1 of the Lagrangian. That is why there is no requirement of homogeneity in the statement of Proposition 3.1. REMARK 4.2 It is easy to confirm that the invariance of the action integral (18) under the group of translations in physical space will provide the Euler-Lagrange equations, i.e. Eqs. (23) and (24). 4.2.
Invariance under the scaling group
In this case we will use the following one-parameter group of scalings in material and physical space XA : XA -~ EXA,
~ = x,~ - Ex~.
(34)
Thus, invoking the relations (10) we obtain ZA = XA,
(¢t = --X~,
(35)
which in turn are substituted into the invariance identity (14) to yield the following result. LEMMA 4.2 The action integral of a thermoelastic body without dissipation is invariant under the transformation group (34) /f and only if its Lagrangian fulfils the identity OL XA -- 2 OXA
Xa,A + 4L = 0.
(36)
We remark that for a homogeneous thermoelastic body, namely when its Lagrangian does not depend explicitly on XA, Eq. (36) becomes 0L OXot,A
x~,a = 2L,
(37)
which is nothing but Euler's identity (equation) of degree 2. Consequently, for a homogeneous thermoelastic body all homogeneous Lagrangians of degree 2 meet the invariance identity for the scaling group. Among them of practical importance are the quadratic Lagrangians because they provide linear constitutive relations. Hence, the following results hold. LEMMA 4.3 Let us assume that a homogeneous thermoelastic body without dissipation admits linear constitutive relations deduced from (3). Then its Lagrangian satisfies the invariance identity for the scaling group. PROPOSITION 4.2 Let the motion and the temperature functions xi and 0 satisfy the field equations (23-24)for a homogeneous thermoelastic body without dissipation through linear constitutive relations, in the domain f2 x [tl, t2]. Then the following
CANONICAL FORMULATION AND CONSERVATION LAWS
381
conservation law also holds on f2 x [tl, t2].: D --[(L~KL -t- TLiXi,K -- SLflK)XK -t- (rLiXi -- SLO)t -t- TLiXi -- SLOt] DXL -t- - ~ [ - -
where
e = e(X,
(~PRfCi2i + e ) t -- (PR2iXi, K -l- rlflK)XK -- PR2iXi -- rlOt] = o,
(38)
t) is the internal density function per unit volume.
Proof: The linearity and the homogeneity of the thermoelastic body imply that Lagrangian meets the invariance identity (36). Thus, for the case under discussion, the action integral is invariant under the group (34). That means we can invoke the general form for the conservation law given by Eq. (15). Due to the fact that the group (34) has only one parameter the quantity (16) will take the form OA "-- .
(
. LXA
.
OLx~,BxB .
OXot,A.
OL
OXot,Ax~
)
(39)
and the conservation law corresponding to the symmetry (34) will be
DOA DXA
-
D (
DXA
- 0 = ,
LXA-
OL X~,BXB
OXot,A
OL x~)=O.
OXot,-----A
(40)
Eq. (40) can be analyzed as follows,
D
DXL
OL LXL -- --(X~,MXM + 2~t) -- OL x.') OXa,L
OXc~,L /
Ox~OL(X~,MXM +2fl)-- ~OLx~,I = 0 + --~O ( L t - 77-. OL OL OL OL ) D LXL -- - - ( X i , M X M -t- )git) -- -~'-"-(OtMXM -~- dtt) xi - - - o t DXL OXi,L OOtL OXi,L OOt,L -~D ( __OL OL OL OL ) + Lt -- ~Xi(Xi. ,MXM + ]tit) -- -~-~(OtMXM -~- &t) -- --xi02i - -~7daot = O. (41) Taking now into account that the internal energy is related with the free energy function by e=~p+O0 and recalling the relations (3) and (17), we can easily obtain the required conservation law (38) from (41). []
382
V.K. KALPAKIDES and G. A. MAUGIN
4.3.
Invariance under the group of rotations In this section we will examine the invariance of action integral (18) under rotations of spatial variables. This is the group SO(3) in physical space which is given by the equations:
Y~A = X A ,
A = 1,2,3,4, i, j = 1, 2, 3,
Xi -~ Q i j x j ,
(42)
:74 = x4, where Q is an orthogonal matrix with detQ = 1. The corresponding infinitesimal group is given by XK =XK, iCi = Xi
~I~4.~-X4,
if- eiwj~wXj,
-~4 = x4,
i, j, K, w = 1, 2, 3.
(43)
So, we take for the quantities given by Eqs. (10) Z ~ --~ Z ~ -~- O,
~o = eiwjXj,
~
= O.
(44)
Inserting Eqs. (44) into the invariance identity (14) we obtain the following result. LEMMA 4.4. The action integral of a thermoelastic body without dissipation is invariant under the transformation group (42) if and only if its Lagrangian fulfils the identity OL eiwj OXi, K X j, K = O.
(45)
Invoking the constitutive relations (4) we easily obtain
ewij TKiXj,K = 0.
(46)
Consequently, one can easily prove the following proposition. P R O P O S I T I O N 4.2. Let the motion and the temperature functions xi and 0 satisfy the field equations ( 2 3 - 2 4 ) f o r a thermoelastic body without dissipation through constitutive relations (3) on the domain ~ × [tl, t2]. Moreover, let the Lagrangian fulfil the identity (45). Then the following conservation law also holds on f2 × [tl, t2], D D DXL (eiwjxjTxi) - - ~ (pReiwj/CiXj) = 0. (47)
REMARK 4.3. Eq. (47) is the balance of angular momentum for dissipationless thermoelasticity. Certainly, this equation together with equation of momentum (23) can provide us, as usual, the symmetry of the tensor TLixj,L. But this is erroneous because it is an assumption for the Proposition 4.2 (see Eq. 46) and not a consequence of it. 5.
Material balance laws So far, we have presented conservation laws for the field equations of dissipationless thermoelasticity. From the point of view of material mechanics, it is interesting
383
CANONICAL FORMULATION AND CONSERVATION LAWS
to focus on what can be called material balance laws, that is these corresponding to conservation laws which are nonhomogeneous equations. To obtain such equations we must allow for the presence of sources in the already derived equations. This in turn can be done by relaxing the assumptions we have imposed in order to obtain them. In this way, for every conservation law, having found the conserved quantity, we can obtain a balance law. Depending on the particular equation, we expect these nonhomogeneous terms to be the so-called material forces or moments of such forces. We will apply this procedure to conservation laws (27) and (38). 5.1.
The canonical m o m e n t u m balance law
We proceed now to the above mentioned procedure for Eq. (27) by removing the homogeneity of the Lagrangian from the assumptions of Proposition 4.1. Let us assume that the Lagrangian (17) depends explicitly on material variables, that is
OL OXL
--50,
L = 1,2,3.
In this case no invariance with respect to space translations can be secured. In spite of this, the Euler-Lagrange equations still hold and it is possible to produce a balance law by calculating the expression
DO~
w = 1,2,3, A = 1,2,3,4, DXA' which certainly does not vanish identically any more. Indeed, calculating this term we obtain
DO~ DXA
--
--
D DXA
OL OX w
L~wA
OL
.,--Xa OXot,A
w_ ]
OL D Oxot,aXot,aw -+" ~ + b- A
( OL x OL O ~k-OXa,A - } ,~,w "~- -~Xa,a D X A (Xot,w)
xo,w.
(48)
The last term in the right-hand side of (48) vanishes due to the Euler-Lagrange equations, thus we conclude aL D X ~ - - aXw" (49) But the left-hand side of (49) has already been calculated in Proposition 4.1, hence equating the right-hand side of Eqs. (30) and (49) and taking into account Eq. (31a), we can write
D D OL , (L~KL -}- TLiXi,K - SLflK) -- "-~t (lOR-~iXi,K "~- OflK) -DXL OXK
--
K , L = 1,2,3,
(50)
which is the expected material balance law referred to by some authors [20] as canonical momentum equation or pseudomomentum equation.
384 5.2.
V. K. KALPAKIDES and G. A. MAUGIN
The scalar moment of canonical momentum balance law
In the same way, we can develop the conservation law (38) related to the scaling symmetry. Hence, removing the assumptions of linearity of constitutive relations as well as the homogeneity of the Lagrangian with respect to material variables we obtain directly the balance law
OL ~____~A(LXA_OLx~,,XS_oX~,Ax~)
DOA DXA
OXot,A
@L XL OXL
_
--
(51)
X~,A + 4L,
ct, A = 1, 2, 3,4,
L = 1,2,3.
DOA The term --~-~a has been calculated in the proof of Proposition 4.2 (see Eq. (38)), thus Eq. (51) takes the form
D D X L [(L~KL -1- TLiXi,K -- S L f l K ) X K -]- (TLi-~i -- SLO)t -I- ZLiXi -- SLCt]
°[
q- - - ~
--
pRJCiJQ+ e
)
t - (pRfQXi,K + 17flK)X K -- pRXiXi -- rlct _
@L XL OXL
-- 2
(&)
]
X~,A +4L.
(52)
We remark that Eq. (52) is valid for nonlinear, nonhomogeneous thermoelasticity in the framework of GN theory. Hence, for linear thermoelasticity, implementing Lemma 4.3, the balance law originated by the scaling symmetry will become D
DXL [(Lt~KL "~ ZLiXi,K --SLflK)XK "~ (TLiJQ -- SLO)t Jr ZLiXi -- SLOt] D
1
"~ N [ - - ( ~ p R J C i J Q q - e ) t - (pRxiXi,K 21- l~flK)X K -- pRJQXi -- Oct] :
3L
OXL X L. (53)
Eq. (53)holds for linear, nonhomogeneous thermoelasticity and represents a balance law for scalar moment of pseudomomentum or canonical momentum. The corresponding balance equation in physical space is not often used because it does not play any role in the description of the equilibrium or the motion of a body as does, for instance, the momentum or angular momentum equation. In the case of physical space, the factors that balance the rate of scalar moment of momentum are referred to as scalar moments or virials. So the right-hand side term of (53) is a sort of material scalar moment or material virial. 5.3.
Comparisons with previous results and some remarks
The obtained above results can be compared to previously published work of other researchers. We must especially refer to the work of Dascalu and Maugin
CANONICAL FORMULATION AND CONSERVATION LAWS
385
[6] for GN thermoelasticity and Maugin [20] and Fletcher [8] for elasticity. Let us return to Eq. (27) which represents the canonical momentum conservation law. It can be written in the form D ((1 ) ) D DXL -~PR2i2i - - ~r S K L ~ - TLiXi, K - - a L l K - - "-~(PR2iXi,K q- Of K) : O, or Div
PR-~- - 0
I + TF - S ® f
- ~-~ (pRFrV + Off) = 0.
(54)
Eq. (41) coincides with the corresponding one derived through a vectorial approach in [6]. The conservation laws (27), (28) and (38) restricted to the case of elasticity are in full agreement with these given by [20] and [8]. To obtain a clearer comparision we introduce the following definitions (compare to the general definitions (A.16), (A.17), (A.14) and (A.15) in [23] ' ~ L~ : = --(pR:~f,F -I- rift L) :
b th := {bLK :
pnh
--(Lt~LK nt- TLiXi, K
OL
1
0X
2
:. . . .
- - p " F - ~fl = emech _ t/E, -- SLflK)},
O0 i . i V R p R -- - -
0X'
1
(55)
(56) (57)
7-/:= ~ p e i " i + e,
(58)
Q := T~ - S0,
(59)
for canonical momentum of the present approach, canonical material stress tensor (Eshelby stress tensor), material force of true inhomogeneities, Hamiltonian density (total energy density) and material energy flux (Umov-Poynting vector), respectively. Also, we have defined the mechanical canonical momentum by pmech :__ _ p . F. The presence of fl in 79L ~ may appear strange on a first look. However, it happens that the additional contribution 0fl in the definition (55) is supported by other theories. In a variational approach to the modelling of liquid helium it was shown by Lhuillier et al. [16] that the quantity -Off is nothing more than the internal momentum (accounting for the relative motion of normal and superfluid components) of the superfluid. But in a Lagrangian variational formulation the canonical momentum is nothing more than the pseudomomenturn and thus this internal momentum is truly a part of the pseudomomentum. After definitions (55)-(59), the balance equations (50) and (53) and the energy equation (28) can be written in vectorial form as follows -
DT:,th Div b th + - - finh, Dt
D i v ( - b th • X
+ Qt + T. x -
(60) Sot) + -~O (-7-/t + T ~th • X - p . x - t/ot) = finh. X, L) t
(61)
386
v . K . KALPAKIDES and G. A. MAUGIN
and
DH Dt
-
-
Q = 0,
VR.
(62)
respectively. We recall that Eq. (61) holds for nonhomogeneous but linear thermoelasticity. Under this restriction and in the framework of elasticity it can be compared with Eq. (4.89) of [20]. If we assume, as in standard continuum thermodynamics, that entropy and heat flux are related by the usual relation q
S = ~, we have
(63)
a~
y q = - 0 m0/~
(64)
and Eq. (62) takes the classical form of the energy conservation equation [26]
DH Dt
- -
-
VR • (T. v -
q) = 0.
(65)
REMARK 5.1. After some rearrangements we can write the canonical momentum equation (60) (note that Curlfl = 0; T =transposed) in the form Dpmech D i v b mech =
Dt
rlVRO + S" (VR~) T +
firth,
(66)
where ~meeh = bth _ S ® ft. Also, the energy equation (28), by virtue of constitutive relations (3) and the momentum equation (23), can be written as
___DSL 0 DXL
-
SrOL +
SLt~L + 170 - _O- - - ( T O ) Dt
= O,
or, taking into account (63), we can write
D(Orl_____~)+ VR "q = 7/0 + S. ~. (67) Dt This equation is interesting by itself because of its structure---especially the right hand side--which is similar to that of (66), time derivatives replacing material space derivatives. REMARK 5.2. Under some conditions, a remarkable identity can be produced by scrutinizing Eq. (61) (or equivalently Eq. (53)). We remind that Eq. (37) is a necessary condition for Eq. (61) (or Eq. (53)) to hold. Assuming moreover material homogeneity and developing the terms including stress, entropy flux, momentum and entropy density we can write Eq. (61) in the form
387
CANONICAL FORMULATION AND CONSERVATION LAWS
Div(-b th.X+Qt)+
(~th.x_~t)+ +
(
DivT-
o0)
-DivS--D-~-
Dt
.x
oe-p./~-rl&-T'F-S.~=O
Inserting into the above equation the momentum and entropy balance equations and using the constitutive relations as well as the definition of the Lagrangian, we obtain D i v ( - b th. X + Qt) +
_~
(,pth. X - ~ t )
3L
3L.
3L
3L
= -~--f~ + ~--~o~+ 3---F : F + ~-~./~.
To conclude we have just to recall that the right-hand side of the above equation is equal to 2L due to identity (37), hence one can write D i v ( - b th . X + Qt) 4- ~ t (T~th "X-7-/t) = 2L,
(68)
consequently, we have proved that /f the Lagrangian fulfils Eqs. (26a) and (37), then (68) holds. Summing up, from the Lagrangian density (17) we have deduced all field equations, balance laws and constitutive relations for the theory of material inhomogeneous, finitely deformable, thermoelastic conductors of heat. As a matter of fact, Eqs. (23), (47) and (65) are the local balance equations of linear momentum, angular momentum (in physical space) and energy, respectively. This is completed by the balance equation of mass which here trivially reads
3pR 3t
- - 0.
(69)
These are all formally identical to those of the classical thermoelasticity of conductors (e.g. as recalled in [26]). Apart from the functional dependence of 7t the equation of angular momentum, i.e. Eq. (47), it is also formally identical to the classical counterpart. Only the equation of canonical momentum (60) differs from the one originally obtained by Epstein and Maugin [7] in material thermoelasticity. But, abstraction being made of material inhomogeneities, it is the same as the one obtained by direct algebraic manipulations by Dascalu and Maugin [6] in the "dissipationless" formulation of thermoelasticity. Indeed, canonical momentum (55) is made of two parts, a strictly mechanical part--which is nothing more than the pull back, changed of sign, of the physical momentum--and a purely thermal part given by the constitutive behaviour. In addition, the canonical stress tensor (56)--also called Eshelby stress tensor---contains a contribution of 1~, because from its very definition, it captures material gradients of all fields. One should note that the source term in Eq. (60) has no dissipation contents. Furthermore, contrary to common use, even the entropy equation (24) is source free. This means that in the absence of material inhomogeneities, all equations obtained are strict conservations laws, hence the qualification of "dissipationless theory". In this rather strange--we
388
v.K. KALPAKIDES and G. A. MAUGIN
admit it--approach, the entropy flux and heat flux are derived from the free energy, on the same footing as the entropy density, and stress (Eqs. (3) and (64)).
6.
Relationship to the classical theory
Although the equation of energy looks like the corresponding classical one (compare Eq. (65) with Eq. (4.67) of Maugin [20] or with Eq. (5.3) of [27]), there are essential differences stemming from the functional dependencies of the free energy function 7t in the two cases. Moreover, the equation of canonical momentum (60) deviates from the classical one even more due to the explicit presence of fl in it. Due to the utmost importance of these equations in fracture and phase-transitions front studies [26], we will try to explore their relation to the classical theory; actually we will examine under which conditions a passage to the classical equations is possible. We start from Eqs. (50) and (28) which can be written in another form by the insertion of the entropy and momentum equation (Eqs. (23)-(24)) into them,
DL Dflg OL a tTL,iXi,KL -PRJCiVi,K : __OX-~ SL atrIO'K qOX---~' D X-----~ DL TL'iVi'L Dt PR.~iJfi=O, LSL + 00,
(70)
(71)
where the fact that V × fl = 0 holds is taken into account. Next, we need to isolate the contribution of fl in order to get some "classical" limit: to this end we expand the free energy function in a Taylor series with respect to fl at f l = 0 ,
7~(x, F, 0, #) = ~(X, F, 0, # = 0) + ~-~M(X, F, 0, # = 0)/~M + 0(# 2) and retaining the first-order terms we can write 7t(X, F, O, fl) = ~(X, F, O) + ~M(X, F, O)flM,
(72)
where ~ denots the free energy at the limit and ~PM = O~/OflM[~=O. AlSO, we define the entropy flux at the limit as SM(X, t) = --TiM(X, F, 0),
(73)
following the general definition of entropy flux, i.e. Eq. (4b). With the aid of (72) we compute all the relevant terms in Eqs. (70) and (71) as follows:
Dap DX~-
D~t DaPMo D X~L + "-D-~LpM +
DflM ~M-DXL
D~ DTtM n -- DXL -+ -D-~LpM --
DflM S M -DXL -
D~ (01]?M OI~M OF O;M 0 0 ) DflM = DX----~+ \ OXL + O---F-OZ---~+ - - OXL tiM -- ~SMDXL '
(74)
389
CANONICAL FORMULATION AND CONSERVATION LAWS
T=0~ z
~-F 6
OF o, _
O0
SL--
a• abe o4/
~
]~M
a6
o,M
80
O0
(75)
"~-~ - ~ ~M,
(76)
OU
(77)
- - ~ L = sL, o~M
(78)
3XL
ax~ + 5-~ ~ '
D~p Dt
D6 DTtM ~ D~M + --bT-pM + ~v D---7Dt
-- D---7-+
Dflm
F+
(79)
0 tiM + ~PM Dt
where hereafter the hat over any quantity denotes this quantity at the limit. Inserting Eqs. (74)-(79) into Eqs. (70) and (71) we obtain
oL
^
aL
DX--~ + TL,iXi,KL -- pRXiVi,K = OO,K + OX---K' fL'il)i'L
DL
(80)
PR-~iXi 00,
Dt
(81)
=
respectively. We demand now that the momentum and entropy balance equations as well as the relation between the heat and entropy flux at the classical limit hold, thus DTLi DXK
D Dt (pRJci) = 0,
DSL DO ~SLe,L --DXL + - ~ + --O -- 0,
qL SL = --.O
(82)
Inserting now (82a) and (82) into (80) and (81), respectively, we obtain D
D
DXK
oL
(L + ~Lix~,KL) -- -~(;R~iXi, K) = OO K + OX----K'
O (f,~,~ _ 4L) - O DXL ~-~ (K + g) = 0,
(83) (84)
where ~ is the internal energy at the limit and its standard relation with the free energy function (thus 6 = ~ - 00) was used. Eqs. (80) and (81) are the ones of pseudomomentum and energy for the classical thermoelasticity as given by Epstein and Maugin [7]. Hence, we have proved the following proposition. PROPOSITION 6.1. Let the field equations, the canonical momentum equation and the energy equation for dissipationless thermoelasticity and the balance equations for momentum and entropy of classical thermoelasticity hold, then we can pass from
390
v . K . KALPAKIDES and G. A. MAUGIN
the canonical momentum and energy equations of dissipationless thermoelasticity to the corresponding ones of the classical thermoelasticity by assuming the form for the free energy function given by Eq. (72). REMARK 6.1. We must underline that the described above passage from the dissipationless thermoelasticity to the classical one is accompanied by a drawback; this is the relation (73) providing the entropy flux at the limit. The functional dependence of the entropy flux is unacceptable from the viewpoint of thermodynamics. What is really missed is a dependence of S on the temperature gradient. Maybe one could assume a less smooth function for the free energy function 7z with respect to /], especially for fl = 0. This could provide the necessary freedom for the free energy function at the classical limit, a perspective that, in any case, needs more investigation in the framework of convex analysis. REMARK 6.2. Eq. (83) can be written in the form Dpmech
Div bmech
=
fina + fth,
(85)
fth := ~VR0.
(86)
Dt where b mech :
- ( [ , I g + T . F),
The last introduced quantity is the material thermal force of quasi-inhomogeneity clearly defined by Epstein and Mangin [7] in their general theory of material uniformity and inhomogeneity. Thus (85) recovered its "classical" form the quotation marks here emphasize that, in fact, although this equation is "classical" from our viewpoint, practically unknown to most people, it is the one on which thermoelastic generalizations of J-integral of fracture could be based (cf. the review by Maugin and Berezovski [26]; Maugin and Trimarco [27]). 7.
Conclusions
We have effectively formulated a canonical theory of the thermoelasticity of conductors. All field equations, balance laws and constitutive equations follow from it. The relationship with the "classical" formulation was examined. To proceed further, one must envisage the case where nonthermal dissipative processes (e.g. anelasticity) are present. Considering the theory of internal variables of state in order to describe these phenomena is a sufficiently general approach as demonstrated in a recent book [24]. The only a priori change would be accounting for the dependency of the free energy 7z on a new set of variables collectively represented by the symbol ~b. The corresponding equations of state reads ep + (0ap/0~b)= 0. The main problem, however, is to built the evolution equation of q~, normally a relationship between ~b and the thermodynamical force qb constrained by the second law of thermodynamics. Thus the very presence of q~ is related to dissipative processes and a priori not amenable by means of a canonical variational formulation (~p(X, t) is not a classical field; it has neither inertia, nor is its gradient introduced). But it was recently shown how the variables ~b and 0 could play parallel roles in a certain reformulation of the
CANONICAL FORMULATION AND CONSERVATION LAWS
391
anelasticity of thermoconductors [22, 25]. This is the trend to follow for it is shown that Eqs. (66) and (67) are simply modified by the addition of terms ~ . (VR~b)~ and q~.~, respectively, respecting thus the already noticed space-time symmetry. This hints at considering pmech and 07 as the dual space-time canonical momenta conjugated to space-time coordinates X and t (compare to the nondissipative case in [14]). Acknowledgement The authors are grateful to the anonymous reviewers, whose constructive remarks led to some improvements upon an earlier version of the paper. Part of this work was completed during a stay of VKK at LMM-UPMC with financial support of the European TMR 98-0229 <