viscoelasticity 89 M. Fabrizio
M. Ciarletta * S. Chiritat
1. - Introduction
The study of the equations of dynamical linear thermoelasticity backward in time was initiated by Ames and Payne [1] in order to obtain stabilizing criteria for solutions of the boundary-final value problem. It is well known that this type of problem is ill posed. We recall that the backward in time problems have been initially considered by Serrin [2] who established uniqueness results for the Navier- Stokes equations. Explicit uniqueness and stability criteria for classical Navier- Stokes equations backward in time have further established by Knops and Payne [3] and Galdi and Straughan [4] (see also Payne and Straughan [5] for a class of improperly posed problems for parabolic partial differential equations).
The boundary-final value problems associated with the linear theory of thermoe- lasticity have also been considered by Ciarletta [6] for establishing uniqueness and continuous dependence results upon mild requirements concerning the thermoelastic coefficients.
The spatial behaviour of the thermoelastic processes backward in time has been studied by Ciarletta and Chiri(;a [7]. A time-weighted volume measure is used for establishing a first-order partial differential inequality which implies the spatial esti- mate describing the spatial exponential decay of the thermoelastic process backward in time.
In this paper we consider the boundary-final value problem associated with the linear theory of thermoelasticity. The final data are given at t = 0 and then we are interested in extrapolating to previous all times. We study the temporal behaviour of the thermoelastic processes backward in time. In this aim we introduce the Cesaro means of various parts of the total energy and then, by means of some auxiliary Lagrange-Brun identities [8,9], we establish the relations describing the asymptotic behaviour of the mean energies, provided some mild restrictions are imposed on the backward in time process.
'Department of Information, Engineering and Applied Mathematics, University of Salerno, Salerno, Italy
^Faculty of Mathematics, University of Iasi, Iasi, Romania
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2. - Linear thermoelasticity backward in t i m e
Throughout this paper we shall denote by B a bounded regular region of the physical space £3, whose boundary surface is dB. Identified £3 with the associated vector space, an orthonormal system of reference is introduced, so that vectors and tensors will have components denoted by Latin subscripts ranging over 1,2,3.
Summation over repeated subscripts and other typical conventions for differential operations are implied such as a superposed dot or a comma followed by a subscript to denote partial derivative with respect to time or the corresponding cartesian coordinate.
Throughout this paper we suppose that B is filled by an anisotropic and inho- mogeneous thermoelastic material. We consider the boundary-final value problems associated with the linear theory of thermoelasticity on the time interval (—oo, 0] . Thus, in the absence of the supply terms, the fundamental system of field equations consists [10] of the strain-displacement relation
(1) e-ii = g (ui,j + uj,i) in B x ( - c o , 0], the thermal gradient-temperature relation
(2) 9i = 0,i in B x ( - o o , 0 ] , the stress-strain-temperature relation
(3) Sij = Cijklekl + Mij6 in B x ( - c o , 0 ] , the heat conduction equation
(4) qi = -Kijgj in B x ( - o o , 0 ] , the equations of motion
(5) Sjij = pui in B x (—oo, 0], and the energy equation
(6) -qiti + OoMijeij = cO in B x (-oo,0].
In the above relations we have used the following notations: «j are the components of the displacement vector, 6 is the temperature variation from the uniform refer- ence temperature 0O > 0, ey- are the components of the strain tensor, gt are the components of the thermal gradient vector, S^ are the components of the stress tensor and q, are the components of the heat flux vector. Further, p is the mass density, Ci^i are the components of the elasticity tensor, My- are the components of the stress-temperature tensor, c is the specific heat and ify are the components of the conductivity tensor. In what follows we assume that the density p and the specific heat c are continuous functions of x on B. We also assume that the elastic- ity tensor, the stress-temperature tensor and the conductivity tensor are continuous differentiable functions of x on B and they satisfy the symmetry relations
(7) (-'ijkl — ^klij — ^jiklt
(8) Ma = M^
(9) Kti = Kn.
In what follows we consider the boundary-final value problem (V) defined by the relations (1) to (6), the final conditions
(10) ui(x,0) = u?(x), ui( x , 0 ) = « ? ( x )> 0(x,O) = 0°(x), x e f i , and the homogeneous boundary conditions
Ui(x,t) = 0 on Ei x (-oo,01, Si(x,t) = 0 on E2x ( - o o , 0 ] , (11)
0 ( x , t ) = O on E3x ( - o o , 0 ] , q ( x , i ) = 0 on E4 x (-oo,0], where u°, M° and 0° are prescribed functions and
(12) Si(x,t) = 5 ^ ( x , i ) nj( x ) , ?(x,t) = ft(x,i)nj(x),
rij are the components of the outward unit normal vector to the boundary surface and Ei, E2, E3, E4 are subsurfaces of dB so that E i U S2 = E3U E4 = dB, E ! n E2 = E3 n E4 = 0.
3. — T h e t r a n s f o r m e d b o u n d a r y - i n i t i a l value p r o b l e m . S o m e a u x i l i a r y i d e n t i t i e s
We use an appropriate change of variables and notations convenably chosen in order to transforme the boundary-final value problem (V) into the boundary-initial value problem (V*) defined by the following equations
(13) (14) (15) (16) i n f i x (17) (18) i n f i x
[Or
(0, oo),
oo),
eij
&ij
%i
- 1( \
9i — 0,i!
= Cijkieki + MijB,
Qi **-ij9ji
jij P^lii
+ QoMijiij - c0, , with the initial conditions
(19) u«(x,0) = u?(x)1 «i(x,0) = u?(x)) 0(x,O) = 0°(x), x£B, and the boundary conditions
Ui(x,t) = 0 on Ei x [0, oo), s,(x, t) = 0 on E2 x [0, oo), (20)
0(x,t) = 0 on E3 x [0,oo), q(x,t) = 0 on E4 x [0,oo).
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By a solution of the boundary-initial value problem (V) we mean an ordered array IT = [ui,eij, Sij,0,gi,qi] with the following properties:
(i) Ui, iii, Hi, (uij + Ujj) and ( u y + iijj) are continuous on B x [0, oo);
(ii) ey- is a continuous symmetric tensor field on B x [0, oo);
(iii) Sij and Sjij are continuous on B x [0, oo);
(iv) 8, Qj, 8 are continuous on B x [0, oo);
(v) ft are continuous on B x [0, oo);
(vi) qi and q^ are continuous on B x [0, oo), and which meets the equations (13) to (20).
We proceed now to establish some auxiliary identities concerning the solutions of the boundary-initial value problem (V*). These identities constitute the essential ingredients in our analysis concerning the temporal behaviour of the solutions of the boundary-initial value problem (V*). •
LEMMA 1 Let -K — [itj, e ^ S y , 0 , g i , q i \ be a solution of the boundary-initial value problem (V*). Then, for all t e [0, oo), we have
I f c • - / [pui{t)Ui(t) + Cijkleij{t)eki{t) + — 9{t)2]dv
z JB <7O
( 2 1 ) =\( [pui(0)«i(0) + Cijkieij{0)ekl{0) + ^-8{0)2}dv
l JB !7o
+ Jo JB ^Kijgi{s)gj{s)dvds.
P R O O F . The relations (7), (8), (13) and (17) imply that (22) pui(s)ui(s) = [Sji{s)ui{s)]j - Si:i{s)ei:j(s), so that, by means of the relations (15) and (18), we get
s-i 1
^-{o[P«i(s)"i(s) + Cijkleij{s)ekl{s) + ^-0(s)2}}
(23) ds 2 1 1
= [Sji{s)ui(s) + Tqj(s)0(s)]j - Tqj{s)gj{s).
VQ (70
Finally, we substitute the relation (16) into (23) and then integrate the result over B x [0,i\. Thus, we get the identity (21) and the proof is complete. I
LEMMA 2 Let n = [iii,eij,Sij,0,gi,qi] be a solution of the boundary-initial value problem (P*). Then, for all t G [0, oo), we have
2 / QUi{t)ui{t)dv - — / Kij / gi{z)dz / gj(z)dzdv
JB C7Q JB JO Jo
(24) = 2 / / {SUi{s)Ui{s) - [Cijklei:j(s)ekl{s) + +~9{s)2]}dvds
Jo JB VQ
+2 J QUi(Q)ui{Q)dv -2 I [ 0(s)[Myey(O) - £-9{0)}dvds.
JB JO JB VQ
PROOF. We start with the following identity
(25) ~-[gUi(s)ui(s)} = gui(s)ui(s) + £Uj(s)u;(s), so that, by an integration over [0, t] , we get
(26) gui(t)ui(t) = gui(0)v,i(0) + / [QUi(s)ui(s) + QUi{s)ui(s)]ds.
Jo
In view of the relations (7), (8), (13) and (17), we get (27) gui{s)ili(s) = [Sji{s)ui(s)]j - % ( s ) e y ( s ) , and therefore, by means of the relation (15), we obtain
(28) gui(s)ui{s) = [5ji(s)Mi(s)]j - Cijkieij{s)eki{s) - Miieij{s)e{s).
now, we integrate the equation (18) over [0, t], to obtain
1 /•' c
(29) Mijeij(t) = - — / qu{z)dz + -6{t) + rj0,
fo •'o fo
where
(30) Vo = Myey(O) - f f l ( 0 ) . By combining the relations (28) and (29), we get
1 fs c
QUi(s)ui(s) = [Sji(s)ui{s) + —0(s) / qj{z)dz]j - [Cijfc,e„(s)efci(s) + -z-0(s)2]
VQ JO t)0
1 ["
(31) -r]o0(s) + —Kijgtis) / g5{z)dz,
UQ JO
where an use was made by the relations (9) and (16).
Finally, we substitute the relation (31) into (26) and integrate the result over B and then we take into account the divergence theorem and the boundary conditions (20). Thus, we are led to the identity (24) and the proof is complete. I
L E M M A 3 Let 7v — [UJ, ejj, Sij, 6, gi, qi\ be a solution of the boundary-initial value problem (V*). Then, for all t € [0, oo), we have
2 / gui(t)ui(t)dv - — I Ki:i / gi{z)dz / gj{z)dzdv
JB O0 JB JO JO
(32) = f e[ui(2t)ui(0) + Ui(2t)ui(0)]dv + f f r)o[0(t + s) - 6{t - s)]dvds.
JB JO JB
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P R O O F . Obviously, we have
—{g[ui(t + s)ui{t - s) + v,i{t + s)ui(t - s)}}
(33) = o[ui{t - s)Ui{t + s) - Ui(t + s)ili(t - s)], so that, by an integration over [0, t] with respect to s , we get
2QUi{t)Ui{t) = Q[ui{2t)ui(0) + Ui(2t)ui(0)]
(34) + / Q\ui(t + s)v,i(t - s) - Ui(t - s)ili(t + s)]ds.
Jo
On the other hand, by using the relations (7), (8), (13) and (17), we obtain g[ui(t + s)ili(t - s) - Ui(t - s)ili(t + s)] — [Sj,(t - s)Ui(t + s) - Sji(t + s)ui(t - s)]j (35) +[Sij(t + s)ei:i(t - s) - Sij{t - s)eij{t + s)].
Further, we use the relations (7) and (15) to deduce (36) Sij(t + s)eij(t - s) - Sij(t - s)eij(t + s)
= 0(t + s)Mijeij(t - s) - 9{t - s)Miieij(t + s), and, by means of the relations (16) and (29), we obtain
Sij(t + s)eij(t -s)~ Sij(t - s)eij{t + s) = q0[0(t + s) - 0{t - s)}
(37) +V ^( i _ s )I S<li(z)dz--6(t + s)Jo 'qi(z)dz]}t
1 rt+s rt-s +-K-[9i(t - s)Kij gj(z)dz - gt(t + s)Ktj gj{z)dz\.
(70 ' O J0
Now, we substitute the relation (37) into (35) and the result into the relation (34).
Then we integrate the result over B and use the divergence theorem and the bound- ary conditions (20). Thus, we are led to the identity (32) and the proof is complete. | COROLLARY 1 Let n = [ui,ey,Sy,0,ft,</j] be a solution of the boundary-initial value problem (V*). Then, for all t S [0, oo), we have
2 / / {QUi(s)ui(s) - [Cijkieij(s)eki(s) + --0(s)2]}dvds
JO JB UQ
(38) = - 2 / gui{0)ui(0)dv + [ ^ ( 2 ^ ( 0 ) + Ui{2t)uM]dv
JB JB
+ f f 7?o[20(s) + 0(t + s) - 9(t - s)]dvds.
Jo JB
P R O O F . A combination of the relations (24) and (32) implies the identity (38) and the proof is complete. I
4. - A s y m p t o t i c partition
In this section we derive the relations which exhibit asymptotic partition of the energy provided only that the thermoelastic process is constrained to lie in a set M.
i.e., in the set of all thermoelastic processes -K = [«;, e^, SV,-, 9, gi, qi] defined on B x [0, oo) which satisfy
(39) f f ^Kij9i{s)gj{s)dvds < M2, Vt e [0, 00 JO J B UQ
For later convenience we shall asume in this section that meas E3 ^ 0.
Let 7r = [ui, eij, Sij, 9, g^ g,] be a solution for the boundary-initial value problem (V*) and let associate with it the following Cesaro means
(40) Kc{t) = — / / gui{s)ui{s)dvds,
It Jo JB
(41) Sc(t) = T r r / / Cijkieij(s)eki{s)dvds, At Jo JB
(42) ^ 4 I 7 B ^ 2 ^
(43) Vc{t) = - f [' [ -Kijgi(z)gj{z)dvdzds.
t Jo Jo JB fo
We observe that if meas Si = 0, then there exists a family of rigid motions and null temperature which satisfy the equations (13) to (18) and the boundary conditions (20). For this reason, we decompose the initial data u° and ii° as follows
(44) u? = uj + ^ , u° = u* + U°,
where u* and it* are rigid displacements determined in such a way that J gU°dv = 0 , / 0eUkXjUjdv = 0,
JB JB
(45) / gU°dv = 0 , J 0£ijkXjUf> = 0,
JB JB
where £„•* represents the alternating symbol.
Let us introduce the following notations:
C1^ ) ^ { v = ( t )1, »!, »s) , «16 C1( B ) : «i = 0 on Ex
and if measEi = 0, then / gvidv = 0 , / ^eykXjVkdv = 0 j ;
&{B) = {jeC1(B): 7 = 0 on E3 };
W i ( B ) = the completion of Cx(i?) by means of || • ||wi(B)!
W I ( B ) = the completion of C ^ B ) by means of || • \\wi(B) •
In the above relations C1 (B) represents the set of scalar functions which are continuous and continuously difTerentiable on B. Moreover, Wm(B) represents the familiar Sobolev space [11] and Wm(B) = [Wm{B)f.
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We note that because CyM is a positive definite tensor it follows that the following inequality [12] holds
T / Cijki{vi:j + vjti)(vkti + Vitk)dv > mj / v^dv, ni! = const. > 0, Vv e W i ( B ) . (46) B
Moreover, the boundary value condition (20) coupled with the fact that meas
£3 ^ 0 and the positive definiteness of the conductivity tensor, imply that the following Poincare inequality holds
(47) / Ku") fl Av >m2 72efo, m2 = const. > 0, V7 e W ^ B ) .
JB ' JB
If meas £1 = 0, then we shall find it is a convenient practice to decompose {MJ, 8} as follows
(48) in = u* + tut + vt , 0 = 7,
where {v, 7} (E W j (B) x Wi (B) represents the solution of the boundary-initial value problem (V*) in which the initial conditions (19) are substituted by (49) «i(x,0) = l/?(x), it( x , 0 ) = [)°(x), 7( x , O ) = 0 ° ( x ) , x e B ,
We further introduce the total energy associated with the solution" ir — [ui, etj, Sij, 9, gu qi] by
(50) £(t) = \f [pui(t)ui(i) + Czjkieij{t)ekl(t) + ^6(tf]dv.
Z JB UQ
We have now assembled all the preliminary material needed to derive the asymp- totic partition in terms of the Cesaro means defined by the relations (40) to (43).
T H E O R E M 1 Let IT = [ui,eij,Sij,d,gi,qi] be a solution of the boundary-initial value problem (V*) residing in the class M defined by the relation (39). Then, for all choices of the initial data u° € Wi(B), u° G W0(-B) and 8° G W0(B), we have
(51) lim Tc(t)=0-
t-KX>
Further, we have:
(i) if meas £1 ^ 0, then
(52) lim £c( t ) = lim 5c( t ) ,
t - f o o t->oo v '
(53) l i m 5 c ( t ) = i f ( 0 ) + ilimPc(t);
(ii) if meas £1 = 0, then
(54) lim / Cc( t ) = l i mlSc( t ) + i / BK u >!
(55) lim 5b(t) = \ £(0) + \ j gu*u*dv + \ lim 2>c(t).
' - » ° o I 2 JB 2 t - » o o
P R O O F . We first note that the Lemma 1 and the relation (50) give (56) £(t) = £(0) + f J ^Kiigi{s)gi(s)dvds, t > 0.
JO JB UQ
By taking into account the relations (40) to (43), from the relation (56) we deduce that
(57) ICc(t)+Sc{t)+Tc(.t)=£(0) + Vc{t), for all t > 0.
On the basis of the relations (39), (42) and (47) it results Tc(t) < Ui maxc(x)] J* JB 0(s)2dvds
(58) < 2 S ^ [m|x c(x) ] Jo JB ^Kjjgi(s)gj(s)dvds
< ^ [ m a x c ( x ) ] , t > 0,
and hence by making t to tend to infinity, we get the relation (51). Thus, the relation (57) implies that
(59) lim £c( t ) + lim «Sc(t) = £(0) + lim Vc(t).
On the other hand, from the relations (38), (40), (41) and (42), we get JCc{t) - Sc{t) - Tc{t) = ~ ( QUi{0)ui(0)dv
2.1 JB
(60) +-J- / ' J r]Q[2e{s) + + 0(t + s) - 0(t - s)]dvds At Jo JB
+ 4£ / S[ui{2t)iii(0) + Ui(2t)ui{0)]dv, t > 0.
Further, the relations (39), (47) and (56), give (61) / gui{s)ui{s)dv < 25(0) + 2M2,
JB
(62) / 0{sfdv < — \ - [ U{afdv < - ^ W f ( O ) + M2].
JB maxc(x) JB 00 maxc(x)
Thus, by using the Schwarz's inequality and the relations (51), (61) and (62) into the relation (60), we obtain
(63) lim £c( t ) - lim 5c( t ) = l i m { - ^ / 0Uj(O)ui(2t)dv}.
t—>oo t—• oo t—voo 4t JB
Let us first consider the point (i). Since meas Ei / 0 and u e W i ( B ) , from (46), (50) and (56), we deduce that
(64) J ui(s)ui{8)dv < —£{s) < —[£(0) + M2],
JB mi mi
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(67) / Vi(s)vi(8)dv < —£(s) < —[£(0) + M2},
JB 77ii TTl\
so that, by means of the Schwarz's inequality, we get (65) lim{-i- J gui{0)ui(2t)dv} = 0.
t->oo At JB
Therefore, the relations (63) and (65) lead to the relation (52). The relation (53) results now from the relations (52) and (59).
Let us further consider the point (ii). Since meas Ei = 0, then the decomposi- tions (48) and the relation (45) give
— / QUi(0)ui(2t)dv
At JB
(66)
= 1 / Qii*u*dv + 1 / Q(U* + Uf)vi(2t)dv + 1/ 8u*u*dv.
At JB At JB I JB
On the other hand, the relations (46), (50), (56) and (39) imply
2 „ , N 2
, _ — £ * < — .
IB mi mi
so that, the relation (66) leads to
(68) l i m { - / gui{0)ui(2t)dv} = - f Qu*u*dv.
t->ao At JB 2 JB
Therefore, if we substitute the relation (68) into (63), then we obtain the relation (54). The relation (55) follows then by coupling the relations (54) and (59). Thus, the proof of theorem 1 is complete. I
5. - Concluding remarks
We first note that the procedure presented in the above Theorem 1 can be extended to cover the case when meas E3 = 0, but a detailed analysis of various situations which can appear must be considered.
The relations describing the asymptotic partition of energy for the thermoelastic processes forward in time have been established in [13] without any kind of con- straint restrictions upon the class of processes. The constraint restriction used to establish the theorem 1 is to be expected in view of the results on the uniqueness and continuous dependence results obtained by Ames and Payne [1] and Ciarletta [6]. On the other hand, in view of the identity (56), the constraint restriction (39) can be substituted by the following one
£{t) < M2, Vt > 0, where £{t) is defined by the relation (50).
References
[1] K.A. Ames and L. E. Payne, Stabilizing solutions of the equations of dynamical linear thermoelasticity backward in time, Stab, and Appl. Anal, of Continuous Media, vol. 1, pp. 243-260, 1991.
[2] J. Serrin, The initial value problem for the Navier-Stokes equations, in Proc.
Symp. Non-Linear Problems, Madison: Univ. Wisconsin Press, pp. 69-98,1963.
[3] R. J. Knops and L. E. Payne, On the stability of solutions of the Navier-Stokes equations backward in time, Arch. Rational Mech. Anal, vol. 29, pp. 321-335, 1968.
[4] G. P. Galdi and B. Straughan, Stability of solutions of the Navier-Stokes equa- tions backward in time,Arch. Rational Mech. Anal, vol. 101, pp.107-114, 1988.
[5] L. E. Payne and B. Straughan, Improperty posed and non-standard problems for parabolic partial differential equations, in G. Eason and R. W. Ogden (eds), Elasticity: Mathematical methods and applications, Ellis Horwood Limited, 1990. The Ian N. Sneddon 70th Birthday Volume, pp. 273-300, 1990.
[6] M. Ciarletta, On the uniqueness and continuous dependence of solutions in dynamical thermoelasticity backward in time, Journal of Thermal Stresses, submitted.
[7] M. Ciarletta and S. Chirit,a, Spatial behaviour in linear thermoelasticity back- ward in time, Fourth International Congress on Thermal Stresses THERMAL STRESSES 2001, June 8-11, 2001, Osaka, Japan, pp. 485-488.
[8] L. Brun, Sur l'unicite en thermoelasticite dynamique et diverses expressions analogues a la formule de Clapeyron, C. R. Acad. Sci. Paris, vol. 261, pp.
2584-2587, 1965.
[9] L. Brun, Methodes energetiques dans les systemes evolutifs lineaires. Premiere Partie: Separation des energies. Deuxieme Partie: Theoremes d'unicite, Journal de Mecanique, vol. 8, pp. 125-166, 167-192, 1969.
[10] D. E. Carlson, Linear thermoelasticity, in C. Trusdell (ed), Handbuch der Physik , vol. VI a/2, Springer-Verlag, Berlin, 1972.
[11] R. A. Adams, obolev spaces, Academic Press, New York, 1975.
[12] I. Hlavacek and J. Necas, On inequalities of Korn's type, Arch. Rational Mech.
Anal, vol. 36, pp. 305-334, 1970.
[13] S. Chiri^a, On the asymptotic partition of energy in linear thermoelasticity, Quarterly of Applied Mathematics, vol. XLV, pp. 327-340, 1987.
Internal parameters and superconductive phase in metals
V. A. Cimmelli* A. R. Pace*
1. — Introduction
Superconductivity, i.e. the nondissipative current transport in metals, is a low temperature phenomenon, due to quantum effects which become apparent at the macroscopic scale. After the pioneering work of London brothers [1], its description was based on the celebrated Ginzburg-Landau theory [2], where special attention is paied to the transition from the normal to the superconducting phase. In modern nonequilibrium thermodynamics such a fascinating phenomenon motivated a wide literature, [3,4,5]. Let us quote the series of papers by Fabrizio and co-workers [6,7,8], where a nonlocal theory of superconductivity is developed by modifying the classical London's model. In the last decade several authors approached the problem in the framework of internal variable thermodynamics, by introducing a complex internal variable which models the phase effect of quantum mechanics [9,10].
In the present paper instead we introduce in the constitutive equations a real in- ternal variable together its gradient in order to account for weakly nonlocal effects, both in space and time, which characterize a superconductive state. A similar point of view has been applied by Kosinski and Cimmelli in the description of the prop- erties of the superfiuid helium II, [11]. We develop a phenomenological model of a rigid electromagnetic solid which is able to describe the main features of the super- conducting phase, i.e. the nondissipative current transport and the expulsion of the magnetic induction from the conductor (Meissner effect). According with London's approach [1], we split the total current density J into a sum of a normal current J„
and a supercurrent Js, namely:
(1) J = J „ + JS.
Our main hypothesis consists in assuming the vector Js proportional to the gradient of the internal variable. By design Js is compatible with the system of Maxwell equations and, moreover, it does not dissipate energy inside the conductor.
In Section 2 we sketch the main properties of the superconducting systems in order to point out the experimental starting points of the theory.
In Section 3 the complete set of evolution equations for a rigid electromagnetic solid 'Department of Mathematics, University of Basilicata Contrada Macchia Romana,
85100 Potenza - Italy
e-mail: [email protected], [email protected]
43
with an internal variable is derived. The local form of second law of thermodynamics is obtained as well.
In Section 4 we postulate a Fourier's type constitutive relation between the super- current vector Js and the gradient of the internal variable and show that the basic properties of Js are traduced by a suitable initial and boundary value problem whose mathematical structure is analyzed.
In Section 5 the compatibility of the form of Js with both the system of Maxwell equations and the Clausius-Duhem inequality is investigated. Such a compatibility yields a set of thermodynamic restrictions on the response functions together with some additional constraints on the material functions characterizing the model. We close the paper by a final discussion and a comparison with a different theory pro- posed recently in the literature.
2. - A s p e c t s of t h e superconducting phase
The superconducting phase was first observed in 1911 at the Kamerlingh Onnes Laboratory of the Univesity of Leiden. Onnes itself discovered that in some metals if the temperature is lowered until a given critical value 6C, the electrical resistance suddenly drops to zero and an electrical current can flow in the absence of dissipation.
That property allows some important applications, for instance the generation of magnetic fields without the need to remove the Joule heating due to the current creating the fields [13]. In 1933 Meissner and Ochsenfeld measured the magnetic induction B outside a superconductor as it is cooled in an applied field. They found that the strength of B immediately outside the superconductor increased while its normal component on the boundary appeared to be zero, indicating the vanishing of B inside the specimen and the existence of a perfectly diamagnetic state of the superconductor which caused the internal field to be expelled. The final state of the superconductor was found to be independent of whether it was cooled through 6C and then placed in a field or viceversa, since all the flux was expelled from the superconductor in either case. The phenomenon is referred to as Meissner effect.
If the magnetic field H increases at a constant temperature below 0C the metal remains superconductor until a given critical field Hc at which the normal behaviour is restored. The transition from the superconductive to the normal phase at constant temperature and magnetic field Hc represents a first order phase transition since the experiments show the existence of a nonvanishing latent heat [14].
The London brothers were the first to suppose the existence of superconductive electrons flowing without resistance. In the presence of an electric field E these electrons obey the equation of motion:
(2) rav, = eE,
where m, e and vs are the mass, the electric charge and the mean velocity of the electrons. Under the hypothesis:
(3) Js = nsev.