Models of electromagnetic materials

MATHEMATICAL AND COMPUTER MODELLING PERGAMON

Mathematical and Computer Modelling 34 (2001) 1431-1457 www.elsevier.com/locate/mcm

M o d e l s of E l e c t r o m a g n e t i c Materials M. FABRIZIO Universitg, D i p a r t i m e n t o di M a t e m a t i c a P i a z z a di P o r t a S. D o n a t o 5, 40126 Bologna, I t a l y

A. MORRO Universitg, D I B E v i a O p e r a P i a l l a , 16145 Genova, I t a l y

Abstract--A

thermodynamic framework is provided for the modelling of smart materials acted upon by electromagnetic fields. The description is based on the notion of state and process which, jointly, determine the response of the material. The view is taken t h a t only the balance equations which are compatible with thermodynamics represent physically-admissible behaviours. Concerning simple materials, linear, anisotropic materials with memory are considered in detail by disregarding the effects of motion. Necessary and sufficient conditions are determined for the relaxation functions to be compatible with the second law. Next, ferromagnetic and ferroelectric materials are considered. Liquid crystals are described both as materials with microstructure of a micropolar form and as materials within a generalized form of the director model. The modelling of nonsimple materials requires t h a t the second law is considered in a global form. This is exemplified by considering dielectrics with quadruples and spatially-dispersive crystals. (~) 2001 Elsevier Science Ltd. All rights reserved.

Keywords--Electromagnetic tions.

solids, Thermodynamic restrictions, Nonlocal constitutive equa-

1. I N T R O D U C T I O N This paper has a threefold purpose: first, to provide a thermodynamic framework for the modelling of s m a r t materials where the electromagnetic field plays a central role; second, to elaborate specific models of material behaviour which are of interest in m a n y applications; third, to establish a useful connection with papers of this issue where models of smart materials are developed in detail. Concerning the general framework, we find it convenient, if not imperative, to describe the pertinent variables in terms of state and process. In essence, the process determines the time evolution of the state. The process and the state jointly are enough to determine the response of the material. Also, state and process variables are subject to the balance equations. T h e y are at least Maxwell's equations for the electromagnetic field. Other equations have to be considered if the material is deformable or a microstructure enters the model. In any case, the constitutive functions, namely the functions characterizing the response, are required to obey the Second Law of Thermodynamics. We take the view that only the set of functions which are compatible with the second law represent physically-admissible behaviours.

0895-7177/01/$ - see front matter (~ 2001 Elsevier Science Ltd. All rights reserved. PII: S0895-7177(01)00139-X

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1432

M. FABRIZIO AND A. MORRO

Really, as with the balance equations, the form of the second law is related to the kind of material under consideration. To be specific, the material may be deformable, and hence, also motion and forces enter the balance equations. The occurrence of a microstructure requires additional variables and terms. For nonsimple materials, we need to introduce an additional entropy flux in the expression for the second law or even leave the second law in a global form. In connection with specific models, linear, anisotropic materials with memory are considered in detail by disregarding the effects of motion. Also, because of the linearity, it is equivalent to examine the behaviour for time-harmonic dependence. In such a way, necessary and sufficient conditions are determined for the relaxation functions to be compatible with the second law. Next, an outline is given of ferromagnetic and ferroelectric materials. Liquid crystals are described both as materials with microstructure of a micropolar form and as materials within a generalized form of the director model. The modelling of nonsimple materials requires that the second law is considered in a global form or, possibly, with a generalization of the Clausius-Duhem inequality where an additional entropy flux has to be determined. The connection with other articles of this issue is given in the appropriate sections. 2. E L E C T R O M A G N E T I C

PROCESSES

AND

STATES

We consider a body or system B which models an electromagnetic solid. The body B occupies a region T~ in the three-dimensional point space. Any point of B is labeled by the position vector x E T~. Henceforth, it is understood that the statements are relative to any fixed point x E 7~. Also, V denotes the set of vectors, II~ the set of reals. The symbols E, H, D, B, J, q denote the electric field, the magnetic field, the electric displacement, the magnetic induction, the electric current density, and the free-charge density. These fields depend on x E T~ and time t E ]~. They satisfy Maxwell's equations which, in the rationalized MKSA system of units, take the form ~ 7 . D =-q,

~TxE+B=0,

~7.B =0,

~7×H-D=J,

where V. and V × stand for divergence and curl while a superposed dot means time differentiation. As a consequence, Poynting's theorem holds in the form E.D+H.B+J.E+V.

( n X H ) = 0.

An electromagnetic simple system is the set 1 (H, E, ~,/~} where l=I denotes the space of electromagnetic processes 15 given by

P(t) = (E'(t),H'(t),O'(t),g'(t)),

t E [0, db),

is the absolute temperature and g = ~70. Also, Z is the set of states 5, and writing 0 (30, 15) = means that 3 is the state obtained through the process P by starting from the (initial) state 30. The response function/~ is the set of constitutive quantities /~ = (D, B, J, h, q). By way of example, the response function may be given by the set of relations D(t) = eE(t),

B(t) = till(t),

h = a . E(t) + b - H ( t ) +

c~(t),

J(t) = a E ( t ) , q(t) = -tog(t),

1The superposed bar is merely a reminder that thermal effects are considered.

Models of Electromagnetic Materials

1433

where a, b are vectors and c is a scalar. Here, ~(t) = (E(t), H(t), O(t)) and 0 is defined by t

E(t) = _[ E'(~)d~ + E(0), J0

o(t) =

H"(~)d~+H(0),

H(t) =

fo

0'~(~) d~ + 0(o).

Another example involves the simultaneous dependence on electromagnetic variables and temperature in the form D(t) = I ) ( E t,0t),

B(t) = I3 ( H t,0t),

h = h ( E t, H t, 0 t, E(t), H(t), t~(t)) ,

J(t) = J (E t,0t),

q(t) = Cl (E t, n t, 0 t, g(t)).

Hence, ~(t) = (E t , H t,0t), while ~ is defined as before. A further example, which models a dielectric with heat conduction, is expressed by D(t) = e°E(t) +

/5

e'(u)Et(u) du,

J(t) = 0,

B(t) = ~ ° H ( t ) +

/0

e'(u)Ht(u) du,

~'(u)gt(u) du,

q(t) = ~0g(t) +

where e ~, ~', ~' E LI(R+), and a prime denotes differentiation with respect to u E R +. The heat flux might be taken to depend on the integrated history ~t rather than on gt where ~t (u) =

g(~) d~. U

For, upon the observation that gt(u) = -

d(gt ( u )

du , an integration by parts gives

Hence, because ~t(0) = 0, the relation

~'(u)gt(u) du =

~"(u)~t(u) du

holds if ~" exists and e~r(c~) = 0. To save writing, henceforth, we omit the overline on the pertinent symbols. DEFINITION. The state s E ~ of an electromagnetic system is said to be controllable (or attainable) from all of F~if, for any initial state s i E ~, there is a process P such that

o(s',P) The system is said to be controllable if any state s E ~ is controllable from all of ~.

1434

M. FABRIZIOAND A. MORRO

DEFINITION. The state space E of an electromagnetic system is said to be reachable from a state s if, whatever any final state s I E E, there is a process P E YI such that 6(s, P) = s y.

If a state s E P~ is controllable, the space 2 need not be reachable from the state s. However, a controllable system makes E reachable from every state. REMARK. The assumption that h is a function of s and P amounts to saying that the rate at which heat is absorbed is a function of the pair (s, P). The balance equation h = - V - q + r then implies that - V • q + r is a function of (s, P). 2.1. T h e r m o d y n a m i c

Laws

A pair (s, P) is called a cycle if L)(s, P ) = s. FIRST LAW OF THERMODYNAMICS. Along any cycle (s, P), the equality j~0dP [ h ( t ) + I)(t) • E(t) + I3(t). H ( t ) + J ( t ) ' E(t)] dt = 0 holds where h, D, I3, J depend on t through (s(t), P(t)) and s(t) = O(s, P[0,0).

As a consequence, we establish the existence of a thermodynamic potential. THEOREM. I f the system is controllable, then the First Law of Thermodynamics impBes the existence of an internM energy w : 2 x II --~ R such that 2 w(s2) - w ( s l ) =

fodÈ

[h(t) + D ( t ) . E(t) + I3(t). H(t) + J ( t ) - E(t)] dt

(2.1)

for every pair of states 81,82 E ~ and P E I I such that Q(S1, P ) = S2.

PROOF. Consider the functional e : P, x II --* N[ defined by e(s, P ) =

f0d" [h(t) + D ( t ) E ( t ) + B ( t ) . H(t) + J ( t ) - E(t)]

dr.

The set E(so,s)

=

:

e(so, P) = s, v P }

consists of a single element. Moreover, because the system is controllable, there exists a process Po E II such that L)(s, Po) = So. Hence, (s0, P * Po) is a cycle and, by the First Law of Thermodynamics, we have e(so, P * Po) = 0. Since e is additive relative to the process, for every process P such that ~)(So, P ) = s, we have e(s0, P) + e(s, Po) = O. This implies that e(so, P ) is independent of P whence, the uniqueness of the value e(so, P).

I

Let s0 be fixed and s = Q(So, P). Hence, the function w(s) := e(so, P)

satisfies condition (2.1). For, if s i , s ~ E 2, then w(82) - w(s1) : e(so,/)2) - e(s0, P1) for every pair Pi, P2 E H such that 6(80, Pi) = 81, O(So, P~) = s2. In particular, letting P2 = Pi * P, where P is any process such that O(si, P ) = s2, by the additivity of w with respect to the process, we have w(s2) - w(s1) --- e(80, P1 * P) - e(So, Pi) = e(so,P1) + e(s1, P) - e(s0,P1) = e ( s 1 , P ) ,

which coincides with (2.1). 2Still h, D, B, J depend on t through ($(t), P(t)) and s(t) = O(s, PIo,t)).

Models of Electromagnetic Materials STRONG FORM OF THE FIRST LAW OF THERMODYNAMICS.

For

1435 every

thermoelectromagnetic

system, there is an internal energy w : E --+ R such that (2.1) holds for every pair $1,$2 C E

and P E H such that 0($1, P) = $2. REMARK. The internal energy w is defined to within an additive constant. It is unique as soon ms we fix the value w(~) at any reference state 3. It follows at once from (2.1) that, if w(s(t)) is differentiable with respect to t, then ~b(s(t)) = h(s(t), P(t)) + J(s(t), P(t)). E(t) + D(s(t), P(t)). E(t) + B ( s ( t ) , P(t)). H(t). Hence, by replacing h, we obtain the first law in the differential form ~b(t) = - V . q(t) + D ( t ) . E(t) + B ( t ) . H(t) + J ( t ) . E(t) + r(t). SECOND LAW OF THERMODYNAMICS.

For every cycle (s, P) of a simple thermoelectromagnetic

system ~o d'" [h(s(t~(t-~(t)) + 02-~ q(s(t),P(t)) . g(t)] dt <_0

(2.2)

holds with s(t) = o(s, PIo,t)) and equality holds if and only if (s, P) is reversible. By replacing h, we can write (2.2) in the form

1

-ff~ [ ~ v ( t ) - J ( t ) . E ( t ) - D ( t ) . E ( t ) - B ( t ) . H ( t ) ]

+0-~q(t).g(t)

1 dt<__O.

(2.3)

As a consequence of (2.2), a thermodynamic potential may exist which is called entropy and is defined as follows. DEFINITION. A function 77 : l) --~ R is called entropy if T) C E is invariant under Q and, for any pair $1,$2 C E and P C H such that 0 ( s 1 , P ) = $2, + 0 - ~ q($(t), P(t)). g(t)

~(s2) - ~(sl) >

dt,

(2.4)

where 8(t) = ~)(Sl, Pt). THEOREM. / / a simple, thermoelectromagnetic system is controllable, then, by the Second Law

of Thermodynamics, there exists an entropy function ~ : ~ ---* ]~ that obeys (2.4). If 77(8(t)) is differentiable, with respect to t, then

i?(s(t)) -

h(s(t),P(t)) 1 O(t) + 0 - ~ q(s(t), P(t)). g(t).

A natural question arises as to the statement of the second law for noncontrollable systems. In this regard, we consider the function I((s,P)

=

l

and let H (s,3) := sup { K ( s , P ) ; P E H, 0 ( s , P ) = 3}. P

We then write the statement as follows.

1436

M. FABRIZIOAND A. MORRO

STRONG FORM OF THE SECOND LAW OF THERMODYNAMICS. For every simple thermoelectromagnetic system, the function H(8, .) is bounded above, name/y, Vs • P., there is a constant H8 such that H (s,~) <_ H s < co, V~ • 2. Moreover, there is a state 8 t, calIed zero state, such that H ( s t , s ) _> 0 for all s • 198t.

We can easily show that the strong form of the second law implies the validity of the second law. If the system is controllable then the two statements are equivalent. THEOREM. The strong form of the second law implies the existence of an entropy function ~m : Z)8* --~ I~. The function "~m is related to any other entropy function 7] : D --* R, subject to D D Dst and r/(8 t) = ~m(St), by the inequality

vm(s) _< v(s),

Vs c z)st.

PROOF. Let ~m($) : : H ( 8 t , 8 ) . Consider two states 81,82:Dst and a process P C H such that 0(81, P) = 82. Moreover, for every e > 0, there is a process Pe such that ~)(st, P~) = 81. Hence, by the definition of 7/m, we have ~?m(sl) < K (s t, P~) + e, ~m(s~) _> K (s t, P~ * P ) , whence, it follows that 7/(82) - ~/(Sl) satisfies the required property for entropy functions. Now, for each process/5 such that 0(s t,/5) = s and each entropy ~/, we have ,(8) _> K ( s t , / 5 ) . The required inequality 7/(8) _> r/re(s) then follows.

|

Within the set of entropy functions, the next theorem singles out a reference function. THEOREM. The strong form of the second law implies that

n(8) = i ~ f { - K ( a , P ) :

P E H}

is bounded below for every s C E and that := R(s)

is an entropy fimction such that ~lM(8¢) = O. The proof is given in [1].

3. L I N E A R

MATERIALS

WITH

MEMORY

Let 0 be constant in space and time. Integration of (2.3) over the duration of a cycle [0, d) yields d

/o E*

°

Standard linear models of simple electromagnetic systems are in fact particular cases of the response functions D(t) = e0EE(t) +

/0

e~(u)Et(u) du + e0,,n(t) +

/0

e,,' (u)Ht(u) du,

(3.1)

Models of Electromagnetic Materials

1437

B(t) = / ~ 0 ~ E ( t ) +

/0

/ ~ ( u ) E t ( u ) du +/~0,H(t) +

/0

/ ~ ( u ) H t ( u ) du,

(3.2)

J ( t ) = er0EE(t) +

fo

a;(u)Et(u) du + a 0 n H ( t ) +

f0°° a ~ ( u ) H t ( u ) du.

(3.3)

Of course, e0~, e~(u),.., are tensors, V -* V, and the dependence on x E T~ is understood and not written. 3 The state s is the pair of histories E t, H t. Hence, a pair (s, P ) is cyclic if the histories E t, H t are periodic. Concerning the dependence on the argument u, as usual we let e ~E, e 'n , . . . , t r , t E LI(R+). Let

i~ = eo~ +

e~(u) exp(i~u) du,

and so on for ~ , , . . . , & , . For time-harmonic fields (o¢ exp(-io~t)), we have D = ~ E + ~HH,

B =/2~E +/2HH,

J = aEE + ~,H.

By definition, e E , . - . , & . are complex-valued tensors. Moreover, lim ~ = eoE, 0 ) --~ O O

lim ~ = eo~ +

e~(u) du =: e~E,

0~'---~0

and so on. To obtain t h e r m o d y n a m i c restrictions, we consider the electromagnetic field E ( x , t) = E l ( x ) sinwt + E2(x) coswt,

H ( x , t) = H i ( x ) sincvt + H2(x) cos cvt,

(3.4)

where w > 0. Correspondingly, we have B = B1 sinwt + B2 coswt,

D = D1 sinwt + D2 cos~vt,

J = J1 sinwt + J2 coswt,

where, e.g., t ' H 2, B1 = (/~0~ + / ~ c ) E 1 + t ~ s E 2 + (/~0,, +/~,,c)H1 +/~,,s 82

=

' ' H 2 -- /~HsH1, ' (/~0~ +/~IEc)E2 --/~EsE1 + (/~0H + ~HC)

and the subscripts c and s denote cosine and sine Fourier transform. Preliminarily we have to ascertain t h a t the field (3.4) Maxwell's equations. In view of (3.4), we let the free-charge density be given by p -- Pl sinwt + P2 coswt. Substitution yields V X El-wB2--0,

V × E2 + wB1

V × H1 +oJD2 = J 1 ,

V X H 2 - ~D1 = J2,

= 0,

V - B 1 = 0,

V . B 2 = 0,

(3.5)

V.D1 =pl,

V.D2=p2.

(3.6)

In conclusion, at any point of the body we can choose field (3.4) with arbitrary values of E l , E2, H I , H2 provided only t h a t the gradients VE1, VE2, VH1, VH2 satisfy (3.5) and (3.6). We can now determine some consequences of inequality (2.3) on the constitutive equations for D, B, J. Substitution of (3.4) in (2.3) and integration over the period [0, d] yields E2 • D1 - E1 • D2 + H2 • B1 - H1 • B2 + w - l ( E t • J1 + E2 • J2) > 0. As w --* oc, by Riemann-Lebesgue's lemma, we have D1 --+ e0~E1 + e0HH1,

D2 --* e0~E2 + e 0 , H 2 ,

3References [2,3] show models where the dependence on x plays a vital role.

(3.7)

1438

M. FABRIZIO AND A. MORRO

and similarly for B and J. Hence, as w --~ oc, (3.7) simplifies to E2" [eoF.- (eoz)T] E1 + H 2 . [ Z o , , - (Zo,,)-r] H1 +E2"

[COl I --

(3.8)

(p0E) T] H1 + H2- [Zon. - (eou) T] E1 > 0.

The arbitrariness of E t, E2, H1, H2 implies that cOF. = (eo~.)T,

Z0,, = (Zo.) T,

co. = (ZoE) T.

(3.9)

Also, letting w --~ 0, we obtain E1

• qooF.E1 + E1 • ~oo.H1

+ E2 • ~oo~.E2 + E2 - qoo.H2

> 0,

whence, it follows that O'oou > 0,

O'oc. ----0.

If, instead, the material is nonconducting then, at the limit w ~ 0, we arrive at (3.8) with e0F.,., Z0E,. replaced by eoo~,., Zo¢~.,.. Using again the arbitrariness of E l , E2, H i , H2 gives CooE = (CooE) T,

Zoo,, = (Zoo,,) T,

Coot, = (ZooF.) T.

(3.10)

By regarding formally the four-tuples of vectors ( E 1 , E 2 , H 1 , H 2 ) as elements of the twelvedimensional Euclidean vector space V12, inequality (3.7) results in the positive definiteness in V12, for every w > 0, of the symmetric tensor

where

N =

sk (tE~c - u g - l o ' ~ s )

L

=

~F.s ' ÷ rM-1

(~r0E + 0"'~¢)

1 [£11s ÷ Z ~ s ÷OJ-1 (O'olI ÷ O.'llC) -~ 'T I e ,I c -- ZF.c -- cO-1 O',s

'

M

=

k skz'l,¢

Z',,~

, - - E' H c ÷ Z ~ cT ÷~-d-1 O'Hs ] 1 'T 03-1 I e , s + Z~s -Jr(o'0, ÷ o',c ) '

and sk means the skew-symmetric part. In particular, it follows that the diagonal terms of N and M are positive definite, namely, ,' + w- 1 (¢roF.+ ¢r,~:) ; e~.~ > 0,

Z .,'s > 0,

Vw > 0.

(3.11)

It is worth observing that, by the first inequality in (3.11), letting w be as small as we please, we conclude that ~o~ + a ' ~ ( o ) > o, unless ~r0~ = 0 and er~(u) = 0, Vu E ]R+. THEOREM. The constitutive equations (3.1)-(3.3) meet the Second L a w of T h e r m o d y n a m i c s if and only if (3.9) and the positive definiteness of A hold. PROOF. The state and the process are the pairs s(t) = (E t, Ht), P = (E P, HP). The "only if" part has just been proved. To prove the "if" part, we consider a periodic state function 8 of period d. Accordingly, we represent s, namely, the histories E t, H t by Fourier series as (X)

E t (x, u) = ~ Ekl (x) sin kw(t - u) + Ek2 (x) cos kw(t - u), k=O

M o d e l s of E l e c t r o m a g n e t i c M a t e r i a l s

1439

oo

Ht(x, u) = E

Hk~(x) sin kco(t - u) + Hk~(x) cos kw(t - u),

k=0

where w = 2~r/d. Upon suitable dependences on x, any four-tuple of values Eke, Ek2, Hkl, Hk2 is compatible with Maxwell's equations. The function D(t) is then given by oo

D(t) = E

[Dhl sin kcvt + Dh~ cos kwt],

h=0

where Dhl = [£0~ + e~c(hw)] Eh~ + e~s(hW)Eh2 + [co,, + e',c(hco)] Hhl -- e~,s(hw)nh2, Dh~ = [eo~. -- e',(hco)] Eh~ + e'~(hco)Eh~ + leo,, - e',,,(haa)] Hh~ + e ' , c ( h w ) n h > 'The functions B(t) and J(t) are obtained by simply replacing e with > and a , respectively. The process P is taken as the corresponding pair ( E , H ) on [0, d]. The work W(~r, P) along the cycle is given by W(cr, P) = w

E

h{ [Dhl cos hwt - Dh2 sin hwt]. [Ekl sin kwt + Ek2 cos kwt]

h=0 k=l

+[Bhl cos hoot - Bh2 sin hwt] • [Hkl sin kwt + Hk2 cos kcot]} dt +

E

[Jhl sin ha~t + Jh2 sin haJt] • [Ekl sin kc~t + Ek2 cos kant] dt.

h=O k = 0

Term by term integration, as t C [0, d], shows that the only nonzero terms are those with h = k. Further, integration yields a common factor d/2 = 7r/w. Hence, letting '='k be the four-tuple of vectors (Ekl, Ek2, Hkl, Hk2), in view of (3.9), we have oo

lg(~r, P ) = rr E

k'~k. A(kco)~k -F wrr {E01 • [~r0E q- ~r~c(0)] E01 + E02" [~r0r. q- ~ c! ( 0 ) ] Eo2}. ! 03

k=l

':['he positive definiteness of A, and hence, of a0z + er~(0), yields the desired conclusion W(cr, P) > 0 for every nontrivial cycle, which is the content of the second law of thermodynamics. Consider the relaxation functions ~ , ( u ) = ~0~, +

F

~(~) <,

~(u)

= ~,0~ +

d0

/]

/z,, (u) =/*0,, +

~ ' ( ~ ) d~,

fO u

/z~({) d{.

Since e'~,, crF.,'/*,,' E LI(R+), the limit values e o ~ , ~ro~, /~o~,, of e~(u), cr~.(u), /&, (u), as u --+ oo, hold finite. It is convenient to consider also ~(u) LEMMA.

If~,

= e~(~)

- e~,

~(~)

= a~(~)-

~ ,

p,,(u) = g,,(u) - g~,,.

E LI(R+), then 1"o,, - I~,, ( u ) < 0

and

f0 ~ /i,,c(w) dw = lZo,, - lZoo,,. PROOF. By the Fourier inverse transform, we have 2 f0 ~ ~,,, (u) = 7r -

p/., (aa) sin wu dw,

1440

M. FABRIZIOAND A. 1V[ORRO

whence, by integration, 2 f o o 1--cos03u tt,,(u) - tt0,, = -~r Jo 03 U',,s(03) d03. Since tt,~ ' > 0, then # , ( u ) > # 0 , . Now, for any 03 > 0, an integration by parts yields ~,,~(03) = __l #,,,~(03) < 0, 02

and hence,

tt,,(u) - / t o , , = - -

2// 7r

(1 - coswu)fZ,,c(03)dw.

The limit as u --~ ~ and Riemann-Lebesgue's lemma yield the second result. As a consequence of this lemma, we have /zoo. >/-tO. ; namely, the equilibrium magnetic permeability p o o . is greater than the instantaneous magnetic permeability it0.. An integration by parts yields !

and hence, (3.11) becomes 1 e's(03) + o ~ s + - a o ¢ ~ > 0. 03

Application of the Fourier inverse transform and integration with respect to u yields e~(u)

-

eo~

>

--

2 f 0 ~ 1 - c o- s 0 3 u 7(

O2

( (r~s

1

+ -

aoo~

)

d03.

03

If, further, the solid is nonconducting--and hence, a ~ = 0 - - t h e n e~s'(w) > 0 and 2 ~0 c¢ 1 - coswu e~ s(w) d03 > O. ~(u)

-

~o~

=

~

03

Taking the limit as u ---, oc yields £ o o E > EOr~.

For nonconducting systems, the equilibrium electric permittivity eoo~ is greater than the instantaneous electric permittivity e0~If e . , #~, ¢r. are zero, then we let e, #, ~r stand for en, # . , a% and write the conditions tt's > 0, 4.

FERROMAGNETIC

AND

/

% + #~ + 0 3 - 1 a ~ > 0. FERROELECTRIC

MATERIALS

It is a common feature of ferromagnetism and ferroelectricity that they are based on nonlinear constitutive equations. Also, both phenomena are characterized by the occurrence of hysteresis. The essence of hysteresis shows in connection with a cyclic loading. Despite the periodicity of the load, the responses in loading and unloading differ markedly. Moreover, hysteresis is a phenomenon where nonlinearity and memory effects occur at the same time. Hysteretic phenomena abound in nature, the more familiar ones being the stress-strain curve, the magnetic hysteresis of hard ferromagnets, and the electric hysteresis of ferroelectric crystals. Though some properties are common to all hysteretic phenomena, for definiteness, attention is restricted to ferromagnetism and ferroelectricity.

Models of E l e c t r o m a g n e t i c Materials

1441

4.1. F e r r o m a g n e t i c M a t e r i a l s The modelling of hysteresis may proceed through different approaches. The first one is to determine the physical laws of the system by starting from the modelling (differential) equations of the iron-core inductor or relay hysteresis. Once the elementary inductor is modelled in a satisfactory way, a crucial difficulty is inherent in the account and interaction of the physical processes involved. The second one is purely macroscopic. It is based on the introduction of suitably general constitutive functions (or functionals) which are based on experimental data and necessarily embody nonlinearity and memory effects. Then it proceeds by applying to the model the general properties which are required to hold. This approach traces back to Volterra [4]. A third approach postulates certain idealized behaviour of elementary domains, and then the mathematical model is obtained by assuming a distribution function of such domains. Such is the case for the Preisach model [5]. Within the theory of irreversible processes, it is natural to write the evolution equation l~I=- ~ (H- Hr), T

where H r is the reversible part of the magnetic field while H is the present value. Somewhat equivalently, it can also be written in the form of Bloch equation, 1VI = _ 1 ( M - M~). T The constant T is usually called relaxation time. If M is regarded as the independent field and H is the response, then the evolution equation is considered in the form H + ~ - H = -v 1VI, X which is the analogue of Maxwell's model in viscoelasticity. A direct integration allows H to be expressed by a memory functional on the history of M. A generalization is given by the magnetic analogue of the standard model of viscoelasticity, thus, letting M + 7-1M = x ( H + T2H) • All these models, however, cannot describe the fact that hysteretic materials show different responses in loading and unloading. Necessarily, the sign of the time derivative of the loading field must enter the model. Let H, B E R be the input and the output and let a < /~ be threshold values for H. The admissible values for (H, B) are given by the hysteresis region A/[ = { ( H , B ) : (~ < H < fl, hL(H) < B < hu(H)} along with (H, hL(H)) and (H, hu(H)) as H < a and H > ~, respectively. Within the hysteresis region the trajectory (H(t), B(t)), as t varies, is determined by two families of curves, one family for increasing H(.) and one family for decreasing H(.). It was perhaps Duhem [6] who first focused on the fact that the output jumps from one family to the other when the input changes direction. In 1971, Bouc [7] used a relation H --+ B which is a particular case of

B+a

[-I g(H,B)=b[-I.

While a, b are appropriate parameters, a convenient choice for g is g(H, B) = B - be(H) where ¢ is piecewise linear. Some articles by Coleman and Hodgdon [8] and Hodgdon [9] show that such a model is very useful in applications because it is characterized by a few parameters which

1442

M. FABRIZIOAND A. I~IORRO

are determined by fitting experimental data. The paper by Gentili [10] provides a detailed investigation of ferromagnetism as modelled by the more sophisticated evolution equation

2(/I(t) = a ~ (H(t), [-I(t), M(t)) U [~ (H(t), [I(t), M(t))] [-I(t) + g (H(t), [-I(t)) , where a is a positive constant, U is the Heaviside function, /)(t) is the latest value of H(t) at w h i c h / : / h a s changed sign,

.T (H(t), [-I(t), M(t)) = a (H(t), [-I(t), M(t)) [f(H(t)) - M(t))] and

a (H(t),[-I(t),M(t)) =

a (H(t) - [-I(t)). 1. -1,

hL(H(t)) < B < h,j(H(t))), B(t) = h,.(H(t)), B(t) = h.(H(t)).

A model elaborated by Chua and Stromsmoe [11] is based on a differential equation which, in our notation, reads

J~ = h(B)g(H - f(B)). The hysteresis properties are shown to be described and the functions f , g, h are related to experimental data. The Preisach model is the archetype of the third approach and dates from 1935 [5]. The body is regarded as a superposition of simple inductors as

B(H(t)) = / / # ( a , / 3 ) . P ~ ( g ( t ) )

dad/3,

where # is a distribution function and Fa~ is the operator describing the single inductor with thresholds a
04 'q= -0-o'

04 0D"

E-

For simplicity, let the system be one dimensional in character in that only E and D in a common direction are nonzero. Hence, we let E = / 3 D + ~D 3 + ,-,/D5, where /3, ~, 7 are temperature-dependent parameters. As /3 < 0, the D - E curve shows a hysteretic behaviour. In the simplest way, let 3' = 0. Letting ¢ = - % we can write ¢=¢o+

1

4

/3D2+~ D ,

Models of Electromagnetic Materials

1443

where ¢0 is a function of 8. The observation that D E > 0 at small values of D (or E) as 8 is above a critical temperature 80 suggests that we set ~ = ~(8 - 8o). Hence, if 0 > 80 then D approaches zero as E does so. Also, at a fixed temperature, the dependence D ( E ) is roughly linear for small values of D while D cx E 1/3 for large values. If, instead, 0 < 0o, then, as E = 0, we have D=+

i ~ (80 ~- 8) ,

which means that there is a spontaneous polarization which is proportional to v / ~ o - 8. Meanwhile 1 = ~0 - - c~D~. 2 In the two cases, 8 > 0 or 8 < 0, we have

~=~0

O:2

or

= 70 -

(80 - 8)

as D approaches zero. This implies that the specific heat eo-~0suffers a jump (a2/2~)80 between the two cases (transition of the second kind). More involved models are based on additional variables which are governed by appropriate evolution equations. For instance, j and m are introduced so that [14] el(15+uj)=V×m, #1 (ill + u a m ) = - V x P. So m is regarded as an internal magnetic field and j as a current density which is driven by the difference by an equilibrium electric field ]~(P) and the electric field E, i.e., dj

dt

Hence, the polarization P is found to satisfy

//--2 [1:~_~_(El~tl)--1~ X (~7 X l:)) ] _~_/]o:i~ : " f (U - ]~(P)) .

iq.i.o.

s n s As W I T H MICROSTRUCTURE

There are similar but different approaches to the modelling of liquid crystals. First, we look at liquid crystals as materials with a microstructure. A material particle is a collection of aggregate molecules with an orientational order. The rigid body nature of each particle is made apparent by letting three mutually-orthogonal rigid directors I1, I2, I3 be attached to the particle. Owing to rotation of the particle, {Ij}(x) are mapped to {Xk}(x, t). For each x, the function Xh(X, t) gives the dependence of Xh on time for the particle at x; for each t, Xh(X, t) gives the field of directors. A rotation matrix field R(x, t) is then defined to be given by Rhk = Xh " Ik. In terms of R, we define the gyration matrix

.(x,t) =/~TR, in the fixed basis {Ik}, and hence, the intrinsic angular velocity as 1 02 l ~ - - - - ~ ~ l h k l 2 h k •

1444

M. FABRIZIOAND A. MORRO

Let ft be the matrix defined

as

[~jk : Wj,k, whence,

1

~ j k =- ---~ ejpqlZpq,k"

The dynamics of the material particles is described in terms of the mass density p, the velocity v, and the intrinsic angular momentum, per unit mass, 1. The balance of mass and linear m o m e n t u m are written in the standard forms D+pV.v=0, p~? = V . T +

b,

where T is the Cauchy stress tensor and b is the body force. Quite often, the restriction •. v = 0 is adopted; this condition is not essential for later developments. The balance laws of angular m o m e n t u m and energy are established as follows. For any region 1) of the body, we let d--t

p(x×v+l)

xxbdv+

dv=

cdv+

xxTnda+ )2

mda, 12

where c is the body couple density and m is the surface couple density. By arguing as in the Cauchy theorem for stress, we find that there is a couple stress tensor r such that m ( x , n, t) = r ( x , t)n. Hence, in view of the balance of linear momentum, we have fli= C+U.T+~/,

where ~p = epqrTTq. The rotational kinetic energy is Tl = (1/2) pco • l, and the power on l) due to c and m is written as

~ c " uJ dV + fo

w "T n

By the divergence theorem, we find that the power per unit volume is c . w + w . ( V . ~') + r .

~.

The balance of energy is then taken in the form

d~

p e+~

+

.

.

.

.

. +io

v

(v. Tn+w.rn-

q.n)da,

where e is the internal energy, q is the heat flux, 7" is the heat supply per unit mass. The balance equations for linear m o m e n t u m and angular m o m e n t u m and the continuity of the integrand allow us to write p~ = H . B + E . D + J . E + T . L ~ +T.W+~. fl- V.q+r, where L ~ is the symmetric part of the velocity gradient L and W , such t h a t Wqr = (1/2) (LqT L r q ) - £qrpaJp is the relative vorticity. This form of the energy balance is similar to that in [15]. The second law can be expressed by the Clausius-Duhem inequality

p~+v

-~_>o.

Depending on the constitutive equations, the material might be complex enough t h a t the second law needs a generalization such as the occurrence of an entropy extra flux F. Here we disregard this possibility. Hence, in terms of the free energy ~ = e - 0r/, we have ~)<-0r/+H.B+E.I)+J.E+T.LS+T.W+'r.fft

-

~+F

.KTO+V.(0F).

Models of Electromagnetic Materials

1445

T h e director theory, which traces back to Oseen and Franck, describes each particle by a direction, say d. In such a case, d is taken to have a fixed magnitude, for definiteness d . d = 1. Moreover, no dissipation is taken to arise from the gradient of intrinsic angular velocity w. The particle is then regarded as a small rod so that 1 = I w i where w± = o~-- (w • d)d. In a more realistic scheme, the director is allowed to vary in magnitude and the liquid crystal is said to have a variable degree of orientation. Here we outline the approach developed by Leslie [16] and Ericksen [17]. For convenience, we keep d a unit vector, but then we need a quantity to describe the degree of orientation. T h e orientation of the particle is described through a second-order, traceless, symmetric tensor Q. Isotropic configurations correspond to Q = 0, and all eigenvalues Qn vanish. If, instead, two of the three eigenvalues are equal then Q=s

d®d-gl

,

d.d=l,

-5-

-

and the liquid is said to be nematic. The scalar S is the degree of orientation while d is the director. The case where the three eigenvalues are all distinct corresponds to biaxial nematic configurations. In addition to the standard equation for linear momentum, a scalar and a vector equation are considered to govern the evolution of S and d. The scalar equation involves a surface force t and the analogues of a body force, G i, G e of internal and external origin. Analogously, the vector equation involves a stress tensor "r and body forces gi, ge. Next the balance of energy is taken in the form d-t

p

~+

v2

dv=

/o(

v. Tn+cl.~-n+:~t.n-q.n

)

da

where ~ is the energy density which also accounts for the kinetic energy associated with S and cl. Hence, the analysis of the entropy inequality provides the restrictionson the constitutive equations. T h e corresponding developments and conclusions are strictly related to the evolution, or balance, equations and to the form of the entropy inequality. For definiteness, we mention the model investigated by Calderer [18] in connection with the stability of shear flows. T h e scalar and vector equation are taken as 2~2(S)S = - @ ' ( S ) - 2 ~ l ( S ) d . LSd, Vl(S)d x a = 71(S)d x L a d

-

V2(S)d X Lad,

where ~b is the free energy. The stress tensor T is taken in the form T = - p l + a L s + a2 (d ® LSd + LSd ® d) + 2a~(d • LSd)d ® d

+

(a® d + d

L~d ® d -

d ® L a d ) + 213,,5'd® d,

where p denotes the pressure while a, oL1, OL2, ill, ~2, "~1, ~/2 are constitutive functions of S. Further, it is assumed that a approaches a nonzero limit as S --~ 0, while a l to Y2 approach zero as S - ~ 0. 6.

NONLOCAL

MATERIALS

Much research on thermodynamics of continua is based upon the Clausius-Duhem inequality as the expression of the entropy inequality. In such a case, q / 0 is regarded as the entropy flux and

1446

M. FABRIZIOAND A. MORRO

this identification looks appropriate in thermostatics and for the wide class of simple materials. This requirement appears to be much too restrictive in connection with materials modelled by constitutive equations where spatial interaction, or a dependence on suitable gradients, occurs. In electromagnetic theory, there are phenomena such as the broadening of the absorption lines in the Doppler effect, the behaviour of the electric current due to electrons in metals, and the superconductivity which call for nonlocal models. This means that the response, at a point x, is influenced by causes placed at points x ~ in a suitable neighbourhood, possibly covering the entire body, of the point x. For generality, also memory effects are considered. Hence, the response of the body, at time t, at the point x, is a functional of suitable state functions in the space-time domain. With this motivation, we regard the laws of thermodynamics as expressed in a global form. At every point x E B, the state s and the process P are assumed to exist. The domain of the process P is the interval [0, dp); the restriction of P to [0, t) C [0, dR) is denoted by P[0,t). Still o denotes the state-transition function that associates to the pair (s ° (x), P ( x ) ) , of the initial state 8 o and the process P, the state s at time t, at the same place x,

s(x, t) = e (s°(x), PI0,,)(x)). In particular, s(x, dR) = Q (8°(x), P ( x ) ) . For every x 6 ft, the constitutive equations are expressed by the functionals D(x, t) = I~)(s(x, t), P ( x , t)), B ( x , t) = 13(s(x, t), P ( x , t)), J ( x , t) = J($(x, t), P ( x , t)),

h(x, t) =

(6.1)

t), P(x, t)),

q(x, t) = ~l(S(x, t), P ( x , t)). It is worth remarking that the state s and the process P, at the place x and time t, may depend on the value of pertinent fields at the same place x and time t, but also at places x ~ ~ x and at times t ~ prior to the time t. The statements of the laws of thermodynamics are now given for the whole b o d y / 3 by having in mind that the whole body is not affected from outside. DEFINITION. A body B (or a subbody A C B) is called self-consistent if the constitutive relations (6.1) relative to any point x E B (x E A), at any time t, is unaffected by the fields outside B ( A ) .

DEFINITION. A pair (s°(x), P ( x ) ) is called a cycle if ~o(s°(x), P ( x ) ) = 8°(x), Vx E B; a cycle is said to be constant on Oft if g(s°(x), P[o,t)(x)) is constant Vx E OFt and t E [O, dp), Let f denote the integration on t E [0, dp) where (s°(x), P ( x ) ) is a cycle. FIRST LAW OF THERMODYNAMICS. Ill3 is a self-consistent body, then the condition

(6.2) + E(x, t ) . I~(s(x, t), P ( x , t)) + J ( s ( x , t), P ( x , t)). E(x, t)] dv dt = 0 holds for every state s°(x), and process P ( x ) such that (s°(x), P ( x ) ) is a cycle which is constant on Oft.

The global relation (6.2) can be replaced by a seemingly stronger statement by letting a flux ,I~ of (s(x), P ( x ) ) exist such that, for every pair (s(x), P ( x ) ) , which determines a nonconstant cycle

Models of Electromagnetic Materials

1447

on 0~, we have

/ / ~ [h(8(x,t),P(x,t)) + H ( x , t ) . fi(s(x,t),P(x,t)) + E(x, t ) . ~ ( s ( x , t), P(x, t)) + J(s(x, t), P(x, t)). E(x, t)]

: Cf

dv dt

(6.3)

~(s(x,t),P(x,t)). n(x)dadt,

J J0

where n is the unit outward normal to 0~2. It is easy to show that (6.2) is a consequence of (6.3). Conversely we can show that a flux • exists which satisfies (6.3) for meaningful, particular cases. Letting 0 be the (absolute) temperature and g the temperature gradient, g = ~Y0, we state the second law as follows. SECOND LAW OF THERMODYNAMICS. 1[]3 iS a

/f~

self-consistent body, then the inequality

"h(s(x,t),P(x,t))O(x, t) + ~t(s(x,t),P(x,t)).g(x,t)]O2(x, t)

holds for every field s°(x), and process P(x) such that on O~t.

(s°(x), P(x))

dvdt
(6.4)

is a cycle which is constant

For cycles which are not constant on the boundary 0~, the existence of a flux F is allowed such that the second law takes the form

[h(s(x,t),P(x,t)) + ~t(s(x,t),P(x,t)). /J L 02(x,t)

g(x,t)-

dv dt

(6.5)

<- f foa F(s(x,t),P(x,t)).n(x)dadt. Let 0 be constant, in space and time, in which case the processes are said to be isothermal. In view of (6.3), inequality (6.5) takes the form

.f ~ [H. B + E. EI + J . E + V . N] dvdt > O,

(6.6)

where N = 0F - O. Hence, there is no need of separate, independent, nonlocal contributions @ and F to the balance of energy and entropy. Accordingly, without any loss in generality, for isothermal processes, we let • = 0. Let 0 be a function of x and t. By following the lines of [19], we may assume that there exist two state functions, namely the internal energy e and the entropy 7/, per unit mass, such that, tbr every smooth process P, (6.4) and (6.5) become

+ E(x, t). b ( s ( x , t), P(x, t)) + J(s(x, t), e ( x , t)). E(x, t)]

dv (6.7)

- -~-f~ ~(s(x, t), P(x, t)). n(x) da,

a fo 7)(s(x, t)) dx > / £(s(x,t),P(x,t)) 0(x, t) +

d-t

- foa F(s(x, t), P(×, t)). n(x) aa.

02 (x, t)

dv

(6.8)

1448

M.

FABRIZIO AND A.

MORRO

In the particular case of isothermal processes, the free energy %6 -- e - 0r/ and use of (6.7) and (6.8) provide

(6.9) + J ( s ( x , t), P(x, t))- E(x, t) + V . N(s(x, t), P(x, t))] dv. In terms of the free enthalpy ( = ~# - H • B - E - D, it follows from (6.9) that d (6.10) + J ( s ( x , t ) , P ( x , t ) ) . E ( x , t ) + V - ~ q ( s ( x , t ) , P ( x , t ) ) ] dv. DEFINITION. A body B is said to be separable if every subbody A C 13 is self-consistent. REMARK. In general, a body with constitutive equations given by functional (6.1) is not separable. PROPOSITION. If B is separable then equations (6.1), relative to any subbody `4, can be obtained as the restriction to ,4 of equations (6.1) for the whole body 13. As a consequence, for separable bodies, relations (6.7),(6.8) hold for every subbody `4 C 13. Hence, the arbitrariness of `4 implies that the relations

~(x,t) = h(x,t) + H ( x , t ) . B(x,t) + E ( x , t ) . D ( x , t ) + J ( x , t ) . E(x,t) - V . ~ ( x , t ) , h(x,t) q(x,t) f/(x, t) > 0(x, t------~+ t02(x, - - - - - ~ " g(x, t) - V . f ( x , t), hold at every x E f~. Henceforth, B ( x , t ) , D(x,t), and J(x, t) are meant as the value of the response functions (6.1). Comparison of ~ a n d / / y i e l d s

0(x, t)0(x, t) _> ~(x, t) - H(x, t). B(x, t) - E(x, t). D(x, t) - J(x, t). E(x, t) 1 +V. ~(x,t) + ~ q ( x , t ) , g(x,t) - 0 ( x , t ) V . F ( x , t ) .

(6.11)

The occurrence at the same time of two fluxes, • and F, is due to the assumption that both the first and the second law must be formulated in a nonlocal manner. Indeed, while there are different formulations of the second law, the local formulation of the first law is well established for separable bodies. In this sense, we let • = 0. In terms of the free energy g) = e - Or/, use of (6.11), with ¢ = 0 and N = OF, gives ~(x, t) _< -r/(x, t)0(x, t) + H(x, t). B(x, t) + E(x, t). I)(x, t) + J(x, t). E(x, t) 1 - - q(x, t)- g(x, t) - O(x, t ) V . F(x, t).

(6.12)

0(x,t)

Hence, for isothermal processes, it follows that ~(x, t) _< - H ( x , t ) • B ( x , t ) - l~(x,t) - D(x, t) + J ( x , t ) • E(x, t) + V . N(x,t).

(6.13)

The separability property may be regarded as a weak form of locality. It allows the local form of the balance laws provided the effect of the flux N is considered. The nonloeal character of the theory is then confined to the occurrence of the flux N. If, though, N = 0, then (6.13) reduces to the standard local form of the Clausius-Duhem inequality for isothermal processes ~(x, t) _< - H ( x , t ) . B(x, t) - E ( x , t ) . D(x, t) + J ( x , t ) . E(x, t).

(6.14)

Models of Electromagnetic Materials

1449

REMARK. Inequalities (6.12) and (6.13) hold when the body is separable, namely when, for any pair ($(x, t), P(x, t)) which is compatible with the whole body/~, the restriction of (8(x, t), P(x, t)) to the subbody A C B is compatible too. If, instead, the body is nonseparable, then only the global inequality holds, i.e., d

~(s(x, t))

dv <_/~ [ - H ( x ,

t). I3(s(x, t), P(x, t)) - E(x, t). D(s(x, t), P(x, t)) + J ( s ( x , t), P(x, t)). E(x, t) + V . i~l(s(x, t), P(x, t))]

dv.

In such a case, the local form is possible through a localization residual r, i.e., <_ - H - B - I ~ . D + J . E + V . N + r , where

/ rdv=O. This in turn shows that two additive terms may occur in the second law. The occurrence of N is strictly analogous to that of ~I' (in [20]) or of 0p (in [21]) in the Clausius-Duheln inequality and need not be nonlocal in character. The occurrence of r, instead, is fully related to nonlocality. 7. D I E L E C T R I C

BODIES

WITH

QUADRUPLES

In this section, we examine constitutive relations such that nonlocality occurs through second gradients of the electric field. For definiteness, by analogy with [22,23], we model dielectrics with quadruples in the form D ( x , t ) = e0(x)E(x,t) - V ( e l ( x ) V . E(x,t)) - V . (e2(x)VE(x,t)), B(x,t) = #(x)H(x,t),

(7.1)

J(x, t) = 0, where V E = V E - (1/3)(V • E)I. Moreover, co, el, e2, and # are scalar-valued functions of x and, in indicial notation, [~- (e2VE)]j = (~2Ej,k),k. The state s is given by s = (E, VE, H), while the process P is given by

all quantities being considered at the same place and time. The Second Law of Thermodynamics is considered in form (6.14) so that the enthalpy ~ is also involved. We first observe that the constitutive equation (7.1) for D is not compatible with the local version (6.14) of the dissipation principle. The proof parallels that developed by Gurtin [24] within the thermomechanical framework. Let N - 0, and hence, consider inequality (6.14). In view of (7.1), we have ~(x, t) <_ - H ( x , t). # ( x ) H ( x , t) - E ( x , t).

t) - V(

I(X)V. E ) ( × , t) - V . (

2(x)VE(x, t ) ) ] .

The evaluation of the time derivative of ~(E, ~ E , H) and some rearrangement yield

VE<" E + VVE(" VI~ + VH<" H _< -#H. H - eoE. I~+ (E. Vel) V.

E

+ ~IE. V ( V . E) + l~. ( r e 2 . V)E + e21~- AE.

1450

M. FABRIZIO

AND

A. MORRO

Owing to Maxwell's equations and the constitutive equations, at any point x and may regard V E and V H as constrained values while V(V - E) and A E are arbitrary. find that q and e2 must vanish. Accordingly, though the constitutive equations (7.1) separability property, they are not compatible with form (6.14) of the second law. If, is allowed to be nonzero, then we can write the second law in the form

VE<" E ~-V ~ .

_[_( l ~ . ~ l ) ~ . E _

time t, we Hence, we ensure the instead, N

v E -]-VV.E
~ . . ~ I ~ . E _ ~E.e2~/E_.}_~. [~£I]~V.E_~_,!.2~:,]~E+N],

(7.2)

where the identity

is applied. Of course, N depends on x and t through the pair (s(x, t), P ( x , t ) ) .

Now, at any

point x and time t, we may regard E, v E , V. E, H as arbitrary. This implies that inequality (7.2) holds if and only if VH¢

VE4

---- - # H ,

=

-eoE,

VV.E~

=

--fflV• E,

and N = - [ q E v . E + c2VEI~] . Hence, to within an inessential additive constant, we find that ( = -2

1

[ # H - H + e 0 E - E + c l ( V . E) ~ + e2VE. V E ] ,

where #, co, el, (2 are arbitrary functions of the position x. A more compact constitutive equation (for D) involves the derivatives of E through V × E, namely D ( x , t ) = c0(x)E(x,t) + V × (~(x)V × E ( x , t ) ) . Owing to the identity

E.

V ×

× E ) : v . [ 4 v × E) × E] +

× E. v ×

we can write the entropy inequality in the form _< - # H . H - ~ 0 E . E - ~ V x E . V x E + V .

[ N - c(V x E) × E l .

Hence, we find that, again, the inequality holds as an equality if ((E,V ×E,H)=-2

1 [¢0E. E + # n . H + ¢(V × E ) . (V × E)],

N (V X E,E) = e(V × E) × E.

(7.3)

(7.4)

REMARK. If V × E = - # H is viewed as a constraint, then we have ~U'xE~

~-

__~7 X E

and w(E, V X E, H) • V X E _< 0,

w := VH¢

+

pH.

Now, tile inequality shows that w . V x E has a maximum at V x E = 0. Hence, we have w(E, O, H)

=

O,

whence, it follows that VH (E, O, H) = - # H , while no definite property follows for VH~(E, V × , H). Of course, (7.3) and (7.4) are sufficient for the validity of the second law.

Models of Electromagnetic Materials

8. S P A T I A L L Y - D I S P E R S I V E

1451

CRYSTALS

The main problem of crystal optics is the investigation of monochromatic waves in terms of the fi'equency w and of the wave vector k. Within linear electrodynamics, a constitutive equation of the form D = eE seems quite natural. However, the Fourier components D(cz, k), E(w, k), such that

E(w,k) = f~ f E(x,t)exp[-i(k. and the like for D, are related by D(w, k) =

e(w,k ) E ( ~ , k).

T h e dependence of e on w and k means that the crystal experiences m e m o r y effects in time and space. Within wave propagation, the dependence of e on w and k models temporal and spatial dispersion. Often k) is taken as a polynomial in k with coefficients parameterized by w. Developments and conclusions can be obtained by arguing within the w - k domain. However, often the polynomial in k is taken as a motivation for a model in the space variable x where k and its powers are associated with the gradient and analogous operators (cf. [25, Section 2.1.3]). It looks more appropriate, or at least equally well motivated, to start from a model in the spatial domain, and hence, to proceed by pertinent mathematical methods such as Fourier analysis. Accordingly, we start from constitutive equations in the space-time domain and let D be determined by the value of E and of the (spatial) derivatives ~ 7 E , ~ T V E , . . . . The analogous assumption is made for B in terms of H. Next, observe that the skew part of ~TE and V H may be written through V X E and V × H. Hence, we let

e(w,

D = eE + o~V x E + f~VE + ' 7 ~ V E + . . . , B = t t H + AV x H + v V H +

~VH

+...

,

where e, tt, c~, A are second-order tensors, f~ and v are third-order tensors (symmetric relative to the second and third indices), 7 and ~ are fourth-order tensors. Also,/3 and v are symmetric with respect to the second and third indices, i.e., ~kpq = ~kqp" For definiteness, we restrict attention to derivatives up to second order also because no conceptual difficulty is associated with higher-order derivatives. Owing to Maxwell's equations, V × E and V × H cannot be regarded as independent on the other values, at the same place x and time t. It is then convenient to soon replace ~ × E with - B and ~7 × H with D. Hence, we write the constitutive equations as D + a B = eE + f~VE + ")'VVE, B - XI9 = t t H + v,V H + ~ V ~ H .

W i t h these constitutive equations, the state s and the process P are given by s = (E, H , D , B , ~ E , V H ) ,

P = (l~, H , r E , v H , W E ,

UVH).

We now apply the constitutive equations, and hence, derive thermodynamic restrictions, in the case of monochromatic waves. Let E(x, t) = E l ( x ) sinwt + E2(x) coswt and similarly for H , D, B. Hence, B (x, t) = wB 1 (X) COSw t

--

wB2 (x) sin

wt

1452

M. FABRIZ]O AND A, MORRO

and similarly for D. Of course w e may regard the tensors e, tt . . . . . "~, ~ as parameterized by the frequency w. Substitution and the identical validity with respect to time yield D2 + waB~ = eE2 + fiVE2 + 7 V V E 2 , D1 - ~ a B 2 = e E l + fl~TE1 + 7~TVE~, B2 - wAD~ = ttH2 + r , ~ H 2 + ~7~7H2, B~ + wAD2 = / , H ~ + u V H 1 + / ~ V V H 1 . Now, D2 + w a B x minus w a times B1 +w)kD2 and B1 +w.XD2 minus wX times D2 +wc~B1 give 02

=

A_ IcE2 + ~ E 2

+ T Y P E 2 - w ~ ( / t O l + v~TH1 + ~ 7 H 1 ) ] ,

B1 = A [/~H~ + v V H ~ + ~ V U H ~ - wA(eE2 + / ~ V E 2 + "7~VE2)], where A_ --- ( 1 - tzJ2C~) - 1 , Similarly, by means of D1 - ~ B 2

( 1 - .:2Ac~) -1

A_ :

and B= - wAD1 we have

D1 = A+[eE1 + fiVE1 + ~/~TVE1 + ~vc~(/zH2 + v~TH2 + ~¢V~TH2)], B2 = A+[/zH2 + v~TH2 + ~ V V H 2 + ~vA(eE1 + fiVE1 + 7~7VE1)], where A+ = (1 + w2otA)-1 and A+ = (1 + w2Aa) -1. We now consider the entropy inequality

dt
H-B+I~.D-V.N

For definiteness, we let d = 2~r/w. Integration over [0, 2~r/co] provides d

0_> fo

[H.B+I~-D-~7.N]

dt:H1.B2-H2.B1

where

+E1.D2-E2.DI-~7.1

~I,

= 2 [2~1~ --

N

Q'J ta o

dt.

Upon replacing B1, B2, D1, D2 and making some rearrangement, we obtain H I ' [A+/~ - (A_/~) T] H2 + H 1 . A + v V H 2 - H 2 - A _ v V H 1 +E,.

[A e - ( A + e ) - c ] E 2 + E 1 . A _ / ~ v E 2 - E 2 ' A + f ~ V E 1

+ ~7H2- [A ~; - (A+~;) T] VH1 + ~7E2. [A+7 - (A_"/) T] ~TE1 + w [ H 1 • A+XeE1 + H2 • A_AeE2 + H1 • A+A/Y~7E1 + H2 • A_AflVE2 - VH1 • A+A'y~TE1 - ~ H 2 • A A T V E : + VE1 • A _ a ~ T H 1

+ ~7E2 • A + a ~ V H 2

-E1 • A _ a t t H 1 - E2 • A+c~tIH2 - E1 • A _ a v V H 1 - E2 • A+av~YH2] + V • [HIA+~ETH2 - H 2 A ~;ETH1 + E I A _ T V E 2 - E2A+7~TE1] - ~ • 1Q +w~7 • [HIA+A'),~TE1 + H2A+A"/VE2 - E 1 A a ~ 7 H 1

- E2A+czgXTH2] < 0.

The validity as w --~ 0 and the arbitrariness of H1, H2, E l , E2 and of their derivatives imply that A + / t = ( A _ / * ) ~,

A_g=(A~)

~,

A e = ( A + e ) T,

A+7=(A-~f) ~

Moreover,

1Q = H 1 A + ~ H ~

- H 2 A _ ~ V H 1 + E1A_"/E7E2 - E2A+'yVE1 + w[H1A+A~/VE1 + H 2 A . X T V E 2 - E I A _ a ~ ; V H 1 - E~A+a~;VH2].

Models of Electromagnetic Materials

1453

9. N O N L O C A L D I E L E C T R I C S W I T H M E M O R Y In this section, we consider materials such that the state is the history of the field F = (E, H) in a proper subregion occupied by the material. Let ~tx be the set of points r such that x + r E f~; the origin belongs to ft× for every x G f~. The symbol O will denote a spherical neighbourhood of the origin. Also, O~ = {r C gt×; Irl < e}. It is convenient to regard the dependence on F as split in a dependence on the present value, at the point of interest, F(x, t), a nonlocal dependence on the present value Fx(-, t) in the domain ftx, with F×(r,t) = F(x + r , t ) , a local dependence on the history Ft(x, -) in R +, with Ft(x,~) = F ( x , t - 4), and an effective nonlocal dependence on the history r'x(-, .) of the field F in the domain f~x x N +, with F t ( r , ~ ) = F(x + r , t - ~). Hence, the state s is the set

s(x,t) = r~ := (r(x,t), rx(.,t),r~(x, .),rx(-,-)), while the process P ( x , .) is given by

P(x,t)=(F(x,t),Fx(.,t)),

t e [0, d . ) .

Hence, if 9c denotes the response functional, we can write 7 ( x , t) = J" (r'x, P ( x , t ) ) . Relative to isothermal processes, consider the set of constitutive equations

~(x, t) = ~ (r'x), C(x, t) = d (r~x),

(9.1)

J(x, t) = J if'x),

(9.3)

(9.2)

where G stands for the pair (D, B). As a simple example of nonlocal material with memory, we may consider the linear dielectric characterized by the constitutive equations

D(x,t)=eoE(x,t)+

/o[ x

B(x, t) = ~(x)H(x, t),

T(r)E(x + r,t) +

/0

v ( r , ~ ) E ( x + r , t - ~) d~

] dv~,

J ( x , t ) = 0, where e0, tt and the values of % v are second-order tensors; the subscript r in dvr is a reminder that we are integrating over r E ftx. These equations are the generalization to materials with memory of the dielectric with attenuating neighbourhood [22]. Denote by D(~) the domain of definition of the functional (9.1) and by D(G, J) the domain of functionals (9.2),(9.3). The two domains may be unequal, in which case we have :D(~) C D ( G , J). REMARK. It is worth emphasizing that, as is generally the case for materials with memory, the free enthalpy ~ proves to be nonunique. Hence, the functional ~ occurring in (9.1) is meant as one of the admissible free enthalpies of the material. Suppose that the space D(~) is normed and that the norm I]" II is given by [26]

Ilr'xll = IF(x, t)l + Ilrx(., t)llx + Ilr'(x, )11' + Ilr~x( , )ll'x,

(9.4)

where l" I denotes the absolute value in R 6, while II' IIx, I1" II~, and I1" II~x are the norms relative to the space of functions defined on f~x, R +, and f~,, x R +, with values in II~a and there absolutely continuous with respect to Lebesgue's measure. The norm of D(~) is required to satisfy the following properties.

1454

M. FABRIZIO AND A. MORRO

CONSTITUTIVE ASSUMPTIONS. (i) The functionals ~, G, and 3 are continuous on 23(¢) which is taken to be a Hilbert space; (ii) the free enthalpy is continuously differentiable on D( ();

(iii) if

{(._s) E~(r's) =

(e 0,

L e~ (~1 + w~) - ~ ] , (r, s) e O×,e x [0, el, s)w2, (r, s) e (~x \ Ox,~) × [0, 4, (r, ~) e ( ~ × [~, oo),

(9.~)

where ~ G [0,a), wl, and ~o2 are two arbitrary vectors of •6 and (gx,e = tOE A ~x, then lim~--+0 U~[[ -- 0. Continuous differentiabili W of ~ implies that, for any regular process,

where ($r~, ~ r ~ , and ~ r ~ are the Pr~chet differentials of ~ in the direction of P×, F t, and P~. We assume that, for every point x and time t, it is always possible to choose arbitrarily F*x in ~D(() such that there is at least one process that generates the history F~ and satisfies Maxwell's equations and the constitutive equations. The dissipation principle, in form (6.13), for materials obeying the constitutive equations (9.1)(9.3) provides the inequality

- J (rtx) • E(x, t) - Vrlql (r~, P(x, t)). v r - V/~IQ (r~, P(x, t)). VF

- ~ x N (rL V(x, t); Vrx) - ~;,~¢ (rL P(x, t); v r ~)

(9.6)

-ar~N (rL p(x, t); vr~) - ~ r N (r'x, P(x, t); Vrx) _<0. Since the Fr~chet differentials are linear and hounded, by the Riesz-Pr~chet representation theorem, we have , c0Fx Under smoothness assumptions for ~ and lql,we can assume that the partial derivatives, like ~qFx oq are bounded with respect to r = x ~ - x. Given any history field Fix, w e consider history fields

7

a~x,. : r~x + ~x,., parameterized by e, where E~ is defined by (9.5). In view of the constitutive Assumption (iii), the fields A~,e are elements of D((), and hence, also Atx,~ has to satisfy inequality (9.6). Observe that

•

-=~,~(r, s) =

{

[r[ 2 -Ty ( w l + w 2 ) - w l ,

(r,s) eOx,~ x (0,s),

w2, 0,

(r, s) c (~x \ (-gx,e) x (0, e), (r, s) COx x (e, oo),

2(~ - s)

e----y--r ® (wl + ¢o2),

Vr=-x,e(r, s) = {

0,

(r, s) ¢ Ox,e x (0, ~), 2

v.~x,~(r, s) =

-Tr®gol+~O~), [ o,

(r, s) e Ox.e x (0, e),

(r,s)~Ox,~×(0, e), (r, s) ¢ o,,,~ × (o, ~),

Models of Electromagnetic Materials

1455

where r ® co denotes the dyadic product between r and co. By requiring that (9.6) hold for the t E, we obtain history fields A×, At

At

t . + ~x,~)+ aA:,~¢(AL;rt + ~,x,~) +a,.,e (Ax,~,rx _

t t J ( A x,~)" E ( x , t ) - VA×.~lXl ( a t e P ) . V F - V3,~.N- (Ax,~,P) • VF t - 6A.,,lq (Ax,~, P; VPx + V~x , ~)

-aa~, lq ( Axt , , e ; v r t

_

t aA~.oiQ(Ax,~, P,. v r t +

V~x

(9.7)

,e )

+ V*x~) , - a x ~ N - (Atx,~,P;VFx + V * x ~ ) <_ 0.

The constitutive assumptions and the boundedness of the F%chet differentials allow us to say that, for every e --+ 0, (9.7) requires that the inequality

- a ( r t ) . E(x, t) - V , N (rL P ) . v r - V~N (rt, e ) . W _ arxN (rL P; Vrx) - aF,N (rL P; v r ~)

_

aF,J~ ,px, ( ~ P,. v r ^:~ - a' ~ . , x N f x(, t P ; V e x / <_ 0

hold for all OOl, ~o2 E R 6. This implies the vanishing of the coefficient of F(x,t) - c o 1 and the vanishing of at.C, which is linear with respect to (Fx + w2). As a consequence, we have

(2 (rt)= -Vr¢(r:),

arx¢ (r:;e~(t)) =0.

(9.8)

Restriction (9.8)2 means that the functional ~ cannot depend on Px. The same conclusion holds for the functional G as a consequence of (9.8)1. Yet we are unable to prove that the analogous restriction holds for the functional J. Accordingly, we can say that functionals (9.1) and (9.2) are independent of r x but may depend on F, r t, and F t. REMARK. Within smoothness assumptions for the constitutive functionals, the Second Law of Thermodynamics, as expressed by condition (6.13), is not compatible with a nonlocal dependence on the present values of P. 10.

STRICTLY-NONLOCAL

The state at any point domain fL It may then as split in a dependence (P(x, t), F×(., t)) and the

MATERIALS

x is the field of F = (E, H), at the present time t, relative to the whole be convenient to regard the dependence on the field of F at x and t on F(x, t) and on the relative field F×(.,t). Hence, the state 8(x,t) = process P(x) map any t E [0, dR) into

P(x,t)=(~(x,t),rx(.,tl). Any constitutive relation is then written as Y:(x, t)

fi" (P(x, t), Fx(', t), F(x, t), Fx(', t ) ) .

(10.1)

Some examples of such relations are given in [22]. In particular, a simple nonlocal model for dielectrics is described by system (9.4) in the limit case v = 0, namely, when memory effects are absent. The constitutive relations become r ( [ r l ) E ( x + r, t) dv,.,

D(x, t) = e0(x)E(x, t) + [_ x

B(x, t) = lz(x)H(x, t), J(x, t) = 0,

(10.2)

1456

M. FABRIZIO AND A. MORRO

where e0, it, and the values of r are second-order symmetric tensors. Model (10.2) is investigated in [22,23]. 4 By analogy with a procedure for elastic [24] and viscoelastic materials [27], we can show that the compatibility of (10.2) with the Second Law of Thermodynamics, as expressed by the local form (6.14) of the Clausius-Duhem inequality, implies that v(.) = 0. Quite naturally, though, a nonlocal model of dielectric has to be framed in a nonlocal thermodynamic scheme. Since the material is nonseparable, we have to apply the second law 5 in form (6.10). Indeed, substitution of (10.2) in (6.10) yields d

/o

-/~

<-/o

(10.3)

/~ ~'(]r])E(x+r,t).

E(x,t) dvrdv+ ~ V.N(s(x,t),P(x,t))dv.

Hence, it follows that (6.10) holds as an equality with N = 0, while the global free enthalpy is given by ~(s(x,t))dv=

- -~

[e0(x)E(x,t).E(x,t)+ tt(x)H(x,t)-H(x,t)] dv (10.4)

2

r(Jrl)E(x + r, t). E(x,

t) dvr dv.

REFERENCES 1. M. Fabrizio and A. Morro, Dissipativity and irreversibility of electromagnetic systems, Math. Models Methods Appl. Sci. 10, 217-246, (2000). 2. J.A. Sherwin and A. Lakhtakia, Nominal model for structure-property relations of chiral dielectric sculptured thin films, Mathl. Comput. Modelling, (this issue). 3. M. McCall, Axial electromagnetic wave propagation in inhomogeneous dielectrics, Mathl. Comput. Modelling, (this issue). 4. V. Volterra, Theory of Functionals and Integral and lntegro-Differential Equations, Dover, New York, (1959). 5. F. Preisach, 0 b e r die magnetische Nachwirkung, Zeit. Phys. 94, 277-302, (1935). 6. P. Duhem, Die dauernden Aenderungen und die Thermodynamik, I, Z. Phys. Chem. 22, 543-589, (1897). 7. R. Bouc, ModUle mathematique d'hyst~r~sis, Aeustica 24, 16-25, (1971). 8. B.D. Coleman and M.L. Hodgdon, A constitutive relation for rate-independent hysteresis in ferromagnetically soft materials, Int. J. Engng Sei. 24, 897-919, (1986). 9. M.L. Hodgdon, Applications of a theory of ferromagnetic hysteresis, I E E E Trans. Magn. M A G 2 4 , 218-221, (1988). 10. G. Gentili, A history-differential model for ferromagnetic hysteresis, Mathl. Comput. Modelling, (this issue). 11. L.O. Chua and K.A. Stromsmoe, Lumped-circuit models for nonlinear inductors exhibiting hysteresis loops, I E E E Trans. Circuit Theory C T 1 7 , 564-574, (1970). 12. A. Visintin, Differential Models of Hysteresis, Springer, Berlin, (1994). 13. B.D. Coleman and E.H. Dill, Thermodynamic restrictions on the constitutive equations of electromagnetic theory, Z A M P 22, 691-702, (1971). 14. J.M. Greenberg, R.C. MacCamy and C.V. Coffman, On the long-time behavior of ferroelectric systems, Physiea D 134, 362-383, (1999). 15. A.C. Eringen, A unified continuum theory of electrodynamics of liquid crystals, Int. Y. Engng. Sci. 35, 1137 1157, (1997). 16. F.M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal. 28, 265-283, (1968). 17. J.L. Ericksen, Liquid crystals with variable degree of orientation, Arch. Rational Mech. Anal. 113, 97-120, (1991). 18. M.C. Calderer, Stability of shear flows of polymeric liquid crystals, J. Non-Newtonian Fluid Mech. 43, 351-368, (1992). 19. B.D. Coleman and D.R. Owen, A mathematical foundation for thermodynamics, Arch. Rational Mech. Anal. 54, 1-104, (1974). 4Really, it may happen t h a t nonlocality effects occur in a suitable neighbourhood of the pertinent point x through functions vanishing outside a region ]Ex C ~ around x. In such a case, the space integrals are over a fixed region around the origin, for the variable r -- x' - x. 5The material described by the constitutive relations (10.2) does not satisfy the definition of separable material, and hence, the formulation (6.13) of the dissipation principle does not apply.

Models of Electromagnetic Materials

1457

20. I. Miiller, On the entropy inequality, Arch. Rational Mech. Anal. 26, 118, (1967). 21. A.E. Green and P.M. Naghdi, On thermodynamics and the nature of the second law, Proc. R. Soc. Lond. A 357, 253, (1977). 22. G.A. Maugin, Non local theory or gradient-type theories: A matter of convenience?, Arch. Mech. 31, 15, (1979). 23. C.B. Kafadar, The theory of multipoles in classical electromagnetism, Int. J. Engng. Sci. 9, 831, (1971). 24. M. Gurtin, Thermodynamics and the possibility of spatial interaction in elastic materials, Arch. Rational Mech. Anal. 19, 339-352, (1965). 25. V.M. Agranovich and V.L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons, Springer, Berlin, (1984). 26. M. Fabrizio, Una teoria fenomenologica per la termodinamica del campo elettromagnetico, Boll. U.M.L 4, 708, (1971). 27. A. Morro, Thermodynamics and spatial interaction in viscoelastic materials, Boll. UMI 10, 355, (1974).