Thermodynamics of deformable magnetic materials with memory

ht. 3. Engng Sci.. Vol. 12, pp. 45-60. Pergamon Press 1974. Printed in Great Britain

THERMODYNAMICS OF DEFORMABLE MAGNETIC MATERIALS WITH MEMORY

MATTHEW F. MCCARTHY National University of Ireland, University College, Galway, Ireland

(Communicated by H. F. TIERSTEN)

Abstract-The theory of deformable heat conducting magnetic materials with fading memory is developed. The case of saturated ferrogmagnetic type materials is treated in detail. The restrictions placed on the response functionals by the Clausius-Duhem inequality are derived. Relaxation is discussed and integrated dissipation inequalities are derived. Finally, the theory of deformable paramagnetic materials with memory is briefly developed.

1. INTRODUCTION THE STUDY of interaction of electromagnetic fields with deformable continua has at- tracted the attention of many authors during the past decade. We are concerned here with such interactions when the electromagnetic field is such that magnetic effects dominate. Among the earlier contributions in this area may be cited the works of Tiersten[ 1,2] and Brownl31. More recently the subject has been treated by Akhiezer, ~r’yakhtar and Peletminskii 141, Kaliski 151, Maugin and Eringen [6] and Tiersten and Tsai [7].

Apart from the brief treatments given in [ 11, [5] and [6], very little attention has been paid by these authors to the study of dissipative effects in deformable magnetic media. Our object here is to extend the foregoing results to the case when the material under study has a fading memory.

In section 2 the equations which govern the motion of a magnetically saturated deformable heat conducting material which does not conduct electricity are presented. The electromagnetic field is assumed to be quasi static. We lay down our constitutive equations and smoothness assumptions in section 3. We assume that the specific free energy, the stress tensor, the surface exchange tensor, the local magnetic induction field, the specific entropy and the heat conduction vector are functionals of the histories of the deformation gradient, magnetization per unit mass, the temperature and functions of the present value of the temperature gradient.

We assume that the response functionals of the material obey the principle of fading memory@]. Each admissible local thermodynamic process is assumed to obey the Clausius-Duhem inequality. The consequences of the Clausius-Duhem inequality are explored in section 4. Our main new result is the establishment of the fact that the specific free energy is independent of the temperature gradient and that the stress, surface exchange tensor, local magnetic induction field and specific entropy are deter- mined by derivatives of the specific free energy.

Sections 5 and 6 are devoted to a study of relaxation and a derivation of integrated dissipation inequalities. In section 7 the consequences of the principle of objectivity are discussed.

Finally, in section 8, the theory of deformable heat conducting paramagnetic materials with memory is briefly outlined.

4.5

46 MATTHEW F. MCCARTHY

2. BASIC EQUATIONS AND FORMULAE

The motion of a material body C?4 may be described by specifying x, the spatial position of a material point X of the body at time t, in terms of t and the position X in some reference configuration 5% :

x=x(X, t).

The gradient of x(X, t) with respect to X i.e.

(2.1)

F = F(X, t) = Vx(X, t) (2.2)

is called the deformation gradient at X relative to the configuration $?I. We assume that the function x(X, t) is always smoothly invertible in X so that the inverse F-’ of F exists i.e. det IFI+O.

Explicit dependence on X is not exhibited throughout the paper: thus we simply write G = G(t) in place of G(X, t).

The motion of a deformable magnetically saturated ferromagnetic body is governed by the equations [ 1,7]:

(2.3)

(2.4)

$ I

x x t(n)d9’+ (pxxf+I.BxM)dY IT _ pxxvdY=

I .Y I I’

-$ I 5

T-‘Mdv = I

Mxt’“‘(n)dY+ I

Mx(B+LB)d’Y, 9 -t.

d dtv I

p(fv*+E)dY= (t(n).v+pP’(n).fi -q.n)dY

(2.5)

(2.6)

+ (pB./i++f.v+pr)d‘V, I (2.7) C’

where Yf is a material volume bounded by a surface 9 with unit normal n. These equations are the mathematical statements of the principles of balance of mass, momentum, moment of momentum, magnetic spin angular momentum and energy, respectively. The symbols occurring in equations (2.3)-(2.7) have the following physical significance:

p = Mass density in current configuration, v = Velocity,

t(n) = Surface traction, f = Total body force per unit mass,

Thermodynamics of deformable magnetic materials with memory 47

LB = Local magnetic induction field, M = Magnetization per unit volume,

t(%r) = Surface exchange field, B = Maxwellian magnetic induction field, E = Specific internal energy, q = Heat flux vector, r = Heat supply per unit mass,

P = M/P,

I’( -c 0) = Gyromagnetic ratio.

It is clear from the balance laws (2.5), (2.6) that, without loss of generality, we may assume that

LB . p = 0, fi. t”‘(n) = 0. (2.8)

For non electrically conducting media in which magnetic effects dominate, the Maxwell equations, in the quasi static approximation, are

f (E+vxB,c).dx=-b$j- B.dY

% 9

# H.dx=O,

I B.dY=O, (2.9)

Y %

where 9 is an open material surface bounded by a curve %. Here E is the electric field and H =B-4rM is the Maxwellian magnetic field. In equations (2.9), and throughout this paper, we use Gaussian electromagnetic units.

Since the material is assumed to be magnetically saturated 1~1 is a constant:

P. or_ = & = constant (2.10)

and, consequently

p./i=O, fi.F&+l&=O, (2.11)

where

We put

t(n) = Tn, t’“‘(n) = F, (2.12)

where T and A are the stress and surface exchange tensors, respectively. We note that, in view of equation (2.8),, ATp =O. We assume that

f=?+(p.gradB)= (2.13)


where f is the extrinsic body force per unit mass. Of course, the magnetic body force per unit mass cc. (grad B)= may also be written in the form

p . (grad B)T = b div TM, (2.14)

where

T”=&{H@B-i(B’-SrM.B)I} (2.15)

is the Maxwell stress tensor. When a standard argument is applied to equations (2.3)-(2.7), and (2.9), these equa-

tions yield the following local forms:

pJ = po, J = det IFI,

pv=divT+pf,

(2.16)

(2.17)

T*=$piB r\p,aAb=a@b-b@a, (2.18)

(2.19)

pi = tr (TLT) - pJ3 . /.i + tr (AF’TET,,,) - div q + pr, (2.20)

curlE=-fiB,curlH=O,divB=O, (2.21)

where L = fiF_’ and T* denotes the skew symmetric part of T. A superposed dot denotes material time differentiation: f = @(X, t)/at. The operators grad, div and curl refer to spatial derivatives.

In deriving equation (2.19) the condition

A(grad ~r)~ = (grad p)AT (2.22)

has been imposed in order to satisfy the saturation condition (2.10). For future reference, we record the forms of the equations of balance which apply

at a propagating surface of discontinuity 2:

[PUI = 0, (2.23)

[p&l = [T + T”]n, (2.24)

[pUPpI = [P x Aln, (2.25)

ipv2+pe+&B2-M.B U = v.T+p.A-q-&e+&B2v >I [ 1 .n, (2.26)

c[E]xn+V[B]=O; [H]xn=O, [B].n=O. (2.27)


Here 4 is the magnetic scalar potential: H = -grad (6, V is the speed of displacement of the surface I: and U = V-v . n is its local speed of propagation.

Finally, we introduce the specific free energy I++ through the definition

dJ = E - 077, (2.28)

where n is the specific entropy. It follows from equation (2.20) that

4 = tv(T*8’)“rtr(A*~~,)-=B.(I;L)--7fe-_eri -jdivq+r, (2.29)

where

T” = f T(F-‘)T, A* = f A(F-‘)T. (2.30)

3. CONSTITUTIVE EQLJATIONS AND SMOOTHNESS ASSUMPTlONS

A local process at point X in P73 is a collection,

(3.1)

of functions defined for all t E(- m. ~1, where g =grad 8. Let f(t) be a function on (- cc, =). Then the function f’ defined by

f” =f’(s> =f(t - s), s E [O, x), (3.2)

is called the history of f up to time t. A material is characterized by constitutive equations which limit the class of local

processes possible at any point of a body of the material. We confine our remarks to simple homogeneous materials which are described by the constitutive equations

q(r) = S(F’, I%), P’, 0’; g(t)),

T”(t) =*(I? F:,,, P: 8’; g(t)),

A*(t) =&F’, Ff,,, EL’, 8’; g(t)),

LB(t) = d(F’, Fir,, pt, 6’; g(t)),

7?(f) = ri@*, FL pf, 8 ‘; g(t)),

q(t) = il(F’> IL, pi, 0’; g(t)). (3.3)

A local process is said to be admissible at the point X if it is compatible with the balance laws (2.16)-(2.21) and the constitutive equations (3.3).

We now assume that in each admissible local process in $B the Clausius-Duhem inequality

1 pi -I+-div(Gl0)aO &J P

(3.4)

SO MATTHEW F. MCCARTHY

holds. Of course, the inequality (3.4) will place restrictions on the response functionals (3.3) which characterize the material. Our primary object in this paper is to determine these restrictions when the response functionals obey the principle of fading memory [8].

Let A denote the ordered quadruples (K, L, a, A), where K and L are second order tensors, a a vector and A is a scalar. The space of all A is a 22-dimensional vector space Vcz2, with norm llhll defined by

ljAll*=tv(KK3+tv(LLTI+aZ+h2, (3.5)

The set of vectors

I-’ = (F, Fw, Y, 0) (3.6)

with F an invertible tensor and 8 a positive scalar form a cone % in -r/;,,,. Clearly, if G denotes any of the response functionals (3.3), we may write

G = G(I-‘; g). (3.7)

We assume that the response functionals (3.3) obey the principle of fading memory. Let h(s) be a positive, monotone decreasing, continuous, integrable function on [0, m). We define the norm j/J?‘j\ of the history I? by

llr’ 112 = llr’ (o)ll’ + llr* ilk (3.8)

where

(3.9)

Corresponding to each function h(s), the space of histories for which \\I?\[<~ forms a Hilbert space S(h). q(h) is a cone in the Hilbert space Y(h) formed by functions mapping 10, 5)) into YSzl.

We assume that the common domain of the response functionals (3.3) is %(h) x &), where r/;,, is a 3-dimensional vector space. The response functionals (3.3) are assumed to be continuous and continuously differentiable over their common domain. Conse- quently, for all functions L”E Z(h), fi E ‘V(h), with I”+s7t E %(h) we have

(3.10)

where the functional dG(*; ~1.) is linear in Q and is continuous in all its arguments. Furthermore, if g, u, g+u E V(S), we have

(3.11)

where a,G E V,7), is continuous in all its arguments. A local admissible process is said to be regular at X at time t if (a) r and g are

Thermodynamics of deformable magnetic materials with memory

time-differentiable at time t, (b) I” is an absolutely continuous function in (e(h), (c) the function I% = p(s) = - dI?(t - s)lds is contained in ‘V(h).

Since a knowledge of the total history I? on [U, a) is equivalent to a knowledge of the past history r: on (0, m) and f’(0) = r, we may rewrite (3.7) in the form

G = G(IYS; r, g). (3.12)

Our smoothness hypotheses quarantee the existence and continuity of the partial derivatives I&G and 6G where

6G(-; -, +.w = $3(r: + a, ; r, g)lY=O. (3.13)

Equation (3.23) holds for all iCtE%” (22jt provided r+ (n E %‘, while (3. 13j2 is valid for

each dr, which is the past histury of some function @E V(li) which is such that I’ +QE S’(h). Both I&G and SG are continuous functions of their arguments and 6G is linear in a,. The instantaneous derivative G,G has component form

wherre

(3.15)

It follows from our smoothness assumptions that in every regular process the following chain rule holds [S, 91:

4. CONSEQUENCES OF THE CLAUSIUS-DUHEM INEQUALITY

It follows from equations (2.20), (2.29) and (3.16) that, in any regular process, the Clausius-Duhem inequality (3.4) may be written


In view of our assumption that the material is magnetically saturated, we must take cognisance of equations (2.11) when determining the consequences of the inequality (4.1). Thus, we must introduce the undetermined Lagrange multipliers w, a scalar, and 1E “Ir,,,, multiply (2.1 lh by w, (2.1 1)2 by 1 and add to the left hand side of (4.1). Conse- quently, we have

8y = tr{(- DFljl + %)l?‘} + tr{(- DF<.+)tj + d - /_@l)&}

+ a,$. g - 6$5(l? ; g[P:) - 4. g/pe 2 0. (4.2)

When a standard argument of the type employed by Coleman [8] is used in (4.2), we are led to the following

Theorem: It follows from the Clausius-Duhem inequality (3.4) that

(a) $ is independent of g:

(b) the functionals ?, A, L6, ?j are related to I$ by the equations

(c) when r’ = (F’, Fir,, t.~-‘, 13’) is a regular total history, the inequality

(4.3)

(4.4)

(4.5)

where

CT = -$ Slj((r’; f:) (4.6)

is the internal dissipation, must hold for all g. When the expressions (4.4) for i and & are substituted into equations (2.30) and

(2.8), we find that

(4.7)


so that equations (4.4) may now be written

53

(4.8)

It is now clear from equations (4.8) that we have proved that the response functionals ‘k, A., & and +j are independent of g.

Equations (4.8) are equivalent to the following

(4.9)

17 = -D&

If we set g = 0 in (4.5) we have the internal dissipation

US-O.

5. RELAXATION

inequality

(4.10)

Given a total history FE S’(h) and a positive scalar 6, we can define a new total history

The history I?‘+’ is called the steady state continuation of the history I? by amount S[8,9]. We note that f“‘“(s)=O, for s eIO,6).

Since

$(T) = ~r(D~~tir(~))+ trcD,,,,~,BT,,(7>,+D,~. #(~)fDet,&&) + si,6(T)f’>, (5.2)

it follows, from (4.10), that during the steady part of a steady state continuation of a regular total history, i.e. for 7 ~(t, t + S), we have

lj(7) = - &T(T) G 0. (5.3)


Consequently, the specific free energy cannot increase during the steady part of a steady state continuation so that

lj@‘“s) 6 I&r’). (5.4)

Let

rt = l?(s) = I-, s E [O, m), (5.5)

denote the constant function with value I”. The equilibrium response functions of the material are defined to be

IL = d&w) = $m,

T* = @e-t) = %*(r),

A* = &rt> = A*(r),

(5.6) a = .I?i(rt) = d(r),

q = grt ; 0) = izj(r).

A material for which

lim /jr'+*-rtji=o S-W (5.7)

is said to possess the relaxation property[9]. In view of (5.7) and the assumed continuity of the response functionals we see that

lim G(F+“) = G(r), s-7j (5.8)

where G denotes any of the response functionals (3.3). When (5.4) and (5.8) are combined, it follows, in the usual way[8,9], that the specific

free energy is a minimum when the past history is a constant so that we have

@(I-?. I 44 = 0,

624m I+,dd 20, (5.9)

where d, is an arbitrary element of q(h). Equations (4.9), (5.7) and (5.9), are together equivalent to the assertion that the

equilibrium response functions are related by the formulae


.ti = - ~,~-Fw(~~,,,rl;)*~ +$ (EL. &&P

6 = - a,sj* (5. IO)

Equations (5.10) may be used to describe the response of a material in any steady state process. In such a process the inequality (4.5) reduces to

W;g).g~O (5.11)

and it follows, from an argument of CoIeman and Noll[lO~, that if q(I’; g) is a differentiable function at 0 in V(,, then

gr; 0) = 0.

6. INTEGRATES DISSIPATfON INEQUALITIES

It follows, from equations (2.26), (4.9) and (5.2) that

(5.12)

JI=tr($.rLT)+tr(~~Lf,,)-.~.~-,e-e,

where Lt,) = fiCL1,F-‘.

(6.1)

Consider an admissible local thermodynamic process at a point X at various times t E (- 03, ~0). Let t, and t2( < t,) be two times and define the integral

9 tt,, tz) = 1” (rr (&j W)LT(t)) + tr (-& A(t)Ll,ct)) - 8(t) . i;(t) - q@jd(t)j dt. f,

Equation (6.1) implies that (6.2)

W)-$W=~(t,, tz)- (’ Nt)c+(t)dt, (6.3)

and, in view of the dissipation

An inequality of the type (6.4)

inequality (4.1 1), we have

9P(t,, &)a $w*)- +(tl).

is called an integrated dissipation inequality. If the fields {F(t), F,,,(t), p(t), 0(t)) = i?(t) are such that

I‘, a constant, t < tl, I’(t) = a smooth function of t, t E [t,, til,

r, t = t*,

then, on using the inequality (5.4), it follows that

(6.4)

(6.5)

set,, tZ) = ~~[tr(~T(“)LTit))+tr(~ A(t)L:,,(t))-LB(t). /i(t)-&)&)] dt 20.

(6.6)


If I’(t) is further restricted so that 8 = g, a constant, for t E [t,, tz], then (6.6) leads to the inequality

In words, the inequality (6.7) says that in any smooth isothermal process which starts from a state of equilibrium and returns F, F(,, and p to their original values, the integral

of ~~(~TL~)+~~(~AL~~)- LBf & around a closed path is non negative.

In the case of a process in which E,, and cr. are constant for t E [t,, t2], we have

l’i tr (h T(t)LT(f)) dt 20, fi

(6.8)

which is the original result given by Coleman [8]. If F is constant for t f: It,, t2], then

h AttY.& > -LB(t). C;.(t)} dt 20. (6.9)

We note that, in particular, the inequality (6.9) is always valid in any isothermal process, in a rigid magnetically saturated body, which starts from equilibrium and returns Fm and p to their original values.

7. CONSEQUENCES OF THE PRINCIPLE OF OBJECTIVITY

The principle of objectivity [ 1 I], states that a process which is admissible must re- main admissible after a change of reference frame. A change of frame can be characterized by at each instant by the mapping x + ii, with

Z(t)=c(t)+R(t)[x(t)-al (7.1)

where c(t) is an arbitrary time dependent vector, R(t) is a time dependent proper orthogonal tensor and a is an arbitrary vector which need not depend on t. The scalars 8, 4, n are unaltered by a change of frame, but F, F (@jr p, g, q, J3, T and A transform in the following manner:

F -+ RF, F,,, + RS,,, P + RP, g -+ Rg (7.2)

q-+Rq,,B+RLB,T-+RTR-‘,A-+RAR-’

Notice that, as is usual in non relativistic treatments of deformable magnetic media[l, 71, we assume that JJ behaves as a three dimensional vector as far as considerations of objectivity are concerned.

The transformations (6.2) imply that

q(RF”, RF:,,, RP’, 8’; Rg) = R(O)q(F’, FL,, P’, e’),


rB(RF’, RF:,,, Rp’, 0’) = R(O),B(F’, F;,,, pt, 0’),

T(RF’, RF;,,, Rp *, 8’ ) = R(O)T(F’, F:,,, /..t *, 8’ )R-‘(0),

57

(7.3) A(RF’, RF;,,, Rp’, 0’) = R(O)A(F~, F:,,, y’, @‘)R-‘(0),

for all r’={F”, Ff,,, p’, 03 n %(h) and all proper orthogonal R(e). In order that the identities (7.3) be identically satisfied there must exist functionals

4, 6, 3, T’, A, such that

3, = & (C', W’, N’, 8’ ),

FTq = i&Z’, W’, N’, 8’; FTg),

FTJB = &C’, W’, N’, 8’; FTg),

F’TF = F(Ct, W’, N’, f33,

FTAF = d(C’, W’, N’, 8’ ),

(7.4)

where C = F”F, W = FTF,,,, N = FTp. Now, equations (7.4) are the most general forms which the constitutive functionals

may assume if the principle of objectivity is to be satisfied. However, it remains for us to ensure that equations (2.22):

A (grad p)‘= (grad p)A’ (7.5)

are satisfied. It follows, by an argument of the type used by Coleman[8], that

and, on substituting from (7.6) into equations (4.9h and (7.5), we find that there must exist a functional $ such that

9 = t&C’, C:,,, N', 0’) (7.7)

where Ccp) = F&E;,,. Since

it fohows from equations (4.7) that I = 0. Consequently, equations (4.9) reduce to the forms

T = 2pFDcsjFT+ pp &J DN&FT,

‘A = 2~F<,,Dc,,t,8F=,

LB = -FD&+-+.FD,$)p,


g= -De& (7.9)

We notice that while the condition (2.22) requires that $, q, LB, T and A be functionals of the histories of C, Ccp), N and 8, it places no additions restrictions on q beyond those required by the principle of objectivity.

Of course, the forms of the response functionals of the material will be further restricted by whatever material symmetries the medium under consideration may possess. The analysis of such restrictions is straightforward but, even in the simplest case of an isotropic material, the constitutive relations which one obtains as a resuit of applying symmetry transformations are so complicated that they are unlikely to be of any great practical value. For this reason we do not present such results here. However, we study the consequences of material symmetry in a forthcoming work on the finite linear theory of deformable magnetic materials with memory.

8. DEFORMBLE PARAMAGN~TIC MATERIAL WITH MEMORY

In the case of a paramagnetic material magnetic spin and magnetic exchange interactions are of no significance. Thus, in this case, the surface exchange tensor A vanishes and the motion of the body is governed by the local equations (2.16), (2.17), (2.18) and (2.21) together with the equations

B+l,B=O, (8.1)

p.G=tr(TL*)+pB.k-divq+pr. (8.2)

If we define the specific free enthalpy of the material by the relation

then it follows from (8.2) that

where T* = T(F-‘)TIp. We assume that

+= &F’, B’, 8’; g(t)),

T” = *(IF’, B’, 0’; g(t)).

P= P(F’, B’, 8’; g(t)),

q = 7j(F*, B’, 0’; g(t)),

q = il(I?, B’, 8’; g(t)),

(8.3)

(8.4)

(8.5)

where the response functionals have the same smoothness properties as the corresponding functionals (3.3).

Proceeding as in section 4 we have the following:


Theorem: It follows from the Clausius-Duhem inequality (3.4) that

(a) 4 is independent of g:

4 = &F’, B’, 8’ ), (8.6)

(b) the functionals f, $ and 4 are independent of g and are related to 4 by the equations

$’ = D&,

r;= -D&2 (8.7)

6=-D&

(c) when 5’ = (F’, B’, 0’) is a regular total history, the inequality

y=P-&q.g>o, (8.8)

where

is the internal dissipation inequality, must hold for all g. It follows from the principle of objectivity that 4 = &C’, N’, 0’3, where N = F%‘.

Thus, we have

T = 2pFDc&F=- pB @ /.L (8.10)

/_L = -FD,& (8.11)

IJ = -De& (8.12)

Analogous results to those of sections 5 and 6 may also be derived in this case.

REFERENCES [l] H. F. TIERSTEN, J. Math. Phys. 5, 1298 (1964). [23 H. F. TIERSTEN, J. Math. Phys. 6, 779 (1965). [3J W. F. BROWN, Jr., Magnetoelastic Interactions. Springer (1%6). [4] A. I. AKHIEZER, V. G. BAR’YAKHTAR and S. V. PELETMINSKII, Spin Waves. North-Holland

(1968). [5] S. KALISKI, Proc. Vibr. Probl. 2, 10, 113 (1969). [6] G. A. MAUGIN and A. C. ERINGEN, J. Math. Phys. 13, 143 (1972). [7] H. F. TIERSTEN and C. F. TSAI, J. Math. Phys. 13, 361 (1972). [8] B. D. COLEMAN, Arch. ration. Me&. Analysis 17, 1 (1964). [9] B. D. COLEMAN and V. J. MIZEL, Arch. ration. Me& Analysis 27, 255 (1967).

[lo] B. D. COLEMAN and W. NOLL, Arch. ration. Mech. Analysis 13, 187 (1%3). [l l] C. TRUESDELL and W. NOLL, The Nonlinear Field Theories (Edited by S. FLUGGE), Handbuch der

Physik, 111/3. Springer (1%5).

(Received 29 November 1972)


R&sum&La theorie des materiaux magnetiques, deformables, conducteurs de la chaleur et avec une memoire qui s’atttnue, est developpee. le cas des materiaux du type ferromagnttique satures est trait& en detail. les restrictions ajoutees aux fonctionnels de rtponse par IflinCgalitC de Clausius-Duhem sont deduites. Finalement, la theorie des materiaux paramagnetiques deformables avec m&moire est brievement developpee.

Zusammenfassung-Es wird die Theorie deformierbarer wkmeleitender magnetischer Stoffe mit schwinden- dem Gedachtnis entwickelt. Der Fall des gesattigten ferromagnetischen Stofftyps wird in Einzelheiten behan- delt. Es werden die Einschrlnkungen abgeleitet, die auf die Ansprechfunktionale durch die Ungleichheit nach Clausius-Duhem ausgetibt werden. Es wird Entspannung besprochen und integrierte Zertreuungsungleic- hheiten werden abgeleitet. Schliesslich wird die Theorie deformierbarer paramagnetischer Stoffe mit Gedschtnis kurz entwickelt.

Sommaria-Quest0 articolo sviluppa la teoria dei materiali magnetici, conduttori de1 calore, deformabili e con memoria evanescente. Tratta in dettaglio il case dei materiali di tipo ferromagnetic0 saturato. Deriva le restrizioni poste sui funzionali di responso dalla disuguaglianza de Clausius-Duhem. Discute il rilassamento e deriva le disuguaglianze della dissipazione integrata. Sviluppa infine brevemente la teoria dei materiali paramagnetici deformabili con memoria.

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Documents

Thermodynamics of deformable magnetic materials with memory