Internal dissipation, relaxation property, and free energy in materials with fading memory

JournalofElasticity 40: 107-122, 1995. 107 © 1995 KluwerAcademic Publishers. Printed in the Netherlands.

Internal Dissipation, Relaxation Property, and Free Energy in Materials with Fading Memory

M. FABRIZIO 1, C. GIORGI 2 and A. MORRO 3 1Mathematics Department, Piazza di Porta S. Donato 5, 4012Z Bologna, Italy 2Department of Electronics and Automation, University of Brescia, 25060 Brescia, Italy 3DIBE, University, Via Opera Pia l la, 16145 Genova, Italy

Received 12 April 1995

Abstract. This paper examines some features of the standard theory of materials with fading memory and emphasizes that the commonly-accepted notion of dissipation yields unexpected consequences. First, application of the Clausius-Duhem inequality to linear viscoelasticity shows that there is a free energy functional such that the so-called internal dissipation vanishes in spite of the dissipative character of the model. Second, upon the choice of a suitable function norm, the relaxation property is proved not to hold for viscoelastic solids. Finally, the particular case is considered when the relaxation function is a superposition of exponentials. Different descriptions of state are then possible which prove to be inequivalent as far as the free energy is concerned.

1. Introduction

Mathematical theories of materials with fading memory have been developed to a great extent in the last three decades. In this regard, we mention the books [1-3] for expository accounts of the subject. In spite of the considerable progress about modelling and mathematical development, the analysis of relatively simpler models, such as linear viscoelasticity, has shown the need for revisiting the theory.

Some papers by Fichera on non-existence or non-uniqueness of the solution to problems in linear viscoelasticity [4, 5] show that even the constitutive properties of the body are affected by the choice of the norm for the underlying fading memory space. Also, recent investigations of ours [6] on linear viscoelasticity have shown that some aspects may be better understood by reexamining carefully the notion of fading memory and relaxation property along with the content of the second law.

More generally, Coleman and Mizel [7] say that a theory of materials with memory which does not rest upon L p spaces has the advantage of 'eliminating the arbitrary numbers p and deemphasizing the influence function k which are not experimentally determinable and appear in a physical theory as analytical encumbrances rather than as aids to understanding'. In addition, the occurrence of a subjective influence function in the norm and hence in the statement of general properties and principles, such as the second law for approximate cycles, makes the pertinent statements of dubious meaning. Nevertheless, to our mind little has been

108 M. FABRIZIO ET AL.

performed toward a successful theory which is free from an influence function or whose consequences are unaffected by the function norm.

Some views or prejudices about irreversibility and dissipation provide a further motivation for the present investigation. Let A denote a suitably defined entropy production. According to Ericksen [8], 'thermodynamicists interested in irreversible processes use A as a good measure of dissipation.' 'There seems to be some prejudice that only reversible processes have the property A = 0.' The oper- ative meaning of A = 0 is investigated in this paper in connection with materials with fading memory. We show that A fails to be a good measure of dissipation and relate this failure to the non-uniqueness of thermodynamic potentials, in the particular case of viscoelastic solids.

Another purpose of this paper is to clarify via an example that modelling the material behaviour, as the relaxation property, in terms of function norm is highly deceptive. A third purpose is to emphasize that, when the relaxation function is a superposition of exponentials, different state descriptions are possible and they are inequivalent as far as the free energy is concerned.

2. Materials with Fading Memory

For any quantity f, the symbol f t denotes the history of f up to time t, viz. f t (u) = f(t - u), u E IR +. The material under consideration is characterized by specifying a set of variables, say g, in terms of the history of a set of variables, say X. For the purpose we have in mind it is enough to write

g =

The histories X t are elements of a space • which contains L°°(I~+). Let T (~) denote the translation operator such that

[

The domain/I~ is required to be closed under the translation operator T (a). As a peculiar feature of fading memory, the influence of X is taken to be less for

the values occurring longer ago. This idea is rendered mathematical through the relaxation property. Let X1 and X2 be two functions which are equal after some time to,

, '~l(t) = ,~,2(t), t ~ to.

The relaxation property means that

lim [g~(.),~) - ~(X~)] = O. (2.1) t--+oo

INTERNAL DISSIPATION AND FREE ENERGY IN MATERIALS WITH FADING MEMORY 109

The property (2.1) was stated by Volterra [9], as the principle of dissipation of hereditary action, by saying that changes of g(t) due to changes of A(r) in the interval ( - ~ , to) can be made as small as we please by letting (to, t) be sufficiently large.

Though the literature shows a variety of definitions, mathematically the property of fading memory amounts to some degree of smoothness (e.g., continuity, differentiability) of the response functional ft. In this regard we need the structure of a Banach space for • with a norm which reflects the fading influence.

In linear (isothermal) viscoelasticity the pertinent variable )~ is the infinitesimal strain tensor E while g is the Cauchy stress tensor T. For such a case, a thermodynamic analysis has been developed [3] (cf. also [6]) and here we take it for granted. Rather, we emphasize some conceptual drawbacks which occur if we follow some seemingly obvious views about the second law and the relaxation property.

3. Free Energies

Let ~r be the space of restricted histories rE t, namely, for every E t E ~ then rEt(s) = Et(s), 8 E ll~ ++. It is customary to say that a functional Sr on • = Sym x ~r is a free energy if

(1) ~" is continuous on 4, differentiable with respect to the present value E of E t and

7" = DE~(E , rEt);

(2) for each value of~ such that l~(t + ~) is continuous, $" satisfies the inequality

7(Et+ ) < E(t +

(3) the functional F is minimal in correspondence with constant histories in that, if E t is any history ending at the value E then

7(E ) > j:(wt),

where Et is the constant history, equal to E at each time.

Let T be the standard functional of linear viscoelasticity, viz.

5 T(t) = T ( E t) = GoE(t) + G'(s)E(t - s) ds. (3.1)

Property (1) of U and the form (3.1) of T yield

~ ( t ) : ½E(t)GoE(t) + E( t ) . G'(s)E(t - s) ds + ~'(rEt),

where .~ is an arbitrary functional on Or. However, (2) and (3) are not sufficient to determine the functional b t'. Indeed (cf. [10, 3]) there is a convex set of admissible


free energies with properly defined minimal and maximal free energies. Moreover, the maximal free energy .TM is given by

~'M(E t) = .T(E(0) t) + for 'T(E~) • 1~(~) d~,

whence

~M(t) = a '(Et) • E(t).

This fact is not surprising. Non-uniqueness of thermodynamic potentials in materials with memory was first shown by Day [11 ] in his theory of heat conductors with memory.

Let G be the relaxation function,

£ G(u) = Go + G'(~) d~.

It is known that, as a consequence of the second law of thermodynamics [12], the sine transform G'8 satisfies

.~G',(~) < o, w # o.

The solid character of the body is expressed by letting Goo = lim~,-,oo G(u) be positive definite, Goo > 0. For simplicity, we let G(u) be symmetric for any value ofu E ~+.

Consider the functional ~ defined by

1/0T ~I/(E t) = 1E( t ) . Goo~(t ) + ~ [Et(81) - ]E(t)]

• G12(lsl - ,2I)[Et(s2) - E(t)] dSl d82, (3.2)

where the subscripts l, 2, of G denote differentiation with respect to Sl, s2, so that G12 = oq2G/OqSl (9s2. We can write

1 [~°[°°Et(s,) "Glz(Is,- saI)Et(s2)d*l 082 ~(E t) = ½E(t). GooE(t) + 2 J0 J0

J077 - E ( t ) • C-12( Is l - s2l)Et(s2)ds1 d82

+lE(t). [~°°~°°G12(,81 - 82[,d81 d82] F,(t ,.

Because

/oY Os (Is2 - 811) dsl ds2


substitution yields

• (E') = ½E(t/.CoE(t)+ E(t).

dO "/0

Hence we have 0 ~ / 0 E = T, namely O(E t) satisfies the property (1). Now we evaluate the time derivative

O(E ~) = T . E + E(t). fo~C'(8)E~(8) d8

+ f0c~0°° l~.t(81)G12(182 - - 811)Et(82)dSl d82.

Considering the last integral, integration by parts gives

f0~0°° l~t(81)" G12(182 - sli)Et(82)ds2 dsl

CO " f O0

= f0 d81E (81)" [-G/(81)E(t)+ / G1(182-81])]~t(82)d82]

The first integral in the right side cancels. The second integral vanishes because

0 OG (Isl - s21) = sgn(s1 - s2)G'(IsI - s2l).

Accordingly, ~(E t) = T • !~ and hence ~(E t) satisfies the property (2) as an equality.

The last step is to show that the functional 9 is minimal at constant histories. To this purpose we need the following result.

PROPOSITION. Let W E L2(]R +) and K E LI(~+); W has values in Lin and K maps Lin into Lin. The integral

fo ~ fot K(t J = W(t) - r )W(r) dr dt

is positive for every nonzero W if and only if the cosine transform Kc is positive definite in Lin

Proof. If W, V E L2(~ +, Lin), by the Parseval-Plancherel theorem we have

/0 1// W * ( t ) . V ( t ) d t = ~ W~(~) • Vr(, ,) d-,, o o


where the subscript F denotes Fourier transform and • denotes the complex con- jugate. Letting

f0 t V(t) = K(t - r ) W ( r ) d r dr,

we have VF = K F W F . Moreover, since W, K vanish on ~ - , the cosine transforms Wc(~;), Kc(~) are even functions while the sine transforms Ws(w), K~(oa) are odd functions. Hence substitution yields

/ F W~-(w) • VF(W) dw = [W~(w) • K~(w)W~(w) o o o o

+W~(~) • K~(~)W~(w)] dw.

If I~(w) is positive definite for every w E ~+ then J > 0 for every non-zero function W. Conversely, if J > 0 for every non-zero function then We- I~Wc + W~. K~W~ > 0 for every W~ and W,, which implies that K~ is positive definite in I~ + . [ ]

Now observe that

G12(lSl - s21) = -2~(Sl - a2)Gt(lSl - s2l) - G"(ISl - 821),

and hence

fo °~fo °~ W(s l )" G12(Isl - s21)W(s2) do2 dsl

/0T = - 2 W(sl).i~(81-82)Gt(ls1-82[)W(82)d82dal

- w ( 8 ~ ) • G"(I , , - ,21)w(82) ds~ d,2

/o = - 2 W(s) • G~W(8) ds - 2 W ( t ) . G " ( t - r ) W ( r ) d r d t

fo fo = - 2 w ( t ) . [G"(t - r ) + 6 ~ ( t - ~ ) ] w ( r ) dr dr.

According to the proposition stated above, GI2 is a positive definite kernel if and only if the cosine transform of G"( t - r) + G~6(t - r ) is negative definite. Now, an integration by parts gives


G~(~) ~0 °° = c o s . ¢

= G'(() cosaJ(I ~ + ~o G'(() sin~o( d(,

whence

G~(o.,) + G~ = wG's(w ) < 0.

Hence, because

0°°[G"(~) + G'(~)6(~)] cosoJ~ d~ = G~'(w) + G~,

it follows that the quadratic form

f0°c~0°° W(81) • G12(lSl - s21)W(82) d82 dsl

is positive definite and hence vanishes only at W = 0. On taking W(8) = Et(8) - E(t), the minimality of the functional at constant histories follows. This proves that k~(E t) satisfies the property (3) of a free energy functional.

4. Clausius-Duhem Inequality and Dissipation

Now that • can be regarded as a free energy, we examine aspects which show how the classical view of the second law of thermodynamics may lead to wrong conclusions.

Denote by 0 the (absolute) temperature, r 1 the entropy, e the internal energy, and p the mass density. Moreover, let L be the velocity gradient and denote by w = T • L the stress power. For formal simplicity we let/7 be uniform. Hence one way of stating the second law of thermodynamics is through the Clansius- Duhem inequality which, because 8 is uniform, reduces to the Clausius-Planck inequality

A := 0¢1- ( ~ - p ) _> 0, (4.1)

A being called internal dissipation [2], p. 112. Let F be the deformation gradient and

1 1 4 M. FABRIZIO ET AL.

In terms of the free energy ¢ = e - Or/, we have

A = 7 - . ~ - g).

Consider the constitutive equations for ¢ and 7- in the form

¢(t) = 7-( t )=

Differentiability of the functional 3 c yields

A = (7- - DA.T ) • A + 7),

where D~ is the derivative with respect to the present value of the history )it and

-7) = d~()~t[~ t) is the Fr6chet derivative at the history )it in the direction of

the restricted history ~jt . Along with the continuity of T this implies that (cf. [2], Lect. 3) ~" and 7- satisfy the inequality (4.1) for arbitrary values of )t if and only if

7- = D ~ , (4.2)

7) >__ 0. (4.3)

By (4.2) we can write

and say that the generalized power 7- • J~ consists of two contributions. The first one is the recoverable power in that it results in the time derivative of the free energy. The second one is then nonrecoverable and by (4.3) is nonnegative for every process. One might then be tempted to say that 7) is the effective measure of dissipation and that 7) = 0 for every process characterizes nondissipative bodies. In particular, since viscoelasticity models dissipative bodies we expect that 7) > 0 for viscoelastic solids. Otherwise, 7) = 0 is expected to correspond to elastic solids.

Though these conclusions may seem obvious, in materials with memory they are inconsistent. We show that such is really the case for linear viscoelasticity where 0 is a parameter and

7- -~ T, ~ X ~ E .

Letting ¢( t ) = ff/(E t) we have

A = 7 9 = 0 ,

INTERNAL DISSIPATION AND FREE ENERGY IN MATERIALS WITH FADING MEMORY 1 15

namely the internal dissipation vanishes. Meanwhile, for the s a m e viscoelastic body we are able to provide an infinite set of free energies for each of which A # 0. For instance, let G"(u) _> 0, Vu E/~+, and take the functional

~G(E t) = ½E( t ) .G~E( t )

l f 0 ~ - ~ [ E ( t ) - E ( t - s)]. G ' ( s ) [E( t ) - E ( t - s)] ds.

This functional is a generalization of a (free energy) functional determined by Graffi [13]. It obviously satisfies the properties of a free energy and, moreover, yields

f0 ° A = [E(t) - E ( t - s)]. G"(s)[E(t) - E ( t - s)] ds,

and hence A > 0 for non-constant histories E t. Examples of free energy function- als, with different values of A, are given in [6]. This shows that A depends on the free energy functional.

This conclusion makes the interpretation of A, as a measure of the dissipation of the body, meaningless. Also, this makes clear that A = 0 does not mean that the solid is elastic. The measure A, associated with the functional (3.2), vanishes for every viscoelastic material element whose relaxation function G is compatible with thermodynamics and has derivatives up to second order which are integrable. At least for materials with memory whose state is the whole history, the crucial condition for the vanishing of A is the choice of the (maximal) free energy ~P(Et), not the constitutive properties through the relaxation function. It is worth remarking that the existence of a free energy which amounts to A = 0 has been found by Lucchesi [14] for elastic-plastic material elements.

5. Relaxation Property

Let H(~v) = -~G's(w); H maps ~++ into the set of symmetric positive definite elements of Lin(Sym, Sym). Since, by assumption, G'~ is continuous, summable and vanishing at infinity, it follows that H is bounded on ~+. Consider the space 7-/of functions (histories) on I~ +, with values in Sym, defined as

7-/ = {(E( t ) , rE t) E Sym × L2(~++),

j0

I 16 M. FABRIZIO ET AL.

where Etc and Ets denote the half=range cosine and sine Fourier transforms of E t. The space 7-[ becomes a pre-Hilbert space by defining the inner product of two elements E~, E~ as

t t (El,E2) = 1El(t)-GooE2(t )

1 o~ + ~ f 0 [ (E~(w) E~- t ) ) 'H(w) (E~ , (w) E ~ t ) )

+E~(w)- H(w)Ei~(w) ] dw.

We denote by II • the corresponding norm. Reference [61 shows that the functional

~01(E t) = ½E(t) -G~E(t)

+ I f0 °° [(Ets(W) - E ( t ) ) . H(w ) (Ets(w)_ E(t ) )

+Eta(w) • H(w)Etc(w) ] cl~

is a free-energy functional and, moreover, is the free energy (3.2) in terms of sine and cosine transforms. It is of interest to remark that the free-energy functional in (3.2) determines a norm through the relation

IIEtll 2 = q~l(E t) = ~(Et).

By means of the Parseval-Plancherel theorem, the stress functional of linear viscoelasticity,

T(t) = GccE(t) + G'(s)[Et(s) - E(t)] ds,

can be written in the form

As shown in [6], this functional is continuous with respect to the norm 1[ • I Iw t a t For any history E E 7-/, consider the translated history T( )E. We find that

= Et,(w)sinwa + Et,(w)coswa,

(T(a)E')c(w) = Etc(w)coswa - Ets(w) sinwa.

INTERNAL DISSIPATION AND FREE ENERGY IN MATERIALS WITH FADING MEMORY 1 1 7

Accordingly,

IIT(~)Et lI~ = {Et,(oJ). H(oJ)E~(~) + Etc(~). H(~)Etc(~)} d~.

The norm I]T(~)Et[[7~ is then independent of a and hence does not approach zero as a --+ e¢, i.e.

=lina IIT(')E*II~ # 0.

In the standard theory of fading memory, cf. e.g. [15], the norm ]]. II involves an influence function b. In such a case the relaxation property is taken in the form

lira IfT(=)Eql = 0 g ---+OO

for every history E t in the corresponding function space. In this sense we might say that the relaxation property does not hold for linear viscoelastic solids.

A direct substitution gives

~r(T(°)E ~) = ~ G ' ( u ) E ' ( u - a ) d u = G'(~+a)E~(~)d~.

Letting G~(() = G ' ( ( + a), we apply the Parseval-Plancherel theorem to write

/7 1/; OO

Hence we have

G~(~o) = G ' ( ( + a) exp(ia~() d(

= / '~ G'(r/)exp(iau/)exp(-ia~a) dr/= exp(-taa)G~(oa). (2,O

Substitution yields

Z /; oo 1 exp(-iwa)G~(~)[E~(~)]* dw. 6°(~)E~(~) ~ = ~ o~

If GF[EF]' t • E LI(IR) then the Riemann-Lebesgue lemma allows us to conclude that

lim 17"(T(=)Et)I = 0. (~ "-'+ OO

The linear viscoelastic solid has the relaxation property in the sense of (2.1) with

E~ = T(~)Et,E~ = 0).


As a comment, by applying (2.1) we conclude about the possible validity of the relaxation property by merely considering the constitutive assumption. In the standard theory the conclusion is related to the function norm chosen for histories, and hence may depend on a subjective choice.

Sometimes, cf. [16], the validity of the relaxation property is shown to be a consequence of a pointwise convergence (called asymptotic rest property). Here T(~)E t is pointwise convergent to the zero history but [IT(~)Etlln is non-zero and independent of a. In other words, the norm II • does not satisfy the sequential Fatou property whose validity is an axiom of the standard theory of materials with memory. To our mind, there is no physical motivation for the validity of the sequential Fatou property.

6. State and History for Exponential-type Relaxation Functions

An important feature of materials with memory occurs when the relaxation function is a linear combination of exponentials, viz.

G'(~) = - A ~ akgk exp(-ak~), (6.1) k

where A E Lin(Sym) is positive definite and c~k > 0; summation over h, k is meant from 1 to the number n of exponentials. Of course, all previous results hold for functions of the form (6.1) as a particular case. However, the occurrence of exponentials allows the state a to be expressed by (n + 1)-tuples of vectors and not necessarily by the history E t. When such (n + 1)-tuples are involved, the state a does not contain superfluous variables and hence we say that the minimality condition is satisfied; minimality means that from two distinct states at least two distinct pairs of strain-stress paths originate.

First, let

Tk(t) = Ayk [E(t) - ~k foo°°exp(-c~k~)E(t - ~) d~] .

Hence, by (3.1) and the observation that

Go = Goo + A ~ gk, (6.2) k

it follows that

T(t) = GooE(t) + ~ T k ( t ) . k

We may view the state, at time t, as the (n + 1)-tuple E, T1,. . . ,T,~, at the same time t. If n = 1, Day's free energy is given by the functional

~/Day(E t ) = ½E(t)-GooE(t)

1 + ~ {a(Go - G~o) '/2 fo°° e x p ( - a ~ l [ E ( t ) - E(t - ~)] d~} 2 ,

iNTERNAL DISSIPATION AND FREE ENERGY IN MATERIALS WITH FADING MEMORY 119

which, in terms of the pair (E, T), becomes

6(E,T) : ½E. GooE + l{(G0 - Goo)-m/2[T- GooE]}2.

The function ~(E, T), though, is not a free-energy function in that

0~(E,T) - GooE + Goo(G0 - Goo) - l [T - GooE] # T.

0E

In this sense, if n > 1 we are not able to determine a free energy function (such that 0~b/0E = T). Alternatively, let

Ek(t) = exp( -ak( )E( t - () d(

whence

T(t) = G~E(t ) + A ~ g ~ [ E ( t ) - ~kEk(t)]. k

The state may then be viewed as the (n + 1)-tuple E, E l , . . . , En and free-energy functions exist. Look for one, or more, free-energy functions in the general form

1 6 ( E , E , , . . . , E n ) = ½E(t)" GooE(t) + ~ y~ Chk[E(t)- ahEh(t)]

h,k

• A[E(t) - akEk(t)], (6.3)

where (Chk) is a symmetric n x n matrix. Incidentally, the variables Tk involve both the present value E(t) and the past history of E through the integral of exp(-ak~)E(t - ~). The variables Ek do not involve the present value E(t) and this is the essential difference between the two representations.

The parameters A, gk are required to comply with the requirement of the second law of thermodynamics. Observe that

gkO~k G'~(~) = - A ~ ~2 ~2 + ~ .

k

A direct check shows that

gkO~k Ek gkOq~ IIh#k(W 2 + a 2) P2(n_l)(Ca)

k ~2 + .~ - 1-i~(~2 + .~,) = 1-i~(~: + .~,),

where P2(n_i)(w) is the 2(n - 1)th degree polynomial in w given by

\ k / h


The polynomial is positive if all coefficients are non-negative (and not all zero). Accordingly, the thermodynamic condition Gts < 0 holds if A is positive definite and

E gkak >_ O, k

-> 0, k h#k

° ° . . . . . ° . . . . , . . , ° . . . . . . . . . , , ° . . . . . . . . .

h

If all gk's are positive (as with the theological models) the thermodynamic condition holds.

We now show that £5(E, E l , . . . , En) is a free energy, provided that appropriate conditions on (Chk) hold. The property (1) holds, in that

0£5 = A ~ chk[E -- akEk] 0E h,k

equals T, provided that P~h ¢hk = gk. For the validity of the property (3) we consider a constant history Et and obtain

ZE, £5(E, 1E.Go E. C~k

Hence (3) holds if and only if (ehk) is positive definite. Finally, we ascertain the validity of (2). Straightforward calculations yield

£5(E, E1,... ,E~) = T. E - ~ ~ ( ~ h + ~k)Chk[E- ~hEh] h,k

• A[E - akEk], (6.4)

and hence (2) holds if the matrix (Fhk), rhk = (ah + ak)Chk, is positive semidefinite. Altogether, ~b is a free energy provided that

E Chk = gk, h

(chk) positive definite,

(I'hk) positive semidefinite. (6.5)


By way of example, the general expression of Day's free energy would be (cf. [3], ch. 3)

~Day(E t) = 21-E(t).GooE(t)

I ( ( G 0 _ G ~ ) _ I / 2 oo E(t d~} 2, +2 fo G'(~)[E(t)- - ~)]

and hence we have

~Day(E, EI , . . . ,En) = ½E "GooE + - -

by (6.3) we have

ghgk Chk -- ~ j g j"

1 {~h gh[E--Eh] } 2 E j gj

Hence (6.5) holds if and only if n = 1. It is known (cf. [6]) that 9Day(E t) is a free energy only if G ~ is made of a single exponential. Now, consider an expression that we ascribe to Dill (cf. [17]), namely

~DilI(E t) = 1E(t). G~E(t) + ~ [E( t ) - E ( t - ~1)]

• G'(~I q- ~2)[E(t) - E(t - ~2)] d~Cl d~2.

As shown by Graffi [13], the function (6.1) makes ~Din a free energy only if gk > 0 for every value ofk. Moreover, we say that if the state is the (n + 1)-tuple (E, E l . . . , En) then

1 CDin(E, E1,.. . ,E~) = ½E. G~E + ~ ~-~'~ gk[E - akEk] • A[E - akEk].

k

Incidentally, we can view ~Dill as the sum of free energies in Day's form, with a single exponential. Further, we have

(Chk) = diag(gl, . . . ,gn).

In conclusion, no free-energy function ¢ of the form (6.3) satisfies the dissipation inequality (4.3) as an equality. Indeed, the last term in (6.4) vanishes for any (n + 1)-tuple (E, E1,.. . ,En) if and only if Fhk, and hence Chk, is the zero matrix. Of course, for exponential-type relaxation functions the functional q(E t)

122 M. FABRIZIO El" AL.

still is a free-energy but is not a free energy as a function of the state variables (E, E1 . . . ,E~).

Acknowledgements

The authors gratefully acknowledge useful discussions with W. Kosiliski.

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Internal dissipation, relaxation property, and free energy in materials with fading memory

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