Irreversible thermodynamics of nonlocal systems

ht.

J. Engng

Sci.,

1974, Vol.

12, pp. 607-631.

Pergamon

Press.

Printed

in Great

IRREVERSIBLE THERMODYNAMICS Center

for the Application

Britain

OF NONLOCAL SYSTEMS

DOMINIC G. B. EDELEN of Mathematics, Lehigh University,

Bethlehem,

Pennsylvania,

U.S.A.

Abstract-Global laws of balance of momentum, moment of momentum and energy, together with local conservation of mass are reduced to point statements involving localization residuals. A fundamental functional inequality is then obtained which reduces to the Clausius-Duhem inequality for local theories. For nonlocal theories, the functional inequality states that the total internal production of heat of a material body at any given time is non-negative. This inequality is reformulated in terms of a functional inequality on a Hilbert space of ordered collections of L2 functions. The general solution of this functional inequality is obtained and this leads to all admissible constitutive relations. The existence of dissipation potentials and symmetry relations are established for material bodies with admissible constitutive relations. Nonlocal analogues of Maxwell’s reciprocity relations are also obtained as well as a proof of consistency with the results of thermostatics. Satisfaction of nonlinear forms of Onsager’s reciprocity relations are shown to be equivalent to the requirement that the operators generating the admissible constitutive relations be potential operators. It is also shown the certain functionals of curves in a function space are odd under time reversal if and only if the nonlinear form of Onsager’s reciprocity relations are satisfied. Thus, Gurtin’s results [IO] for processes which may be approximated by linear departures from equilibrium are extended to all processes with admissible constitutive relations. Similar results are established for significantly less restrictive sets of histories than those used by Gurtin and for a wide class of generalizations of the time reversal operator on such histories. Indications are given that satisfaction of invariance under superimposed rigid body motions implies satisfaction of the zero mean conditions for all localization residuals.

1. INTRODUCTION:

DECOMPOSITION OF OPERATORS INEQUALITIES

AND FUNCTIONAL

theorems for finite dimensional vector-valued functions established in [I] have provided a simple and direct means of answering a number of important questions in the thermodynamics of irreversible processes associated with local theories of continua. Theorems on the existence of dissipation potentials and symmetry relations are direct consequences of the decomposition [ 1,3]. A full characterization of ‘fluxes’ in nonlinear irreversible thermodynamics has been obtained [2] together with the foundations of several of the variational principles in current use in linearized theories. Complete solutions of the Clausius-Duhem inequality have also been obtained [3] since the decomposition theorems can be used to convert the C-D inequality into an integrable differential inequality involving a single scalar-valued function. From this has emerged a unified thermodynamic basis for non-history dependent constitutive relations which model general, nonlinear, elastic, visco-elastic, fluid, and perfect plastic behavior in the presence of heat conduction, chemical reactions and diffusion[4,5]. Detailed analysis of the asymptotic approach to equilibrium states has also been obtained [5]. Nonlocal theories of material behavior (see, for instance [6-lo]), in contrast with local theories, cannot be formulated in terms of finite dimensional vector spaces. The reason for this is twofold. First, the balance equations are formulated in terms of statements for whole bodies, and this introduces localization residuals when such formulations are reduced to equivalent systems of point equations[6,9]. The localization residuals introduce dependences on the values of state variables and their derivatives throughout the body. Second, the constitutive relations which obtain in nonlocal theories give dependences of stress, heat fux, etc. on the values of strain, temperature, THE DECOMPOSITION

607

608

D. G. B. EDELEN

etc. throughout the body. Thus, functional rather than function dependences are involved and the natural setting becomes a function space rather than a finite dimensional vector space of values of a finite number of functions at a point in their mutual domain of definition. Proofs of the decomposition theorems for operators on function spaces and solutions of the functional inequalities required in the analysis of irreversible processes in nonlocal continuum theories have been obtained in reference [ 111. The essential results are summarized here for the convenience of the reader and for reference in later sections. We use the notation, conventions and definitions given by Vainberg [ 121 without further note. Theorem 1. Suppose the following conditions are satisfied: D,. The operator F(x) is a mapping from a Banach space p into the conjugate space P*. D2. F(x)

has a linear Gateaux diflerential 6F(x, h) at every point of a ball D : jlx -

x01\< r. D3. The functional (6F(x, h,), hz) is continuous in x at every point of D. There exists a unique functional f(x) on D, for given f (x,) = fO, and a unique continuous mapping U(x) from /3 into p*, with the properties

(U(x), x -x,1 = 0, (U(xo), h) = 0,

(1.1)

such that

(F(x), h) = (sf(x),

h)+(U(x),

for every x E D and every h E p. The mappings f(x) f(x)=fo+L’(F(

h) and U(x)

(1.2) are given by

x,, + h (x - x,)), x - x,) dh,

U(x) = 1’ h{W( xo + A (x - x,), x - x,) - G*F(xo + A (x - xo), x - x0)1 dh, 0 where 6*F(x,

(1.3)

(1.4)

h) is defined by (6F(x,

h,), h,) = (6*F(x,

hz), hJ.

(1.5)

Definition 1. An operator F(x) is said to have the D-property if it satisfies the conditions D, through Do of Theorem 1. Thus, every operator F(x) with the D-property admits the unique decomposition F(x) = SF(x) + U(x) where U(x) satisfies the conditions (1.1). The operator 6 * introduced by (1 S) provides a basis whereby an operator analogue of exterior differentiation can be defined. Definition 2. The mapping curl F(x, y) of D x p into /3* is defined for any operator F(x) with the D-property by

(curl F(x, Y 1, h) = (6F(x, y) - 6*F(x,

y), h).

(1.6)

Irreversible

We thus

thermodynamics

systems

609

have (curl F(x, h), 8) = (6F(x,

when

of nonlocal

(1.5) is used,

h), g) - (@Xx, g), h)

and hence

curl F(x, Ah) = A curl F(x, h), (curl F(x, h), h) = 0. In particular,

(1.7)

formula

(1.8)

(1.4) can be written U(x) = 1 curl F(xo + A (x - x0>, A (x - x0>) dh.

Lemma 1. Let F(x) have the D-property. Then the following statements are equivalent: 1. F(x) is a potential operator, that is F(x) = 6f(x), 2. (curl F(x, g), h) = 0, 3. For any curve L which lies in D, the curvilinear integral JL. (F(x), dx) is independent of path. In view of the uniqueness of 6f(x) and U(x) in the decomposition F(x) = Sf(x) + U(x) established in Theorem 1, we shall say that U(x) and 6f(x) are the associated operator and the associated gradient of F(x), respectively. Lemma 1 then gives the following immediate result. Lemma 2. Either of the following conditions is necessary and suficient for the associated operator U(x) of F(x) to be the null operator: 1. (curl F(x, g), h) = 0. 2. For any curve L which lies in D, the curvilinear integral JL (F(x), dx) is independent of path. Theorem 2. If F(x > has the D-property, then (F(x),

x - xo) = (6f(x),

x - 4,

(curl (F - U)(x, g), h) = 0,

(1.10) (1.11)

where U(x) and 6f(x) are the associated operator and gradient of F(x). Further, suppose that the following conditions are satisfied: (i) p(x) is a given functional on /3 which has a continuous second Gateaux differential for all x in D and is such that p(x,J = 0: (ii) V(x) has the D-property and is such that (V(x,), then among all operators

h) = 0,

(V(x),

with the D-property (G(x),

x -x0) = 0; there is an operator G(x)

x -xo) =

P(X),

(curl G(x, g), h) = (curl V(x, g), h). Theorem

3. Let F(x)

have the D-property.

(1.12) for which (1.13) (1.14)

There exists a unique mapping f(x, y) of

610

D. G. B. EDELEN

D x D into R, for g(x - y) a given functional D x D into B* such that (F(x),

h) = (kf(x.

Y),

on D, and a unique mapping U(x, y) of

h)+(&f(x,

y), h)+(U(x,

y), h)

(1.15)

for all x, y, in D. where

(U(x, y), y - xo) = 0, (U(x, xd, h) = 0.

(1.16)

The mappings f(x, y) and U(x, y) are given by

f(x>~)=g(x-Y)+

UC.&Y1 =

(1.17)

‘(F(x+(A-l)(y-xo)),y-xo)dh,

1,’A curl

F(X + (A - l)(y - x,,), Y - xo)

dA.

(1.18)

Theorem 4. Let 9 denote the collection of all operators on a Banach space with the D-property and let Xdenote the class of functionals on /3 with a continuous first Gateaux digerential. All pairs {F(x), k(x)}, F(x) E 9, k(x) E .!I”,which satisfy (F(x),

x - x,,) + k(x) 2 0

(1.19)

6f(x) + U(x),

(1.20)

for each x E D are given by F(x)

=

f(x) = f(x0) + I:,’ ( p ( xo + A (x - xo)) - k(xo + A (x - x,)))$,

(1.22)

k(xo) = P (xo), where

p(x)

(1.21)

ranges over all elements of Yf such that (1.23)

P(X)20

and where U(x)

ranges over all continuous

( U(xo), h) = 0,

operators

such that

(U(x ), x - x,,) = o.

(1.24)

Corollary 1. If U(x) E 9 and k (x) has a continuous second Gateaux diflerential, then the symmetry relations (1.25) curl F(x, h) = curl U(x, h) hold whenever p(x) has a continuous second Gateaux differential. Corollary 2. If k(x) = k(x,,) for all x ED, then k(x) 20.

Irreversible thermodynamics

Corollary

611

of nonlocal systems

3. If k(x) = 0, then p (xo) = 0,

(F(xa), h) = 0

and f(x,) is the absolute minimum of f(x) for all x E D. Corollary 4. A necessary and suficient condition for (1.26)

(F(xo), h) = 0 is that (ap(xo),

Corollary

h) = (Sk(xo), h).

5. The following conditions

(1.27)

are equivalent:

(U(x), h) = 0,

(1.28)

(curl F(x, h,), hZ) = 0,

(1.29)

I (F(x), dx) =

(1.30)

1.

4(L)

is independent of the path L. Theorem 5. Let 9 and X be the same as in Theorem F(x) E 9, k(x) E Yt, which satisfy (F(x),x

4. All pairs {F(x),

k(x)},

-xu)+k(x)zO

(1.31)

for each x in a connected set R CD are given by equations p(x) ranges over all elements of 3%such that

(1.20) through (1.22) where

(1.32)

p(x)rO

for each x E 1R, and U(x) 2. GLOBAL

satisfies the same conditions

LAWS OF BALANCE

as those of Theorem

AND THE FUNDAMENTAL

4.

INEQUALITY

The purpose of this section is to obtain a formulation of continuum mechanics whose structure is such that the results of the previous section becomes applicable. The development follows the basic lines of thought presented by Edelen and Laws [6], Edelen et al. [71, and Eringen and Edelen [8] up to the point where thermodynamic,restrictions on the constitutive relations are obtained. The constitutive theory presented here stems from an inequality which differs from the Clausius-Duhem inequality whenever there are nonlocal effects present. Consider an open, simply connected 3-dimensional region of 3-dimensional Euclidean space 8, which is referred to a Cartesian coordinate cover (X”). Let B denote the closure of this region with respect to the Euclidean topology of ZZxand let dB denote the

612

D. G. B. EDELEN

boundary of B. We assume that each material body under consideration occupies such a region B at a given time t = 0. Furthermore, we identify the ‘points of the body’ with the coordinates (X”) of the coincident mathematical points at t = 0. A motion of the body is given by the continuous, differentiable and invertible mapping

xi = Xi(XA,

t).

(2.1)

of B into 8,. We use the notation

anxi= dx’(XB, t)/ax^,

21’= ax’(X”,

t>/at

and demand that J = det (8,~~) be strictly positive for all t. Let pO(XA) be the mass density of the material body at time t = 0, and let p(x’, t) be the mass density in the configuration defined by the motion (2.1). We assume that the mass of any measurable subset of the body is conserved under the motion (2.1): i.e. we assume the local conservation of mass statement

,oJ = po. The remaining mechanical postulates material description, where

are given in terms of global statements

pTiA%,xj = pot”,

and the notation of Truesdell Balance of linear momentum d -1 dt Balance

B

d 1 5.

phAaAx’ = p,h i

for the

(2.3)

and Toupin[ 131 is adopted throughout.

p,v’dV(X)-]aB

T+dS,(X)-j

B

pof’dV(X)=O.

(2.4)

of moment of momentum

d 1 p,xCkvildV(X)ZB Balance

(2.2)

[a_ x’“Til”d&(X)

- lB p,x[kf’f’ldV(X) = 0.

(2.5)

of energy

Po(~ +fviv’)dV(X)

- [

r)B

(viTiA + hA)dSa(X)

- I, po(uf’ + q)dV(X)

= 0.

(2.6)

The localization of these statements will involve localization residuals, as first noted by Edelen and Laws[6]. For example, the balance of linear momentum gives

I

B {pod’ - dAT’^ - pof i }d V(X)

and hence

the local form

equivalent

to this statement

= 0 is given

by

Irreversible

thermodynamics poti’

where

pi is the nonlocal

daT’A

-

or mutual

of nonlocal - POf’ = pOpi

body

force

Similar

arguments

based

(2.7)

(localization

I po_?

d V(X)

B

613

systems

residual)

which

satisfies

(2.8)

= 0.

on (2.5) and (2.6) yield aAx [kT’lA = po(lk’ + x’“fi’),

the localization

(2.10)

- a,hA - poq = po(cj - dj),

~0; - T”aaui where

(2.9)

1*i and 4 satisfy

residuals

1cxi)_ - 0,

I

B

IB

(2.11)

p,l kid V(X) = 0,

pocj dV(X)

(2.12)

= 0;

that is, 1ki is the mutual couple and 4 is the mutual energy source function. interpretations of these residuals is straightforward. Let P denote any sub-body then (2.7) gives

IP since

(p0v i - aaTiA - p&d

~“1’ dV(X)

d j-

ZP

,mi

= I, P,,fi dV(X)

V(X)

= - I,_,

The of B,

PO.? dV(X)

= 0. We thus have

dV(X)-lap

TiA d&(X)-jp

pof dV(X)=-lB_P

PO_?dV(X)

with the obvious implications. Similar results are indicated by (2.9) through (2.12). Clearly, (2.7), (2.9) and (2.10) reproduce the standard local equations when the localization residuals ji, fii and 4 vanish. It is convenient at this point to introduce the Helmholtz free energy by E

so that the local energy

equations

=*+eq,

takes

the form

p&j + 0;l + r/i) - T:&,u~ A standard

sequence

of manipulations

(poil_a,(h_J_~)=_

(2.13)

- aahA -poq

then

= p,,(tj - vi{).

(2.14)

gives

. .

po(r/8 +l+b)+po(cj

-jd)-t

TAaAui

+!$Mx (2.15)

614

D. G. B. EDELEN

Now,

the Clausius-Duhem

inequality

for local theories

is given

by (2.16)

and this can be multiplied

by 0 to obtain

an equivalent

inequality (2.17)

since 8 > 0. The two statements have quite different physical meanings, however: (2.16) states the non-negativity of the internal production of entropy per unit volume at any given time while (2.17) states the non-negativity of the internal production of heat (uncompensated heat of Clausius) per unit volume at any given time. An even larger apparent disparity comes about when (2.16) and (2.17) are integrated over the whole body: (2.16) gives Osl,

(po;, -a,($)-y)dV(X) d porl d V(X) - r ydS,(X)-/B =dt I R _)ilE3

while

ydV(X),

(2.18)

(2.17) gives

=- d 1 P0e77 dV(X)-j-B dt s

h” dSn(X)-/e

(~0~ + +j -!$M))dV(X).

(2.19)

Since local theories demand that (2.18) or (2.19) hold in unaltered form for every measurable sub-body, we recover the equivalent forms (2.16) and (2.17) from (2.18) and (2.19), respectively. On the other hand, (2.18) and (2.19) are inequivalent statements if they are required to hold for the whole body B but not necessarily for any proper sub-body. The nonlocal theory presented here is based upon the axiom,

I, e(pui)-a,(~)-~)dV(X)eO;

(2.20)

that is, the total internal production of heat of any material body at any instant of time is non-negative.? We make this choice for two reasons. The first is because the origins of tIf memory

dependent

materials

are considered,

(2.20) should

be replaced

by the weaker

postulate (2.20a)

If the function space structure introduced in the next section is modified to include functions with domain B x(-m, t] rather than functions with domain B, a full theory of history dependent materials is easily obtained along the same lines as those developed in this paper.

Irreversible

thermodynamics

of nonlocal

615

systems

classic thermostatics of homogeneous phases are heat statements (&AT 2 AQ) rather than entropy statements (AT 2 AQ/&). The second is because (2.12), (2.15) and (2.20) give

(2.21)

that is, 4 drops out since (dldt) JB pO$ dV(X) which

Js p04 dV(X) = 0 and the $-dependence can be analyzed in terms of the Gateaux

d - 1 po$(w(t))dV(X) dt B

= $ (I,

p&w(t)

+ tw(t))dV(X)ll

takes the form differential

lEO.

In other words, (2.21) can be obtained directly from the global statement of balance of energy and the localized statements of balance of momentum and conservation of mass. It is also worth noting at this point that, in practice, the local form of the C-D inequality is usually manipulated into the equivalent form h*

. . - p0(1,//+ T/O) +

Tf'dav' +---da0

2

8

0

before mining the constitutive theory. In the final analysis, the ultimate arbitrator of physical postulates is their relevance in predicting or accounting for the various phenomena recorded in the laboratory. We thus ask the reader to reserve judgment on our choice of the fundamental inequality until after its consequences are developed in the remainder of this paper. 3. STATEMENT

Our purposes al inequality

AND

SOLUTION

in this section

11B

OF THE

are twofold.

FUNDAMENTAL

INEQUALITY

The first is that of recasting

-p,(ljr+_rl~+~v’)+T;4a,u’+~~q~

I

the fundament-

dV(X)rO

(3.1)

as a functional inequality on appropriately defined function spaces. The second is that of obtaining the general solution of the functional inequality by means of the results established in [l l] and summarized in section 1. Let L2(B) denote the usual Hilbert space? with

Ilu(X t)ll’ = (u(XA,

t), v(X”,

I, u WA, t>’d V(X)

t)) = I,

tThis choice of the Hilbert space is primarily number of other choices are possible.

u(X”,

t)v(X”,

one of convenience.

t)dV(X); It will be clear from the context

that a

616

D. G. B. EDELEN

that is, the t-dependence is considered as parametric. ‘Y&‘,,(B) of ordered IO-tuples of L(B)-functions

We then form the Hilbert

lT$x”, t) = {e(x^, t), Xi(XA, t>, &xi(XA, t)} and 93,6(B) of ordered

16-tuples

of L(B)-functions?

which

spaces

(3.2)

includes

9(X”, t) = {B(X^, t), u’(X^, t), a,u’(X^, t), &8(X”, c

the elements* t)}.

(3.3)

We use 9 to denote the neutral elements of these spaces (1j(J]= 0); the scalar products being the lo-fold and 16-fold sums of (J and the squares of the norms being the IO-fold and 16-fold sums of the squares of the L(B) norms. Thus, for instance, if x(~)(t) is a functional on “ur,,(B), then we use the notation (with (Jr0 as inner product in w,“(B))

=

I

B {g(XA, t)&x + gi(XA,

Our first constitutive

assumption

*(%)(r)

t)&x

+ g’R(XA, t)&,.,‘x}dV(X).

is that

= (1, PO&) = 1

*

pn(XA)&)(XA,

r)dV(X)

(3.4)

is a mapping from “ur,,(B) into 92 with continuous Gateaux (Frechet) differentials, where $ = 6(%)(X*, t). This assumption amounts to the usual one that the Helmholtz free energy is determined by 0, xi, and a,xi. We use the notation $(e)(XA, t) to denote the fact that $(X*, t) can depend on functionals of 0, xi, a,xi as well as the values of e(X^, t), xi(XA, t), &x’(X^, t). Thus, $ = &($)(XA, t) denotes the provision for arguments such as kk”(y)(XA, t) =

J

g',"(X",

ZA,

$‘X”, t), c$ZA, t)dV(Z), (3.5)

k?($)(XA,

t) = i,

k:‘(c$(X*,

I,

g,(*I(X*, ZA, YA, %(X”. t), ?(Z^,

t) = .r, g?(X”,

Frtchet differentiation khZ’(XA, t)), is easily variations[l4, 151 and

ZA, w(X”,

t), $ZA,

r), e(YA,

t))dV(Z)dV(Y),

t), kb;7’(%)(ZA, t))dV(Z).

of (1, p,,$), where $(e)(XA, t) = 2(XA, r(XA, handled by the methods of the nonlocal elementary extensions thereof. Cases where

(3.6)

t), k’,“(X*, t), calculus of $($)(X^, t) =

tWith %4,(B) fixed, different choices are available for B\,(B). In fact, the analysis goes through with quite general choices of the Hilbert spaces “N/‘,“(B) and B,,(B). *The inclusion of these elements in Q,,(B) suggests the use of appropriate Sobolev spaces. This is clearly called for in order to establish certain existence theorems which we do not take up in this paper. It is a reasonably direct matter to reformulate our results in terms of Sobolev spaces and we leave this to the reader should the need arise.

Irreversible

thermodynamics

of nonlocal

617

systems

Z’(X^, %(X^, t), k%‘($)(XA, t)) can be handled with the methods developed by Bhatkar [16]. More general functional dependences than those indicated by (3.5) or (3.6) must be investigated individually. On the other hand, if I,!Jdoes not depend on functionals of e!, we write $ = $(e(XA, t)) in (3.4). It is now a trivial matter to see that (3.4) implies

(3.7) We now eliminate J p&d V(X) = J pO$(e)(XA, damental inequality (3.1) so as to obtain

Thus,

if we define

the fundamental

the operator

inequality

F(jJ; w) from

(3.8) becomes

cgj;

t)d V(X) between

B16(B) x ?Yrlo(B) into

the functional

(3.7) and the fun-

Bjh(B)

by

inequality

$!I? jyl6~0~

(3.10)

where (,),, is the inner product in Blh(B). This accomplishes the first purpose. Clearly, the specification of a specific form for the left-hand side of (3.9) is equivalent to assigning constitutive relations for the quantities on the right-hand side of (3.9). In fact, the assumption that c = EQ; e) is equivalent to the constitutive assumption that the quantities on the right-hand side of (3.9) depend on the quantities (6, ui, &vi, &0; 8, xi, a,xi> in view of (3.2) and (3.3). An inspection of (3.3) shows that some of the entries in Y_are spatial derivatives of other entries. The first impulse would thus be to consider variations of some of the entries of y as dependent on variations on other entries as is usually done in a = variational context involving both functions and their derivatives. This however, is a fatal error in constitutive theory as Edelen and Laws have shown [ 171. What is required in constitutive theory is a variation process for which variation and spatial differentiation do not commute. To obtain this is simplicity itself: all that is required is to replace j r by an arbitrary element y= of Bt6(B), solve (FQ: on Bj,,(B)

for fixed but independent

IJES Vol. 12, No. 7-E

$!)? J$ro y, and then

evaluate

the f(y

=

; 2)

so obtained

618

D. G. B. EDELEN

for y = j? In fact, this is exactly what Edelen and Laws [ 173 have shown to be a correct pro
(C(y_; ‘?)? J$620

on 93,6(B) x ‘W,,(I3), where the elements of 7&“,,(B) are considered as independent parameters. Definition 4. The collection of all admissible constitutive relations consists of all admissible preconstitutive relations which are then restricted by y = 2. Remark. An inspection of (3.2) and (3.3) shows that an element of “ur,,)(B) determines a unique corresponding element j of BIIh(B). The restriction y = 9 thus correlates the entries of y with the entries of 4 by the indicated differentiation brocesses so that all entries of 9 and y are obtained from the underlying variables 0(X*, t), xi(X”, t>. The functional inequality (3.11) is of exactly the form covered by Theorem 4 and 5 with k(y) = 0. We thus obtain the following conclusions when Definitions 3 and 4 are used and we make the identifications x0 = G, !Blh(B) = 93, and D is all of %‘1h(B). Theorem 6. All admissible constitutive relations have the form

(3.13) where (i) p(y; e) ranges over all functionals on !43,4B) x WlO(B) which are continuous in y,and y, h&e a continuous Fre’chet diflerential with respect to l and are such that

PQ;~)~o~

(3.14)

p($~)=o

and where &J(y ; 5,) ranges over all operators with a linear Gateaux differential such that (3.15) Remark. Clearly,

the collection

{ti(X^,

of all elements

t>, Wi(XA, t), a,wi(X^, t), a4u(XA, t)}

of LZZI16(B) forms a subspace, a, of 93I316(B),and hence thus be used to replace the requirements

fl is connected.

Theorem

5 can

(3.16)

Irreversible

Thus,

thermodynamics

of nonlocal

systems

619

since (3.17)

by (1.17) and (1.21) (recall that k(;;

CLJ)= 0), this has the effect of replacing

(c(y*;_r e), y*),hzo =

(3.11) by (3.18)

v y*En. r

Hence (t(y; $), y)l, need be non-negative only for those elements of c+B,~(B) which arise from ‘some dlement CLJ*of W,,(B). The condition (3.18) is still more restrictive than (< (2 ; e,), g),, 2 0 since (3.18) must hold for all % E W,,,(B), I* E Sz rather than just those for which 9 and x are correlated by the requirement y* = 2. Note that satisfaction of the correlation condition y= * = 2 is obtained by u(X”, i) = 0(X”, t), w i(XA, t) = i-(x”, t) = vi(XA, t). 4. PROPERTIES

OF

ADMISSIBLE

CONSTITUTIVE

RELATIONS

If we set f(y; $) = (l,f(y; $))(= J$(y ; y)(X^, t)dV(X)), a straightforward combination of (3.3),=(3.9), (3.12) kd (3.15) givk the following explicit forms of admissible constitutive relations:

(4.1)

(4.2)

from

(3.12); (U(~;~),e)+(u,(~;y),vi)+(UA(4;~),

aAui)+(U”(~;~),aAe)=O,

UC?; e) = CL@; $) = UAQ; y) = U”Q;

(4.5)

e’) = 0

(4.6)

from (3.15); and f(jj; y) is given by (3.13) subject to the conditions (3.14) on the functional p (y ; 13) E=X. If & depe;ds on e but not on functionals of 9, then E&,6= $I,& Similarly, if f(P ; to) depends on 2 and e but not on functionals event, the admissible constitutive relations become

Tq = These relations

involve

p,&&l$ + a,,,$ +

the familiar

terms

u:,

$$(o)

II* =

of 2 and %, then

sf = $f

In this

ea,,Bf+ WA.

of local thermodynamic

theories.

The

620

D. G. B. EDELEN

less

familiar

terms

sf(j;

9,)

=

are

also

known,

. ye) for heat efflux and derivatives

&(ky0

at

least

of the Rayleigh

in

linear dissipation

problems; function

for Aviscous stress. The existence and physical interpretations of the markedly unfamiliar terms F((j; e) have been given in 111 and 131 for corresponding finite dimensional theories, to ghich the reader is referred. If G(e) depends on 9 through functionals of the form ki(X^, t) given by (3.5) then [6,7]

where (PO+)* = j-B &^)-$(Z”.

CL

r)gh”(Z“‘, XA, y(Z”,

r), e(X^,

t))dV(Z).

In this case, the terms in (4.1) through (4.3) involving &,wp& replicate the results given by Edelen and Laws [6], Edelen et al. [71 and by Eringen and Edelen[9]. We refer tof(l; G) as the dissipation potential? since (l.lO), (1.12) and (1.13) give c (C($

$j), +=

(GY.f($ 2)* +=

p(Y; 9). z

(4.7)

In fact, the results quoted in section 1 and (4.7) give the following result. Corollary 6. Every material body with admissible constitutive relations has a dissipation potential f(E; y) which is uniquely determined by c(g ; $‘) to within a functional

of y alone. In particular

Clearly, LJ(i; e) is the nondissipative part of c(z; y) since the total internal generation of heat due to LJ($ ; CLI)is given by (U(E ; 2); $)lh, as follows from the equality of (t(j; y), $),h and the left-hand side of (3.1), and thii functional vanishes for all i and $ by (4.5) (by (3.15)). Corollary 4 together with the following theorem thus con&tute a full extension of the results of the local (finite dimensional) case established in [l]. Theorem 7. Every material body with admissible constitutive relations obtained from a functional ~(2; y) with a continuous second Gateaux differential admits the symmetry relations curl c((j; $), h) = curl y((i; 9), r_t), (4.9) I6 I6 where c? is held fixed as indicated by the notation curl. Proof. The results are an immediate consequenckh of Corollary 1. Remark. The constitutive relations (4.1) through (4.3) can also be solved for &,p,,&. The nonlocal form of Maxwell’s reciprocity relations then follow from the fact that curl ({Yr, 5) = 0 since ‘I”($) is assumed to have a continuous second Gateaux differen10 tial. tThis

definition

corresponds

with that given

in [I] for the finite dimensional

case.

Irreversible

thermodynamics

of nonlocal

systems

621

The symmetry relations (4.8) and the operator ‘curl’ are such an integral part of the theory that we pause at this point in order to derive explicit representations. A straightforward combination of (1 S) and (1.6) gives

(curl Ih (4.10) If C depends on y, but (curl F((;; $!,)>y, csh6 Ih

where tions

a,, = alay,,.

Since this statement

(4.8) are equivalent &(FQ;

which

were

established

to the local y) - wx;

!

k

is of class ew>

z;

=

=

$,)I

(4.11)

t))dV(Z),

(4.12)

kX

by

t); $!,(x, t), $,(Z, t)>

i.e. interchange X* and Z*, then (curl F((Y; 0~). h), sh6 =

rela-

the dependence

C’ in y(X, t), y(Z, t), and we introduce = =

=

16 =

y) - UIQ;

Z: x(X, t), x(-Z. t); %(X, t), y(Z,

~CX t>, $z

the symmetry

rel&ns

y)) = &\mQ:

in [l]. If c exhibits

F4x; $!‘)= B k(X, where

holds for all (&, g) E 930), symmetry

&(Z, x; y,(Z,

(4.13)

(4.10) gives

=

(4.14)

D. G. B. EDELEN

622

We thus have cu$ CA((~; %‘),k) = &,;(2

r)-$$j]h~(X, aR.4

+

t)dV(Z)

3

(4.15)

a&

3yr(Z, t) - 8y,(X, t) 1hlG

t)d v(z)

in this case, and in particular, c$

$(@;

$!), ?) = jB ( ay;(% t)-

aya(% &l-(X,

3

A

9

t)dV(Z) (4.16)

f

a%

a&

~YlGz

t) -

ay4x,

Since a(& + &Qlayr(X, t) = a(&, + I?:X)/ayr(Z, t), can also rewrite (4.14) in the equivalent form (c$ c(Q; y), k), r;b =

III3 B

bs(X,

t)kr(X, t) + g,(Z, t)k,(X,

f)

and

yr-(Z, t )d V(Z).

(E, + I?)* = I?, -t 2~

we

t,]

(4.17) (R,+&?-

ay2\ &

.

,+kr- t- k?)}dV(Z)d

Thus, (curl F((x; %), II), g),e = 0 for all %,gz% E %(B) 16

V(X).

if and only if 4 RF).

Accordingly,

if v(x; v) can be represented

U.2Q: %) = f

B

then the symmetry

ay,(;

fik(X^, ZA l

$X”

in a form similar to (4.12)

= , f>, VU*,

f>: %‘(X‘+,

relations (4.8) assume the equivalent ,!(&

+ & - $2, - ii?) = av,

,

(4.18)

-1

f), ytZA,

t))dV(Zh

(4.19)

local form

(“xt)(Kr + RF - I%- A?>. .

(4.20)

Definition 5. The set S of all thermostatic states consists of all elements of vlo(B) for which 5 = {constant, xi(X”), Clearly S is a subspace of ‘w,,(B). Lemma

3. The set S is characterized

by

&xi(X”)} “E ys,.

Irreversible thermodynamics

623

of nonlocal systems

(4.21)

S=G = for all t. Proof. Equations

(4.2) and (4.21) give

&X^, t>= ii(X^, t> = a,mi(x”, t> = a&(x”, whose

solutions

for all t are 8 = constant,

t) = 0,

x i(XA, t) = xi (X”),

and a,xi(X”,

t) =

&jxi(X^).

Theorem

8. The admissible constitutive relations reduce to

(4.22)

for every thermostatic state and f(jJ; $1 attains its absolute minimum with respect to jt only in thermostatic states. Proof. The results (4.22) are a direct consequence of Corollary 3. This corollary also shows that f(y_; %) has its absolute minimum with respect to 2 (0;’ fixed) at y, = 9.

However,

Lemma 3 shows that x = 9 only when $ = ySt. In view of (4.22), this theorem proves that the theory presented here is consistent in all respects with the predictions of thermostatics; simply assume that 1+6(%)is an ordinary function of $ so that &1,6= &I+& Remark.

Theorem relations :

9. The following conditions are equivalent for all admissible constitutive (9

(ii) c(y =; z> are potential operators ; that is (cu$ c(Q;

(iii)

IL

(c(x;

%), &y &>M = 0, CC; ; y) = Gyf(;; +j!),

(4.24)

(4.25)

$), dy) = 4(L)

is independent of path. Proof, The results are an immediate

consequence of Lemma 1 and Corollary 5. By direct analogy with the results established in [3,4] for local theories, it is clear that admissible constitutive relations provide descriptions of both local and nonlocal material bodies which exhibit: elastic, viscoelastic, fluid and plastic behavior in the presence of heat conduction. Extension to more complex situations involving multiple component bodies which experience chemical reactions in the presence of electromagnetic interactions is reasonably straightforward. 5. SYMMETRY,

TIME REVERSAL,

AND THE ONSAGER

PROPERTY

We showed in the last section that the symmetry relations (4.24) are satisfied, for c:
(5.1)

624

Further,

D. G. B. EDELEN

if F(y:

y) has the form

(4.12), then

(4.24) holds

if and only if

(5.2) Since (4.24) imply (LJ(y ; 2), h),, = 0 (Theorem (3.12), reduce to =

9), the admissible

constitutive

relations,

(5.3) When the equivalent forms (4.1) through (4.4) are used, (5.1) are seen to be the nonlinear forms of Onsager’s reciprocity relations which were first introduced by Gyarmati [ 181 and rediscovered in [ 191 and [20]. Thus, fluxes 4(x ; %)(= c(r : e)) in a local theory are Onsager fluxes if and only if J(y ; 2) is a potential operator ?$(yc; e). The similarity of this result with the results established in Theorem 9 moticates the following definition. Definition 6. Admissible constitutive relations have the Onsager property if and only if c(x; 9’) is a potential operator on L%,,(B). Theorem 9 thus has the following corollary. Corollary 7. Admissible constitutive relations have the Onsagerproperty if and only if any one of the following equivalent conditions is satisfied: (9 (YQ:

(ii) is independent

%), r-r>16 = 0,

lL(FQ;

$1, dxh6 = 4(L)

(5.4) (5.5)

of the path L.

(5.6) (iii) (curl F((y; w). h,), &)M = 0. Ih = = = = The importance of the Onsager property, from the mathematical point of view, is the large literature of results on potential operators which is thereby applicable (see, for instance, Chapters III and V of [12]). In fact, if (c(y ; $), Y),~ >O for y + c, then Theorem 16.2 of [12] and the previous assumptions cokerni
%&(y ; $) = /.Ly.\

(5.7)

have a continuum of solutions whose norms exceed any given numberprovided only that ((93:)) is a completely continuous linear operator on B,,(B) into %I,h(B). Needless to say, results of this nature are extremely useful when it comes down to solving the equations of balance with specific admissible constitutive relations. The original papers of Onsager[21] showed that invariance of the underlying microscopic systems under time reversal implied satisfaction of the Onsager reciprocity (symmetry) relations for resulting linear macroscopic systems. Recently, Gurtin [23] has derived similar results under linear approximations of the constitutive relations for materials with memory. We shall now show that admissible constitutive relations admit comparable results, in which there is no restriction to linear approximations.

Irreversible thermodynamics

of nonlocal systems

625

As before, we use S to denote the subspace of w,,(B) which is comprised of all thermostatic states. Let a and b be real numbers with a < b and let (Y and /3 denote arbitrary real numbers such that (Y< p, a (min(a,-P), Let y,,, denote a continuous

b rmax(p,-a).

(5.8)

mapping from [a, b] into Ur,,(B) (a curve) with

+(XA, t) = %$‘(XA) for a 5 t 5 a, (5.9)

and such that the corresponding curve y(t) = j(t) obtained under the identifications (3.2) and (3.3) is contained in %,6(B). S&e 2 i%defined by (3.3), the conditions (5.9) become

(5.10) j(X”,r)=c c

for

Pstlb.

We use yz6 to denote the corresponding curve {j(t), $(t)} in %16(B) X “Mr,,(B) which maps (9; %I:‘) onto (0; ez’). The collection of all such curves yz6, where %$’ and $2’ range throughout S,& denoted by I. Finally, let Aylo denote the time reversal of ylo defined by hCO(XA, t) = $(X^, -t) and let AyZ6 denote the corresponding identifications (3.2) and (3.3); i.e. fq

= {-

(5.11)

curve in CB16(B)x ‘V,,(B) that is induced by the

b(X^, - t>, - vi(XA, - t>, - i3AUi(XA, - t>, a*e(XA, - t)}.

In view of the inequalities (5.8) it is easily seen that A is a mapping of I onto I. In particular yZ6is a curve connecting {CJ;$$‘(X*)} with {CJ;%?(X”>} and AyZ6 connects {g; I} with (9; $j?(XA)}. We have based our constitutive theory on the postulate that the total internal production of heat of any material body at any instant of time is non-negative, for time running forward of course. If time runs backward, the total internal production of heat should be of fixed sign. Satisfaction of this condition of macroscopic theories is taken as one of the fundamental postulates of macroscopic (Onsagerin) irreversible thermodynamics of Meixner [22] and many others. The following theorems show that this expection is realized for admissible constitutive relations with the Onsager property and that a partial converse also holds. Theorem 10. If an admissible system of constitutive relations with associatedgradient &f(x;

y) has the Onsager

property and if $_,f(jj; CIJ)exists, then the functional e

626

D. G. B. EDELEN

on 316(B) X wlo(B) X [a, hl is odd under time reversal for all -yznE r; that is

v yx

5(Y26)= - 5(Ayx) Remark. Most previous theories c = c(x), (3.13) and (3.12) give

so that (=8-f
have

not

r.

allowed

i $!)y@)IO= (cwf(y), $)IO=&f(o). j-‘ ($uf(~)&9w=/y26 YX

E

Since

(5.13) ; 9)

to depend

y10 begins

and ends

c(y

=

on y. If

in S,

$f(o)=f(@)-f($:‘)

while

since AyZh starts assigned

at 22’ and ends at o’,:‘. However,

arbitrarily.

We may thus

view

the functional

the additional

f(%) in (3.13) can be

(&fQ; %I,deh as

term

I YZh compensating for the arbitrary functional f($) when c = F(y). In the general case, this additional term compensates for the fact that e is now allowid to vary, in contrast to all of our previous considerations in which all calculations of variations associated with f occurred with e fixed. Proof of Theorem 10. Since E(T; 01’) is assumed to have the Onsager property, c(y;= 0) = g,f(y; CLJ),and hence (5.12) gives {(&f(& 2)> df)lfi + (&f(j; S(YX) = j726

$,)> d+jhol (5.14)

= I

Y2h$f(&

+t

since yZh starts at (0; %5?> and ends However, (3.13) shows that

=f(G; @)-f($

&I:‘)

at (G; 9::‘).

f@ $I= f(%,> and hence

(5.15)

(5.14) gives S(rzn) =f(c$‘)-f(y::‘).

(5.16)

Similarly,

(5.17)

since hyZ6 starts at (0; %I?‘) and ends establishes the theorem.

at (0; $Li:‘). A combination

of (5.16) and (5.17)

Irreversible thermodynamics

627

of nonlocal systems

The essential ingredient of the proof is that the functional Lj(yTh) is independent of the path, as shown by (5.14). In fact (5.14) shows that 5(yz6) = - E(h*yz6) where A* is any mapping of I onto I which interchanges end points of elements of I. The reason why the curves I start and stop in B’,,(B) at 9 (see (5.10)) is because time reversal does not exchange end points of curves in Bt6(B) unless the end points are both the element c. It is also clear that a much wider class of curves can be employed than those which comprise I. We thus have established the following result. Corollary 8. Let 9 be the subspace of 6!ZI16(B)whose elements are dynamically correlated with elements of W&B) by 2 = 3, and let I’* be the collection of all curves -& in P? x W,“(B) which are diferentiable in phrametric form. If an admissible system of constitutive relations with associated gradient $f (x, . T) has the Onsager property and &f(y_; 2) exists, then the functional t(yg) defked by (5.12) satisfies

5($6) = - m**) for all ~2 E r* and for any continuous mapping A* of points of elements of T*. Further, the functional E(y$) and the families of curves I’* in $7 x W,,(B) can be W,,(B) upon eliminating the restriction y = j. The converse of Theorem 10, in the mori general easily obtained. Theorem 11. If the functional

(5.18) I’* onto T* which exchanges end is independent of the curve r&, replaced by curves in B16(B) x setting

given

in Corollary

8, is

(5.19) on all smooth curves 72 in 93,,(B) x W,,,(B) is independent of the curve only on the end points of y$), then the admissible constitutive relations associated gradient 6,f(y; e) have the Onsager property. Proof. The hypothesrs concerning the independence of 5(y2*6) on the satisfied if and only if there exists a differentiable functional g(y; %) E W,,(B) such that 5(G) Hence,

= 1

(i.e. depends c(y ; e) with c curve $$, is on 9$,6(B) x

* d&y_; %I. 726

(5.19) gives

(5.20)

Since

this is to hold for all smooth

curves

in 93,,(B) x W,<,(B), we have (5.21) (5.22)

628

D.G.B.EDELEN

The associated gradient and associated vector of c(y ; CO)being unique, since c(y ; r) satisfies the hypotheses of Theorem 1 from the fact that c(x ; $) gives admissible=constitutive relations, 6,gQ; CIJ,> = i$f(;; y). Hence c(x; 5) has the Onsager property. Remark. This theorem provides the extension of the results of Theorem 9 to situations in which y is also allowed to vary; i.e. the extension from paths in cB,~(B) with fixed 9 E W,,(B) to paths in CB,6(B) x W,,(i3). Corollary 9. If the functional (5.12) is odd under time reversal for all yZhE r, then the admissible system of constitutive c(z; y) with associated gradient GYf(g; e) has the Onsager property on the subspace P. Proof. The collection r restricted to $B,6(B) contains all possible smooth closed curves in the subspace ?? given by y = j and all smooth curves in Wlo(B) and hence every two points in LP can be connecfed b=y a piece of an infinite number of element of r and every two points in cUrlo can be connected by an infinite number of elements of r. Thus, since [(Y?~) = - [(A%J implies &(yZ6) + c(hy2J = [(yZh + Ayz,) = 0, where yZ6+ A yZ6 is the ordered curve t E [a, b ] on yZh and t E [b, 2b - a] on A yZhr we have that the integrals around all closed curves r + AT vanish. This implies that t(yz6) is independent of path in B x W,“(B) and the result follows from Theorem 10. Although the results established above provide agreement with previous theories based upon the properties of systems under time reversal, they do not provide a means whereby the Onsager property can be checked in the laboratory. This is because time reversal of a macroscopic, nonconservative system can not be generated by any concatenation of techniques currently available in the laboratory. However, if we set d$C = $= dt, dw= = ~j dt in (5.12), we have c y(t)),

j(%+(&f(t(t);

9(t)),

+)),o}dt

(5.23)

where the curve y26 is parameterized by the time variable t. If the curve yZh starts at t = to in an equilibrium state WI:’ and terminates at t = t, in an equilibrium state w$‘, the above analysis gives t-(YX)[to, t,1 = f($‘)

if F has the Onsager property. Further, a constant functional on the subspace

since f(w) can be specified, we can take f(w) as of static states. This gives

t(Y*h)rtO,

It is then a direct consequence property if and only if

- f(w6:‘)

of Theorems

t,1 = 0.

(5.24)

9 and 10 that F(F; e) has the Onsager CZ

(5.25)

holds for every time dependent process x(t), %,(t) which connects all pairs of equilibrium states WI:’ and ~2’ at times to and t,, respectively. Verification of this condition is at least a possibility in the laboratory since all quantities which enter in (5.25) occur in a form which is potentially measurable.

Irreversible thermodynamics 6. REMAINING

629

of nonlocal systems QUESTIONS

The thermodynamics of material bodies with global laws of balance is now complete. There are, however, two remaining problems which must be resolved before a complete theory of such bodies is obtained. The first of these problems is that of securing invariance of the theory under superimposed rigid body motions. For theories in which c(y ; 2) and $(e) depend on functionals K,, and kbl’, the results obtained by Edelen et al. [17],Eringen and Edelen 191 and Eringen [lo] can be used. On the other hand, if more general functional dependences are involved, there are no specific concrete results which are available in the literature. The second problem is that of obtaining satisfaction of the zero mean conditions (2.8), (2.11) and (2.12) for the localization residuals $, Iki and 4. We note that Iki is by admissible constitutive reladetermined by (2.9) since fi and TiA are determined tions, while 4 is determined by the local energy equation (2.14) when 4 = 0. The zero mean conditions would appear to place restrictions on the choice of f(y ; e) and y(y ; e) in the admissible constitutive relations. This is not all together the Ease if we ha& already solved the problem of invariance under superimposed rigid body motions as is shown by the results of Edelen et al. ([17], section 4). It would also appear that similar results obtain under quite general functional dependences of & and CQ; y). e.g., (3.7) can be written (1, PO+) = (s,$, If this equation

is localized,

+>10.

(6.1)

then

residuals where vR($ : d), r,g(cg: vi) and raAxi($ : a,vi) are localization (functionals) of the arguments following the colon: that is (1, rR(% : 6)) = (1, r-2(% : v’)) = (1, r,,,i(y Now, if 6 is invariant consider a superimposed give

under superimposed constant translation

which

are linear

: a,u’)> = 0.

rigid body motions, then velocity a’, the invariance

(6.3) so iz 4. If we of I,!Jand (6.2)

a ‘6,fp0~ + a irxt($ : 1) = 0 in view of the linearity B and (6.3) yield

of r,, (we : ~1i, in vi. Since (6.4) holds for all a’, an integration (1, SxapO$) = - (1, I;#(+ : 1)) = 0

When the admissible (2.8), we obtain

o=

constitutive

I

B

(6.4)

relations

p,$ dV = (1, p$)

are substituted

= - (1, &p,$)

V

5

over

(6.5)

into the zero mean condition

- (1, SJ~) - (1, Ui).

(6.6)

630

D. G. B. EDELEN

. Thus, (6.5) shows the invariance of 4 under superimposed rigid body motions implies that the contribution - (1, G,fpo&) to (6.6) from $ vanishes identically. It appears that invariance of f(j; y) and (E, x),6 under superimposed rigid body motions guarantees satisfaction of th=e remaining parts of the zero mean conditions. The calculations in the general case are of such length and sufficiently delicate that we leave conclusive proof of this important conclusion to a future paper. [1] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

REFERENCES D. G. B. EDELEN, Arch. ration. Mech. Analysis 51, 218 (1973). D. G. B. EDELEN, Znt. J. Engng Sci. (in press). D. G. B. EDELEN, Znt. J. Engng Sci. (in press). D. G. B. EDELEN, Arch. Mech. (Warzawa) (in press). D. G. B. EDELEN, Adv. Chem. Phys. (in press). D. G. B. EDELEN and N. LAWS, Arch. ration. Mech. Analysis 43, 24 (1971). D. G. B. EDELEN, A. E. GREEN and N. LAWS, Arch. ration. Mech Analysis 43, 36 (1971). A. E. GREEN And N. LAWS, Arch. ration. Mech. Analysis 43, 45 (1971). A. C. ERINGEN And D. G. B. EDELEN, Znt. J. Engng Sci. 10, 233 (1972). A. C. ERINGEN, Znt. J. Engng Sci. 10, 561 (1972). D. G. B. EDELEN, On decompositions of operators and solutions of functional inequalities (to be published). M. M. VAINBERG, Variational Methods for the Study of Nonlisear Operators, Holden-Day, San Francis0 (1964). C. TRUESDELL and R. A. TOUPIN, Handbuch der Physik ZZZ/l. Springer (1960). D. G. B. EDELEN, Nonlocal Variations and Local Invariance of Fields. American Elsevier (1969). D. G. B. EDELEN, Znt. J. Engng Sci. 8, 517 (1970). V. P. BHATKAR, Znt. J. Engng Sci. 9, 871 (1971). D. G. B. EDELEN and N. LAWS, J. Math. Anal. Appl. 38, 61 (1972). I. GYARMATI, Period. Polytech. 5, 219 (1961; ibid. 5, 321, 1961). G. OSTER, A. PERELSON and A. KATCHALSKY, Nature, Lond. 234, 393 (1971). D. G. B. EDELEN, Znt. J. Engng Sci. 10, 481 (1972). L. ONSAGER, Phys. Rev. 37, 405 (1931; ibid, 38, 2256, 1931). J. MEIXNER and H. REIK, Handbuch der Physik ZZ1/2. Springer (1959). M. E. GURTIN, Arch. ration. Mech. Analysis, 44, 387 (1972). (Received

12 September

1973)

R&sum&Les lois g&&ales d’bquilibre de la quantitC de mouvement, du moment de la quantite de mouvement, et de l’tnergie, de mtme que la conservation locale de la masse, sont ramenCes a des descriptions locales comportant des valeurs rtsiduelles de localisation. Une inegalitC fonctionnelle fondamentale est alors obtenue qui se ramene ?t l’intgalitt de Clausius-Duhem pour les theories locales. Pour les theories non locales,l’intgalitC fonctionnelle stipule que la production interne totale de chaleur d’un corps matCriel g n’importe quel moment est non-negative. Cette inCgalitC est formulCe g nouveau en termes d’une inCgalitt fonctionnelle sur un espace de Hilbert de collections ordonnCes de fonctions L2. La solution g&n&ale de cette inCgalitt! fonctionnelle est obtenue et ceci conduit a toutes les solutions constitutives admissibles. L’existence de potentiels de dissipation et des relations de symBtrie sont ttablies pour des corps mattriels avec des relations constitutives admissibles. Les relations analogous non-locales des relations de rCciprocitC de Maxwell sont Cgalement obtenues de m&me qu’une preuve de cohtrence avec les rtsultats de la thermostatique. II est montrC que si les formes non-IinCaires des relations de rCciprocitC d’onsager sont satisfaites, cela Cquivaut a la condition que les operateurs gCntrant les relations constitutives admissibles soient des operateurs potentiels. II est ainsi montrC que certains fonctionnels de courbes dans un espace de fonctions sont impairs dans une inversion temporelle si et seulement si les formes non-lintaires des relations de rCciprocitC d’onsager sont satisfaites. Ainsi, les resultats de Gurtin pour des procCdts qui peuvent &tre approximCs par des &arts IinCaires B partir de l’tquilibre sont ttendus g tous les pro&d&s avec des relations constitutives admissibles. Des rtsultats semblables sont Ctablis pour des systbmes historiques sensiblement noins restrictifs que ceux utilists par Gurtin et pour une large classe de gCnCralisations d’optrateurs d’inversion temporelle sur des suites historiques semblables. Des indications sont donntes sur le fait que la satisfaction de I’invariance sous l’effet de mouvements superposCs d’un corps rigide implique la satisfaction des conditions moyennes de ztro pour toutes les valeurs rCsiduelles de localisation. Zusammenfassung-Globale lokaler Massenerhaltung, einbeziehen. Es wird dann

Gesetze von Momentgleichgewicht, Impulsmoment und Energie, zusammen mit werden auf Punktfeststellungen reduziert, die LokalisierungsriickstHnde eine funktionelle Ungleichheit erhalten, die sich auf die Clausius-Duhem’sche

Irreversible

thermodynamics

of nonlocal

systems

631

Ungleichheit fur lokale Theorien reduziert. Fur nicht-lokale Theorien fordert die funktionelle Ungleichheit, dass die totale interne Warmeerzeugung eines Materialkorpers zu jeder bestimmten Zeit nicht-negativ ist. Diese Ungleichheit wird in Ausdriicken einer functionellen Ungleichheit an einem Hilbert-Raum geordneter Sammlungen von L, Funktionen neu formuliert. Die allgemeine Losung dieser funktionellen Ungleichheit wird erhalten und dies ftihrt zu allen zuhissigen konstitutiven Beziehungen. Das Bestehen von Zerstreuungspotentialen und Symmetriebeziehungen wird ffir Materialkijrper festgelegt, die zulassige konstitutive Beziehungen haben. Nicht-lokale Analoge der Maxwell’schen Reziprozitatsbeziehungen werden such erhalten, wie such Bestltigung der Konsistenz mit den Resultaten der Thermostatik. Es wird gezeigt, dass die Zufriedenstellung nicht-linearer Formen von Onsager’s Reziprozitatsbeziehungen der Forderung gleichwertig ist, dass die Operatoren, die die zulassigen konstitutiven Beziehungen erzeugen, Potentialoperatoren sind. Es wird such gezeigt, dass bestimmte Funktionale von Kurven in einem Funktionsraum unter Zeitumkehrung ungerad sind, wenn, und nur wenn, die nicht-lineare Form von Onsager’s Reziprozitatsbeziehungen befriedigt wird. So werden Gurtin’s Resultate (10) fur Verfahren, denen man sich durch lineare Abweichungen vom Gleichgewicht annahern kann, auf alle Verfahren mit zulassigen konstitutiven Beziehungen ausgedehnt. Ahnliche Resultate werden fur bedeutend weniger einschrankende Satze von Geschichten aufgestellt, als die von Gurtin verwendeten, und fur eine breite Klasse von Verallgemeinerungen des Zeitumkehroperators an solchen Geschichten. Es werden Andeutungen gegeben, dass Befriedigung von Invarianz unter iiberlagerten Starrkorperbewegungen die Befriedigung der Nullmittelbedingungen fur alle Lokalisationsriickstande bedeutet. Sommario-Le leggi globali de1 momento, de1 moment0 de1 moment0 e dell’energia, insieme con la conservazione locale della massa vengono ridotte a dichiarazioni in un punto implicanti i residuali della localizzazione. Viene quindi ottenuta un’ineguaglianza funzionale fondamentale the si riduce all’ineguaglianza Clausius-Duhem per le teorie locali. Per le teorie locali l’ineguaglianza funzionale stabilisce the la produzione interna totale di calore di un corpo materiale in qualunque istante dato e non negativa. Questa ineguaglianza viene riformulata come una ineguaglianza funzionale su uno spazio Hilbertiano di collezioni ordinate di funzioni L1. Viene ottenuta la soluzione generale di questa ineguaglianza funzionale e cib porta a tutte le relazioni costitutive ammissibili. Viene stabilita l’esistenza di potenziali di dissipazione e di relazioni di simmetria per corpi materiali con relazioni costitutive ammissibili. Vengono inoltre ottenuti gli analoghi non locali delle relazioni di reciprocita di Maxwell come prova della consistenza coi risultati della termostatica. Viene dimostrato the il soddisfare le forme non lineari delle relazioni di reciprocita di Onsager a equivalente al requisito the gli operatori generanti le relazioni costitutive ammissibili siano operatori potenziali. Viene pure dimostrato the certi funzionali di curve nello spazio di una funzione sono dispari su inversione de1 tempo se e soltanto se la forma non lineare delle relazioni di reciprocitl di Onsager e soddisfatta. I risultati di Gurtin (10) per processi the possono venir approssimati da deviazi oni lineari dalla posizione di equilibrio vengono estesi a tutti i processi con relazioni costitutive ammissibili. Risultati simili vengono stabiliti per insiemi di storie significativamente meno restrittive di quelle usate da Gurtin e per una vasta classe di generalizzazioni dell’operatore d’inversione de1 temp in tali storie. Vengono date indicazioni the il soddisfare l’invarianza con superimposti moti di un corpo rigid0 implica il soddisfare le condizioni medie di zero per tutti i residuali della localizzazione. 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