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Theory of structured continua I. General consideration of angular momentum and polarization$ BY J. S. DAHLER AND L. E. SCRIVEN Department of Chemical Engineering, University of illinnesota, AMinneapolis,Minnesota (Communicatecl by 111. J. Lighthill, F.R.8.-Received 12 AMarch1963) Classical continuum theory is generalized to provide a systematic procedure for treating in successively greater detail the macroscopic manifestations of subcontinuum events without sacrificing the convenience of the field approach. The point of departure from traditional theory is the general angular momentum principle for continua. Internal angular momentum is associated with configurational and kinematic structure of the continuum 'particle' and that structure is represented by moment expansions in polyadic polarization densities. Each step in the theoretical development is analyzed both in terms of the statistical mechanics of polyatomic media and from the standpoint of continuum physics. Accurate accounting of content and flux is found to require introducing the concepts of proper densities ant1 currents. Among topics examined in detail are asymmetric states of stress, equations of change for dynamical variables, and equilibrium conditions in structured continua, including alinement of internal angular momentum with the principle axes of local mass distributions. The central problems in devising continuum descriptions of the behaviour of real materials seem to be these: (a) Translating the basic principles of elemental physics into the language of the continuum. The key lies in the choice of system, that is. in distinguishing system from surroundings, classifying the possible interactions of the two, and picturing the interactions in terms of intermediary fields. (b) Representing macroscopic properties which originate in the internal structure of subcontinuum units, for example, in molecular configurations and changes of configurations. A solution is provided by assigning point structure to the continuum. (c) Characterizing those aspects of macroscopic behaviour which arise in the ordering or lack thereof in aggregations of subcontinuum units, be they atoms, molecules, or larger. The solution here lies in assigning to the continuum a spatial structure specified, for example, by a group of symmetry operations or by a collection of topological properties. (d) Formulating admissible constitutive relatioils among the variables that emerge as describing the static and, especially, the dynamic states of macroscopic systems. Here physical experience abstracted in the familiar second law of thermodynamics and in the still unfolding general principles of modern rheology (Truesdell & Toupin 1960) are the keys. Finally, there is the problem of evaluating the phenomenological coefficients in the constitutive relations either by statistical calculations or, more often, by experimental measurements. - .; This research wab supported in part by grants from the Sational Science Foundation. [ 504 ] T h e o r y of structured c o n t i n u a . I 505 We are here concerned with the first two problems. They are not separate problems; for unless the fine structure of matter is properly taken into account, the translation of basic principles cannot be complete. From the conventional, undiscriminating viewpoint continuous media are dense collections of point masses-concentrated, infinitesimal masses devoid of internal structure. Inherent in this viewpoint are drastic limitations on the extent to which continuum descriptions of macroscopic behaviour can successfully mirror the fine structure of matter. The limitations become more acute as more resned, more complete descriptions of material behaviour are sought. They can be overcome by commanding a higher point of view, from which continuous media are sets of structured particles-elementary distributions whose internal structure can be accounted for with point functions of state. The dipole moment densities of continuum electrodynamics and the material directors of rod and shell theory foreshadow the way in which knowledge of microscopic structure can be incorporated into the continuum theory of macroscopic systems. The possibility of formally extending the classic notions of continuous media seems to have been noted first by Duhem in 1893 and by the Cosserats shortly thereafter, as mentioned by Ericksen & Truesdell (1958). With the evolution of molecular and statistical physics in the intervening years the physical basis an6 the practical justification for doing so have become clearer. We outline a part of the extension here. Our procedure is to advance the theory of structured continua by steps, pausing to examine each in the light of statistical descriptions of the underlying molecular phenomena. The degree to which the approaches of continuum and statistical physics illuminate and complement one another is striking. For example, the former suggests useful classifications that might escape the statistician, while the latter heeds the more abstruse physical processes that the classicist might overlook. The molecular phenomena referred to are, for the sake of concreteness, those occurring in a system of chemically stable polyatomic molecules. To minimize notational complexity we shall focus attention on diatomic fluids in particular. The generalization to more complex molecular species can be accomplished in a non-trivial but straightforward manner. ,4 statistical description of polyatomic fluids is provided by the theory presented previously (Dahler 1959) u-hich is equally valid for more general molecular models, and which is no less applicable when the molecular dynamics is governed by quantum as by classical mechanics. For simplicity we adopt the usage of classical mechanics. Both approaches, continuum and statistical, yield the same principles of macroscopic behaviour, regardless of the nature of the molecular and submolecular particles of which the physical system is composed. Our point of departure is the linear momentum principle for continua, which we review in the next paragraphs. I n 2 we proceed to a fully general statement of the angular momentum principle for continua, heretofore lacking even though the rudiments are to be found in old writings of Maxwell. Gibbs, and -notably-the Cosserats. Partial statements had been given by later writers on continuum mechanics, as noted below. A complete statement was recently formulated in the language of statistical mechanics by one of us (Dahler 1959), and a preliminary 506 J. S. Dahler and L. E. Scriven account of its translation has been given elsewhere (Dahler & Scriven 1961). From this principle we arrive a t the equation of change of internal angular momentum of continua, and thence the full set of conditions under which the usual hypothesis, that the material stress dyadic is symmetric (or that its tensor is, which is equivalent), is justified. I n § 3 we associate internal angular momentum with point structure of the continuum, and in particular with an internal moment of inertia and an internal angular velocity of that structure, as suggested by molecular considerations.$ We propose that the point structure can be represented b jr essentially geometric moment expansions. Where the idealization afforded by the pointwise structureless continuum fails, the first step of generalization is consideration of a pointwise polarized continuum. We introduce a comprehensive conception of polyadic polarizations for representing the macroscopic manifestations of the successive moments of any elementary distribution-of mass, the monoinorphic scalar measure, of charge, the dimorphic scalar measure, of moving mass and moving charge and so forth. For the lower-order polarization densities we develop equations of change taking into account the internal angular velocity of point structure. For illustration we apply the theory to continua that are structured by subcontinuum mass and charge distributions, deriving general expressions for proper mass density, mass flux density (mass current), and moment of mass flux (angular momentum) on the one hand, for proper charge density, charge flus density (electric current), and moment of charge flus (magnetization) on the other. We leave considerations of energy and general thermodynamics to a later communication. I n 5 4 we remark on the nature of equilibrium, the role of vorticity, the formulation of constitutive relations, the comparison of mass and charge, the origin of excess moments, and the outstanding dynamical features of structured continua. Mathematically rigorous formalizations of continuum mechanics are still under construction (Noll 1958, 1959). The traditional application of Kewton's linear momentum principle to continua is as follows. I n the usual way we define our system to be the constant inertial mass assigned to the region enclosed by an arbitrary, closed control surface located in and convected with a continuous mediuni in motion. Accepting the existence of the mass density as a point function.$ me have by the conservation of mass t I t is necessary to distinguish two kinds of internal angular momentum: that which can be identified with molecular structure and hence with an internal moment of inertia; and that which is ' intrinsic ' in the sense of being ' submolecular '. Into the second category fall the electron and nuclear spin angular momenta. An explicit treatment of these intrinsic and submolecular spins has been presented elsewhere (Dahler 1963). I n the present communication \I-e shall usually be thinking in terms of the angular momenta associatecl with rotational motions of molecular or similar mass distributions. Horn-ever, so many of the notions and conclusions expressed here are equally applicable tjo the second category that we shall frequentl?. employ the word ' spin' as a generic term for internal angular momentum rather than reserl-c it for the designation of intrinsic electron and nuclear angular momenta in particular. $ If mass and momentum are regarded as a non-negative absolute scalar measure and an absolute vector measure, respectively, of a set of material particles, then the condition thac mass and momentum measures be absolutely continuous functions of volume evidently ensures the existence of mass and momentum densities with respect to volui~le(cf. No11 1959). Theory of structured continua. I 507 whence by Reyi~olds'stransport theorem, the divergence theorem, and the arbitrariness of the control surface we have the equation of change of mass density The velocity u, with which an element of control surface would move, is the ratio of mass flux density to mass density; i.e. the mass average velocity. The interactions of this constant mass system with its surroundings can be classified according as the interaction is with distant or contiguous surroundings. I n the case of linear momentum the first class consists of body forces and the second consists of surface ('contact') forces. Following the customary procedure we have a t every point within the system the vector density of linear momentum, P. Likewise we have volume and surface densities of forces exerted on the system by its surroundings: F is the body force (momentum source), and t = n. t is the surface force (normal flux of momentum), n being the unit outward normal to an arbitrary surface element and t the ordinary stress dyadic (negative of the momentum flux density). The latter can be resolved into symmetric and antisymmetric parts: thus we have where U is the idemfactor and A the pseudovector of the stress, i.e. Then we have by the conservation of linear momentum whence by the usual arguments we have the equation of change of linear momentum density ii -(P)+V.(uP) = F+V.t(s)++VxA. (1.5) ?t I n application this equation must be supplemented by three or more constitzctise relations connecting dynamical to configurational and kinematic variables. Stress requires what has come to be called a rheological constitutive relation, body force requires a dynamic constitutive relation, and momentum requires what might be j: Our vector and dyadlc notatlon 1s that of Gibbs, except in one detall: the matter of multlple Internal products betmeen two dyadics or polyadics. TThereas Glbbs would put ( a b ) 2 (cd) = ( a .c ) ( b x d ) , n e follom the ' nesting convention' of S. Chapman and T. C4. Cowllng, T h e ?~zathematlcaltheory of non-unzfortn gases, Cambridge (1953) and put ( a b ) x (cd) = ( a . d ) ( b x c). I n terms of Cartesian base vectors and components, then, A.B=e,e,A,,B,,, A:B=A,,B,,, A2B=e,e,,,A,,BhZ, AyB=e,e,,,A,,B,,, etc., also U = e,e,6, and E =eze3ehe %,,. Certaln products of the isotropic triadic are sometimes called duals: d u a l ~ = e . ~ = - U xand ~ dualA=-qe:A=$UxA. Incidentally, dual (dual t) = t ' ~ ' . A summary of this and the following section, in Cartesian tensor notation, has been published elsewhere (Dahler & Scriven I 96 I ) . J. S. Dahler and L. E. Scriven 508 called a kinematic constitutive relation. (Situations in which momentum is not simply proportional to velocity are conceivable, although me know of none that has received detailed attention in the literature. Ericksen Gc Truesdell (1958) have pointed out that such a situation will arise in the exact treatment of the dynamics of rods and shells; that it indeed does is clear from a recent analysis of the dynamics of strata by Eliassen and one of us (1963).) Let us examine (1.5) in the light of the statistical theory for polyatomic materials, presented previously (Dahler 1959) as a natural generalization of earlier studies of monatomic systems by Irving & Kirkwood (1950) and by Irving & Zwanzig (1951). Equations of change are obtained by taking appropriate moments of the Liouville equation in a manner reminiscent of that used by &!taxwell in treating transport phenomena in dilute gases. For any dynamical variable a Here the brackets denote averaging in the statistical ensemble to give the ensemble expectation value (macroscopic average), i.e. <a) = with ~''(2~) S f(-")(q,p;t ) 4%P) dqdp, the ensemble distribution function; and with the q,'s and p,'s a complete set of co-ordinates and corresponding conjugate momenta for the molecular system. Choosing a = C m,S(R, - W) where m, is the v mass of the vth molecule, R, the position of the vth molecule, R the running position vector, and S(x) the Dirac delta f~~nction, we once inore obtain the continuity equation 2,0/2t f V. (pu) = 0 provided we define density and mass flux in terms of molecular mass and centre-ofmass motion, respectively, as follows: 1 V p(R,t ) and pu(R, t) - 2 (m,S(R, - R)) = 2, <n~,d'~) (1.7) C (rn,fi,~(R,- w)) = z,, ( ~ 1 2 , f i , ~ , , ) . (1.8) r v=1 AT v=l A: (Hereafter we adopt the abbreviations 2, = C and 6, = S(R, - R) for compactness.) u=l Choosing now a = C,P,G(R,-R) where P, is the linear momentum of the vth molecule, we once more obtain the protoequation of motion, this time with stress expressed in terms of the pressure dyadic p = - t ~(IP)I?~+V.(U =PF)- 8 . p provided we make the following definitions: (1.6) Theory of structured continua. I and p = 509 pK + p,, where (1.11) PK = C, (m,l(P, - P) (P, - P) S,), - Here R,, RA- R, is the displacement of the centre of mass of the hth from that of the vth molecule, UAvis the potential of the pair-interaction force between them, and D,, is the operator (cf. Dahler 1959) a?~-l I = Jo1dz [1 - n ~ , , .v + ... + ---- (-Rhu.V)'L-l+ ... . (n- I)! (The origin of this operator has been discussed by Irving & Kirkwood (1950).) With P, = m,k,, the specific momentum density P/p is simply the mass average velocity u, and (1.5) reduces to the familiar form sometimes called Cauchy's momentum principle. The kinetic pressure dyadic pK represents the flux of momentum arising from the chaotic thermal motions of the molecular mass centres. It is always symmetric. The remaining contribution to the pressure dyadic is p,, which is associated with the transfer of momentum by the action of intermolecular forces. These interactions of the molecules provide an overwhelmingly effective mechanism for the transfer of linear momentum through condensed phases. I n phases of low density, however, it is the simpler diffusive mechanism represented by pK that is of dominant importance. Now the most striking characteristic of p, is that it need not be symmetric. I n fact it is symmetric only in the event that the molecular interactions are central, i.e. only if L:, depends solely on IRA,/. Otherwise p, has an antisymmetric part which corresponds to the antisymmetric part of the stress in (1.3) and which may be written as pg)= - t ( a ) = - Ig. A, 2 (1.13) Thus asymmetric states of stress appear quite naturally as the rule rather than the exception in the statistical theory of real materials. P e t the existence of antisymmetric states of stress is 'disproved ' as a matter of course in almost every treatise on elasticity and hydrodynamics. The founders of these branches of continuunl physics were far less dogmatic than their orthodox followers, who have been pilloried in a very recent encyclopaedic review in which the question of stress symmetry is re-opened (Truesdell & Toupin 1960). The fallacies in the conventional continuum treatments become obvious upon consideration of the angular momentum of matter. 510 J. S. Dahler and L. E. Scriven The treatment of angular momentum of continua parallels that of linear momentum described above. We accept the existence of the pseudovector density of angular momentum, L, a t every point within the system (angular momentum may be regarded as a pseudovector measure of a set of material particles), and of body couples and contact couples representing interchange of angular momentum between the system and its surroundings by mechanisms that involve no macroscopic forces. Thus we have the volume density of body couples (angular momentum source) G and the surface density of contact couples (normal flux of angular momentum) c = n .c. I n the latter c is the couple stress dyadic (angular momentum flux), analogous to the ordinary stress dyadic. It too can be resolved into symmetric and antisymmetric parts, a step that is unnecessary here (but which will figure in investigations of the symmetry of the couple stress). Then for the system under consideration we have by the conservation of angular momentum From this, the continuity equation, and the transport and divergence theorems it follows that at every point a ( ~ ) p t + v . ( u=~R) X F + G - v . ( ~ x R ) + v . c . (2.2) This equation of change of total angular momentum is to be compared with that for the moment of linear momentum, which is simply the first monient of (1.5) Rx?(P)/at+RxV.(uP)= R x F + R x V . t , Moment of momentum, or external angular momentum, is conserved only when the stress dyadic happens to be symmetric; for otherwise its vector A does not vanish and external angular momentum is consumed or produced throughout the system. Upon subtracting (2.3)from (2.2)we obtain an equation for the difference between total angular momentum and the moment of linear momentum, M = L - R x P : c'(M)/at+V.(uM) = G + V . c + A . (2.4) This equation of change of internal angular momentum M makes clear that externcrl angular nzomentunz may be transformed to internal angular momentunz, or vice versa, by the working of an asymmetric state of stress. I n application this equation, like that for linear momentum, must be supplemented by constitutive relations for the dynamic variables. Long ago the Cosserats (1909, p. 137) developed a version of the internal angular momentum equation which lacked the accumulation and convection terms that appear on the left of (2.4). Revived in a footnote by Truesdell (1952) and discussecl a t length by Giinther (1958)~their version has been taken up by writers on the elastic dielectric (Toupin 1956), the exact theory of rods and shells (Ericksen & Truesdell 1958), and the continuum theory of lattice defects (Kroner 1960). I t Theory of structured conti?zua. I 511 has also been rediscovered in the controversy over the Laval-Le Corre-Raman theory of crystal elasticity (Tiffen & Stevenson 1956; Joel & TYooster 1960; Jaffe & Smith 1961). In each of these contexts the possibility of asymmetric states of stress has been recognized but, as in the Cosserats's equation, the connexion with internal angular momentum seems not to have been formulated. However, from the concluding section of the paper by Ericlisen & Truesdell (1958) on the statics of rods and shells it may be inferred that a discrepancy between angular momentum and moment of linear momentum will arise in an exact treatment of the dynamics. More recently Ericksen (1960) in developing a theory of what are called anisotropic fluids has arrived a t a version of the internal angular momentum equation lacking only the body-couple term. The same version had been advocated somewhat earlier by Grad (1952) by continuum arguments grounded in statistical mechanical considerations. Following Grad the emergence of an independent equation of change of angular momentum in the kinetic theory of real fluids has been discussed in several places [Hirschfelder, Curtiss & Bird (1954), p. 506; Curtiss (1956) gave a more complete and rigorous development of the angular momentum equation for a dilute gas than the original kinetic theory derivation by Ishida ( I 9 I 7), kellanzy (1958)~ Valleau (1958)~ Waldmann (1958)~cf. de Groot & Mazur (1962), chap. 12 ; Sather & Dahler (1962)l. The complete equation of change of internal angular momentum seems to have been first derived by the methods of statistical physics. By choosing in the statistical theory, with M, the internal angular momentum of the vth molecule, we obtain (2.4) provided we make the following definitions and c = C, + c,, where the kinetic flux of internal angular monlentum is and, in the case of diatomic molecules, the flux arising in molecular interactions is Here r, = R,, - R,, is the internuclear separation in the idealization of a diatomic molecule as a pair of mass points bound together. As in the case of the ordinary stress dyadic, c, is of prime importance a t high densities whereas c, is at low densities (Dahler 1959). Because the stress dyadic is so often assumed to be symmetric, it is instructive to inquire under just what conditions this assumption is justified. The common 'proof' that stress t (or its negative, the pressure dyadic p) is symmetric amouilts to the tacit assumptions that the density of internal angular momentum is steady and uniform throughout the system, that the surroundings exert no body couples, and that there is no net couple stress within the system (cf. Truesdell & Toupin 5. S. Dahler and L. E. Scriven 1960,footnotes on p. 546). If any of these conditions is violated the usual argument is fallacious, as a glance a t (2.4) shows. On the other hand, if the stress dyadic should happen to be symmetric then both internal angular momentum and moment of momentum would be separately conserved, as Grad (1952) has observed. Since asymmetry of the stress dyadic stems solely from the intermolecular interaction term p, in (1.5) and (1+12),the coupling between the two forms of angular momentum becomes progressively smaller as the density of the system decreases. Thus in a dilute gas the angular momentum delivered by applied torques remains segregated as moment of momentum of the resulting macroscopic motion, whereas the angular momentum pumped into the internal degrees of freedom from external sources remains isolated within the spin system. In contrast the interaction of the spin and centre-of-mass systems may become strong enough to produce striking effects in dense gases and particularly in condensed phases. Evidently the Barnett and Einstein-de Haas effects (Barnett 1935) are estal)lished examples; other possibilities have been suggested elsewhere (Dahler & Scriven 1961). It should be emphasized that the concept of internal angular momentum or spin is by no means restricted to the case of polyatomic molecules and their internal rotational motions. I n developing the continuum theory there is no need for explicit statements concerning the origins of the internal angular momentum density. In the development of the statistical theory by Dahler (1959)the only real limitation is that the non-central interactions of the molecular or atomic units which support the spins be pair-additive. Wikh this understanding we see that the equation of change for spin momentum, (2.4),may be applied not only to molecular rotations in fluids and solids, but also to the transfer of intrinsic electron or nuclear spin. In the case of solids c, would then be identified as the diffusional flux of this intrinsic angular momentum by virtue of various spin-coupling mechanisms. The means by which external fields can be used to pump angular momentum into the systjem will, of course, be governed by the precise nature of the spin system. The special case of polar, diatomic molecules is most closely related to the procedure we are following here, however. Only when the continuum is regarded, not as a dense aggregate of point masses, but as a collection of infinitesimal regions or 'particles ' each containing a host of far smaller entities within which mass is distributed on a subcontinuum, say molecular or smaller, scale-only then does internal angular momentum fit into the continuum picture. The density of linear momentum represents the average of the centre-of-mass momenta of all the molecules (or equivalent mass distributions) within a 'particle ': the internal angular momentum density represents the average of the angular momenta associated with rotations of the molecules (or equivalent distributions) about their mass centres (cf. footnote, p. 506). But to each subcontinuum mass distribution there corresponds a moment-of-inertia dyadic, and to the rotation of each an angular velocity. Thus, in the continuum picture, it is reasonable to postulate the existence of two more continuous functions whose product gives the internal angular momentum density: an effective spin velocity, Theory of structured co.iztinua. I 513 w,; and a symmetric moment-of-inertia density, I , representing the average of the inertia dyadics of all the subcontinuum units contained within an infinitesimal continuum particle. Then M = l .w,, w, I-I.M. (3.1) - This step implies that a structure, a mass distribution giving rise to a moment of inertia, can be assigned to every point in the continuum, i.e. to every infinitesimal particle. I n the statistical theory this structure is introduced by considering the variable, u = ?;,I,S(R, - R ) , where I, is the moment of inertia of the 11thmolecule. Then and M(R, t ) = 2, (I,.W,~,), (3.3) where w, is the rotational velocity of the vth molecule (referred to a space-fixed frame of reference). The effective spin velocity is not the ensemble expectation value rather it is defined by (3.1) even in the statistical approach. of o,; It is important to note that although the molecular inertia dyadic I, may be singular (as it is in the case of idealized diatomic molecules), the ensemble average I(R,t) generally possesses an inverse. The only exceptions would be entirely pathological situations involving perfectly alined arrays of linear molecules. As we have seen, body couples are excluded from the conventional picture of the continuum. Now, however, the body couple density arises naturally as the average of the torques exerted by distant surroundings on all sf the n~olecules within a continuum particle. These torques, whatever the nature of the interactions with the surroundings, depend on the structure and orientation of the individual molecules. For example, the torque in a non-uniform inertial field depends on the molecular mass distribution and its orientation; that in an electric field, on the molecular charge distribution and its orientation; and so on. Furthermore, the surface density of contact couples, except for the contribution from diffusion, arises naturally as the average of short-range intermolecular torques and as such depends on intramolecular distributions. I n every case the essential feature of the distribution is simply its geometric configuration, that is, its structure. It is this subcontinuum arrangement that is to be represented by assigning structure to each continuum point or 'particle'. The representation arises naturally. We consider a scalar quantity such as mass or charge that is distributed on a subcontinuum scale, and examine the macroscopic manifestations of the distribution first in an isolated system. If the ordinary density of the quantity is p , the total in the system and the corresponding configurational moment are In continuum mechanics, however, the position vector is that of the centre of mass of the particle. Thus an isolated system may possess an excess configurational 514 J. S. Dahler and L. E. Scriven moment that is independent of the origin of co-ordinates and therefore qualifies as an internal moment, or vector polarization. We emphasize the distinction between con$qu,rational moments under consideration here and the kinenzutic moments taken up below. If the polarization density is P , the total first moment is Similarly an isolated system may possess an excess second moment (a symmetric dyadic) that qualifies as an internal second moment, or dyadic polarization of density B, giving for the total second moment Excess higher moments may be defined in the same way. Now it is reasonable to replace the internal moments by equivalent macroscopic distributions. That this is always possible follows from the following identity and others, of the same sort, given in the appendix P = (VR).P = V.(PR)-RV.P. (3.8) Then for the entire isolated system the moments are while for any portion A V of a system the true contents are The proper scalar density within any portion of a system having configurational polarization must be independent of the origin of co-ordinates; it is given by Whereas the ordinary density accounts for everything assigned to a given differential volume, the proper density accounts for only what is actually contained Theory of structured contirzua. I 515 within the volume. The difference can of course be interpreted as a surface contribution, e.g. crX = n ( - 9 + 4V.52 - ...). Likewise, the proper density of vector polarization is 9" = 9 - & B . O + ... (3.15) . and similarly for dyadic, and higher-order polarization densities. I n the language of ;\lason & Weaver (1929)and follolr-ing them King (1945), who confronted the same problem of terminology in a more limited context, our proper densities would be called 'essential ' densities. Spin of individual molecular distributions alters their spatial orientations. This effect is mirrored in the rate of rotation w ox 9 of the vector polarization of a continuum particle for which the effective spin velocity is coo. This accounts for one kind of change of polarization density. The change arising from conduction is characterized by a polarization flux vector J and its dyadic density J. Finally, the sum II+ Q is the effective source density for polarization generated 'spontaneously' within the system. One source of this apparently spontaneous change is orientational diffusion due to subcontinuum motions of the Brownian type. Whereas w ox 9 represents a mean rate of change, II stands for the effects of dispersion about that mean; the latter term complements the former just as the diffusive flux density J complements the convective flux density UP. Q accounts for remaining mechanisms which can lead to the appearance of net polarization. Therefore, included in Q could be the rate of polarization arising from vibrations of the subcontinuum distribution. More generally, Q describes the rate of polarization due to any distortions of the subcontinuum distributions that cannot be identified with the rigid body motions of these distributions. It is obvious that the distortions included in Q and the rotations covered by II can arise in response to interactions of the subcontinuum units with neighbouring elements of the system or with externally imposed fields. However, these fields will not appear explicitly in the equation of change for the configurational polarization since the velocities and not the accelerations of the distributions are involved. To further clarify the situation we draw upon the statistical theory and a vector configurational polarization 9 = X u {PUS,)such that 9, is a linear vector function of the configurational variof the vth molecule. The equation of change for P is then, ables ti, where X u ((db,/dt) 8,) = C,C, ( t i , . (a9,/25i,) 8,) cuci ((mu X g i v + g i v ~ i v / ~ i v(ZPi/?giu) ). 8,) = 00 x 9 + u' ((mu- 0 0 ) x 9v'u) + cijei (CiY(5ilj/'tZV) 1 ?9uI2gip 18,). = For this case the second and third terms on the right-hand side clearly correspond to the 'spontaneous generation' terms II and Q, respectively. Furthermore, thib example illustrates unambiguously the rotational dispersion and vibrational motions which give rise to these generation terms. Both mechanisms will obviously contribute to the electric polarization of a fluid composed of deformable polar molecules: a simple model of this sort is considered belo~v. J. S. Dahler and L. E. Scriven 516 The sum of the various separate rates of change is simply the time rate of change of polarization. Thus for the convected system defined previously i1, Pdv = IYr w , x P d ~ + / 9Q ~ V + /I I~d v + SY n. J d 9 , a ( q j a t + v . ( ~=~ W ) ,XP+Q+II+V.J. (3.16) (3.17) This is the equation of change of the density of any vector configurational polarization of the continuum. It should be emphasized that the rate of rotation of the polarization vector is given by the internal angular velocity, or spin rate of the internal structure of the particle rather than by the local angular velocity of the centre-of-mass system, which is equal to one-half the vorticity (&Vx u). The distinction between these two rotational velocities will of course disappear in those cases for which w, = curlu, as in equilibrium systems (see below). The treatment of polyadic configurational polarization densities is similar. The rate of rotation of a dyadic polarization density, or of the moment-of-inertiadensity; is just o,x Q- B x w, (Milne 1948). (Because this quantity vanishes when the polarization is isotropic, i.e. Q cc U , it may a t times be convenient to resolve the dyadic into isotropic and deviatory parts: B -= U U : Q + (Q - UU :B). The deviatory part is one-third of what is often called 'quadrupole moment'.) Thus the equation of change for any dyadic configurational polarization of the continuum is a ( a ) j a t + v . ( u ~=) ~ , x B - B ~ w ~ + Q + I ' I + V . S . (3.15) + + The geometric configuration of intramolecular distributions may thus be represented by a hierarchy of polarization densities. A second essential feature of the subcontinuum distribution is the rate of change of its configuration. To represent certain macroscopic manifestations of molecular motion it is necessary to assign additional structure to each continuum point. That is, motion on a subcontinuum scale of a distributed quantity such as mass or charge induces additional, kinematic structuring of continuum particles. Because this structure is associated with excess flux moment on the macroscopic scale, it is pertinent to study flux, or 'current '. We examine the rate of change with time of the total of a scalar measure of a system; first, an isolated system For a system that is not isolated, but is closed in the sense of being of constant mass, the rate of change of any content can be equated to the total flux of the corresponding quantity across a surface enclosing the system. For such a system the total flux of mass itself vanishes. Flux, or current, is defined in terms of change in content of an appropriate reservoir. Flux and surface go together, but since total flux is independent of the shape of the enclosing surface it can be expressed in terms of local flux density in the usual way dq, d pdY = -h n. J d Y . Theory of structured continua. I 517 The flux density J may be called 'free current'; it is often referred to loosely as 'flux ,. For any portion of a closed system or body the reckoning is more difficult. A s we have seen, if there are excess moments of distribution, the proper density locally is given by (3.15);thus for any surface enclosing only a part of the system we have Because density and configurational polarizations are particle f~mctionswe consider the change in content of a convected volume and hence the flux through a convected control surface, as before. By Helmholtz's theorem for convected surfaces (Milne 1948) we obtain where I n the convected, constant-mass portion of the system the ordinary density, or 'free density', is still identified with the free current From (3.22) and (3.23) it follows that the flux density relative to a convected surface is According to Reynolds's transport theorem the equivalent relation for an arbitrary fixed volume is TVe conclude that the flux density relative to a fixed surface is given by the bracketed vector: convection of free density plus free current plus polarization currents plus convection of polarization densities. I n electrical parlance, the total flux consists of convection current, conduction current, polarization currents (relative to the motion), and 'dielectric convection currents' (cf. Truesdell Ss Toupin 1960). It is convenient to display the flux density in alternative form 518 J. S. Dahler and L. E. Scriven The first three terms correspond t o the flux density commonly derived in electrodynamics, as in the contemporary treatment by Toupin (Truesdell & Toupin t*). An isolated system may possess an excess moment of flux density that is in. dependent of the origin of co-ordinates and therefore qualifies as an internal moment, a kinematic polarization. If the dyadic density of this polarization is M', the total first moment of flux density is P P electro- The system may likewise possess an excess second moment, and so on. We merely allude here to the higher moments and focus on the vector of the first moment which is of primary importance. The vector moment is the dual of the corresponding dyadic moment; hence r r where &' = - +E: M'. It is again reasonable to replace the internal moment by an equivalent macroscopic distribution. That this is always possible follows from an identity given in the appendix. Then for an isolated system whereas for any portion of a system or body t,he true kinematic moment is The proper flux density, which must be independent of the origin of co-ordinates, is for a system having kinematic polarization. -4proper surface flux density can be discerned in the surface contribution to the moment; thus K* = n x ( -A' + ...). The vector density of kinematic polarization &' represents flux moment in excess of the moment of macroscopic currents. It is useful to subdivide this excess into two parts, A' = d*+A?*, the first comprising any excess moment d*= + ... that accompanies non-convective macroscopic transport, i.e. diffusion, of structured particles. As mentioned below, an example is the macroscopic charge circulation accompanying dipole diffusion. The second part of the excess moment, then, is the remainder .M* = & + ... that cannot be identified with any mode of macroscopic transport, convective or otherwise. Thus we may write the proper flux density as + v x (d...)+V x ( J d - ...) 519 Theory of structured continua. I It is sometinles convenient to define an apparent kinematic polarization density as follows Spin of individual molecular distributions does noh alter the circulation of those distributions. Hence kinematic polarization of a continuum particle is not locked into the structure of the particle as the configurational polarization necessarily is. With the exception of the effect of internal spin, then, kinematic polarization is subject to the same kinds of change as configurational polarization As explicit examples we consider distributed mass and charge, the basic measures of classical mechanics. I n applying the statistical theory we again refer to a system of chelnically stable diatomic molecules for the sake of concreteness. The proper mass density is the expectation of finding entire molecular masses within a given element of volume; hence -\- p*(R, t ) = C (m,,S(R,, v=l -R) + nzl,6(Rl, - R)), (3.36) where Ri, is the position of the ith atom of the vth molecule. To elucidate the conilexion with vector and dyadic polarization densities we expand the delta functions and retain the first three terms of each series where ri, = Ri, - R, is the displacement of the ith mass mi, fro111 the centre of mass of the molecule. Then (3.38) JiV= 8, - V (riY 8,) + VV : . follows because VR,8(R, - R ) = - V8(R, - R ) and ri, is a constant so far as the ordinary macroscopic gradient operator V is concerned. Furthermore, the gradient operator commutes with the sunilnation and integration operations so that, p*(R, t ) = xu ((m,, + m1,) 8,) - V. C, ((7n2,r,, + m,,r,,) 8,) + VV : xu (*(m2vr2v r 2 v + mlv r l v r1,) 6,). (3.39) With the definitions we obtain the version of (3.14) that gives proper mass density in the present example of a structured continuum J. S. Dahler and L. E. Scriren 520 By choosing a = 2, Quavin (1.6) we recover in similar fashion a sinlpler version of (3.18) for the change of dyadic polarization, provided we make the following identifications: (3.42) Q(R, t) = C, ( F , P V ~ , 6 , ) , Here Elv = U : Q, is the scalar mass quadrupole moment and F, the unit vector parallel to the internuclear separation r,. The equation of change for the lnoment of inertia density follows directly fro111 these results and the connexions, I -Q:UU-Q, a -I+V.(ul) at = 1, - a , : U U - Q , : a,xI-Ixao+Q,+II+V.3,, QI = Q : U U - Q , with =3:UU-3. (3.45) (3.46) (3.47) Equation (3.45) nlay be combined with (2.4)and (3.1)to yield an equation of change for the local spin velocity a,. We proceed next to the proper flux density for nlass which corresponds to the expectation of finding entire molecular mass fluxes within a given volunle element By decomposing the velocities of the atonls in the manner and using the same expansions of delta functions ar before, this becomes Here, di(R,t) = 2, (AvSv); dlv= +(mlvrlvx fI,)+ ~ n z v rxz fzv). v (3.51) The latter is the molecular Inass circulation vector; it is simply one-half the angular momentum when, as in the present case, sublnolecular intrinsic spins are excluded. Since the mass dipole 9, is identically zero we can omit the second and fourth terms of (3.50). If the equation of change for Q is then invoked we obtain, = Jkj,. A term-for-term comparison of this result with (3.33) indicates where that the vector of the excess mass flux density, A , is identical with one-half of the internal angular nlomentum density M and that the diffusive part d of the excess flux moment is equal to - +E : (V. 3'). Theory of structured continua. I 52 1 I n the absence of submolecular angular nlomentunl the equation of change (3.35) for the kinematic polarization associated with mass is simply the internal angular momentum equation (2.4) multiplied through by a factor of +,with An analogous set of equations applies to electrical charge in a structured continuum. With the definitions where qivis the charge located on the ith 'atom', we obtain the version of (3.14) appropriate to proper charge density in this structured continuum p: = pe-V.P,+&VV: Q, = pe-V.9:. (3.15b) By choosing cc = ~ v ~ , vin8(1.6) v we recover (3417)for the change of electric polarization, provided that we make the identifications: With a now obvious choice of variable in the statistical theory and identification of terms we can recover (3.18) for the change of dyadic electric polarization. Furthermore, with the definitions we obtain the version of (3433)that gives the proper charge flux density or electric current for the present example where d: E E : (Je- iV.3:). The unfamiliar term V x d,* represents the current associated with Brownian dispersional motions of the molecules; cf. (3.61) and 522 J. S. Dahler and L. E. Scriren (3.64). As usual, replacing the average of a product by a product of averages (here for example (P,,x P,) by P, x P) requires introducing a complementardispersional term (here d,) to account for the di~crepa~ncy. A glance back through the preceding paragraphs reveals many instances of this sort. The last equation can be recast in terms of a magnetization vector, 9:, defined as an apparent kinematic polarization density in (3.34) Again, with suitable choice of variable and identification of terms, we can recover (3.35)for the change of internal magnetic moment, At',. It bears emphasizing that the molecular moments P,,, a,,, &lev,etc., are b) definition reckoned about the centre of mass of the molecule; for in general. and especially when the molecule carries a net charge, such moments vary with the centre about which they are taken. Thus if r' is the position of some other point relative to that occupied by the centre of mass, the moments about that other point are related to those about the centre of nzass as follows: There are cases, for example the elementary dipole, in which a centre can be found about which the electric quadrupole moment and charge circulation vanish. Although this centre might be convenient in purely electrostatic considerations it is not appropriate in any dynainical context-unless it should fortuitously coincide with the centre of mass. Corresponding moments of mass and charge can be related, if desired, through polyadic electro-inertial and gyromagnetic coefficients. A particularly simple set of relations obtains in the case of the elementary dipole, for which - where ,LA, = mIvm2,/m, is the reduced mass and a, jqivl (m2,- m,)/2nz,m2, is the scalar gyromagnetic ratio (cf. the e/2mc of electrodynamics). Considerations of dyadic polarization have appeared in the literature of electrodynamics, for example, in the work of Mazur & Nijboer (1953))who themselves follow Rosenfeld (195I). It appears, however, that there has been no comprehensive treatment of configurational and kinematic electric polarizations. Maxwell's equations for the macroscopic force fields to be associated with the configurational and kinematic aspects of charge distribution in a structured continuum retain their familiar form provided proper densities are employed Theory of structured continua. 1 523 4. CLOSINGREMARKS Above we remarked that a t equilibrium the distinction between spin velocity w, of internal structure and one-half the vorticity 3V x u vanishes. This can be simply demonstrated by adapting and enlarging a scheme employed by Landau KLifshitz (1958, g 10). We regard a continuum as a set of structured particles, having mass as the absolute scalar measure, and energy, linear momentum, internal angular momentum, and moment-of-inertia measures with densities relative to A A A mass E , P , M, and respectively. The corresponding density of thermodynamical A h A internal energy is E-&P2-$M.Ikl.M. A A Since the entropy is a function of internal energy, the total entropy of the set may be expressed as If the continuum constitutes an isolated system, the total energy, linear momentum. and angular momentum are constants S S dd~= n constant, Pdrn * = constant, (' (R x P + M) dm = constant. J A A (4.1) Under these cotstrfints the entropy of the isolated system, viewed as a functional of the fields 8, P, M, is maximum a t equilibrium. The method of Lagrange multipliers is apt for solving the variational problem to obtain the thermal and dynamical characteristics of the equilibrium state. Thus we seek unrestricted extremals of S 1= S [8+a.fi+b.(~xfi+ii)+cd]d~n, where a , b, and c are constant Lagrange multipliers. The Euler equation for variation with respect to E is - and leads to the expected conclusion that the thermodynamical temperature field is uniform T = - l / c T,,.. (4.3) The Euler equation for variation with respect to P is from which it follo~vsthat u =f' = T a + T b x R -u,,,+TbxR. The third Euler equation, for variation with respect to M, provides the relation w, - A A I-l.M from which we conclude that u = u,,. = T b -= we,., $ we,. x R. 524 J. S. Dahler and L. E. Scriven Since u,,. and we,. are constants, this equation implies that the only allowed niacroscopic motions of an isolated system in thermodynamical equilibrium are solid-body translation at uniform velocity and rotation at uniform angular velocity. Moreover, by (4.7) the internal angular velocity o, is uniform and equal to the external angular velocity o,,. of the entire system at equilibrium. Finally, taking the curl of (4.8) and looking back a t (4.7), we find that at equilibrium & V x u = a , (=oe4.). (4.9) It is clear that equilibration of internal and external angular velocities cannot be expected when the constraint requiring constant angular momentum is relaxed; in the presence of an external couple field, as contemplated above, a steady state can be anticipated in which the difference o, - +V x u is related to the strength G of the couple field. I n general the local value of the difference o, - &Vx u, because it is a measure of the departure from rotational equilibrium, can be expected to govern, partly at the least, the net rate A of conversion of external moment of momentum to internal angular momentum: cf. (2.3), (2.4). Interestingly, before Lamb named V x u the 'vorticity ' in 1916, the vector +V x u was in use, and was called 'molecular rotation' by some authors,$ although as is well known Stokes had early identified it as the local angular velocity of the continuum. We see now that vorticity actually does mirror internal angular velocity, although it does so faithfully only at equilibrium. I n a structured continuum vorticity stands generally as a potential for transformation of external to internal angular momentum according to present indications. Hence it might be expected to give a strange performance when cast as a density of internal angular momentuni. (As suggested earlier (Dahler & Scriven 1961), the peculiar system of spin-excitations in liquid helium 11 might be more conveniently described in terms of intrinsic angular momentum density than in terms of vorticity singularities.) Complete equilibrium of a structured continuum is distinguished by an additional feature which has a parallel in solid-body mechanics. Unlike translational kinetic energy, which depends only on the magnitude of momentum, the rotatioiial kinetic energy depends not only on the magnitude of angular momentum, but also on its direction relative to the principal axes of the moment-of-inzrt;la dyadic. The rotational kinetic energy can be written & ~ ? ~where f, 9 -= M . I - l . Q / 1112. The scalar 9 can range between l/hmi,. and l/hm,,., thereciprocals of the minimum and maximum eigenvalues of the moment-of-inertia dyadic, and in general only for rotation about any one of the principal axes does l/f coincide with the scalar moment of inertia. Thus for complete equilibrium the entropy of an isolated system must also be a maximum with respect to the field f of local orientation-u~zder the constraint, however, that 1 1 2 f ( R ) 2 -(4.10) 14,lax.(R) throughout the system. There is no relative maximum, since the variation of entropy with respect to 2, --Sfdm The term survives in C. E. V7eatherburn, Advanced v e c t o ~analysis (London: Bell, 1924) and in L. 31. Alilne-Thomson,Theoretical hydrodynamics (4th edn., London: Macrnillan, 1960). Theory of structured continua. 1 shows S to be monotone increasing with decreasiny 9. The maximum is therefore attained when 2 = l/h,,,. throughout, which corresponds to perfect local alinement of the interhnal spin vector o,parallel (or, equivalently, antiparallel) to the principal axis of I having the greatest moment of inertia. The parallel case of rotating bodies in solid nlechanics seems by no means as obvious as implied by Landau & Lifshitz (1958, § 26). The formula for rotational kinetic energy on which these authors base their reasoning is in fact valid only when rotation is already about a principal axis. Their conclusion, that 'equilibrium rotation of the body takes place about the axis about which the moment of inertia is a maximum ', is, however, confirmed by the reasoning of the preceding paragraph. For if it rotates about any other axis we see that the body possesses excess kinetic energy of rotation, + $ 2 ( f - l/h,,,.), which is available for conversion, a t constant total energy and angular momentum, to thermal energy with an accompanying increase of entropy. The same interpretation applies to continuum particles having internal structure. We remark that 2 constitutes a gyrostatic coefficient representing a relative orientation of kinematic and configurational polarizations associated with mass. Further accounting of energy and entropy of structured continua, particularlin non-equilibrium situations, we postpone to future communications. Suffice it here to point out that considerations of energy and entropy are central to the formulation of constitutive relations connecting fluxes to configurational and kinematical variables. Evidently all of the fluxes appearing in the various equations of change must ultimately be related to essentially geometric variables that are in some sense observable, whether as measures of means of microscopic position, velocity, etc., as in mechanics, or as measures of dispersions of microscopic velocity, position, etc., as in thermodynamics. Just to apply the equations of change associated with distributed mass, for example, constitutive relations are required for momentum (in terms of velocity, say), body force (e.g. position and velocity), stress (e.g. positional strain and strain rate), angular momentum (e.g. spin velocity), body couple (e.g. configurational and kinematical polarization), couple stress (e.g. orientational strain and strain rate), moment of inertia (e.g. in terms of polarizations), and the succession of polarization fluxes and sources. The second law of thermodynamics and the invariance requirements of rheology are keys to the formulation of admissible relations. There is a nearly complete analogy of continuum structure induced by distributed charge with that induced by distributed mass. Mass assumes the dominant role, however, in the dynamics of a structured continuum. The reason is the convenience of adopting the centre-of-mass convention for position, as above; classical dynamics including electrodynamics is based on the kinematics of mass not charge. The special place of mass can be traced to the evanescence of dipole moment of any monomorphic scalar measure, which stands in marked contrast to the persistence of dipole moment of charge and other dimorphic and polymorphic measures. The analogy that does exist between the roles of mass and charge raises a point about the assemblage of force fields associated with them which is said to have been first noticed by Paraday himself: the electric force is assigned to charge configuration, J. S. Dahler and L. E. Scriven the gravitational force field-to-mass configuration; the magnetic force field is assigned to charge kinematics, but-outside of general relativity theory-no need seems to have arisen yet to associate an analogous force field with mass kinematics.$ The pivotal postulates of the theory of structured continua are those of the existence of excess moments, of mass or of charge, for example. I n cases where the subcontinuum units are actually macroscopic their configuration might be determinable by direct observation, dissection or some other manipulation-but even with their anatomy known the weighting to be assigned to the functioning of their constituent parts would have to be determined by additional operations. For in general the existence of excess moments must be inferred fro111 the nature of the interaction of the system with its surroundings; more simply, with another system or body; most simply, with elementary test bodies of appropriate kinds. Excess moments can be expected to arise whenever adoption of a continuum representation corresponds to an averaging over centroids of subcontinuum distributions, whether they be molecular, microscopic, or macroscopic. We have chosen systems of molecules for guidance and for illustration, but the elementary units might as well be larger objects or regions. I n some contexts the excess moments can be neglected, the classical structureless continuuin affording an adequate approximation to physical behaviour of interest, as in much of fluid mechanics and elasticity. But where more accurate descriptions of macroscopic material behaviour are sought, as first of all in electromagnetism, the structured continuum, with its fields of polarization densities by which excess moments are represented, provides the means for successively refining the approximation while retaining the advantages of a continuum mechanics approach. Such refinements can be built around the framework that has been raised here, a skeletal theory originally designed to accommodate the angular momentum principle in its entirety. Prom general considerations of its angular momentum, a t both molecular and macroscopic levels, several remarkable features of the structured continuum are evident. I n its dynamical behaviour i t is distinguished by (a)internal angular momentum that is, apart from quantum spin contributions, equivalent to excess moment of mass flux; (b) state properties that are, by virtue of their directedness, vector-valued (and polyadic-valued) point functions; (c) interpenetrating, interacting translational and rotational subsystems which both appear to fill the same physical space; (d) asymmetric states of stress, by which these subsystems are directly coupled; (e) capacity for internal storage of ordered rotational kinetic energy. Elsewhere (Dahler & Scriven 1961) we mentioned a few of the fields to which these features seem pertinent; more detailed discussion of applications is left to subsequent communications. The authors are indebted to R,.Aris for helpful discussions of several points. $ Compare Bridgman (1952, pp. 17, 30). J.T.Fraser has Bindly called to attention the considerations of the fourth force fielcl by half-a-dozen authors over the past fourteen years; see his paper (Fraser I 961 ). Tlzeory of structured co.rzti.rzua. I 527 5 . APPENDIX I n 5 3 certain fundamental polarization identities are used to replace internal moments by equivalent macroscopic distributions. Of whole families of these identities which can be established we record here four that pertain to vector and dyadic moments. The notation is Cartesian tensor 9i -- 8i j P3. r R . . 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