Download Dahler and Sciven 1963

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 . . P 3
(5.1)
2,j
&(aij
+ aji)= $(RiGicji-QkiRj),k- &(RiRjalkJl)
+ $RiRjQlk,lk,
,k
i
- QsijNkk
= (Ri
i ( ~ i j k ~ ~ ~ k l r n &cijkRj(Eklna
~l),rn
JLm,l),
(5.2)
(5.3)
JG~,,),
- QsijRlXkl),
- Q(Riejrrn
Rlekml
- QRi'jlm Rl(emkT
&T,
sk)
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