Clockwork, ephemeral and hybrid continua Текст научной статьи по специальности «Математика»

y^K 539.3

Clockwork, ephemeral and hybrid continua

M. Brocato, G. Capriz1

University Paris Est, Ecole nationale superieure d’architecture Paris-Malaquais, Paris, 75006, France

1 University of Pisa and Accademia dei Lincei, Dipartimento di Matematica, Pisa, 56127, Italy

On the occasion of the centenary of the publication by the Cosserat brothers of the treatise “Theorie des corps deformables”, a medley of ideas was listed which were inspired by that treatise and by the papers of the many Authors who refer to it. The preference was accorded to the topics which, in our opinion, merit attention for greater potential of further achievements in the description of events in complex bodies.

Keywords: Cosserat continua, continua with microstructure, multi-field theory, ephemeral continua

1. Preamble

In their account of the well-known work of the Cosserat brothers [1, 2], Truesdell and Noll praise the careful study, through an action principle, of the influence of torques (beyond couples of forces) on the behaviour of generalized continua [3]. Then they speculate: “The final end the Cos-serats seem to have had in mind for their three-dimensional model was an aether and radiation theory, replacing and uniting the Fresnel-Cauchy theory of elasticity and the MacCullagh-Maxwell aether theory“. The evidence for the surmise is only circumstantial, however, and might have been strengthened by a reflection “a posteriori“ on the events and preoccupations, within physics, at the beginning of the XX century.

Certainly Truesdell and Noll’s point of view is a licit interpretation of the treatise: still, that interpretation is largely ignored by researchers of the second half of the century. On the whole, they quote the book, but while they extend concepts and developments found there on rods, shells and three-dimensional continua and adapt them to various material, not spatial, manifolds: the aim is a multifield theory rather than a multivariable one.

The dichotomy seems to us still worth exploring: (i) is it the ambience (i.e., an Euclidean space) where we place the images B of bodies that needs expansion, i.e. do we need an appropriate ether as the space within which we work so as to lead to a multivariable local theory? or (ii) is the Euclidean space sufficient again as in the classical analyses and what we need is more fields for the narration of com-

plex structures, events, processes occurring in the deeper recesses of bodies? (iii) Finally, are there phenomena which require both broadenings of the modelling tools simultaneously?

To some extent, these questions above overlap with the query regarding the limits within which, in modelling a material, one can accept the stipulation of perfect identifi-ability of material elements and their intricacies; when the stipulation is acceptable the dichotomy fades, at least formally, and might be (and actually is) ignored.

In fact, a deeper hypothesis presumes that, during any process, any material element is not only composed of the same population of specks (grains, molecules, etc.), but, besides, all specks within an element share instantaneously the same stance (direction of each stick, shape of each blob, adjustment of each array of gears and levers, etc.). These two assumptions justify the title of the first chapter: Clockwork continua. Then, as a reference stance is available or conceivable, it could be placed within an “ether“ and the present stance obtained by bijection within that ether.

If the assumption of identifiability is abandoned, some radical changes are required in the theory. The approach can only be local: it is the Euclidean space E, within which we think the body placed, which is now imagined split into loculi (representative volume elements) of such small dimension that, at the ordinary level, each can be confused with a place x in E. But, at an appropriate available magnification, separate evidence appears of very many specks within each loculus and of their rates of change of place.

© Brocato M., Capriz G., 2011

The ether is now six-dimensional; the dimension six implies that each speck is taken to be as a shapeless point y with its loculus. Actually, sometimes, in the sense made clear in Sect. 2.2, very short time intervals (contuiti) are also expandable minimally though sufficiently to allow the measure of rates.

In papers published over the years we happened to explore, separately, instances of all three environments: book [4] covers an instance of type (ii) (an extensive view of the matter together with many appropriate citations is available in treatise [5]); paper [6], with some indulgence, would be an example of type (i); book [7] an example of the hybrid type (iii).

One must be indulgent, on reading here Chapter 3, titled Ephemeral Continua, and book [8] to allow for minor infidelities. The first, of setting: true, beyond the overall Euclidean space, each of its points is expanded into its own local Euclidean space, thus creating a six-dimensional ether; however (as already mentioned and contrary to what is allowed in multipolar theories; see, e.g., [9]) only a small region in each fibre, the loculus, is physically significant. The second infidelity is the whole contravening strive away from a multivariable theory and towards a multifield theory issuing from invented and, in principle, artificial motions (and ensuing one-to-one mappings), achieved through statistical estimates; the latter largely already enabled, however, by the kinetic theory of gases.

As an alternative to the inquiry of Sect. 2, one could reaffirm the assumption of identifiability of material elements, but not in its deeper version. Specks belonging to an element are supposed again to be always the same, but not to share necessarily the stance. Rather they are though to disperse in classes of stance in accordance with a distribution function, thus possibly leading to a multivariable theory. Such hybrid canon (Sect. 4) might either involve an evolution equation of Boltzmann type and proceed again to a more complex multifield theory, or, rather, mimic a sort of mesoscopic version of molecular dynamics, the specks taking the place of the molecules of that dynamics.

2. Clockwork continua

2.1. Multifield theories

As mentioned above, many researchers were inspired by the proposals of the Cosserat brothers to introduce multifield theories; however, whereas the Cosserat consider rigid trihedra attached to each point of the Euclidean space as a purely geometrical assumption to reflect the possible nature of physical space, later followers (and ourselves) have contrived the idea to employ the added structure to reflect the nature of a specific class of materials and thus to study the behaviour of bodies when their gross processes are influenced essentially by the local minute details of their makeup, details which can be ascertained only by a magnified view of each material element.

As in the standard treatises a body is imagined to span in three dimensions over a fit region B of the ordinary Euclidean space E made up of points x. Still, bodies are also viewed at a lower scale and, whereas the phenomena observed at the gross scale are mainly described through the usual tools (e.g., placement gradient F, Cauchy stress T, etc.), for events lower down (i.e., “inside“ each element) keener agencies are invoked. Some fencing in of subdivisions is needed to achieve significant inferences; finally only one descriptor of shape and bearing is presumed needed for each speck and, by the deeper hypothesis already mentioned in the preamble, the same descriptor for all specks in the element. In fact, in our approach, we propose to restrict attention to cases where an «-ple of real numbers va (a = 1, ..., «), to be interpreted as coordinates of a point v in a manifold M, is a sufficient descriptor (gauge connections do not seem to have found yet pertinence here). Thus a trivial fiber bundle is the overall canvas over which the body is depicted, with the space E as the base and M as fibre; in the Cosserat special case, M is the group Orth+. So it might seem that our proposal, despite being addressed to a different wider goal, be simply an obvious, perhaps even only formal, generalisation; in fact, it requires some novel steps left covert in the special case. For instance, a tacit hypothesis is that M be Riemannian; in particular, that a metric be consequently available for M. Its physical import must be checked case by case. Besides, the vastness of the class of manifolds that can be drawn upon allows one to model very many technically relevant materials. Just to quote, incidentally, an instance pursued by many authors repeating the original steps, the trihedra may be taken as not necessarily rigid (leading to the multipolar family of continua); even not necessarily at a fixed stance relative to gross directions (gyrocontinua [10, 11]). But on such special cases we will return later on.

One foundational issue, that must be tackled, is the choice of the «-ple of real numbers describing the understructure; it must be appropriate to the modelling task, of course, but it must also be minimal. Chosen one, generally, physical circumstances tolerate any other into one-to-one (and sufficiently smooth) correspondence with it, so that relevant quantities depending on the stance of the body must be indifferent to a relabeling of the coordinates of M. It is such indifference which can be invoked to obtain, from an expression of the appropriate action, the balance laws governing our generalized continuum (rather than simply the indifference to orthogonal changes, as in the Cosserat analysis).

The next essential modelling task is thus the choice of the expression of the action, a pursuit which might seem easy, implying an apparently straight broadening of the proposals of the two pioneers. In fact, one deep implication of a fairly direct use of those proposals is the acceptance of a substantial indifference (analytical details apart) of the spa-

tial and of the material approach in the modelling of an actual material. Such indifference is rooted on the preconception that the specks of the cluster inhabiting any niche in space at a certain instant cling together, however intricately, forever without loss or gain to contrive a material element. The preconception, adhered also here within this chapter, is often acceptable, though with clamorous exceptions: mixtures, for instance, and even perfect gases, which, consequently, must be rather classified as ephemeral.

The second divide is in the necessary recourse to concepts and procedures from elementary Lagrangian rather than Newtonian (point or solid) mechanics, with consequent choices of expressions of inertia, kinetic energy and forces. To realize the transfer a cardinal hypothesis must declare the uniqueness of meaning for the operation of taking gradients; hence the existence of one and only one physically significant connection for the manifold. The failure of the hypothesis would open the way to striking developments: microstress and internal agency would loose distinct identity, i.e., the separation between action at local distance and local contact action would presumably founder.

Microstructural inertia (p, density) per unit volume

P

^dx&

dv

dx

dv

is linked directly with kinetic coenergy per unit mass X(v, v) the Legendre transform of which

K = _d2av a

dv a

is the kinetic energy per unit mass, as asserted in textbooks (repeated indices imply summation).

The distinction between the two functions has scarce relevance generally, but might be essential in some exotic circumstances. For instance, if x were linear in v: X = (v)va, k would vanish but inertia need not; it would

be of Coriolis type

(

P

dvP dvc

>P

a, P = 1,..., n,

and hence powerless. This strange case may be relevant when almost furtive mechanical effects are of magnetic origin [12]; consequent general issues do not seem to have been explored.

Of course, when k and x are taken, as is often the case, to be quadratic forms in v, they necessarily coincide and their coefficients offer the natural choice to characterize the metric to be associated with M. Consequently, the Levi-Civita connection deduced from the metric becomes the legitimate candidate to employ for covariant derivatives. But k and x might be distinct, though functions of | v |2 only, and not of v, when a physical limit be present for | v |, as happens when v is used to describe a dislocation density and, so to speak, the first speed of sound imposes a limit on the second. Again, a general scrutiny seems to be missing

as yet, though a Finsler geometry is already at hand to offer help [13].

In their treatise, the Cosserat investigated the conservative case when the contribution to the action a due to both internal and external agencies can be expressed in terms of potentials and we follow them here, at first:

T

a = J Jydx»dT,

®, 0

12

y = 2 P(x + x) - pe(x, F, v, gradv) - p^(x, v)

with manifest meaning of variables (e is the potential of internal agencies; ^ is the potential of external ones).

The conditions of extremality of a, under appropriate assumptions of regularity for the sought functions of t and x* (or x), are the usual Cauchy equation

p x = pb + div T, or p» x = p»b + Div P, (1)

plus an equation of evolution for v

p

3v

W J

dx

dv

&

or

p*

where

w

dx

dv

&

= pP-C + div S,

= p*P--S+ DivP,

P = (detF) TF~T =p* , b = |i,

d F a x

(2)

dv

P = (det F) S F ~T = p

de

d(Gradv) ’

ft = (detF)Z = p* , P = |^,

dv dv

T and b have the usual status; S and P respectively are interpreted similarly at the microlevel, whereas Z would then account for internal microactions (which do not necessarily sum up to zero).

Actually, the action principle suggests, formally, an even more compact version of the balance equations, with a merging of time- and space-variables into a higher-dimensional space-time; a version exploited by the Cosserats to transit from equilibrium to evolution problems. However, the longer explicit variant was preferred above, to glide easily towards special instances. In particular, and crucially, the format of the equations, as written, is accepted even when, because of prevalence of attrition and consequent power losses, the right-hand sides cannot be derived from an action, as spelled, and T, S, Z, b and P, cannot be expressed through derivatives of potentials.

Contrariwise, many accessory, even critical, comments need complement the rather formal rendering above. For one, the dot over a symbol indicates everywhere, according to usual practice, total time-derivative; hence a unique scale of time intervals is implied. Yet, under some circum-

stances, the time-rates of change at the macro- and microlevel diverge markedly, so that, together with the separation implied by recourse to a spatial fibre bundle, a similar splitting of gross time and rhythm within a “contuitus“ must be envisaged. Such subtle scheme might even seem to be more in consent with the quoted manipulations of the Cosse-rat, when they suggest to treat time as an extra coordinate. We have suggested some time ago the first steps that, consequently, would need to be made; but a thorough approach is still wanting and the matter is barely introduced in the next section.

Another serious matter is the consequence, on formal definitions of Z and S, of relabelling of members of M. Physical circumstances may intrude to prefer, even impose, a choice of labels denying relabelling; in which case the va appearing above must be chosen accordingly. Otherwise, as already hinted, an intrinsic version of operators must be broached, with recourse to a physically significant choice of connection for M, to allow for covariant derivatives, indifferent to relabelling.

Finally a word must be said about the boundary conditions which may be associated with the balance equations. Direct reference to current practice would suggest foremost extended strong conditions of place, i.e. the assignment of macrodisplacement and microstructure at dB*. The difficulties associated with proofs of existence, uniqueness and regularity of even standard boundary problems of place are well-known. But here, even if all desired smoothness of boundary and initial data are assured, there is still another catch: suppose that values assigned to v at the boundary be continuous; map them on M to form there a sphere; if it does not belong to the identity of the second homothopy group on M, a singularity must be present within B*. Its classification may be found via the topological theory of defects [14]; there may be a sheet of phase transition, line and point defects. Actually, experiments show, particularly in liquids and semisolids, that full regions of quasicrystall-inity or glassy state may occur where the value of v is random. Then a quite different mathematical model is required and we will turn to it later.

Boundary conditions of traction (when a component of T and of S are assigned on dB) are, usually, more difficult to realise in a laboratory. Even in a strictly mechanical interpretation of the Cosserat analysis, it is difficult to imagine devices able to impose torques at the boundary. Actual conditions for the microstructure at the boundary are better modelled by imagining an energy imbedded at the boundary with a surface density depending on v and n (n, unit normal vector). The action principle must be correspondingly modified. T is altered by the addition of a term representing surface tension, whereas S n is ruled by the derivative of the surface energy density with respect to n.

The freedom allowed to the global shape of B is, at times, curtailed, as occurs for large solid crystal shapes.

The form taken by small semisolid blobs and liquid drops as a consequence of the presence of microstructure may be decided by the appropriate construction of Wulff’s set (see, e.g., [4, § 33]).

Boundary phenomena are multifarious already for standard bodies. They become even more intricate when microstructure is present; the relevant literature is accordingly vast (see, e.g., [15]).

2.2. The structure of time

From the two main general themes of the Cosserat treatise (the addition of an extra space for parameters covering local details and the suggestion to use time not differently from space macrocoordinates) the inkling arises spontaneously to introduce a mesotime to expand what we tend to call the present instant. Curiously the matter of the definition of the concept of “present“ is already brought up by Saint Augustine over fifteen hundred years ago. In the eleventh book of his Confessions [16] he observes with dismay that time, a concept apparently so clear, is in fact elusive: on the one hand, time present seems to be the only reality, because past is only present memory of things in past and future is only present expectation of things in future. On the other hand an essential quality of time is to admit of a measure and, if we imagine “present“ as the element of separation between past and future, it has no duration being a singleton.

However, Saint Augustin uses for present time a pregnant Latin substantive, contuitus, translated in English as “sight“, in Italian as “visione“, malappropriately in our opinion, because the original latin word has a different semantic imprint, leading to a conclusion opposed to the Augus-tinian one. Contuitus is a subjectively collective term, even temporally collective. It has a duration, which can be measured and which depends on the time constants of our sensory system and brain. The fact is well-known to television and communication engineers and, since millennia, to conjurers and jugglers.

Contuitus, as the medley of “subtuitive“ phenomena, has obvious citizenship in the study of materials (or circumstances) where at least two time constants of a different order of magnitude have a decisive role. Actually, said this way, the matter may appear even trivial and not in need of a philosophical preamble: the Knudsen number enters often physical problems; a sequence of time constants may even occur, nested one inside the next. Actually, the latter case has already the attention of many authors on adiabatic time.

On the other hand, the Augustinian emphasis suggests the need for a review of our otherwise straight modelling of the time variable and leads one to propose an approach to that variable perhaps more utilitarian than profound but then perhaps of some interest for the engineer. The variable time is to be read on a one-dimensional oriented manifold on

which a metric is exactly specified (at least, if we want to study physical rather than psychological and even physiological phenomena; see, e.g., [17]), but it is not necessary to realize the manifold as a straight line.

Let us then suppose, for simplicity, that we deal with bodies reacting to involve two time constants of different order of magnitude. The gross response is scanned by the set of contuiti (each defined by a value of the variable T, say), but such response cannot be specified if the muddle of events is not disentangled, and those events are scanned, within each contuitus, by a second time-variable T2. Then it may be convenient to imagine the manifold of events as a helix traced on a cylindroid T, so that each loop is distinguished by the variable T, whereas the details along one thread are singled out by the second variable t2 . If the Knudsen number is very large, the pitch Tj is very small compared with the radius T2; then it may be convenient (for an analysis of intermediate accuracy) to “homogenize“ the helix (to smooth the screw) and to think of an instant as a point on the cylindroid, and of time as filling the two-dimensional manifold T.

Notice that we used the word cylindroid rather than cylinder because the quantity T2 might come to depend on the intensity of other stimuli as experiments have shown. Indeed, it seems that sometimes the exact order of subtuitive events be irrelevant: the impulse due to subtuitive events and acting contuitively should be evaluated as though the temporal order within the contuitus be dictated by bare intensity.

From now on, within this section, we denote time t as an element of T, though in reality time “flows“ along the helix. Contuiti are fixed by Tj and contuitive phenomena are characterized by mean values calculated on each single thread. Actually, one can imagine to move on the cylinder even on paths variably inclined on the generatrix. Such motion would correspond to sort of “stroboscopic“ vision of the phenomena. Thus, the gradient of any time dependent variable ^ (dependent, that is, on time t = (t1, t2 )) could be termed stroboscopic time-gradient. We will choose the symbol vel for such time gradient.

One important remark must be stated in advance of any specific development on the whole of T. There is a generatrix, a sort of “date line“ of change of contuitus crossing a “Pacific Ocean“, scarcely crowded by events. On such date line, a discontinuity ~ of any |x must be admitted such that

1 tj, 1

t2

^ = — dT2 =-

l2 0 dT2

-№(^,0)],

if T2 is counted from the date line, square brackets indicates jump, T2 is the duration of a contuitus, and ~ is the impulse rate of ^ over a contuitus.

On the other hand, one can imagine that a sampling of any time dependent variable |x through some filter (smooth

or not, but anyway causal) allows us to distinguish the slower from the fastest effects, the former affecting the actual perception }! of the variable in the contuitus, the latter being “subtuitive“ and given by ^ - ^ as fluctuations about the contuitive part

i T1

~(T1) = — J^© MT -£)d^

T2 t1 -T2

^ being any appropriate measure.

The splitting of time into a contuitive and a deep-seated component may be convenient when dealing with phenomena of intermittency, of “cut and paste”, as in the tearing and mending of substructures, such as, in Couette flow, the formation-destruction-reformation of chains of molecules or of molecular chains and columns for electrorheological fluids.

If we call the dependent substructural variable v and we want to describe its intermittent abrupt change (together with a longer term drift), then x being the kinetic co-energy, e its substructural “elastic“ energy, vel a time gradient on T and divel the corresponding divergence, the evolution equation becomes

p

divel

__________dx&

d(vel v) dv

dx

a de de

: pP - — + div ----------------.

dv 3(grad v)

(3)

Earlier developments suggest to us the splitting of our problem into two simpler subproblems: one obtained by “homogenization” of any function Z(T , T2) over the con-tuitus

-T- Jce^ T2)dT2 = Z(T1) (4)

T2 0

the second, obtained by subtracting the homogenized version of (4) term by term from its full version (3):

_d_

3t1

d(X - X) 3(d^/3T1)

_d_

3t2

d(X - X) 3(3^3t2)

d(X - X) 3v

(5)

thus following essentially only the subtuitive perturbation and the dots being there to indicate impulses still to be explored.

In particularly simple cases where a separation of effects is justified, Eq. (5) may involve only average values over a contuitus although necessarily at least the subtuitive impulse rate

1

3k

3(3v/3t2)

(brackets denote jumps) affects the balance of such averages. One important discriminating issue is the possible dependence of T2 on t1; a case when the exchange of derivation and averaging is directly denied.

Under opposite special circumstances, however, derivations and averaging may be interchangeable. In any case,

the total impulse rate due to substructural events must be added. Actually, that essential term shows that the temporal succession of substructural events is irrelevant: one could change their temporal order without changing the gross effect; in fact, such is the unexpected result of experiments on man [17].

One case when such subtuitive events have a dramatic role is when they describe a configurational shock. Body shocks are not usually studied, but they will become relevant, e.g., to the study of electrorheological fluids. Then the electric control of rheological properties may be achieved at speeds of electronic circuitry. The formation of molecular chains and columns can be achieved hardly less fast.

Also the time constants of chemical reactions are generally far smaller than the mechanical ones. Actually, as happens within the theory of continua with substructure, the multivariable approach is, at times, much more involved and less effective than a multifield approach. If such circumstances prevail also here, and if the class of accessible subtuitive processes is narrow so that each of them is specified by the choice of finite number of parameters (e.g., harmonic oscillation of period T2 and variable amplitude, or a sawtooth oscillation specified by two slopes), then an evolution equation in t1 only would suffice. Subtuitive phenomena may often interest only a thin layer in space and analysis may be drastically reduced in difficulty by introducing a shock surface where a continuous impulse rate must be applied, determined by an appropriate simplified model of subtuitive phenomena.

2.3. Observer independence

One set of issues was left out of account so far; it concerns the constitutive options available to specify individual classes of materials: say, liquid crystals, to quote a paramount instance. The horizon is vast; our interest for special cases was primarily due to the chance the offer to test the scope of the general theory.

Anyhow, some general rules must be obeyed to insure the validity of a tender. They stem form the need to insure observer independence latu sensu: beware, as already mentioned, one might unconsciously accept as legitimate a particular instancing of M without checking the objective significance of the implied metric and connection.

In any event the reading of v may be affected by a change of stance of the observer, though hardly by a translation. Rather, the group Q = {r} acting on members of M need be detailed through the rule connecting the vector q of macrorotation and the change possibly afflicted on v by that rotation, leading v to V( q) = T(q)v. From an operational point of view, the infinitesimal generator A{v} of the local action of the group is the central tool:

v(q) =v+Aq+o(| q|).

Consequently A occurs in the embodiment of many prerequisites of objectivity: for instance, the potential e may depend on v and gradv only in such a way that there be no

effect of a change of v into V( q). Conceivably, as variables in e, the potential appearing within the action, one may pursue the goal of finding and using only observer-independent measures of macro and microstrain. The pursuit was already successful in many special cases, conspicuously for nematics, and the success is accepted by implication in many analyses where the operator A is not mentioned even in a footnote. In any event, under appropriate circumstances of smoothness the condition requested on e is most easily expressed by

de

dq

= 0.

q =0

One quantity which is conditioned strikingly by enforced neutrality to observer motion is the power of internal actions

-(T • grad x +Z-v + S • grad v).

(6)

For it the requirement leads to the pertinent alternative to balance of moment of momentum

eT = A T Z + (gradA T) S (7)

(e is the Ricci alternating tensor). Thus T may be non symmetric, generally.

It should be evident that the reading of v, implied by the choices above, is imagined as effected by an observer external to the body. One glaring outcome emerges conspicuous when the most elementary expression is chosen for the total kinetic energy per unit mass:

1 x +k(v,v), k = ^v“Qap(v)vP.

Then the kinetic energy for the whole body during a rigid rotation, both macro and micro, with angular velocity w turns out to be

-2 w • (J + M )w, where

J :=Jp(x21 - x ® x) d(vol)

B

is the classical inertia tensor, whereas

My := Jp^«pAaAP d(vol)

B

is the contribution to inertia due to the microstructure.

One must accept the conclusion that the classical expression follows only if entrainment of the microstructure is, perhaps artificially, excluded. The contribution of “epiro-tation“ is given by the second term.

One may question the wisdom of our choice for v and come out strongly for a native separation of effects of en-trainment from genuine microkinematics. The difficulties, easily brushed aside when dealing only with a totally rigid rotation, stems with the ambiguity of the notion of entrain-ment, in general. Is it really to be linked with skw (gradx) only, as can happily be done for nematics, say, or with the whole L = gradx, as seems reasonable for microcrystalline materials, or even with higher gradients ofx, as may

be necessary when continuous distributions of dislocations are studied? The answer is still open.

Mention of entrainment suggests a devious interpretation of condition (7): during rigid rotation, entrainment occurs without expense of power. The emphasis was on the extremely simple type of motion, but we would fabricate a mirror argument: during any motion the microstructure is entrained, locally, as decreed by skwL, without loss of power though at the expense of added macrostress; significant derangement of microstructure may occur, over and above, but generally at a cost.

Such view of the motion occasions a broader reflection. Suppose now that the entrainment be linked to the whole L: there is a group {A} on M such that V(= A(F)v is the value reached by the variable from v as a consequence of the change in the gross placement measured by the placement gradient F. Calculate the ensuing infinitesimal generator L

v^F) = v“ + JL^y(Fik -Gik) + o(|Fik -Gik\ X where G is the shifter, so that, during pure entrainment,

■a 3V?F) • a v a=_i_! Fik + LOjFjkLij.

3F-,

ik

Throughout any motion, real or virtual, split v into the sum of a term, v, due to entrainment and another, v*, peculiar (in the sense embraced within the kinetic theory of gases). Insert the sum into (6) and separate contributions due to the two types of time rates.

Assume that the first contribution, due to entrainment, be always null, for any virtual L. Remark that v* would not be obtained by a locally rigid motion, in fact not even by an affine one. Ultimately one infers that the power density (6) must vanish for any choice of L even if v is substituted simply by v

TjLj + J aL^,hFJkLg + ^aLikFjkLij = 0 VL.

Hence a more circumstantial qualification for T, now totally bound to microactions

Tij = -^aLakFjk - JaLlk,hFjk.

Some, or even most, of the premises leading to this result may appear precarious; however, they might hold at least in special cases, when the gross distortions are driven by changes in the microstructure, such as when plastic flow is essentially engendered by the movement of dislocations.

These reflections lead one to conceive further an instance where the coefficients of k depend also on the gross placement gradient F (or, at least, on its invariants, such as detF). It would follow that the metric on M if delivered by k, would be influenced, at least, by gross density as occurs in general relativity.

2.4. Gyrocontinua

The title chosen for this second chapter will be deemed more deserved if we allow us a brief detour and illustrate a

case of micromachinery embodied in gyrocontinua [10, 11]. The topic might seem too special, too technical, even incongruous here. On the the contrary, in our opinion, it might be viewed also as germane, as it evidences (granted its, perhaps offending, robot-like features) interactions somehow akin to those exerted, within a purer context, by some magnetic coercions in an ether-type interpretation of Cosserat’s proposals. Gyrocontinua are ideal, lank fabrics made of a flexible frame bearing a large number of very small gyroscopes spinning about axes that are connected to the frame. The movement of the latter changes the moment of momentum of the gyroscopes, thus possibly giving rise to torques of inertia that, from these devices and through their connections to the frame, ultimately influence the gross movement.

More precisely, each gyroscope is supposed to be pin-fixed through controllable gimbals to a capsule — a small part of the frame, in such a way that their centres of gravity always coincide. Being a small part of the frame, the capsule is fully entrained by the motion of the latter and can be assumed to undergo affine deformations only. The gyroscope is only partly entrained by the capsule, depending on the linkage between them, and, in any case, animated by a spin about its axis that is assigned relative to the capsule.

A continuum model can then be set where the frame is smoothed out and material elements are capsules with gyroscopes, whose angular momentum is controlled by some external source. The description of the entrainment of the gyroscope, when the capsule where it is embedded follows the macroscopic motion of the body, is the main issue of the model. It gives an example where the ambiguity of the notion of entrainment can be removed adding specifications on how the microelements are linked to make a main structure.

In fact our study of gyrocontinua has shown the importance, in a mesomechanics theory, of these details, which are instead conceivably irrelevant when modelling deeper, perhaps electromagnetic, effects. In one elementary instance the issue becomes glaring: i.e., when studying the motion of a chain each rigid link of which contains a gyroscope; an additional asset of that instance is that it can be easily envisioned and, perhaps, experimentally implemented.

Let us take, in the reference placement of a capsule, the pin-to-pin direction g*, a particular plane through g*, of which the normal is the unit vector a*, and the normal to g* within this plane. A macroscopic motion of gradient F modifies these directions as follows (the —T exponent denotes the inverse of the transpose ofF, x is the vector product, and \ v\ is the norm of v)

„ Fg* „

g:=----------, a:=

\ Fg*\ ’

F~T a* „ „

\F-t \ , a xg. \ F a*\

(8)

The entrainment rotation of the capsule Q, i.e. the orthogonal tensor which moves g* into g and the plane normal to a* into the plane normal to a, is thus completely deter-

mined by the macroscopic motion (<8> denotes the tensorial product: (v ® w)j = vtwj)

Fg *

Q =

+

\ Fg *

) Z7

a*

F ~T a*

\ F ~Ta*\

&

a* \

Fg* \ Fg*

(9)

The total rotation of the gyroscope is given by Q = GQ, where G is the relative rotation between gyroscope and capsule.

The current inertia torque per unit mass in the present placement is given by (Jw), where Jis the inertia tensor of the gyroscope per unit mass in the same configuration ( a1 and 2 positive real numbers)

J := a1g ® g + a2(I - g ® g), g = Gg, and w is the total spin sum of relative and entrainment spins. The relative spin can be written in terms of a relative angular speed ro about g and a relative velocity of precession jxp (p any unit vector orthogonal to g), whereas the spin of entrainment is a function of the macroscopic velocity gradient L that can be made explicit starting from its expression in terms of Q and its time derivative (see [10] for details):

1 T

w = rog + |ag x p - — (e + h )L.

Here the third order tensor h accounts for the fact that the gyroscopic effect is not only due to the rigid rotations of the capsule bearing the gyroscope (as it can be the consequence of a local rotation of the body), but also to the deformations of this capsule when the pin-to-pin direction is not fixed in the deformation process: h := -g ® eg - [(eg) ® g]T+a ® ea+

+ [(ea) ® a]T - 2sym(a ® g) ® (<5xg).

It should be noticed that, due to h, the entrainment term differs from one that can be expressed through the orthogonal tensor issuing from a polar decomposition of F. To recover this simpler and common expression of the entrain-ment one should, at least in the present instance, conceive a different kind of linkage, with a rigid spherical shell of negligible mass included in the capsule and housing the gyroscope, joined to the capsule via elastic bonds (unstressed in the reference placement). In this case, and under the assumptions of quasistatic elasticity, we have shown that the minimization of the bond potential entails Q to be the above mentioned orthogonal component of F [10]. Notice also that the difference between this particular expression of the entrainment and the general one depends on terms of first order in the stretch.

For the ensuing balance laws and consequent remarks, we refer to the papers quoted [10]. We just mention, as a sample, one very special but striking occurrence: vibrations of a beam with embedded microgyroscopes can be con-

trolled through these devices diverting energy from one mode to another [11].

Consider a straight, elastic, simply supported beam with a continuous distribution of identical gyroscopes along its axis, say x1, all gyroscopic axes being along xv The speed of rotation of the gyroscopes is used as a control field, assuming that an external agent can affect it without sensible — or at least notwithstanding — retroactions. Furthermore, take this control to be simply of bang-bang type, i.e. rotors can only be switched on/off and neglect transient conditions due to this switching. Assume the beam oscillating naturally in the x1 - x3 plane, x2 and x3 being two principal axes of inertia of the cross section of the beam. It is possible to define a sequence of switch times for the beam to oscillate, at the end of it, in the plane x1 - x2. According to this sequence, the x1- x3 oscillations will have their amplitude gradually reduced, though not monotonically, toward zero, while the orthogonal oscillations are amplified, still not monotonically.

3. Ephemeral continua

3.1. Mean field theory

Already in the Preamble we mentioned models conceived to portrait the behaviour of bodies where the permanence of material elements was still assured, though the stance of specks within each could be amiss. Mixtures primarily but also many other media, granular assemblies to wit, fail to satisfy even the first axiom of permanence.

An inquiry on the global behaviour of granular media is a vast, even prodigious, enterprise. Even a neat breakdown into separate stages of their flows (dense to rarefied or quasistatic to fast) is still wanting to a large extent and, besides, often one phenomenon (a landslide, say) spans across any plausible, conceivable spectrum. Though the contours remain fuzzy, it is fair to expect that fast, sparse flows share some at least of the features of a molecular gas flow. Even then the order of magnitude of several quantities stays jarring: for instance, the peculiar speeds of grains cannot match those of molecules and, consequently, directly imported quasithermal measures end up by being grossly observer-dependent; an obnoxious event already evidenced even in the kinetic theory of gases, when that theory is pushed to high-order approximation via the Enskog procedure.

Hence the efforts expended to accommodate matters the least but enough to avoid awkward corollaries (e.g., by inserting explicitly apparent forces, as in many papers on extended thermodynamics). The heart of the matter appears to be the ascertainment of a field of local preferred frames with origin and orientation such that the densities of peculiar momentum and of peculiar moment of momentum both vanish (“peculiar” meaning here “evaluated with respect to the local frame”).

The route starts with the following epitome of reality [6]. The region B occupied at time t by the body (i.e., the set of all places where specks can be found) is fancied as split into loculi (i.e., representative volume elements) with a diameter indicative of the lower scale. At the gross scale each loculus e is labelled by its place x in B. However, each loculus is also observed at a magnification such that subplaces z = x © y can, in principle, be discerned in it; behold the notation: it reflects the crucial understanding that x and y belong to different Euclidean spaces. Indeed, we summon here a trivial vector bundle with the space of x as a base and the space ofy as fibers. In our earlier analyses one grain was imagined posted in each subplace; here, instead, as we shall see, an entire family of grains is conjectured to belong to y with a spectrum of velocities w.

In its most complete format, the scheme sees the body (of mass density p) subject to macrodeformations as in the standard narrations (x, place; v(x, t), velocity at x and at time t; L = gradv), but besides, affected, at the level of each loculus e (x) (i.e., of a representative volume element centred at x), by an affine mesodistortion, with rate B, which causes changes in a local mesoinertia tensor Y and in a tensor moment of momentum K (both per unit mass), while safeguarding the standard affine link between the two:

K = YBT.

Furthermore suffusion of specks from/to loculi might be present at a rate a, due to discrepancy between the two disfigurements

a = tr(L - B),

beyond an otherwise equilibrated cross-transit of specks at a rate measured by the correlation tensor of peculiar velocities, the Reynold’s tensor H (this name is used rather than “Maxwell pressure tensor”, to promote its kinetic rather than dynamic role here). Actually, in view of the supposed indistinguishability of the specks (all assumed to have the same mass ^), the tensor H could provide, rather, an appraisal of the frequency and direction of collisions among the specks.

Indeed, from B a tensor G can be deduced, the time rate G of which satisfies an equation slightly more complex than an expected one from analogies with L, precisely

Gg - +1 al = B,

2

just because of the presence of suffusion.

The fields p, v, Y, K, H arise, actually, from the possibly disorderly, stirring of specks within each loculus e via averages with a mesodensity ^0, if 0 is number density of specks at y (a subplace within e) and ^ is the mass of one speck (presuming that all specks have the same mass, as already pledvged).

In fact, 0 itself is a derived quantity, the fundamental entity being the distribution 0 which, at time t within e (x), expresses the numerosity of the crowd of specks transiting through the subplace y with velocity W. Verbatim

0(t, x; y) = J 0(t, x; y; w)dw,

V

where V is the full vector space of possible velocities. The use of 0 presupposes the availability of an absolute reference with respect to which w, in particular, is evaluated.

Notice that the distribution 0(t, x; w), which enters the standard account of phenomena by the kinetic theory, is the total distribution over e

0(t, x; w) = J 0(t, x; y; w) dy.

e

Thus, whereas 0 declares a property at the microlevel involving only specks crossing instantaneously the same subplace y, 0 lifts that property at the mesolevel merging the whole lot of specks within e into the spot x, though still keeping them arranged into batches in accord with their velocity; hence it preservevs the statistical character of 0, contrary to what occurs to 0 which tales rather, as we said, the attributes of a mesodensity, so that the value of the field p at x is given by

p(T, x) = ^roS-3, ro = J 0 dy.

e

The average velocity at y, w, is given obviously by

w = 0-1 = J 0wdw

V

and the value of the field v at x by v(t, x) = ro-1 J 0rody.

e

Similarly

Y = ro-1 J0y®ydy, K = ro-1 J 0y®rody

ee

and

H = ro-1 J 0c ® ~dy with ~ = w - v - By.

e

There is a cascade of peculiar velocities, including ~. The deepest one is the peculiar random velocity c

c = w - w or c = c - c

with

c = w - v - By.

Kinetic energy tensors per unit mass ensue: an exhaustive one W, involving all details of motion

W =1 ro-1 J J0w ® w dydw =1 ro-1 J0w ® w dw

e V

V

and a mesoscopic “reduced” one W:

~ 1 _1 r~~ ~

W = — ro J 0w ® w dy,

2e

for which an explicit expression is easily derived W = 2(v ® v + BYBT + H).

The difference

W = W - W =1 ro-1 J

ro J 0c ® c dc

V

(10)

(11)

is not accessible via the density 0 and the ensuing multifield theory alone; thermodynamic postulations must be brought to bear.

Let us define the “coshaping“ time-derivative of any tensor, say e.g. Y, as

Y = G (G ~lYG ~T) GT

(12)

and thus equal to Y& - BY - YBT. The balance equations involve the coshaping time-derivatives of Y, K, and H: dp

3t

+ div(pv) = ap

or

or

or

p + p tr B = 0,

p\dYL+ (gradY)v + aY - BY - YB | = 0

Y + aY = K + KT,

p| — + Lv + av | = pb + divT,

(dT % p \-dK- + (gradK) v + aK - BK - KBT & -

- pH = pM - A + divm

p(YBT - H) = pM - A + divm,

) dH T

p -----+ (gradH)v + aH - BH - HBT

3t

V

= pJ - Z + div j,

(13)

(14)

(15)

(16)

(17)

(18)

(19)

skw T = skw A sometimes, more deeply,

T = - AT.

Equation (18) substitutes, within the context, the classical requirement of symmetry for T.

To make the set into a forecasting tool, constitutive laws must be devised, appropriate for a special material in hand, for T, m, j and for Z and A (the last one obeying rule (18)), so that they all become pertinent functions of shape and kinetic state, possibly of L, B, Y and their gradients.

3.2. Steps towards an alternative deduction of balance laws

As we will recall explicitly in the next section, a theorem of reduced kinetic energy may be deduced from the balance equations (13)—(19), a theorem which obviously involves only fields at the mesoscopic level. However, the evolution within each loculus implies the deployment also of deeper microscopic energies not accounted for, explicitly, by the multifield theory; though they could at least steal in, indirectly, later through their possible influence on constitutive laws, not yet discussed. As already mentioned, the opening in [8] calls upon the distribution 0(t, x; y, w), though

later the strive is only on what can be achieved, in the account of events, wvith the bare use of the derived number density of specks 0. We have already recalled, in this vein, that a grave consequent handicap was evidenced when the full kinetic energy tensor W of specks was sought.

There are also minute discrepancies in the other balance laws, due to the discarding of terms involving gradients, within the loculus, of 0 and c, so that, strictly, the laws are correct only if the total in e of devy[0(By + c)] can be, in approximation, evaluated as the total of 0trB.

In general, because the dependence on y of neither 0 nor c can stem simply through the multifield theory, that blemish cannot be healed without an even subtler analysis of flows within e. The pretence of Molecular Dynamics is to pursue the niftiest route of following the path of each speck (supposing that there is only one such at any place y). An intermediate step could be a microcontinuum analysis within each e(x) based on the density ^0 and the relative velocity c, both, for each x, as functions of t and y.

Still an alternative option, advanced already within specific settings (see, e.g., [18] within the theory of nematics) is to brave the task of proposing an appropriate version of the Boltzmann equation to rule the evolution of 0 and pursue the goal of finding solutions. In principle, within the context, such aims appear daunting; beyond the so widely attended hurdles, here the stage widens with many new facets: for one, the interactions among specks cannot be restricted to collisions; coherence might even prevail over repulsion. Even conditions for 0 at the boundary of B should be specified and pressed to service.

A much more modest pursuit mimics developments within the kinetic theory of gases and tends to reach an alternative satisfactory deduction of balance laws already declared in Sect. 3.1.

Thus we confine ourselves here to a brief scrutiny of an equation of Boltzmann type, which is supposed to rule the evolution of the distribution 0 and which could have been really placed at the very beginning of Sect. 3.1, in particular before any mention of the equation of balance of mass. The question is delicate in view of the immense literature and the depth of opinions on the matter. We propose only minimal changes in the standard formulation of the equation, changes which take into account the peculiarities in our context: the emerging of distinct places within the loculs e (which adds the variable y in the distribution 0) and a boundary de within which y may roam (with consequent possible diffusion of specks) plus the effected splitting of the velocity w into the sum of entrainment velocity v + By, average peculiar velocity cc and the truly random ingredient c.

In principle, when the distribution 0(t, x; y; w) is available, it is possible to determine the average velocity v(t, x) of all specks in e(x), the average velocity w(t, x; y) of specks aty, the “best“ rate B(t, x) of affine distortion in e(x)

around x, and the average peculiar velocity c(t, x; y) = = wc - v - By at y, leaving yet, as a difference between w and w, the random velocity c:

w = v + By + c + c, c = c + c. (20)

To propose the Boltzmann equation possibly relevant here we follow and adapt the procedure illustrated in the treatise [19]: we presume that the time rate of change in 0 be as would occur if the specks “retrogressed“ from the present instant in accordance with the fabricated affine motion with the supervenient effects of interactions among specks (urging some of them belonging to a velocity class to change allegiance) left to be rendered by an addendum 0Z, as yet unspecified, intended to be a functional of the authentic motion:

30 30 30 30

— + — • v + —• (w - v) + —• f + 0q = 0, (21)

ot ox oy ow

where f is external force per unit mass on specks at y.

Accepting the format, presently we follow [19, Ch. V (iv)] and offer and exploit an interpretation of (21) alternative to that which led to its origin; in it we take x, y, w (together with t) to be independent variables (rather than functions of time, as up to now) on which 0 is contingent; we take f to be a known function ofx and y only and v and Z to be functionals of 0, precisely ) &-1

v = II0 =II0®

(eV % eV

(hence a function of t and x only) and Z still free to be chosen constitutively up to some conditions to be quoted below. In conclusion, then, we interpret (21) as a functional-partial differential equation for 0(t, x; y; w). An immediate corollary ensues by integration of the left-hand side over V (using index notations, for ease)

30 30 30(wi- - vt) „

— +------vt +——--------— = 0,

3t dxi dyi

(22)

a corollary which rests on two conditions, standard in allied theories,

l|^ = 0, I0Z = 0,

;3w J

(23)

Vdw V

of which the first is based on the assumption that 0 be continuous over the whole space V and tend to zero sufficiently rapidly when | w | ^ ^ (an assumption accepted with ease under “normal“ circumstances, but failing in critical cases); whereas the second restricts the choice of Z.

We may now accept (22) per se, retracing somehow our path in reverse, interpret in it 0 as a still indeterminate function of t, x and y taking the latter as independent variables on which also wc depends (whereas v is a function of t and x only as already mentioned). Then (22) becomes, in such our last view, the partial differential equation for 0. In particular, further integration over e leads to the stipulation of conservation of mass, i.e. to the first of (13).

Returning now to (21) and following the same train of thought which led to (22), but after preliminary multiplica-

tion of all addenda in (21) by w (and taking also again advantage of independence, at this stage, of x, y, w), by integration over V we get

30 wi 30 wi

i + i

3t

d ~ v j + I

axj v

3[0( w, - v,)wi] -

j 1 - 0fi = 0, (24)

dy,

provided that we add to (23) the sharper conditions

Ifs. = 0, l0Zwi = 0.

V dwj V

Further integration of (24) over e, an operation which gains meaning after a change of interpretation as before, allowing, in particular, the replacement

3[0(w, - v,)w;] r3[0(Bjkyk + c,)wi]

dy

,

=I

V

dy,

= 0^st

3c,- & Bjj +^^

dy,

leads to

3mvi- 3mvi-------- +-------- v

3t dx,

= f,

provided, as tacitly assumed in [6, Sect. 7], that c be sole-noidal, and finally to the law of balance of momentum (13).

Actually, to achieve the proposed cast of the right-hand sides of the balance laws of momentum and of moment of momentum as a sum of three addenda (one pertaining to actions due to the exterior of the body, one representing the influence from the interior of each loculus — absent in the first law — and the last from the interior of the body at the macroscale — in the form of a divergence), two old lemmata of Noll [20] can be called upon, appropriately adapted, to justify the qualification of the latter two.

These lemmata are also sufficient, at least under some prominent circumstances; precisely, when the internal actions derive from two potentials, 9* and 9 respectively, the first one ruling the interaction between specks within each loculus and the second one that between populations in different loculi. Then the combined force per unit volume on all specks in e (x) due to specks within all other e(x + z), with x + z e B, amounts to

I

x+zeB

d9

d| z 1

ro(T, x) ro(T, x + z) — d(vol).

If such self-action decays to zero sufficiently fast with increasing distance | z | and for z sufficiently far from the boundary dB and/or m(T, x + z) goes appropriately fast to zero when x + z approaches the boundary, then the integral above can be extended, without grave error, to all space (i.e., z to all vector space V): d9

divT = I

x+

“I

x+zeB d| z d9

-ro(T, x) ro(T, x + z)— d(vol) ■

V

d| z 1

ro(T, x) ro(T, x + z) — d(vol).

The lemmata lead to the following explicit expression of stress T(t, x), such that div T is exactly equal to the ex-

tended integral above

T(t, x) = — I z O z d9 v(t, x; z) d(vol)

V

d| z 1

with

v := I ro(T, x + Xz) ro(T, x - (1 - X)z) dX.

0

In fact

di=11iv d(vol),

dx, 2 v | z | d| z | dx,

and, on the other hand,

dv r ( 3<d(t, x + Xz)

dx,

w(t, x - (1 - X)z) +

+ ro(T, x + Xz)

3ro(T, x - (1 - X)z)

&

dx,

dX.

But

3ro(T, x + Xz) 3ro(T, x)

3X

3xk

-zk,

3ro(T, x - (1 - X) z) 3ro(T, x)

3X

3xk

-zk,

hence

z, = 1^- (®(t, x + Xz)ro(T, x - (1 - X)z)) dX =

dx, 0 dX

= (ro(T, x + z) - ro(T, x - z)) ro(T, x) and, finally,

^Tt = 1I jEl d 9 x

dx, 2 v I z| d| z |

x (ro(T, x + z) - ro(T, x - z)) ro(T, x) d(vol), as announced.

Similarly, for the balance of tensor moment of momentum, a third-order tensor m needs be found, such that

divm = I z O d9 ro(T, x)ro(T, x + z)^— d(vol),

x+zeB d| z 1 1 z 1

and the route followed above, though with a stronger obvious assumption regarding 9, assures us that the tensor

m

=11 z O z O -d9-v(T, x; z) d(vol)

2 v I z I d| z |

answers the need.

Besides, if 9*(x; y), as announced, is now the potential of actions internal to e(x), the vector

I-j9— c(T, x; o) 0(t, x; y)-y-d(vol)

e d|y1 1y1

vanishes, and, therefore, there is no extra term in the law of balance of momentum, whereas

ro

-11

Iy O 0(T, x; o) 0(t, x; y)^-d(vol)

y

d| y I IyI

is, generally, not null and must be accepted as an explicit version of the tensor A.

Remarks. There remains, in all results above, the unresolved doubts arising from the condition of rapid decay of actions within distance and from the trouble possibly occurring nevertheless near the boundary. Some vague similarity here with possible consequences of disregarding ancillary terms in earlier developments.

Evidence should still be brought forth for the tensors Z and j appearing in the last balance equation. The matter appears to be less easy and is left open here.

3.3. Giving extended mechanics a thermodynamic role

The developments of Sect. 3.1 and the suggestions of Sect. 3.2 have also energetic consequences that mimic some usually classified under a thermodynamic heading.

To put the matter into appropriate evidence, we must premise a theorem of kinetic energy associated with the balance equations (13)—( 19). For any fit region 6 belonging to B and under appropriate smoothness conditions, it reads, in tensorial form (i.e., as a virial theorem), as follows

I p{W - sym[B(B7BT) + H ] + aj~}d(vol) =

= Ip

sym(v O b + BM) + — J

d(vol) +

+ I jsym[vOTn + B(mn)] +1 jnVd(area)-

96 X 2 J

-I

— Z + sym(LT + BA + bmt)

d(vol).

(25)

The sum of the first two addenda under the integral sign within the left-hand term expresses the coshaping time derivative of W (see Eq. (12))

— 1 —

W = - (v O v) + G[G- (W - v O v)G ~T ] GT;

besides, n is the unit vector normal to d6, the exponent t means minor right transposition when applied to a third order tensor, namely (bmt), = b^m^.

The tensor power of internal actions has density

sym ("2 Z + LTT + BA + bmt %; (26)

it is invariant under observer motion if and only if

T = - AT,

under the restriction for Z to depend on the kinetic quantities only if they are intrinsic (such as a, for instance); the remark is the basis for the last balance equation (19).

The scalar power, i.e. the trace of (26) is similarly invariant under the already mentioned looser condition (18), again on the proviso that trace of Z be independently invariant.

Peculiarly, the tensor J does not affect the power (26); consequently we will assume later that it does not react to the enforcement of any internal constraint. Similarly, the absence of the deviator of Z persuades us that it cannot

contain, under internally constrained circumstances a reactive additive component. Of course, the matter may be contentious and, again, some later developments, based on the mentioned convictions, may possibly need amendment.

Formally one can write a further equation for the sole portion

W* = W -1 v <8> v = 1(BYBT + H)

of the kinetic energy W, portion due only to the deeper motions accounted for merely by the multifield theory. Under conditions of sufficient regularity and in view of the arbitrariness in the choice of b, that equation can be written in differential form

p Y W* - 2sym

B -1 o! |W*

=p

1

sym BM + — J

+ sym div (Bm) +

+ —div j -

-2-Z + sym (BA + bm‘)

Taking the trace of both sides and using the notation

k* = trW *,

X = p tr[B(2W * + M)] + ^tr(pJ - Z) + atrW *,

hi = -Bjkmkji - ^ Jjfi,

we obtain a scalar equation in a format convenient for thermodynamic speculations

pK* + B • A + b • ‘ m = pX- div h.

(27)

In fact, from a purely formal point of view, (27) seems to tally exactly with standard enunciations of the first principle of thermodynamics, provided the deeper kinetic energy be deemed to count for all available internal energy, the energy flux due to the torques and agitation to be taken as heat flux and a similar congruence with heat generation rate be forced on X. To call (27) a principle, rather than a theorem, is to twist a misnomer, lifting the margin between dynamic and thermal effects just below standard continuum dynamics and pushing mesoscopic events into an (in fact, unwarranted) maelstrom together with really chaotic peculiar events. Actually, if these latter events can be energetically of scarce relevance and, consequently, their effects disregarded, one may dream to treat “thermodynamic“ phenomena on the basis of (27) with appropriate constitutive choices. If our interpretation of their scheme is not improper, Rubi and Vilar [21] seem to us to propose to steer just along such a course, with many claimed advantages: no need to restrict matters to the immediate neighbourhood of equilibrium nor to make amends for irreversibility.

A grave matter is still left out of account: to choose an allied notion of “temperature“, a notion workable at least within the context. That apparently obvious concept is spoiled by a degree of ambiguity, outside the special realm

of perfect gases. There the equation of state relates absolute temperature to pressure (and even here, to wit, the “stress“ pH could be linked with a sort of anisotropic temperature 0:

H = —k0----------,

p meas(e)

k is the Boltzmann constant), but, besides, absolute temperature is the essential unique parameter deciding the distribution of energy among molecules. To approach a parallel goal within the present context remember that

k* = trW * = — f ~(| Byl2 + ~2)d(vol).

2® e

Find the subset e*(£) of e where

|-(lByl2 + ~2) <m£K*

Then the derivative y(£) of m-1 f 0 d(vol) is such that

e-(£)

Y(£) d £ gives the value of the fraction of specks with energy within the interval [£k*, (£ + d£)K-].

Within the kinetic theory of perfect gases, under conditions of quasistaticity, y(£) depends exponentially on £, the opposite of the derivative of the logarithm of y is a constant and, apart from the factor

2 meas (e) p 3k<d

the constant is exactly the absolute temperature. But the dependence of Y on £, even if still exponential, might be such that the conclusion fails because an extra factor intervenes (a factor which is equal to 1 only in the special sterling case); the factor might even be negative, if the support of Y is compact.

Ultimately y(£) need not be an exponential function of £ at all; it might even be non monotonic. As Rubi writes, the medium might be at the same time hot and cold! All avowed rules of heat flow must be mulled over anew. It is sobering thought that conditions flippantly assumed on Y when deducing the law of energy balance from the Boltzmann equation (continuity, sufficiently fast vanishing when £ ^ ro, etc.) might fail.

4. Hybrid continua

4.1. A population of deviant specks — fluctuations around the mean

It was already mentioned toward the end of Sect. 2.1 that it might not always be possible to satisfy strong boundary conditions of place for the microstructure on dB with smooth fields within B; in a sense, the requirement of perfectly smooth order within B issuing from the strict conditions on dB is denied by the straightjacket of the bare three dimensions available for B. Corresponding experiments display mushy zones inside the cell bounded by dB. To achieve a satisfactory model of reality without abandoning the tenet that each material element be always made up of

the same specks or molecules, one must at least admit that not all specks share the same attitude.

Thus the theory must be modified to allow for partial order, up to complete disorder. The most direct approach calls for an extension of the multifield theory to comprise parameters measuring the proportion and type of order left.

To begin with, think again the material element as made up of many specks of matter, each distinguished now by a feature measured by a different value of the microstruc-tural variable. If the values, though not all the same, are nevertheless sufficiently congruent, some average may appear obvious, it prevails physically, thus it can still be taken as a sufficiently descriptive of a slightly fuzzy reality and the scheme proposed in Sect. 2 still applies.

Otherwise and similarly to suggestions in Sect. 3.2 for ephemeral continua, the next step is to count within each element, at x say, the proportion y(v, x)dv of specks for which the value of the microstructure is in the immediate neighbourhood of v; thus, because v takes now the role of an independent variable, one begins, if timidly, to open up the route towards multivariable theories. Then, in mathematical terms, one needs to establish the physical laws regulating such deep distributions and finally, at each time, face the problem of determining the mapping of B into some appropriate space of functions Y with domain M and codomain in the interval [0, 1]: in a sense, the range of the microstructure becomes infinite-dimensional; in an alternative view the spread of the variables of place soars over E X M (multivariable approach). Before going to such extremes, one can seek again some expanded average within the now unavoidably very fuzzy context and endeavour to state the presumably simpler laws regulating such averages, leading to a sort of mean field theory.

To pursue first just the latter matter, the simplest course is to resort to the “easy“ theorem of Withney, which assures us of the existence of an embedding of M in a linear space of dimension 2m + 1. Actually, as mentioned already in Sect. 2.1, it befits one to assume of M to be Riemannian, so as to have access to a metric on M ; then, as an alternative to Withney’s, a deeper theorem of Nash [22] comes handy, a theorem which proves the existence of an embedding of M in a Euclidean space of dimension 2m + 1 even under the condition of isometry. Thus it is feasible to interpret M as an m-dimensional surface M in such a space J. Though our use of those theorems is here naive, as a hint for progress rather than as precise tool, for each of two examples we will construct an embedding explicitly and find it physically significant. Remark that the proposed embeddings are not unique and there are instances when they can be achieved in a space of dimension lower than 2m + 1; indeed the prototypes quoted above (where m is equal to 2 and 3 respectively) display one such downgrading (with J of dimension 5 in both cases, see [10, 23]).

Ultimately, physical discernment must decide the choice of embedding, and we will call m the dimension of the

consequent J. The essential message is that J exists and, in it, averages can be evaluated in a straightforward manner. We will call ^ a point in J. so that, when the probability measure on M is absolutely continuous with respect to the m-dimensional area of M , as broached above leading to the assignment of Y, the following normalization condition is satisfied

Jy(^, x)dC~) = 1 (28)

M

and the average at x is given by <^(x)}JVy(^, x)d(M).

(29)

M

Averages fall, generally, outside the image M of M in J. and fill altogether the convex hull H of that image (see the related “Biscari cone“ for uniaxial nematics).

Any point of J which falls in H may be taken to represent an enlarged view of the microstructural features within each element; thus, at the cost of an increase in dimension to m, we achieve the aim of a satisfactory portrayal of averages, with the subsidiary advantage that the space we work in, finally, is Euclidean. The ease of the success has a cost: many different distributions y(M-, x) lead to the same value of (m-(x)}; at least a conjecture can be broached: if the dimension of J is minimal, the points of dH which belong to M are points of strict convexity and, when interpreted themselves as the result of averaging, they derive from a unique distribution which is a delta-function and correspond to conditions of perfect order within the element. Contrariwise, complete disorder is represjnted by the average ^ 0 of a uniform distribution y0 on M:

^o = Yo JVW (30)

M

or, by the normalization condition, ^0 = (meas M~)-1 J^d(M~).

(31)

M

Notice that ^ 0 is a global quantity, which depends on M and on the embedding (which leads to M) only.

In principle there are no difficulties in dealing with the enlarged model of the body; formally one needs only to write ^ instead of v wherever the latter appears in our report of Sect. 2. Constitutive laws may not be so easily formulated. One preliminary step is the extension, call it G, of the group G, and also that, call it r, of the homomorphism r and, finally, the extension A of the infinitesimal generator A, so that they now apply to elements of J. The role of G, r and A is essential in expressing the extended law of balance of moment of momentum and in formulating the partial differential equation

—IA = 0 (32)

d|l

V /

(the symbol £ denotes again the potential of internal agencies, but the function is now different than in the previous occurrences of that symbol as we are here dealing with ave-

rages), which must be satisfied by functions e(^) which are invariant under any change of observer characterized by the orthogonal tensor Q:

e( ~q №)) = e(M) (33)

Condition (32) may be interpreted as a system of three linear partial differential equations for a function e of as many coordinates as are the dimensions m of J. Of these equations, all tree are independent if the characteristic of A is 3; otherwise the independent restrictions imposed are fewer (equal in number to the characteristic k of A), in fact even none if A vanishes.

By known classical theorems the system (32) admits m - k independent integrals, which may be taken to form together an invariant basis for |x. The invariants are the essential variables in the constitutive laws and lead, as we shall see in the examples, to convincing definitions of order/disorder measures.

In conclusion, through the embedding and averaging, the wide population of microstructural events occurring within each material element is reduced to a finite set of parameters, actually a single element |x of a Euclidean space J of finite dimension. Then the question arises as to the constitutive nature of the interactions between neighbouring material elements and the evolution of the “mean“ micro-structural variable. Within the limits of multifield theories it is assumed that both the interactions and the evolution depend on the usual fields describing the motion of the continuum (say, the deformation gradient F) and the field of “mean“ microstructure ^ e H c J and its gradient. In this kind of models, once the “mean“ microstructural parameters are chosen, their statistical origin from a distribution of truly microstructural events is forgotten. The only field playing a role within the model is the field of “mean“ microstructure: the model follows all steps described for the case of continua with microstructure, but for the choice and physical meaning of the manifold J, as we have already mentioned.

Averages, though, might not offer an adequate portrait of events; one might need to press an investigation to reach explicit knowledge of the “presence“ y of the “phase“ v, or the deep density p(T, x) y(t, x; v) within the “ether“ E x M; an ether which now provides the inescapably whole stage, not conveniently shelved into base and fibers.

The need for such penetrating view arises, in particular, when a non-local “domino“ impact can be expected to affect even distant (in E) material elements when the stance v is the same or nearly so. Such urgency may require recourse to a quantity mathematically mimicking standard stress, though quite differently motivated, as a dual to flux of presence, for instance.

The homomorphism r may be summoned up also in the proper objective assignment of the distribution y; rather than envisioning the latter as a function of v (or, rather, of |x on M), one can define first the vector a = ^ - ^0 (which

vanishes at complete disorder), then, for any Q e Orth+, the modified vector

a = (gQV) (34)

taking finally for Q the local macrorotation at x

Q( x) = {F (FTF) -l2}x.

If y is expressed as a function of any dependence on the observer is avoided. The variant (34) is of particular relevance only for solids; the advantages or otherwise in the use of a instead of ^ or a need still to be explored.

Now our view of events, in formulating problems, can be split in two stages. First we could pursue the task of finding the rules obeyed by averages (^( x)) or, equivalently, by the vectors (^(x))-^, taking into appropriate account what happens in the neighbourhood of x, on the basis of balance equations of the type (2) and (7) and the associated boundary conditions, as explained in Sect. 2. Next, we could go, separately, one stage below; assume that, for immediate purposes, a survey of events exclusively within each material element at x suffices and seek rules of evolution of y as a function of the vector a, the connecting kingpin between such separate mesoscopy and the microscopy being the property

a = (|a( x)) - |a 0 = J a Y(a) d(area).

M

The significance of the first stage per se is promoted further by looking into it along an alternative line as follows. The permanence of the metric under the embedding of M into J, assured by Nash’s theorem, allows one to maintain the implicit physical significance to scalar products. Suppose that there exists an element on M, say 9(x), such that the distribution y be locally always Maxwellian, i.e.

) &-1

Y = kexp(9a), k = Jexp(9a)d(area)

M M

Thus one can seek conditions such that not only at any choice of 9 in M there corresponds an average (^), but also vice versa, given ^(x) in H, one can determine uniquely 9(x) leading to |x(x) as average. Finally, a parallel can be drawn with the classical kinetic theory: 9 can be attributed the character of a microstructural “temperance“ as a compact and, under the circumstances, exhaustive describer of the disorder [24].

Pursuing the line of thought suggested in the opening lines of this section and developed for ephemeral continua in Sect. 3.2, one is induced to assume that, at each place x, and time t, the convected time-derivative of y be determined by some constitutive functional E of y itself

!T+Ix+fY~i=wy'^ .i.

(35)

Next one introduces the idea that the influence of Y(a') on the values of E at a be weakly non-local; i.e., that it be primarily dictated by the values taken by y when a' in the

immediate neighbourhood of a. Hence E may be replaced, without great loss of accuracy, by a function of Y(a) and some of its gradients evaluated at a. The proof of the validity of such replacement was offered, within different contexts, by many authors; the pioneering work of Colemann and Noll is evoked again in [25]; it suggests the estimate

Ea'eM__ [Y(a'); M “ a(Y(a); ^) +

+ 6(Y(a); ^o) IgradaY|2 +c(Y(a); ^.)^Y (36)

leading, by substitution in (35), to a differential equation of the Fokker-Plank type. Naturally, the solutions one seeks must satisfy the side-conditions (28) and (29).

Thus, in the twin-stage approach sketched above, a boundary-value problem in the Euclidean space controls the field of averages and a separate evolution equation (where the average enters as a parameter, whereas the primary independent variable a runs over a manifold) directs the choice of the distribution y. So one avoids what appears to be the main flaw of somewhat similar mesoscopic proposals where y is thought as a function of ^ directly, rather than indirectly through ^0 and a (thus, by the way, skipping, perhaps unjustifiably, worries about possible lack of objectivity) and a forthright influence of space gradients of the average is dismissed. Consequently, in those proposals, there is no need to resort to equations of balance of microactions such as (2); on the other hand, the possibility of dealing persuasively with boundary-value problems in physical space is fortified.

Doubtlessly the twin-stage approach grounded on an appropriate extension of (2), (7) and on (35) or (36) is complex; so much so that no instance of solution of a special problem based on it seems to have been sought; in fact, even explicit satisfactory constitutive proposals, for the functions a, b, c in (36), say, seem to be missing as yet. Besides, the potential of the approach in gaining additional information may be deemed to be modest, because the intercourse between the mean field and the field of distributions is presumed to be, essentially, one-way, the latter being subservient to the former. In statics the approach leads to a restriction on the choice of y within the class of functions which satisfy (28), (29) and are such that the right-hand side of (35) (or (36)) vanishes; in a sense it requires y to be canonical under the circumstances.

Of course, bare knowledge of y is relevant, for instance, in the evaluation of the consequences of certain processes of fabrication; then the befitting constitutive modelling of the drastic dependence of E also on macroscopic strain and stress becomes the critical issue. On the whole, the dearth of precise corollaries leaves the matter contentious; in any case no satisfactory (let alone simpler) alternative is at hand; rather, the even more complex deep space approach was proposed and pursued.

The macroscopic stress-strain rate relation is written in terms of y the following way. Ideally, the macroscopic stress

is the mean value obtained through y from some microscopic distribution of stress. The microscopic stress-strain rate relation being supposedly known, the former assumption leads to a relation giving the macroscopic stress as a functional of the distribution of microscopic strain rate and of the distribution of orientation. Then the question of how the microscopic distribution of strain rate can match a particular macroscopic value of the strain rate is a matter of homogenization procedure. Most often, following Taylor’s assumption, a uniform strain rate is taken through the material element, so that the localization procedure is simply an identity.

Anyway the macroscopic description of material elements is thus based on macroscopic parameters and on the microscopic distribution represented by y, while the evolution ofY is obtained on the basis of two assumptions: property (28) preserving the total number of individuals in the population of grains packed in the material element, and the evolution of lattice orientations as given by integration of the lattice spin. This spin is derived as a function of the present lattice orientation (a microscopic variable) and of the overall strain rate of the element (a macroscopic factor).

In conclusion, in the study of polycrystals, a mild fluctuation around the mean field is considered, though in a very restricted sense it enters the constitutive equation of the stress only.

4.2. An example: nematic liquid crystals

In the case of nematic liquid crystals, for which M is the manifold of directions, hence of dimension 2, Whitney’s embedding can be realised in a linear space of dimension 5. Each direction is put first into one-to-one correspondence with the tensor n 8 n -1/31, where n is any one of the two unit vectors having the required direction, I is the identity and, in the context, ^0 is equal to 1/3 trI. All those tensors belong to the linear space (with dimension 5) of the symmetric traceless tensors; one of them, say N, will be the average when the element contains molecules with varying degree of orientation

N = J| n 8 n -—I |yd(area), 1 = Jyd(area),

J2( % J2

where J2 is the unit sphere.

The principal axes of N provide the frame upon which details regarding the distribution of orientations can be assigned. Still, already the eigenvalues of N, call them Xi determine two parameters which describe essential traits of the distribution: the degree of prolation s (called also, by Ericksen, degree of orientation) in [-1/2,1]:

) i 3 y/3

— (37)

=3

and the degree of triaxiality (or optical biaxiality) in [0, 1]:

3 1/3

p = 31/221/3 n(Xi -Xi+1) . (38)

i=1

Perfect ordering corresponds to the values s = 1, P = 0; optical uniaxiality occurs where P = 0; “melting“ of the liquid crystal occurs when both parameters vanish (see [23]).

Remark 1. The eigenvalues Xt or the parameters s and P determine also the variance of the distribution:

2

J

Y-N

d(area) =

= 3- (X1 + X2 +X3).

Obviously the variance is the greatest when both s and P vanish and vanishes when s = 1 (and, necessarily, P = 0). Suppose the distribution be Maxwellian

Y = k exp

k =

B\ n 8 n-—I 3

J exp B In 8 n -11 | d(area)

_J2 LI 3 JJ

-1

with B (the nematic temperance tensor) symmetric and trace-less. Then N is determined by B; the principal directions of N and B coincide and, in particular, N = 0 when B vanishes. Thus one can work with the tensor of temperance (and its inverse, the temperature tensor) instead of N. In the case of optically uniaxial distributions (when two eigenvalues Xi coincide and hence P = 0) one can express explicitly a scalar temperature as a function of s and one finds that it has the sign of s, tending to 0+ when s — 1 and to 0- when s —— -1/2 and going to ± <» when s — 0“ [24].

Remark 2. A very special corollary of theorems in [24] is that H consists here of all symmetric tensors N such that the ordered spectrum Xj < X2 < X3 satisfies

X3 < —, X2 +X3 < —,

3 3 2 3 3

X1 + X 2 + X3 = 0.

Remark 3. To proceed beyond the mean field theory for nematics, as an alternative to a direct study of the map of distributions y, Muschik and coworkers have recommended an ulterior extension of the multifield theories to embrace alignment tensors up to an order higher than 2, such as J Yn 8 n 8 n d(area), etc. It is not clear if such proposal be M

workable here, though there is the far-fetched analogy with extended thermodynamics where interesting theorems were proved involving moments of any order.

The evolution equations for N are, formally, of the type (1), (2) and (7), though, now, v is a symmetric tensor N. Actually, N must also be traceless; in analogy with the standard nematic case (when the vector n is unimodular), either equation (7) is modified by an additive term which is the product of a Lagrange multiplier by, here, (trN)I or that equation is written in terms of another Lagrange multiplier for skwN.

Again there is only scarce experimental evidence of inertia effects. Thus, mainly quasistatic phenomena are stud-

ied; alternatively the simplest expression, quadratic in N, is accepted for the kinetic energy. Any complexity of dynamic behaviour is attributed to the multitude of viscosity parameters which necessarily enter into play.

Relatively simpler is the setting of equilibrium problems, when one needs choose only the expression of the elastic energy of (here, partial) orientation; mostly, the Landau - de Gennes proposal is accepted:

-K |VN|2 +atrN2 -btrN3 + ctrN4,

where k, a, b, c are appropriate constants.

4.3. A second example: orientation distribution in polycrystals

The qualitative description of polycrystals as clusters of crystallites, each showing a perfect lattice, pertains to a particular range of observation scales. Actually, at the scale of most engineering applications (when structural sizes are much larger than those of the largest grain), material elements must be taken which comprise each a population of thousands of crystallites and are characterized by a whole distribution of lattice orientations. No account is taken of shapes and boundaries of grains (i.e. of clusters of equally oriented crystallites) and such polycrystalline material element is described simply through an orientation distribution function.

Consider a cluster of such crystallites, each uniquely identified through a proper orthogonal tensor N, modulo rotations of n about any m(i); call M c SO(3) the subgroup of such rotations.

Represent each crystallite uniquely with the tensor

3

£m(i) 8 m(i) (39)

i=1

belonging to

M ~<M e Sym|trM = 1,trM2 =X(m(i))4,

i =1

(40)

det M = n (m'i)) i=1

Any tensor M of that subset has three distinct eigenvalues (m(i))2, with the corresponding eigenvectors identifying the direction of the vectors m(i). The spectral decomposition of such a tensor M is thus

| m(1)| 0 0

M ({m(i)}) = N ??0 | m'2)| 0

_0 0 | m'3)

NT

(41)

and one can suggest the embedding in the affine space of symmetric tensors with unitary trace:

M — J = {M e Sym| trM = 1}, (42)

or, as in the previous Sect. 4.2, the embedding of Q = M --1/31 in the linear space of symmetric traceless tensors.

A distribution of crystals can be given a first rough representation through the “mean“ crystallite defined as:

M:= Jy(N)M (N) d(SO(3)). (43)

SO(3)

If the distribution is one of perfect order, with all crystals oriented as specified by some N, then y = §(N and M = M (N) has three distinct eigenvalues, equal exactly to (m(1))2, (m'2))2, (m'3))2, and the corresponding eigenvectors represent the axes of the crystallite and we return to the circumstance examined in Sect. 4.2. Contrariwise, if the disorder is complete, then M is spherical and no preferred axis can be assigned to the average. Intermediate conditions are possible, with the “optical“ properties of the aggregate cSorresponding to the number of distinct eigen values of M.

Introducing the eigenvalues Xt of M -1/31 ordered as in Sect. 4.2, two characteristic parameters ~ and P can be similarly defined, the latter proportional to the root 1/3

of the product n (Xi- -Xi+1) and the former, again, mea-

i=1

suring a degree of orientation, given by

13 X &1/3

n-

V

i=1 | m

(i) 2

-1/3

Remember that

1

— < (m(1))2 — <X, <0 <X3 < (m'3))2 — <-

3 3 1 3 3

so that (m(1))2— 1/3 implies (m(2))2 — 1/3, (m(3))2 — 1/3, and (m(3))2 —1/3 implies (m(1))2— 1/3, (m(2))2— 1/3. Also Xj — 0 implies the vanishing at the limit of X2 and

X3 and X3 — 0 implies the vanishing at the limit of X1 and X 2. Perfect order occurs when |~ | = 1, the sign distinguishing prolate from oblate crystallites; perfect disorder leads to the vanishing of ~.

Remark 1. As per Remark 2 of Sect. 4.2 the convex hull H consists now of all symmetric matrices such that

12

21

X3 < (m'3))2 —<-,

3 3 3

X2 +X3 < (m(2))2 + (m(3))2 --<-,

2 3 3 3

X1 +X 2 +X 3 = 0.

The balance equations can be written following the developments of § 21 of [4] for a continuum with affine microstructure M, taking into account also the properties of symmetry and of constant trace of M. Together with the non necessarily symmetric Cauchy stress T, the equilibrated internal force Z and the microstress s need be introduced as, respectively, a symmetric second order tensor and a third order tensor enjoying the property of minor left symmetry.

The balance of moment of momentum is then

skw'T -ZMT -MTZ + stgrad(MT) + grad(MT)st) = 0 and the balance equations for the microstructure in statics

(44)

is written in a particularly expressive form after introduction of the following decomposition of the microstresses into rotational and non-rotational components:

W = ZMT + MTZ - (st)grad(MT) -

- grad(MT) st = skwT,

w = t ((MT) s) + (MT) s,

X = ntZN - grad( NT )(st )N -

- ((NT s)t )(grad( NT ))T, x = (NT (st) N )t

(N is the orthogonal tensor defined above through the spectral decomposition (41)), so that two orthogonal conditions can be written, for rotational equilibrium and for the equilibrium of texture:

pW + skwT + div w = 0, pX + X + div x = 0

(W and X are an external torque and an external action on texture per unit mass). In particular the second of (44) can be written in terms of deviators as a pure equation for the active parts of the stresses of texture, the reactions to the constraint trM = const affecting the spherical parts of X and divx only.

5. Deep space dynamics

5.1. Multivariable theories

So far the microstructural entity was brought into play, primarily, to secure an additional field on B with the aim of a more truthful portrait of the body; however, already as an accessory in the spelling out of averages, it took rather the alternative role of independent variable, lending explicit content to a mean field theory. Then, it is as though each body element, which, by its very nature, is devoid of Eucli-dean’s dimensions, spread out in the fictitious space offered by M ; this metaphor is pursued by multivariable theories, carefully, both because of the physical subtlety of the ensuing model and because many convenient attributes of the Euclidean space may not be available on M. Clearly the primary field defined on E X M roves to be the distribution y, placed in a space of functions with the properties (28) and (29). Consequently, at the outset, multivariables theories address a restricted problem of accessing to the distribution y, assigning laws which link it to the gross flow and rule its evolution. For such laws one may find inspiration either in the theory of mixtures (the problem being thus transferred into that of prescribing the term which expresses the rate of creation of each component at the expense of others) or in the kinetic theory of gases (substituting the collision operator with an alternative functional). A phenomenon which could be then studied analytically is the separation occurring in the flow of a gas through the Ranque-

Hilsch vortex tube leading to a realization of a Maxwell demon [26].

An even deeper approach may be suggested, where a distinct dynamics is created for the evolution of subelements (i.e., specks in an element with different values v). Then an additional term must be added to the expression of the kinetic energy, a term due to changes in the distribution y of the population of subelements (echoing the term H in the theory of ephemeral continua [6]). Such term is usually understood to depend on thermal parameters in such a way to exclude inertia effects, but with the unfortunate consequent exclusion of a sharper second sound and the imposition of an infinite velocity in the spread of deep perturbations. Arguments for or against such addition are perhaps only partially relevant here, as we have in mind principally the study of quasistatic phenomena. But the ingress here of an additional balance equation (containing or not inertia terms) is of overwhelming relevance; it must involve a distribution stress urging on a reduced value of the variance or other significant action.

The need for such penetrating view arises, in particular, when a nonlocal domino impact can be expected to affect even distant (in E) material elements provided their stance v is nearly the same. Such urgency must be modelled by a quantity mathematically mimicking standard stress through quite differently motivated.

Therefore, going beyond the model of Sect. 2 but using the same background (Euclidean space E and manifold M), the body is imagined now as a set of subelements. All subelements lying in a particular place x form together an element. It is fair to repeat that each subelement has no distinct subplace within the element, contrary to what was here allowed in Sect. 3; it is characterized by the value of its substructure but that value is not exclusive. Subelements with the same value of v are wholly indistinguishible; the subelements forming an element might even have all the same substructure. Only the fractional number density

Y(t, x; v) is relevant; y(v, x) is the presence of subelements of type v at x. It must satisfy (28), whereas equation (29) determines again the average ,. Equations of the type (7) and (2), together with (1), might still be invoked to determine averages (v being formally substituted there by |x) as fields over B. An equation of type (35) must be decided to determine y; it is interpreted now as an equation of conservation of mass. In it E is interpreted as the actor of changes in the number density of subelements of type v at the expense of others; the integral of E over M must vanish. Finally, a distinct dynamics for the evolution of subelements must be created.

Unfortunately, a general approach to such deep-space dynamics is missing as yet (hints are given in [27]). The only special case studied in detail so far is that of a polycrystalline materials. The bare essentials of that analysis are recalled in the next section.

5.2. An example: polycrystals

A polycrystal is an aggregate of grains, each possibly divided into subgrains; each subgrain displays a crystalline structure with identifiable directions, and can thus be attributed a substructure taking — for instance — the lattice orientation angles as the coordinates on M . At the boundary of grains and of subgrains the lattice jumps across interfaces that are more or less sharp depending on circumstances. Sometimes, relatively thick interfaces might exist between grains, where no lattice can be defined at all. Often the lattice bends within grains or subgrains, making the experimental identification of crystalline directions tricky. The superior complexity of nature notwithstanding, to begin with, we will imagine a grain as a small part of the body having a regular crystalline structure and admit the polycrystal to be an assembly of such crystallites.

Thus, at first, the body can be imagined as in the multifield approach: it occupies at a given instant a fit region B c E and a function v valued in M is given on B. To cope with the even simplified polycrystalline nature, the model comprises a thorough portrayal of the grain pattern, with discontinuities of the function v delineating the grain boundaries. Clearly a coarsening of this representation is suitable for many applications and the path depicted in Sect. 4 can be followed for this purpose. Then a population of grains is represented by an average substructure and an evolution equation for the presence function, but the effect of the latter on the former is, as we have already mentioned, disregarded.

To achieve a model viable at the macroscale a large number of grains must actually be included in a material element, but the interactions between grains in the element cannot be neglected. Whence the need of a model where subelements are distinguishable, even though less distinctly than by a refinement of the subdivision of the image of the body in the Euclidean space.

Our proposal adapts ideas given in [28] and disregards information on the spatial distribution of heterogeneities within material elements. In addition, we make the identity of a grain depend only on the orientation of lattice within it; let grains merge or split, shrink or grow at the expenses of their orientational neighbours.

That process is supposed to be driven by interactions depending primarily on the relative orientation of crystallite’s lattices and being greater when the differences are smaller. The differences can be objectively measured by the geodesic distance on M . Thence we define an integral operator that allows us to gather some of the information content of refined multifield model described above (we will call micro), based on the taken of a test substructure.

Let be a fit function derivable with continuous derivative as many times as needed, let 8M (,, v) be the geodesic distance between any two elements |x and v of M; let us define an “observation“ function O, of the substruc-

r-

tures from the test substructure , by composition of these two functions

V(,,v)eMxM O,(,,v):=<;1(8m(,,v)).

Notice that M being a subgroup (eventually proper due to crystalline symmetries) of the rotations of the Euclidean space, it can be represented by a map R(v) in Orth+. Thanks to the transitive property of rotations, the tangent and the cotangent spaces at any v e M will be represented through the tangent space at the identity, which is the set of skew symmetrical second order tensors; these spaces are {Y | YR(v)T e Skw} = {Y | R(v)TYe Skw}.

Then we can show the following transport properties of the derivatives of O.

,

dO,

3R(v)

RT (v) = R(M-)

I dO,, &

T dO,

RT (v) ,

I dO,

3R(v)

3R(,)

&t

3R(,)

R(,).

Similarly, we can choose a real valued derivable function c2 and define the observation function of positions in the Euclidean space Ox:

V(x, y)e ExE Ox(x, y):=c2dx-y |), with the transport property

grad xOx (x> y) = - grad yOx (x, y).

Let E be a linear space, f any field B — E, and v the field of substructures B — M, we call “globalisation“ the integral transform (here the superposed S has a different meaning than in Sect. 3)

f(^ v) := J f(y) O, (, v(y)) Ox (x, y) d(vol)y ,

B

whose result is the field f defined on the “deep space“ E X M with values on the same linear space than the original function E and having the same variance with respect to changes of the observer. The choice of c and c2 can be made such that:

J f d(vol)x d,= J f d(vol)y .

BXM B

The following theorems ensue, allowing one to exchange differential operators and globalizations:

p~ =3M + div, (p f R(,)RT ) + div, (p~x), (45)

I

div yT = div ,

&

-R(,)gradyR (,)T

+ div xT, (46)

V J

where the divergence with respect to , is defined as:

(divub)i :=-

dk,

ijk

-Rkh (,).

dRjh (,)

If we now assume that all balance laws must be independent of the choice of the space where the body is placed — an assumption that can be made operative through the newly

introduced globalization operator — we get the following expressions (to shorten them, we define N := R(v)).

(i) Mass conservation, the definition of the velocity fields in the deep space and the definition of the fluctuations about these fields:

+ div x (~u) + div„ (~X) = .

(47)

3p

3t _

u :=i~, uf := x-u, X := pNN , Xf := Nnt -X. p p

(ii) Linear momentum conservation, the definition of the stresses, Td and t, in the deep space and the definition of material derivatives in such a space:

div xTd + div, t + pb = ~u,

(48)

Td := T -puf 8 uf,

t := - N (grad yNT )T - puf 8 Xf,

r\f

f (x, ,, t) := —— + gradxf • u + grad, f • X. ox

(iii) Substructural momentum conservation (the definition of the corresponding inertia forces is rather complex is a quadratic form is not adopted for the kinetic energy of the substructure; we prefer to skip the difficulty here writing the equation in statics) and the related microstresses and microforces:

Zd + div xs d + div ,S = 0 (49)

Zd := Z, sd := S S :=-N(gradyN )s.

Similar results can be obtained for the first and second principle, but their scrutiny goes beyond our scopes here.

Observing these results one can admit the following approach. The body B is imagined as a set of subelements represented by a (sufficiently regular) region of E X M (often comprising all of M) , each subelement occupying a distinct place x in a fit region B c E and having a substructure v chosen in M. All subelements lying in a particular place x form together an element. Whereas each element occupies a distinct place, the subelements forming an element might even have all the same substructure. It is fair to repeat that each subelement has no distinct place within the element (there is no such place available!); it is characterized by the value of its substructure but that value is not exclusive; subelements with the same value of v are wholly indistinguishable.

The basic assumption is the identification of material particles with their trajectories in the deep space as a function of time; it is then always possible to choose a reference stance of the body B* c E X M and describe its motion through functions (T a chronology) B* X T — B c E X M.

Taking (x», M») e B», with M» := R(,»), and (x, M) e e B, M := R(,), the local property of the transplacement B* XT — B are described by the double vectors/tensors:

dx-

Fxx := -——ei ® ej e Lin,

dx,

*j dxi

fxt": dM.

-M*Haei ® eJ ® eH e Lin(Skw, ^),

*Ja

dMia

Mjaei ® e;- ® eH e Lin(^, Skw),

(50)

*H

dMia

■нр

-MjaM*Kpei ® ej ® eH ® eK e Lin(Skw)

(ei- is the unit vector in the i direction; capital latin indices denote elements in the reference stance, lower latin indices for the actual one and greek indices for the reference substructure).

The balance equations of mass and momentum are (p mass density in the present placement for the volume measure taken on E X M):

-dX + div x (px) + div, (pX) = .

div xT + div,t + pb = px,

A + div xc + div , D + pB = pM skw T - A = 0,

dx

dM

dx

dM

&

where T, t, c and D must be given a constitutive description based on objective functions of the kinematic variables (50).

We refer to [7] for further comments and analyses of specific problems.

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Поступила в редакцию 20.12.2010 г.

CeedeHun 06 aem0pax

Brocato Maurizio, Prof., University Paris Est, France, maurizio.brocato@paris-malaquais.archi.fr Capriz Gianfranco, Prof., University of Pisa and Accademia dei Lincei, Italy, gianfranco.capriz@mac.com