Micropolar theory with production of rotational inertia: a Farewell to material description Текст научной статьи по специальности «Физика»

УДК 539.3

Micropolar theory with production of rotational inertia: A farewell

to material description

W.H. Müller1, E.N. Vilchevskaya2'3, and W. Weiss1

1 Institute of Mechanics, Technische Universität Berlin, Berlin, 10587, Germany

2 Institute for Problems in Mechanical Engineering RAS, St. Petersburg, 199178, Russia

3 Peter the Great Saint-Petersburg Polytechnic University, St. Petersburg, 195251, Russia

This paper takes a new look at micropolar media. Initially the necessary theoretical framework for a micropolar continuum is presented. To this end the standard macroscopic equations for mass, linear and angular momentum are complemented by a recently proposed kinetic equation for the moment of inertia tensor containing a production term. The main purpose of this paper is to study possible forms of this production term and its effects. For this reason two examples are investigated. In the first example we study a continuum of hollow particles subjected to an external pressure and gravity, such that the number of particles does not change. In the second example a continuous stream of matter through a crusher is considered so that the total number of particles will change. In context with these examples it will also become clear that the traditional Lagrangian way of describing the motion of solids is no longer adequate and should be superseded by an Eulerian approach.

Keywords: micropolar media, spatial description, characteristics, transport equations

Микрополярная среда с изменяющимся тензором инерции: отказ от материального описания

W.H. Müller1, E.H. Вильчевская23, W. Weiss1

1 Берлинский технический университет, Берлин, 10587, Германия

2 Институт проблем машиноведения РАН, Санкт Петербург, 199178, Россия

3 Санкт-Петербургский политехнический университет Петра Великого, Санкт Петербург, 195251, Россия

В статье представлен новый подход к описанию микрополярных сред. В рамках предложенной теории в систему стандартных уравнений для микрополярного континуума, включающую в себя балансы массы, количества движения и момента количества движения, вводится дополнительное балансовое уравнение с источниковым членом для тензора инерции. Целью данной работы является исследование различных форм источникового члена и его влияния на инерционные характеристики среды. Приведены решения двух модельных задач. В первом примере рассмотрены сферические оболочки под действием внешнего давления и гравитации. Момент инерции оболочек меняется во времени и пространстве, но их число остается неизменным. Второй пример посвящен прохождению материала через дробильную установку с изменением числа частиц. Показано, что традиционный лагранжевый подход не позволяет адекватно описывать подобные ситуации и должен быть заменен эйлеровым описанием.

Ключевые слова: микрополярные среды, пространственное описание, характеристики, уравнения переноса

1. Introduction

Most recently generalized continuum theories (GCTs) have gained the attention of the materials science community, the reason being the making of high performance materials with an inner structure for large and small scale

applications ranging from light-weight aerospace and automotive panels down to micromechanics and microelectronic gadgets. One of the GCTs is the so-called micropolar theory,

which emphasizes the aspect of inner rotational degrees of freedom of a material [1—3]. Hence this theory seems particularly promising for applications to soils, polycrystalline and composite matter, granular and powder-like materials, and even to porous media and foams. It should be noted that traditionally the tensor of the moment of inertia of a continuum particle, J, plays a role only in context with rotations, specifically the angular velocity vector, ra, as-

© Müller W. H., Vilchevskaya E.N., Weiss W., 2017

signed to the continuum element, and that it is conserved and known a priori. However, more recently, it has been emphasized by Ivanova, Vilchevskaya [4], that the moment of inertia tensor deserves to be treated as an independent field variable just like the inertia linked to linear momentum, namely the field of mass density p. In fact, they were the first to propose a balance equation for J, which contains a production term % J of moment of inertia due to "structural transformations" as they called it. This is supposed to mean that the moment of inertia will change due to combination or fragmentation of particles during mechanical crushing or chemical reactions. In addition a production may arise because of phase transitions or physical property changes, such as electric magnetization or polarization.

We therefore need to extend the original goals of micropolar theory. In fact, in what follows we will initially formulate the corresponding equations in the most general manner possible, such that they are ready for all kinds of future investigations of different problems, may they be static or dynamic, and with or without coupling of linear and angular momentum. After that we shall successively specialize them to the needs of this paper. In this spirit we state as follows.

Mathematically speaking, in its most general form the obj ective of micropolar theory is to determine the following primary fields: (a) the scalar field of mass density p(x, t); (b) the vector field of linear velocity v(x, t); (c) the symmetric, second rank, and positive definite specific moment of inertia tensor field J(x, t) in units of m2; (d) the specific moment of inertia coupling tensor field B(x, t) in units of m; and (e) the spin (also known as angular velocity) field ra(x, t) in all points x and at all times t within a region of space B, which can be either a material volume, i.e., it consists of the same matter at all times, or be a region through which matter is flowing.

The determination of these fields relies on the field equations for the primary fields. The field equations are based on balance laws and need to be complemented by suitable constitutive relations. In regular points these macroscopic balances read as follows:

balance of mass §p

St

+ pV-v = 0,

balance of momentum g

p—(v + B • ra) = V- o + pf + p%B • ra

ot

(1)

(2)

balance of moment of inertia and coupling moment of inertia tensors

J T

—+J x ra - ra x J = % J, ot

SB

—+B x ra - ra x B =% b ,

ot

balance of spin

(3)

p—(J • ra + v • B) = V - ^ + ox + pm -

5t

- vxB-ra + p(v• %b + %j • ra). (4)

We denote by

f -M+WO (5)

ot dt

the substantial (material) derivative of a field quantity. Moreover, o is the (non-symmetric) Cauchy stress tensor, f is the specific body force, %J (a second rank symmetric tensor) and %B are the productions related to the moment of inertia tensor J and to the coupling moment of inertia tensor B, respectively; ^ is the couple stress tensor, ox: = £ • o is the Gibbsian cross applied to the (non-symmetric) Cauchy stress tensor (where " • " is supposed to denote the outer double scalar product), £ being the Levi-Civita tensor, and m are specific volume couples.

In context with Eqs. (3) several comments are in order. It has already been mentioned that in its present form this equation can only be found in a recent paper by one of the authors [4]. There is a precedent to the equation for the inertia tensor J namely what is called "conservation of microinertia" in [1, p. 15]. However, that equation does not contain a production term % J. On the macroscopic continuum level this new term must be interpreted as a constitutive quantity. One of the purposes of this paper is to specify this constitutive relation in context with two illustrative problems (see below). However, due to the production of moment of inertia, we face another problem, as follows. Continuum mechanics of solids is typically formulated in the Lagrangian form, also known as material description, which is based on the concept of an indestructible "material particle." This particle is identifiable by its reference position vector X, which can then be used in a bijective mapping for describing uniquely the motion x = x(X, t) of the particle through three-dimensional space in time. Note that this requires the neighboring material particles to remain "close" to each other during the motion.

Furthermore note that a material particle in the continuum sense is composed of myriads of atoms or molecules, so that statistical fluctuations play no role in a macroscopic continuum. Moreover, there is no exchange of atoms and molecules between material particles. The mass of a material particle is simply conserved.

Traditionally, micropolar theory was not any different in this respect. One may say that the corresponding material particle consists of a statistically significant number of subunits on a mesoscopic scale, which are often also called "particles," a term that may give rise to confusion. Now, if the Lagrangian idea of a material particle is followed, the material particles must stay together during the motion and there should be no exchange of subunits between them. Particles are neither destroyed nor generated. In other words, this concept cannot handle production. Also note that within the material description of a micropolar conti-

nuum, each material point is phenomenologically equivalent to a rigid body, such that its moments of inertia do not change. Consequently, under such assumptions the field equations of mass and of the inertia tensor, Eqs. (1) and (3)t, can be integrated in time. Following [3, p. 33] we may write:

P =

Po

pJ =

Po Jo

det F det F

F: = Vxx(X, t),

(6)

where all functions of the current configuration depend on (X, t), and the ones in the reference placement (identifiable by the subscript 0) on (X, t0). However, as indicated before, there is a catch: A granular medium (say) is frequently milled. This affects the material particle, because its subunits will be crushed. They will change their mass and their moment of inertia and, what is more, during the milling process there might even be an exchange of crushed subunits between neighboring material particles, which are then no longer material in the original sense. Consequently, on a macroscopic scale the moments of inertia will change as well and all of this gives rise to the production term % J. This is why the authors in [4] have decided to depart from the Lagrangian description and turned to the Eulerian perspective (spatial description) instead.

Originally the Eulerian description stems from fluid mechanics. It does not impose strict constraints on the motion of mass-conserved material points. Rather it embraces the idea of an open system, allowing a priori for exchange of mass, momentum, energy, moment of inertia, etc., between and within the cells of the Eulerian grid.

A second remark concerns Eqs. (2) and (4). Note that the specific moment of inertia coupling tensor B is different from zero unless we choose the center of mass within a spatial volume element as a reference point. By coupling linear and angular velocities, i.e., v • B • ro, it contributes toward the kinetic energy present in that element. This way B also couples the linear balance of momentum and the spin balance, Eqs. (2) and (4), which is also a new feature when compared to the usual formulations found in (for example) [1, p. 15, 3, p. 30]. However, in [4] it was shown that if a geometrically isotropic medium is considered (in the sense that particles have a random orientation and they are homogeneously distributed across the representative volume) then, due to symmetry considerations, the symmetric tensor J must be spherical and the antisymmetric tensor B must be zero. In this paper, we are not going to consider any kind of anisotropy, so there is no need for a coupling inertia tensor field B(x, t).

Moreover, in this paper we are also not concerned with the determination of the angular velocity field ro(x, t). In fact, it is our intention to show that a balance for rotational inertia and hence the production term in Eqs. (3) are physically meaningful by themselves, independent of the angular velocity. Hence right now we put it equal to zero. Thus instead of Eqs. (1)—(4) we are left with the following

simplified set of equations for v and J:

d v „ —+V-(pv) = 0, p—+pv•Vv =

at ' ' Mat M

„ d J __

= V- o + pf , — + v-VJ =X j .

dt

We shall now proceed and illustrate the theory by two examples. We shall consider, first, ensembles of pressurized hollow particles expanding and contracting under external pressure, and second, the problem of continuous crushing of particles. By the first example it is intended to show what happens to the moment of inertia if the number of particles does not change. The corresponding production term will first be derived based on two models for a single pressurized particle, the so-called membrane and the Lame model. The second example shows what happens if the number of particles does change, for example due to the presence of a crusher. In fact the corresponding form for the production term, a linear population growth model, has been proposed in [4] before. However, in that paper only the time dependence in a homogeneous medium was studied, whereas now we shall consider a proper initial-boundary value problem.

2. A homogeneous continuum of hollow particles subjected to an external pressure

Consider the situation depicted in Fig. 1. Hollow spherical particles containing "air" are homogeneously distributed in space, which is also filled by some gas. Initially the inside and the outside pressure are both equal at a fixed level p0. Then the outside pressure changes in time, p = =p(t) 1. It is assumed that the particles react immediately and adjust to the pressure change volume-wise. Their mass stays constant, however, their dimensions will change. Consequently their moment of inertia is a function of time. This change will be investigated in the present section on the basis of Eq. (7)3. Since we assume that the particles are spherical, we have J = JI. Hence Eq. (7)3 reduces to dJ

dt

= Xj .

(8)

Note that right now the term v • VJ vanishes for two reasons: The medium does not move and it is homogeneous.

In the following two subsections we will first present two mesoscopic models, which allow calculating the production term XJ based on the dependence of J on the outside pressure as a function of the external action so-to-speak. This production term can later be used for a solution of more complicated problems with inhomogeneous distribution of particles and nonuniform pressure (Subsect. 2.3).

2.1. Membrane model

We assume that the material dividing the inner region of the spherical particle from the outside is an incom-

1 In Subsect. 2.3 it will change in time t and space z.

Fig. 1. A homogeneous assemblage of pressured particles

pressible membrane of fixed volume V0. We denote the initial and the current volume of the air fillings by V0air and V air(t ), respectively. The incompressibility condition for the membrane reads

in ( r3, 3 ( r°

"Г5):

V0,

(9)

rx and ro being the current inner and outer radii of the hollow spheres, respectively. Initially the inner and outer pressure will be the same: The membrane will not carry any tangential stress. Then the outer pressure will change p(t) and tangential stresses will develop within the membrane. The change in pressure is accomplished quasista-tically (we neglect inertial forces of the membrane) at a constant temperature T. Then the ideal gas equation provides the following relation:

PoC = mRT = p(t )V air(t ), M

(10)

because we assume isothermal conditions, ^=8.314 kJx x(kg • K)-1, being the ideal gas constant, M is the molecular weight of the filling gas, and m is the total gas mass. Mutual insertion of these relations leads to

i(t )=

V/3

v4 ny

С

p0

1/3

Vo P (t)

r°(t ) =

5Vo

4 n

13

V air 1 + Vo-

Po

Vo P(t)

1/3

(11)

Recall that the specific moment of inertia of a homogeneous hollow sphere is given by

y(t) = 2 fr)-^)

(12)

5 rO(t) - r{(t)

This is also the moment of inertia of a particle if the filling is neglected. This in mind we obtain

J (t ) = Jo

1+

VT Al Vo P(t)

5/3

Vo P(t)

5/3

1+

Vo

53

VoT

Vo

53

where

Л-

4n

2/3

V air

1 + Voo-V

513

С

Va

Following Eq. (8) we find for the production

X j

5 Jo

V ai 1 + ^

Po

Vo P(t)

23

Vo

o__Po_

Vo P(t)

V air

1 + VV-V

53

V

53

Po dP

Vo p (t) dt

(13)

513'

(14)

23

(15)

1.0 1.5 p(t)/p0

J/J0- \b_

1.4-

1

1.2-

l.o-

0.8- ^ ^

0 1 1 1 1 2 3 1 4 at

Fig. 2. Variation of moment of inertia with increasing pressure

X

Figure 2 illustrates the variation of the moment of inertia J(t) with changing pressurep(t). Since the outer radius must decrease if the outside pressure is increased (or vice versa), J must decrease as well. The effect is more pronounced for large filling ratios (V0air/vo = 10-° (1), 1 (2), and 0.1 (3), Fig. 2, a). In Fig. 2, b it was assumed that the pressure increases or decreases linearly in time, p(t ) = p0(1 ± at ) with a = 0.1 s1. The curve 1 holds for +a and the curve 2 one for -a.

Figure 3 shows the behavior of the production of moment of inertia normalized by %0 : = J0a with linearly increasing (curve 1) or decreasing (curve 2) pressure. If the pressure is increased, xJ goes to zero. Then the holes in the particles are closed more and more, and there is no further change of moment of inertia, i.e., xJ subsides. If the pressure decreases the particles increase their volumes faster and faster because the membrane does not present a strong barrier, at least not for the chosen parameter of a filling ratio of 10.

2.2. Linear-elastic shell model based on Lamé's solution

Recall the following solution for the radial displacement and (nonvanishing) stresses in a linear elastic hollow, totally radially symmetric sphere of initial inner and outer radii r{(t = 0) =: R and ro(t = 0) =: Ro, respectively, which is due to Lamé (see, for example, [5, Sect. 6.9])

Ur(r) =

,+4 r

j

= 3kA - 4ц

r

Goo = G

B

(16)

ФФ

= 3kA + 2ц—, R < r < Ro,

where k and | are the bulk and the shear moduli, respectively, A and B are constants of integration, which have to be adjusted to boundary conditions, which read

"P0'

tfrr ( r = Ri) The result is

A = -Po__ P(t)-Po_

3k 3k 1 -R3

( r = Ro) = - p (t ).

1

(18)

Fig. 3. Variation of the production of moment of inertia with linearly increasing pressure

B = _ p(t) - Po 1

R3 1 -p3'

where P: = R / Ro. This could now be inserted into Eq. (16).

However, the result for the radial displacement can be written in more concise form if we make use of the in-compressibility assumption. In other words, the bulk modulus tends to infinity k ^ ~ and

Ur ( Ri):

Ur ( Ro) =

p0

1

4ц 1 -ß3

Po ß3

4ц 1 -ß3

P(t) _ p0

" PO

Po

1

1

Rf

Ro.

(19)

In general we have r{/o = R/o + ur (R/o). Hence within a consistent linear approximation we find for the moment of inertia shown in Eq. (12):

J (t ) = 1 - 5 p,

J

4ц

P(t ) Po

-1

ß3(1 -ß2)

:5Ч2

(1 -ß5)

with

ß5

Jo = f R2T-ß3

(20)

(21)

Then according to the left hand side Eq. (8) one finds for the production

5 R3(1 -R2)dp(t)

XJ = Jo

(22)

'4^ (1 -P5)2 dt which, in contrast to Eq. (15), turns into a time-independent constant, if we assume that the pressure increases or decreases linearly in time, i.e., p(t) = p0(1 ± at):

= m5p0 P3(1 _P2)

J0 a

(23)

4^ (1 _P5)2

Figure 4 shows graphical representations of the temporal development of the moment of inertia shown in Eq. (20)

(17)

for the same choice of parameters for V0air /vo and a as in Fig. 2, and p0/ |w = 0.9. Qualitatively the progressions are identical. However, notice that J now varies linearly with time. Figure 5 shows the dependence of xJ with regard to the filling ratio V0air /vo = R3/(1 -R3). This dependence is nonlinear. As it is already indirectly visible in Fig. 3, the higher the filling ratio, the greater the production will be.

2.3. Non-homogeneous case

We wish to generalize our models of pressurized spheres to a nonhomogeneous, yet one dimensional case allowing for a closed-form analytical result and shall illustrate the procedure by using the linear-elastic Lamé solution. For this purpose imagine the spheres to be initially homogeneously immersed in a stationary isothermal atmosphere of height H (Fig. 6). This atmosphere shall serve as an external reservoir and provide a position-dependent boundary condition for the external pressure acting on the spheres. The mass of the atmosphere within height H is a constant, p h H = const, p h being the mass density of

1.2 p/p o

-0.5

Fig. 4. Variation of moment of inertia with increasing pressure

the atmosphere if it were thought distributed homogeneously, with no gravitational acceleration g being present. However, because of gravity it is not homogeneous and follows the stationary isothermal barometric pressure distribution instead:

p (z) = p (0)exp(-az),

phomgh Z a.= ^gH (24)

p (0):=

z

-—, a := H

1 - exp(-a Y H kT

where ^ is the mass of one particle constituting the atmosphere, k is the Boltzmann constant, and T is absolute temperature.

We assume that the pressurized spheres are able to adapt instantaneously to the pressure at the (normalized) height z of their very stratum. We also assume that they start moving under the influence of gravity and pass the ground z = 0 without being stopped. Moreover, their movement is not influenced by the presence of the atmosphere which also remains unaffected. In other words friction is no issue. In order to describe the temporal and spatial change of the associated fields we turn to Eqs. (7), which we specialize to the one-dimensional case:

dP + dpVz

dt dz

„ dvz dvz ■ 0, —^ + vz—z- = ' dt z dz

dJ dJz

¥+v-t = x J '

(25)

0.0 0.5 1.0 Fig. 5. Variation of moment of inertia production on filling ratio

The balance of momentum assumes this simple form, if we assume that the pressurized particles are "dust" for which the stress tensor vanishes. Moreover, we have f = -ge z. Note that the density cancels out and a true field equation for the velocity vz (z, t) remains. The method of characteristics (see Appendix) can be used to solve this partial differential equation:

i t \

Vz (Z, t) = -gt + 9

- J vz (z, Odt , (26)

t'=0 y

where is an arbitrary function. It can be determined from initial conditions. Recall that we require the pressurized particles to be initially at rest. Hence = 0 and v z (z, t) = v z (t) = - gt, i.e., the particles fall like a rigid body.

In fact this becomes even more evident if we now turn to the partial differential equations for mass and particle density, Eqs. (25)x 3, and start solving them based on the final result for the velocity. They read

gt ^ dt s dz

and the method of characteristics (see the Appendix for a compilation of handy formulae) then leads to:

' 1 2 ^ 2 (28)

= 0,

(27)

P(z, t) = po

z+2 gt

Fig. 6. Notations in context with the barometric equation

p0 (•) being the initial distributions of mass density of the pressurized spheres, which we assume to be constant: p( z, t = 0) =: P0 (z) =

0, if z e (H, + ~), = < p0 = const, if z e [0, H], (29)

0, if z e (-~,0).

Finally we turn to the balance for the moment of inertia, Eq. (25)3. For the production %J on the right hand side we obtain from Eqs. (22) and (24) (^ = E/3): 15 p3(1 -P2)d^(z(t))

XJ = Jo

gJ0,

5ч2

4E (1 -ß5)

dt

ytexp(-oz)

2 H

with the dimensionless constant

э3/1 п2\

Y : = a

15 ß3(1 -ß2) p(o)

(30)

(31)

2 (1 -p5)2 E For numerical analysis it is useful to define further dimensionless quantities as follows:

t := 1, T := T

- J(?, t ) X := Tx(z, t )

J , KJ

J

o

J

(32)

o

Then the balance for the normalized moment of inertia reads:

dJ _-9J _ - , _4 — - 2t— = Xj = -Yt exp(-az). д t oZ

(33)

The method of characteristics leads to the following solution:

J(z, t ) = Jo(z + t2) -

Y2 —— exp(-az ) [1 - exp(-at2)],

2a

(34)

J0 (•) being the initial distribution of the moment of inertia field.

According to Eqs. (20) and (24) it is given by

J0(z) =

o, if z e (1, + ~),

1+

Y

2a

1 - _pi°)exp(-az )

Po

.if ze[o,1],

o, if z e (-~,o).

(35)

z/H

—I-1-1—1-1-r

0 2 4 6

Normalized moment of inertia

—I-1-r-1-1—

0 2 4 6

Normalized moment of inertia

T-1-T-1-10 2 4 6 Normalized moment of inertia

z/H

—i-1-1-1-1—

0 2 4 6

Normalized moment of inertia

z/H

1.2

0.4

t/T= 0.161611

z/H

1.2

O.i

0.4

0.0

t/T= 0.959596

Ж

0 2 4 6 0 2 4 6

Normalized moment of inertia Normalized moment of inertia

Fig. 7. Spatial and temporal development of the moment of inertia

Note that this expression is different from zero only where there is matter. At positions with vanishing particle density, see Eq. (29), we simply have to put J0.

In what follows we present a few simulations using a = 2.5 and y = 7.3. Figure 7 illustrates Eq. (34). Initially, as expected, we find that the higher the location of the pressurized particles the greater their moment of inertia, because the pressure is lower at great heights and the particles can expand. Then they start to fall down. Particles with a large moment of inertia move into regions of higher pressure. This means that their moment of inertia has to decrease as shown. The reason why at a fixed height the moment of inertia decreases in time is due to the competition of falling velocity, which increases linearly in time (see the solution to Eq. (26) in the surrounding text), and the exponential decrease of pressure so that higher-up layers are more effectively compressed as time goes on. In fact this is also visible in the differential equation (33) and in the solution (34). In this context note the production, which is linearly dependent on time and decreases exponentially with height.

3. Analysis of a one-dimensional crusher

Consider the situation depicted in Fig. 8. We consider the following problem in infinite one-dimensional space,

< x < A continuous flow of spherical particles is coming in from the left and keeps moving to the right at a constant, prescribed speed v0 like ore transported on a conveyor belt. On its way it enters a region - 8 < 0 < + 8, symmetrically arranged around the position x = 0, where it is continuously crushed to form smaller and smaller particles. Hence we assume that the momentum balance shown in (7) is identically satisfied and the balance of mass and of moment of inertia read:

dp dp . dJ dJ

+ v0 = 0, — + v0 — = %J. dt dx dt dx

(36)

For the production of moment of inertia we postulate the following relationship:

X J(x, t) =

0, if - x <-8,

-a[ J(x, t) - J*], if - 8 < x < xs,

0, if xs < x <+8,

0, if +8< x <+<»,

(37)

where a is a positive rate constant, J* indicates the moment

Fig. 8. Idealization of the crusher problem

of inertia pertinent to the minimum crushing size of the particles, and xs is the location of the incoming shock front of the to-be-crushed particles. Note that the front will eventually leave the crusher area. In this case the third case differentiation in Eq. (37) becomes obsolete.

At this point a few comments regarding the status of the production relation (37) are in order, in particular in context with the nature of the constants J* and a. Recall that the production terms of partial mass and partial momentum balances are traditionally interpreted as pure constitutive quantities (see, e.g., [6, pp. 70, 173]). However, in the case of the production of the inertia tensor 1 J, the case is not as clear cut as we shall venture to explain. For example, imagine the incoming ore to be composed of grains that are attached to each other by some kind of glue. Then we may want to interpret the quantity J* as an inertial characteristic of the size of these grains. We would not relate it to the sieve size of the crusher, which has no inertial characteristics. As such it is a constitutive quantity. Analogously, we may want to interpret the coefficient a as a measure of the dynamic resistance of the glue, in other words it could be related to its viscosity. It might also depend on the state of stress or rather stress rate in terms of an equivalent measure, just like yield stress, hence "converting" the action of the crusher blades to a material response of the glue's resistance or dynamic strength. But here is the catch, in that way it is also related to the effectiveness or to the rotational speed of the crusher blades and transmission of its exerted forces. The authors of this paper hence conclude that a depends on both, material as well as process (the crusher's) characteristics, and so does the production

1J.

The problem for the mass density can be solved in closed form by using the method of characteristics for initial value problems on the infinite domain (see appendix). We find p( x, t) = p0( x - v0t), (38)

where p0 is the initial mass density distribution, which, for simplicity, we shall assume as piecewise constant function, as follows:

p(x, t = 0) =: p0(x) =

p0 = const, if - ^ < x < -8, = <0,if -8 < x <+8, (39)

0, if +8< x <+^.

In other words, the solution is a step function of height p0 steadily and uniformly advancing from the left to the right (Fig. 9).

We will now find the solution for the moment of inertia, Eq. (36)2. Recall that we have to distinguish various cases in Eq. (37), and therefore the infinite interval < x < is dissected in three regions, where the boundary value at x = -8 is known. Consequently, the method of characteristics as pertinent to a boundary value problem (see Appendix) must be used. The solution process is also detailed in the

:o.4

I

o.o--1.0

cl

-0.5

0.0

x/l

vo о

0.4

Q_

o.o--1.0

-0.5

-та

0.5 1.0

0.0

x/l

(N Ö

:o.8-

0.4

cl

o.o-

-1.0 -0.5

0.0

x/l

oo о

; oi

0.4

Q_

0.5 1.0

o.o-

-1.0 -0.5

0.0

x/l

Ж

0.5 1.0

Тё

СП ö

cl

"0.8 -> 0.4

Lo.o -1.0

-0.5

0.0

x/l

cl

0.5 1.0

ч 0.8

, 0.4

Lo.o -1.0

-0.5

0.0

x/l

Fig. 9. Spatial and temporal development of the moment of inertia

1С

0.5 1.0

^Z

0.5 1.0

Appendix and the final result reads (J0 = const is the moment of inertia corresponding to the particles with density

P0):

- left of the crusher region, i.e., at positions < x < -8 and at times 0 < t < ^

J( x, t) = J0, (40)

- crusher region at times 0 < t < 28/v0 for -8 < x < xs

J(x, t) = J* + (J0 - J*)exp

-—(x + 5)

v0

and for xs < x < +5 J (x, t) = 0,

and at times 25/v0 < t < ^ for -5 < 0 < +5

a

J(x, t) = J* + (J0 - J*)exp

-(x+5)

(41)

(42)

(43)

whilst the shock front moves at a constant velocity dxs/dt = v0,

- right of crusher region at times 0 < t < 28/v0 for 8 < x < ^ J (x, t) = 0, (44)

and at times 28/v0 < t < ^ for 8< x < v0t - 8

T, ^ T , T T \ ( 2a8

J( x, t) = J* + (J0 - J*)exp

(45)

and for Vn t -5 < x < ^

J (x, t) = 0, (46)

and for - 8 < x < ^

J (x, t) = 0. (47)

These results are visualized in dimensionless form in Fig. 10. To this end we define the following dimensionless variables

Я 0.8H ö

II

0.4-1

0.0-1.0

2 1

1! 3

-0.5 0.0 0.5

00 0.8 ö

11

0.4

0.0-

-L0

IE

m

-0.5 0.0 0.5

P0.8

0

и

0.4

к

14

0.0

-1.0 -0.5 0.0 0.5

0.4^

14

1Ч

o.o-_ -1.0

n

f

>4*« ^.Hi.yt'U'k.

4 \ \\

-0.5 0.0 0.5

Fig. 10. Spatial and temporal development of the moment of inertia

¿0.8-p I 0.4- sä ^0.0--1 1*1 1 2 •n

0 -0.5 0.0 0.5 л

tor in I 0.4£ o.o--1 I \ \ \ 4. ___ [d_

.0 -0.5 0.0 0.5 л

Fig. 11.

f- l 0.4- sä s o.o--1 \ \ \ \ l 1 [Ji

.0 -0.5 0.0 0.5 x/l

I»,: ö 1 0.4- sä o.o--1 \ \ \ \ V»__ \L

.0 -0.5 0.0 0.5 x/l

and temporal development of the

Я 0.8- СП I 0.4- sä o.o --l I \ \ \ \ V» 1 l£ dl

.0 -0.5 0.0 0.5 ;

i f

X0.8-p \ \

l 0.4- \ \

sä i

o.o-

-|--1--г

-1.0 -0.5 0.0 0.5 x/l of inertia

J(x, t): = J(X,t), t : =at, x := —, (48)

Jo l

where l is an arbitrary length-scale parameter, and we choose J* IJ0 = 0.25, 8/1 = 0.25, v0/al = 12.75.

It is clearly visible that for this choice of parameters the time during which the particles remain within the crusher region is too short in order to reduce their size down to the level pertinent to J*. We therefore reduced the input velocity by choosing the parameter v0/al = 1/15 so that they remain longer within the crusher region and obtain an optimal milling. The result is illustrated in Fig. 11.

Moreover, note the dots (3) in Fig. 10. They are the result of a first numerical simulation based on the FiPy software (a finite volume partial differential equation solver using python) provided by NIST [7, 8]. The result is sobering because simple finite volume techniques have great difficulty to capture a moving sharp interface. Actually the shock front broadens with increasing time. This is well known and in this context one is reminded of the famous quote by [9]: "The ultimate embarrassment is to be unable to solve the simplest of equations du(x, t) + du(x, t) = o dt dx

accurately by numerical methods on a fixed grid." Most recently, adaptive mesh solvers have been developed, which are capable to capture our analytical result with greater accuracy, especially at higher time. However, the investigation of this phenomenon and in-depth numerical analysis is left to future research.

4. Conclusions and outlook

This paper was dedicated to a reinvestigation of micropolar theory in terms of rotational inertia production, which required us taking the view of a spatial (Eulerian) description. In particular the following tasks have been accomplished.

Initially the general balance equations of micropolar theory were stated. However, in contrast to classical micropolar theory an extension was made, which allows us to study the development of rotational inertial characteristics, namely the specific moment of inertia tensor J and, in principle, also the coupling inertia tensor B in addition to the translational measure of inertia, the mass density, p. It is noteworthy that the balances for J and B contain production terms characteristic of structural transformations.

As a consequence, the concept of an indestructible material particle became obsolete and required us using a spatial (Eulerian) description when solving the resulting field equations from the very beginning on. The whole formulation is Eulerian based, and this includes balances, field equations, and mathematical solution techniques.

The set of general balance equations was then specialized to the case of the primary fields mass density p, specific moment of inertia tensor J and translational velocity v. This was done in order to demonstrate that rotational inertia J is a field of its own right and may develop without an angular velocity ro being present.

For illustration of the new concepts two examples were studied analytically. First, an ensemble of pressurized hollow particles expanding and contracting under external pressure, homogeneous or inhomogeneous-wise. In this case rotational inertia is produced because of change of the outer pressure leading to variations of the particle radii. However, the number of particles does not change. Second, the problem of continuous crushing of particles was investigated. In this case the production of rotational inertia results from increasing particle numbers due to the crushing process.

Analytical solutions to the corresponding field equations were found based on the method of characteristics. A first numerical investigation of the developing shock fronts, based on finite volumes, was also given.

A detailed numerical study is left to future work, just like a full-scale investigation of the coupling between linear

and angular momentum including the form and effects resulting from the coupling inertia tensor B.

Appendix: The method of characteristics

In this section we will compile some important formulae, based on the method of characteristics, which allow to obtain analytical solutions for the transport equations of interest to this paper. It is a hands-on approach far from mathematical stringency. For more details the reader is referred to the pertinent literature, for example [10, Chapter II, §3, p. 75], the nomenclature of which we adapt, at least in part.

Illustration of the general method

Consider the following partial differential equation (PDE) of first order in two variables x and t for the unknown field u = u(x, t):

n du ~ . du _. . a(x, t, u) — + b(x, t, u)— = c(x, t, u) dt dt

(A.1)

with three known functions a(x, t, u), b(x, t, u), c(x, t, u).

For a solution of the problem we map the PDE to a system of ordinary differential equations (ODEs) by means of the two so-called characteristic variables a and s, such that x = x(a, s), t = t(a, s) and u = u(a, s). To this end we compute the total differential

d u = du d t + du d x d s dt d s dx d s

(A.2)

and by comparison of both equations we obtain the following relations:

dt w ^ dx -, ,

— = a (a, s, u ), — = b (a, s, u ), d s v ' d s V ;

du

— = c (a, s, u ). ds

(A.3)

Prescription of suitable initial and/or boundary conditions now allows to find a solution in parameterized form, i.e. x = x(a, s), t = t(a, s) and u = u(a, s), the first two of which can be used to obtain a = a(x, t), s = %(x, t), which inserted into the third solution yield u = u(x, t).

We proceed to illustrate this method for the problems relevant to this paper.

Applications

We first consider pure initial value problems posed on the infinite domain, i.e., we assume that an initial condition u(x, t = 0) = u0(x) is prescribed and known on -«> < x < < The ODEs (A.3) are then solved under observation of the following "initial conditions:" A(a): t (a, s = 0) = 0, x(a, s = 0) = a, u(a, s = 0) = u0(a).

In the case of Eq. (25)2we assign a ^ 1, b = u ^ vz, c ^ -g, and u0(a) ^ 0. A straightforward integration of Eqs. (A.3) under observation of Eqs. (A.4) then yields:

(A.4)

g 2

t(a, s) = s, x(a, s) = -s +a,

(A.5)

vz s) = -gS,

or simply, vz = vz (x, t) = -gt.

Next we turn to Eq. (27). Here we assign a ^ 1, b ^ -gt, and c ^ 0. Similarly as before integration of the characteristic ODEs yields:

t(a, s = 0) = s, x(a, s = 0) = - ^2s2 + a, P(a, s) = Po(a),

(A.6)

(A.7)

or, more simply, by assigning x ^ z, p = p(z, t) =

= P0( z + g/212).

The next example concerns Eq. (36)15 and so we put a ^ 1, b ^ v0, and c ^ 0. This yields: t(a, s = 0) = s, x(a, s = 0) = v0s + a,

P(a, s) = Pa(a), or more to the point, P = P(x, t) = P0(x - v0t).

As a remark on the side, note, that the characteristic curves in the first two examples are bundles of parabolae and in the third example a bundle of straight lines, x(t) = = -g/2t2 + a and x(t) = v0t + a, respectively.

We now finally turn to the initial boundary value problem posed by Eqs. (36)2 and (37). Note that a /-field can exist only where there is mass, i.e., the mass density is not zero. We have just demonstrated by the last example that the mass density propagates to the right, first through the crusher region, -8 < x < +8, and then on to +8 < 0 < We expect the /-field to move accordingly. However, because of the crushing it will not stay constant unlike the mass density if we accept the initial condition as shown in Eq. (39). Rather its initial value J(x = -8, t =0) =: J0 must decrease during its move through the crusher, because of a > 0 and J* < J0 which characterizes the finest milling size possible (see Eq. (37)). But the velocity at which the / front moves is identical to that of the mass density front. The speed of such shock fronts can be obtained from the so-called Rankine-Hugoniot relations. Following [11] we may draw conclusions from the balance in regular points to the jump as follows:

If aP+a^ = 0, then - vs[[p]] + [[v0P]] = 0 (A.8) at ox

with the jump brackets [[A]] := A+ - A~, A± being the field values to the right and to the left of a jump. Since v0 is a constant, we conclude that the velocity of the shock front vs must also be a constant of equal value and the shock propagates uniformly to the right:

dxs _

^s ^-77 = ^0 ^ xs = v0t-8, dt

(A.9)

xs being the position of the shock front, which initially at time zero resides at x = -8, the crusher entrance position. Note that the /-front will leave the crusher region at t = 28/ v 0 after which its intensity will cease to decrease

and stay at the level it had when it passed the crusher exit at

x = +8.

We now return to the initial boundary value problem posed by Eqs. (36^ and (37). We find for the characteristic ODEs (A.3) in the crusher region filled with matter, i.e.,

for -8 < x < xs:

Vo, J = -a(J - J*), (A. 10)

dt , dx - = 1, - = Vo

ds ds ds

which we solve formally:

f(CT, s) = s + Cj, £(ct, s) = s + C2,

•7(ct, s) = J* + C3 exp(-as ). Now we impose, in contrast to Eqs. (A.4), boundary-value like conditions,

(A.11)

B(a): t(a, s = 0) =—, x(a, s = 0) = -8,

Vo

./(a, s = 0) = J0 = const and find for the three constants of integration:

C1 = , C2 = -8, C3 = J0 - J»•

Vo

By mutual insertion we finally arrive at x = x(a, s) = v0t - a - 8,

. x+8+a t = t(a, s) =-,

and most important:

J = J(x, t) = J» + (J0 - J») exp

( x + 8)

Vo

(A.12)

(A.13)

(A.14)

(A. 15)

The latter holds within the crusher area wherever there is matter to be crushed, i.e., at positions -8< x < xs during the time of passage through the crusher, which is 0 < < t < 28/ v 0 • Beyond that position, i.e., at xs < x < + 8 < ^ we have J = 0. Recall that the shock front at xs will reach the crusher exit at position x = +8 at time t = 28/v0 and the intensity of J at the exit position is almost obviously now:

Jmin = J(X = 8, t = 28/ V0 ) = r î t T \ 2a8

= J* + (J0 - J») exp

(A. 16)

After that time the solution (A. 15) continues to hold in the crusher region and to the right of it we have:

J = J( x, t) =

Jmin,if + 8< x < ^

0, if xs < x < +«>.

(A.17)

Acknowledgments

Support of this work by a grant from the Russian Foundation for Basic Research (16-01-00815) is gratefully acknowledged. The authors also want to thank Mr. Navneet Singh for most valuable help when converting the TeX file into MS-Word.

References

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IV. - London: Academic Press, 1976.

2. Eringen C. Nonlocal Continuum Field Theories. - New York: Springer,

2002.

3. Eremeyev V.A., Lebedev L.P., Altenbach H. Foundations of micropolar mechanics. - Heidelberg: Springer, 2012.

4. Ivanova E.A., Vilchevskaya E.N. Micropolar continuum in spatial description // Cont. Mech. Thermodyn. - 2016. - V. 28. - No. 6. -P. 1759-1780.

5. Eringen A.C. Mechanics of Continua. - Robert E. Krieger Publishing Company, 1980.

6. Müller I. Thermodynamics. - Boston: Pitman, 1985.

7. NIST FiPy Package. - 2017. - http://www.ctcms.nist.gov/fipy/ index.html.

8. Guyer J.E., Wheeler D., Warren J.A. FiPy: Partial differential equations with python // Cont. Mech. Thermodyn. - 2009. - V. 11. - No. 3. -P. 6-15.

9. Mitchell A.R. Recent developments in the finite element method // Computational Techniques and Applications: CTAC. V. 83. - New York: Elsevier North-Holland, 1984. - P. 6-15.

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Поступила в редакцию 23.01.2017 г.

Сведения об авторах

Wolfgang H. Müller, Prof. Dr., Institute of Mechanics, Chair of Continuum Mechanics and Materials Theory, Technische Universität Berlin, [email protected]

Elena N. Vilchevskaya, Cand. Sci. (Phys.-Math.), Assoc. Prof., Senior Researcher, Sci. Secr., Institute for Problems in Mechanical Engineering RAS, Assoc. Prof., Peter the Great Saint-Petersburg Polytechnic University, [email protected]

Wolf Weiss, Priv.-Doz. Dr., Institute of Mechanics, Chair of Continuum Mechanics and Materials Theory, Technische Universität Berlin, [email protected]