A Geometric Chiral-Continuum Theory of Elasticity

7/31/2019 A Geometric Chiral-Continuum Theory of Elasticity

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On a Geometric Theory of Generalized Chiral Elasticity with

Discontinuities

By

Indranu Suhendro

Abstract

In this work we develop, in a somewhat extensive manner, a geometric theory of chiral

elasticity which in general is endowed withgeometric discontinuities (sometimes referred

to as defects). By itself, the present theory generalizes both Cosserat and void elasticitytheories to a certain extent viageometrization as well as by taking into account the action

of the electromagnetic field, i.e., the incorporation of the electromagnetic field into the

description of the so-called microspin (chirality) also forms the underlying structure ofthis work. As we know, the description of the electromagnetic field as a unified

phenomenon requires four-dimensional space-time rather than three-dimensional space asits background. For this reason we embed the three-dimensional material space in four-

dimensional space-time. This way, the electromagnetic spin is coupled to the non-electromagnetic microspin, both being parts of the complete microspin to be added to the

macrospin in the full description of vorticity. In short, our objective is to generalize the

existing continuum theories by especially describing microspin phenomena in a fully

geometric way.

1. Introduction

Although numerous generalizations of the classical theory of elasticity have beenconstructed (most notably, perhaps, is the so-called Cosserat elasticity theory) in the

course of its development, we are somewhat of the opinion that these generalizationssimply lack geometric structure. In these existing theories, the introduced quantities

supposedly describing microspin and irregularities (such as voids and cracks) seem to

have been assumed from without, rather than from within. By our geometrization ofmicrospin phenomena we mean exactly the description of microspin phenomena in terms

of intrinsic geometric quantities of the material body such as its curvature and torsion. Inthis framework, we produce the microspin tensor and the anti-symmetric part of the stress

tensor as intrinsic geometric objects rather than alien additions to the framework of

classical elasticity theory. As such, the initial microspin variables are not to be freelychosen to be included in the potential energy functional as is often the case, but rather, at

first we identify them with the internal properties of the geometry of the material body. In

other words, we can not simply adhere to the simple way of adding external variables that

are supposed to describe microspin and defects to those original variables of the classical

elasticity theory in the construction of the potential energy functional without firstdiscovering and unfolding their underlying internal geometric existence.

Since in this work we are largely concerned with the behavior ofmaterial points such as

their translational and rotational motion, we need to primarily cast the field equations in a


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manifestly covariant form of the Lagrangian system of material coordinates attached to

the material body. Due to the presence of geometric discontinuities (geometricsingularities) and the localnon-orientability of the material points, the full Lagrangian

description is necessary. In other words, the compatibility between the spatial (Eulerian)and the material coordinate systems can not in general be directly invoked. This is

because the smooth transitional transformation from the Lagrangian to the Eulerian

descriptions and vice versa breaks down when geometric singularities and the non-orientability of the material points are taken into account. However, for the sake of

accommodating the existence of all imaginable systems of coordinates, we shall assume,

at least locally, that the material space lies within the three-dimensional space of spatial

(Eulerian) coordinates, which can be seen as a (flat) hypersurface embedded in four-

dimensional space-time. With respect to this embedding situation, we preserve thecorrespondence between the material and spatial coordinate systems in classical

continuum mechanics, although not their equality since the field equations defined in the

space of material points are in general not independent of the orientation of that localsystem of coordinates.

At present, due to the limits of space, we shall concentrate ourselves merely on the

construction of the field equations of our geometric theory, from which the equations ofmotion shall follow. We shall not concern ourselves with the over-determination of the

field equations and the extraction of their exact solutions. There is no doubt, however,

that in the process of investigating particular solutions to the field equations, we might

catch a glimpse into the initial states of the microspin field as well as the evolution of the

field equations. Wed also like to comment that we have constructed our theory with arelatively small number of variables only, a characteristic which is important in order to

prevent superfluous variables from encumbering the theory.

2. Geometric structure of the manifold 3 of material coordinates

We shall briefly describe the local geometry of the manifold 3 which serves as the

space of material (Lagrangian) coordinates (material points)i ( )3,2,1=i . In general, in

addition to the general non-orientability of its local points, the manifold 3 may containsingularities or geometric defects which give rise to the existence of a local materialcurvature represented by a generally non-holonomic (path-dependent) curvature tensor, a

consideration which is normally shunned in the standard continuum mechanics literature.

This way, the manifold 3 of material coordinates, may be defined either as a continuumor a discontinuum and can be seen as a three-dimensional hypersurface of non-orientablepoints, embedded in the physical four-dimensional space-time of spatial-temporal

coordinates 4 . Consequently, we need to employ the language of general tensoranalysis in which the local metric, the local connection, and the local curvature of the

material body 3 form the most fundamental structural objects of our consideration.

First, the material space 3 is spanned by the three curvilinear, covariant (i.e., tangent)

basis vectors ig as 3 is embedded in a four-dimensional space-time of physical events


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3

4 for the sake of general covariance, whose coordinates are represented byy

( )4,3,2,1= and whose covariant basis vectors are denoted by . In a neighborhood

of local coordinate points of 4 we also introduce an enveloping space of spatial(Eulerian) coordinates Ax ( )3,2,1=A spanned by locally constant orthogonal basisvectors Ae which form a three-dimensional Euclidean space 3 . (From now on, it is tobe understood that small and capital Latin indices run from 1 to 3, and that Greek indices

run from 1 to 4.) As usual, we also define the dual, contravariant (i.e., cotangent)

counterparts of the basis vectors ig , Ae , and , denoting them respectively asig ,

Ae ,

and , according to the following relations:

=

=

=

,

,

,

ABB

A

i

kk

i

ee

gg

where the brackets denote the so-called projection, i.e., the inner product and where

denotes the Kronecker delta. From these basis vectors, we define their tetrad

components as

y

xe

y

x

AAA

iii

A

i

A

ii

A

==

==

==

,

,

,

e

g

eg

Their duals are given in the following relations:

A

BB

A

i

kk

iA

k

i

A

ee

=

==

(Einsteins summation convention is implied throughout this work.)

The distance between two infinitesimally adjacent points in the (initially undeformed)

material body 3 is given by the symmetric bilinear form (with denoting the tensorproduct)

ki

ikg ggg =


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called the metric tensor of the material space, as

ki

ik

ddgds =2

By means of projection, the components of the metric tensor of 3 are given by

kiikg gg ,=

Accordingly, for 3,2,1, =ba , they are related to the four-dimensional components of the

metric tensor of 3 , i.e., vG , = , by

kikiab

b

k

a

i

kiik

kkbkG

Gg

++=

=

)(2

where the round brackets indicate symmetrization (in contrast to the square brackets

denoting anti-symmetrization which we shall also employ later) and where we have set

44

44

4

G

GGb

tck

a

a

iii

iii

=

==

==

Here we have obviously put tcy =4 with c the speed of light in vacuum and t time.Inversely, with the help of the following projective relations:

ng

g

nii

ii

+=

=

we find that

nngG ikki

+=

or, calling the dual components of g , as shown in the relations

=

=

GG

ggk

i

kr

ir

we have


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nnii =

Here 1= and

n and n respectively are the contravariant and covariant componentsof the unit vector field n normal to the hypersurface of material coordinates 3 , whose

canonical form may be given as 0, = ki where k is a parameter. (Note that the

same 16 relations also hold for the inner product represented by AA ee .) We can write

2/1

2/1

=

yyG

yn

Note that

=

==

nn

enn ii 0

Let now gdenote the determinant of the three-dimensional components of the material

metric tensorikg . Then the covariant and contravariant components of the totally anti-

symmetric permutation tensor are given by

ijkijk

ijkijk

g

g

2/1

2/1

=

=

where ijk are the components of the usualpermutation tensor density. More specifically,

we note that

ig k

ijkj gg =

where the symbol denotes exterior product, i.e., ig ( ) = ijjijg .In the same manner, we define the four-dimensional permutation tensor as one withcomponents

2/1

2/1

=

=

G

G

where GG det= . Also, we call the following simple transitive rotation group:

n=


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6

where

=

=

ijk

kji

kjiijk

n

n

6

1

Note the following identities:

( ) ( ) ( )

r

s

ijr

ijs

p

j

q

i

q

j

p

i

pq

ij

pqr

ijr

p

k

q

j

q

k

p

j

r

i

r

k

p

j

p

k

r

j

q

i

q

k

r

j

r

k

q

j

p

i

pqr

ijk

pqr

ijk

=

==

++==

where pqrijk andpq

ij represent generalized Kronecker deltas. In the same manner, the

four-dimensional components of the generalized Kronecker delta, i.e.,

=

det

can be used to deduce the following identities:

=

=

=

=

6

2

Now, for the contravariant components of the material metric tensor, we have

nngG

kkbkGg

ikv

ki

kikiabk

b

i

a

ik

+=

++= )(2

where

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44

1

G

GGb

tck

ai

a

ii

ii

=

==

=


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Obviously, the quantities

t

i

in ik are the contravariant components of the local

velocity vector field. If we choose an orthogonal coordinate system for the background

space-time 4 , we simply have the following three-dimensional components of thematerial metric tensor:

kiabk

b

i

a

ik

kiab

b

k

a

iik

kkGg

kkGg

+=

+=

In a special case, if the space-time 4 is (pseudo-)Euclidean, we may set 1== .However, for the sake of generality, we shall not always need to assume the case just

mentioned.

Now, the components of the metric tensor of the local Euclidean space of spatial

coordinates Ax ,BAAB

h ee ,= , are just the components of the Euclidean Kronecker

delta:

ABABh =

Similarly, we have the following relations:

ik

k

B

i

AAB

A

k

A

iAB

B

k

A

iik

ghhg

===

Now we come to an important fact: from the structure of the material metric tensor alone,

we can raise and lower the indices of arbitrary vectors and tensors defined in 3 , and

hence in 4 , by means of its components, e.g.,

.,,,, etcBGBBGBAgAAgA vvk

ikik

iki

====

Having introduced the metric tensor, let us consider the transformations among the

physical objects defined as acting in the material space 3 . An arbitrary tensor field T

of rank n in 3 can in general be represented as

......

......

......

...

...

...

...

...

...

=

=

=

T

T

T

DC

BA

AB

CD

lk

ji

ij

kl

eeee

ggggT

In other words,


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...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

..........

............

............

AB

CDBA

DCij

klji

lk

DC

BAij

kl

l

D

k

C

B

j

A

i

AB

CD

lk

jiAB

CD

D

l

C

k

j

B

i

A

ij

kl

TeeeeTT

TeeeeTT

TTT

====

==

For instance, the material line-element can once again be written as

( ) ( ) dydyyGdxdxddgds BAABkipik ===2

We now move on to the notion of a covariant derivative defined in the material space

3 . Again, for an arbitrary tensor field T of 3 , the covariant derivative of thecomponents of T is given as

.........

...

...

...

...

...

...

...

...

......

... +++

= ijkr

r

lp

ij

rl

r

kp

ir

kl

j

rp

rj

kl

i

rpp

ij

klij

klp TTTTT

T

such that

.........

... =

= lkjiij

klpppT gggg

TT

where

r

r

ikk

i gg

=

Here the 273 =n quantities ijk are the components of the connection field , locally

given by

k

A

ji

A

i

jk

=

which, in our work, shall be non-symmetric in the pair of its lower indices ( )jk in order

to describe both torsion and discontinuities. If i represent another system of coordinates

in the material space 3 , then locally the components of the connection field are seento transform inhomogeneously according to

jk

p

p

ip

rsk

s

j

r

p

ii

jk

+

=

2


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i.e., the ijk do not transform as components of a local tensor field. Before we continue,

we shall note a few things regarding some boundary conditions of our material geometry.

Because we have assumed that the hypersurface 3 is embedded in the four-dimensionalspace-time 4 , we must, in general, have instead

ngg

ikr

r

ikk

i K+=

where ikikik nK == ng , are the covariant components of the extrinsic

curvature of 3 . Then the scalar given byds

d

ds

dKK

ki

ik

= , which is the Gaussian

curvature of 3 , is arrived at. However our simultaneous embedding situation in whichwe have also defined an Euclidean space in 4 as the space of spatial coordinatesembedding the space of material coordinates 3 , means that the extrinsic curvature

tensor, and hence also the Gaussian curvature of 3 , must vanish and we are left simply

with rr

ikk

i gg

=

. This situation is analogous to the simple situation in which a plane

(flat surface) is embedded in a three-dimensional space, where on that plane we define a

family of curves which give rise to a system of curvilinear coordinates, however, with

discontinuities in the transformation from the plane coordinates to the local curvilinear

coordinates and vice versa.

Meanwhile, we have seen that the covariant derivative of the tensor field T is again atensor field. As such, here we have

E

AB

CDE

p

D

l

C

k

j

B

i

A

ij

klpx

TT

=

...

......

... ......

Although a non-tensorial object, the connection field is a fundamental geometricobject that establishes comparison of local vectors at different points in 3 , i.e., in theLagrangian coordinate system. Now, with the help of the material metrical condition

0= ikp g

i.e.,

kipikpp

ikg +=

where rkpirikp g = , one solves fori

jk as follows:


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[ ] [ ] [ ]( )srjkssrkjsriijkjkr

r

jk

k

rjrii

jk gggggg

g ++

+

=

2

1

From here, we define the following geometric objects:

1. The holonomic (path-independent) Christoffel or Levi-Civita connection,

sometimes also called the elastic connection, whose components are symmetric in

the pair of its lower indices ( )jk and given by

{ }

+

=

j

kr

r

jk

k

rjrii

jk

gggg

2

1

2. The non-holonomic (path-dependent) object, a chirality tensor called the torsiontensor which describes local rotation of material points in 3 and whosecomponents are given by

[ ]

==

j

A

k

k

A

ji

A

i

jk

i

jk

2

1

3. The non-holonomic contorsion tensor, a linear combination of the torsion tensor,

whose components are given by

[ ] [ ] [ ]( )

{ }

=

=

+=

A

r

r

jkk

A

ji

A

A

jk

i

A

s

rjks

s

rkjs

rii

jk

i

jk gggT

~

which are actually anti-symmetric with respect to the first two indices i and j .

In the above, we have exclusively introduced a covariant derivative with respect to the

holonomic connection alone, denoted by p~ . Again, for an arbitrary tensor field T of

3 , we have

{ } { } { } { } ......~ ...........................

......

... +++

= ijkr

r

lp

ij

rl

r

kp

ir

kl

j

rp

rj

kl

i

rpp

ij

klij

klp TTTTT

T

Now we can see that the metrical condition 0= ikp g also implies that 0~

= ikp g ,A

r

r

ik

A

ik T =~

, and 0= Aik .


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Finally, with the help of the connection field , we derive the third fundamentalgeometric objects of 3 , i.e., the local fourth-order curvature tensor of the material space

lkj

i

i

jklR ggggR = .

where

i

rl

r

jk

i

rk

r

jll

i

jk

k

i

jli

jklR +

=

.

These are given in the relations

( ) [ ] irrjkrrijkikjjk FFRF = 2.

where iF are the covariant components of an arbitrary vector field F of 3 .

Correspondingly, for the contravariant components iF we have

( ) [ ] irrjkrirjkikjjk FFRF = 2.

The Riemann-Christoffel curvature tensor R~

here then appears as the part of the

curvature tensor R built from the symmetric, holonomic Christoffel connection alone,whose components are given by

{ } { } { }{ } { }{ }irlr

jk

i

rk

r

jl

i

jkl

i

jlk

i

jklR +

=

.

~

Correspondingly, the components of the symmetric Ricci tensor are given by

{ } ( ) { } ( ) { }{ }rsksirse

s

ikik

er

ikr

r

irkik

ggRR

+

==

2/12/12

.

loglog~~

where we have used the relations

{ } ( ) kkiie

k

ik

g=

=

2/1log

Then the Ricci scalar is simplyi

iRR

.

~~= , an important geometric object which shall play

the role of the microspin (chirality) potential in our generalization of classical elasticitytheory developed here.


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Now, it is easily verified that

( ) rrijkikjjk FRF .~~~~~ =

and

( ) rirjk

i

kjjkFRF

.

~~~~~=

The remaining parts of the curvature tensor R are then the remaining non-holonomic

objects J and Q whose components are given as

irl

rjk

irk

rjll

i

jk

k

i

jlijkl

TTTTTTJ +

=

.

and

{ } { } { } { }irl

r

jk

i

rl

r

jk

i

rk

r

jl

i

rk

r

jl

i

jklTTTTQ +=.

Hence, we write

i

jkl

i

jkl

i

jkl

i

jkl QJRR ....~

++=

More explicitly,

i

rl

r

jk

i

rk

r

jl

i

jkl

i

jlk

i

jkl

i

jklTTTTTTRR ++=

~~~..

From here, we define the two important contractions of the components of the curvaturetensor above. We have the generalized Ricci tensor whose components are given by

r

r

ikik

s

rk

r

is

r

ikrik

r

irkikTTTTRRR ++==

~~~.

where the 3=n quantities

[ ]k

ik

k

iki T == 2

define the components of the microspin vector. Furthermore, with the help of the

relations [ ]is

ks

iki

rs

rs gTg == 2 , the generalized Ricci scalar is

ikj

ijk

i

i

i

i

i

i TTRRR == ~

2~

.


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It is customary to give the fully covariant components of the Riemann-Christoffel

curvature tensor. They can be expressed somewhat more conveniently in the following

form (when the ikg are continuous):

{ } { } { } { }( )sjlriksjkrilrsikjl

jl

ik

il

jk

jk

ilijkl g

ggggR +

+

=

2222

2

1~

In general, when the ikg are continuous, all the following symmetries are satisfied:

klijijkl

ijlkjiklijkl

RR

RRR

~~

~~~

=

==

However, for the sake of generality, we may as well drop the condition that theik

g are

continuous in their second derivatives, i.e., with respect to the material coordinates i

such that we can define further more non-holonomic, anti-symmetric objects extracted

from R such as the tensor field V whose components are given by

==

k

A

l

ii

A

l

k

l

A

r

rikik RV

.

The above relations are equivalent to the following ( ) 3121 =nn equations for the

components of the material metric tensor:

( )jiklijkllij

kk

ij

lRR

gg+=

which we shall denote simply by klijg , . When the ikg possess such discontinuities, we

may define the discontinuity pontentialby

{ } ( )i

ek

iki

g

==

2/1log

Hence we have

k

i

i

k

ikV

=

From the expression of the determinant of the material metric tensor, i.e.,


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kjiijk gggg 321=

we see, more specifically, that a discontinuum with arbitrary geometric singularities is

characterized by the following discontinuity equations:

ksrrsjiijk

jsrrskiijk

isrrskjijksrrs

ggg

ggg

gggg

3

22

21

2

22

31

1

22

32

22

+

+

=

In other words,

( ) ( )

( ) jirskkrsijk

kirsjjrsijkkjrsiirsijksrrs

ggRR

ggRRggRRg

2133

31223211

22

+

++=

It is easy to show that in three dimensions the components of the curvature tensor R

obey the following decomposition:

( ) RggggRgRgRgRgWR jlikjkililjkjkilikjljlikijklijkl +++=2

1

i.e.,

( ) RRRRRWR jlikjkililjkjkilikjljlikijklijkl +++=2

1......

where ijklW (andij

klW.. ) are the components of the Weyl tensor W satisfying 0. =r

irkW ,

whose symmetry properties follow exactly those of ijklR . Similarly, for the components

of the Riemann-Christoffel curvature tensor R~

we have

( ) RRRRRWR jlikjkililjkjkilikjljlikijklijkl~

2

1~~~~~~...... +++=

Later, the above equations shall be needed to generalize the components of the elasticity

tensor of classical continuum mechanics, i.e., by means of the components


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( ) Rjkiljlik~

2

1

Furthermore with the help of the relations

[ ] [ ] [ ][ ] [ ] [ ]

+++

+

+

=++ rjk

i

rl

r

lj

i

rk

r

kl

i

rjk

i

lj

j

i

kl

l

i

jki

ljk

i

klj

i

jkl RRR

2...

we derive the following identities:

[ ] [ ] [ ]( )

[ ] [ ]

ijk

r

r

ij

r

s

s

ri

ikikik

i

ijrl

r

pkijrk

r

lpijrp

r

klijpklijlpkijklp

RRgRgR

RRRRRR

....22

1

2

+=

++=++

From these more general identities, we then derive the simpler and more specialized

identities:

0~

2

1~~

0~~~~~~

0~~~

=

=++

=++

RgR

RRR

RRR

ikik

i

ijpklijlpkijklp

iljkikljijkl

often referred to as the Bianchi identities.

We are now able to state the following about the sources of the curvature of the material

space 3 : there are actually two sources that generate the curvature which can actuallybe sufficiently represented by the Riemann-Christoffel curvature tensor alone. The first

source is the torsion represented by [ ]i

jk which makes the hypersurface 3 non-

orientable as any field shall in general depend on the twisted path it traces therein. As we

have said, this torsion is the source of microspin, i.e., point-rotation. The torsion tensor

enters the curvature tensor as an integral part and hence we can equivalently attribute the

source of microspin to the Riemann-Christoffel curvature tensor as well. The secondsource is the possible discontinuities in regions of 3 which, as we have seen, render the

components of the material metric tensorAB

B

k

A

iikg = discontinuous at least in their

second derivatives with respect to the material coordinates i . This is explicitly shown in

the following relations:


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The relations we have been developing so far of course account for arbitrary non-

orientability conditions as well as geometric discontinuities of the material space 3 .

Consequently, we see that the holonomic field equations of classical continuummechanics shall be obtained whenever we drop the assumptions of non-orientability of

points and geometric discontinuities of the material body. We also emphasize thatgeometric non-linearity of the material body has been fully taken into account. A material

body then becomes linear if and only if we neglect any quadratic and higher-order terms

involving the connection field of the material space 3 .

3. Elements of the generalized kinematics: deformation analysis

Having described the internal structure of the material space 3 , i.e., the material body,

we now move on to the dynamics of the continuum/discontinuum 3 when it is subjectto an external displacement field. Our goal in this kinematical section is to generalize the

notion of a material derivative with respect of the material motion. We shall deal with the

external displacement field in the direction of motion of 3 which brings 3 from its

initially undeformed configuration to the deformed configuration 3

. We need to

generalize the structure of the external displacement (i.e., external diffeomorphism) toinclude two kinds of microspin of material points: the non-electromagnetic microspin as

well as the electromagnetic microspin which is generated, e.g., by electromagnetic

polarization.

In this work, in order to geometrically describe the mechanics of the so-called Cosserat

continuum as well as other generalized continua, we define the external displacementfield as being generally complex according to the decomposition

iii iu +=

where the diffeomorphism 33

is given by

iii +=

Here

i

u are the components of the usual displacement field u in the neighborhood ofpoints in 3 , and

i are the components of the microspin point displacement field satisfying

0=+ kiik

which can be written as an exterior (Lie) derivative:

0=ikgL


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This just says that the components of the material metric tensor remain invariant with

respect to the action of the field . We shall elaborate on the notion of exteriordifferentiation in a short while.

The components of the displacement gradient tensorD are then

( ) ( )

( ) ( ) ( )

ikik

kiikkiikkiik

kiikkiik

ikik

iuuuu

D

+=

+++=

++=

=

2

1

2

1

2

1

2

1

2

1

Accordingly,

( ) =+= kiikik uu2

1 ( )ikik

lin

ik ggg =

2

1uL

2

1

are the components of the linear strain tensorand

ikikik +=

are the components of the generalized spin (vorticity) tensor, where

( )kiikik uu =2

1

are the components of the ordinarymacrospin tensor, and

( )kiikik i =2

1

are the components of the microspin tensor describing rotation of material points on their

own axes due to torsion, or, in the literature, the so-called distributed moment. At this

point, it may be that the internal rotation of material points is analogous to the spin of

electrons if the material point themselves are seen as charged point-particles. However,

we know that electrons possess internal spin due to internal structural reasons while thematerial points also rotate partly due to externally induced couple stress giving rise to

torsion. For this reason we split the components of the microspin into two parts:

iii Ae+=


19/44

19

where i describe non-electromagnetic microspin and iAe describe pure electron spin

with e being the electric charge and iA , up to a constant of proportionality, being the

material components of the electromagnetic vector potential A :

AqAii =

where q is a parametric constant and A are the components of the four-dimensional

electromagnetic vector potential in the sense of Maxwellian electrodynamics. Inversely,we have

( ) nNAq

A ii +=

1

where AnqN = . The correspondence with classical electrodynamics becomes

complete if we link the electromagnetic microspin tensor f represented by the

components

( )kiikik AAeif =2

1

to the electromagnetic field tensor = FF through

F

ef kiik =

where2

2

1eqi= . The four-dimensional components of the electromagnetic field

tensor in canonical form are

=

=

00

0

0

123

132

231

321

BBEBBE

BBE

EEE

y

A

y

AF

where ( )321 ,, EEE=E and ( )321 ,, BBB=B are the electric and magnetic fields,

respectively. In three-dimensional vector notation,

=r

r

tc

A1E and Acurl

r=B ,

where ( ) ,Ar

== A . They satisfy Maxwells equations in the Lorentz gauge

0=Adivr

, i.e.,


20/44

20

0

14

41

2

=

=

==

=

B

EB

E

jBE

div

curltc

div

ccurl

tc

e

where j is the electromagnetic current density vector and e is the electric charge

density. In addition, we can write

j

cF

4=

i.e., AA jc

F

4= and ej =

4 . The inverse transformation relating ikf to ikF is

then given by

Ffe

F ikki +=

where

( )

nFnFnF

nFnFnF

==

=2

This way, the components of the generalized vorticity tensor once again are

( ) ( ) ( )

[ ] ( )

Fe

AeiiuAA

eiiuu

AAeiiuu

kiikik

rrr

r

iki

k

k

i

i

k

k

i

i

k

k

i

kiikkiikikikik

++=

++

+

+

=

++=

2

1

2

1

2

1

2

1

2

1

2

1

where

( )kiikik i =2

1


21/44

21

are the components of the non-electromagnetic microspin tensor. Thus we have now

seen, in our generalized deformation analysis, how the microspin field is incorporatedinto the vorticity tensor.

Finally, we shall now produce some basic framework for equations of motion applicable

to arbitrary tensor fields in terms of exterior derivatives. We define the exterior derivative

of an arbitrary vector field (i.e., a rank-one tensor field) of 3 , say W , with respect tothe so-called Cartan basis as the totally anti-symmetric object

[ ]ki

ki WU ggWU = 2L

where U is the velocity vector in the direction of motion of the material body 3 , i.e.,

tU

ii

=

. If we now take the local basis vectors as directional derivatives, i.e., the

Cartancoordinate basisvectors iii =

=

g and ii dg = , we obtain for instance, in

component notation,

( ) kikikk

iiUWWUW +== UU W LL

Using the exterior product, we actually see that

UWU =L

UWWUW =

Correspondingly, foriW , we have

( ) ikki

k

kii UWWUW == UU W LL

The exterior material derivative is then a direct generalization of the ordinary material

derivative (e.g., as we know, for a scalar field it is given byi

iU

ttD

D

+

= ) as

follows:

(UU +

=+

=

t

WW

t

W

tD

WD ii

iiL ) kikik

ki

i UWWUt

W++

=V

(UU +

=+

=

t

WW

t

W

tD

WDi

iii

L ) kiki

k

ki

iUWWU

t

W+

=V


22/44


23/44

23

with the help of the metrical condition 0= ikp g . Similarly, we also find

( )kiikikik

UUt

gtD

gD +

=

Note also that

0=tD

D ik

The components of the velocity gradient tensor are given by

( ) ( )kiikkiikikik UUUUUL ++==21

21

where, following the so-called Helmholtz decomposition theorem, we can write

( )jkkj

ljk

ilii gU +=2

1

for a scalar field and a vector field . However, note that in our case we obtain thefollowing generalized identities:

[ ]( )

[ ] ll

ij

ijk

kl

l

ijl

l

kij

ijk

gradcurl

AURcurldiv

=

= 22

1.U

which must hold throughout unless a constraint is invoked. We now define the

generalized shear scalar by

( )

[ ] kllijijkllkijijk

kijji

ijki

i

i

i

R

U

+=

+==

.2

21

2

1

In other words, theshear now depends onthe microspin fieldgenerated by curvature and

torsion tensors.

Meanwhile, we see that the contravariant components of the local acceleration vector

will simply be given by


24/44

24

t

U

UUUUt

U

tD

UDa

i

i

k

ki

k

kii

i

=

+

==

However, we also have

( ) kkiikii

i UUUt

U

tD

UDa ++

==

for the covariant components.

Furthermore, we have

kikiki

i Ut

g

tD

gD

t

Ua

+

=

Now, define the local acceleration covectorthrough

( )

tD

gDU

t

U

UUUt

Ug

aga

ik

k

i

k

k

ii

k

kik

i

iki

=

++

=

=

such that we have

tD

gDUaa

ik

k

ii =

Hence we see that the sufficient condition for the two local acceleration vectors to

coincide is

0=tD

gD ik

In other words, in such a situation we have

( )kiikik UUt

g+=


25/44

25

In this case, a purely rotational motion is obtained only when the material motion is rigid,

i.e., when 0=

t

gik or, in other words, when the condition

( ) 0=+== kiikikik UULgUL

is satisfied identically. Similarly, a purely translational motion is obtained when [ ] 0=ikL ,

which describes a potential motion, where we have iiU = . However, as we haveseen, in the presence of torsion even any potential motion of this kind is still obviously

path-dependent as the relations [ ] 0 ll

ij

ijk hold in general.

We now consider the path-dependent displacement field tracing a loop l , say, frompoint 1P to point 2P in 4 with components:

( )

+==2121

..

PP

ki

lk

li

k

i

k

PP

iidd

Let us observe that

( )

l

i

l

Ai

A

l

Ai

A

Ai

Al

l

i

AA

l

i

AA

k

k

l

A

ki

A

ki

kl

k

=

=

=

=

=

Now since ii = , and usingii

ff

=

for an arbitrary function f , we have

0=

=l

iB

lB

Ai

A

i

kl

k

x

x

and we are left with

( )

+==2121

..

PP

ki

k

i

k

PP

iidd


26/44

26

Assuming that the ik are continuous, we can now derive the following relations:

( ) ki

lk

l

i

kk

lk

ii

kl

ki

kl

kk

i

kik

i

ddd

ddD

==

==

.

..

With the help of the above relations and by direct partial integration, we then have

( ) lkPP

i

lk

i

kl

P

P

ki

k

PP

ii dd

==21

2

1

21

.

.

..

where

( )jkiikjjikik = ...

It can be seen that

ji

jkl

i

lk

i

kl Z ... =

where we have defined another non-holonomic tensorZ with the components

jki

liljkjl

ik

ikjl

ijklZ += ...

Now, the linearized components of the Riemann-Christoffel curvature tensor are given by

+

=

ik

jl

jl

ik

il

jk

jk

illin

ijkl

ggggR

2222

2

1~

Direct calculation gives

( )jlikikjljkililjklinijkl ggggR +=21~

However, ikikg = , and hence we obtain

( )jlikikjljkililjk

lin

ijklR +=2

1~

In other words,


27/44

27

ijkl

lin

ijkl RZ~

2 =

Obviously the ijklZ possess almost the same fundamental symmetries as the components

of the Riemann-Christoffel curvature tensor, i.e., ijlkjiklijkl ZZZ == as well as the

general asymmetry klijijkl ZZ as

( ) ( ) ( ) ( )

[ ] [ ] [ ] [ ]( )jkrrliilrrkjjlrrikikrrjljr

r

kli

r

likrl

r

ikj

r

jikir

r

lkj

r

kljrk

r

jli

r

ijlklijijkl RRRRRRRRZZ

+++

+++++++=

2

........

When the tensor Z vanishes we have, of course, a set of integrable equations giving riseto the integrability condition for the components of the strain tensor, which is equivalent

to the vanishing of the field . That is, to the first order in the components of the straintensor, if the condition

0~

=ijklR

is satisfied identically.

Finally, we can write (still to the first order in the components of the strain tensor)

( )

( )

=

+==

21

2

1

21

2

1

21

..

..

~

2

1

PP

klji

jkl

P

P

ki

k

klj

PP

i

jkl

P

P

ki

kPP

ii

dSR

dSZd

where

ikkiik dddS =

are the components of an infinitesimal closed surface in 3 spanned by thedisplacements d and in 2 preferred directions.

Ending this section, let us give further in-depth investigation of the local translational-

rotational motion of points on the material body. Define the unit velocity vector by

ds

d

gU

i

lk

kl

i

i

==

44

4

such that


28/44

28

( )ds

dg

t

ilk

kl

ii

2/1

444 =

=

i.e.,

( )

( )dtU

dtUU

dt

gds

i

i

l

l

ki

ik

=

=

=

2/1

2/1

44

Then the local equations of motion along arbitrary curves on the hypersurface of material

coordinates 43 can be described by the quadruplet of unit space-time vectors

( )nWVU ,,, orthogonal to each other where the first three unit vectors (i.e., WVU ,, )are exclusively defined as local tangent vectors in the hypersurface 3 and n is the unit

normal vector to the hypersurface 3 . These equations of motion are derived bygeneralizing the ordinary Frenet equations of orientable points of a curve in three-

dimensional Euclidean space to four-dimensions as well as to include effects ofmicrospin generated by geometric torsion. Setting

n

Ww

Vv

Uu

i

i

i

i

i

i

=

==

==

==

n

gW

gV

gU

we obtain, in general, the following set of equations of motion of the material points on

the material body:

ws

n

nvs

w

ukwsv

vks

u

=

+=

=

=

where the operator


29/44

29

== uU

si

i

represents the absolute covariant derivative in 43 . In the above equations we havedefined the following invariants:

2/1

2/12/1

=

==

=

=

s

n

s

n

G

s

VVUn

s

vvu

s

U

s

Ug

s

u

s

uGk

kji

ijk

ki

ik

In our case, however, the vanishing of the extrinsic curvature of the hypersurface 3 means that the direction of the unit normal vector n is fixed. Consequently, we have

0=

and our equations of motion can be written as

lki

kl

ii

k

k

lki

kl

iii

k

k

lki

kl

ii

k

k

UWTVWU

UVTUkWVUUUTVkUU

~

~

~

=

==

in three-dimensional notation. In particular, we note that, just as the components of the

contorsion tensori

jkT , the scalar measures the twist of any given curve in 3 due to

microspin.

Furthermore, it can be shown that the gradient of the unit velocity vector can be

decomposed accordingly as

ikikikikik AUhU

4

1 +++=

where


30/44

30

( ) ( )( )

( ) ( ) [ ]

s

UA

U

UThhUUhhUUhh

UThhUUhhUUhh

UUgh

ii

i

i

l

l

rs

s

k

r

irssr

s

k

r

irssr

s

k

r

iik

l

l

rs

s

k

r

irssr

s

k

r

irssr

s

k

r

iik

kiikik

2

1~~

4

14

1

2

1~~

4

1

4

1

=

=

==

+=+=

=

Note that

0 ===k

ik

k

ik

k

ik UUUh

Setting ( ) 2/144

= kiikg such thatii UU = , we obtain in general

( ) ( )

l

lkik

ikii

k

ki

l

likikki

rssr

s

k

r

irssr

s

k

r

iik

UUU

s

UU

s

UU

s

UU

sUUUg

sgU

UUhhUUhhU

+

++++

++=

3322

4

1

2

1

4

1

4

1

4

1

2

1

4

1

4

1

Again, the vanishing of the extrinsic curvature of the hypersurface 3 gives 0=s

.

Hence we have

( ) ( )

l

lkik

ii

k

l

likki

rssr

s

k

r

irssr

s

k

r

iik

UUUs

UU

s

UUUgU

UUhhUUhhU

+++

++=

22

4

1

2

1

4

1

2

1

4

1

4

1

for the components of the velocity gradient tensor.

Meanwhile, with the help of the identities

( ) ( ) ( ) ( ) ( )( ) [ ] il

jl

jk

j

l

l

ijkikjjk

j

i

j

jkiki

j

jkij

j

kijk

j

UUUURUU

UUAUUUUUU

2

. =

==

we can derive the following equation:


31/44

31

( ) ( ) [ ] klilikkiikikkii

iUUUURUU

s

U

s

2

+

=

Hence we obtain

( ) ( ) ( ) [ ] klilikkiikeii

i

k

k

i

i

i UUUURs

UUU

s

U

s+

=

2log2

for the rate of shear with respect to the local arc length of the material body.

4. Generalized components of the elasticity tensor of the material body 3 in thepresence of microspin and geometric discontinuities (defects)

As we know, the most general form of the components of a fourth-rank isotropic tensor is

given in terms of spatial coordinates by

BCADBDACCDABABCD CCCI 321 ++=

where ,, 21 CC and 3C are constants. In the case of anisotropy, ,, 21 CC and 3C are no

longer constant but still remain invariant with respect to the change of the coordinate

system. Transforming these to material coordinates, we have

jk

il

jl

ikkl

ijijkl CCggCI 321.. ++=

On reasonably relaxing the ordinary symmetries, we now generalize the components ofthe fourth-rank elasticity tensor with the addition of a geometrized part describing

microspin and geometric discontinuities as follows:

( ) ( )jkiljlikjkiljlikklijijkl ggC +++=..

where

R

AA

AA

ki

ki

ki

ik

ik

ki

ki

ki

~

2

1

10

1

10

1

15

1

15

2

....

....

=

=

=

where is a non-zero constant, and where


32/44

32

( )jkiljlikklijijkl ggA ++=..

are, of course, the components of the ordinary, non-microspin (non-micropolar) elasticity

tensor obeying the symmetries ijklijlk

jikl

ijkl AAAA ........ === . Now if we define the remaining

components by

( ) RB jki

l

j

l

i

k

ij

kl

~

2

1.. =

withij

kl

ij

lk

ji

kl

ij

kl BBBB ........ === , then we have relaxed the ordinary symmetries of the

elasticity tensor. Most importantly, we note that our choice of the Ricci curvature scalar

R~

(rather than the more general curvature scalarR of which R~

is a component) to enterour generalized elasticity tensor is meant to accommodate very general situations such

that in the absence of geometric discontinuities the above equations will in general still

hold. This corresponds to the fact that the existence of the Ricci curvature tensor R~

is

primarily due to microspin while geometric discontinuities are described by the full

curvature tensor R as we have seen in Section 2.

Now with the help of the decomposition of the Riemann-Christoffel curvature tensor, we

obtain

( )( )

ij

kl

i

l

j

k

j

k

i

l

i

k

j

l

j

l

i

k

ij

kl

j

k

i

l

j

l

i

kkl

ijij

kl

RRRRRWggC..........

~~~~~~+++++=

for the components of the generalized elasticity tensor. Hence for linear elastic

continua/discontinua, with the help of the potential energy functional F , i.e., the onegiven by

kl

ijkl

ij DDCF ..2

1=

such that

ijij D

F

=

i.e.,

( )

[ ] ijij

ijij

F

F

=

=


33/44

33

we obtain the following constitutive relations:

kl

kl

ijijDC

..=

relating the components of the stress tensor to the components of the displacementgradient tensor D . Then it follows, as we have expected, that the stress tensor becomes

asymmetric. Since 0)..( =kl

ijB , we obtain

ij

k

kij

kl

kl

ijkl

kl

ijij

g

ADC

2.

..)..()(

+=

==

for the components of the symmetric part of the stress tensor, in terms of the components

strain tensor and the dilation scalari

i. = . Correspondingly, since [ ] 0.. =kl

ijA , the

components of the anti-symmetric part of the stress tensor are then given by

[ ] [ ]

( ) ( )( ) ( )

R

RRRW

RDRDRDRDRW

BDC

ij

ik

k

jjk

k

ikl

kl

ij

kl

ij

ik

k

jjk

k

iik

k

jjk

k

i

kl

ij

kl

ij

kl

kl

ijkl

kl

ijij

~

~~2

~~

~~~~~~

......

........

....

=

+=

++=

==

in terms of the components of the generalized vorticity tensor. We can now define thegeometrized microspin potentialby the scalar

( )ikjijkiiii TTRRS +++== ~

2~

Then, more specifically, we write

[ ]

++=

Fe

S jiijijij

From the above relations, we see that when the electromagnetic contribution vanishes, we

arrive at a geometrized Cosserat elasticity theory. As we know, the standard Cosseratelasticity theory does not consider effects generated by the electromagnetic field. Various

continuum theories which can be described as conservative theories often take intoconsideration electrostatic phenomena since the electric field is simply described by a

gradient of a scalar potential which corresponds to their conservative description of force

and stress. But that proves to be a limitation especially because magnetic effects are stillneglected.


34/44

34

As usual, should we consider thermal effects, then we would define the components of

the thermal stress t by

Tgt ikTik =

where T is the thermal coefficient and T is the temperature increment. Hence thecomponents of the generalized stress tensor become

TgDCikTrs

rs

ikik = ..

Setting now

( )

23

1

+=T

T

we can alternatively write

( )TgDC rsTrsrs

ikik+= ..

Finally, we shall obtain

( ) RgTikikikT

r

rik

~2

. ++=

i.e., as before

( ) ( )

[ ] RSR

gT

k

ijkijij

ijijT

r

rij

~

2

1~

2.

==

+=

wherekS are the components of the generalized vorticity vector S .

We note that, as is customary, in order to accord with the standard physical description ofcontinuum mechanics, we need to set

( )

=

+==

21

2

12

G

EG

where G is the shear modulus, E is Youngs modulus, and is Poissons ratio.


35/44


36/44

36

( ) ( )( ) ( ) ( )

( ) ( )

( ) ( )

( ) Rggggg

RgggggRggggg

RgggggRggggg

ggggggggggggggg

gggggggggggggK

knilnlikjm

knjlnljkimjkiljlikmn

jminjnimkllmknnlkmij

knilnlikjmknjlnljkimjkiljlikmn

jminjnimkllmknnlkmijmnklijijklmn

~

2

1

~

2

1~

2

1

~

2

1~

2

1

5

43

21

654

321

+

++

++

++++++

++++=

where we have set 6105846342211 ,,,,, ====== BBBBBB and where, for

constant 521 ,...,, , the five quantities

RBK

RBK

RBK

RBK

RBK

~

2

1

~

2

1

~

2

1

~

2

1

~

2

1

5115

494

373

252

131

==

==

==

==

==

form a set ofadditional microspin potentials. Hence we see that in the non-linear case, atleast there are in general sixmicrospin potentials instead of just one as in the linear case.

Then the constitutive equations are readily derivable by means of the third-order potential

functional

mn

kl

ijkl

mnijkl

ijkl

ij DDDKDDCF ....*

3

1

2

1+=

through

( ) ( )ijijijij

D

F

21*

+=

=

where ( )1 indicates the linear part and ( )2 indicates the non-linear part. Note that this istrue whenever the ijklmnK in general possess the above mentioned symmetries. Direct, but

somewhat lengthy, calculation gives


37/44

37

( )

( )( ) ( ) ( )( )( ) ( )RDRD

RgR

DDK

ikik

k

jjkjk

k

i

ijij

k

kij

kl

kl

kl

kl

k

k

mn

kl

kl

mnijij

~2

~2

~22

~2

56.45.

3243.12

2

.1

..

2

++++

++++++=

=

Overall, we obtain, for the components of the stress tensor, the following:

( )

( )( ) ( ) ( )( )( ) ( )RDRD

RgR

RgT

ikik

k

jjkjk

k

i

ijij

k

kij

kl

kl

kl

kl

k

k

ijijijT

k

kij

~2

~2

~22

~2

~2

56.45.

3243.12

2

.1

.

++++

+++++++

++=

5. Variational derivation of the field equations. Equations of motion

We shall now see that our theory can best be described, in the linear case, independently

by 2 Lagrangian densities. We give the first Lagrangian density as

( ) ( ) ( )

++= kmk

ki

ii

iT

kl

ij

ij

klikik

ikUfUTDDDCDgL ...

2

1

where m is the material density and f is a scalar potential. From here we then arrive at

the following invariant integral:

( ( )( ) [ ]( )

( ) ( )) dVUfUT

BAI

k

m

k

ki

ii

iT

kl

ij

ij

kl

kl

ij

ij

klikik

ik

vol

ikik

ik

+

+++=

.

....2

1

2

1

where 321 dddgdV = .

Writing LgL = , we then have

( ) ( ) 0=

+++= dVLLLLI vol kiki

ikik

ikik

ikik

Now

( )( )

( ) ( )

( )

=

=

vol

k

ki

i

k

kivol

i

vol

k

ki

iki

vol ki

dVL

dVL

dVL

dVL


38/44


39/44

39

( )( ) ( ) kiimkimikiikiiiki

ki

i UUUUUfUfL

++=

Define the extended shear scalar and the mass current density vector, respectively,through

( )i

m

i

i

i

UJ

Ufl

=

=

Now we readily identify the force per unit mass f and the body force per unit mass b ,

respectively, by

( )( )

++=+=

==

imi

m

ik

k

i

m

i

ii

k

ki

Ut

flUJflb

tUUUf

11

11

where we have used the relation

i

imm UtD

D=

i.e., ( ) 0=+

imi

m Ut

, derivable from the four-dimensional conservation law

( ) 0* = Um where ( )cUU i ,= .

Hence we have (for arbitrary k )

( ) 0=+ dVfb kvol

k

m

k

m

ik

i

i.e., the equations of motion

( )kkmiki bf =

Before we move on to the second Lagrangian density, lets discuss briefly the so-calledcouple stress, i.e., the couple per unit area also known as the distributed moment. We

denote the couple stress tensor by the second-rank tensor field M . In analogy to the

linear constitutive relations relating the stress tensor to displacement gradient tensorD , we write


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rs

rs

ikikNDM

..=

where

ijklijklijkl FED +=

are assumed to possess the same symmetry properties as ijklC (i.e., ijklE have the same

symmetry properties as ijklA while ijklF , representing the chirality part, have the same

symmetry properties asijkl

B ).

Likewise,

( ) [ ] ikikikikik YXNNN +=+=

are comparable to ( ) [ ] ikikikikik DDD +=+= .

As a boundary condition, let us now define a completely anti-symmetric third-rank spin

tensor as follows:

[ ] ikliklikl JJ ==2

1

where is a scalar function such that the spin tensor of our theory (which contains boththe macrospin and microspin tensors) can be written as a gradient, i.e.,

ijk

ijki RS ==~

such that whenever we desire to subject the above to the integrability condition

0= kjijk S , we have [ ] 0= ll

jk

ijk S , resulting in 0=ikY .

In other words,

+=kij

ijk dR ~0

where 0 is constant, acts as a scalar generator of spin.

As a consequence, we see that


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( )ik

pqk

p

i

q

k

q

i

p

pqpql

ikl

l

ikl

l

iklikl

l

R

R

R

SJ

~

~

2

1

~21

2

1

2

1

=

=

=

==

i.e.,

[ ]ikikll J =

Taking the divergence of the above equations and using the relations

[ ] [ ] ll

ikik = 22 , we obtain the following divergence equations:

[ ][ ] rr

kl

iklik

k S=2

1

coupling the components of the spin vector to the components of the torsion tensor.

Furthermore, we obtain

( )[ ]

( )k

eik

r

r

kl

iklik

k

RSR

=

~log~

2

1

We now form the second Lagrangian density of our theory as

( ) ( ) ( )

++= km

k

ki

irsi

ki

k

rs

kl

ij

ij

klikik

ik VIShSUJSNNDNSMgH ...2

1

where h is a scalar function (not to be confused with the scalar function f ), I is the

moment of inertia, andiV are the components of the angular velocity field.

Hence the action integral corresponding to this is

( ( )( ) [ ]( )

( ) ( ) ( )) dVVIShSUJS

YYFXXEYSMXSMJ

k

m

k

ki

irsi

ki

k

rs

kl

ij

ij

kl

kl

ij

ij

klikik

ik

vol

ikik

ik

+

+++=

.

....2

1

2

1

As before, writing HgH = and performing the variation 0=J , we have


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t

VVU

ii

k

ki

==

and the angular body force per unit mass by

( )

+= ikk

ii

m

i VJIt

ShSl

1

where ( )ii Uhl = .

We have

[ ]( ) 0. =+ dVSIM kvol

km

km

rskrs

iki

Hence we obtain the equations of motion

[ ] kkm

rsk

rs

ik

i IM += .

6. Concluding remarks

At this point we see that we have reproduced the field equations and the equations of

motion of Cosserat elasticity theory by our variational method, and hence we havesucceeded in showing parallels between the fundamental equations of Cosserat elasticitytheory and those of our present theory. However we must again emphasize that our field

equations as well as our equations of motion involving chirality are fully geometrized. In

other words, we have succeeded in generalizing various extensions of the classicalelasticity theory, especially the Cosserat theory and the so-called void elasticity theory by

ascribing both microspin phenomena and geometric defects to the action of geometrictorsion and to the source of local curvature of the material space. As we have seen, it is

precisely this curvature that plays the role of a fundamental, intrinsic differential

invariant which explains microspin and defects throughout the course of our work.


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A Geometric Chiral-Continuum Theory of Elasticity

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