A Micromorphic Electromagnetic Theory

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Nov 16, 2013 (3 years and 4 months ago)

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A Micromorphic Electromagnetic Theory



James D. Lee, Youping Chen and Azim Eskandarian

School of Engineering and Applied Science

The George Washington University

801 22
nd

street, NW, Academic Center

Washington, DC 20052


Email: jd
lee@gwu.edu

Fax:
202
-
994
-
0238




Abstract


This work is concerned with the determination of both macroscopic and microscopic
deformations, motions, stresses, as well as electromagnetic fields developed in the
material body due to external loads of thermal, mechanical, and
electromagnetic origins.
The balance laws of mass, microinertia, linear momentum, moment of momentum,
energy, and entropy for microcontinuum are integrated with the Maxwell’s equations.
The constitutive theory is constructed. The finite element formulation

of micromorphic
electromagnetic physics is also presented. The physical meanings of various terms in the
constitutive equations are discussed.


Keyword:
micromorphic, electromagnetic, balance laws, constitutive relation, finite
element formulation.


1 Int
roduction


Optical phonon branches exist

in all crystals that have more than one atom per primitive
unit cell. Under an electromagnetic field it is the optical modes that are excited. Optics is
a phenomenon that necessitates the presence of an

electromagne
tic field.


While classical continuum theory is the long acoustic wave limit, lattice dynamics
analysis has shown that micromorphic theory yields phonon dispersion relation similar to
those from atomistic calculations and experimental measurements
(Chen et

al [2002a]).

It

provides up to 12 phonon dispersion relations, including 3 acoustic and 9 optical
branches. The optical phonons in micromorphic theory describe the internal displacement
patterns within the microstructure of material particles in consisten
t with the internal
atomic displacements in the optical modes.


The physics of mechanical and electromagnetic coupling is hence related to the optical
vibrations, and the continuum description of electrodynamics naturally leads to a
micromorphic electroma
gnetic theory.


International Journal of Solids and Structures
, submitted for publication

2 Physical Picture of Micromorphic Theories


Micromophic Theory, developed by Eringen and Suhubi [1964] and Eringen [1999],
constitutes extensions of the classical field theories concerned with the deformations,
motions, and electromagnetic

(E
-
M) interactions of material media, as continua, in
microscopic time and space scales. In terms of a physical picture, a material body is
envisioned as a collection of a large number of deformable particles,
each particle
possesses finite size and micro
structure. The particle
has the independent degrees of
freedom for both stretches and rotations (micromorphic),

and for rotations only

(micropolar), in addition

to the classical translational degrees of freedom of the center,
and

may be considered as a po
lyatomic molecule, a primitive unit cell of a

crystalline
solid, or a chopped fiber in a composite, et al. As shown in Fig.2
-
1,
a generic particle
P

is represented by its position vector
X

(mass center of
P
) and by a vector


attached to
P

representing the microstructure of
P

in the reference state at time
t
= 0. The motions that
carry
(,)
P
X

to
(,,)
P t
x
ξ

in the spatial configuration (deformed state) at time
t

can be
expressed as


(,),
k k
x x t

X


(2
-
1)


(,)
k kK K
t
 
 
X
. (2
-
2)


It is seen that the macromotion, eqn. (2
-
1), accounts for the motion of the centroid of the
particle while the micromotion, eqn.(2
-
2), specifi
es the changing of orientation and
deformation for the inner structures of the particle. The inverse motions can be written as



(,)
K K
X X t

x

, (2.3)


(,)
K Kk k
t
 
 
x

,

(2.4)


where



Kk
,
kK Kl kl kL KL
     
 

. (2.5)













Fig. 2
-
1 the macro
-

and micro
-
motion of a material particle



P(X,

)

X



P(x,

,, t)

x



3. The E
-
M Balance Laws


The balance

laws of the micromorphic electromagnetic continuum consist of two parts:
the thermomechanical part and the electromagnetic (E
-
M) parts. The E
-
M balance laws
are the well
-
known Maxwell’s equations that can be written as


e
q
 
D

or

e
k
,
k
q
D

, (3.1)


1
0
c t

  

B
E

or
0
1




t
B
c
E
e
i
j
,
k
ijk
, (3.2)


0
 
B

or
0

k
,
k
B
, (3.3)


1 1
c t c

  

D
H J

or
i
i
j
,
k
ijk
J
c
t
D
c
H
e
1
1




, (3.4)


where
D
is the dielectric displacement vector,
B

the magnetic flux vector,
E

the electric
field vector,
H

the magnetic field vector,
J
the current vector,
e
q
the free charge density.
The divergence of eq. (3.4) with the use of eq. (3.1) leads to


0
e
q
t

  

J

, (
3.5)


which is the law of conservation of charge. The divergence of eq. (3.2) gives



( ) 0
t

 

B


which is a duplicate of eq. (3.3).


The polarization vector,
P
, and the magnetization vector,
M
, are defined as



P D- E

, (3.6)




M B- H

. (3.7)


It is noted that the

E
-
M
vectors,
D,E,P,B,H,M,J
, are all referred to a fixed laboratory
frame
C
R
.
The Galilean transformations of inertial frames form a
group that consists of
time
-
independent spatial rotations and pure Galilean transforms, i.e.,


*
i ij j i i
x R x Vt b
  

, (3.8)


where


ij
kj
ki
jk
ik
R
R
R
R




and
det( ) 1

R
. (3.9
)


The requirement of the form
-
invariance of the Maxwell’s equations under the Galilean
transformations leads to the following transformations (Eringen and Maugin [1990])


e
*
e
q
q


,

(3.10)


*
e
q
 
J J v

, (3.11)


*
P = P

, (3.12)


1
c
 
*
M = M v P

,

(3.13)


1
c
 
*
E = E v B

, (3.14)


1
c

*
B = B v×E

, (3.15)


1
c

*
D = D
ν
×B

, (3.16)


1
c

*
H = H
ν
×D

, (3.17)


where the quantities,
*
,
e
q
* * * * * * *
J,P,M,E,B,D,H
, are referred to a co
-
moving frame
G
R

with material particles of the body having velocities,
v
. A typical nonrelativistic
feature of these transformations, eqs. (3.10
-
3.17), is the asymmetry between eq. (3.12)
and eq. (3.13), which says, according to Galilean relativity, a polarized moving body will
appear to be magnetized whereas a magn
etized moving body will not appear to be
polarized. Although it is this lack of symmetry that started to stimulate the study of
relativistic electrodynamics in the early 20
th

century, it should be remarked that few
observable conclusions can be made due to

the difficulty of obtaining sufficiently high
velocities for material media. The fully symmetric relativistic laws replacing eqs. (3.12,
3.13) may be found in Jackson [1975].


4. The Thermomechanical Balance Laws


The thermomechanical balance laws were or
iginally obtained by Eringen and Suhubi
[1964] by means of a “microscopic space
-
averaging” process. Later, Eringen [1999] re
-
derived the balance laws by starting with the following expression for the kinetic energy
of a material particle


1
( )
2
i i kl ik il
K v v i
 
 

, (4.1)


and, after the energy balance law is obtained, by requiring it to be form
-
invariant under
the generalized Galilean transformation to yield the balance laws of linear momentum
and moment of mom
entum. Recently, Chen et al. [2002] identified all the instantaneous
mechanical variables, corresponding to those in micromorphic theory, in phase space;
derived the corresponding field quantities in physical space through the statistical
ensemble averagin
g process; invoked the time evolution law and the generalized
Boltzmann transport equation for conserved properties to obtain the local balance laws of
mass, microinertia, linear momentum, moment of momentum, and energy for
microcontinuum field theory. In
the case that the external field is the gravitational field,
the balance laws obtained by Chen et al. [2002] in a bottom

up approach agree perfectly
with those obtained by Eringen and Suhubi [1964] and Eringen [1999] in a top
-
down
approach.


The balance la
ws of micromorphic continuum with E
-
M interactions can be expressed as


0
 
  
v

or
,
0
i i
v
 
 
, (4.2)


d
dt

  
i
i
υ +υ i

or
kl
km lm lm km
di
i i
dt
 
 
,

(4.3)


( ) 0

    
t f v F

or
,
( ) 0
ji j i i i
t f v F

   
, (4.4)


( ) 0

 
   
λ+t - s l - σ L

or
,
( ) 0
klm k ml ml lm lm lm
t s l L
  
     
, (4.5)



* *
:( ):
e h w
 

         
λ υ t v s - t PE BM υ q


or


* *
,,,
( )
klm lm k kl l k kl kl l k l k lk k k
e t v s t E P M B q h w
   
        

, (4.6)


where

,
v
,
i
,
υ
,
t
,



s s
,

λ
,

e
,

q
are

the mass density, velocity, microinertia,
microgyration, Cauchy stress, microstress average, moment stress, internal energy, heat
input, respectively;
f
,
l
,

h

are the body force, body moment, heat source of

mechanical
origin, respectively; the spin inertia
σ
is defined as


( )
kl ml km kn nm
i
  
 

; (4.7)


and the body force, body moment, energy source of E
-
M origin are given as (Eringen
a
nd Maugin [1990], Eringen [1999], De Groot and Suttorp [1972]):


* * * *
1
( ) ( ) { ( ) ( )}
e
q
c
              
F E P E B M J P P v P v B

, (4.8)



* *
L= PE + M B

, (4.9)


W
     
* * * *
E (P + P v) M B+ J E


. (4.10)


The second law of thermodynamics, also referred to as the Clausius
-
Duhem inequality, is
written as

( ) 0
h
   
  
q

, (4.11)


where


is the entropy de
nsity and


is the absolute temperature. Now the generalized
Helmholtz’s free energy


is introduced as



e
  
   
*
E P

. (4.12)


Then the Clausius
-
Duhem (C
-
D) inequ
ality can be expressed as


* *
,,
* * * *
,
( ) ( )
1
0
ijk jk i ij j i ij ij j i j i ji
i i i i i i i i
t v s t E P M B
q PE M B J E
   


       
    

. (4.13)


5. Constitutive Relations


The fundamental laws of micromorphic electromagnetic continuum consist of a system of
27 partial differential equations, eqs. (3.1
-
3.4, 4.2
-
4.6), and one inequality
, eq. (4.13).
There are 83 unknowns:

,
kl
i
,
k
v
,
kl

,

,

,
e
,
kl
t
,
kl lk
s s

,
klm

,
k
q
,
e
q
,
k
E
,
k
P
,
k
B
,
k
M
,
k
J
,

considering that the body force, body moment, and heat source are
given. Therefore 56
constitutive relations are needed to determine the dynamics of the thermomechanical
-
electromagnetic system.


The generalized Lagrangian strain tensors of micromorphic theory are defined as


KL
Lk
K
,
k
KL
x






,

(5.1)


KL kK kL KL LK
    
  

, (5.2)


,
KLM Kk kL M
  


, (5.3)


and the strain rates can be obtained as


,,
( )
KL l k lk k K Ll
v x
  
 


, (5.4)


( )
KL kl lk kK lL LK
     
  

, (5.5)


,,
KLM kl m Kk lL m M
x
   


. (5.6)


It can be easily proved that the Lagrangian stra
ins and their material time rates of any
order are objective, and hence they are suitable for being employed as independent
constitutive variables in the development of a constitutive theory. In the same spirit,
define the Lagrangian forms of the electric
field vector and the magnetic flux vector as


K
,
k
*
k
*
K
x
E
E


, (5.7)


K
,
k
k
K
x
B
B


, (5.8)


and their material time rates are obtaine
d


* * *
,,
( )
K k l l k k K
E E E v x
 

, (5.9)


,,
( )
K k l l k k K
B B Bv x
 

, (5.10)


The generalized 2
nd

order Piola
-
Kirchhoff stress tensors of micromorphic theory are

defined as


lL
k
,
k
kl
KL
X
jt
T



, (5.11)


2
Ll
Kk
kl
KL
js
S




, (5.12)


Ll
kK
m
,
M
mkl
KLM
X
j






, (5.13)


where
,
det( )
k K
j x


is the jacobian of the deformation gradient. It is straightforward to
show


,( ),
{ ( ) }
KL KL KL KL KLM KLM kl l k lk kl kl klm lm k
T S j t v s
     
     

, (5.14)


which means
{ }
T,S,
Γ
are the thermodynamic conjugates of
{ }
α,β,γ
. Similarly, the
Lagrangian form
s of the heat input, polarization, magnetization, and current are defined
as


k
,
K
k
K
X
jq
Q


, (5.15)


,
K k K k
P jP X


, (5.16)


* *
,
K k K k
M jM X


, (5.17)


* *
,
K k K k
J jJ X


. (5.18)


Now, the Clausius
-
Duhem inequality (4.13) can be rewritten as


* * * *
,
( )
1
0
o m m
KLM KLM KL KL KL KL
K K K K K K K K
T S
Q P E M B J E
    


    
    

, (5.19
)


where


l
*
k
*
l
k
kl
m
kl
B
M
E
P
t
t




, (5.19)


* *
m m
kl kl k l l k lk
s s M B M B s
   

, (5.20)



,
m m
KL kl K k lL
T jt X



, (5.21)


2
Ll
Kk
m
kl
m
KL
js
S




, (5.22)


where the superscript ‘
m
’ refers to the mechanical parts, i.e., if there is no E
-
M
interaction, then
m
kl
kl
t
t

. If it happens that
P

is proportional to
*
E
and

*
M
is proportional

to
B
, then there is no need to add terms to the Cauchy stress
t

and microstress average
s
. This situation prevails in the case of isot
ropic fluids in both magnetohydrodynamics
and electrohydrodynamics. However, the general case is needed for ferroelectric and
ferromagnetic materials, and also for materials affected by intrinsic spin and/or electric
quadruples (Dixon and Eringen [1965]).
It is remarked that the mechanical part of the
microstress average
m
S

is also symmetric.


In this work, the independent and dependent constitutive variables are set to be


 

*
Y = {
α,β,γ,,,E,B,X}

, (5.23
)



m m * *
Z = {T,S,
Γ,,,Q,P,M,J }

, (5.24)


and, following the axiom of equipresence, at the outset the constitutive relations are
written as


Z = Z(Y)

. (5.25)


It is noted t
hat there are 56 dependent constitutive variables in
Z
and both
Y
and
Z
are
presented in Lagrangian forms and, hence, the axiom of objectivity is automatically
satisfied. Substituting eq. (5.25
) into the C
-
D inequality (5.19), it follows


,
,
* *
*
* *
,
( ) ( ) ( )
( ) ( ) ( )
1
0
m m
K KL KL KL KL
K KL KL
KLM KLM K K K K
KLM K K
K K K K
T S
P E M B
E B
Q J E
   
       
   
  
   



   
      
   
  
      
  
  

. (5.26)


Since the inequality (5.26) is linear in


,



,
α
,
β
,
γ
,

*
E
,

B
, it holds if and only if


(,,)
  

*
α,β,γ E,B,X

, (5.27)










, (5.28)


m





T
α

,

(5.29)


m





S
β

, (5.30)







Γ
γ

, (5.31)


*



 

P
E

,

(5.32)


*



 

M
B

, (5.33)


*
0
 
    
*
Q J E

, (5.34)


these constitutive relations, eqs. (5.27
-
5.34), are further subjected to the axioms of
material inva
riance and time reversal. It may be stated as: the constitutive response
functionals must be form
-
invariant with respect to a group of transformations of the
material frame of reference
*
{ }

X X
and microscopic time reversal
}
t
t
{


representing the material symmetry conditions and these transformations must
leave the density and charge at
(,)
t
X
unchanged (Eringen and Maugin [1990]). It is
noticed that magnetic symmetry properties of solids cannot be discussed r
ationally by
means of three
-
dimensional point groups only since magnetism is the result of the spin
magnetic moment of electrons, which changes sign upon the time reversal. In other words,
diamagnetic and paramagnetic crystals do not exhibit any orderly di
stribution of their
spin magnetic moments, therefore they are ‘time symmetric’ and the crystallographic
point group is enough for the discussion of their material symmetries; on the other hand,
for ferromagnetic, ferrimagnetic and antiferromagnetic materia
ls, which are characterized
by an orderly distribution of spin magnetic moment, an additional symmetry operator is
needed to take care of the time reversal. For a complete account of this subject, interested
readers are referred to Shubnikov and Belov [196
4], Kiral and Eringen [1990].


6. Finite Element Formulation


The energy equation (4.6) can now be written as


* *
,
0
K K K K
Q h J E
 
   

, (6.1)


Multiply eq. (6.1) by the variational temperature


a
nd then integrate over the
undeformed volume, it gives


*
,
q
K K
V V V S
dV Q dV dV Q dS
   
   
   

, (6.2)


where


*
K
*
K
E
J
h






, (6.3)


and

*
K K
Q Q N


,

(6.4)


is the heat input specified at
q
S
, part of the surface
S

that enclosing the volume
V
and
K
N
is the outward normal to
S
. It is noted that
S
S
S
q



, where

S
is part of the
surface on which the temperature is specified.


The balance laws of linear momentum and moment of momentum, eqs. (4
-
4, 4
-
5), can be
expressed in the Lagrangian for
ms as


,
( ) ( ) 0
m
KL Li K i i
T f v
 
  

(6.5)


,
( ) ( ) ( ) 0
m m
LMK Li jM K ji ji ij ij
j t s l
  
     

(6.6)

where



}
)
B
M
E
P
(
F
{
f
f
~
j
,
i
*
j
*
i
j
i
i
i





. (6.7)


Multiply eq. (6.5) and eq
. (6.6) by the variational velocity vector
i
v

and the variational
microgyration tensor
ij

,
respectively, and then integrate the sum over the undeformed
volume, it leads to



* *
{ } { }
{ }
t
m m
KLM KLM KL KL KL KL i i ij ij
V V
i i ij ij i i ij ij
V S S
T S dV v v dV
f v l dV T v dS dS
     
    

    
    
 
  

,

(6.8)


where


K
Li
m
KL
*
i
N
T
T



, (6.9)


K
jM
Li
LMK
*
ij
N






, (6.10)


are the surface load and surface moment specified at
t
S
and

S
, respectively. It is noted
that


t v
S S S S S


   

,

(6.11)


where the velocity and the microgyration are specified on
v
S
and
S

, respectively. In this
finite element formulation no restrictive assumption has been made to the magnitude of
any independent constitutive variables. The results are valid for coupled
thermomechanical
-
electromagnetic phenomena. It is seen, from eqs. (6.1
, 6.8), that to
proceed further one needs the explicit constitutive expressions for the entropy

, the heat
input vector
Q
, and the generalized 2
nd

order Piola
-
Kirchhoff stress tensor
m
T
,
m
S
, and
Γ
.


7. Linear Constitutive Equations


To derive the linear constitutive equations for micromorphic electromagnetic continuum,
first, let the Helmholtz’s free energy density, eq.(5.27), be expanded as a polyno
mial up
to second order in terms of its arguments


*
2 1 2 3 4 * 5
1
2
1 4 5 1 * 1
1
2
2 6
1
2
/
o o o o o o o o o o
KL KL KL KL KLM KLM K K K K
o o
KL KL KL KL KLM KLM K K K K
KLMN KL MN KLMN KL MN KLMNP KL MNP KLM KL M KLM KL M
KLMN KL MN KLMNP KL
T T S P E M B
T T a T a T a T a TE a TB
A A A B E C B
A A
     
   
       
  
      
     
    
 
2 * 2
3 3 * 3
1
2
1 * * 2 3 *
1 1
2 2
MNP KLM KL M KLM KL M
KLMNPQ KLM NPQ KLMN KLM N KLMN KLM N
KL K L KL K L KL K L
B E C B
A B E C B
D E E D B B D E B
  
   
 
  
  

, (7.1)


where
0
T

is the reference temperature,


T
T


0

,
0
0

T
,
0
T
T

,

(7.2)


1
1
MNKL
KLMN
A
A


, (7.3)


2
2
2
2
KLNM
LKMN
MNKL
KLMN
A
A
A
A




, (7.4)


3
3
NPQKLM
KLMNPQ
A
A


, (7.5)


4
4
KLNM
KLMN
A
A



, (7.6)


6
6
LKMNP
KLMNP
A
A


, (7.7)


0
0
LK
KL
S
S


, (7.8)


2
2
LK
KL
a
a


,

(7.9)


2
2
LKM
KLM
B
B


, (7.10)


2
2
LKM
KLM
C
C


, (7.11)


1
1
LK
KL
D
D


,

(7.12)


2
2
LK
KL
D
D


. (7.13)


Then eqs. (5.28
-
5.33) leads to


0
5
4
3
2
1
0
0







}
B
a
E
a
a
a
a
{
T
T
K
K
*
K
K
KML
KLM
KL
KL
KL
KL








, (7.14)


M
KLM
*
M
KLM
MNP
KLMNP
MN
KLMN
MN
KLMN
KL
KL
m
KL
B
C
E
B
A
A
A
T
a
T
T
1
1
5
4
1
1
0











, (7.15)


M
KLM
*
M
KLM
MNP
KLMNP
MN
MNKL
MN
KLMN
KL
KL
m
KL
B
C
E
B
A
A
A
T
a
S
S
2
2
6
4
2
2
0











, (7.16)


N
KLMN
*
N
KLMN
NP
NPKLM
NP
NPKLM
NPQ
KLMNPQ
KLM
KLM
KLM
B
C
E
B
A
A
A
T
a
3
3
6
5
3
3
0













, (7.17)


L
KL
*
L
KL
LMN
LMNK
LM
LMK
LM
LMK
K
K
K
B
D
E
D
B
B
B
T
a
P
P
3
1
3
2
1
4
0











, (7.18)


* 0 5 1 2 3 2 3 *
K K K LMK LM LMK LM LMNK LMN KL L LK L
M M a T C C C D B D E
  
      

, (7.19)


where
0

,
}
{
0
0
0
Γ
,
S
,
T
,
0
P
,
0
M

are the initial entropy, stresses
, polarization,
magnetization, respectively;
1
a
,
2
a
,
3
a
are the thermal stresses moduli;
4
a
,
5
a

are the
pyroelectric and pyromagnetic moduli;


is the heat capacity;
i
A
(
i
=1,2,…6) are the
generalized elastic moduli;
1
B
,
2
B
,
3
B
are the generalized piezoelectric moduli;
1
C
,
2
C
,
3
C
are the generalized piezomagnetic moduli;
1
D
is the dielectric susceptibility;
2
D
is the magnetic susceptibility;
3
D
is the magnetic polarizability.


Now, in view of

the Clausius
-
Duhem inequality (5.34), the linear constitutive equations
for the heat input and the current can be obtained as


*
L
KL
L
,
KL
K
E
H
H
Q
3
1




, (7.20)


L
,
KL
*
L
KL
*
K
H
E
H
J


4
2



,

(7.21)


where
1
H

is the heat conductivity,
2
H

is the electric conductivity,
3
H

indicates the
Peltier effect,
4
H
indicates the Seebeck effect. If Onsager postulat
e is followed, then
there exists a dissipation function
(,)




*
E

which is nonnegative with an absolute
minimum at
*
0

  
E
and yields


(/)
K
Q



 

, (7.22)


*
K
*
K
E
J





, (7.23)

This implies


1
1
LK
KL
H
H


,
2
2
LK
KL
H
H


, (7.24)


KL
LK
KL
G
H
H


4
3

,
(7.25)


and

1
2


H G
H
G H

(7.26)


is positive definite. All the above
-
mentioned material moduli may be functions of the
Lagrangian coordinate
X

and the reference temperatur
e
o
T
.


On the other hand, from eqs.(5.23
-
5.25), it is seen that in general
Q

and
*
J

are functions
of the three generalized Lagrangian strains, temperature, temperature gradient, el
ectric
field, magnetic flux, and the Lagrangian coordinate. For isotropic material in a simpler
case, i.e., neglecting the effect of the third order strain tensor
γ
, the current and the heat
input can be rewritten as




*
Q= Q(
ε,β,θ,E,B,S,X)
, (7.27)



*

* *
J = J (
ε,β,θ,E,B,S,X)
, (7.28)


where

1
2
( )
KL KL LK KL LK
    
   

, (7.29)


1
[ ]
2
K KLM LM
S e



. (7.30)


It should be remarked that (1)
KL


will be reduced to
(,)
K L
u

-

the classical macro
-
strain
tensor in the case of small deformation, and (2)
B

and
S

are axial vectors and
transformed as



det( ) , det( )
B = RB R S = RS R

, (7.31)


while the absolute vectors
*
E

and



are transformed as



,
X

 

   

* *
R E = RE

, (7.32)


where


X = RX
. (7.33)


Now, according to Wang’s representation theorem for is
otropic functions (Wang
[1970,1971]), it follows



* 2 * 2
1 2 3 4 5 6
* * *
7 8 9 10 11 12
* *
13 14 15 16
* *
17 18
* *
19 20
[( ) ( ) ] [( ) ( ) ]
[( ) ( ) ] [( ) (
c c c c c c
c c c c c
c c c c
c c
c c
  
  
 
 

  
  
  
         
       
*
2 2
Q = E + +
εE + ε + ε E + ε
+
βE + β + β E + β + B
× E + c B×
+ S × E + S × + (
εβ - βε)E + (εβ βε)
B B E B E B B B B B
S S E S E S S S S
* *
21 22
* *
23 24
* *
25 26
* *
27 28
* *
29 30
) ]
[ ] [ ]
[ ] [ ]
[ ] [ ]
[ ] [ ]
[ ] [
c c
c c
c c
c c
c c




 
 
         
       
       
       
  
S
(B E )S (S E )B (B
θ)S (S )B
+
ε(E B) (εE ) B + ε( B) (ε θ) B
+
ε(E S) (εE ) S + ε( S) (ε θ) S
+
β(E B) (βE ) B + β( B) (β ) B
+
β(E S) (βE ) S +
]
 
    
β( S) (β ) S

, (7.34)



* * 2 * 2
1 2 3 4 5 6
* * *
7 8 9 10 11 12
* *
13 14 15 16
* *
17 18
* *
19 20
[( ) ( ) ] [( ) ( ) ]
[( ) ( ) ] [( ) (
d d d d d d
d d d d d d
d d d d
d d
d d
  
  
 
 

  
  
   
         
       
*
2 2
J = E + +
εE + ε + ε E + ε
+
βE + β + β E + β + B
× E + B×
+ S × E + S × + (
εβ βε)E + (εβ βε)
B B E B E B B B B B
S S E S E S S S
* *
21 22
* *
23 24
* *
25 26
* *
27 28
* *
29 30
) ]
[ ] [ ]
[ ] [ ]
[ ] [ ]
[ ] [ ]
[ ]
d d
d d
d d
d d
d d




 
 
         
       
       
       
  
S S
(B E )S (S E )B (B
θ)S (S )B
+
ε(E B) (εE ) B + ε( B) (ε θ) B
+
ε(E S) (εE ) S + ε( S) (ε θ) S
+
β(E B) (βE ) B + β( B) (β ) B
+
β(E S) (βE ) S +
[ ]
 
    
β( S) (β ) S

, (7.35)


where
i
c

and
i
d

(
1,2,,30
i
   
) are functions

of

,
X
, and the invariants made from
two 2
nd
order symmetric tensors
ε
and
β
, two absolute vectors
*
E
and


, two axial
vectors
B
and
S
. The constitutive functions,
and
i i
c d

, are subjected to the Clausius
-
Duhem inequality (5.34) .


From eqs.(7.34, 7.35), the Peltier effect

electric fiel
d producing heat flow
-

and the
Seebeck effect

temperature gradient producing current
-

are clearly seen and, also, the
second order vectorial effects are noticed: (1)
3 5 7 9
,,, and
c c c c

indicate that strains
produce an anisotropic Peltier effect,
(2)
12
c

shows that heat flows perpendicular to
and


B
, which is the Righi
-
Leduc effect, (3)
11
c

shows that heat flows perpendicular
to
*
and
B E
, which is the Ettingshau
sen effect, (4)
4 6 8 10
,,, and
d d d d

indicates that
strains produce an anisotropic Seebeck effect, (5)
11
d

gives the Hall effect


current flows
perpendicular to
*
and
B E
, and (6)
12
d
gives the Nernst effect


current flows
perpendicular to
and


B
. Further more, it is seen that the axial vector
S
, which is
equivalent to the anti
-
symmetric strain tensor representing the difference between t
he
macro
-
motion and the micro
-
motion, has a similar effect as the magnetic flux vector
B
.
It is interesting to see that if micromorphic theory is reduced to classical continuum
theory, then eqs.(7.34, 7.35) become



* 2 * 2
1 2 3 4 5 6
*
11 12
* *
17 18
* *
23 24
[( ) ( ) ] [( ) ( ) ]
[ ] [ ]
c c c c c c
c
c c
c c
  

 

  

         
       
*
Q = E + +
εE + ε + ε E + ε
+ B× E + c B×
B B E B E B B B B B
+
ε(E B) (εE ) B + ε( B) (ε θ) B

, (7.36)

* * 2 * 2
1 2 3 4 5 6
*
11 12
* *
17 18
* *
23 24
[( ) ( ) ] [( ) ( ) ]
[ ] [ ]
d d d d d d
d d
d d
d d
  

 

  

         
       
*
J = E + +
εE + ε + ε E + ε
+ B× E + B×
B B E B E B B B B B
+
ε(E B) (εE ) B + ε( B) (ε θ) B

. (7.37)


The difference between eq.(7.34) and eq.(7.36) and between eq.(7.35) and eq.(7.37) are
the effects due to the micro
-
structure and the micro
-
motion.


Acknowledgment


The suppo
rt to this work by National Science Foundation under Award Number CMS
-
0115868 is gratefully acknowledged.


Reference


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