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A gauge field theory of dislocations and disclinations with irreversible processes
0020-722.5/92 $5.00 + 0.00 Copyright @I 1992 Pergamon Press plc
Int. J. Engng Sci. Vol. 30, No. 2, pp. 231-236, 1992 Printed in Great Britain. All rights reserved
A GAUGE FIELD THEORY OF DISLOCATIONS AND DISCLINATIONS WITH IRREVERSIBLE PROCESSES K. M. DENG’,‘, ’ Center for Fundamental * International
C. W. LUNG* and K. L. WANG’
of Science and Technology of China, Hefei, Anhui, P.R. China Center for Material Physics and Institute of Metal Research, Academia Sinica, Shenyang, P.R. China Physics,
University
Abstract-We, basing on a variation principle, proposed a gauge field theory of dislocations and disclinations with irreversible processes. The linear momentum equation, the dislocation and disclination balance equation, the energy balance equation, and the entropy balance equation have been derived. It shows that the internal production of entropy is due to three parts, i.e. heat conduction, plastic deformation and the motions of dislocations and disclinations, and viscous dissipation in general situation.
1. INTRODUCTION
The gauge field theories of dislocations and disclinations have well been developed to some extent, since many researchers [l-8] have devoted their efforts to the subject. Most of them are discussed through treating the material body with dislocations and disclinations as elastic body with continuous distribution of dislocations and disclinations and the irreversible effects of the defects in the material body are hardly approached. Only Edelen [9] has ever discussed the irreversible effects through establishing the various balance equations of the system with the conventional thermo-dynamical method. It is obvious that the motion and deformation of the material body with dislocations and disclinations usually have the irreversible processes. Generally speaking, the system should be regarded as a non-conservation system and the irreversible processes in the material body with dislocations and disclinations should be considered in its dynamical theory. On the other hand, a variation principle is an efficient and precise method to be used to establish the dynamical theory of a system, with which the dynamical equations of the system would be immediately derived provided the Lagrangian density function and some other parameters were given. In the light of above viewpoint, we, basing on a variation principle, proposed a gauge field theory of dislocations and disclinations with irreversible processes. In Section 2, we give a brief introduction to the parameters which will be used in this paper. Some content of this section has first been presented at [8], to which the reader may refer for the details. We derive the dynamical equations of the system and expound the specific significance of them in the next section. Last section, we discuss the differences between this paper and the previous theories and their consistency.
2. THE
DETERMINING
PARAMETERS
In order to describe the motion and deformation of the material body with dislocations and disclinations, we have used the reference state, the elastic deformed state and the final state [S] which are referred as the r-state, the e-state and the f-state respectively. The r-state is an idealized state of the material which can be regarded as aggregation of a number of small pieces of the idealized material. Let f be the coordinate with base vectors e, in Euclidean space E,. So the metrical tensor in the r-state is hiB=e,*eB where the capital letters A etc. run from 1 to 3. When the material body has undergone an elastic deformation, ES3or2-H
231
(2.1) it changes from the r-state to
232
the e-state. relations
K. M. DENG
Let X’ as coordinate
et al.
with base vectors
Xi = Xi(XA, t)
and
ei in the Euclidean
space E,.
XA= XA(Xi, t)
We have
(2.2)
and the metrical tensor of the e-state is h;j = ei * TV,
(2.3)
where the small letters i, j run from 1 to 3. The length of two ajoint points in the e-state relating to Xi and X’ + dXi should be ds’ = 6, dXi dX’ = b,d~X’ 3,X’ dXA dXH =h,,dXAdXB
(2.4)
where aAX’ is the partial differential of Xl. It should be noted that we use rectangular coordinate in the r-state and the e-state, i.e. h:B = 6,, and hij = 6, for convenience. If curvilinear coordinate systems were used, the similar results of this paper would be obtained as well. After elastic deformation, the material body further undergoes a plastic deformation, and the dislocations and disclinations may be created and motion in the material body. The f-state then is reached. Let y’ represent a local coordinate of the f-state. The length of two neighboring points of the f-state is dsr2 = b, dy’ dy’ = 6,&,&
dXA dXB (2.S)
=&,dXAdXB
where the small letters i, j run from 1 to 3, the vielbein c$> be called distortion tensor, and g,, consists the metrical tensor of non-Riemannian space of the plastic manifold [lo]. The strain tensor of the material body is E AB
which can be decomposed plastic strain tensor Ef,:
=;
k,,
-
bAB)
(2.6)
into two parts, one is the elastic strain tensor E&,
EzB =; (hAB -
6AB).
ES’,
=;
(gAB
-
hA,j
the other the
(2.7)
Since each torn small piece of the material body in the f-state can be rotated freely in the r-state, we intuitively regard it as an internal symmetry of the material body. So we choose the W(3) group as the gauge group [8]. Considering both the vielbein connection and the gauge connection and adopting usual assumptions in non-Riemannian geometry that the total covariant derivative of the vielbein QAi are identically zero, we easily get
where I& is the vielbein connection and wii the gauge connection. The torsion tensor and the curvature tensor of the non-Riemannian space can be formulated with the vielbein connection. Using the corresponding relations between the torsion tensor and dislocation density tensor, the curvature tensor and the disclination density tensor, we can obtain the dislocation density tensor:
(2.9) and the disclination density tensor: (2.10)
A gauge field theory of dislocationsand disclinations
233
where @sAcr, etc. are the permutation symbol derived by I& and g = det(gAB), and the attached subscript symbol [C, D] means the antisymmetric part with respect to those indices. We consider xi, @Aiand WLi as the determining parameters of the system which determine the behaviour of the material body, and the entropy another important determining parameter. A set of tensors Ltpj are also regarded as the determining parameters, which characterize the physical and geometrical properties of the material body in the r-state. 3. A VARIATION
PRINCIPLE
A variation principle of a system with irreversible 6
processes can be expressed [ll]:
Zddzdt+6w+6w*=O
(3.1)
where 6p is the Lagrangian density, V is an arbitrary domain associated with the material body, dr is a volume element dt = dx’ dx* dX3; The functional 6w is determined for an arbitrary domain Vt when 9 and 6w* are known. The assigment of 6w on the boundary of the domain Vr leads also to the boundary condition. The functional SW” is a prescribed function and contains terms which take into account the variations of entropy and heat flux, this allows us to obtain the proper energy equation for irreversible processes. The equation (3.1) is the basic relation when the variations of the determining parameters are arbitrary. The Lagrangian density 9 depends on the dete~ining parameters mentioned in Section 2. The variations of the determining parameters can be defined as: 6xi = X’i(XA, t) i
-
&A,
t)
s@Ai
=
$&(X”P
t)
-
@Ai(x”s
t)
sfYl&,
=
Oj@,
t)
-
dii(f,
t)
(3.2)
which are based on constant coordinate f and t. For the specific purpose of this paper, define the variations at constant coordinate xA and different time t, i.e. &xi = Xli(XA, t) - &A,
we
t)
= axi + vi at b#Ai=
@,h(XA, =
61&i= 1
b#Ai
t) +
We@,
-
$Ai(?,
t)
d/dt~Ai 6t t) - fIIi,i(XA,t)
= SOLi + dfdt Eli 6t
(3.3)
where, and as in what follows, the symbol 6%will denote the total variation and the symbol 6 the variation at constant time. Varying the first term in equation (3.1), we get Sdrdt=
234
K. M. DENG
et al.
where x,‘B, $Ai,B and &i,B mean the partial differential of xi, GAi and c$,, to xB; a,$,; and a,c&i the differential to time t; and n, are the components of the normal to the surface Z and
6J.f -=_L
dLst?
d5?
se
~dti,B
3, ad,,,
as----
hi&;
aoli,,
(3.5)
We, for general situation, define
- QABcc$~(~cb$Aj -
12
=
II{ v fl
pe 6s + (I; + a,$)
+ QcBA#$#cj
&I&) + NSt 6x’
+ Q “““&i$+,;)]
f*[~~,xi + Q”““c#& 6,@,Jn, IIx fl
where p is the mass density temperature.
dt dt
SOi,, + (dCQABC - Q”“)@b
+ [ev’ + N + d&f++
-
@Aj
of the material
body,
d@Ai
St) dr dt
da dt
8 has the meaning
(3.6) of the absolute
4, QAE, QABC, N are certain generalized forces and stresses, which determine the external actions and internal irreversible effects. Considering arbitary non-vanishing variations on the boundary, we may derive the expression for
We obtain various dynamical equations from (3.1), (3.4), (3.6) and (3.7): (3.8) (3.9) -
62%
+ QcBAc#&@ci = 0
(3.10)
z+pt3=0
(3.11)
SC&i
235
A gauge field theory of dislocations and disclinations
$ ~_~,i-_-
(
a2
a2
at4Ai - -&&,,,
a+,40
+~v’+N+aB +
[(8 I
Q”““t$~ -
)
a,ci&
-$)vi-~at4i
g) at$Ai] =0
(3.12)
A1.B
(3.8) is the linear momentum equation, (3.9) and (3.10) dislocation balance equation and disclination balance equation; (3.11) and (3.12) are energy and entropy equations. If we assume the Lagrangian density is equal to the difference between the kinetic and internal energies
where
(3.13)
A!?=T-W the linear momentum equation (3.8) will become the ordinary form +i=&(~+$)+F;
(3.14)
It can make us sure that d and r$ have the meaning of the components of the stress tensor, while E have the meaning of the components of the body force vector. Making use of equations (3.8), (3.9) and (3. ll), we can derive the entropy balance equation from (3.12) p8
ds z
=
N + QABnAB + QABCacnAB - QcBAc#&&-iatd~i +
$aBvi
(3.15)
where the entropy increase connected with N, by definition, shall be regarded only due to the heat flux g and N satisfies the eqUdity N = - div q, nAB = #f‘,&$Ai Let us denote by (I the internal production of entropy p6 dsldt and isolate o from the right side of (3.15) in the following way:
02 =
QABnAB i- QABCdcnAB - QCBA$&$q6’,wi,i
03 =
$aBd
(3.16)
where the quantity u1 represents the irreversible effects due to heat conduction, a2 that due to plastic deformation and the motions of dislocations and disclinations of the material body, and a3 that due to viscous dissipation. ui, u2 and u3 may be positive or negative, but the quantity u satisfies the inequality u 2 0 according to the second law of thermodynamics.
4. DISCUSSION
We have brought up a gauge field theory of dislocations and disclinations with irreversible processes in a general situation. It should be pointed out that we will get different results for different specific models, i.e. the results depend not only on the forms of the Lagrangian density and the prescribed function &v*, but also on the domain concerned and its relevant boundary condition. For example, we have, for the case of small deformation, assumed: Z=T-W
where T is the kinetic of the components of f-state and the r-state, The coefficient of the
(4.1) energy, and W is the internal energy which consists of a quadratic form the elastic strain tensor E_& of the entropy difference (s -s,-,) of the and of components of the dislocation density and disclination density. quadratic forms correspond to the parameters Lo,) of Section 2 in the
236
K. M. DENG
et al.
general theory. And
- Q”““$~l,(acs4,,
- $Aj 6W{~i)- div g at] dr dt
(4.2) From those, we can get the similar equations of (3.8)-(3.12) in Section 3. In order to solve those equations, we also need consider the domain we concerned and its relevant boundary conditions, i.e. at
t=t,
at
t=t2
(4.3) With the discussion of above, we know that the domain, you choose, can be arbitrary as long as you know its boundary conditions. It contrasts that of the previous theories, in which the domain usually is considered infinite and the boundary effects are often neglected. It can be regarded as one difference of the theories. Another most important difference is that the irreversible processes of heat conduction, the plastic deformations and the motions of the dislocations and disclinations in the material body are considered, which is just a main improvement needed to be made in the previous theories. The results of [8] can easily be got in this paper provided the irreversible effects are neglected, i.e. only Lagrangian term in (3.1) is considered. It shows the consistency of the theories. The last thing which should be kept in mind is that the Lagrangian density or the internal potential need preserving invariant under the gauge transformation, so we select the internal energy as W = W(E:,, dE, oAB. g,,) (4.4) where E& a?, OABand g,, are all scalar tensors in Lie algebraic space. We have reason to assume the entropy is also a scalar tensor in Lie algebraic space, so the theory is still gauge invariant. REFERENCES [l] A. KADIC and D. G. B. EDELEN, A Gauge Theory of Dislocation and Dklinafions. Springer, Berlin (1983). [2] Y. S. DUAN and 2. P. DUAN, Znt. J. Engng Sci. 24 (4), 513 (1986). [3] B. K. D. Gairola, Gauge invariant formulation of continuum theory of defects. In Continuum Models of Discrete System 4 (Edited by 0. BERLIN and R. K. T. HSICH), North-Holland, Amsterdam (1981). [4] H. GUNTHER, Ann. Phys pp. 55. 40,291 (1983). [5] H. KLEINERT, Letr. Nuouo. Cimento 34, 103 (1982). [6] H. KLEINERT, Whys. Left. A97, 251 (1983). [7] E. KRONER, On gauge theory in defects mechanics. In Proc. 6th Symp. on Trends in Application of Pure Malhematics. Lecture Notes in Physics 249 (1986). [S) K. M. DENG et al., Znr. J. Engng Ski. 29, 79-86 (1991). [9] D. G. B. EDELEN, Znf. /. Engng Sci. 17 (1979). [lo] RAAG Memoirs, Vol. 3. Dir. D. Gakajntsu Bunken Fukyukai Tokoy (1962). [ll] L. I. SEDOV and V. L. BERDITCHEVSKI, A dynamic theory of continual dislocations, In Mechanics of Generalized Continua (Edited by E. KRONER ) (1967). (Received
18 April
1991; accepted
21 Ma?, 1991)
Int. J. Engng Sci. Vol. 30, No. 2, pp. 231-236, 1992 Printed in Great Britain. All rights reserved
A GAUGE FIELD THEORY OF DISLOCATIONS AND DISCLINATIONS WITH IRREVERSIBLE PROCESSES K. M. DENG’,‘, ’ Center for Fundamental * International
C. W. LUNG* and K. L. WANG’
of Science and Technology of China, Hefei, Anhui, P.R. China Center for Material Physics and Institute of Metal Research, Academia Sinica, Shenyang, P.R. China Physics,
University
Abstract-We, basing on a variation principle, proposed a gauge field theory of dislocations and disclinations with irreversible processes. The linear momentum equation, the dislocation and disclination balance equation, the energy balance equation, and the entropy balance equation have been derived. It shows that the internal production of entropy is due to three parts, i.e. heat conduction, plastic deformation and the motions of dislocations and disclinations, and viscous dissipation in general situation.
1. INTRODUCTION
The gauge field theories of dislocations and disclinations have well been developed to some extent, since many researchers [l-8] have devoted their efforts to the subject. Most of them are discussed through treating the material body with dislocations and disclinations as elastic body with continuous distribution of dislocations and disclinations and the irreversible effects of the defects in the material body are hardly approached. Only Edelen [9] has ever discussed the irreversible effects through establishing the various balance equations of the system with the conventional thermo-dynamical method. It is obvious that the motion and deformation of the material body with dislocations and disclinations usually have the irreversible processes. Generally speaking, the system should be regarded as a non-conservation system and the irreversible processes in the material body with dislocations and disclinations should be considered in its dynamical theory. On the other hand, a variation principle is an efficient and precise method to be used to establish the dynamical theory of a system, with which the dynamical equations of the system would be immediately derived provided the Lagrangian density function and some other parameters were given. In the light of above viewpoint, we, basing on a variation principle, proposed a gauge field theory of dislocations and disclinations with irreversible processes. In Section 2, we give a brief introduction to the parameters which will be used in this paper. Some content of this section has first been presented at [8], to which the reader may refer for the details. We derive the dynamical equations of the system and expound the specific significance of them in the next section. Last section, we discuss the differences between this paper and the previous theories and their consistency.
2. THE
DETERMINING
PARAMETERS
In order to describe the motion and deformation of the material body with dislocations and disclinations, we have used the reference state, the elastic deformed state and the final state [S] which are referred as the r-state, the e-state and the f-state respectively. The r-state is an idealized state of the material which can be regarded as aggregation of a number of small pieces of the idealized material. Let f be the coordinate with base vectors e, in Euclidean space E,. So the metrical tensor in the r-state is hiB=e,*eB where the capital letters A etc. run from 1 to 3. When the material body has undergone an elastic deformation, ES3or2-H
231
(2.1) it changes from the r-state to
232
the e-state. relations
K. M. DENG
Let X’ as coordinate
et al.
with base vectors
Xi = Xi(XA, t)
and
ei in the Euclidean
space E,.
XA= XA(Xi, t)
We have
(2.2)
and the metrical tensor of the e-state is h;j = ei * TV,
(2.3)
where the small letters i, j run from 1 to 3. The length of two ajoint points in the e-state relating to Xi and X’ + dXi should be ds’ = 6, dXi dX’ = b,d~X’ 3,X’ dXA dXH =h,,dXAdXB
(2.4)
where aAX’ is the partial differential of Xl. It should be noted that we use rectangular coordinate in the r-state and the e-state, i.e. h:B = 6,, and hij = 6, for convenience. If curvilinear coordinate systems were used, the similar results of this paper would be obtained as well. After elastic deformation, the material body further undergoes a plastic deformation, and the dislocations and disclinations may be created and motion in the material body. The f-state then is reached. Let y’ represent a local coordinate of the f-state. The length of two neighboring points of the f-state is dsr2 = b, dy’ dy’ = 6,&,&
dXA dXB (2.S)
=&,dXAdXB
where the small letters i, j run from 1 to 3, the vielbein c$> be called distortion tensor, and g,, consists the metrical tensor of non-Riemannian space of the plastic manifold [lo]. The strain tensor of the material body is E AB
which can be decomposed plastic strain tensor Ef,:
=;
k,,
-
bAB)
(2.6)
into two parts, one is the elastic strain tensor E&,
EzB =; (hAB -
6AB).
ES’,
=;
(gAB
-
hA,j
the other the
(2.7)
Since each torn small piece of the material body in the f-state can be rotated freely in the r-state, we intuitively regard it as an internal symmetry of the material body. So we choose the W(3) group as the gauge group [8]. Considering both the vielbein connection and the gauge connection and adopting usual assumptions in non-Riemannian geometry that the total covariant derivative of the vielbein QAi are identically zero, we easily get
where I& is the vielbein connection and wii the gauge connection. The torsion tensor and the curvature tensor of the non-Riemannian space can be formulated with the vielbein connection. Using the corresponding relations between the torsion tensor and dislocation density tensor, the curvature tensor and the disclination density tensor, we can obtain the dislocation density tensor:
(2.9) and the disclination density tensor: (2.10)
A gauge field theory of dislocationsand disclinations
233
where @sAcr, etc. are the permutation symbol derived by I& and g = det(gAB), and the attached subscript symbol [C, D] means the antisymmetric part with respect to those indices. We consider xi, @Aiand WLi as the determining parameters of the system which determine the behaviour of the material body, and the entropy another important determining parameter. A set of tensors Ltpj are also regarded as the determining parameters, which characterize the physical and geometrical properties of the material body in the r-state. 3. A VARIATION
PRINCIPLE
A variation principle of a system with irreversible 6
processes can be expressed [ll]:
Zddzdt+6w+6w*=O
(3.1)
where 6p is the Lagrangian density, V is an arbitrary domain associated with the material body, dr is a volume element dt = dx’ dx* dX3; The functional 6w is determined for an arbitrary domain Vt when 9 and 6w* are known. The assigment of 6w on the boundary of the domain Vr leads also to the boundary condition. The functional SW” is a prescribed function and contains terms which take into account the variations of entropy and heat flux, this allows us to obtain the proper energy equation for irreversible processes. The equation (3.1) is the basic relation when the variations of the determining parameters are arbitrary. The Lagrangian density 9 depends on the dete~ining parameters mentioned in Section 2. The variations of the determining parameters can be defined as: 6xi = X’i(XA, t) i
-
&A,
t)
s@Ai
=
$&(X”P
t)
-
@Ai(x”s
t)
sfYl&,
=
Oj@,
t)
-
dii(f,
t)
(3.2)
which are based on constant coordinate f and t. For the specific purpose of this paper, define the variations at constant coordinate xA and different time t, i.e. &xi = Xli(XA, t) - &A,
we
t)
= axi + vi at b#Ai=
@,h(XA, =
61&i= 1
b#Ai
t) +
We@,
-
$Ai(?,
t)
d/dt~Ai 6t t) - fIIi,i(XA,t)
= SOLi + dfdt Eli 6t
(3.3)
where, and as in what follows, the symbol 6%will denote the total variation and the symbol 6 the variation at constant time. Varying the first term in equation (3.1), we get Sdrdt=
234
K. M. DENG
et al.
where x,‘B, $Ai,B and &i,B mean the partial differential of xi, GAi and c$,, to xB; a,$,; and a,c&i the differential to time t; and n, are the components of the normal to the surface Z and
6J.f -=_L
dLst?
d5?
se
~dti,B
3, ad,,,
as----
hi&;
aoli,,
(3.5)
We, for general situation, define
- QABcc$~(~cb$Aj -
12
=
II{ v fl
pe 6s + (I; + a,$)
+ QcBA#$#cj
&I&) + NSt 6x’
+ Q “““&i$+,;)]
f*[~~,xi + Q”““c#& 6,@,Jn, IIx fl
where p is the mass density temperature.
dt dt
SOi,, + (dCQABC - Q”“)@b
+ [ev’ + N + d&f++
-
@Aj
of the material
body,
d@Ai
St) dr dt
da dt
8 has the meaning
(3.6) of the absolute
4, QAE, QABC, N are certain generalized forces and stresses, which determine the external actions and internal irreversible effects. Considering arbitary non-vanishing variations on the boundary, we may derive the expression for
We obtain various dynamical equations from (3.1), (3.4), (3.6) and (3.7): (3.8) (3.9) -
62%
+ QcBAc#&@ci = 0
(3.10)
z+pt3=0
(3.11)
SC&i
235
A gauge field theory of dislocations and disclinations
$ ~_~,i-_-
(
a2
a2
at4Ai - -&&,,,
a+,40
+~v’+N+aB +
[(8 I
Q”““t$~ -
)
a,ci&
-$)vi-~at4i
g) at$Ai] =0
(3.12)
A1.B
(3.8) is the linear momentum equation, (3.9) and (3.10) dislocation balance equation and disclination balance equation; (3.11) and (3.12) are energy and entropy equations. If we assume the Lagrangian density is equal to the difference between the kinetic and internal energies
where
(3.13)
A!?=T-W the linear momentum equation (3.8) will become the ordinary form +i=&(~+$)+F;
(3.14)
It can make us sure that d and r$ have the meaning of the components of the stress tensor, while E have the meaning of the components of the body force vector. Making use of equations (3.8), (3.9) and (3. ll), we can derive the entropy balance equation from (3.12) p8
ds z
=
N + QABnAB + QABCacnAB - QcBAc#&&-iatd~i +
$aBvi
(3.15)
where the entropy increase connected with N, by definition, shall be regarded only due to the heat flux g and N satisfies the eqUdity N = - div q, nAB = #f‘,&$Ai Let us denote by (I the internal production of entropy p6 dsldt and isolate o from the right side of (3.15) in the following way:
02 =
QABnAB i- QABCdcnAB - QCBA$&$q6’,wi,i
03 =
$aBd
(3.16)
where the quantity u1 represents the irreversible effects due to heat conduction, a2 that due to plastic deformation and the motions of dislocations and disclinations of the material body, and a3 that due to viscous dissipation. ui, u2 and u3 may be positive or negative, but the quantity u satisfies the inequality u 2 0 according to the second law of thermodynamics.
4. DISCUSSION
We have brought up a gauge field theory of dislocations and disclinations with irreversible processes in a general situation. It should be pointed out that we will get different results for different specific models, i.e. the results depend not only on the forms of the Lagrangian density and the prescribed function &v*, but also on the domain concerned and its relevant boundary condition. For example, we have, for the case of small deformation, assumed: Z=T-W
where T is the kinetic of the components of f-state and the r-state, The coefficient of the
(4.1) energy, and W is the internal energy which consists of a quadratic form the elastic strain tensor E_& of the entropy difference (s -s,-,) of the and of components of the dislocation density and disclination density. quadratic forms correspond to the parameters Lo,) of Section 2 in the
236
K. M. DENG
et al.
general theory. And
- Q”““$~l,(acs4,,
- $Aj 6W{~i)- div g at] dr dt
(4.2) From those, we can get the similar equations of (3.8)-(3.12) in Section 3. In order to solve those equations, we also need consider the domain we concerned and its relevant boundary conditions, i.e. at
t=t,
at
t=t2
(4.3) With the discussion of above, we know that the domain, you choose, can be arbitrary as long as you know its boundary conditions. It contrasts that of the previous theories, in which the domain usually is considered infinite and the boundary effects are often neglected. It can be regarded as one difference of the theories. Another most important difference is that the irreversible processes of heat conduction, the plastic deformations and the motions of the dislocations and disclinations in the material body are considered, which is just a main improvement needed to be made in the previous theories. The results of [8] can easily be got in this paper provided the irreversible effects are neglected, i.e. only Lagrangian term in (3.1) is considered. It shows the consistency of the theories. The last thing which should be kept in mind is that the Lagrangian density or the internal potential need preserving invariant under the gauge transformation, so we select the internal energy as W = W(E:,, dE, oAB. g,,) (4.4) where E& a?, OABand g,, are all scalar tensors in Lie algebraic space. We have reason to assume the entropy is also a scalar tensor in Lie algebraic space, so the theory is still gauge invariant. REFERENCES [l] A. KADIC and D. G. B. EDELEN, A Gauge Theory of Dislocation and Dklinafions. Springer, Berlin (1983). [2] Y. S. DUAN and 2. P. DUAN, Znt. J. Engng Sci. 24 (4), 513 (1986). [3] B. K. D. Gairola, Gauge invariant formulation of continuum theory of defects. In Continuum Models of Discrete System 4 (Edited by 0. BERLIN and R. K. T. HSICH), North-Holland, Amsterdam (1981). [4] H. GUNTHER, Ann. Phys pp. 55. 40,291 (1983). [5] H. KLEINERT, Letr. Nuouo. Cimento 34, 103 (1982). [6] H. KLEINERT, Whys. Left. A97, 251 (1983). [7] E. KRONER, On gauge theory in defects mechanics. In Proc. 6th Symp. on Trends in Application of Pure Malhematics. Lecture Notes in Physics 249 (1986). [S) K. M. DENG et al., Znr. J. Engng Ski. 29, 79-86 (1991). [9] D. G. B. EDELEN, Znf. /. Engng Sci. 17 (1979). [lo] RAAG Memoirs, Vol. 3. Dir. D. Gakajntsu Bunken Fukyukai Tokoy (1962). [ll] L. I. SEDOV and V. L. BERDITCHEVSKI, A dynamic theory of continual dislocations, In Mechanics of Generalized Continua (Edited by E. KRONER ) (1967). (Received
18 April
1991; accepted
21 Ma?, 1991)