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Boundary tractions in the gauge theory of dislocations and disclinations
hr. 1. Engng Sci. Vol. 24, No. 6, pp. 933-937. Printed in Great Britain.
1986
002~‘X5/86 0 1986 -on
$3.00 + .OO Pres Ltd.
BOUNDARY TRACTIONS IN THE GAUGE THEORY OF DISLOCATIONS AND DISCLINATIONS DIMITRIS C. LAGGUDAS Center for the Application of Mathematics, Lehigh University, Bethlehem, PA 18015, U.S.A. (Communicated by D. G. B. EDELEN) Abstract-A null Lagrangian is added to the Lagrangian of Elasticity in order to include boundary tractions and initial values for the linear momentum. The Yang-Mills minimal replacement construct is then applied, in order to restore the invariance of the Lagrangian under the inhomogeneous action of the gauge group W(3) D T(3). This results in the introduction of compensating fields, which describe defects in elastic materials. The field equations show that dislocations are driven by appropriately defined effective stresses and linear momenta, and disclinations are driven by effective couplestresses and angular momenta. IN A PREVIOUS work [l] the homogeneous boundary value problem for an elastic material with dislocations was modified to include specified boundary tractions and initial values for linear momentum. This problem was shown to be well posed in [2]. The purpose of this paper is to extend these ideas in order to include disclinations as well. The formalism of the Exterior Calculus [4] will be used throughout the paper. The explicit component form of the field equations will also be given for the convenience of the reader. The same null Lagrangian is chosen as in [I]:
7 = dp = d(x’dQJ = dx’ A dQi = ~~i8~Q4b~.
(1)
When it is added to the Lagrangian of Elasticity, it does not change the Euler-Lagrange equations, but it does change the natural Neumann data, in order to give the appropriate boundary tractions and initial values for linear momentum. Here xi = x’(P) denote the mapping functions from the reference to the current configuration and Qi is a 2-form
Qi = iQYbPab,
Qyb = -Qf“,
dQi = a&h.
(2) (3)
Theindicesi,j, .a* andA,B,C, a*. runfrom 1 to3anda,b,c, ... rtmfiom 1 to4. XA, A = 1,2, 3, denote the spatial reference configuration coordinates, and X4 is the time coordinate. Further, ~1= d.J? A a2 A dx3 A dT is the 4dimensional volume element, P,, 1 5 a 5 4, form a conjugate basis for exterior forms of degree 3 and P&, I s a < b zs 4, form a conjugate basis for 2-forms (see [4], pp. 100-104). The null Lagrangian (1) must be invariant under the homogeneous action of the gauge group GO= SO(3) t> T(3) of rigid body rotations and translations, This requires the following transformation law for Qi: -Q = QR-‘,
(4)
-x = Rx + b,
(5)
when
and we have defined Q = [Qr ,Q2, Q3J, x = [x ‘, x2, x3]? The 3 X 3 orthogonal matrix R which gives rise to rotations and the vector b which translates x do not depend on (P); they move all of the material particles the same way (homogeneously). The breaking of the homogeneity of the action of GO, which comes about when R and b depend on (X”), requires the Yang-Mills minimal replacement construct, in order that 933
DIMITRIS C. LAGOUDAS
934
invariance of the Lagrangian be restored (i.e. in order to restore balance of linear and moment of momentum by Noether’s second theorem) d_,+ t+ Bi = Dxi + 4’ = hi
+
wy;jxj
+ @i,
dQi I+ DQi = dQi - Qj A w*yj,i.
(6) (7)
The l-forms w”( are the compensating potentials for the inhomogeneous rotations, while the l-forms 4’ compensate for the inhomogeneous translations. The matrices ru form a basis for the matrix Lie algebra of the representation of the rotation group SO(3) on V(3). The Lagrangian for an elastic material with dislocations and disclinations is given after Yang-Mills minimal coupling construct is applied, by Lp = (JR(&) - SAL,- s~L~)/J+ B’ A DQie
(8)
Here Lo is the Lagrangian of Elasticity and L,, L, are the coupling Lagrangians of the compensating fields $J’, wa (see [3] for their explicit evaluations). The magnitudes of the couplings are determined by the values of the coupling constants sl and s2. We now form the action integral
A[x’, #, w”] =
s
ByLp.
(9)
After standard manipulations [4], vanishing of the first variations of A give the following Euler-Lagrange equations and boundary conditions: D(Z + DQ) = -2R A 8,
(10)
DR = f(Z + DQ),
(11)
DG = 43,
(12)
at all interior points of the body, and x = f(P)
Z+DQ=O RA&$=O (G + Ryx) A 6w = 0
on aB6,
(13)
on dB!j,
(14)
on aB4,
(15)
on aB4.
(16)
Here, c3B: is the support of the Dirichlet data, dB? is the support of the Neumann data and the symbols in the above equations are defined through the constitutive relations
(17)
Ri
=
;
I
Rybpab = - -
2
aL
aDLbpab’
(18) (19)
5 = -(2Ri - Qi)y$ where
A
B’Fy,,
(20)
Boundary tractions in gauge theory
935
D;b = a,& - d&L + r$( Wgd$ - wB~{) + F~~~~jX’,
(22)
8 = fF&:srJX” A Mb.
(23)
It is essential at this point to observe that the integrability conditions for eqn (12) are nontrivial. Using the special way that the Lagrangian depends upon D’ and F”, these conditions reduce to Z A 7o,B = Q A r,D.
(24)
Equations ( IO)-( 16) and (24) have the following explicit evaluations:
From (17) we see that Zj’ = -u: = - Piola-Kirchhoff stress, 24 = pi = linear momentum. These relations together with the antisymmetry of r;j in i and j reduce (24a) to gf&Lj
= -)QybDJ&j.
Wb)
The presence of the terms on the right-hand side of (24b) is of particular significance. The interaction of Qfb fields with the torsion (D = E) gives rise to a torque per unit volume. This destroys the symmetry of the Cauchy stress tensor, as expressed by the relations aABj I
A
-
-
ufB:,
(25)
which obtain from (24b) when the right-hand sides vanish. The above can probably be better understood if we parallel the breaking of the homogeneity of the action of the rotation group SO(3), with the introduction of the deformable directors by Toupin [5] (they characterize the homogeneous deformations of the micromedium, defined at each material point of the global continuum). We should also require that the directors undergo only rigid body rotations, those of a Cosserat Continuum, in order for the correspondence to be complete. The introduction of the directors gives rise to the antisymmetric part of the Cauchy stress tensor, and so does the inhomogeneous action of SO(3). We will now attempt to give an answer to the question: What drives the dislocation and disclination fields? First we define the “elastic” Fiola-Kirchhoff stresses and couple-stresses, linear momenta and angular momenta:
(26) (27) (28) (29)
DIMITRIS C. LAGOUDAS
936
Equations ( 1 la), (12a) reduce to
with Jj’f =
2R”‘B’l
Jt4 J
-2RA+B’l [J
=
[I
(30)
0)
(31)
A-
It is clear from (1 lb) that (a: - S:) is the effectivestress and (pi - Pi) is the efictive linear momentum. We define (Jy - My) to be the efictive couple-stressand (Jj“ - Qj4) to be the effective angular momentum. From (1 lb) we see that the effective stresses and effective linear momenta give rise to dislocation fields Rpb, while from (12b) we conclude that the effective couple-stresses and effective angular momenta give rise to disclination fields Gzb. The elastic stresses, couple stresses, linear and angular momenta are solutions to the elastic problem, with the same boundary tractions that apply to the true stresses and same initial conditions that apply for the true momenta:
ULNA= SfNA = Ti(XB)
on aBy,
(32)
M”NA = 0 J
on dBj ,
(33)
pi = Pi = py (X’)
at
T= To,
(34)
at
T= To.
(35)
Qj4
=
0
They satisfy the following equations DDQ=-QAe, D(Qr,AB)=Dw,AB+Q*/,AD.
(36) (37)
The above relations written out exphcitly (using the definitions (26-29) as well as the antisymmetry of rkj in i and j), take the form
aAs: = &Pi - &,(Qj’bw~r’,i),
(364 (374
The balance of moment of momentum configuration are
equations for a Cosserat Continuum in the current
jm&k - jt,ij, + jl,jj, + P[iij] = Q~J.
(38)
After being pulled back in the reference configuration, they can be placed into a 1- 1 correspondence with (37a). This justifies the definition of M,j’ as the elastic couple-stress and Qj4 as the elastic angular momentum. The interpretation of the term -iQffD% in (24b) is also correct, because it corresponds to jlriil which is a couple per unit undeformed volume [5]. Equations (36a), (37a) together with the boundary and initial conditions (32)-(35) form the elastic problem of a Cosserat Continuum. Their solutions will give the elastic stresses, couple-stresses, linear and angular momenta, and their difference with the true stresses, couple-stresses, linear and angular momenta will generate the dislocation and disclination fields. Of course eqns (36a), (37a) cannot be solved independently, since they explicitly contain x, Cp,w.
937
Boundary tractions in gauge theory
The presence of disclinations has the effect of replacing dQ by dQ - Q A yw in the null Lagrangian, hence fixing Q is required in addition to dQ. This is done, however, by the boundary conditions (33), (35) and for all interior points by the requirement that -$Q$“#$, corresponds to a torque per unit volume applied to the material. If we relax (33), (35) and make them inhomogeneous, we can control the values of wb on the boundary. The analogy between the disclinations and the Cosserat elastic continuum serves as a possible explanation as to why it is so difficult to create and sustain disclinations in a material. A useful insight is finally gained if we look at the equations the other way around. Defining the effective dislocation field as
R=R-;Q,
(34)
eqns (10-12) take the form DZ = -2R A 8,
(1Oc)
;Z,
(1 w
2
(1W
DR= DG=‘j
7
where j is formed from a in accordance with (20). These equations are the same as the corresponding ones that solve the homogeneous boundary value problem, with the only replacement given by eqn (34). The true stresses have as a potential the effective dislocation fields and the interaction of the effective dislocation fields with the distortion B gives rise to disclinations. Acknowledgement-The author wishes to thank Professor D. G. B. Edelen for the numerous discussions of several aspects of the problem. REFERENCES
[I] D. G. B. EDELEN, Lett. Appl. Engng Sci. 20, 1049 ( 1982).
[2] D. G. B. EDELEN, Int. J. Engng Sci. 21,463 (1983). [3] A. KADIC and D. G. B. EDELEN, A Gauge Theory ofDislocations and Disclinutions.Lecture Notes in Physics, No. 174, Springer-Verlag, Berlin (1983). [4] D. G. B. EDELEN, AppIiedExterior Calculus.John Wiley & Sons, New York (1985). [5] R. A. TOUPIN, Arch. NationalMech. Anal. 17,85 (1964). (Received 12 June 1985)
1986
002~‘X5/86 0 1986 -on
$3.00 + .OO Pres Ltd.
BOUNDARY TRACTIONS IN THE GAUGE THEORY OF DISLOCATIONS AND DISCLINATIONS DIMITRIS C. LAGGUDAS Center for the Application of Mathematics, Lehigh University, Bethlehem, PA 18015, U.S.A. (Communicated by D. G. B. EDELEN) Abstract-A null Lagrangian is added to the Lagrangian of Elasticity in order to include boundary tractions and initial values for the linear momentum. The Yang-Mills minimal replacement construct is then applied, in order to restore the invariance of the Lagrangian under the inhomogeneous action of the gauge group W(3) D T(3). This results in the introduction of compensating fields, which describe defects in elastic materials. The field equations show that dislocations are driven by appropriately defined effective stresses and linear momenta, and disclinations are driven by effective couplestresses and angular momenta. IN A PREVIOUS work [l] the homogeneous boundary value problem for an elastic material with dislocations was modified to include specified boundary tractions and initial values for linear momentum. This problem was shown to be well posed in [2]. The purpose of this paper is to extend these ideas in order to include disclinations as well. The formalism of the Exterior Calculus [4] will be used throughout the paper. The explicit component form of the field equations will also be given for the convenience of the reader. The same null Lagrangian is chosen as in [I]:
7 = dp = d(x’dQJ = dx’ A dQi = ~~i8~Q4b~.
(1)
When it is added to the Lagrangian of Elasticity, it does not change the Euler-Lagrange equations, but it does change the natural Neumann data, in order to give the appropriate boundary tractions and initial values for linear momentum. Here xi = x’(P) denote the mapping functions from the reference to the current configuration and Qi is a 2-form
Qi = iQYbPab,
Qyb = -Qf“,
dQi = a&h.
(2) (3)
Theindicesi,j, .a* andA,B,C, a*. runfrom 1 to3anda,b,c, ... rtmfiom 1 to4. XA, A = 1,2, 3, denote the spatial reference configuration coordinates, and X4 is the time coordinate. Further, ~1= d.J? A a2 A dx3 A dT is the 4dimensional volume element, P,, 1 5 a 5 4, form a conjugate basis for exterior forms of degree 3 and P&, I s a < b zs 4, form a conjugate basis for 2-forms (see [4], pp. 100-104). The null Lagrangian (1) must be invariant under the homogeneous action of the gauge group GO= SO(3) t> T(3) of rigid body rotations and translations, This requires the following transformation law for Qi: -Q = QR-‘,
(4)
-x = Rx + b,
(5)
when
and we have defined Q = [Qr ,Q2, Q3J, x = [x ‘, x2, x3]? The 3 X 3 orthogonal matrix R which gives rise to rotations and the vector b which translates x do not depend on (P); they move all of the material particles the same way (homogeneously). The breaking of the homogeneity of the action of GO, which comes about when R and b depend on (X”), requires the Yang-Mills minimal replacement construct, in order that 933
DIMITRIS C. LAGOUDAS
934
invariance of the Lagrangian be restored (i.e. in order to restore balance of linear and moment of momentum by Noether’s second theorem) d_,+ t+ Bi = Dxi + 4’ = hi
+
wy;jxj
+ @i,
dQi I+ DQi = dQi - Qj A w*yj,i.
(6) (7)
The l-forms w”( are the compensating potentials for the inhomogeneous rotations, while the l-forms 4’ compensate for the inhomogeneous translations. The matrices ru form a basis for the matrix Lie algebra of the representation of the rotation group SO(3) on V(3). The Lagrangian for an elastic material with dislocations and disclinations is given after Yang-Mills minimal coupling construct is applied, by Lp = (JR(&) - SAL,- s~L~)/J+ B’ A DQie
(8)
Here Lo is the Lagrangian of Elasticity and L,, L, are the coupling Lagrangians of the compensating fields $J’, wa (see [3] for their explicit evaluations). The magnitudes of the couplings are determined by the values of the coupling constants sl and s2. We now form the action integral
A[x’, #, w”] =
s
ByLp.
(9)
After standard manipulations [4], vanishing of the first variations of A give the following Euler-Lagrange equations and boundary conditions: D(Z + DQ) = -2R A 8,
(10)
DR = f(Z + DQ),
(11)
DG = 43,
(12)
at all interior points of the body, and x = f(P)
Z+DQ=O RA&$=O (G + Ryx) A 6w = 0
on aB6,
(13)
on dB!j,
(14)
on aB4,
(15)
on aB4.
(16)
Here, c3B: is the support of the Dirichlet data, dB? is the support of the Neumann data and the symbols in the above equations are defined through the constitutive relations
(17)
Ri
=
;
I
Rybpab = - -
2
aL
aDLbpab’
(18) (19)
5 = -(2Ri - Qi)y$ where
A
B’Fy,,
(20)
Boundary tractions in gauge theory
935
D;b = a,& - d&L + r$( Wgd$ - wB~{) + F~~~~jX’,
(22)
8 = fF&:srJX” A Mb.
(23)
It is essential at this point to observe that the integrability conditions for eqn (12) are nontrivial. Using the special way that the Lagrangian depends upon D’ and F”, these conditions reduce to Z A 7o,B = Q A r,D.
(24)
Equations ( IO)-( 16) and (24) have the following explicit evaluations:
From (17) we see that Zj’ = -u: = - Piola-Kirchhoff stress, 24 = pi = linear momentum. These relations together with the antisymmetry of r;j in i and j reduce (24a) to gf&Lj
= -)QybDJ&j.
Wb)
The presence of the terms on the right-hand side of (24b) is of particular significance. The interaction of Qfb fields with the torsion (D = E) gives rise to a torque per unit volume. This destroys the symmetry of the Cauchy stress tensor, as expressed by the relations aABj I
A
-
-
ufB:,
(25)
which obtain from (24b) when the right-hand sides vanish. The above can probably be better understood if we parallel the breaking of the homogeneity of the action of the rotation group SO(3), with the introduction of the deformable directors by Toupin [5] (they characterize the homogeneous deformations of the micromedium, defined at each material point of the global continuum). We should also require that the directors undergo only rigid body rotations, those of a Cosserat Continuum, in order for the correspondence to be complete. The introduction of the directors gives rise to the antisymmetric part of the Cauchy stress tensor, and so does the inhomogeneous action of SO(3). We will now attempt to give an answer to the question: What drives the dislocation and disclination fields? First we define the “elastic” Fiola-Kirchhoff stresses and couple-stresses, linear momenta and angular momenta:
(26) (27) (28) (29)
DIMITRIS C. LAGOUDAS
936
Equations ( 1 la), (12a) reduce to
with Jj’f =
2R”‘B’l
Jt4 J
-2RA+B’l [J
=
[I
(30)
0)
(31)
A-
It is clear from (1 lb) that (a: - S:) is the effectivestress and (pi - Pi) is the efictive linear momentum. We define (Jy - My) to be the efictive couple-stressand (Jj“ - Qj4) to be the effective angular momentum. From (1 lb) we see that the effective stresses and effective linear momenta give rise to dislocation fields Rpb, while from (12b) we conclude that the effective couple-stresses and effective angular momenta give rise to disclination fields Gzb. The elastic stresses, couple stresses, linear and angular momenta are solutions to the elastic problem, with the same boundary tractions that apply to the true stresses and same initial conditions that apply for the true momenta:
ULNA= SfNA = Ti(XB)
on aBy,
(32)
M”NA = 0 J
on dBj ,
(33)
pi = Pi = py (X’)
at
T= To,
(34)
at
T= To.
(35)
Qj4
=
0
They satisfy the following equations DDQ=-QAe, D(Qr,AB)=Dw,AB+Q*/,AD.
(36) (37)
The above relations written out exphcitly (using the definitions (26-29) as well as the antisymmetry of rkj in i and j), take the form
aAs: = &Pi - &,(Qj’bw~r’,i),
(364 (374
The balance of moment of momentum configuration are
equations for a Cosserat Continuum in the current
jm&k - jt,ij, + jl,jj, + P[iij] = Q~J.
(38)
After being pulled back in the reference configuration, they can be placed into a 1- 1 correspondence with (37a). This justifies the definition of M,j’ as the elastic couple-stress and Qj4 as the elastic angular momentum. The interpretation of the term -iQffD% in (24b) is also correct, because it corresponds to jlriil which is a couple per unit undeformed volume [5]. Equations (36a), (37a) together with the boundary and initial conditions (32)-(35) form the elastic problem of a Cosserat Continuum. Their solutions will give the elastic stresses, couple-stresses, linear and angular momenta, and their difference with the true stresses, couple-stresses, linear and angular momenta will generate the dislocation and disclination fields. Of course eqns (36a), (37a) cannot be solved independently, since they explicitly contain x, Cp,w.
937
Boundary tractions in gauge theory
The presence of disclinations has the effect of replacing dQ by dQ - Q A yw in the null Lagrangian, hence fixing Q is required in addition to dQ. This is done, however, by the boundary conditions (33), (35) and for all interior points by the requirement that -$Q$“#$, corresponds to a torque per unit volume applied to the material. If we relax (33), (35) and make them inhomogeneous, we can control the values of wb on the boundary. The analogy between the disclinations and the Cosserat elastic continuum serves as a possible explanation as to why it is so difficult to create and sustain disclinations in a material. A useful insight is finally gained if we look at the equations the other way around. Defining the effective dislocation field as
R=R-;Q,
(34)
eqns (10-12) take the form DZ = -2R A 8,
(1Oc)
;Z,
(1 w
2
(1W
DR= DG=‘j
7
where j is formed from a in accordance with (20). These equations are the same as the corresponding ones that solve the homogeneous boundary value problem, with the only replacement given by eqn (34). The true stresses have as a potential the effective dislocation fields and the interaction of the effective dislocation fields with the distortion B gives rise to disclinations. Acknowledgement-The author wishes to thank Professor D. G. B. Edelen for the numerous discussions of several aspects of the problem. REFERENCES
[I] D. G. B. EDELEN, Lett. Appl. Engng Sci. 20, 1049 ( 1982).
[2] D. G. B. EDELEN, Int. J. Engng Sci. 21,463 (1983). [3] A. KADIC and D. G. B. EDELEN, A Gauge Theory ofDislocations and Disclinutions.Lecture Notes in Physics, No. 174, Springer-Verlag, Berlin (1983). [4] D. G. B. EDELEN, AppIiedExterior Calculus.John Wiley & Sons, New York (1985). [5] R. A. TOUPIN, Arch. NationalMech. Anal. 17,85 (1964). (Received 12 June 1985)