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Geometric Theory of Defects
M. O. Katanaev Steklov Mathematical Institute, Moscow
Katanaev, Volovich Ann. Phys. 216(1992)1; ibid. 271(1999)203 Katanaev Theor.Math.Phys.135(2003)733; ibid. 138(2004)163 Phisics ­ Uspekhi 48(2005)675.

Notations

R

3 i

- continuous elastic media = Euclidean three-dimensional space

xi , y

i = 1, 2, 3 - Cartesian coordinates



ij

- Euclidean metric Elasticity theory of small deformations

u i ( x) - displacement vector field ij = 1 ( iu j + j ui ) - strain tensor 2



ij

- stress tensor

i ij + f j = 0

- Newton's law

ij = ij

k

k

+ 2 ij - Hooke's law

f i ( x) - density of nonelastic forces ( f i = 0) 1 , - Lame coefficients


Differential geometry of elastic deformations

x

y

xi = y i + u i ( x)

y i xi ( y ) - diffeomorphism: R3 3 yi xi ij gij
ij

y k y l ij - iu j - j ui = ij - 2 gij ( x) = i j kl x x

- induced metric



ijk

= 1 ( i g jk + j gik - k gij ) 0 - Christoffel's symbols 2
jk l

Rijk l = i xi =- Tij k =
jk k i

-
k

ik

m



jm

l

- (i j ) = 0 - curvature tensor

x jx -

- extremals (geodesics)
k

ij

ji

= 0 - torsion tensor
2


Dislocations
Linear defects:
x

2

x

2

b
x Edge dislocation
3

x

1

b x3 Screw dislocation b - Burgers vector

x1

Point defects:

Vacancy

is continuous = elastic deformations u i ( x) is not continuous = dislocations

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Edge dislocation
x
2

b
x1

C



dx u i =- dx y i = -b
C

i

(*)

x , = 1, 2, 3 - arbitrary curvilinear coordinates y i ( x) - is not continuous !

C

y i - outside the cut i e ( x) = i lim y - on the cut
C

- triad field (continuous on the cut)

(*) bi =



dx e i =


S

dx dx ( e i - e i ) - Burgers vector in elasticity
- torsion

T i = e i - ij e j - ( ) R ij = ij -

ik




i

k

j

- ( ) - curvature



ij

=-



ji

SO(3)-connection

bi =



dx dx T

- definition of the Burgers vector in the geometric theory

Back to elasticity: if

R ij = 0 then



ij

0

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Disclinations
Ferromagnets

ni ( x) - unit vector field ni = n0j S j i ( )

i n0 - fixed unit vector

Si j SO(3) - orthogonal matrix ij =- ji so(3) - Lie algebra element (spin structure) 1 i = ijk jk - rotational angle 2 ijk - totally antisymmetric tensor (123 = 1)
Examples
x
2

x

2

ij =
x1

C



dx
jk

ij

x1

i = ijk

- Frank vector (total angle of rotation)

C

C

= 2

= 4

= i

i
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Model for a spin structure:

i ( x) so (3) k Si j = i j cos +
l
i
j ki j

- basic variable

i j sin + 2 (1 - cos ) SO(3),
- trivial SO(3)-connection (pure gauge)

= ii

= ( S -1 )i k S

k

j

l



ij

=0

- principal chiral SO(3)-model

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Frank vector

ij ( x)

- is not continuous ! - SO(3)-connection (continuous on the cut)

ij - outside the cut ij ( x) = ij lim - on the cut

ij =



dx



ij

=



dx dx ( ij - ij )

ik

- the Frank vector

R ij = ij -



k

j

- ( )

- curvature

ij =



dx dx R

ij

- definition of the Frank vector in the geometric theory
2

Back to the spin structure: if

n

then

SO(3) SO(2)

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Summary of the geometric approach (physical interpretation)
Media with dislocations and disclinations

=

3

=

with a given Riemann-Cartan geometry

e i - triad field Independent variables ij - SO(3)-connection
T i = e i - ij e j - ( ) R ij = ij -

ik

- torsion



k

j

- ( )

(surface density of the Burgers vector) - curvature (surface density of the Frank vector)

Elastic deformations: Dislocations: Disclinations: Dislocations and disclinations:

R ij = 0, R ij = 0, R ij 0, R ij 0,

T i = 0 T i 0 T i = 0 T i 0
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The free energy

S = d 3 x eL, e = det e

i

L = R - 1 Tijk ( 1T 4

ijk

+ 2T
klij

kij

+ 3T j ik )

+ 1 Rijkl ( 1R 4

ijkl

+ 2R

+ 3 R ik jl ) +

Tij k = e i e jT k , ... - transformation of indices , 1, 2 , T j = Tij i - trace of torsion
Rik = Rijk j - Ricci tensor R = Ri
i

3

1, 2 , 3 ,

- coupling constants

R = 0, T 0 - only dislocations equations of equilibrium Postulate: R 0, T = 0 - only disclinations admit solutions R = 0, T = 0 - elastic deformations
The result:

- scalar curvature

L = R - R[ij ] R[

ij ]

R (e) - the Hilbert-Einstein action R[ij ] (e, ) - antisymmetric part of the Ricci tensor

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Elastic gauge

(1 - 2 ) ui + i j u j = 0

- the elasticity equation - Poisson ratio

=

2( + )

ei i - u

i

- the linear approximation - the elastic gauge (fixes diffeomorphisms)

(1 - 2 ) ei + i e = 0

Lorentz gauge





ij

=0

- the Lorenz gauge

(fixes SO(3)-invariance)
i
j

If there are no disclinations

R ij = 0 , then

=l

i

j

= ( S -1 )i k S
pure gauge

k

j

l



ij

= 0 - principal chiral SO(3)-model

10


y

Wedge dislocation in elasticity theory

r , , z - cylindrical coordinates ui = {u (r ), v(r ) , 0} - displacement vector
R z
-2

x

Boundary conditions:
= 2

ur |r =0 = 0, u | =0 = 0, u |

= -2 r , r ur |r

=R

=0

v(r ) = - r
u r (r r u ) - = D - elasticity equations r 1 - 2 D =- , - Poisson ratio 1- D c u = r ln r + c1r + 2 , c1,2 = const - a general solution 2 r 1 - 2 r 2 2 1 - 2 r 1 2 ln dr + r 1 + ln + dl = 1 + 1- 1- 1- R R induced metric 1, r R



- deficit angle

d

2

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Wedge dislocation in the geometric theory
y

R

x

- deficit = 1+
-2

angle

dl 2 =

1



2

df 2 + f 2 d 2 - metric for a conical singularity
Where is the Poisson ratio

(exact solution of 3D Einstein eqs.)



???

The elastic gauge: For
2

(1 - 2 ) ei + i e = 0

e i = u i it reduces to elasticity equations: (1 - 2 )ui + i j u j = 0

dl =

r R

2( n -1)

2 2r 2 2 dr + 2 d - exact solution of the Einstein n equations in the elastic gauge
2

- + 2 2 + 4(1 + )(1 - ) n= 2(1 - )

12


Comparison of the elasticity theory with the geometric model

r dl 2 = R

2( n -1)

2 2r 2 2 dr + 2 d n

- the geometric model



1,

n 1+

1 - 2 2(1 - )

1 - 2 r 2 2 1 - 2 r 1 2 dl 2 = 1 + ln dr + r 1 + ln + d - the elasticity R R 1- 1- 1- theory
The result of the elasticity theory is valid only for small deficit angles 1 and near the boundary r R The result of the geometric model is valid for all and everywhere



lnduced metric components define the deformation tensor and can be measured experimentally

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Conclusion
Geometric theory of defects
?????

Elastic gauge

Lorenz gauge

Elasticity theory

Principal chiral SO(3)-model

1) The geometric theory of defects in solids appears to be a fundamental theory of defects. 2) It describes single defects as well as continuous distribution of defects. 3) It provides a unified treatment of defects in media (dislocations) and in spin structures (disclinations). 4) In the absence of defects it reduces to the elasticity theory for the displacement vector field and to the principal chiral SO(3)-model for spin structures.

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