Geometric theory of defects in solids

R
i
3
i
x , y i=
1,2,3
δ
u
ε
σ
ij
i
( x)
= ( ∂ u +∂ u )
1
ij 2 i j j i
ij
Notations
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.
- continuous elastic media = Euclidean three-dimensional space
- Euclidean metric
- Cartesian coordinates
- displacement vector field
- stress tensor
- strain tensor
Elasticity theory of small deformations
i
ij
∂ σ + f =
j
0
ij ij k ij
k
σ = λδ ε + 2με
- Newton’s law
- Hooke’s law
f
i
( x)
λ,
μ
- density of nonelastic forces
- Lame coefficients
i
( f = 0)
1

Differential geometry of elastic deformations
y
x
i i i
x = y + u ( x)
i
y → x ( y)
i
- diffeomorphism:
R
i
y
δ
3 3
ij
→
x
g
i
ij
k
l
∂y
∂y
gij ( x) = kl ij iuj jui ij 2
i j
ij
x x δ ≈ δ −∂ −∂ = δ − ε
∂ ∂
1
Γ
ijk = ( ∂ ) 0
2 ig jk +∂ jgik −∂k gij
≠
- Christoffel’s symbols
- induced metric
l l m l
R = ∂Γ −Γ Γ
−( i↔ j)
= 0
ijk i jk ik jm
- curvature tensor
x
i i j
jk
=−Γ
x x
- extremals (geodesics)
k
k k k
T = Γ
−Γ = 0
ij ij ji
- torsion tensor
2

Dislocations
Linear defects:
2
x
2
x
3
1
x
x
Edge dislocation
b
3
x
b
1
x
Screw dislocation
b
- Burgers vector
Point defects:
Vacancy
i
u
( x)
⎧
⎨
⎩
is continuous = elastic deformations
is not continuous = dislocations
3

C
(*) ⇒
2
x
∫
b
1
x
∫∫
Edge dislocation
∫
C
μ
i μ i i
μ
C
dx ∂ u =− dx ∂ y =−b
μ
x μ , μ = 1,2, 3
i
y
( x)
i i i
b = dx μ e = dx μ ∧ dx ν ( ∂ e −∂ e i )
μ μ ν ν μ
C
S
i i ij
μν =∂μ ν −ωμ ν j −( μ ↔ν)
- torsion
T e e
R
ij ij ik j
μν μ ν μ ν k
e
i
μ
( x)
i
⎧∂ ⎪ μ y
= ⎨
⎪⎩ lim∂
=∂ ω −ω ω −( μ ↔ν)
μ
y
∫
- arbitrary curvilinear coordinates
- is not continuous !
i
- outside the cut
- on the cut
-curvature
- Burgers vector in elasticity
ω
ij
μ
=−ω
ji
μ
(*)
- triad field
(continuous on the cut)
SO(3)-connection
i
∫∫
μ ν i
μν
b = dx ∧ dx T
- definition of the Burgers vector
in the geometric theory
Back to elasticity: if
ij
ij
μ
R μν = 0 then ω → 0
4

Ferromagnets
i
Disclinations
n ( x)
n0
i j i
- unit vector field - fixed unit vector
n = n0 Sj
( ω)
j
Si
∈SO(3)
ij ji
ω =−ω
∈so(3)
1 jk
ωi
= εijkω
2
- orthogonal matrix
i
- Lie algebra element (spin structure)
- rotational angle
( ε = 1)
εijk
- totally antisymmetric tensor 123
2
x
Examples
1
x
2
x ij ij
1
x
Ω =
∫
C
Θ = ε
i
ijk
dx μ ∂μω
Ω
jk
- Frank vector
(total angle of rotation)
C
C
Θ = 2π
Θ = 4π
i
Θ = ΘΘ i
5

Model for a spin structure:
i
ω ( x) ∈ so(3)
- basic variable
k j j
j j ki
i
i
S = cos + ωε
sin + ωω
i i (1 − cos ) ∈ SO(3),
=
2
i
δ ω ω ω ω ω ω
ω ω
l
j 1 k j
μi
= ( ∂μS
− ) i Sk
- trivial SO(3)-connection (pure gauge)
μ ij
l μ
∂ =
0
- principal chiral SO(3)-model
6

Frank vector
ij
ω ( x)
- is not continuous !
ω
ij
μ
( x)
ij
⎧∂ ⎪ μω
= ⎨
ij
⎪⎩ lim∂μω
- outside the cut
- on the cut
- SO(3)-connection
(continuous on the cut)
ij μ ij μ ν ij ij
∫
μ ∫∫
μ ν ν μ
Ω = dx ω = dx ∧ dx ( ∂ ω −∂ ω )
ij ij ik j
R μν μ ν μ ν k
=∂ ω −ω ω −( μ ↔ν)
- the Frank vector
-curvature
ij
Ω =
∫∫
dx
μ
∧
ν
dx R
ij
μν
- definition of the Frank vector
in the geometric theory
Back to the spin structure: if
2
n∈ then SO(3) → SO(2)
7

Summary of the geometric approach
(physical interpretation)
Media with dislocations and disclinations =
3
with a given Riemann-Cartan geometry
Independent variables
=
⎧
i
⎪
⎨
⎪⎩ ω
e μ
ij
μ
- triad field
- SO(3)-connection
i i ij
μν μ ν μ ν j
T =∂ e −ω e −( μ ↔ν)
R
ij ij ik j
μν μ ν μ ν k
=∂ ω −ω ω −( μ ↔ν)
- torsion (surface density of
the Burgers vector)
- curvature (surface density of
the Frank vector)
Elastic deformations:
R
ij
μν
i
μν
= 0, T = 0
Dislocations:
R
ij
μν
i
μν
= 0, T ≠ 0
Disclinations:
R
ij
μν
i
μν
≠ 0, T = 0
Dislocations and disclinations:
R
ij
μν
≠
i
μν
0, T ≠ 0
8

The free energy
∫
3
S = d xeL, e=
dete μ
i
1
4
1
4
ijk kij j ik
1 2 3
L= κR− T ( βT + β T + β T δ )
ijk
ijkl klij ik jl
1 2 3
+ R ( γ R + γ R + γ R δ ) +Λ
ijkl
k μ ν
ij = i j
i
j = Tij
T e e T
T
R
ik
R
=
=
R
R
ijk
i
i
Postulate:
j
k
μν
The result:
,...
- trace of torsion
- Ricci tensor
- scalar curvature
equations of equilibrium
admit solutions
- transformation of indices
L= κR
−γR R
Re ()
R
[ ij]
(, e ω)
[ ij]
⎧R= 0, T ≠ 0
⎪
⎨R
≠ 0, T = 0
⎪
⎩R
= 0, T = 0
[ ij]
κ, β , β , β
1 2 3
γ , γ , γ , Λ
1 2 3
- the Hilbert-Einstein action
- coupling constants
- only dislocations
- only disclinations
- elastic deformations
- antisymmetric part of the Ricci tensor
9

Elastic gauge
(1− 2 σ ) Δ u +∂ ∂ u = 0
e
i i j
σ =
≈ δ
j
λ
2( λ + μ)
−∂
μi
μi
μ
u
i
- the elasticity equation
- Poisson ratio
- the linear approximation
μ
(1− 2 σ ) ∂ e +∂ e = 0
μi
i
μ
μ
- the elastic gauge
(fixes diffeomorphisms)
Lorentz gauge
μ ij
ω μ
∂ = 0
- the Lorenz gauge
(fixes SO(3)-invariance)
If there are no disclinations
ij
R μν =
0, then
j j −1
ω ( )
k j
μ i = l μ i = ∂ μ S i Sk
μ ij
μ
∂ l = 0 - principal chiral SO(3)-model
pure gauge
10

y
Wedge dislocation in elasticity theory
R
z
−2πθ
x
r, ϕ,
z
ui
- cylindrical coordinates
{ u(),() r v r ϕ,0}
= - displacement vector
Boundary conditions:
u | = 0, u | = 0, u | =−2 r, ∂ u | = 0
r r= 0 ϕ ϕ= 0 ϕ ϕ= 2π πθ r r r=
R
vr ()
u
∂r( r∂ru)
− = D - elasticity equations
r
1−
2σ D =− θ , σ - Poisson ratio
1 − σ
D
c2
u = rln r+ cr 1 + , c1,2
= const - a general solution
2
r
2 ⎛ 1−2σ
r ⎞ 2 2⎛ 1−2σ
r 1 ⎞ 2
dl = ⎜1+ θ ln ⎟dr + r ⎜1+ θ ln + θ ⎟dϕ
⎝ 1−σ R⎠ ⎝ 1−σ R 1−σ
⎠
induced metric θ 1, r ∼ R
= −θ r
θ - deficit angle
11

Wedge dislocation in the geometric theory
R
y
x
−2πθ
θ
- deficit angle
α = 1+
θ
2 1 2 2 2
dl = df + f dϕ - metric for a conical singularity
2
α
(exact solution of 3D Einstein eqs.)
Where is the Poisson ratio σ ???
The elastic gauge:
μ
(1− 2 σ ) ∂ e +∂ e = 0
μi
i
μ
μ
e
i
μ
=∂
u
i
For it reduces to elasticity equations:
μ
(1− 2 σ ) Δ u +∂ ∂ u = 0
i i j
j
2( n 1) 2 2
2 ⎛ r ⎞
− ⎛
2 α r 2⎞
dl = ⎜ ⎟ ⎜dr + dϕ
2 ⎟
⎝R
⎠ ⎝ n ⎠
- exact solution of the Einstein
equations in the elastic gauge
n
=
2 2 2
− θσ + θ σ + 4(1 + θ)(1 −σ)
2(1 −σ
)
12

Comparison of the elasticity theory
with the geometric model
2( n 1) 2 2
2 ⎛ r ⎞
− 2 α r 2
dl = ⎛ dr dϕ
⎞
⎜ ⎟ ⎜ +
2 ⎟
⎝R
⎠ ⎝ n ⎠
- the geometric model
1−
2σ
θ 1, n ≈ 1 + θ
2(1 −σ
)
2 ⎛ 1−2σ
r ⎞ 2 2⎛ 1−2σ
r 1 ⎞ 2
dl = ⎜1+ θ ln ⎟dr + r ⎜1+ θ ln + θ ⎟dϕ
⎝ 1−σ R⎠ ⎝ 1−σ R 1−σ
⎠
- the elasticity
theory
The result of the elasticity theory is valid only for small
deficit angles and near the boundary
θ 1
r ∼ R
The result of the geometric model is valid for allθ
and everywhere
lnduced metric components define the deformation
tensor and can be measured experimentally
13

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.
14