Geometric theory of defects in solids
Geometric theory of defects in solids
Geometric theory of defects in solids
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The free energy<br />
∫<br />
3<br />
S = d xeL, e=<br />
dete μ<br />
i<br />
1<br />
4<br />
1<br />
4<br />
ijk kij j ik<br />
1 2 3<br />
L= κR− T ( βT + β T + β T δ )<br />
ijk<br />
ijkl klij ik jl<br />
1 2 3<br />
+ R ( γ R + γ R + γ R δ ) +Λ<br />
ijkl<br />
k μ ν<br />
ij = i j<br />
i<br />
j = Tij<br />
T e e T<br />
T<br />
R<br />
ik<br />
R<br />
=<br />
=<br />
R<br />
R<br />
ijk<br />
i<br />
i<br />
Postulate:<br />
j<br />
k<br />
μν<br />
The result:<br />
,...<br />
- trace <strong>of</strong> torsion<br />
- Ricci tensor<br />
- scalar curvature<br />
equations <strong>of</strong> equilibrium<br />
admit solutions<br />
- transformation <strong>of</strong> <strong>in</strong>dices<br />
L= κR<br />
−γR R<br />
Re ()<br />
R<br />
[ ij]<br />
(, e ω)<br />
[ ij]<br />
⎧R= 0, T ≠ 0<br />
⎪<br />
⎨R<br />
≠ 0, T = 0<br />
⎪<br />
⎩R<br />
= 0, T = 0<br />
[ ij]<br />
κ, β , β , β<br />
1 2 3<br />
γ , γ , γ , Λ<br />
1 2 3<br />
- the Hilbert-E<strong>in</strong>ste<strong>in</strong> action<br />
- coupl<strong>in</strong>g constants<br />
- only dislocations<br />
- only discl<strong>in</strong>ations<br />
- elastic deformations<br />
- antisymmetric part <strong>of</strong> the Ricci tensor<br />
9
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