Geometric theory of defects in solids

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
)
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