Geometric theory of defects in solids

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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
Page 1 and 2: R i 3 i x , y i= 1,2,3 δ u ε σ i
Page 3 and 4: Dislocations Linear defects: 2 x 2
Page 5 and 6: Ferromagnets i Disclinations n ( x)
Page 7 and 8: Frank vector ij ω ( x) - is not co
Page 9 and 10: The free energy ∫ 3 S = d xeL, e=
Page 11: y Wedge dislocation in elasticity t

Geometric theory of defects in solids

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