Geometric theory of defects in solids

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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
Page 1 and 2: R i 3 i x , y i= 1,2,3 δ u ε σ i
Page 3: Dislocations Linear defects: 2 x 2
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 and 12: y Wedge dislocation in elasticity t
Page 13 and 14: Comparison of the elasticity theory

Geometric theory of defects in solids

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