The continuum limit of distributed dislocations Cy Maor Institute - PowerPoint PPT Presentation

The continuum limit of distributed dislocations Cy Maor � Institute of Mathematics, Hebrew University � � Conference on non - linearity, transport, physics, and patterns � Fields Institute, October 2014 


Di ff erent Models for Dislocations

Di ff erent Models for Dislocations • A single dislocation

Di ff erent Models for Dislocations • A single dislocation • V olterra ( ~1900 )

Di ff erent Models for Dislocations • A single dislocation • V olterra ( ~1900 ) • Riemannian manifold with singularities.

Di ff erent Models for Dislocations • A single dislocation • V olterra ( ~1900 ) • Riemannian manifold with singularities. • Burgers vector

Di ff erent Models for Dislocations • A single dislocation • Distributed dislocations • V olterra ( ~1900 ) • Riemannian manifold with singularities. • Burgers vector

Di ff erent Models for Dislocations • A single dislocation • Distributed dislocations • V • Nye, Bilby, etc. ( ~1950 ) olterra ( ~1900 ) • Riemannian manifold with singularities. • Burgers vector

Di ff erent Models for Dislocations • A single dislocation • Distributed dislocations • V • Nye, Bilby, etc. ( ~1950 ) olterra ( ~1900 ) • Riemannian manifold • Smooth manifold with with singularities. a torsion field ( =Burgers vector • Burgers vector density ) .

Di ff erent Models for Dislocations • A single dislocation • Distributed dislocations • V • Nye, Bilby, etc. ( ~1950 ) olterra ( ~1900 ) • Riemannian manifold • Smooth manifold with with singularities. a torsion field ( =Burgers vector • Burgers vector density ) . How to bridge between the descriptions? What kind of homogenization process yields a torsion field from singularities?

Di ff erent Models for Dislocations • A single dislocation • Distributed dislocations • V • Nye, Bilby, etc. ( ~1950 ) olterra ( ~1900 ) • Riemannian manifold • Smooth manifold with with singularities. a torsion field ( =Burgers vector • Burgers vector density ) . How to bridge between the descriptions? What kind of homogenization process yields a torsion field from singularities? A new limit concept in di ff erential geometry!

Continuum Limit of Dislocations

Continuum Limit of Dislocations • Overview:

Continuum Limit of Dislocations • Overview: • What is an edge - dislocation?

Continuum Limit of Dislocations • Overview: • What is an edge - dislocation? • Construction of manifolds with many dislocations.

Continuum Limit of Dislocations • Overview: • What is an edge - dislocation? • Construction of manifolds with many dislocations. • Dislocations become denser — what does converge?

Continuum Limit of Dislocations • Overview: • What is an edge - dislocation? • Construction of manifolds with many dislocations. • Dislocations become denser — what does converge? • Connection to the classical model of distributed dislocations.

An edge - dislocation p + p − p − $ $ $ $ 2 θ 2 θ d

An edge - dislocation • Remove a sector of angle 2 𝜄 , and glue the edges ( a cone ) . p + p − p − $ $ $ $ 2 θ 2 θ d

An edge - dislocation • Remove a sector of angle 2 𝜄 , and glue the edges ( a cone ) . • Choose a point at distance d from the tip of the cone, cut a ray from it, and insert the sector into the cut. p + p − p − $ $ $ $ 2 θ 2 θ d

An edge - dislocation • Remove a sector of angle 2 𝜄 , and glue the edges ( a cone ) . • Choose a point at distance d from the tip of the cone, cut a ray from it, and insert the sector into the cut. • A simply connected metric space, a smooth manifold outside the dislocation line [ p - ,p + ] . p + p − p − $ $ $ $ 2 θ 2 θ d

The building block b A D d p − p + a a + ε B C b

The building block • Encircle the dislocation line with four straight lines with right angles between them, obtaining a “rectangle”. b A D d p − p + a a + ε B C b

The building block • Encircle the dislocation line with four straight lines with right angles between them, obtaining a “rectangle”. • Denote the lengths of these lines by a , b , b , and a+ ε , where is the dislocation magnitude . ε = 2 d sin θ b A D d p − p + a a + ε B C b

Manifolds with many dislocations b a / n + ε / n 2 a / n + 2 ε / n 2 a / n + 3 ε / n 2 a / n + ε / n · · · . . . . . . a a + ε . . . . . . · · · b / n b / n b / n b / n b

Manifolds with many dislocations • Glue together n 2 building blocks, such that: b a / n + ε / n 2 a / n + 2 ε / n 2 a / n + 3 ε / n 2 a / n + ε / n · · · . . . . . . a a + ε . . . . . . · · · b / n b / n b / n b / n b

Manifolds with many dislocations • Glue together n 2 building blocks, such that: • Each with the same cone angle 2 𝜄 and with dislocation magnitude ε /n 2 . b a / n + ε / n 2 a / n + 2 ε / n 2 a / n + 3 ε / n 2 a / n + ε / n · · · . . . . . . a a + ε . . . . . . · · · b / n b / n b / n b / n b

Manifolds with many dislocations • Glue together n 2 building blocks, such that: • Each with the same cone angle 2 𝜄 and with dislocation magnitude ε /n 2 . • The boundary consists of straight lines of lengths a , b , b , and a+ ε . b a / n + ε / n 2 a / n + 2 ε / n 2 a / n + 3 ε / n 2 a / n + ε / n · · · . . . . . . a a + ε . . . . . . · · · b / n b / n b / n b / n b

Manifolds with many dislocations • Glue together n 2 building blocks, such that: • Each with the same cone angle 2 𝜄 and with dislocation magnitude ε /n 2 . • The boundary consists of straight lines of lengths a , b , b , and a+ ε . • The rectangular properties of the blocks ensure us that the gluing lines and corners are smooth. b a / n + ε / n 2 a / n + 2 ε / n 2 a / n + 3 ε / n 2 a / n + ε / n · · · . . . . . . a a + ε . . . . . . · · · b / n b / n b / n b / n b

Metric Convergence

Metric Convergence How do these manifolds M n look like when n →∞ ?

Metric Convergence How do these manifolds M n look like when n →∞ ? Theorem: The sequence M n converges in the Gromov - Hausdor ff sense, to M , a sector of a flat annulus whose boundary consists of curves of lengths a , b , b , and a+ ε .

Metric Convergence How do these manifolds M n look like when n →∞ ? Theorem: The sequence M n converges in the Gromov - Hausdor ff sense, to M , a sector of a flat annulus whose boundary consists of curves of lengths a , b , b , and a+ ε . b a / n + ε / n 2 a / n + 2 ε / n 2 a / n + 3 ε / n 2 a / n + ε / n · · · . . . . . . a a + ε . . . . . . b · · · b / n b / n b / n b / n b a + ε a ε / b b

Metric Convergence Gromov - Hausdor ff convergence: � GH � � � � if there exist bijections � M n → M − − T n : A n ⊂ M n → B n ⊂ M � between δ n - nets A n and B n ( δ n → 0 ) such that b a / n + ε / n 2 a / n + 2 ε / n 2 a / n + 3 ε / n 2 a / n + ε / n · · · . . . . . . a a + ε . . . . . . b · · · b / n b / n b / n b / n b a + ε a ε / b b

Metric Convergence Gromov - Hausdor ff convergence: � GH � � � � if there exist bijections � M n → M − − T n : A n ⊂ M n → B n ⊂ M � between δ n - nets A n and B n ( δ n → 0 ) such that dis T n = sup | d M n ( x, y ) − d M ( T n ( x ) , T n ( y )) | → n →∞ 0 x,y ∈ A n b a / n + ε / n 2 a / n + 2 ε / n 2 a / n + 3 ε / n 2 a / n + ε / n · · · . . . . . . a a + ε . . . . . . b · · · b / n b / n b / n b / n b a + ε a ε / b b

What else converges? b a / n + ε / n 2 a / n + 2 ε / n 2 a / n + 3 ε / n 2 a / n + ε / n · · · . . . . . . a a + ε . . . . . . b · · · b / n b / n b / n b / n b a + ε a ε / b b

What else converges? • A n consists of geodesics ( straight lines ) . b a / n + ε / n 2 a / n + 2 ε / n 2 a / n + 3 ε / n 2 a / n + ε / n · · · . . . . . . a a + ε . . . . . . b · · · b / n b / n b / n b / n b a + ε a ε / b b

What else converges? • A n consists of geodesics ( straight lines ) . • B n does not. b a / n + ε / n 2 a / n + 2 ε / n 2 a / n + 3 ε / n 2 a / n + ε / n · · · . . . . . . a a + ε . . . . . . b · · · b / n b / n b / n b / n b a + ε a ε / b b

What else converges? • A n consists of geodesics ( straight lines ) . • B n does not. � Or does it? b a / n + ε / n 2 a / n + 2 ε / n 2 a / n + 3 ε / n 2 a / n + ε / n · · · . . . . . . a a + ε . . . . . . b · · · b / n b / n b / n b / n b a + ε a ε / b b

What else converges? b a / n + ε / n 2 a / n + 2 ε / n 2 a / n + 3 ε / n 2 a / n + ε / n · · · ( ∂ r , r − 1 ∂ ϕ ) . . . . . . a a + ε . . . . . . b · · · b / n b / n b / n b / n b a + ε ( ∂ x , ∂ y ) a ε / b b

What else converges? • A n consists of geodesics w.r.t. the canonical ( Levi - Civita ) parallel - transport on M n . b a / n + ε / n 2 a / n + 2 ε / n 2 a / n + 3 ε / n 2 a / n + ε / n · · · ( ∂ r , r − 1 ∂ ϕ ) . . . . . . a a + ε . . . . . . b · · · b / n b / n b / n b / n b a + ε ( ∂ x , ∂ y ) a ε / b b