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


Different Models for Dislocations

• A single dislocation

Different Models for Dislocations

• A single dislocation
• V
• lterra (~1900)

Different Models for Dislocations

• A single dislocation
• V
• lterra (~1900)
• Riemannian manifold

with singularities.

Different Models for Dislocations

• A single dislocation
• V
• lterra (~1900)
• Riemannian manifold

with singularities.

• Burgers vector

Different Models for Dislocations

• A single dislocation
• V
• lterra (~1900)
• Riemannian manifold

with singularities.

• Burgers vector
• Distributed dislocations

Different Models for Dislocations

• A single dislocation
• V
• lterra (~1900)
• Riemannian manifold

with singularities.

• Burgers vector
• Distributed dislocations
• Nye, Bilby, etc. (~1950)

Different Models for Dislocations

• A single dislocation
• V
• lterra (~1900)
• Riemannian manifold

with singularities.

• Burgers vector
• Distributed dislocations
• Nye, Bilby, etc. (~1950)
• Smooth manifold with

a torsion field ( =Burgers vector density).

Different Models for Dislocations

• A single dislocation
• V
• lterra (~1900)
• Riemannian manifold

with singularities.

• Burgers vector
• Distributed dislocations
• Nye, Bilby, etc. (~1950)
• Smooth manifold with

a torsion field ( =Burgers vector density).

Different Models for Dislocations

How to bridge between the descriptions? What kind of homogenization process yields a torsion field from singularities?

• A single dislocation
• V
• lterra (~1900)
• Riemannian manifold

with singularities.

• Burgers vector
• Distributed dislocations
• Nye, Bilby, etc. (~1950)
• Smooth manifold with

a torsion field ( =Burgers vector density).

Different Models for Dislocations

How to bridge between the descriptions? What kind of homogenization process yields a torsion field from singularities? A new limit concept in differential 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− d

2θ 2θ

An edge-dislocation

p+ p− $ $ $ $ p− d

2θ 2θ

• Remove a sector of angle 2𝜄, and glue the edges (a cone).

An edge-dislocation

p+ p− $ $ $ $ p− d

2θ 2θ

• Remove a sector of angle 2𝜄, 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.

An edge-dislocation

p+ p− $ $ $ $ p− d

2θ 2θ

• Remove a sector of angle 2𝜄, 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
• utside the dislocation line [p-,p+].

The building block

A B C D p+ p− d a b b a + ε

• Encircle the dislocation line with four straight lines with

right angles between them, obtaining a “rectangle”.

The building block

A B C D p+ p− d a b b a + ε

• Encircle the dislocation line with four straight lines with

right angles between them, obtaining a “rectangle”.

The building block

A B C D p+ p− d a b b a + ε

ε = 2d sin θ

• Denote the lengths of these lines by a, b, b, and a+ε,

where is the dislocation magnitude.

Manifolds with many dislocations

b b a a + ε

b/n b/n b/n

· · ·

b/n a/n + ε/n2 a/n + 2ε/n2 a/n + 3ε/n2

· · ·

a/n + ε/n

. . . . . . . . . . . .

Manifolds with many dislocations

• Glue together n2 building blocks, such that:

b b a a + ε

b/n b/n b/n

· · ·

b/n a/n + ε/n2 a/n + 2ε/n2 a/n + 3ε/n2

· · ·

a/n + ε/n

. . . . . . . . . . . .

Manifolds with many dislocations

• Glue together n2 building blocks, such that:
• Each with the same cone angle 2𝜄 and with

dislocation magnitude ε/n2.

b b a a + ε

b/n b/n b/n

· · ·

b/n a/n + ε/n2 a/n + 2ε/n2 a/n + 3ε/n2

· · ·

a/n + ε/n

. . . . . . . . . . . .

Manifolds with many dislocations

• Glue together n2 building blocks, such that:
• Each with the same cone angle 2𝜄 and with

dislocation magnitude ε/n2.

• The boundary consists of straight lines of lengths a, b,

b, and a+ε.

b b a a + ε

b/n b/n b/n

· · ·

b/n a/n + ε/n2 a/n + 2ε/n2 a/n + 3ε/n2

· · ·

a/n + ε/n

. . . . . . . . . . . .

Manifolds with many dislocations

• Glue together n2 building blocks, such that:
• Each with the same cone angle 2𝜄 and with

dislocation magnitude ε/n2.

• 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 b a a + ε

b/n b/n b/n

· · ·

b/n a/n + ε/n2 a/n + 2ε/n2 a/n + 3ε/n2

· · ·

a/n + ε/n

. . . . . . . . . . . .

Metric Convergence

Metric Convergence

How do these manifolds Mn look like when n→∞?

Metric Convergence

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

b b a a + ε

b/n b/n b/n

· · ·

b/n a/n + ε/n2 a/n + 2ε/n2 a/n + 3ε/n2

· · ·

a/n + ε/n

. . . . . . . . . . . .

ε/b b b a a + ε

Metric Convergence

b b a a + ε

b/n b/n b/n

· · ·

b/n a/n + ε/n2 a/n + 2ε/n2 a/n + 3ε/n2

· · ·

a/n + ε/n

. . . . . . . . . . . .

ε/b b b a a + ε

Gromov-Hausdorff convergence:

• if there exist bijections
• between δn-nets An and Bn (δn→0) such that

Tn :An⊂ Mn → Bn⊂ M Mn

GH

− − → M

Metric Convergence

dis Tn = sup

x,y∈An

|dMn(x, y) − dM(Tn(x), Tn(y))| →n→∞ 0

b b a a + ε

b/n b/n b/n

· · ·

b/n a/n + ε/n2 a/n + 2ε/n2 a/n + 3ε/n2

· · ·

a/n + ε/n

. . . . . . . . . . . .

ε/b b b a a + ε

Gromov-Hausdorff convergence:

• if there exist bijections
• between δn-nets An and Bn (δn→0) such that

Tn :An⊂ Mn → Bn⊂ M Mn

GH

− − → M

What else converges?

b b a a + ε

b/n b/n b/n

· · ·

b/n a/n + ε/n2 a/n + 2ε/n2 a/n + 3ε/n2

· · ·

a/n + ε/n

. . . . . . . . . . . .

ε/b b b a a + ε

What else converges?

• An consists of geodesics (straight lines).

b b a a + ε

b/n b/n b/n

· · ·

b/n a/n + ε/n2 a/n + 2ε/n2 a/n + 3ε/n2

· · ·

a/n + ε/n

. . . . . . . . . . . .

ε/b b b a a + ε

What else converges?

• An consists of geodesics (straight lines).
• Bn does not.

b b a a + ε

b/n b/n b/n

· · ·

b/n a/n + ε/n2 a/n + 2ε/n2 a/n + 3ε/n2

· · ·

a/n + ε/n

. . . . . . . . . . . .

ε/b b b a a + ε

What else converges?

• An consists of geodesics (straight lines).
• Bn does not.
• Or does it?

b b a a + ε

b/n b/n b/n

· · ·

b/n a/n + ε/n2 a/n + 2ε/n2 a/n + 3ε/n2

· · ·

a/n + ε/n

. . . . . . . . . . . .

ε/b b b a a + ε

b b a a + ε

b/n b/n b/n

· · ·

b/n a/n + ε/n2 a/n + 2ε/n2 a/n + 3ε/n2

· · ·

a/n + ε/n

. . . . . . . . . . . .

What else converges?

ε/b b b a a + ε

(∂x, ∂y) (∂r, r−1∂ϕ)

• An consists of geodesics w.r.t. the canonical

(Levi-Civita) parallel-transport on Mn.

b b a a + ε

b/n b/n b/n

· · ·

b/n a/n + ε/n2 a/n + 2ε/n2 a/n + 3ε/n2

· · ·

a/n + ε/n

. . . . . . . . . . . .

What else converges?

ε/b b b a a + ε

(∂x, ∂y) (∂r, r−1∂ϕ)

• An consists of geodesics w.r.t. the canonical

(Levi-Civita) parallel-transport on Mn.

• Bn consists of geodesics w.r.t. a non-canonical one

(i.e. with torsion) — ∂r and r-1∂φ are parallel.

b b a a + ε

b/n b/n b/n

· · ·

b/n a/n + ε/n2 a/n + 2ε/n2 a/n + 3ε/n2

· · ·

a/n + ε/n

. . . . . . . . . . . .

What else converges?

ε/b b b a a + ε

(∂x, ∂y) (∂r, r−1∂ϕ)

• An consists of geodesics w.r.t. the canonical

(Levi-Civita) parallel-transport on Mn.

• Bn consists of geodesics w.r.t. a non-canonical one

(i.e. with torsion) — ∂r and r-1∂φ are parallel.

b b a a + ε

b/n b/n b/n

· · ·

b/n a/n + ε/n2 a/n + 2ε/n2 a/n + 3ε/n2

· · ·

a/n + ε/n

. . . . . . . . . . . .

What else converges?

ε/b b b a a + ε

(∂x, ∂y) (∂r, r−1∂ϕ) Do we have convergence of the parallel-transport?

b b a a + ε

b/n b/n b/n

· · ·

b/n a/n + ε/n2 a/n + 2ε/n2 a/n + 3ε/n2

· · ·

a/n + ε/n

. . . . . . . . . . . .

ε/b b b a a + ε

Convergence of the parallel-transport

(∂r, r−1∂ϕ) (∂x, ∂y)

Tn : An → Bn can be extended to a smooth embedding

b b a a + ε

b/n b/n b/n

· · ·

b/n a/n + ε/n2 a/n + 2ε/n2 a/n + 3ε/n2

· · ·

a/n + ε/n

. . . . . . . . . . . .

ε/b b b a a + ε

Convergence of the parallel-transport

(∂r, r−1∂ϕ) (∂x, ∂y)

Tn : An → Bn can be extended to a smooth embedding

b b a a + ε

b/n b/n b/n

· · ·

b/n a/n + ε/n2 a/n + 2ε/n2 a/n + 3ε/n2

· · ·

a/n + ε/n

. . . . . . . . . . . .

ε/b b b a a + ε

Convergence of the parallel-transport

Fn : Mn → M (∂r, r−1∂ϕ) (∂x, ∂y)

Tn : An → Bn can be extended to a smooth embedding

b b a a + ε

b/n b/n b/n

· · ·

b/n a/n + ε/n2 a/n + 2ε/n2 a/n + 3ε/n2

· · ·

a/n + ε/n

. . . . . . . . . . . .

ε/b b b a a + ε

Convergence of the parallel-transport

Fn : Mn → M (∂r, r−1∂ϕ) (∂x, ∂y) Parallel-transport operators converge:

Tn : An → Bn can be extended to a smooth embedding

b b a a + ε

b/n b/n b/n

· · ·

b/n a/n + ε/n2 a/n + 2ε/n2 a/n + 3ε/n2

· · ·

a/n + ε/n

. . . . . . . . . . . .

ε/b b b a a + ε

Convergence of the parallel-transport

Fn : Mn → M (∂r, r−1∂ϕ) (∂x, ∂y) Parallel-transport operators converge:

lim

n→∞

Z

Fn(Mn)

|dFn(∂x, ∂y) − (∂r, r−1∂ϕ)| = 0

Tn : An → Bn can be extended to a smooth embedding

b b a a + ε

b/n b/n b/n

· · ·

b/n a/n + ε/n2 a/n + 2ε/n2 a/n + 3ε/n2

· · ·

a/n + ε/n

. . . . . . . . . . . .

ε/b b b a a + ε

Convergence of the parallel-transport

Fn : Mn → M (∂r, r−1∂ϕ) (∂x, ∂y) Components of the covariant derivative do not converge! Parallel-transport operators converge:

lim

n→∞

Z

Fn(Mn)

|dFn(∂x, ∂y) − (∂r, r−1∂ϕ)| = 0

b b a a + ε

b/n b/n b/n

· · ·

b/n a/n + ε/n2 a/n + 2ε/n2 a/n + 3ε/n2

· · ·

a/n + ε/n

. . . . . . . . . . . .

ε/b b b a a + ε

Uniqueness

(∂r, r−1∂ϕ) (∂x, ∂y)

Metric limit: unique by properties of GH convergence.

b b a a + ε

b/n b/n b/n

· · ·

b/n a/n + ε/n2 a/n + 2ε/n2 a/n + 3ε/n2

· · ·

a/n + ε/n

. . . . . . . . . . . .

ε/b b b a a + ε

Uniqueness

(∂r, r−1∂ϕ) (∂x, ∂y)

Metric limit: unique by properties of GH convergence. Is the limit parallel-transport well-defined? 
 Does it depend on the choice of the embeddings Fn and the parallel frame fields (∂x,∂y)?

b b a a + ε

b/n b/n b/n

· · ·

b/n a/n + ε/n2 a/n + 2ε/n2 a/n + 3ε/n2

· · ·

a/n + ε/n

. . . . . . . . . . . .

ε/b b b a a + ε

Uniqueness

(∂r, r−1∂ϕ) (∂x, ∂y)

Uniqueness

Fn : Mn → M Theorem: Let (Mn,gn,Πn), (M,g,Π) be manifolds endowed with

path-independent parallel-transport operators (equiv. flat cov. derivatives) such that there exist embeddings such that:

Uniqueness

• 1. Fn are asymptotically surjective.

Fn : Mn → M Theorem: Let (Mn,gn,Πn), (M,g,Π) be manifolds endowed with

path-independent parallel-transport operators (equiv. flat cov. derivatives) such that there exist embeddings such that:

Uniqueness

• 1. Fn are asymptotically surjective.
• 2. The distortion of Fn tends to zero.

Fn : Mn → M Theorem: Let (Mn,gn,Πn), (M,g,Π) be manifolds endowed with

path-independent parallel-transport operators (equiv. flat cov. derivatives) such that there exist embeddings such that:

Uniqueness

• 1. Fn are asymptotically surjective.
• 2. The distortion of Fn tends to zero.
• 3. Fn are asymptotically rigid on the mean.

Fn : Mn → M Theorem: Let (Mn,gn,Πn), (M,g,Π) be manifolds endowed with

path-independent parallel-transport operators (equiv. flat cov. derivatives) such that there exist embeddings such that:

Uniqueness

• 1. Fn are asymptotically surjective.
• 2. The distortion of Fn tends to zero.
• 3. Fn are asymptotically rigid on the mean.
• 4. There exist Πn-parallel frame fields En and a Π-parallel

frame field E such that

Fn : Mn → M Theorem: Let (Mn,gn,Πn), (M,g,Π) be manifolds endowed with

path-independent parallel-transport operators (equiv. flat cov. derivatives) such that there exist embeddings such that:

Uniqueness

• 1. Fn are asymptotically surjective.
• 2. The distortion of Fn tends to zero.
• 3. Fn are asymptotically rigid on the mean.
• 4. There exist Πn-parallel frame fields En and a Π-parallel

frame field E such that

Fn : Mn → M Theorem: Let (Mn,gn,Πn), (M,g,Π) be manifolds endowed with

path-independent parallel-transport operators (equiv. flat cov. derivatives) such that there exist embeddings such that:

lim

n→∞

Z

Fn(Mn)

|dFn(En) − E| = 0

Uniqueness

Then (M,g,Π) is defined uniquely, that is, independent of the choice of embeddings and frame fields.

• 1. Fn are asymptotically surjective.
• 2. The distortion of Fn tends to zero.
• 3. Fn are asymptotically rigid on the mean.
• 4. There exist Πn-parallel frame fields En and a Π-parallel

frame field E such that

Fn : Mn → M Theorem: Let (Mn,gn,Πn), (M,g,Π) be manifolds endowed with

path-independent parallel-transport operators (equiv. flat cov. derivatives) such that there exist embeddings such that:

lim

n→∞

Z

Fn(Mn)

|dFn(En) − E| = 0

What else can we construct?

What else can we construct?

• Essentially, any compact 2-manifold with

boundaries endowed with a (metric) path- independent parallel-transport.

What else can we construct?

• Essentially, any compact 2-manifold with

boundaries endowed with a (metric) path- independent parallel-transport.

• Even if the manifold itself is non-flat!

What else can we construct?

• Essentially, any compact 2-manifold with

boundaries endowed with a (metric) path- independent parallel-transport.

• Even if the manifold itself is non-flat!
• For example, the torus with the direction of the

meridians and the parallels as a parallel frame-field.

What else can we construct?

• Essentially, any compact 2-manifold with

boundaries endowed with a (metric) path- independent parallel-transport.

• Even if the manifold itself is non-flat!
• For example, the torus with the direction of the

meridians and the parallels as a parallel frame-field.

Conclusion

• W

e proved that a sequence of manifolds with edge- dislocations converges, as the dislocations get denser, to a non-singular manifold.

Conclusion

• W

e proved that a sequence of manifolds with edge- dislocations converges, as the dislocations get denser, to a non-singular manifold.

• The sequence is endowed with the canonical parallel-

transports (equiv. affine connections), which converge to a non-canonical parallel-transport of the limit manifold.

Conclusion

• W

e proved that a sequence of manifolds with edge- dislocations converges, as the dislocations get denser, to a non-singular manifold.

• The sequence is endowed with the canonical parallel-

transports (equiv. affine connections), which converge to a non-canonical parallel-transport of the limit manifold.

• The limit manifold is therefore a manifold with a

parallel-transport that carries torsion.

Conclusion

• W

e proved that a sequence of manifolds with edge- dislocations converges, as the dislocations get denser, to a non-singular manifold.

• The sequence is endowed with the canonical parallel-

transports (equiv. affine connections), which converge to a non-canonical parallel-transport of the limit manifold.

• The limit manifold is therefore a manifold with a

parallel-transport that carries torsion.

• This fits the phenomenological description of a

continuous distribution of dislocations.

Conclusion

• W

e proved that a sequence of manifolds with edge- dislocations converges, as the dislocations get denser, to a non-singular manifold.

• The sequence is endowed with the canonical parallel-

transports (equiv. affine connections), which converge to a non-canonical parallel-transport of the limit manifold.

• The limit manifold is therefore a manifold with a

parallel-transport that carries torsion.

• This fits the phenomenological description of a

continuous distribution of dislocations.

Conclusion

Thank you for your attention!