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/centre for analysis, scientific computing and applications About me Plasticity Dislocations Lucia’s work My work Final slide Dislocations Patrick van Meurs 11 April 2012

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Page 1: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Dislocations

Patrick van Meurs

11 April 2012

Page 2: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Outline

1 About meCV and project

2 PlasticityElasticity vs. plasticityFeatures plasticity

3 DislocationsBasic properties

4 Lucia’s workState of the art

5 My workCurrent plansFuture plans

Page 3: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Outline

1 About meCV and project

2 PlasticityElasticity vs. plasticityFeatures plasticity

3 DislocationsBasic properties

4 Lucia’s workState of the art

5 My workCurrent plansFuture plans

Page 4: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

About me

2006 - 2011: Master’s degree applied mathematics at TU/eAugust 2011: start PhD project in EindhovenAnalysis group of MarkHG 8.34

Page 5: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

About me

2006 - 2011: Master’s degree applied mathematics at TU/eAugust 2011: start PhD project in EindhovenAnalysis group of MarkHG 8.34

Page 6: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Project

Title: Correlating Fluctuations Across the ScalesTeam:

position name and title jobprincipal applicant Prof. dr. Mark A. Peletier mathematicianco-applicant Dr. Adrian Muntean mathematicianco-applicant Dr. Markus Hutter physicistco-applicant Prof. dr. Marc Geers mechanical engineerco-applicant Dr. Ron Peerlings mechanical engineerPhD 1 Patrick van Meurs MSc mathematicianPhD 2 Marleen Kooiman MSc physicist

Dr. Lucia Scardia

Page 7: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Project

Title: Correlating Fluctuations Across the ScalesTeam:

position name and title jobprincipal applicant Prof. dr. Mark A. Peletier mathematicianco-applicant Dr. Adrian Muntean mathematicianco-applicant Dr. Markus Hutter physicistco-applicant Prof. dr. Marc Geers mechanical engineerco-applicant Dr. Ron Peerlings mechanical engineerPhD 1 Patrick van Meurs MSc mathematicianPhD 2 Marleen Kooiman MSc physicist

Dr. Lucia Scardia

Page 8: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Outline

1 About meCV and project

2 PlasticityElasticity vs. plasticityFeatures plasticity

3 DislocationsBasic properties

4 Lucia’s workState of the art

5 My workCurrent plansFuture plans

Page 9: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Elasticity

Also important for (e.g.) designing buildings, like bridgesAccurate models are available (continuum mechanics)

Page 10: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Elasticity

Also important for (e.g.) designing buildings, like bridgesAccurate models are available (continuum mechanics)

Page 11: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Elasticity

Also important for (e.g.) designing buildings, like bridgesAccurate models are available (continuum mechanics)

Page 12: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Plasticity

?

Other examples:bullet through steel plate (CIA)hypervelocity impact on steel plate (Iason)

Plasticity is not easy to model

Page 13: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Plasticity

?

Other examples:bullet through steel plate (CIA)hypervelocity impact on steel plate (Iason)

Plasticity is not easy to model

Page 14: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Plasticity

?

Other examples:bullet through steel plate (CIA)hypervelocity impact on steel plate (Iason)

Plasticity is not easy to model

Page 15: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Length scales

Michael Ortiz

ROME0611

Metal plasticity Multiscale analysis

Lattice defects, EoS

Dislocation dynamics

Subgrainstructures

length

tim

e

mmnm µm

ms

µs

ns

Polycrystals

Engineeringapplications

Quantum mechanical or atomistic

Discrete or linear elastic

Continuum

Objective: Derive ansatz-free,

physics-based, predictive models

of macroscopic behavior

Me: dislocation structureupscaling−−−−−→ grain level

Page 16: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Length scales

Michael Ortiz

ROME0611

Metal plasticity Multiscale analysis

Lattice defects, EoS

Dislocation dynamics

Subgrainstructures

length

tim

e

mmnm µm

ms

µs

ns

Polycrystals

Engineeringapplications

Quantum mechanical or atomistic

Discrete or linear elastic

Continuum

Objective: Derive ansatz-free,

physics-based, predictive models

of macroscopic behavior

Me: dislocation structureupscaling−−−−−→ grain level

Page 17: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Outline

1 About meCV and project

2 PlasticityElasticity vs. plasticityFeatures plasticity

3 DislocationsBasic properties

4 Lucia’s workState of the art

5 My workCurrent plansFuture plans

Page 18: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Schematic pictures

THE CONCEPT OF CRYSTAL DISLOCATIONS 7

b b

A A

(c)(b)(a)

B

Fig. 1.4. (a) A perfect simple-cubic crystal. (b) Displacement of two half-crystalsalong cut planeA by lattice vector b results in two surface steps but does not alterthe atomic structure inside the crystal. (c) The same “cut-and-slip” procedurelimited to a part of cut plane A introduces an edge dislocation ⊥.

(a) (b) (c)

B

bb

Fig. 1.5. (a) An edge dislocation created by inserting a half-plane of atomsB. (b) Ascrew dislocation created by a “cut-and-slip” procedure in which the slip vectoris parallel to the dislocation line. The slipped area of the cut plane is shown indark gray and the un-slipped area is shown in light gray. The dislocation line ismarked by the solid line. (c) A curved dislocation line with an edge orientationat one end (on the left) and a screw orientation at the other end (on the right).

Such a “cut-and-slip” procedure produces a permanent (plastic) deformation of thecrystal, but does not yet create a dislocation. A dislocation appears when twohalf-crystals are displaced not over the whole cut plane A, but only over a partof it marked by the solid line in Fig. 1.4(c). The remaining “un-slipped” part ofthe cut plane is marked by the dashed line. The boundary between the slipped andun-slipped parts of the cut plane is a dislocation line. This line runs perpendicularto the plane of the paper and is marked by symbol ⊥ on Fig. 1.4(c).

A very similar structure is created by inserting an extra half plane ofatoms into a perfect SC crystal from above. Shown in Fig. 1.5(a), the new

Page 19: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Edge dislocations: movement

Imposed restriction: straight edge dislocations2D model

Linear drag law: v = BF

Page 20: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Edge dislocations: movement

Imposed restriction: straight edge dislocations2D model

Linear drag law: v = BF

Page 21: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Induced stress field

11/29

/w

Dislocation interactions

incompatibility

repulsion attraction→ annihilationCallister (2007)

x

y

Linear elasticity: F (x , y) = Kxx2 − y2

(x2 + y2)2

Page 22: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Outline

1 About meCV and project

2 PlasticityElasticity vs. plasticityFeatures plasticity

3 DislocationsBasic properties

4 Lucia’s workState of the art

5 My workCurrent plansFuture plans

Page 23: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Setting

x0 = 0 x1 x2 x3

τ

τ

h

h

Specific configuration: walls of dislocations (→ 1D)Find xi , i = 1, . . . ,n such that F = 0 for all dislocations

Page 24: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Setting

x0 = 0 x1 x2 x3

τ

τ

h

h

Specific configuration: walls of dislocations (→ 1D)Find xi , i = 1, . . . ,n such that F = 0 for all dislocations

Page 25: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Nondimensionalization + scalingPosition of walls: xn ∈ Rn, xn

i < xni+1 for all i = 1, . . . ,n

En(xn) =βn

n

n∑k=1

n−k∑i=0

V(nβn(xn

i+k − xni ))

︸ ︷︷ ︸interaction

+1n

n∑i=1

xni︸ ︷︷ ︸

external stress

Motivation scaling: keep xni = O(1), En = O(1)

1 dimensionless parameter:βn, the aspect ratioA minimizer of Enis a stable equilibriumEn convex ⇒ ∃! minimizer

0 2 40

2

4

6

s

∼ s e−s

∼ −ln s

Page 26: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Nondimensionalization + scalingPosition of walls: xn ∈ Rn, xn

i < xni+1 for all i = 1, . . . ,n

En(xn) =βn

n

n∑k=1

n−k∑i=0

V(nβn(xn

i+k − xni ))

︸ ︷︷ ︸interaction

+1n

n∑i=1

xni︸ ︷︷ ︸

external stress

Motivation scaling: keep xni = O(1), En = O(1)

1 dimensionless parameter:βn, the aspect ratioA minimizer of Enis a stable equilibriumEn convex ⇒ ∃! minimizer

0 2 40

2

4

6

s

∼ s e−s

∼ −ln s

Page 27: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Nondimensionalization + scalingPosition of walls: xn ∈ Rn, xn

i < xni+1 for all i = 1, . . . ,n

En(xn) =βn

n

n∑k=1

n−k∑i=0

V(nβn(xn

i+k − xni ))

︸ ︷︷ ︸interaction

+1n

n∑i=1

xni︸ ︷︷ ︸

external stress

Motivation scaling: keep xni = O(1), En = O(1)

1 dimensionless parameter:βn, the aspect ratioA minimizer of Enis a stable equilibriumEn convex ⇒ ∃! minimizer

0 2 40

2

4

6

s

∼ s e−s

∼ −ln s

Page 28: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Nondimensionalization + scalingPosition of walls: xn ∈ Rn, xn

i < xni+1 for all i = 1, . . . ,n

En(xn) =βn

n

n∑k=1

n−k∑i=0

V(nβn(xn

i+k − xni ))

︸ ︷︷ ︸interaction

+1n

n∑i=1

xni︸ ︷︷ ︸

external stress

Motivation scaling: keep xni = O(1), En = O(1)

1 dimensionless parameter:βn, the aspect ratioA minimizer of Enis a stable equilibriumEn convex ⇒ ∃! minimizer

0 2 40

2

4

6

s

∼ s e−s

∼ −ln s

Page 29: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Nondimensionalization + scalingPosition of walls: xn ∈ Rn, xn

i < xni+1 for all i = 1, . . . ,n

En(xn) =βn

n

n∑k=1

n−k∑i=0

V(nβn(xn

i+k − xni ))

︸ ︷︷ ︸interaction

+1n

n∑i=1

xni︸ ︷︷ ︸

external stress

Motivation scaling: keep xni = O(1), En = O(1)

1 dimensionless parameter:βn, the aspect ratioA minimizer of Enis a stable equilibriumEn convex ⇒ ∃! minimizer

0 2 40

2

4

6

s

∼ s e−s

∼ −ln s

Page 30: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

n→∞

Motivation limit: micro→ meso scaleEn

Γ−→ EΓ-limit: ensures the convergence of minimizersE depends on how βn scales with n

Page 31: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

n→∞

Motivation limit: micro→ meso scaleEn

Γ−→ EΓ-limit: ensures the convergence of minimizersE depends on how βn scales with n

Page 32: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

n→∞

Motivation limit: micro→ meso scaleEn

Γ−→ EΓ-limit: ensures the convergence of minimizersE depends on how βn scales with n

Page 33: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

n→∞

Motivation limit: micro→ meso scaleEn

Γ−→ EΓ-limit: ensures the convergence of minimizersE depends on how βn scales with n

Page 34: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

βn � 1/n

0 0.05 0.1 0.15 0.2 0.250

50

100

150

200

250

300

350

400

450

500

Page 35: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

βn ∼ 1/n

0 0.1 0.2 0.3 0.4 0.50

10

20

30

40

50

60

70

Page 36: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

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About me Plasticity Dislocations Lucia’s work My work Final slide

1/n� βn � 1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

5

10

15

20

25

Page 37: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

βn ∼ 1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

1

2

3

4

5

6

Page 38: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

1� βn

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Page 39: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Outline

1 About meCV and project

2 PlasticityElasticity vs. plasticityFeatures plasticity

3 DislocationsBasic properties

4 Lucia’s workState of the art

5 My workCurrent plansFuture plans

Page 40: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Finite grain size

Main goal: extend Lucia’s results

x0 = 0 Lx1 x2 x3

τ

τ

h

h

Page 41: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

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About me Plasticity Dislocations Lucia’s work My work Final slide

Positive and negative dislocations

0 Lx1 x2 x3y1 y2 y3

τ

τ

h

h2

h2

h2

Interaction potential nonconvex

Page 42: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Positive and negative dislocations

0 Lx1 x2 x3y1 y2 y3

τ

τ

h

h2

h2

h2

Interaction potential nonconvex

Page 43: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

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About me Plasticity Dislocations Lucia’s work My work Final slide

Dynamics

xn(t) = −B∇En(xn(t)

)

Page 44: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Thank you for your attentionEven more questions are more than welcome

Page 45: Dislocations - win.tue.nl · About me Plasticity Dislocations Lucia’s work My workFinal slide Outline 1 About me CV and project 2 Plasticity Elasticity vs. plasticity Features plasticity

/centre for analysis, scientific computing and applications

About me Plasticity Dislocations Lucia’s work My work Final slide

Thank you for your attentionEven more questions are more than welcome