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About me Plasticity Dislocations Lucia’s work My work Final slide
Dislocations
Patrick van Meurs
11 April 2012
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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
/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
/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
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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
/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
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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
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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
/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)
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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)
/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)
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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
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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
/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
/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
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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
/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
/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
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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
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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
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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
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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
/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
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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
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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
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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
/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
/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
/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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
/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
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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
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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
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About me Plasticity Dislocations Lucia’s work My work Final slide
Dynamics
xn(t) = −B∇En(xn(t)
)
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About me Plasticity Dislocations Lucia’s work My work Final slide
Thank you for your attentionEven more questions are more than welcome
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About me Plasticity Dislocations Lucia’s work My work Final slide
Thank you for your attentionEven more questions are more than welcome