Tutorial I: Mechanics of Ductile Crystalline Solids

1
Tutorial IMechanics of Ductile Crystalline
Solids

• Alberto M. Cuitiño
• Mechanical and Aerospace Engineering
• Rutgers University
• Piscataway, New Jersey
• cuitino_at_jove.rutgers.edu

IHPC-IMS Program on Advances Mathematical
Issues in Large Scale Simulation (Dec 2002 - Mar
2003 Oct - Nov 2003)
Institute of High Performance Computing
Institute for Mathematical Sciences, NUS
2
Hierarchy of Scales
SCS test
ms
Grains
Single crystals
time
µs
Microstructures
ns
Force Field
nm
µm
mm
length
3
The Role of Dislocations
ELASTICITY
Initial
PLASTICITY
area swept by dislocation
Dislocation Motion
Dislocation Generation
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Anatomy of a Dislocation Loop
Edge
Mixed
Screw
Mixed Segment
Edge Segment
b burgers vector
Glide Plane
Screw Segment
Area swept by dislocation loop
Animations from http//uet.edu.pk/dmems/
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Dislocation Motion and Arrest
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Tracking Dislocation in ONE Plane
(Ortiz, 1999)
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Overview
Effective Dislocation Energy

• Core Energy
• Dislocation Interaction
• Irreversible Obstacle Interaction

Equilibrium configurations

• Closed form solution at zero temperature.
• Metropolis Monte Carlo algorithm and mean field
  approximation at finite temperatures.

Macroscopic Averages
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Effective Energy
Elastic interaction
Core energy
External field
where
with
m
Displacement jump across S
Slip Surface S
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Elastic interaction

• Displacement field
• Elastic distortion
• Elastic interaction
• Green function for an isotropic crystal

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Elastic Interaction
with
A1
R
A2
(Hirth and Lothe,1969)
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External Field
with
applied shear stress
forest dislocations
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Core Energy
d inter-planar distance
Ortiz and Phillips, 1999
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Phase-Field Energy
Minimization with respect to ? gives
core regularization factor
elastic energy
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Phase-Field Energy
Elastic energy
Core regularization
Core regularization factor
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Energy minimizing phase-field
Unconstrained minimization problem
if
with
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Irreversible Process and Kinetics

• Irreversible dislocation-obstacle interaction
  may be built into a variational framework, we
  introduce the incremental work function

incremental work dissipated at the obstacles

• Primary and forest dislocations react to form a
  jog

• Updated phase-field follows from

• Short range obstacles

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Irreversible Process and Kinetics
Kuhn-Tucker optimality conditions
Equilibrium condition
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Solution Procedure
and compute the reactions

• Stick predictor. Set
• Reaction projection
• Phase-field evaluation

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Closed-form solution
Calculations are gridless and scale with the
number of obstacles
Dislocation loops
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Macroscopic averages

• Slip

• Dislocation density

• Obstacle concentration

• Shear stress

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Forest Hardening

• Obstacle distribution

Parameters
BOUNDARY CONDITIONS Periodic OBSTACLE STRENGTH
Uniform, f 10 G b2 PEIERLS STRESS tp 0
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Monotonic loading
Evolution of dislocation density with strain.
Stress-strain curve.
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Dislocation Patterns
Evolution of dislocation pattern as a function of
slip strain
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Interaction with Obstacles
Detail of the evolution of the dislocation
pattern showing dislocations bypassing a pair of
obstacles
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Fading memory
a
b
d
c
Stress-strain curve.
e
f
Three dimensional view of the evolution of the
phase-field, showing the the switching of the
cusps.
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Cyclic loading
a
b
c
Stress-strain curve.
d
e
f
Evolution of dislocation density with strain.
i
g
h
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Irreversibility/Cyclic Loading
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Poisson ratio effects
Evolution of dislocation density with strain.
Stress-strain curve.
Stress-strain curve
Dislocation density vs.strain
b
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Obstacle density
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Multiple Glide
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Some Concluding Remarks

• The aim of this study is to develop a phase-field
  theory of dislocation dynamics, strain hardening
  and hysteresis in ductile single crystals.
• This representation enables to identify
  individual dislocation lines and arbitrary
  dislocation geometries, including tracking
  intricate topological transitions such as loop
  nucleation, pinching and the formation of Orowan
  loops.
• This theory permits the coupling between slip
  systems, consideration of obstacles of varying
  strength, anisotropy, thermal and strain rate
  effects.

Ortiz,1999
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Summary

• Phase-field model.
• Closed form solution at zero temperature.
• Temperature effects.
• Strain rate effects.
• Dislocation structures in grain boundaries.