Measurement and Prediction of Dislocation Density Development during Plastic Deformation

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Measurement and Prediction of
Dislocation Density Development
during Plastic Deformation

Bjørn Clausen, Donald W. Brown,

Levente Balogh and Carlos N. Tomé


Los Alamos National Laboratory

Funded by OBES
-
DOE

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Outline


Introduction


Elastic
-
Plastic Self
-
Consistent model


Neutron Diffraction


Development of dislocation density based hardening law for
EPSC


Enable the use of measured peak profile information in model validation


More physically based hardening law


Examples


Uniaxial tension of stainless steel


Uniaxial compression of magnesium alloy


Conclusions

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Modeling Polycrystal Deformation


Continuum mechanics models do not take into account the anisotropy
on the level of the microstructure


Polycrystal Plasticity


Taylor








The polycrystal

behavior is derived

from the grain

properties




Taylor
Taylor
Sachs
Sachs
EPSC
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Modeling Polycrystal Deformation


Continuum mechanics models do not take into account the anisotropy
on the level of the microstructure


Polycrystal Plasticity


Taylor





Sachs


The polycrystal

behavior is derived

from the grain

properties




Taylor
Taylor
Sachs
Sachs
EPSC
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Modeling Polycrystal Deformation


Continuum mechanics models do not take into account the anisotropy
on the level of the microstructure


Polycrystal Plasticity


Taylor


Self
-
consistent


Sachs


The polycrystal

behavior is derived

from the grain

properties



Taylor
Taylor
Sachs
Sachs
EPSC
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Elastic
-
Plastic Self
-
Consistent Model


The polycrystal is regarded as an agglomerate of single crystal grains


Properties of each grain is given by the crystal structure, single crystal
elastic constants, possible plastic deformation mechanisms and their
hardening behavior


No direct grain
-
to
-
grain interactions


Eshelby

theory used to describe interaction

between a single grain and an infinite

homogeneous equivalent medium (HEM)

that it is imbedded in


The properties of the HEM is not known

a priori, but are determined iteratively as

the weighted average over all the grains,

hence the model is ‘Self
-
Consistent’


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Hooke’s law for isotropic materials:




Elastic anisotropy for cubic materials:




Elastic Anisotropy

1
2
2
1
1
12
11
44













C
C
C
C
E
kl
ijkl
ij
ij
kk
ij
ij











2
2
2
2
2
2
2
2
2
2
44
12
11
11
12
11
44
2
2
1
1
2
l
k
h
l
k
l
h
k
h
A
A
S
S
S
S
E
C
C
C
hkl
hkl
hkl

















E
111
E
200
E
220
E
311
E
331
E
420
E
422
E
531
12
11
44
2
C
C
C

GPa
GPa
GPa
GPa
GPa
GPa
GPa
GPa
Molybdenum
0.87
291
363
306
325
301
325
306
313
Tungsten
1.00
389
389
389
389
389
389
389
389
Aluminum
1.22
76
64
73
69
74
69
73
71
Stainless steel
3.77
300
94
194
139
216
140
194
166
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Plastic Anisotropy


Few deformation modes with many possible systems in high symmetry materials, e.g.
FCC and BCC


Plastic anisotropy is moderate as there are many equivalent slip systems

Cubics

FCC:

12 (111)<110> systems

BCC:

12 (110)<111> systems


12 (112)<111> systems


24 (123)<111> systems

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Plastic Anisotropy


Few deformation modes with many possible systems in high symmetry materials, e.g.
FCC and BCC


Plastic anisotropy is moderate as there are many equivalent slip systems


Multiple deformation modes with few possible systems in low symmetry materials,
e.g. HCP


Stronger plastic anisotropy as not all systems are available in all grains


Can lead to extension of the elastic
-
plastic transition region

Cubics

Hexagonals

FCC:

12 (111)<110> systems

BCC:

12 (110)<111> systems


12 (112)<111> systems


24 (123)<111> systems

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Neutron Diffraction


Peak position


Elastic lattice strain from changes in
peak position


Intergranular strains


Peak intensity


Texture change from changes in peak
intensities


Peak width


Depends on defect concentration

and grain size


Generally increases with plastic
deformation

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SMARTS


Spatially resolved measurements


Residual strains in components


In situ

measurements


Strains as a function of stress,

temperature, environment, …


Instrument Scientists:


Bjørn Clausen


Donald W. Brown


Thomas A. Sisneros


S
pectrometer for

MA
terials

R
esearch at

T
emperature and

S
tress

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ND and EPSC Exhibit Comparable Level of Detail


Neutron diffraction


Average lattice data from sub
-
set of grains with a given
hkl

plane normal
along the scattering vector


The determined elastic lattice strain is the average of all grains fulfilling the Bragg
condition for the given reflection along the scattering vector


Grains within the sub
-
set has different immediate surroundings


The measurement does not provide information about individual grains or their
interactions with their neighboring grains


Self
-
Consistent models


We can extract data from selected grains based upon their orientation to
mimic the diffraction experiment


Elastic normal strains along any direction are readily available from the model


Interaction among grains are approximated by the
Eshelby

theory where all
grains interact with a medium with the average properties of the polycrystal


Stress and strain are uniform within each grain


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Symbiotic Development of EPSC and ND


The ‘traditional’ small strain EPSC model and neutron diffraction
measurements


Internal elastic strains as a function of applied load


Turner and Tomé (1994), Clausen et al. (1999), Pang et al. (1998), Holden
et al. (2000)


Improved twinning model


Texture development and back
-
stresses due to twinning


Clausen et al. (2008)


Large strain EPSC model


Kinematics of finite strains and rotations


Neil et al. (2010)


Dislocation density based hardening law


Including peak width information and more physically based hardening law

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Determining Material Behavior

Single Crystal

Elastic & Thermal

Constants

Single Crystal

Slip & Twinning Modes &
Hardening Parameters

Initial Texture

(Discrete Distribution of
Orientations)

POLYCRYSTAL

SIMULATION

Stress/Strain Evolution
in Grains

Stress/Strain Evolution in
Aggregate

Texture Evolution in
Aggregate

Direct Problem

Inverse Problem

Activity of Deformation
Mechanisms

Dislocation Density
Evolution in Aggregate

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Utilize Measured Peak Shape Information


Driver for technique development


Increase experimental basis for model validation


Have dislocation density based hardening model developed for the VPSC
model (Beyerlein & Tomé,
Acta

Mater.
, 2008)


Physically based hardening law instead of the empirical Voce hardening law


Desire to develop line profile analysis for neutron diffraction


Measured data included in model validation


Macroscopic stress strain curve


Elastic lattice strain


Peak intensities/Texture


Peak width/profile


Fitting parameters


Dislocation density based hardening law with 5 physically based parameters
per mode and 2 general parameters


4 extra per mode if temperature dependency is included

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Evolution Law for Statistically Stored Dislocations



















2
1
k
k
removed
storage










Essmann

&
Mughrabi

(1979),
Mecking

&
Kocks

(1981, 2003)

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Evolution Law for Statistically Stored Dislocations



















2
1
k
k
removed
storage










Beyerlein & Tomé (2008)





















0
3
1
2
ln
1











b
D
kT
g
b
k
k
Essmann

&
Mughrabi

(1979),
Mecking

&
Kocks

(1981, 2003)

: dislocation interaction coefficient (Lavrentev,1980)


: normalized, stress
-
independent activation energy (
Kocks

et al., 1975)


: drag stress (Gilman, 2002;
Mordehai

et al., 2003)


: reference strain rate (
Kocks

&
Mecking
, 2003)



g


D
0


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Evolution Law for Statistically Stored Dislocations



















2
1
k
k
removed
storage










Beyerlein & Tomé (2008)







removed










removed
f
1
Annihilation

Substructure








removed
f
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Evolution Law for Statistically Stored Dislocations



















2
1
k
k
removed
storage










Beyerlein & Tomé (2008)










2
k
b
qA
sub
sub



: rate coefficient which reflects how debris can grow from point defects seeded


by the local thermal
-
activated reactions


: temperature dependent scaling factor (incorporating )

q

A

f






removed










removed
f
1
Annihilation

Substructure








removed
f
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Material Parameters for Dislocation Based Model

: Initial Critical Resolved Shear Stress (CRSS)

: Strength of hardening

: Activation energy

: Drag stress

: Rate of substructure generation

: Slip
-
twin interactions

: Interaction coefficient, usually in the range 0.1 to 1.0

: Rate factor for debris generation, usually in the range 1 to 10




C

1
k
q

A

D


0
Temperature dependence adds 4 parameters per mode


g
Per mode

For Voce hardening there is 4 free parameters per mode

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Example: Stainless Steel

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Example: Stainless Steel


317L Stainless Steel


Deformed to a true strain of 30%


Relatively high yield stress and significant hardening


Fully austenitic, even at large deformations


Some relaxation observed during strain
-
controlled hold times for ND measurements

Incident Neutron Beam
+90° Detector
Bank
-
90° Detector
Bank
Q


Q
Tension axis
Incident Neutron Beam
+90° Detector
Bank
-
90° Detector
Bank
Q


Q
Tension axis
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Lattice Strains


Measured lattice strains


Linear elastic region with significant elastic anisotropy


Pronounced plastic anisotropy

Longitudinal

Transverse

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Lattice Strains: Voce Hardening Law


Predicted lattice strains


Excellent agreement in the elastic region


Plastic anisotropy is in qualitative agreement


Transverse direction is off for 111 and 200

Longitudinal

Transverse

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Lattice Strains: Dislocation Hardening Law


Predicted lattice strains


Missing the longitudinal 200 reflection: Hardening matrix is more ‘diagonal’ than the ‘isotropic’
hardening matrix used for the Voce law


Transverse direction is better for the 111 and 200 reflections

Longitudinal

Transverse

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Texture Development


Loading axis is horizontal


Sample cut at an angle to the prior rolling direction


Model predictions using Voce hardening law exhibit stronger texture than measured


Model predictions using Dislocation Density based hardening law are in better agreement
with the measured texture

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Peak Profile Analysis


The measured dislocation density increases almost linearly with strain


Analysis based upon the
Convolutional

Multiple Whole Profile (CMWP) fitting
scheme (
http://www.renyi.hu/cmwp
/
)


T. Ungár, J.
Gubicza
, A.
Borbély
, G.
Ribárik
,

J
. Appl.
Cryst
.
, vol. 34, pp. 298
-
310,
2001


G.
Ribárik
, J.
Gubicza

and T. Ungár,

Mat
. Sci. Eng. A
, vol. 387

389, pp. 343
-
347
, 2004

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Peak Profile Analysis


The measured dislocation density increases almost linearly with strain


The measured
subgrain

size decreases with strain

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Measured and Predicted Dislocation Density


The measured dislocation density increases almost linearly with strain


The predicted dislocation density is ~5x lower than the measured


Model includes only the statistically stored dislocations

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Example: Magnesium Alloy

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Example: Magnesium Alloy

Longitudinal

Transverse

Incident Neutron Beam
+90
°
Detector
Bank
-
90
°
Detector
Bank
Q


Q
Compression axis

Compression of extruded Magnesium AZ31 alloy


Strong initial texture due to the extrusion


Magnesium twins relatively easy on the tensile twin system

86.6
°



2
1
10
1
1
10
Tensile Twin:
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Texture Development


Significant texture change for relative small deformation


Predicted texture is in good agreement with the measured


Dislocation Density model is overpredicting the basal intensity in the loading direction

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Twin Volume Fraction


Measured twin volume fraction
reaches ~70% at 10% deformation

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Twin Volume Fraction


Measured twin volume fraction
reaches ~70% at 10% deformation


The predicted twin volume fraction is
in good agreement with the
measured data

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Twin Volume Fraction


Measured twin volume fraction
reaches ~70% at 10% deformation


The predicted twin volume fraction is
in good agreement with the
measured data


The Dislocation Density hardening
model shows slightly higher twin
volume fraction than the Voce model
due to the increased texture
sharpness

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Lattice Strains


Plateau in macroscopic stress
-
strain curve


Jump in lattice strains at onset of twinning


Slope reversal in transverse non
-
prism reflections

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Lattice Strains


Sample 2 shows actual stress decrease in plateau (true stress)


Sample 2 confirm jumps in elastic strains and slope reversals

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Lattice Strains
-

Model Comparison
-

Voce


Plateau in macroscopic stress
-
strains curve is reproduced


Jump in lattice strains and slope reversals are reproduced


Transverse prism reflections are off

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Lattice Strains
-

Model Comparison
-

DD


Plateau in macroscopic stress
-
strains curve is reproduced


Jump in lattice strains and slope reversals are reproduced


Transverse prism reflections are off

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Predicted System Activity


Parent grains:


Early onset of
Basal

slip followed by
Twinning
,
Prism

and a small amount of
Pyramidal

slip


Note that twinning is taking place in the plateau and into the hardening region


Twins:


Initially deform by
Basal

slip but switches to equal amount of
Basal

and
Pyramidal

slip at higher
strains

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Dislocation Densities


Measured dislocation densities increase approximately linearly with strain


Only <a> type dislocations are registered, but at high strains there may be a few percent <
c+a
>


Calculated dislocation densities are ~20x too low compared to the measured values

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Conclusions


Both neutron diffraction measurements and EPSC modeling can provide
detailed grain level information about the state within a polycrystal


Neutron diffraction measurements can be used to test and validate the
EPSC model, providing information such as:


Internal lattice strains from peak position changes


Texture development from peak intensity changes


EPSC modeling can be used to interpret neutron diffraction results and
provide additional detailed insights, such as:


Active deformation modes and relative activities


Critical resolved shear stresses and hardening behavior for active modes


Work is ongoing to develop dislocation density based hardening model
that will further increase the experimental basis for model validation by
including peak profile information

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Collaborators and Funding


LANL:


Donald W. Brown, Carlos N. Tomé, Thomas A. Sisneros, Sven C. Vogel,
Levente Balogh


University of Virginia:


Sean R. Agnew, James A. Wollmershauser, John C. Neil,
Rupalee

Mulay


Georgia Tech, Lorraine:


Laurent
Capolungo


Funding:


U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials
Sciences and Engineering, Project FWP 06SCPE401 under U.S. DOE Contract No.
W
-
7405
-
ENG
-
36.


This work has benefited from the use of the Lujan Neutron Scattering Center at
LANSCE, which is funded by the Office of Basic Energy Sciences (DOE). Los Alamos
National Laboratory is operated by Los Alamos National Security LLC under DOE
Contract DE
-
AC52
-
06NA25396.

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Texture


Strong initial texture due to the prior hot extrusion processing


No basal peak in the longitudinal direction versus a very strong basal peak in the
transverse direction

0
1
10
1
1
10
2
1
10
0002
0
2
11
3
1
10
0
1
10
0002
1
1
10
2
1
10
0
2
11
3
1
10
Longitudinal

Transverse

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Texture


The texture is
axisymmetric


We can make a full texture measurement on SMARTS

Axial distribution function


Clausen et al., Acta Mater. 2008

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Texture


Tinlu浥ractin


Initially no basal pole intensity at center of pole figure


Determine the twin volume fraction from the integrated area


h

> 45: Parents, and
h

< 45: Twins


Clausen et al., Acta Mater. 2008