Modelling of preferred orientation

1
Modelling of preferred orientation
László S. Tóth Laboratoire de Physique et
Mécanique des Matériaux Université de Metz,
France
2
Outline

• Preferred orientation
• Crystal plasticity
• Lattice rotation
• Ideal orientation
• The case of shear
• Effect of twinning or dynamic recrystallization
  on ideal orientations
• A Taylor type polycrystal computer program

3
Preferred orientation
Y. Zhou, L.S. Toth, K.W. Neale, "On the
stability of the ideal orientations of rolling
textures for fcc polycrystals", Acta Metall. et
Mat., 40, 3179-3193, 1992.
4
Example for preferred orientation
5
Crystal plasticity

• The comprehension of orientation persistence
  requires to know the main elements of crystal
  plasticity.
• Distortion of the lattice by elastic strains is
  neglected in this analysis.
• -Dislocation and slip system
• Strain equations for slip
• Resolution of the strain equations for slip
• Strain rate sensitive slip solution
• The plastic spin
• Lattice rotation
• Role of rigid body rotation in lattice spin

6
Dislocation and slip system
Ball-model of a crystal
7
Strain equations for slip
Displacement field
u
Eulerian velocity field
Velocity gradient
Introducing the Schmid orientation tensor
Eulerian strain rate tensor
8
Resolution of the strain equations for slip

• Equation system to solve
• 5 imposed strain rate components, n unknown slip
  rates.
• !Ambiguity problems when the Schmid law is used
  for n gt 5.

• Resolution of ambiguities
• regularization of the Schmid law
• use of maximum work principle
• random selection of 5 slip systems
• second order plastic work rate criterion
• minimum plastic spin assumption
• use self and latent hardening
• strain rate sensitive slip
• rounding vertices on the yield surface

9
Strain rate sensitive slip solution

• Use of constitutive law
• m strain rate sensitivity index
• Equation system to solve
• 5 equations with 5 imposed strain rate
  components, 5 unknown stress components
• (use of Newton-Raphson technique to solve)
• Good initial guess from m 1 (linear equation
  system)

10
Strain rate sensitive slip solution
From the obtained deviatoric stress state S, the
resolved shear stress is calculated in each slip
system
Then the slip rate is obtained from the
constitutive law
Verification the obtained slip distribution has
to reproduce the imposed strain rates
11
Lattice rotation
Three kinds of rotations have to be
distinguished Rigid body rotation rate
(material/laboratory)
Plastic spin (material/lattice)
Lattice spin (lattice/laboratory)
12
IIlustration The lattice rotation in the sphere
model
13
The plastic spin
The plastic spin polyhedron for 111lt110gt slip
in the crystal reference system. The rotation
vectors are oriented towards the vertices of the
polyhedron. Numbers indicate the index of the
slip system. At least four slip systems are
needed for zero plastic spin. The plastic spin is
zero for linear viscous slip (m1) in f.c.c.
crystals for any imposed deformation. Non-zero
(but small) for hexagonal crystal symmetry.
L.S. Toth, J.J. Jonas, K.W. Neale, "Comparison of
the minimum plastic spin and rate sensitive slip
theories for loading of symmetrical crystal
orientations" (communicated by Rodney Hill),
Proc. Roy. Soc. Lond. A427, 201-219, 1990.
14

• Role of rigid body rotation in lattice spin

plastic spin
lattice spin
rigid body spin
Rigid body rotation zero in rolling, tension,
compression. In general, all cases when the
velocity gradient is symmetric. In those cases,
small lattice spin requires small plastic spin.
Small plastic spin is only possible in multiple
slip (at least 4 in f.c.c., see plastic
polyhedron). Therefore, in rolling, tension and
compression the number of necessary operating
slip systems for zero or very small lattice spin
is four. For simple shear, the rigid body spin is
non-zero and very large, consequently, large
plastic spin is needed to have zero or small
lattice spin. Only one or two slip systems are
operating in ideally oriented crystals.
L.S. Toth, P. Gilormini, J.J. Jonas, "Effect of
rate sensitivity on the stability of torsion
textures", Acta Metall., 36, 3077-3091, 1988.
15
Orientation updating
The lattice spin is obtained from
The orientation of a crystal is described by the
transformation matrix T going from the sample to
the crystal reference system. Its rate of change
is
From this equation
When W is expressed in the crystal axis
When W is expressed in the sample axis
Then, during a small time increment dt
16
Ideal orientation
Y. Zhou, L.S. Toth, K.W. Neale, "On the stability
of the ideal orientations of rolling textures for
fcc polycrystals", Acta Metall. et Mat., 40,
3179-3193, 1992.
17
Ideal orientations in rolling of f.c.c.
polycrystals
Y. Zhou, L.S. Toth, K.W. Neale, "On the stability
of the ideal orientations of rolling textures for
fcc polycrystals", Acta Metall. et Mat., 40,
3179-3193, 1992.
18
Orientation persistence
Parameters describing the orientation
persistence
1. Orientation stability parameter, S
2. Divergence of the rotation field
L.S. Toth, P. Gilormini, J.J. Jonas, "Effect of
rate sensitivity on the stability of torsion
textures", Acta Metall., 36, 3077-3091, 1988.
19
Orientation persistence in simple shear
L.S. Toth, P. Gilormini, J.J. Jonas, "Effect of
rate sensitivity on the stability of torsion
textures", Acta Metall., 36, 3077-3091, 1988.
20
The rotation field in shear
L.S. Toth, P. Gilormini, J.J. Jonas, "Effect of
rate sensitivity on the stability of torsion
textures", Acta Metall., 36, 3077-3091, 1988.
21
The divergence in shear
L.S. Toth, K.W. Neale, J.J. Jonas, "Stress
response and persistence characteristics of the
ideal orientations of shear textures", Acta
Metall., 37, 2197-2210, 1989.
22
Texture evolution in large strain shear
g 2
Key figure for ideal orientations
g 5,5
g 11
Continuous variations in the texture components
in the sense of the rigid body rotation.
L.S. Toth, J.J. Jonas, D. Daniel, J.A. Bailey,
"Texture development and length changes in copper
bars subjected to free end torsion", Textures and
Microstructures, 19, 245-262, 1992.
23
Tilts of the components from ideal positions
g 2
L.S. Toth, J.J. Jonas, D. Daniel, J.A. Bailey,
"Texture development and length changes in copper
bars subjected to free end torsion", Textures and
Microstructures, 19, 245-262, 1992.
24
g 5.5
L.S. Toth, J.J. Jonas, D. Daniel, J.A. Bailey,
"Texture development and length changes in copper
bars subjected to free end torsion", Textures and
Microstructures, 19, 245-262, 1992.
25
g 11
L.S. Toth, J.J. Jonas, D. Daniel, J.A. Bailey,
"Texture development and length changes in copper
bars subjected to free end torsion", Textures and
Microstructures, 19, 245-262, 1992.
26
Rotation field C component
L.S. Toth, P. Gilormini, J.J. Jonas, "Effect of
rate sensitivity on the stability of torsion
textures", Acta Metall., 36, 3077-3091, 1988.
27
Rotation field C component
L.S. Toth, P. Gilormini, J.J. Jonas, "Effect of
rate sensitivity on the stability of torsion
textures", Acta Metall., 36, 3077-3091, 1988.
28
Rotation field C component
L.S. Toth, P. Gilormini, J.J. Jonas, "Effect of
rate sensitivity on the stability of torsion
textures", Acta Metall., 36, 3077-3091, 1988.
29
Conclusions on main characteristics of large
strain shear textures

• Continuous variations in the texture components
• in the sense of the rigid body rotation,
• Tilts of the components from ideal positions,
• Convergent/divergent nature of the rotation
  field
• around ideal positions,
• About 50 of the orientations remain outside of
• the tubes of the ODF (perpetuel
  rotation).

30
Effect of twinning or dynamic recrystallization
on ideal orientations
Rotated cube
Does dynamic recrystallization produce new stable
components?
simulated
Example of shear, measured at 300C, g4
J.J. Jonas, L.S. Toth, "Modelling oriented
nucleation and selective growth during dynamic
recrystallisation", Scripta Met. et Mat., 27,
1575-1580, 1992.
31
Slip activity-versus DRX
Number of active slip systems mapped in the
f245 Euler space section for simple shear
ODF of DRX texture in OFHC copper for simple shear
New DRX components (rotated cube) appear at
positions with high number of active slip systems
(high Taylor factors). These positions, however,
have very low orientation stability, so cannot
form high ODF intensities.
32
Modeling of preferred orientation in DRX
Observation high Taylor factor-increases DRX

• Mechanisms of DRX
• Oriented nucleation and growth (ONG)
• Selective growth into the matrix (SG)

Modeling possibilities 1. Growth of volume
fraction of low Taylor factor positions (ONG) 2.
Creating nuclei by rotation of parent and growth
(SG)
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Simplified modeling of DRX
Selective growth into the matrix (SG) Nuclei are
made by rotating the crystal orientation
according to coincident site lattice or plane
matching criteria. Example 40 rotation around
plane normal 111 in f.c.c. Then transferring
volume fraction from parent to nuclei.
Oriented nucleation and growth (ONG) No change
in orientation, just volume fraction transfer.
A. Hildenbrand, L.S. Toth, A. Molinari, J.
Baczynski, J.J. Jonas, "Self consistent
polycrystal modelling of dynamic
recrystallisation in shear deformation of a Ti-IF
steel", Acta Materialia, 47, 447-460, 1999.
34
Use of volume transfer scheme
Eulerian simulation, fix orientation positions in
grid points.
Variant selection in selective growth Preferred
nuclei pertaining to most active slip system
plane.
35
DRX of IF steel in torsion
Experiment
Simulation
A. Hildenbrand, L.S. Toth, A. Molinari, J.
Baczynski, J.J. Jonas, "Self consistent
polycrystal modelling of dynamic
recrystallisation in shear deformation of a Ti-IF
steel", Acta Materialia, 47, 447-460, 1999.
36
Modeling of preferred orientation in twinning
Does twinning produce new stable components?
Experimental texture in ECAE of silver
These textures cannot be modeled with slip alone,
twinning on 111 planes in lt112gt directions is
necessary.
S. Suwas, L.S. Toth, J.J. Fundenberger, A.
Eberhardt, W. Skrotzki, "Evolution of
crystallographic texture during equal channel
angular extrusion of silver", Scripta Materialia,
49, 1203-1208, 2003.
37
Modeling twinning
Twinning is a large shear deformation. In the
modeling, its contribution to the total
deformation is calculated in the same way as for
slip. When twinning is high, slip activity
reduced and vice-versa. Twinned part of a grain
has new orientation ? use of volume transfer
scheme when all variants are allowed. When only
one twin family is admitted to be significativ ?
use of predominant twinning rule (PTR). Parent
twin coexist. Monte Carlo scheme A parent grain
is completely replaced by its twin orientation
when twinning activity is high enough, and a
random selection is valid. Question does the
twinned part co-rotate with the matrix? In terms
of hardening, there is a strong effect of the
twin lamellas which reduce the mean free path of
dislocation glide (Hall-Petch effect).
38
Simulated twinning activity in silver
Twinning activity map
Twinning activity map made for a relative
critical stress of 1.10. Isolines 0.2, 0.4, 0.6,
0.8, 1.0, 1.2, 1.4, where the values mean the sum
of cristallographic shears on all twinning
systems normalized by the imposed von Mises
equivalent strain rate.
39
A simulation result for twinningslip in ECAE of
silver
Simulation
Experiment
40
Conclusions on simulation of ideal orientation
in DRX or twinning
DRX or twinning does not produce new persistent
ideal orientations. Grain orientations
transferred to new orientation position would
require convergent slip to remain stable at the
new position. Unless the new orientation is not
already a stable position, the grain would rotate
away. Some new texture components may appear,
namely rotated cube, however, they are positions
of limited stability and intensity.
41
A Taylor type polycrystal plasticity code to
simulate texture development
Input Cubicsys.dat or Hcpsyst.dat
Input parameter file Polycr.ctl
Input orientation file
Main program Polycr.exe
Increm.for
Output tauc.out Strength of slip systems/grain
Output stress.out Stress state/grain
Output strain.out Strain state/grain
Output euler.out orientation file
42
Characteristics of the code
A Taylor viscoplastic polycrystal code permetting
to simulate texture evolution for any strain mode
and any strain. It can be run in full and relaxed
constraints conditions with or without hardening.
Elasticity is not taken into account. The slip
constitutive law is
The program solves the non-linear equation system
using Newton-Raphson technique for the deviatoric
stress state
Then the resolved shear stresses and the slip
rates are calculated from the above first two
equations. From the slips, the orientation change
is obtained from the equations presented in slide
no. 15. Hardening is accounted for using self
and latent hardening of the slip systems
according to Bronkhorst et al. (1992)
Bronkhorst, C.A., Kalidindi, S.R. and Anand, L.
Phil. Trans. Royal Soc. London, A341, p. 443.
43
Control parameters - Polycr.ctl
Input parameter file Polycr.ctl
rand1000.eul ! input file
cubicsys.dat
! slip system file, it can only be cubicsys.dat
or hcpsyst.dat Pancake rolling of
fcc ! title of run 20.
! strain rate
sensitivity parameter (1/m) 1
!0 no hardening,
1Bronkhorst type hard. 10
! number of increments
0.02 2 1 1
! strain increment in one step, imode1in von
Mises, 2 control by index, index1,index2 1
! 1 constant inposed velocity
gradient, 2 variable velocity gradient (from
data file velgrad.dat) 1 0 0
! imposed velocity gradient when
it is constant 0 0 0 0 0 -1 0 0
1 ! relaxation matrix 0
0 1 ! put 1 where you want to
relax the strain 0 0 0
! put 0 everywhere for full constraints Taylor
deformation mode
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Control parameters - Polycr.ctl cont.
Line strain control in polycr.ctl
-0.02 2 1 1 ! strain
increment in one step, imode1in von Mises, 2
control by index, index1,index2
The first value is the increment of strain in one
step (deps). Attention it can be negative
(compression) or positive (tension). If the next
parameter is set to 1, the strain increment is
imposed to be a von Mises type strain increment,
which is defined from the strain increment tensor
as follows
If the second parameter is 2, as in the above
example, the strain increment is defined in terms
of a specific strain component only. Then, the
following two parameters define the first and
second index (ij) of the controlled component of
the strain tensor.
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Control parameters - Polycr.ctl cont.
Velocity gradient in polycr.ctl The example in
the file is 1 0 0 0 0 0 0 0 -1 It means a
plane strain compression (or rolling). The value
of 1 means the imposed strain rate, that is 1
s-1. Relaxation matrix in polycr.ctl The
example in the file is 0 0 1 0 0 1 0 0
0 It means pancake and lath relaxed constraints
model in which the shear on plane with normal 3
and in the directions 1 and 2 are relaxed
(meaning will noet be imposed to be 0, they will
be calculated from the crystal plasticity
solution of the problem.
46
Control parameters - Input orientation file
title what you want second title, if
needed random texture generated by RANDTEXT.FOR
(23/01/97) third title 1000 number
of orientations follow 102.74 119.56
33.65 1.0 219.06 36.21 70.51 1.0
166.66 28.59 45.80 1.0 149.74
86.13 38.68 1.0
..
Euler angles are in degrees, in the order fi1,
fi, fi2. Fourth number relative volume fraction
of that orientation.
47
Control parameters - Input slip system file
File Cubicsys.dat Cubic slip systems hardening
parameters 1. 180. 148. 2.25 gam0, h0,
tausat, a in case of use of hardening rule by
Bronkhorst e al. 1. 1.4 1.4 1.4 q1
coplanar, q2 colinear, q3 perpendic. ,q4other
latent hardening parameters (111)lt110gt
.................. 12
type of family, nr. of systems, do
not touch Reference critical resolved shear
stress and shear strain (if twinning)
information for next line, the twinning shear
value must be 0 for slip systems and equal to the
shear associated with twinning, if this is a
twinning family. 16. 0. 1 1 1 -1
0 1 1 2 1 1 -1 1 0 1 3 1 1
-1 1 -1 0 4 1 -1 -1 0 1 -1 5
1 -1 -1 1 0 1 6 1 -1 -1 1 1 0 7
1 -1 1 0 1 1 8 1 -1 1 1 0
-1 9 1 -1 1 1 1 0 10 1 1 1
0 1 -1 11 1 1 1 1 0 -1 12 1 1
1 1 -1 0 (110)lt111gt .................. 12
A second
family, and so on. Reference critical resolved
shear stress and shear strain (for twinning) 0.
0.
48
Organisation of the simulation
The main program is polycr.exe. It reads the
input parameters and input files and sets up the
simulation procedure. The calculation is
incremental in strain. One increment is done for
all the grains in the sub-program increm.for for
one call from polycr.exe. The grain number is
limited to 3000 and the total number of slip (or
twinning) systems is maximum 48. The maximum
number of increments in strain is 2000. When
twinning is used, the orientation of a grain is
replaced by its twin orientation if sufficient
amount of twin occurs (1/3 twin and 2/3 slip) in
a grain. Output files Euler.out contains the
new euler angles of the grains, their volume
fractions, and the stability parameter of that
orientation. Stress.out the (Cauchy) stress
state for each grain. Strain.out the strain rate
state of each grain at the end of the simulation.
Attention NOT the accumulated finite strain! In
full constraints Taylor deformation mode it is
the same for each grain (good for checking that
the calculation was OK.). It is only interesting
if relaxed constraints model is used where the
relaxed components will have non-zero values and
varying from grain to grain. Tauc.out the
updated new reference strengths of the slip
system, in the same order as they are in the
input slip system file, for each grain. It is
only interesting if hardening was taken into
account in the simulation.
49
Making and plotting of pole figures of the
simulated textures
The polycr software package is complemented by
two other softwares for the purpose of
visualizing the obtained simulated textures. They
are POLFIG GRAPH4WIN The POLFIG package is to
calculate a data file containing the pole figure
wanted. This file is ready to be plotted using
the GRAPH4WIN package (other software might also
be possible to use).
50
Making and plotting of pole figures of the
simulated textures cont.
POLFIG package The program is polfig.exe. It can
use directly the output distribution file of the
POLYCR package, named euler.out. Copy first this
file into the //POLFIG directory. Edit the
polfig.ctl parameter file which controls the pole
figure preparation
1 ! input in form of Euler
angles (1) or Miller indices (2) 111
! which projection 100, 110, 111, 0001basal
plane hexagonal 3 1 2 ! sequence of
projection axes middle,right,up 2.992
! radius of pole figure (inch)
Line number one must contain 1, because the pole
figure is made from Euler angle positions. (One
could also plot from Miller indices). Line number
2 defines the type of projection you want. One
type of pole figure (0002) is also possible for
hexagonal structure (not others). Third line
defines how the axis of the reference system
should be positioned. The first number defines
the index of the axis what you want in the middle
of the pole figure, perpendicular to the sheet.
Second number is the index of the axis which is
oriented right, horizontal. The third number is
the axis which is in the vertical direction
(up) on the pole figure. Last row defines the
radius of the pole figure in inches. After
executing the program polfig.exe, you obtain four
data files Polfig.dat, circle.dat, axisx.dat,
axisy.dat. The first contains the positions of
the projected poles of the crystal orientations,
the second is a file containing the contour
circle of the pole figure, the last two ones are
just the horizontal and vertical lines of the two
visible axes of the reference system.
51
Making and plotting of pole figures of the
simulated textures cont.
For plotting of the pole figure you have to copy
four files into the directory //graph4win, they
are Polfig.dat, circle.dat, axisx.dat,
axisy.dat. If you are just re-using the plotting
and did not change the dimension of the pole
figure, only the polfig.dat file is necessary to
copy into //graph4win. To plot the pole figure,
use the execution file graph4win.exe Then open
the polfig.grf file (already prepared for you)
from the opening software window. Your pole
figure will immediately plotted. Then use the
graphical options to change the characteristics
of the appearance (self explanatory, easy to
use). By selecting the whole picture, you can
copy it into word document or into a powerpoint
presentation. Good luck!