Anisotropy part 2: Using LApp Los Alamos polycrystal plasticity

1
Anisotropy part 2Using LApp- Los Alamos
polycrystal plasticity

• 27-750, Fall 2009
• Texture, Microstructure Anisotropy, Fall 2009
• A.D. Rollett, P. Kalu

2
Objective

• The objective of this lecture is to demonstrate
  how to run LApp and obtain useful results in
  terms of texture prediction and anisotropic
  plastic properties.
• LApp calculates the result (in terms of stress
  state) of applying a given strain (increment) to
  a set of orientations (grains). The number of
  grains can be varied from 1 to many thousands.
  The code can be used iteratively to find a
  macroscopic strain state that satisfies a certain
  applied stress state.

3
Principles of LApp

• The principles governing the calculations in LApp
  are described in more detail in subsequent
  lectures.
• This code is based on the Taylor assumption each
  grain/orientation experiences the same strain as
  the macroscopic body being deformed. A relaxation
  of this boundary condition is allowed for
  (relaxed constraints).
• Since the strain (rate) is known for each grain,
  the objective of the calculation is therefore to
  obtain the stress state in each grain that
  permits the given strain to occur. This leads to
  an implicit equation relating strain rate to
  stress state.

4
Input Files

• sxin lists of slip systems (for cubic crystals,
  also lists vertices on the single crystal yield
  surface).
• texin list of orientations Euler angles with a
  weight (sometimes also state parameters).
• bcin boundary conditions (strain and stress).
• propin stress-strain constitutive relations
  (hardening).

5
LApp Flow Chart
outputfiles
rate-sensitivesolution
sxinbcintexinpropin
inputfiles
histlapp.dattexoutanal
sss
newton
preparation
updateorientationof eachgrain
grain, slipgeometry
stop
orient
maxwork
updatehardeningon eachslip system
harden
Bishop-Hill solution
6
sxin slip geometry

• cubic lattices (this is fcc for bcc, LApp gives
  you option to transpose)
• 1 28 nmodes,nvertex. mode nsys ktwin twsh -corr
  (all numbers must appear)
• 1 12 0 0.0 0.0
• 1 1 -1 0 1 1 pk
  -pk
• 1 1 -1 1 0 1 pq
  -pq
• 1 1 -1 1 -1 0 pu
  -pu
• 1 -1 -1 0 1 -1 qu
  -qu
• 1 -1 -1 1 0 1 qp
  -qp
• 1 -1 -1 1 1 0 qk
  -qk
• 1 -1 1 0 1 1 kp
  -kp
• 1 -1 1 1 0 -1 ku
  -ku
• 1 -1 1 1 1 0 kq
  -kq
• 1 1 1 0 1 -1 uq
  -uq
• 1 1 1 1 0 -1 uk
  -uk
• 1 1 1 1 -1 0 up
  -up

fcc slipdirections
fcc slip planes
SlipSystems
7
sxin, contd.
number of active systems

• 28 nvertex
• 8 1
• 2 0 0 0 0
• 2 3 5 6 9 8 11
  12
• 8 33
• 0 2 0 0 0
• 1 15 16 18 19 21 10
  24
• 8 65
• -2 -2 0 0 0
• 13 14 4 17 7 20 22
  23
• 6 97
• 0 0 1 1 1
• 1 2 17 18 7 9 25
  25
• 6 103
• 0 0 1 -1 1
• 1 15 7 20 11 12 25
  25

stress vector
8-fold vertex
IDs of activeslip systems
8
propin strain hardening properties

• Al for Stout's 1100 Al, kond2 for later batch
  (ten,com,chd)
• c 1 lattice, nmodes. MODEs
• 1 - no latent hardening
• mode rs tau tau- h(m,1) h(m,2)
  h(m,3)........
• 1 0.01 1.0 1.0 1.0 1.0 1.0 1.0
  1.0
• STRESS LEVEL AND HARDENING LAWS
• kond RATEref Tref muMPa tau0MPa th0/mu
  tauvMPa th4/th0 kurve
• 1 1.0e-03 300. 25300. 20. 0.005
  30. 0.04 1
• kurve ntaun DISCRETE HARDENING of TAUref, ntaun
  value pairs
• 1 30 taun harn (taun(TAUref-TAU0)/tauv,
  harnth/th0)
• .02 1.00
• .04 .96
• .08 .92
• 1.40 .06
• 1.60 .05

Mode/deformation system
Rate sensitivity
Relative hardening rates on each slip system
9
Hardening parameters

• kond system number
• RATEref strain rate at which properties given
• Tref reference temperature
• muMPa shear modulus (µ)
• tau0MPa yield stress (initial critical
  resolved shear stress)
• th0/mu hardening rate over modulus in Stage II
• tauvMPa Voce stress (saturation, or
  asymptotic flow stress)
• th4/th0 ratio of hardening in Stage IV to that
  in Stage II
• Kurve ID number of discretized hardening rate
  versus stress curve

10
texin initial orientations, grain shape

• texran use any portion (only file when less
  than tetr.cry.sym.)
• Evm F11 F12 F13 F21 F22 F23
  F31 F32 F33
• 0.000 1.000 0.000 0.000 0.000 1.000 0.000
  0.000 0.000 1.000
• KocksPsi Theta phi weight (up to 6 state
  params, f8.2) XYZ 1 2 3
• 158.61 44.96 -161.52 1.0 1. 1.
• 176.88 77.35 -171.43 1.0 1. 1.
• 30.33 72.20 158.06 1.0 1. 1.
• -145.33 59.09 -143.55 1.0 1. 1.
• 130.84 35.92 150.44 1.0 1. 1.
• 99.57 79.29 10.73 1.0 1. 1.
• 105.42 22.61 6.19 1.0 1. 1.

Euler angles
Weight
State Parameters
11
bcin boundary conditions
Test type

• lttencomroltorgt,iplane,iline,evmstep,updt(g.a.),
  RCacc
• 3 3 1 0.02500 0.0
  0.0
• av.strain dir.lt33 (22-11) 223 231
  212gt epstol
• 1.000 1.000 0.000 0.000
  0.000 0.5
• exp'd stress dir.lt33-(1122)/2(22-11)/2233112gt
  ,99 if ?sigtol
• 99.0 99.0 99. 99.0 99.0
  0.05

Stress components
Strain components
s33-(s22s11)/2, (s22-s11)/2, s23, s31, s12
e33, e22-e11, 2e23, 2e31, 2e12
Strain increment
99 means component can take any value
12
LApp dialog
User responses in red

• KRYPTON.MEMS.CMU.EDUgt lapp68
• (C)opyright 1988, The Regents of the University
  of California.
• This software was produced under U. S.
  Government contract by
• Los Alamos National Laboratory, which is
  operated by the
• University of California for the U. S.
  Department of Energy.
• Permission is granted to the public to copy and
  use this
• software without charge, provided that this
  Notice and the
• above statement of authorship are reproduced on
  all copies.
• Neither the Government nor the University makes
  any warranty,
• express or implied, or assumes any liability or
  responsibility
• for the use of this software.
• 
  
• 
  LA-CC-88-6
• Los Alamos Polycrystal Plasticity
  simulation code
• U.F. Kocks, G.R. Canova, C.N. Tome, A.D.
  Rollett, S.I. Wright
• Center for Materials Science
  
• Los Alamos National Laboratory
  
• Los Alamos, New Mexico 87545, USA
  

ltRETURNgt
13
LApp 2

• LApp Version 6.8, 22 Sep 1995
• Needs single crystal deformation modes in
  SXIN,
• kinetics and hardening data in PROPIN,
• grain state data in TEXIN 3
  anglesgrwtstate pars.
• (all must be in prescribed format)
• TEXIN file
• texlat.wts from texlat.write viii
  00
• Enter title (8 chars.)

Enter a (short!) title
14
ksys Deformation System

• Enter KSYS
• 1 for FCC 111lt110gt slip (perhaps w/LH)
• 2 for BCC restricted glide on 110
• 3 for BCC pencil glide
• 4 for FCC card glide
• Enter a number for the lattice type (fcc vs. bcc)
  and the restriction on slip plane (bcc)/
  direction (fcc).
• Typical use 1 for fcc, and 3 for bcc at
  ambient conditions, fcc metals deform in
  restricted glide, whereas bcc metals typically
  deform in pencil glide.

15
ksol Solution procedure

• Enter KSOL
• 0 for Bishop-Hill yield stress only, no
  evolutions
• 1 for BH guess, then rate-sensitive Newton
  solution
• 2 for BH guess on first step only, then
  recursive
• 3 for Sachs guess on first step only, then
  recursive
• 4 for Sachs guess on every step
  
• (recommended 1)
• (need 3 or 4 for Latent Hardening)
• 1
• 0 is the classical Taylor model in the
  rate-insensitive limit.
• 2 and 3 allow for more efficient calculation,
  based on the (reasonable) assumption that the
  previous solution is close to the solution sought
  in the current step.

16
exponent Rate Sensitivity

• PROPIN
• propfe for Salsgiver's Fe-Si,exper.StoutLovat
  o 8/89
• mode rs tau tau- h(m,1) h(m,2)
  h(m,3)........
• Value for max. rate sensitivity exponent
  ltdefault 33gt?
• 33
• kond RATEref Tref muMPa tau0MPa th0/mu
  tauvMPa th4/th0 kurve(or LH)
• 1 1.0e-03 300. 70000. 150. 0.0045
  120. 0.04 1
• The exponent controls the rate sensitivity of the
  single crystal yield surface the lower the
  exponent, the more rounded the SXYS. In general,
  the results are not sensitive to the value of the
  exponent, unless you use a value less than 10.
  

17
kpath type of test

• Reenter TTY input lt1gt, or same as in preceding
  test lt0gt ?
• (Get to choose nsteps and YS-space anyway)
  
• 1 (0 jumps to last question)
• Enter strain path (KPATH)
• 1 many steps in one straining direction (need
  BCIN)
• 2 2-D yield surface probe
• 3 3-D yield surface probe
• 4 Lankford Coefficients R(angle) in the
  3-plane
• 1 (i.e. texture evolution)

18
hardening law

• REFERENCE STRESS AND ITS HARDENING LAW
• Enter 0 for no hardening,
• 1 " " " but stress scale
  (tau0),
• 2 for linear hardening (stage II th0),
• 3 for Voce law (stage III tauv),
• 4 for Voce law plus stage IV (th4),
• 5 for digital hardening according to
  KURVE
• 1 (answer does not affect texture development,
  only hardening)

19
krc, ngrains

• Relaxed Constraints when applicable ltKRC1gt or
  Full Constraints lt0gt?
• 0 (boundary conditions on grain)
• ngrains ltdefault whole file,.le.1152gt
  ?
• 999 (defaults to max. number of orientations in
  texin)

On modern computers, the maximum number of grains
can be easily extended to gt100,000.
20
anal

• Complete file ANAL on the first how many
  lt0,9,ngrainsgt?
• 0 (use for debugging, checks)
• mode,systems 1 12
• n , b , nrs 0.577 0.577 -0.577 0.000
  0.707 0.707 33
• n , b , nrs 0.577 0.577 -0.577 0.707
  0.000 0.707 33
• n , b , nrs 0.577 0.577 -0.577 0.707
  -0.707 0.000 33
• n , b , nrs 0.577 -0.577 -0.577 0.000
  0.707 -0.707 33
• n , b , nrs 0.577 -0.577 -0.577 0.707
  0.000 0.707 33
• n , b , nrs 0.577 -0.577 -0.577 0.707
  0.707 0.000 33
• n , b , nrs 0.577 -0.577 0.577 0.000
  0.707 0.707 33
• n , b , nrs 0.577 -0.577 0.577 0.707
  0.000 -0.707 33
• n , b , nrs 0.577 -0.577 0.577 0.707
  0.707 0.000 33
• n , b , nrs 0.577 0.577 0.577 0.000
  0.707 -0.707 33
• n , b , nrs 0.577 0.577 0.577 0.707
  0.000 -0.707 33
• n , b , nrs 0.577 0.577 0.577 0.707
  -0.707 0.000 33

21
bcin - echo input

• input boundary conditions BCIN
• c lttencomroltorgt,iplane,iline,evmstep,updt(g.a.
  ),RCacc
• c 3 3 1 0.0250 0
  0.000
• c av.strain dir.lt33 (22-11) 223 231
  212gt epstol
• c -1.000 -1.000 0.000 0.000 0.000
  0.50
• c exp'd stress dir.lt33-(1122)/2(22-11)/223311
  2gt,99 if ? sigtol
• c 99.000 99.000 99.000 99.000 99.000
  0.05

22
nsteps

• How many steps? -- Write every ? steps
  
• 40,40
• Thank you, now relax that I take care
• For a step size of 2.5, 40 steps required per
  unit strain if the print interval is less,
  texout will have multiple sets of grains.

23
subroutines

• subroutine graxes(mupt,vfrc,irc1,irc2,rcacc)
• subroutine maxwork(icase,tayfac,ng,sirc1,sirc2)
• subroutine sss(nsys,ksys,smax,niter,evmstep)
• subroutine newton(niter,ksys,nsys)
• subroutine simq(aa,bb,n,ks)
• subroutine sigbc(sdirav,sigtol,itsbc)
• subroutine harden(rlhm,khar,iref,ntaun,klh,namode
  s,emu)

24
subroutines, contd.

• subroutine latent2(h,hq)
• subroutine update(eps,iline,iplane)
• subroutine twinor(ktw,ng,nomen,dbca)
• subroutine orient(iline,iplane)
• subroutine vecpro(k)
• subroutine euler(iopt,nomen,d1,d2,d3,ior,kerr)
• subroutine vectra(q,d)
• subroutine vec5ten

25
output kpath1

• test LApp68 14-Apr-01
• c texlat.wts from texlat.write viii
  00
• Evm F11 F12 F13 F21 F22 F23
  F31 F32 F33 nstate
• 0.000 50.000 0.000 0.000 0.000 1.000 0.000
  0.000 0.000 0.020 2
• c krc, ksys, klh, ksol,nrslim, khar,ngrains,
  iper,lsym, vfRC
• c 0 1 0 1 33 1 999 1 2
  3 0 0.00
• 
  
• Evm 0.000 M 2.55 Svm 394. vfRC0.00 itSbc 0
  Niter 9 0.41 1.02max(devbimod
• Evm 0.025 M 2.54 Svm 392. vfRC0.00 itSbc 0
  Niter 9 0.43 1.02max(devbimod
• Evm 0.050 M 2.53 Svm 391. vfRC0.00 itSbc 0
  Niter 8 0.44 1.02max(devbimod

Strain, Taylor factor, von Mises equivalent
stress, vol frac in RC iterations in sigbc,
ltiters.in sssgt, standard deviation in stress
26
output files

• texout similar to texin contains list of
  orientations corresponding to texin, rotated by
  accumulated slip.
• anal details on a few grains
• hist history of stress and strain
  used/calculated in each step

27
hist history

• c Result of SSS( 9 newton iters.avg.)
• c av strain dir -0.866 -0.500 0.000 0.000
  0.000
• c av strain dev 0.002 0.000 0.000 0.000
  0.000
• c av stress dir -0.821 -0.522 -0.226 0.053
  -0.016
• c av stress dev 0.281 0.412 0.293 0.415
  0.230 avg 0.326
• c 4th momentnor 0.966 1.024 0.876 0.914
  0.949
• c av CA deviatoric stress -0.297 0.039
  -0.314 -0.171 -0.884
• c av CA stress(ii) (SSSmean) 0.094 0.149
  -0.243
• c F 50.000 0.000 0.000 0.000 1.000 0.000
  0.000 0.000 0.020
• c Evm SIGvm TAYav TAYrs GAMav Savdev vfRC asas
  pl LHRlt
• 0.000 393.6 2.55 2.40 0.00 0.33 0.00 4.59
  3.05 1.00
• c Evm nreor atwfr etwfr mode-repartition n( -)
• 0.000 0 0.00 0.00 0.43 0.57

28
texout final orientations

• test texout LApp68 14-Apr-01
• c texlat.wts from texlat.write viii
  00
• c lttencomroltorgt,iplane,iline,evmstep,updt(g.a.
  ),RCacc
• c 3 3 1 0.0250 0
  0.000
• c av.strain dir.lt33 (22-11) 223 231
  212gt epstol
• c -1.000 -1.000 0.000 0.000
  0.000 0.50
• c exp'd stress dir.lt33-(1122)/2(22-11)/223311
  2gt,99 if ? sigtol
• c 99.000 99.000 99.000 99.000
  99.000 0.05
• c propfe for Salsgiver's Fe-Si,exper.StoutLova
  to 8/89
• c mode rs tau tau- h(m,1) h(m,2)
  h(m,3)........
• c 1 0.02 1.00 1.00 1.00
• c kond RATEref Tref muMPa tau0MPa th0/mu
  tauvMPa th4/th0 kurve(or LH)
• c 1 0.1E-02 300. 70000. 150.
• c krc, ksys, klh, ksol,nrslim, khar,ngrains,
  iper,lsym, vfRC
• c 0 1 0 1 33 1 999 1 2
  3 0 0.00
• Evm F11 F12 F13 F21 F22 F23
  F31 F32 F33 nstate
• 1.000117.778 0.000 0.000 0.000 1.000 0.000
  0.000 0.000 0.008 2
• Bungephi1 PHI phi2 ,,gr.wt., tau,
  taustaumodes/tau XYZ1 2 3
• 0.00 70.00 0.00 1.00 150.00 0.00

Re-statement of the input in bcin
29
Output of LApp
Increasing strain

• Figure shows pole figures for a simulation of the
  development of rolling texture in an fcc metal.
• Top 0.25 von Mises equivalent strain 0.50,
  0.75, 1.50 (bottom).
• Note the increasing texture strength as the
  strain level increases.

Graphics wts2pop, then pf2ps
30
r-value calculation

• The next sequence gives an example of how to use
  LApp to calculate r-values based on a given
  texture (no evolution).

31
kpath 4 (r-values)

• Angle increment (degrees lt15gt)
  ?
• 15 (controls direction resolution)
• to what frac.accuracy of stress should I
  iterate?lt0.01gt
• .02 (0.01 minimum practical value)
• What value of RCACC? (use 0 if in doubt)
  
• 0 (trick for exaggerating relaxed constraints
  effect)

32
kpath 4, contd.

• Enforce sample symmetry for property
  calculations?
• 0 no
• 1, 2, or 3 diad on that axis (use 2 or 3 with
  TEXREG)
• 4 orthotropy
  LSYM
• 0 (can add sample symmetry)

33
output (kpath 4)

• ang.fr.X1 r q shears(tension coords)
  tayfavmax(sdevbimod) itsbc
• 0.000 0.727 0.421 0.395 -0.037 0.044
  2.280 0.372 0.962 7
• 15.000 0.480 0.324 0.268 -0.129 0.111
  2.440 0.362 1.002 10
• 30.000 0.299 0.230 0.045 -0.106 0.127
  2.638 0.319 1.058 5
• 45.000 0.233 0.189 -0.250 -0.085 0.050
  2.658 0.312 0.953 13
• 60.000 0.861 0.463 -0.322 -0.004 -0.068
  2.712 0.332 0.973 5
• 75.000 2.109 0.678 -0.183 0.082 -0.045
  2.693 0.385 1.003 6
• 90.000 2.811 0.738 0.094 0.102 0.003
  2.664 0.372 1.017 4
• 
  
• r-bar, as calculated from an average of all
  q-D22/D11 is 0.696

q r/(1r) this output is also recorded in
lapp.dat
34
R-value,q plotted (kpath 4)
Input texture contained high fraction of Goss,
giving rise to maximum in r-value at 90 to the
rolling direction
35
Yield Surface calculation

• The next sequence of slides shows how to
  calculate the locus of points on a yield surface.

36
kpath 2 (2D yield surface)

• Enter strain path (KPATH)
• 1 many steps in one straining direction (need
  BCIN)
• 2 2-D yield surface probe
• 3 3-D yield surface probe
• 4 Lankford Coefficients R(angle) in the
  3-plane
• 2

37
kpath 2, contd.

• Relaxed Constraints when applicable ltKRC1gt or
  Full Constraints lt0gt?
• 0
• ngrains ltdefault whole file,.le.1152gt
  ?
• 999
• Complete file ANAL on the first how many
  lt0,9,ngrainsgt?
• 0

38
kpath 2, contd.

• YS projection (0) or YS section (enter SIGTOL)
  ?
• 0 (typical to assume proj.)
• you want tayfac lt0gt or stress MPa lt1gt
  ?
• 0 (stress proportional to ltMgt)
• Rate dep.on stresses only (0) or also on facets
  (1) ?
• 0 (allows contrast of Bishop-Hill soln. with RS
  solution)

In order to obtain a result for which the only
non-zero stress components (as opposed to strain
components) are the two in the plane of interest
(see later pages for this selection), choose
section instead of projection.
39
kpath 2, contd.

• Angle increment in strain-rate space (gt2
  degreeslt5gt)?
• Enter negative values if you want to scan /-
  range
• 15 (this is coarse choose 2 for high resolution)
• Select one of the indices
• 0 for Cauchy(22) vs (11), with (33)0
• 1 for pi plane -- 2 for S22-S11 vs Sij
• 3 for S11-S33 vs Sij -- 4 for S22-S33 vs Sij
• 5 for S11 vs Sij -- 6 for S22 vs Sij
• 7 for Sij vs S33 -- 8 for Sij vs Skl
  0 (as in most textbooks)

40
kpath 2, contd.

• Enforce sample symmetry for property
  calculations?
• 0 no
• 1, 2, or 3 diad on that axis (use 2 or 3 with
  TEXREG)
• 4 orthotropy
  LSYM
• 0 (again, can compensate for a texture lacking
  the desired sample symmetry)

41
lapp.dat

• KRYPTON.MEMS.CMU.EDUgt more lapp.dat
• xs ys -xs -ys active_sys 2 3 4 5 6 7 8 9
• 0.93894 -4.58022 -0.93894 4.58022 1
  2 3 4 5 6 7 8 9 10 11 12
• 0.68739 -2.21556 -0.68739 2.21556 1
  2 3 4 5 6 7 8 9 10 11 12
• 1.12168 -2.00816 -1.12168 2.00816 1
  2 3 4 5 6 7 8 9 10 11 12
• 1.62587 -1.66510 -1.62587 1.66510 1
  2 3 4 5 6 7 8 9 10 11 12
• 1.80145 -1.44043 -1.80145 1.44043 1
  2 3 4 5 6 7 8 9 10 11 12

stress components, - active slip
systems To plot the complete yield surface, plot
both ys versus xs, and -xs versus -ys (see
example a few slides on from this one).
42
hist (kpath 2)
Output contains pairs of lines

• KRYPTON.MEMS.CMU.EDUgt more hist
• nosort
• c ys HIST LApp68 14-Apr-01
• c dirs.,perp. sub-space RC comps.grainsvfRC
  ksollsym
• c 12 0 1 1 2 0 0 999 0.00
  2 0
• -1.00000 0.00000 0.00000 0.00000
  0.00000 2.83891
• -4.58022 0.93894 -0.21041 0.01443
  -0.19175 4.04711
• -0.96593 0.25882 0.00000 0.00000
  0.00000 2.92495
• -2.21556 0.68739 -0.05986 -0.03260
  -0.08404 0.51476
• -0.86603 0.50000 0.00000 0.00000
  0.00000 2.90462
• -2.00816 1.12168 -0.05832 -0.06920
  -0.06329 0.60977

(5) strain components Taylor factor
(5) stress components standard deviation
43
Yield Surface example (kpath2)
44
Summary

• The interface to the LApp code has been described
  with examples of problems that can be computed.
• LApp is essentially a polycrystal plasticity code
  for solving the Taylor/Bishop-Hill model.
• LApp can be used to compute the anisotropic
  (plastic) properties of textured polycrystals,
  e.g. yield surfaces, r-values.
• Other codes are required for different approaches
  to plastic deformation, e.g. self-consistent
  models, finite element models (incorporating
  crystal plasticity as a constitutive model).