Using LApp Los Alamos polycrystal plasticity

1
Using LApp- Los Alamos polycrystal plasticity

• 27-750, Advanced Characterization and
  Microstructural Analysis,
• Spring 2003

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.

3
Principles of LApp

• The principles governing the calculations in LApp
  are described in more detail elsewhere (some
  information in slides in the 2nd half of this
  lecture.
• 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 (sometimes also
  vertices on the single crystal yield surface).
• texin list of orientations Euler angles with a
  weight (sometimes also state parameters).
• bcin boundary conditions.
• 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

9
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
10
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
11
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
12
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
13
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.

14
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.

15
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.
  

16
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)

17
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)

18
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)

19
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

20
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

21
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.

22
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)

23
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

24
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
25
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

26
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

27
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

28
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).

29
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)

30
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)

31
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
32
R-value,q plotted (kpath 4)
33
Yield Surface calculation

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

34
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

35
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

36
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)

37
kpath 2, contd.

• Angle increment in strain-rate space (gt2
  degreeslt5gt)?
• Enter negative values if you want to scan /-
  range
• 15
• 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 texts)

38
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
  symmetry)

39
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
40
hist (kpath 2)

• 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
41
Yield Surface example (kpath2)
42
Part 2 how does Lapp work?

• The next set of slides is taken from one of the
  lectures on plasticity.
• The basis for rate sensitive (viscoplastic)
  plasticity is explained.

43
Schmids Law
Schmid postulated that

• Initial yeld stress varies from sample to sample
  depending on, among several factors, the relative
  of the crystal lattice to the loading axis.
• It is the shear stress resolved along the slip
  direction on the slip plane that initiates
  plastic deformation.
• Yield will begin on a slip system when the shear
  stress on this system reaches a critical value
  (critical resolved shear stress, crss),
  independent of the tensile stress or any other
  normal stress on the lattice plane.

44
Schmids Law
Resolved Shear Stress
45
Schmids Law
Using Schmids law
46
Rotation of the Crystal Lattice
The slip direction rotates towards the tensile
axis
47
Rotation of the Crystal Lattice in tensile test
of a (fcc) single crystal
48
Rotation of the Crystal Lattice
The slip plane normal rotates towards the
compression axis
49
Burgers vector b
Screw positionline direction//b
Edge positionline direction?b
50
FCC Geometry of Slip Systems
51
Slip steps
Slip steps (from exitof dislocations fromthe
crystal) on thesurface of compressedsingle
crystal of Nb.
Reid deformation geometry for materials
scientists
52
Notation

• Deformation gradient F
• measures the total change in shape (rotations
  included).
• Velocity gradient L
• measures the rate of change of the deformation
  gradient.
• Time t
• Strain rate D
• symmetric tensor.

53
Notation 2

• Plastic spin W
• measures the rotation rate more than one kind of
  spin is used
• Rigid body spin of the whole polycrystal W
• grain spin of the grain axes (e.g. in torsion)
  Wg
• lattice spin from slip/twinning Wc.
• Rotation (small) w

54
Notation 3

• Strain, local Elocal global Eglobal
• Slip direction (unit vector) b
• Slip plane (unit) normal n
• Stress (tensor or vector) s
• Shear stress (usually on a slip system) t
• Shear strain (usually on a slip system) g
• Stress deviator (tensor) S
• Rate sensitivity exponent n
• Slip system index s

55
Notation 4

• Coordinates current x reference X
• Velocity of a point v.
• Displacement u
• Strain, e
• measures the change in shape
• Work increment dW
• do not confuse with spin!
• W infinitesimal rotation tensor

56
Cubic Metals

• In the fcc metals, the slip systems are generally
  confined to 111 slip planes and lt110gt slip
  directions (Burgers vectors) in bcc metals, the
  indices are transposed.
• As a consequence there are only 28 distinct
  stress states (vertices on the single crystal
  yield surface) that activate 5 or more slip
  systems simultaneously.

57
Definitions of Stress states, slip systems

• Deviatoric stress termsA (s22 - s33) F
  s23B (s33 - s11) G s13C (s11 - s22) H
  s12
• Slip systems

Kocks UQ -UK UP -PK -PQ PU -QU -QP -QK
-KP -KU KQ
58
Rate Sensitive Yield

• The standard model in use today is the
  so-called rate-sensitive model.
• The ambiguity in the choice of active slip
  systems is eliminated.
• The flow stress is a non-linear (power law)
  function of the strain rate.
• For typical values of the exponent (15ltnlt200),
  the stress rises steeply with strain rate in the
  vicinity of the critical resolved shear stress.

59
Taylor rate-sensitive model 1

• Symmetric part of the distortion tensor resulting
  from slip
• Anti-symmetric part of Deformation Strain Rate
  Tensor (used for calculating lattice rotations)

60
Taylor rate-sensitive model 2

• Strain rate from slip (add up contributions from
  all active slip systems)
• Rotation rate from slip (add up contributions
  from all active slip systems)

61
Taylor rate-sensitive model 3

• Rotation rate of crystal axes (W)
• Rate sensitive formulation for slip rate in each
  crystal (solve as implicit equation for stress)

62
Taylor rate-sensitive model 4

• The shear strain rate on each system is also
  given by the power-law relation (once the stress
  is determined)

63
Single Slip and Multislip Modes
For FCC materials there are 12 slip systems
Four 111 planes with Three lt011gt directions