Anisotropy part 2:
Using LAppLos Alamos polycrystal plasticity
27-750, Fall 2009
Texture, Microstructure & Anisotropy,
Fall 2009
Carnegie
Mellon
A.D. Rollett, P. Kalu
MRSEC
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
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.
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
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.
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
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).
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
5
LApp Flow Chart
sxin
bcin
texin
propin
input
files
ratesensitive
solution
output
files
sss
newton
hist
lapp.dat
texout
anal
preparation
grain, slip
geometry
maxwork
Bishop
-Hill solution
update
orientation
orient
of each
grain
stop
update
harden
hardening
on each
slip system LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
6
sxin: slip geometry
cubic lattices (this is fcc; for bcc,
transpose)
1 28 =nmodes,nvertex. mode nsys ktwin
must appear)
1
12
0
1
1
-1
0
1
1
1
-1
1
0
fcc: slip
1
1
-1
1
-1
planes
1
-1
-1
0
1
1
-1
-1
1
0
1
-1
-1
1
1
1
-1
1
0
1
1
-1
1
1
0
1
-1
1
1
1
1
1
1
0
1
1
1
1
1
0
1
1
1
1
-1
LApp gives you option to
twsh -corr (all numbers
0.0
1
1
0
-1
1
0
1
-1
0
-1
-1
0
0.0
+pk
+pq
+pu
+qu
+qp
+qk
+kp
+ku
+kq
+uq
+uk
+up
-pk
-pq
-pu
-qu
-qp
-qk
-kp
-ku
-kq
-uq
-uk
-up
fcc: slip
directions
Slip
Systems
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
7
sxin, contd.
28
number of active systems
=nvertex
8-fold
vertex
8
2
2
8
0
1
8
-2
13
6
0
1
6
0
1
1
0
3
33
2
15
65
-2
14
97
0
2
103
0
15
stress vector
0
5
0
6
0
9
0
16
0
18
0
19
21
10
24
0
4
0
17
0
7
20
22
23
1
17
1
18
1
7
9
25
25
1
7
-1
20
1
11
12
25
25………
8
11
12
IDs of active
slip systems
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
8
propin: strain hardening properties
Al : for Stout's 1100 Al, kond=2 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 mu[MPa] tau0[MPa] th0/mu tauv[MPa] 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, harn=th/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
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
9
Hardening parameters
kond
RATEref
Tref
mu[MPa]
tau0[MPa]
th0/mu
tauv[MPa]
th4/th0
Kurve
system number
strain rate at which properties given
reference temperature
shear modulus (µ)
yield stress (initial critical resolved shear stress)
hardening rate over modulus in Stage II
Voce stress (saturation, or asymptotic flow stress)
ratio of hardening in Stage IV to that in Stage II
ID number of discretized hardening rate versus stress
curve
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
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
Kocks:Psi 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
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
11
Test type
bcin: boundary conditions
<ten;com;rol;tor>,iplane,iline,evmstep,updt(g.a.),RCacc
3
3
1
0.02500
0.0
0.0
av.strain dir.<33; (22-11); 2*23; 2*31;
2*12>;
epstol
1.000
1.000
0.000
0.000
0.000
0.5
exp'd stress dir.<33-(11+22)/2;(22-11)/2;23;31;12>,99 if ?;sigtol
99.0
99.0
99.
99.0
99.0
0.05
Strain components
e33, e22-e11, 2e23, 2e31, 2e12
Stress components
s33-(s22+s11)/2, (s22-s11)/2, s23, s31, s12
Strain increment
“99” means component can take any value
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
12
LApp dialog
KRYPTON.MEMS.CMU.EDU> 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
***
*** Please advise Fred Kocks of any errors you find:
***
***
Fax: (1)505-665-2992; Email: ufk@rho.lanl.gov
***
*** GTDA
***
**************************************************************
User
responses
in red
<RETURN>
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
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 angles;grwt;state pars.
(all must be in prescribed format)
TEXIN file=
texlat.wts
from texlat.write [viii 00]
Enter title (8 chars.):
Enter a (short!) title
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
14
ksys: Deformation System
Enter KSYS:
1 for FCC {111}<110> 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.
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
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.
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
16
exponent: Rate Sensitivity
PROPIN:
propfe : for Salsgiver's Fe-Si,exper.Stout&Lovato 8/89
mode
rs tau+ tau- h(m,1) h(m,2) h(m,3)........
Value for max. rate sensitivity exponent <default 33>?
33
kond RATEref
Tref mu[MPa] tau0[MPa] th0/mu tauv[MPa] 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.
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
17
kpath: type of test
Reenter TTY input <1>, or same as in preceding
test <0> ?
(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 3plane
:
1 (i.e. texture evolution)
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
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)
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
19
krc, ngrains
Relaxed Constraints when applicable
<KRC=1> or Full Constraints <0>?
0 (boundary conditions on grain)
ngrains <default = whole
file,.le.1152>
?
999 (defaults to max. number
of orientations in texin)
On modern computers, the maximum number of
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
grains can be easily extended to >100,000.
20
anal
Complete file ANAL on the first how many <0,9,ngrains>?
0 (use for debugging, checks)
mode,systems=
n , b , nrs :
n , b , nrs :
n , b , nrs :
n , b , nrs :
n , b , nrs :
n , b , nrs :
n , b , nrs :
n , b , nrs :
n , b , nrs :
n , b , nrs :
n , b , nrs :
n , b , nrs :
0.577
0.577
0.577
0.577
0.577
0.577
0.577
0.577
0.577
0.577
0.577
0.577
1
0.577
0.577
0.577
-0.577
-0.577
-0.577
-0.577
-0.577
-0.577
0.577
0.577
0.577
12
-0.577
-0.577
-0.577
-0.577
-0.577
-0.577
0.577
0.577
0.577
0.577
0.577
0.577
0.000 0.707 0.707
0.707 0.000 0.707
0.707 -0.707 0.000
0.000 0.707 -0.707
0.707 0.000 0.707
0.707 0.707 0.000
0.000 0.707 0.707
0.707 0.000 -0.707
0.707 0.707 0.000
0.000 0.707 -0.707
0.707 0.000 -0.707
0.707 -0.707 0.000
33
33
33
33
33
33
33
33
33
33
33
33
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
21
bcin - echo input
input boundary conditions BCIN:
c <ten;com;rol;tor>,iplane,iline,evmstep,updt(g.a.),RCacc
c
3
3
1
0.0250
0
0.000
c av.strain dir.<33; (22-11); 2*23; 2*31;
2*12>;
epstol
c -1.000
-1.000
0.000
0.000
0.000
0.50
c exp'd stress dir.<33-(11+22)/2;(22-11)/2;23;31;12>,99 if
?; sigtol
c 99.000
99.000
99.000
99.000
99.000
0.05
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
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.
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
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,namodes,emu)
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
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
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
25
output; kpath=1
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
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. vfRC=0.00 itSbc= 0 Niter= 9 0.41
1.02=max(dev&bimod
Evm= 0.025 M= 2.54 Svm= 392. vfRC=0.00 itSbc= 0 Niter= 9 0.43
1.02=max(dev&bimod
Evm= 0.050 M= 2.53 Svm= 391. vfRC=0.00 itSbc= 0 Niter= 8 0.44
1.02=max(dev&bimod
2
Strain, Taylor factor, von Mises equivalent stress, vol frac in RC
iterations in sigbc, <iters.in sss>, standard deviation in stress
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
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
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
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) (SSS+mean)
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 a#sas #pl
LHR<
=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
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
28
texout: final orientations
test
texout LApp68 14-Apr-01
c texlat.wts
from texlat.write [viii 00]
c <ten;com;rol;tor>,iplane,iline,evmstep,updt(g.a.),RCacc
c
3
3
1
0.0250
0
0.000
c av.strain dir.<33; (22-11); 2*23; 2*31;
2*12>;
epstol
c
-1.000
-1.000
0.000
0.000
0.000
0.50
c exp'd stress dir.<33-(11+22)/2;(22-11)/2;23;31;12>,99 if ?; sigtol
c
99.000
99.000
99.000
99.000
99.000
0.05
c propfe : for Salsgiver's Fe-Si,exper.Stout&Lovato 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 mu[MPa] tau0[MPa] th0/mu tauv[MPa] 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
Bunge:phi1 PHI
phi2 ,,gr.wt., tau,
taus;taumodes/tau;
XYZ=1 2 3
0.00
70.00
0.00
1.00 150.00
0.00
1.07
70.00
1.07
1.00 150.00
0.00
2.21
70.00
2.21
1.00 150.00
0.00
Restatement
of the input
in bcin
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
29
Increasing strain
Output of
LApp
• 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
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
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).
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
31
kpath = 4 (r-values)
Angle increment (degrees <15>)
?
15 (controls direction resolution)
to what frac.accuracy of stress should I
iterate?<0.01>
.02 (0.01= minimum practical value)
What value of RCACC? (use 0 if in doubt)
:
0 (trick for exaggerating relaxed
constraints effect)
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
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)
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
33
output (kpath = 4)
ang.fr.X1; r ; q ; shears(tension coords); tayfav;max(sdev&bimod); 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/(1+r)
this output is also recorded in lapp.dat
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
34
R-value,q plotted (kpath = 4)
Lankford.example.data
3
Lankford coeff. (R), q
2.5
R
q
2
1.5
1
0.5
0
0
20
40
60
80
angle
Input texture contained high fraction of Goss, giving rise to maximum in
r-value at 90° to the rolling direction
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
35
Yield Surface calculation
• The next sequence of slides shows how
to calculate the locus of points on a
yield surface.
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
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
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
37
kpath = 2, contd.
Relaxed Constraints when applicable
<KRC=1> or Full Constraints <0>?
0
ngrains <default = whole
file,.le.1152>
?
999
Complete file ANAL on the first how
many <0,9,ngrains>?
0
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
38
kpath = 2, contd.
YS projection (0) or YS section
(enter SIGTOL)
?
0 (typical to assume proj.)*
you want tayfac <0> or stress
[MPa] <1>
?
0 (stress proportional to <M>)
Rate dep.on stresses only (0) or
also on facets (1) ?
0 (allows contrast of BishopHill 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”.
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
39
kpath = 2, contd.
Angle increment in strain-rate space (>2
degrees:<5>)?
* 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)
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
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)
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
41
lapp.dat
KRYPTON.MEMS.CMU.EDU> 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
0.68739
-2.21556
-0.68739
2.21556
1.12168
-2.00816
-1.12168
2.00816
1.62587
-1.66510
-1.62587
1.66510
1.80145
-1.44043
-1.80145
1.44043
1
1
1
1
1
2
2
2
2
2
stress components, + & -;
3
3
3
3
3
4
4
4
4
4
5
5
5
5
5
6
6
6
6
6
7
7
7
7
7
8
8
8
8
8
9
9
9
9
9
10
10
10
10
10
11
11
11
11
11
12
12
12
12
12
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).
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
42
hist (kpath = 2)
Output contains pairs of lines:
KRYPTON.MEMS.CMU.EDU> more hist
nosort
c ys
HIST LApp68 14-Apr-01
c #dirs.,perp.; sub-space ;RC comps.;grains;vfRC;ksol;lsym
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
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
43
Yield Surface example (kpath=2)
ys
-ys
yield.surf.example.data
3
2
ys
1
0
-1
-2
-3
-3
-2
-1
0
1
2
3
xs
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09
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).
LApp demo, A.D.Rollett, Carnegie Mellon U., 05, updated Sep 09