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Anisotropy part 2: Using LApp- Los Alamos polycrystal plasticity

Anisotropy part 2: Using LApp- Los Alamos polycrystal plasticity. 27-750, Fall 2009 Texture, Microstructure & Anisotropy, Fall 2009 A.D. Rollett, P. Kalu. Objective.

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Anisotropy part 2: Using LApp- Los Alamos polycrystal plasticity

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  1. Anisotropy part 2:Using 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, 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

  9. Hardening parameters kond system number RATEref strain rate at which properties given Tref reference temperature mu[MPa] shear modulus (µ) tau0[MPa] yield stress (initial critical resolved shear stress) th0/mu hardening rate over modulus in Stage II tauv[MPa] 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 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

  11. bcin: boundary conditions Test type <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 Stress components Strain components s33-(s22+s11)/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.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 *** ************************************************************** <RETURN>

  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

  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.

  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.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.

  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 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 <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 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= 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 <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

  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,namodes,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; 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 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. 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 Strain, Taylor factor, von Mises equivalent stress, vol frac in RC iterations in sigbc, <iters.in sss>, 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) (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

  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 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 <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)

  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); 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

  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 <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

  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 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 (>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)

  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.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 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.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

  43. Yield Surface example (kpath=2)

  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).

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