Fast Fourier Transform-based micromechanical modeling of polycrystals with direct input from microstructural images

Transcription

1 Fast Furier Transfrm-based micrmechanical mdeling f plycrystals with direct input frm micrstructural images Ricard A. Lebenshn (Ls Alams Natinal Labratry, USA)

2 FFT-based frmulatin fr plycrystals: fast predictin f mechanical behavir f 3-D plycrystalline materials CP-FEM (mesh, integratins pints) FFT-based methd (vxels) numerically-generated lcal (single crystal) cnstitutive equatin fr each vxel measured (e.g. 3-D EBSD) lcal (single crystal) cnstitutive equatin fr each integratin pint

3 FFT-based frmulatin fr plycrystals: cncept Furier grid {xf} Prpsed by Suquet and c-wrkers, fr linear (Mulinec & Suquet, 1994) and nnlinear cmpsites (Mulinec & Suquet 1998, Michel-Mulinec-Suquet, 2000, 2001), and adapted fr plycrystals (PX): viscplastic (VP) PXs (Lebenshn, Acta Mater. 2001; Lebenshn et al, Acta Mater. 2008, 2009). dilatatinal VP PXs (Lebenshn et al., Phil Mag 2011). Representative Vlume Element (peridic) Slving gverning equatins (equilibrium + cmpatibility): ε(x f ), σ(x f ) elasticity, small-strain EVP ε& (x f ), σ(x f ) viscplasticity F(x f ), P(x f ) finite-strain EVP small-strain elast-viscplastic (EVP) PXs (Lebenshn et al, IJP 2012; Grennerat, Suquet et al. Acta Mater, 2012). finite-strain EVP PXs (Eisenlhr, Diehl, Lebenshn, Rters, IJP, 2013). Based n the equivalence between the micrmechanical respnse f a hetergeneus medium and a hmgenus medium with eigenstrains. Nn-linear prblems are treated similarly t linear prblems, using the cncept f Linear Reference Material.

4 FFT-based apprach fr elastically hetergeneus materials (*) E applied RVE σ( x) = C( x) : ε( x) σ ( x) =σ( x) C = C + C : ε(x) C : ε(x) + C : ε(x) = C e.g. directly frm image Ag-Fe lcal elasticity : ε(x) = : ε(x) +τ(x) stiffness f a linear reference medium τ(x) = C : ε(x) plarizatin (hetergeneity) field functin f slutin requires iterative prcedure t adjust equilibrium: ( x) +τ ( x) Cijkl uk,lj ij, j = 0 peridic bundary cnditins in RVE acrss RVE Green functin Methd: Cijkl Gkm,lj( x x' ) +δimδ( x x' ) = 0 ε ~ ˆ~ ij ij ik, jl kl ˆ 3 R ( ) ( ) ( ) = + τ ε =Γ τ ε =Γˆ x E sym G x x x dx : τ Direct space Upn cnvergence: σ( x), ε( x) Furier space FFT! (*) H. Mulinec & P. Suquet, Cmput. Methds Appl. Mech. Engrg. 157, 69 (1998).

5 FFT-based apprach fr viscplastic plycrystals (VP-FFT) (*) E.g. viscplastic FFT with input frm EBSD: E & Green peratr ε& Schmid (rientatin) tensr, e.g. directly frm EBSD crystal plasticity cnst. eq.: s ( x) =γ& m ( x) plarizatin field: ε& ( x) : σ ( x) s m sgn m σ s s τ ( x) τ (x) =σ x L : ε & iterate ( ) (x) n s ( ( x) : ( x) ) ~ strain-rate field: ˆ~ ( x) =Γ ( L ) τ ε& ( ξ) =Γˆ : τˆ ε& ( x) = E& 1 + FT ( Γˆ : τˆ ) cnvlutin: timecnsuming integral in direct space mere prduct in Furier space stiffness f a linear reference material (*) R.A. Lebenshn, Acta Mater. 49, 2723 (2001). U N C L A S S I F I E D Materials Capability Review 2013 Operated by Ls Alams Natinal Security, LLC fr NNSA

6 FEM-FFT cmparisn [1] Initial micrstructure: peridic cluster f 100 Vrni grains Fields cmparisn: final shape and shear strain FEM and FFT slvers implemented within the same finite-strain framewrk [2] Identical cnstitutive mdel: elast-viscplastic single crystal (fcc) behavir At the same reslutin, FEM and FFT predict similar strain distributins, but FFT strain fields are mre hetergeneus At higher reslutins (unreachable with FEM within decent cmputatin times), FFT predicts very detailed strain lcalizatin pattern Our ability t slve the prblem with higher spatial reslutin allws us uncvering new and relevant features f the micrmechanical respnse [1] Eisenlhr, Diehl, Lebenshn, Rters, IJP, [2] Rters, Eisenlhr, Raabe, et al. DAMASK cde. Bundary cnditin: 30% simple shear Cmputatin time

7 Prsity evlutin in 3-D: synthetic plycrystal vs. istrpic matrix (*) Micrstructural effects n vid grwth: Qualitative cmparisn with experiments φ= 11% shcked Cu (**) hard link n calescence shck dir. ε& x eq ( )/ E & eq 3 Taylr factr hard grain sft link calescence 0 (**) Escbed et al, J. Appl. Phys., 2011 (*) D-VP-FFT: Lebenshn et al. Phil. Mag. (2011); D-VP-FFT + vid grwth: Lebenshn et al, Acta Mater, submitted.

8 VP-FFT calculatin with input frm 2-D data: misrientatin in Cu plycrystal unit-cell cnstructin Initial OIM Lebenshn et al, Acta Mater. 56, 3914 (2008) OIM after 11% tensin average rientatins f largest grains Orientatin dependence f average misrientatins transitin band 111 bins f increasing misrientatin bins f increasing misrientatin (a) OIM - 11% tensin 306 largest grains (b) FFT - 11% tensin 306 largest grains 110 Grains with rientatins near transitin line may be pulled twards different stable rientatins prne t develp higher misrientatins, in average Intragranular misrientatin rientatindependent (highest near [110])

9 Image-based mdelling using 3-D rientatin infrmatin FIB-EBSD vxelized 3-D grain rientatin maps FFT analysis (*) pure Ni VM stress VM strain-rate (*) Rllett, Lebenshn et al., MSMSE, 2010

10 FFT simulatins vn Mises Stress Measured Plycrystal (IN100) (*) Vn Mises Strain Rate Euclidean Distance Maps Stats Stresses vs distance frm features (*) Rllett, Lebenshn et al., MSMSE, 2010 High stresses cncentrate near grain bundaries, triple junctins, etc. Hwever, lw stress pints als cluster near bundaries.

11 In-situ bservatin/mdelling f rientatin fields f 2-D bulk grains (*) One 2-D sectin 0.06% 7% 14% In general, 2-D rientatin fields are stacked tgether t btain a 3D vlume Cnfidence Index (*) R. Pkharel, PhD Thesis, CMU, 2013 Rerientatin f large grains at 14% FFT HEDM

12 Measured vs. predicted Kernel Average Misrientatin (KAM) (*): HEDM initial FFT mid layer KAM [deg] 2-D EBSD: cnsistent with simulatins HEDM 12% strain 2-D sectin frm HEDM FFT mdel 2-D EBSD 3-D HEDM (*) R. Pkharel, PhD Thesis, CMU, 2013