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1 Micromechanics using Spectral Method Interface Decohesion in Polycrystals L. Sha harma, P. Shanthraj, M.Diehl, F. Roters, D. Raabe, R. Peerlings and M. Geers Ou Outline line Motivation DAMASK- material simulation kit The

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  2. Micromechanics using Spectral Method Interface Decohesion in Polycrystals L. Sha harma, P. Shanthraj, M.Diehl, F. Roters, D. Raabe, R. Peerlings and M. Geers

  3. Ou Outline line • Motivation • DAMASK- material simulation kit • The Spectral Solver • Basic scheme • Some applications • Demo: a very simple (1D) implementation using petsc4py • Smeared damage mechanics • Interface decohesion in Polycrystals. • Future Work

  4. Mo Motiv tivation ation • strain localization • tool design • damage initiation • crashworthiness • recrystallization nucleation • component properties • … • …

  5. Mo Motiv tivation ation http://www.virtualexplorer.com.au/special/meansvolume/contribs/jessell/labs/02a.m ov Crystal Plasticity is always a multi-scale problem ! 5

  6. Mo Motiv tivation ation simulation requirements Crystal Plasticity Finite Elemente Method • arbitrary mechanical (CPFEM) boundary value Or problems Crystal Plasticity • continuum mechanics Spectral Method • accounting for crystal (CPFFT) plasticity 6

  7. CP CPFEM/ FEM/CPFFT CPFFT st strat rategy egy solver for • equilibrium • compatibility 𝐆 𝐐 material point model F e deformation constitutive law F M partitioning crystallite • elasticity S & elasto-plasticity • plasticity P homogenization L p 7

  8. Th The sp spectral tral so solv lver A little history  Use spectral method instead of FEM  Solution based on FFT  Much faster than FEM  Small strain framework  Elastic material law  Extended to viscoplastic materials  Large strain formulation  Coupled with DAMASK 8

  9. Sp Spectral ctral me method hod Static equilibrium: Material law: 𝝉 = 𝑫𝜻 div 𝝉 = 0 Split strain: 𝜻 = 𝜻 + 𝜻 Introduce reference medium: Stiffness 𝑫 𝜻 𝑛+1 = 𝜻 𝑛 − 𝜟 ∗ 𝝉 𝑛 𝜻 𝑛+1 = 𝜻 𝑛 − ℱ −1 𝝉 𝑛 FFT 𝜟: −1 with Γ 𝑗𝑘𝑙𝑚 = 𝑙 𝑘 𝑙 𝑚 𝑂 𝑗𝑙 and 𝑂 𝑗𝑙 = 𝑙 𝑚 𝑙 𝑘 𝐷 𝑗𝑘𝑙𝑚 9

  10. Co Comp mparion arion FEM FEM vs vs FFT FFT P. Eisenlohr., M. Diehl, R. A. Lebensohn, F. Roters: International Journal of Plasticity (2013), 37 - 53 10

  11. Ex Experim rimental ental-Numerical Numerical Example: Basal slip in Magnesium F. Wang, S. Sandloebes, M. Diehl, L. Sharma, F. Roters, D. Raabe:Acta Materialia 80 (2014) 77-93 11

  12. Me Mesosc oscale ale me mechan hanics ics • High Resolution Crystal plasticity enabled through robust spectral solvers Shanthraj et al. [IJP, 2015] 12

  13. Demo mo ( 1D elasticity + spectral method using petsc4py ) 13

  14. Int nterface erface decoh ohesion esion (form ormab ability ility li limi miter) er) Void Role e of th the e Int nter erfa face ces Growth/Propagation Sur urroun oundin ding Microst ostru ructur ture Void ? Initiation Compu puta tati tion onal to tool to to model Inte terfa face ce decohesion ion 14

  15. Int nterface erface mo modeli ling ng of of pol olycrystals crystals Interface band Interface elements 15

  16. Ei Eige gen n St Stra rain in Dama mage ge ( Pandolf olfi, i, Ortiz z et al.; ; Menzel el, , Ekh et al., 2002 ) ) • Accomodation by eigen strain. • In an anisotropic way (normal and tangential modes). • (interface-) plane stretching effects. 16

  17. Field problem 17

  18. Dama mage ge re regulariz gularization ation so solv lved us usin ing g FFT FFT • Hetrogenous regularization lengthscale • Utilize Fourier transform • Solved for its roots using Jacobian free Newton method. 18

  19. Te Test st Si Simu mulatio lation 19

  20. Pol olycryst ycrystal al Si Simulation mulation • Res esolut ution on: 256x256x2 • Ran andomly ly orien enta tati tion on FCC • Elas asto to-pl plas astic tic-dam damage age (cryst ystal plasti ticit ity) y) • Int nter erfa face ce Band nd th thickne ness: 4 voxels 20

  21. Polyc ycryst rystal al Si Simu mula latio tion: n: Dama mage e ev evolutio tion 21

  22. Polyc ycryst rystal al Si Simu mula latio tion: n: St Stes ess s Unloadin ding 22

  23. Polyc ycryst rystal al Si Simu mula latio tion: n: Dama mage e vs Plastic sticity ity 23

  24. Fut Future ure wo work rk • Coupling with damage models in the bulk • Monolithic schemes for Multiphysics • Implementation using petsc4py • Time integrators (Fortran support) 24

  25. Acknowl knowledgme edgment nt Düsseldorf Advanced MAterial Simulation Kit, DAMASK  Available as freeware according to GPL 3  Integrates into MSC.Marc and Abaqus (std. and expl.)  Standalone spectral solver  Web: https://DAMASK.mpie.de  Email: DAMASK@mpie.de 25

  26. Thank you. Questions?

  27. Simu Si mulat lation ion ( hexag ally ) xagon onal al polycry cryst stal al loaded ded hori rizo zont ntall 27

  28. Si Simu mulat lation ion 28

  29. St Stra rain in lo localis lisation ation 29

  30. Br Britt ittle le Si Simu mulat lation ion Damage Strain (F11) Stress (P11) 30

  31. Me Mesosc oscale ale me mechan hanics ics • Crystal plasticity. • FFT based Spectral method Shanthraj et al. [2015] 31

  32. Ex Experim rimental ental-Numerical Numerical Example: Basal slip in Magnesium Wang et al. [2014] 32

  33. Ra Rate te in independe endent nt 33

  34. Ra Rate te in independe endent nt 34

  35. Local Damage • 1 for undamaged material • 0 for fully damaged • Monotonously decreases (irreversibility) 35

  36. Normal opening strains 36

  37. Tangential opening strains 37

  38. Stress Integration (the Local problem) 38

  39. Mesh Objectivity • Coarse mesh (10 fourier points in the band) • Fine = 2x coarse 39

  40. Wo Work rk of of se separation ration wi with h band nd thi hickness kness 40

  41. Vox oxelize elized fie ield ld of of the he no norm rmals als • Generator points of • First order cartesian standard voronoi tessellation. moments. [ Libermann et al., 2015 ] 2 3 3 3 3 3 2 2 2 4 4 2 2 4 4 2 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 41

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